Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems [1 ed.] 3030756521, 9783030756529

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Vasile Marinca Nicolae Herisanu Bogdan Marinca

Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems

Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems

Vasile Marinca · Nicolae Herisanu · Bogdan Marinca

Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems

Vasile Marinca Department of Mechanics and Strength of Materials Polytechnic University of Timi¸soara Timi¸soara, Romania

Nicolae Herisanu Department of Mechanics and Strength of Materials Polytechnic University of Timi¸soara Timi¸soara, Romania

Bogdan Marinca Applied Electronics Department Polytechnic University of Timi¸soara Timi¸soara, Romania

ISBN 978-3-030-75652-9 ISBN 978-3-030-75653-6 (eBook) https://doi.org/10.1007/978-3-030-75653-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The analytical investigation of nonlinear dynamical systems is one of the most important but difficult tasks, and the present monograph consists of numerous examples from various domains of engineering and applied sciences. Any problem of motion in nonlinear dynamical systems can be assimilated only by working with differential equations which are applied in concrete examples. All examples presented here and treated from an analytical point of view are accompanied by comparisons with numerical results and sometimes with exact solutions, or with other known results in the literature. The analytical technique presented in this book is an analytical approximation method applicable for highly nonlinear systems independent of the presence of small or large parameters into the governing equations or in the initial/boundary conditions. A good knowledge of different approximate methods, especially the method of harmonic balance, the method of multiple scales, the optimal homotopy asymptotic method, or Krylov–Bogoliubov method, led to a better choice of the so-called auxiliary functions, which are decisive for the technique proposed here. In contrast to the other known techniques, our approach provides us with a simple way to control and adjust the convergence regions of solutions corresponding to nonlinear dynamical systems. The methodology proposed in this book is totally different from all other known analytical techniques, especially regarding the choice of the linear operators and optimal auxiliary functions, as well as the determination of the optimal convergence-control parameters. The success of the present method is an important milestone in any field of exact sciences and techniques. Besides a wide field of applications, the proposed procedure can often be used to provide comparisons with the results obtained by other procedures. The intended readers of this book include undergraduate students and graduate students doing projects on doctoral research in the field of nonlinear dynamical systems; researchers, engineers and university teachers will also find this book useful. The work is based on the results obtained by the authors in the last years of research in the field of nonlinear dynamical systems. New results are illustrated by numerical examples. It is assumed that the reader already has minimal knowledge on how to differentiate and integrate elementary functions. Also, computer skills v

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Preface

would be essential because computer simulation is a powerful tool for examination, confirmation and sometimes for refutation of the obtained results. The book is divided into 31 chapters. The Chap. 1 is the introduction and the Chap. 2 is devoted to the basic ideas of our procedure. The Chaps. 3–28 deals with the first alternative of the optimal auxiliary functions method, solving the first approximate equation. The Chap. 29 treats the second alternative but without solving the equation in the first approximation. The Chap. 31 is devoted to finding the exact solutions and in the Chap. 30, the optimal auxiliary functions method is applied piecewise. Here are treated models from various fields of engineering such as mechanical vibration, thermodynamics, fluid mechanics, astronomy, electrical machine and so on. Most models will demand some independent thinking and are selected in order to illustrate the main ideas of our procedure, which allowed the readers to understand the present material. Timisoara, Romania 2021

Vasile Marinca Nicolae Herisanu Bogdan Marinca

Contents

Part I

A Short Introduction to the Optimal Auxiliary Functions Method

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 8

2

The Optimal Auxiliary Functions Method . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 16

Part II 3

4

The Optimal Auxiliary Functions Method in Engineering Applications

The First Alternative of the Optimal Auxiliary Functions Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Dynamics of an Angular Misaligned Multirotor System . . . . . . . . 3.1.1 The Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Optimal Auxiliary Functions Method for Nonlinear Vibration of Misaligned Multirotor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oscillations of a Pendulum Wrapping on Two Cylinders . . . . . . . . . . 4.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Application of OAFM to a Pendulum Wrapping on Two Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 21

23 31 39 41 44 45 49 60

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5

Contents

Free Oscillations of Euler–Bernoulli Beams on Nonlinear Winkler-Pasternak Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Nonlinear Euler–Bernoulli Beam Model . . . . . . . . . . . . . . . . . . . . . 5.2 OAFM for Free Oscillations of Euler–Bernoulli Beam . . . . . . . . . 5.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 64 65 67 68

Nonlinear Vibrations of Doubly Clamped Nanobeam Incorporating the Casimir Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Nonlinear Equation for Nanobeam . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Galerkin Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Application of OAFM to Eqs. (6.11) and (6.12) . . . . . . . . . . . . . . . 6.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 72 74 75 76 78

7

Transversal Oscillations of a Beam with Quintic Nonlinearities . . . . 7.1 The Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 OAFM for Nonlinear Differential Eq. (7.11) . . . . . . . . . . . . . . . . . . 7.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 80 83 85 85

8

Approximate Analytical Solutions to Jerk Equations . . . . . . . . . . . . . 8.1 OAFM for Jerk Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 88 90 90 90 92

9

Vibration of Nonlinear Nonlocal Elastic Column with Initial Imperfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Application of OAFM to Vibration of Nonlinear Nonlocal Elastic Column with Initial Imperfection . . . . . . . . . . . . . . . . . . . . . 9.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 97 97

10 Nonlinear Vibration of Bernoulli–Euler Beam on a Winkler Elastic Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Discretization and Free Vibration of the Beam Under Study . . . . 10.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 100 101 107 108

6

93 94

11 The Nonlinear Thermomechanical Vibration of a Functionally Graded Beam (FGB) on Winkler-Pasternak Foundation . . . . . . . . . . 109 11.1 The Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

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11.2 Application of OAFM to Eqs. (11.33) and (11.36) . . . . . . . . . . . . . 116 11.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 12 Nonlinear Free Vibration of Microtubes . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Free Vibration of the Microtube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 OAFM for Eqs. (12.16) and (12.18) . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 124 126 127 129 130

13 Nonlinear Free Vibration of Elastically Actuated Microtubes . . . . . . 13.1 Problem Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Free Vibration of the Microtube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Application of OAFM to Elastically Actuated Microtube . . . . . . . 13.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 133 137 138 140 144

14 Analytical Investigation to Duffing Harmonic Oscillator . . . . . . . . . . 14.1 OAFM for Duffing Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 14.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147 148 150 151

15 Free Vibration of Tapered Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 OAFM for Free Vibration of Tapered Beams . . . . . . . . . . . . . . . . . 15.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 154 156 157

16 Dynamic Analysis of a Rotating Electrical Machine Rotor-Bearing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Application of OAFM to the Investigation of Nonlinear Vibration of the Considered Electrical Machine . . . . . . . . . . . . . . . 16.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

160 163 164

17 Investigation of a Permanent Magnet Synchronous Generator . . . . . 17.1 Governing Equations of PMSG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Approximate Solution of Eqs. (17.11) and (17.10) . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167 168 171 176

159

18 Dynamic Response of a Permanent Magnet Synchronous Generator to a Wind Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 18.1 Approximate Solution of the Dynamic Model of the Wind-Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

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19 Axisymmetric Flow and Heat Transfer on a Moving Cylinder . . . . . 19.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Optimal Auxiliary Functions Method for Solving the System (3.17.9)–(3.17.18) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189 197 202

20 Blasius Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 The Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Approximate Solution of the Blasius Problem . . . . . . . . . . . . . . . . 20.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 204 205 207 208

21 Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Governing Equations of Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Approximate Solution of the Eqs. (21.8) and (21.9) . . . . . . . . . . . 21.3 Numerical Example for the First Alternative . . . . . . . . . . . . . . . . . 21.4 Numerical Results by OAFM (The Second Alternative) . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Viscous Flow Due to a Stretching Surface with Partial Slip . . . . . . . . 22.1 The Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Application of OAFM to Viscous Fluid Given by Eqs. (22.8), (22.10), (22.11) and (22.12) . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 186

211 212 213 215 217 220 223 224 225 228 242

23 Axisymmetric MHD Flow and Heat Transfer to Modified Second Grade Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1 The Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 OAFM for Solving the System (23.18), (23.19), (23.21), (23.22) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

250 254 254 258 259 264

24 Thin Film Flow of an Eyring Powel Fluid on a Vertical Moving Belt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1 The Governing Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 24.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267 268 271 271 278 280 284

245 247

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25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium . . . . 25.1 The Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 OAFM for the Steady Flow of a Fourth Grade Fluid in a Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.4 Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.5 Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.6 Case 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.7 Case 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.8 Case 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.9 Case 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.10 Case 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.11 Case 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.12 Case 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.13 Case 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.14 Case 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

291 293 294 294 295 295 295 296 296 297 297 297 298 298 299 309

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 OAFM for Eqs. (26.15) and (26.16) . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 312 315 341

27 Cylindrical Liouville-Bratu-Gelfand Problem . . . . . . . . . . . . . . . . . . . . 27.1 OAFM for Cylindrical Liouville-Bratu-Gelfand Problem . . . . . . . 27.1.1 Case A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.1.2 Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2.1 Case A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2.2 Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2.3 Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

343 345 346 347 349 349 349 351 353

28 The Polytrophic Spheres of the Nonlinear Lane—Emden—Type Equation Arising in Astrophysics . . . . . . . . . . 28.1 The Nonlinear Lane—Emden Equation . . . . . . . . . . . . . . . . . . . . . . 28.2 OAFM for the Polytrophic Spheres of the Lane—Emden Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

285 286

355 356 358 360 360 363

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Contents

Part III Some Variants and Modifications of the Basic Optimal Auxiliary Functions Method 29 The Second Alternative to the Optimal Auxiliary Functions Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.1 Dynamic Response of a Permanent Magnet Synchronous Generator to a Wind Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.1.1 Application of an Alternative of OAFM to the Considered Problem (29.16) and (29.17) . . . . . . . 29.1.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2 Lambert W Function with Application in Electronics and Seismic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2.1 Evaluation of the Lambert W Function by OAFM . . . . 29.2.2 Application of the Lambert Function in Electronics and Seismic Waves . . . . . . . . . . . . . . . . . . 29.3 Nonlinear Blasius and Sakiadis Flows . . . . . . . . . . . . . . . . . . . . . . . 29.3.1 Approximate Solutions for the Blasius and Sakiadis Problems Using the Alternative of the OAFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.4 Poisson–Boltzman (P.B) Equations . . . . . . . . . . . . . . . . . . . . . . . . . 29.4.1 P.B Equation for a Charged Rod in Absence of Added Salt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.4.2 OAFM for P.B given by Eqs. (29.134) and (29.135) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.4.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Piecewise Optimal Auxiliary Functions Method . . . . . . . . . . . . . . . . . . 30.1 The Lane-Emden Equation of the Second Kind . . . . . . . . . . . . . . . 30.1.1 The Nonlinear Lane-Emden Equation of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.1.2 POAFM for the Lane-Emden Equation of Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Some Exact Solutions for Nonlinear Dynamical Systems by Means of the Optimal Auxiliary Functions Method . . . . . . . . . . . . 31.1 Some Exact Solutions for MHD Flow and Heat Transfer to Modified Second Grade Fluid with Variable Thermal Conductivity in the Presence of Thermal Radiation and Heat Generation/Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Some Exact Solutions for Eqs. (31.6)–(31.9) Using OAFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

367 368 370 372 375 380 389 392

394 404 406 408 411 414 417 417 419 421 432 435

436 437 444

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xiii

31.2.1

Case 1. The Flow of a Fourth Grade Fluid Past a Porous Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Case 2. The Flow of a Second Grade Fluid Over a Stretching Sheet with Suction/Injection . . . . . . . . . . . . 31.2.3 Case 3. Thin Film of an Oldroyd 6-Constant Fluid Over a Moving Belt . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.4 Case 4. Viscous Flow Due to a Stretching Surface with Partial Slip . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.5 Case 5. Thermal Radiation on MHD Flow Over a Stretching Porous Sheet . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.6 Case 6. Upper-Convected Maxwell Fluid Over a Porous Stretching Plate . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.7 Case 7. Unsteady Viscous Flow Over a Shrinking Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.8 Case 8. The Flow of a Viscous Incompressible Fluid Over a Porous Stretching Wall . . . . . . . . . . . . . . . . 31.3 Exact Solutions to Oscillations of Some Nonlinear Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.1 Oscillations of a Uniform Cantilever Beam Carrying an Intermediate Lumped Mass and Rotary Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Nonlinear Jerk Equations . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Exact Solutions to Duffing Equation . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Solutions of the Double-Well Duffing Equation . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

444 447 450 452 456 459 461 463 465

465 467 469 475 477

Part I

A Short Introduction to the Optimal Auxiliary Functions Method

Chapter 1

Introduction

A nonlinear system is a set of nonlinear equations—differential, integral, functional, algebraic, difference, or abstract operator equations, or a combination of some of these—used to describe a physical device or process that otherwise cannot be clearly defined by a set of linear equations of any kind. Dynamical system is used as a synonym for mathematical or physical system when the describing equations represent evolution of a solution with an independent variable [1]. The nonlinear systems are used to describe a great variety of engineering and scientific phenomena varying from social, life and physical sciences to engineering and technology. Theory of nonlinear dynamical systems has been applied to a rich spectrum of problems in various engineering disciplines and also in physics, chemistry, biology, medicine, economics, and mathematics. The vast majority of natural phenomena are nonlinear, with linearity being the exceptional but important case. Mathematically, the essential distinction between linear and nonlinear equations is the fact that any two solutions of a linear equation can be added to determine a new solution. This superposition principle is defining for solving any linear problem. In contrast, for nonlinear problems this principle is violated: two solutions of a nonlinear problem cannot be added together to form another solution. The characteristic of the linear problem is the regular motion in space and time (or other independent variable) can be described in terms of well-defined functions. Nonlinear problems show transitions from smooth motion to erratic or random behavior or to chaotic. Then, the response of a linear system to small changes in its parameters or to external stimulation is usually smooth and in direct proportional to the stimulation. On the contrary, for nonlinear problems, a small change in the parameters can produce an enormous qualitative distinction in the motion (for details see [2–6]). Furthermore if we compare the large variety of procedures available in linear problems theory, the techniques for analysis and design of nonlinear problems are limited to some specials cases. On the other hand, in the case of linear problems it is often possible to derive closed form expressions for the solutions of the system equations, but in general in case of nonlinear problems, this is not possible. A linear problem is that problem in which all © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_1

3

4

1 Introduction

the dependence of the current state on previous states can be expressed in terms of a linear combination. A nonlinear problem is a problem in which the dependence of the current state on previous state cannot be expressed entirely as a linear combination. Some of the dependence can be captured in a linear combination of the previous states, but something extra is required to capture all of the dependence. In consequence, it is desirable to be able to make some predictions about the behavior of a nonlinear problem even in the absence of closed form expressions for the solutions, and this lead to an approximate analysis, which is much less important to linear problems. The analysis of nonlinear problems makes use of a large variety of methods and mathematical tools than does the analysis of linear problems. In general, the level of mathematics needed to master the basic ideas of nonlinear problems analysis is higher than that for the linear problems. Although some analytical approaches have been applied over the centuries to cope with systems characterized by nonlinearity, until the advent of numerical procedures offered by computers its study has been relatively limited. In modeling a real-world process, it is important to realize that a natural system that can be described by a linear model in some circumstances must be described by a nonlinear model in others [7]. An example is illustrated by the moving of a pendulum in a plane. For small oscillations, the motion of the pendulum is approximated by a linear model, and the period is independent of its amplitude. But for the full nonlinear equation this results is false. The period depends on the amplitude. In a linear model, a closed-form solution can be obtained and the motion is described analytically for all time. For nonlinear equation, the solution for certain values of the driving frequency becomes chaotic and unpredictable. Linear mathematical systems tend to dominate even moderately advanced university courses. The mathematical intuition so developed equips the student to confront the bizarre behavior exhibited by the simplest nonlinear systems. Yet nonlinear systems are surely the rule, not the exception, not only in research but also in the everyday world [9]. It should also be mentioned [10] that the frontiers of nonlinear dynamical systems are constantly enlarged with new ideas and applications continually appearing on a regular basis in various research publications. At present there is a somewhat “piecemeal” approach to tackling nonlinear dynamical systems, but undoubtedly, as the subject matures, new mathematical techniques and concepts will be discovered and further unification will occurs. Our exploration of the nonlinear world will necessarily be somewhat unknown because the more complicated nonlinear models involve mathematical treatments that are either too lengthy or too complex. Unfortunately the nonlinear differential equations cannot usually be integrated by quadratures, the majority of concrete dynamical models do not admit a qualitative integration by a purely mathematical analysis. This inevitably leads to the use of approximate or numerical methods. It must also be free of unnecessary restriction which, paraphrasing Hadamard, are not dictated by the needs of science but by the abilities of the human mind [8]. Almost no systems are completely linear and linearity is an approximation to reality.

1 Introduction

5

Considering only linear problems lead not only to more quantitative errors, but also deform very often the quality of the phenomena. The engineering science of modern era with their fine instrumentation and advances calculus methods have shown that the establishment of movement laws neglect nonlinear terms lead to inadequate or even false results [11]. Linearity is a convention of classical science. For example, Newton’s laws of physics based themselves on the concept of a constant and static universe [12]. Time and space remained linear concepts. Later, the deterministic and absolute system of knowledge that was grounded in Newtonian mechanics and Euclidian geometry to describe the physical world began to unravel. The questions and problems with classical science suggested by Maxwell in electromagnetism and thermodynamics or Plank’s work in electrodynamics prove to be foundations for a history of nonlinear science, dynamics and complexity. In 1915 Einstein et al. [13] formulated the General Theory of Relativity. Moreover, adding a fourth dimension to the spatial dimensions of classical science, space–time is not flat and constant but rather a dynamic nonEuclidian geometry. John von Neumann directly identified the discoveries in quantum physics as a problem of linearity. The end of the nineteenth century was dominated by Poincare’s works on periodic solutions of ordinary differential equations, which constitute the foundation of the most part of results until now. The Poincare’s contribution to the analytical methods of nonlinear dynamics is the introduction of the small parameter, and the notion of generating solution. He wrote “if we knew exactly the law of nature and the initial situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment”. Poincare discovery of sensitive dependence on initial conditions in what are now termed chaotic dynamical systems. From the other point of view, H. Poincare anticipated the problems of linear methods to explain a nonlinear world as early as 1894 [14]. The history of differential equations is known from around 1690 with Newton and Leibniz. The laws of Newton have provided a foundation for the mathematical modeling of many problems including celestial mechanics. The solutions of the socalled “two body problem” give an interesting explanation of the Kepler’s laws. The works of Poincare stimulated many of the abstract developments in today’s mathematics. Newton and Leibniz were always being mentioned together as the co-inventors of the calculus. Leibniz first conceived his differential and integral calculus around 1675 and 1677 he had a fully developed and workable system. He thought in terms of differentials small increments in the values of the variables. Very strong interactions between theoretical researches and practical implications in engineering or physical systems appear in Russia, starting up to the works of A.M. Lyapunov. He sought information about stability properties of solutions without actually knowing explicit solutions. Lyapunov contributions have lead to methods which yield quantitative and qualitative information about stability and automatic control. Following Poincare’s and Hadamard’s studies of geodesic flows, G.D. Birkhoff showed that near any homoclinic point of a two-dimensional map there is an infinite

6

1 Introduction

sequence of periodic orbits whose periods approach infinity [15, 16]. In the 1920s and 1930s the study of the van der Pol and other radio-engineers have attracted special attention on the so-called relaxation oscillations or on the periodic solutions to nonlinear dissipate equations with a few degrees of freedom. In 1920, Mandelstham formulated the problem of the construction of mathematical tools fitted to the study of nonlinear oscillations in connection with the study of nonlinear systems to radio-engineering. A student of Mandelstham, Andronov established a strong group in dynamics in Soviet Union at Gorki. Together with Pontryagin, introduced structural stability and began a study of local bifurcation [17]. Andronov et al. [18] expanded version of Ref. [17], remains an introduction to nonlinear systems for applied scientists. Lefschetz defined strange attractor and published a text on qualitative theory [19]. The works of Kolmogorov, Anosov, Arnold and Sinai grew out of this “Moscow school” with important work on ergotic theory geodesic flows and billiards [20–23]. Van der Pol’s work was one of the motivations for Cartwright and Littlewood [24] study of the van der Pol equation, which in turn led to that of Levinson [25]. Levinson simplified the problem by replacing the cubic nonlinearity with a piecewise linear function, and he provided a more explicit analysis. Hopf had extended the PoincareAndronov bifurcation that now generally numbers his name to n > 2 dimensions [26]. Peixoto [27] generalized Andronov and Pontryagin’s theory of planar systems to two-dimensional manifolds. After 1950’s Smale [28] applied a topological approach to the study of dynamical systems. Extending Andronov and Peixoto’s work on structural stability to n > 2 dimensions, he defined Morse-Smale systems and conjectured that a system is structurally stable if it is Morse-Smale. Moser [29] gave a beautiful exposition that included explicit criteria for the presence of horseshoes in two-dimensional maps. Independently, Melnikov [30] used regular perturbation methods to prove the existence of transverse homoclinic orbits to periodic motions in ordinary differential equations. Arnold [31] generalized these ideas to produce the first example of what is called Arnold diffusion. In 1959, Chirikov [32] had introduced the first analytical estimate for the onset of chaotic motion in deterministic Hamiltonian systems. In 1956 Malkin [33] published his book about a method of successive approximations in the Lyapunov sense by means of the method of small parameter. Analytical methods were developed by the Krylov–Bogoliubov school from Kiev. Krylov–Bogoliubov method have a close foundation to the van der Pol studies about oscillators and later the asymptotic method due to Mitropolski constitutes an improvement with the use of asymptotically convergence series expansions. It is the same for the averaging method and method of accelerating the convergence [34, 35]. The Hayashi’s school in nonlinear oscillators developed many studies oriented toward electric circuits [36] especially by him and his disciples Kawakami and Ueda. Lorenz’s numerical study [37] of a three-dimensional ordinary differential equation modeling Rayleigh–Benard convection introduced sensitive dependence of initial conditions in meteorology. Ruelle and Takens studied the fluid turbulence [38]. In the last years, efforts were made for collecting the main results concerning dynamical systems in monographs. In this respect, we mention the books of Nayfeh

1 Introduction

7

and Mook [39], Hagedorn [40], Guckenheimer and Holmes [41], Hirsh et al. [42] and so on. Nonlinearities are commonplace in engineering systems. They result from the fluid–structure interaction, finite belt stretching, structural properties which lead to nonlinear elastic force, internal combustion, geometrical and kinematic configurations, inertial effects including rotational loadings, deformations such as curvatures and buckled states, machine tool chatter vibrations, aerodynamic effects, the elastic foundations, systems with elastic deformations [43, 44] and so on. The exploration of various domains of the nonlinear dynamical systems is a vast subject, not only in research, but also in the everyday world. It is a reality that we would all be better if more people realized that simple nonlinear systems do not necessarily posses simple dynamical properties [9]. The study of nonlinear dynamical systems raised great mathematical difficulties. To overcome these difficulties, different tools have been proposed [43], such as the use of Fokker-Plank-Kolmogorov equations, the method of statistical linearization and the perturbation method. Crandall and Mark [44] proposed and applied a perturbation technique, which is an extension to random vibrations of the perturbation technique used for weakly nonlinear deterministic systems. At present, there are a great number of works concerning the theory of linear and nonlinear random vibrations. In the last two centuries has occurred due in large measure to the ability of investigators to respect physical laws in terms of rather simple equations. But the governing equations were not simple, therefore certain assumptions, more or less consistent with the physical situation, were employed to reduce the equations to type more easily solvable. The linear analysis often is insufficient to describe the behavior of physical systems adequately. In the last few years, some interesting results have been obtained for solving various nonlinear problems. There exist many possibilities to solve nonlinear dynamical systems such as: Lindstedt-Poincare perturbation method [45], Differential Transformation Method [46], Adomain Decomposition Method [47], Homotopy Analysis Method [48] Variational Iteration Method [49] Boundary Element Method [50], Optimal Iteration Method [51], Optimal Homotopy Perturbation Method [52], Optimal Homotopy Asymptotic Method [53] and so on. The above mentioned procedures work very well for weakly nonlinear dynamical systems and some of them work even for strongly nonlinear problems. Our goal is to give a glimpse of the nonlinear dynamical systems for solving a nonlinear problem. Our intention is to make the reader familiar and to understand and analyze different nonlinear models that will be presented in our book. For this aim, we deal with the Optimal Auxiliary Functions Method (OAFM) in a proper manner entirely different in comparison with any other techniques. To solve a nonlinear dynamical system, our approach does not need restrictive hypotheses, is very rapid convergent, usually after the first iteration and the convergence of the solutions is ensured in a rigorous way. The cornerstone of the validity and flexibility of our procedure is the choice of the linear operators and the optimal auxiliary functions which contribute to highly accurate solutions. The parameters which

8

1 Introduction

are involved in the composition of the optimal auxiliary functions are optimally identified via various methods in a rigorous way from mathematical point of view. The nonlinear governing equation is reduced to two linear differential equations, which do not depend on all terms of the nonlinear equation.

References 1. G. Chen, Stability of Nonlinear Systems (in Encyclopedia of RF and Microwave Engineering, Wiley, New York, 4891–4896, 2004) 2. D.K. Campbell, Nonlinear science from paradigms to practicalities in theory of nonlinearity. Los Alamos Sci. 15, 218–262 (1987) 3. W.J. Rugh, Nonlinear System Theory (John Hopkins University Press, 1981) 4. L.P. Shilnikov, A. Shilnikov, D.V. Turaev, L.O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part I (World Scientific, 1998) 5. M. Vidyasagar, Nonlinear Systems Analysis, 2th edn (Prentice Hall, 1993) 6. S.M. Boker, Linear and Nonlinear Dynamical Systems (University of Virginia, 1996) 7. D. Campbell, J. Crutchfield, J.D. Farmer, E. Jen, Experimental mathematics: the role of computation in nonlinear science. Commun. ACM 28, 374–384 (1985) 8. J.K. Hale, J.P. LaSalle, Differential equations: linearity vs. nonlinearity. SIAM Rev. 5, 249–272 (1963) 9. R. May, Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976) 10. R.H. Enns, It’s a Nonlinear World (Springer, New York, 2011) 11. V. Marinca, N. Herisanu, The Optimal Homotopy Asymptotic Method. Engineering Applications (Springer, Cham, 2015) 12. K. Burke, Nonlinearity (The history and philosophy of the science (University of Missouri, St. Louis, 2009) 13. A. Einstein, H.A. Lorentz, H. Weyl, H. Minkowsky, The Principle of Relativity: a Collection of Original Memoires on the Special and General Theory of Relativity (Dover Publications Inc., 1952) 14. H. Poincare, Les nouvelles methods de la mecanique celeste, Toms 1, 2 et 3 (Paris, 1892, 1893, 1894) 15. P. Holmes, History of dynamical systems. Scholarpedia 2, 1843 (2007) 16. G.D. Birkhoff, Dynamical Systems (American Mathematical Society, Providence, RI, 1927) 17. A.A. Andronov, L. Pontryagin, Systemes grossiers. Dokl. Akad. Nauk SSSR 14, 247–250 (1937) 18. A.A. Andronov, A.A. Vitt, E.E. Khaikin, Theory of Oscillators (Moscow, 1937) 19. S. Lefschetz, Differential Equations (Geometric theory (Interscience Publisher, New York, 1957) 20. A.N. Kolmogorov, La theorie generale des systems dynamiques et la mecanique classique. Int. Cong. Math. Amsterdam 1, 315–323 (1954) 21. D.V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90, 1–235 (1967) 22. L. Arnold, Random Dynamical Systems (Springer, Heidelberg, 1998) 23. Ya.G. Sinai, Classical dynamical systems with countable multiplicity Lebesgue spectrum. I. Izvestia Acad. Sci. USSR, Mat. 30, 15–68 (1970) 24. M.L. Cartwright, J.E. On nonlinear differential equations of the second order: I.  Littlewood,  the equation y¨ + k 1 − y 2 y + y = bλk cos(λl + α), k large. J. London Mathem. Soc. 20, 180–189 (1945) 25. N. Levinson, A second-order differential equation with singular solutions. Ann. Math. 50, 127–153 (1949)

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26. E. Hopf, Abzweigung einer periodischen Losung von einer stationaren Losung eines differential system. Berichte Math.-Phys. Kl. Sachs. Acad. Wiss. Leipzig Math.-Nat. Kl 94, 1–22 (1942) 27. M.M. Peixoto, Structural stability on two-dimensional manifolds. Topology 1, 101–120 (1962) 28. S. Smale, Differentiable dynamical systems. Bul. Amer. Math. Soc. 73, 747–817 (1967) 29. J.K. Moser, Stable and Random Motions in Dynamical Systems (Princeton University Press, Princeton, 2001) 30. V.K. Melnikov, On the stability of the center for time-periodic perturbation. Trans. Moscow Math. Soc. 12, 1–57 (1963) 31. V.I. Arnold, Instability of dynamical systems with several degrees of freedom. Sov. Math. Dokl. 5, 342–355 (1964) 32. B.V. Chirikov, Resonance processes in magnetic traps. J. Nucl. Energy. Part C Plasma Phys. 1, 253 (1960) 33. I.G. Malkin, Some problems of the Theory of Nonlinear Oscillations (Gostehizdat, Moscow, 1956) 34. N. Bogoliubov, Y. Mitropolski, A.M. Samoilenko, Method of the Acceleration of Convergence in Nonlinear Mechanics (Naukova Dumka, Kiev, 1960) 35. Y. Mitropolski, TThe Average Method in Nonlinear Mechanics (Naukova Dumka, Kiev, 1971) 36. C. Hayashi, Nonlinear Oscillations in Physical Systems (McGraw-Hill, New York, 1964) 37. E.N. Lorenz, Deterministic non-periodic flow. J. Atmos. Sci. 20, 134–141 (1963) 38. D. Ruelle, F. Takens, On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971) 39. A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations (Willey, New York, 1974) 40. P. Hagedorn, Nonlinear Oscillations (Clarendon Press, Oxford, 1981) 41. J. Guckenheimer, P. Holmes, Nonlinear Oscillation, Dynamical Systems and Bifurcations of Vector Fields (Springer, New York, 1983) 42. M.W. Hirsch, S. Smale, R.L. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos, 2nd edn. (Academic Press, Elsevier, San Diego, CA, 2004) 43. V. Marinca, N. Herisanu, Nonlinear Dynamical Systems in Engineering (Some approximate approaches (Springer Verlag, Berlin, Heidelberg, 2011) 44. S.N. Crandall, W.D. Mark, Random vibration in Mechanical Systems (Academic press, New York, 1963) 45. Y.K. Cheung, S.N. Chen. S.L. Lau, A modified Lindstedt-Poincare method for certain strongly nonlinear oscillators. Int. J. Non-Linear Mech. 26, 367–378 (1991) 46. M. Hatami, D.D. Ganji, Differential Transformation Method for Mechanical Engineering Problems (Elsevier, 2016) 47. G. Nhawu, P. Mafuta, J. Mushanyu, The Adomain decomposition method for numerical solution of first-order differential equations. J. Math. Comput. Sci. 6, 307–314 (2016) 48. S.J. Liao, Homotopy Analysis Method in Nonlinear Differential Equations (Springer, 2011) 49. J.H. He, Variational iteration method, a kind of nonlinear analytical technique. Some examples. Int. J. Non-Linear Mech. 34, 699–708 (1999) 50. Y.Y. Wu, S.J. Liao, X.Z. Zhao, Some notes on the general boundary element method for highly nonlinear problems. Commun. Nonlinear Sci. Numer. Simul. 10, 725–735 (2005) 51. V. Marinca, N. Herisanu, An optimal iteration method for strongly nonlinear oscillators. J. Appl. Math, ID 906341 (2012) 52. N. Herisanu, V. Marinca, Optimal homotopy perturbation method for a non-conservative dynamical system of a rotating electrical machine. Z. Naturforsh. 67, 509–516 (2012) 53. V. Marinca, N. Herisanu, An optimal homotopy asymptotic approach to nonlinear MHD JefferyHamel flow. Mathem. Problems Eng., Article ID 169056 (2011)

Chapter 2

The Optimal Auxiliary Functions Method

To apply the Optimal Auxiliary Functions Method (OAFM), we consider the following general nonlinear differential equation. L[u(x)] + N [u(x)] + g(x) = 0, x ∈ D

(2.1)

subject to the boundary/initial conditions   du(x) =0 B u(x), dx

(2.2)

where L is a linear operator, N is a nonlinear operator, g is a known function, x is independent variable, u(x) is an unknown function, D is the domain of interest and B is a boundary operator. It is noteworthy to mention that the linear operator L does not necessarily coincide in its entirely with the linear part of the equation under study. Henceforward, u(x) ˜ will be the approximate solution of Eqs. (2.1) and (2.2) and assume that u(x) ˜ can be expressed in the following form, only with two components u(x) ˜ = u 0 (x) + u 1 (x, Ci )

(2.3)

where Ci are p parameters, unknown at this moment, and p is an arbitrary positive integer number. The initial approximation u 0 (x) and the first approximation u 1 (x, Ci ) will be determined as follows. Inserting Eq. (2.3) into Eq. (2.1), one get L[u 0 (x)] + L[u 1 (x, Ci )] + N [u 0 (x) + u 1 (x, Ci )] + g(x) = 0

(2.4)

The initial approximation u 0 (x) can be determined from the following linear equation © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_2

11

12

2 The Optimal Auxiliary Functions Method

L[u 0 (x)] + g(x) = 0

(2.5)

with the corresponding boundary/initial conditions   du 0 (x) =0 B u 0 (x), dx

(2.6)

It should be emphasized that the linear operator L depends on the boundary/initial conditions (2.2) and the Eq. (2.1), and the function g(x) are not unique. It is clear that the initial approximation u 0 (x) is well-determined from the linear differential Eq. (2.5) with the boundary/initial conditions (2.6). Now, taking into consideration Eqs. (2.5) and (2.6), the first approximation u 1 (x) is obtained from the nonlinear differential equation L[u 1 (x, Ci )] + N [u 0 (x) + u 1 (x, Ci )] = 0

(2.7)

with the boundary/initial conditions   du 1 (x, Ci ) =0 B u 1 (x, Ci ), dx

(2.8)

The nonlinear term of Eq. (2.7) is developed in the form N [u 0 (x) + u 1 (x, Ci )] = N [u 0 (x)] +

 u k (x, Ci ) 1

k≥1

k!

N (k) [u 0 (x)]

(2.9)

where k! = 1 · 2 · 3 · . . . · k and N (k) means the differentiation of order k of nonlinear operator N. To avoid the difficulties that appear in solving the nonlinear differential Eq. (2.7) and to accelerate the convergence of the first approximation u 1 (x, Ci ) and implicitly of the approximate solution u(x), ˜ instead of solving the following equation, obtained from (2.7) and (2.9) L[u 1 (x, Ci )] + N [u 0 (x)] +

 u k (x) 1

k≥1

k!

N (k) [u 0 (x)] = 0

(2.10)

we make the following observations. In general, the solution of linear differential Eqs. (2.5) and (2.6) can be expressed as u 0 (x) =

m1  i=1

ai f i (x)

(2.11)

2 The Optimal Auxiliary Functions Method

13

where the coefficients ai , the functions f i (x) and the positive integer m1 are known. The nonlinear operator N [u 0 (x)] calculated for u 0 (x) may be written in the form N [u 0 (x)] =

m2 

b j g j (x)

(2.12)

j=1

where the coefficients b j , the functions g j (x), and the positive integer m2 are known and depend on the initial approximation u 0 (x) and also on the nonlinear operator N. In what follows, because Eq. (2.10) is very difficult to be solved, we do not solve this equation, but from the theory of differential equations, Cauchy method, the method of influence functions, the operator method and so on [1], it is more convenient to consider the unknown first approximation u 1 (x, Ci ) depending on u 0 (x) and N [u 0 (x)]. More precisely,u 1 (x, Ci ) can be determined from the linear equation L[u 1 (x, Ci )] +

p 

  Ck Fk f i , g j = 0, i = 1, 2, . . . , m 1 ,

j = 1, 2, . . . m 2

k=1

(2.13) where Ck are p unknown parameters and Fk are so-called auxiliary functions depending on the functions fi and gi involved in the Eqs. (2.11) and (2.12) respectively. These functions fi and gi are source for the auxiliary functions Fk . We have a great freedom to choose the values of the positive integer p and of the auxiliary functions Fk . We note that the boundary/initial conditions could be fulfilled by Eq. (2.13) so that finally Eq. (2.3) responds to all boundary/initial conditions given by Eq. (2.2). After using the previous considerations, we can write that for example if fi (x) and gj (x) are exponential functions: u 0 (x) = 3e x + 5e2x + 7e−x ,   a1 = 3; a2 = 5; a3 = 7; f 1 = e x ; f 2 = e2x ; f 3 = e−x , m 1 = 3

(2.14)

  N [u 0 (x)] = 4e x + 6e2x , b1 = 4; b2 = 6; g1 = e x ; g2 = e2x , m 2 = 2

(2.15)

then Fk are exponential functions F1 (x) = e x ;

F2 (x) = e2x ;

F3 (x) = e−x ;

F4 (x) = e−3x ( p = 4)

(2.16)

and therefore Eq. (2.13) can be rewritten as L[u 1 (x, Ci )] + C1 e x + C2 e2x + C3 e−x + C4 e−3x = 0

(2.17)

14

2 The Optimal Auxiliary Functions Method

Another example: if fi (x) and gj (x) are polynomial functions, then Fk are also polynomial functions: u 0 (x) = 11x + 12x 3 + 13x 5

  ai = 10 + i, f i (x) = x 2i−1 , i = 1, 2, 3, m 1 = 3 (2.18)

N [u 0 (x)] = 14x 2 + 15x 3 + 16x 4



 b j = 13 + j, g j (x) = x 1+ j , j = 1, 2, 3, m 2 = 3

(2.19) Fk (x) = x k+2 , k = 1, 2, 3, 4, 5 ( p = 5)

(2.20)

In this case, Eq. (2.13) becomes L u 1 (x, Ci ) + C1 x 3 + C2 x 4 + C3 x 5 + C4 x 6 + C5 x 7 = 0

(2.21)

Now, if for example f i (x) are polynomial functions and g j (x) are exponential functions, then Fk are combinations of polynomial and exponential functions. u 0 (x) = 21x + 22x 2 ,

 ai = 21 + i,

N [u 0 (x)] = 31e2x + 32e3x + 33e4x ,



ji (x) = x i , i = 1, 2; m 1 = 2



(2.22)

 b j = 30 + j, g j = e( j+1)x , j = 1, 2, 3, m 2 = 3

(2.23) Fk (x) = x k e(5−k)x , k = 1, 2, 3 ( p = 3)

(2.24)

such that Eq. (2.13) can be written as L u 1 (x, Ci ) + C1 xe4x + C2 x 2 e3x + C3 x 3 e2x = 0

(2.25)

and so on. In all these examples the auxiliary functions Fk are not unique. Instead of Eqs. (2.16), (2.20) and (2.24) respectively, we can also consider the following expressions F1 (x) = e2x ,

F2 (x) = e4x ( p = 2)

(2.26)

Fk (x) = x k+1 , k = 1, 2, 3 ( p = 3)

(2.27)

Fk (x) = x 2k e(4−k)x , k = 1, 2, 3, 4 ( p = 4)

(2.28)

The approximate analytical solution of Eqs. (2.1) and (2.2) is obtained from Eqs. (2.3), (2.5) and (2.13). Finally, the unknown parameters Ck , k = 1, 2,…, p can be optimally identified via rigorous mathematical procedures such as the least square method, Ritz method,

2 The Optimal Auxiliary Functions Method

15

Galerkin method, collocation method, Kantorovich method and so on. The preferred approach would be to minimize the square residual error by computing the functional [2–21]:

    (2.29) J C1 , C2 , . . . , C p = R 2 x, C1 , C2 , . . . , C p d x (D)

where D is the domain of interest, and the residual R is given by     ˜ Ci ) + N u(x, R x, C1 , C2 , . . . , C p = L u(x, ˜ Ci ) + g(x), i = 1, 2, . . . , p (2.30) and u(x, ˜ Ci ) is obtained from Eq. (2.3). The unknown parameters C1 , C2 ,…,Cp can be identified from the conditions ∂J ∂J ∂J = = ... = =0 ∂C1 ∂C2 ∂C p

(2.31)

In this way, the optimal values of the convergence-control parameters and the optimal auxiliary functions Fk are known. Further, with these parameters known, the approximate solution u(x) ˜ is well-determined. In comparison with any other known methods applied to find approximate analytical solutions for nonlinear dynamical systems, our technique is based upon original construction of the solution using a moderate number of convergence-control parameters Ci (i = 1, 2,…, p) which are components of the so-called optimal auxiliary functions Fk . For the sake of brevity, it is very important to remark that these optimal convergence-control parameters lead to a high precision, when comparing our approximate solutions with exact or numerical solutions or with other known results in the literature. Let us note that the nonlinear differential Eqs. (2.1) and (2.2) are reduced to two linear differential equations, which do not depend on all terms of the nonlinear operator N [u 0 (x)]. Also, the construction of equations which determine the initial approximation (2.5) and the first approximation (2.13) are not unique. We have a large freedom to choose the number p of the optimal convergence-control parameters, the auxiliary functions Fk and some terms from the nonlinear operator N [u 0 (x)]. The accuracy of the results obtained through OAFM is growing along with increasing the number of optimal convergence-control parameters Ci . The values of the convergence-control parameters Ci are determined using rigorous mathematical procedures. OAFM is an iterative approach which often rapidly converges to the exact solution after the first iteration. Our procedure lead to very accurate results, is effective, explicit, simple and provides a rigorous way to control and adjust the convergence of the solutions using only the first iteration, comparing to other procedures which need more iterations to achieve accurate results.

16

2 The Optimal Auxiliary Functions Method

We remark the construction of the linear operator L and of the auxiliary functions Fk . These remarks are in fact the true power of our procedure in solving nonlinear problems without small or large parameters.

References 1. L. Elsgolts, Differential Equations and Calculus of Variations (Mir Publisher Moskow, 1977) 2. N. Herisanu, V. Marinca, Approximate Analytical Solutions to Jerk Equations. Springer Proceedings in Mathematics and Statistics (2015) 3. R.D. Ene, V. Marinca, B. Marinca, Thin film flow of an Oldroyd 6-constant fluid over a moving belt: an analytic approximate solution. Open Phys. 14, 44–64 (2016) 4. V. Marinca, N. Herisanu, Vibration of nonlinear nonlocal elastic column with initial imperfection. Springer Proc. Phys. 198, 49–56 (2018) 5. B. Marinca, V. Marinca, Approximate analytical solution for thin film flow of a fourth grade fluid down a vertical cylinder. Proc. Romanian Acad. Seri. A 19, 69–76 (2018) 6. V. Marinca, N. Herisanu, The nonlinear thermomechanical vibration of a functionally graded beam on Winkler-Pasternak foundation. MATEC Web of Confer. 148, 13004 (2018) 7. V. Marinca, B. Marinca, Optimal auxiliary functions method for nonlinear thin film flow of a third grade fluid on a moving belt. Proc. Romanian Acad. Seri. A 19, 575–580 (2018) 8. B. Marinca, V. Marinca, Some exact solutions for MHD flow and heat transfer to a modified second grade fluid with variable thermal conductivity in the presence of thermal radiation and heat generation/absorption. Comput. Math. Appl. 76, 1515–1524 (2018) 9. V. Marinca, R.D. Ene, B. Marinca, Optimal auxiliary functions method for viscous flow due to a stretching surface with partial slip. Open Eng. 8, 261–274 (2018) 10. N. Herisanu, V. Marinca, G. Madescu, Application of the optimal auxiliary functions method to a permanent magnet synchronous generator. Int. J. Nonlinear Sci. Numer. Simul. 20, 399–406 (2019) 11. N. Herisanu, V. Marinca, Gh. Madescu, F. Dragan, Dynamic response of a permanent magnet synchronous generator to a wind gust. Energies 12, 915 (2019) 12. N. Herisanu, V. Marinca, An effective analytical approach to nonlinear free vibration of elastically actuated microtubes, Meccanica (2021) 13. N. Herisanu, V. Marinca, Free oscillations of Euler-Bernoulli beams on nonlinear WinklerPasternak foundation. Springer Proc. Phys. 198, 41–48 (2018) 14. N. Herisanu, V. Marinca, Analytical Investigation to Duffing Harmonic Oscillator , 26-Th International Conference Noise and Vibration, Nis, 2018, pp. 177–179 15. V. Marinca, N. Herisanu, Free Vibration of Tapered Beams, 26-Th International Conference Noise and Vibration, Nis, 2018), pp. 181–183 16. V. Marinca, N. Herisanu, Optimal auxiliary functions method for thin film flow of a fourth grade fluid down a vertical cylinder. Ro. J. Tech. Sci-Appl. Mechan. 62, 183–189 (2017) 17. N. Herisanu, V. Marinca, C. Opritescu, An approximate analytical solution of transversal oscillation with quintic nonlinearities. Springer Proc. Physics. 25, 41–49 (2021) 18. N. Herisanu, V. Marinca, Analysis of nonlinear dynamic behavior of a rotating electrical machine rotor-bearing system using optimal Auxiliary functions method. Springer Proc. Mathem. Statist. 249, 159–168 (2018) 19. V. Marinca, N. Herisanu, Optimal auxiliary functions method for nonlinear vibration of doubly clamped nanobeam incorporating the Casimir force. Springer Proc. Phys. 251, 41–49 (2021) 20. V. Marinca, N. Herisanu, Nonlinear Dynamical Systems in Engineering. Some Approximate Approaches (Springer, Berlin, Heidelberg, 2011). 21. V. Marinca, N. Herisanu, The Optimal Homotopy Asymptotic Method . Engineering applications (Springer, Cham, 2015)

Part II

The Optimal Auxiliary Functions Method in Engineering Applications

Chapter 3

The First Alternative of the Optimal Auxiliary Functions Method

In this chapter, we will actually solve the Eq. (2.13) from which the first approximation u 1 (x, Ci ) can be determined. In the next chapter, the first approximation u 1 (x, Ci ) is not obtained by solving the Eqs. (2.13) and (2.8).

3.1 Dynamics of an Angular Misaligned Multirotor System Misalignment is one of the most commonly observed faults in rotating mechanical systems [1, 2], being often a major cause of machinery vibration and representing a great concern to designers and maintenance engineers, since perfect alignment cannot be achieved in practice. Even if a perfect alignment is initially realized, in many cases this condition cannot not be maintained for long time during the working regime due to many disturbing factors, such as base foundation movements, improper machine assembly, thermal distortion and lubrication effects [3]. Besides many other possible defects in rotating machines (such as rotor unbalance, rotor bends, cracks, rubs), misalignments generate important dynamical problems which should be avoided, especially when higher speeds and higher loads are envisaged. Obviously, the importance of these aspects is increased for rotating machines having a high capital cost. In principle, there are two types of coupling misalignments: parallel and angular, but often a combination of parallel and angular misalignment is common in the practice of rotating machinery. The problem of misalignment in rotating machines is both analytically and experimentally investigated. Analytical predictions are often useful in order to have a deeper inside into the essence of the phenomenon, especially in the stage of design, while experimental investigations are performed in practice on existent mechanical systems in order to diagnose possible defects. Both analytical and experimental approaches are useful in order to provide better functioning conditions from dynamical point of view. Basically, the analytical and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_3

19

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3 The First Alternative of the Optimal Auxiliary Functions Method

numerical approaches are the most inexpensive ways to analyze and predict misalignment phenomenon. However, the experimental approach remains the way to validate these predictions. A combination of these approaches could be sometimes useful. The need for a complete understanding of the phenomena related to misalignment generates in the last decades a large body of literature. Al-Hussain [4] examined the effect of misalignment on the stability of two rotors connected by a flexible mechanical coupling subjected to angular misalignment. In another work, Al-Hussain and Redmond [5] examined from theoretical and numerical point of view the effect of pure parallel misalignment on the lateral and torsional responses of two rotating shafts revealing that the natural frequencies are excited due to parallel misalignment. Sekhar and Prabhu [6] proposed a higher order FEM model for a rotor-coupling-bearing system which incorporates coupling misalignment reaction forces and moments, which have been determined for both parallel and angular misalignments. By using the proposed model, the vibration responses can be predicted. Beside analytical and numerical approaches, abundant experimental investigations of misalignments using various techniques were reported. Pérez et al. [7] characterized the parallel misalignment in rotating machines presenting the Discrete Time Interval Measurement System (DTIMS) as an effective experimental approach to detect and measure the angular vibration that is produced when a parallel misalignment between coupled shafts of rotating machines occurs. Sinha et al. [8] illustrated a method that can reliably estimate both the rotor unbalance and misalignment from a single machine run-down. The method is demonstrated using experimental data from a machine with two bearings and a flexible coupling to the motor. Experimental investigations on vibration response of misaligned rotors were conducted by Patel and Darpe [9], which efficiently used full spectra and orbit plots to reveal the unique nature of misalignment fault, proposing new misalignment diagnostics recommendations. In the existing literature there are few publications concerned with analytical investigation of misalignment, mainly because of the complexity of the governing equations, which are difficult to be solved analytically without adopting harsh simplifying hypotheses. There are available in the literature many methods intended to analytically solve nonlinear problems, such as the Multiple Time Scales Method [10], the Jacobi elliptic functions method [11], the Adomian Decomposition Method (ADM) [12], the Variational Iteration Method (VIM) [13], the Homotopy Analysis Method (HAM) [14], the Homotopy Perturbation Method (HPM) [15], the Optimal Homotopy Asymptotic Method (OHAM) [16–18]. the Optimal Parametric Iteration Method (OPIM) [19], the Differential Transform Method (DTM) [20], some of them being applicable also for strongly nonlinear problems as is the case of misalignment problems, but no substantial analytical results were reported so far. Starting from this reality, the present section is devoted toward an application of the OAFM to investigate the problem of angular misalignment of two rigid rotors connected by a flexible coupling, the bearings being isotropic in dynamic performances. This system, which is numerically investigated in [21], generates a strongly nonlinear problem, very

3.1 Dynamics of an Angular Misaligned Multirotor System

21

hard to be solved through classical analytical techniques (without considering small parameters). To the knowledge of the authors, no study has presented explicit analytical solutions to such a complex problem without using simplifying hypotheses. In this section we will develop a dynamic analysis using no simplifying hypothesis and avoiding the assumption of small parameters. Highly accurate explicit analytical solutions will be obtained for a very complicated nonlinear dynamical system, which will prove the efficiency of the proposed approach.

3.1.1 The Governing Equations In Fig. 3.1 is shown the model of the multirotor system with flexible couplings used to transmit torque from one rotor to another. We consider that the two rotors are rigid and the bearings are isotropic, m1 and m2 are the lumped masses, k1 and k2 are stiffness of the bearings and kt is the angular stiffness for the flexible couple, the coordinates of the mass centres of discs 1 and 2 are O1 (x1 ,y1 ,z1 ) and O2 (x2 ,y2 ,z2 ), referring to the coordinates system Oxyz which is set up in the static equilibrium position (Fig. 3.1). If x and y are the displacement of the disc 1 at the geometric centre, θ is the angle between rotors,  is the rotating speed, γ is the initial phase angle, and e1 and e2 are the unbalanced masses, then we have x1 = x + e1 cos t y1 = y + e1 sin t z 1 = z C1 = const. and x2 = x + l sin θ cos t + e2 cos θ cos(t + γ)

Fig. 3.1 The model of the angular-misaligned rotor system

(3.1)

22

3 The First Alternative of the Optimal Auxiliary Functions Method

y2 = y + l sin θ sin t + e2 cos θ sin(t + γ) z 2 = z C1 + l(1 + cos θ)

(3.2)

The terms related to e2 could be neglected since the offset h of disc 2 is much larger than e2 . In these conditions, using the Lagrange’s equation after computing the kinetic and potential energy, and considering that m1 = m2 = m, k1 = k2 = k, e1 = e, then the governing equations can be written as [21, 22]: ..

.2

.

2m x¨ + ml θ cos θ cos t − ml θ sin θ cos t − 2ml θ cos θ sin t − ml2 sin θ cos t + 2kx + kl sin θ cos t − mel2 cos t = 0 ..

.2

(3.3)

.

2m y¨ + ml θ cos θ sin t − ml θ sin θ sin t + 2ml θ cos θ cos t − ml2 sin θ sin t + 2ky + kl sin θ sin t − mel2 sin t = 0

(3.4)

..

ml 2 θ +ml x¨ cos θ cos t + ml y¨ cos θ sin t − ml 2 2 sin θ cos θ + klx cos θ cos t + kly sin θ sin t + kl 2 sin θ cos θ + kt θ = 0

(3.5)

Now, substituting x¨ from Eq. (3.3) and y¨ from Eq. (3.4) into Eq. (3.5), after some simplifications we obtain .. .2 1 1 ml 2 θ(1 + sin2 θ) + ml 2 θ sin 2θ − ml 2 2 sin 2θ 2 2 1 2 2 + kl sin 2θ + 2kt θ + mel cos θ = 0 2

(3.6)

From Eqs. (3.6) and (3.3) one can get .2

.

ml x(1 ¨ + sin2 θ) − ml 2 θ sin θ cos t − 2ml 2  θ cos θ sin t .

+ ml 2  θ cos3 θ sin t − ml 2 2 sin3 θ cos t + kl 2 sin3 θ cos t + klx(1 + sin2 θ) − kt θ cos θ cos t − mel2 cos t = 0

(3.7)

and from Eqs. (3.6) and (3.4) it holds that .2

.

ml y¨ (1 + sin2 θ) − ml 2 θ sin θ sin t + 2ml 2  θ cos θ cos t .

− ml 2  θ cos3 θ cos t − ml 2 2 sin3 θ sin t + kl 2 sin3 θ sin t + kly(1 + sin2 θ) − kt θ cos θ sin t − mel2 sin t = 0 (3.8) If r is the radius of rotor and making the transformations

3.1 Dynamics of an Angular Misaligned Multirotor System

y e l x X = , Y = , E = , L = , τ = t, ω0 = r r r r



k , ωt = m2

23



kt dX , X = 2 2 ml  dτ (3.9)

then Eqs. (3.6), (3.7) and (3.8) in the non-dimensional form may be written as X  (1 + sin2 θ) − Lθ2 sin θ cos τ − 2Lθ cos θ sin τ + Lθ cos3 θ sin τ + (ω20 − 1)L sin3 θ cos τ + ω20 X (1 + sin2 θ) − ω2t Lθ cos θ cos τ − E cos τ = 0 (3.10) Y  (1 + sin2 θ) − Lθ2 sin θ sin τ + 2Lθ cos θ cos τ − Lθ cos3 θ cos τ + (ω20 − 1)L sin3 θ sin τ + ω20 Y (1 + sin2 θ) − ω2t Lθ cos θ sin τ − E sin τ = 0 (3.11) θ (1 + sin2 θ) + θ2 sin θ cos θ + (ω20 − 1) sin θ cos θ + 2ω2t θ +

E cos θ = 0 L (3.12)

where the prime denotes differentiation respect to τ. The Eqs. (3.10), (3.11) and (3.12) are second order nonlinear differential ones with variable coefficients and therefore are very difficult to be analytically solved. In what follows, for Eqs. (3.10), (3.11) and (3.12) the Optimal Auxiliary Functions Method is applied to study the nonlinear vibrations of the multirotor system. The accuracy of the results obtained using our procedure is proved by numerical simulation in order to validate analytical results. This approach is independent of any small or large parameters.

3.1.2 Optimal Auxiliary Functions Method for Nonlinear Vibration of Misaligned Multirotor System To solve Eqs. (3.10), (3.11) and (3.12) we consider the following initial conditions X (0) = α, X  (0) = 0, Y (0) = β, Y  (0) = 0, θ(0) = A, θ (0) = 0

(3.13)

For the considered equations, the linear operators can be written in the form L X = X  + ω20 X

(3.14)

LY = Y  + ω20 Y

(3.15)

24

3 The First Alternative of the Optimal Auxiliary Functions Method

Lθ = θ + ω2 θ

(3.16)

where ω is the unknown frequency corresponding to the variable θ, and X = X0 + X1 , Y = Y0 + Y1 , θ = θ0 + θ1 . From Eq. (2.5) we obtain the equations for the initial approximations (g(τ) = 0) L X 0 = 0 X 0 (0) = α, X 0 (0) = 0

(3.17)

LY0 = 0 Y0 (0) = β, Y0 (0) = 0

(3.18)

Lθ0 = 0 θ0 (0) = A, θ0 (0) = 0

(3.19)

X 0 (τ) = α cos ω0 τ

(3.20)

Y0 (τ) = β cos ω0 τ

(3.21)

θ0 (τ) = A cos ωτ

(3.22)

with the solutions

The corresponding nonlinear operators are N (X ) = (X  + ω20 X ) sin2 θ − Lθ2 sin θ cos τ − 2Lθ cos θ sin τ + Lθ cos3 θ sin τ + (ω20 − 1)L sin3 θ cos τ − ω2t Lθ cos θ cos τ − E cos τ (3.23) N (Y ) = (Y  + ω20 Y ) sin2 θ − Lθ2 sin θ sin τ + 2Lθ cos θ cos τ − Lθ cos3 θ cos τ + (ω20 − 1)L sin3 θ sin τ − ω2t Lθ cos θ sin τ − E sin τ (3.24) N (θ) = (2ω2t − ω2 )θ + θ sin θ + θ2 sin θ cos θ E + (ω20 − 1) sin θ cos θ + cos θ L

(3.25)

Within Eqs. (3.23), (3.24) and (3.25), it is necessary to develop the functions sinθ0 and cosθ0 , taking into consideration that θ0 is given by Eq. (3.22), such that   A5 1 3 1 5 A3 θ + .... = A − + + . . . cos ωτ sin θ0 = θ0 − θ0 + 6 120 0 8 192    5  A3 A5 A + − + + . . . cos 3ωτ + + . . . cos 5ωτ + . . . (3.26) 24 384 1920

3.1 Dynamics of an Angular Misaligned Multirotor System

A4 1 1 A2 + + ... cos θ0 = 1 − θ20 + θ40 + . . . = 1 − 2 24 4 64     A2 A5 A4 + − + + . . . cos 2ωτ + + . . . cos 4ωτ 4 48 192

25

(3.27)

Inserting Eq. (3.22) into Eq. (3.23) and having in view Eqs. (3.26) and (3.27), we obtain N (θ0 ) = α0 + α1 cos ωτ + α2 cos 2ωτ + α3 cos 3ωτ + α4 cos 4ωτ + α5 cos 5ωτ

(3.28)

where α0 = α1 = α3 = α4 =

    A2 A2 E E A4 A4 1− − + + . . . , α2 = + + ... L 4 64 L 4 48    2  1 2 3 3 2 5 1 3 1 5 2 A + ... A 2ωt − ω − ω A − ω A + (ω0 − 1) A − A + 2 8 2 12   3 5ω2 A5 1 1 5 − ω2 A3 + (ω20 − 1) A − A3 + A + ... 24 4 2 12 5 E A4 1 2 5 A , α5 = ω A + (ω20 − 1) + ... (3.29) 192L 16 12

To construct Eq. (2.13) in the first approximation θ1 , we consider F1θ (τ) = n(θ0 ) F2θ (τ) = 2(cos 2ωτ)n(θ0 ) n(θ0 ) = α0 + α1 cos ωτ + α2 cos 2ωτ + α3 cos 3ωτ F2θ

(3.30)

These functions are not unique. Alternatively, we can choose the functions F1θ , in the forms F1θ (τ) = 1 F2θ (τ) = n(θ0 ) F3θ (τ) = 2(cos 2ωτ)n(θ0 )

or in another form F1θ (τ) = n(θ0 ) F2θ (τ) = 2(cos 2ωτ)n(θ0 ) F3θ (τ) = 2(cos 4ωτ)n(θ0 )

(3.31)

26

3 The First Alternative of the Optimal Auxiliary Functions Method

F4θ (τ) = 2(cos 6ωτ)n(θ0 )

(3.32)

and so on. Considering only Eq. (3.30), the first approximation θ1 (τ,Ci ) is obtained from Eq. (2.13): θ1 + ω2 θ1 = (C1 + 2C2 cos 2ωτ)(α0 + α1 cos ωτ + α2 cos 2ωτ + α3 cos 3ωτ) θ1 (0) = θ1 (0) = 0

(3.33)

or θ1 + ω2 θ1 = α0 C1 + α2 C2 + [α1 (C1 + C2 ) + α3 C2 ] cos ωτ + [α2 C1 + 2α0 C2 ] cos 2ωτ + [α3 C1 + α1 C2 ] cos 3ωτ + α2 C2 cos 4ωτ + α3 C2 cos 5ωτ + α1 C2 cos 6ωτ (3.34) Avoiding secular terms leads to the condition α1 (C1 + C2 ) + α3 C2 = 0

(3.35)

From Eq. (3.35) and (3.29) one retrieves    [(ω20 − 1)(1 − A2 ) + 2ω2t ]C1 + 2ω2t + (ω20 − 1) 1 − 76 A2 C2   ω = 1 + 23 A2 C1 + 43 A2 C2 2

(3.36)

The solution of Eq. (3.34) with the initial condition θ1 (0) = θ’1 (0) = 0 is given by 1 1 (α0 C1 + α2 C2 )(1 − cos ωτ) + (α2 C1 + 2α0 C2 )(cos 2ωτ ω2 3ω2 1 1 (α3 C1 + α1 C2 )(cos 3ωτ − cos ωτ) + α2 C1 (cos 4ωτ − cos ωτ) + 8ω2 15ω2 1 1 α3 C2 (cos 5ωτ − cos ωτ) + α1 C2 (cos 6ωτ − cos ωτ) − cos ωτ) + 24ω2 35ω2

θ1 (τ) =

(3.37)

From Eqs. (2.3), (3.22), (3.29) and (3.37), we can get the first-order approximate solution of Eq. (3.12) in the form    A2 E A2 E 1 − C − C 1 2 (1 − cos ωτ) ω2 L 4 4ω2 L    

ω20 − 1 A2 E A2 1 2E 1 − C + A3 C 1 (cos 2ωτ − cos ωτ) + − C + 1 2 12ω2 L 3ω2 L 4 32 48ω2

θ(τ) = A cos ωτ +



3.1 Dynamics of an Angular Misaligned Multirotor System   (ω2 − 1)(A − A3 ) A Aω2t A3 C2 (cos 3ωτ − cos ωτ) + + − 0 8 16 8ω2 4ω2

 ω20 − 1 E A2 1 + A3 C2 (cos 5ωτ − cos ωτ) C2 (cos 4ωτ − cos ωτ) + + 60ω2 L 96 144ω2

27

−

(3.38)

To solve Eq. (3.10), we substitute Eqs. (3.20) and (3.22) into Eq. (3.23) and one can get N (X 0 ) = −E cos τ + α6 cos(ω + 1)τ + α7 cos(ω − 1)τ + α8 cos(3ω + 1)τ + α9 cos(3ω − 1)τ + α10 cos(5ω + 1)τ + α11 cos(5ω − 1)τ

(3.39)

where 

  3 3 1 1  2 1 5 1 L ω0 − 1 A − A − Lω2 A3 + Lω2 A5 2 4 32 4 48   1 91 5 − Lω2t A − Lω A + A3 − A 8 192 

 1 1 3 3 1 5 1 L(ω20 − 1) A − A − Lω2 A3 + Lω2 A5 α7 = 2 4 32 4 48   1 91 5 A −Lω2t A + Lω A + A3 − 8 192 

 1 1 1 3 1 L(ω20 − 1) A3 − A5 + Lω2 A3 − Lω2 A5 α8 = 2 4 8 4 96   1 3 43 5 A + A −Lω 8 128 

 1 1 1 3 3 5 1 L(ω20 − 1) A − A + Lω2 A3 − Lω2 A5 α9 = 2 4 8 4 96   1 3 43 5 A + A +Lω 8 128 

1 1 A5 7 L(ω20 − 1) − Lω2 A5 + L A5 α10 = 2 32 96 384 

5 1 1 A 7 2 2 5 5 α11 = L(ω0 − 1) − Lω A − LA 2 32 96 384 α6 =

(3.40)

The optimal auxiliary function F1X (τ) for the first approximation X1 is F1X (τ) = N (X 0 )

(3.41)

Also this function is not unique. The equation of the first approximation X1 is obtained from Eqs. (2.13) and (3.41):

28

3 The First Alternative of the Optimal Auxiliary Functions Method

X 1 + ω20 X 1 = C3 [−E cos τ + α6 cos(ω + 1)τ + α7 cos(ω − 1)τ + α8 cos(3ω + 1)τ + α9 cos(3ω − 1)τ + α10 cos(5ω + 1)τ + α11 cos(5ω − 1)τ

(3.42) with the initial conditions X 1 (0) = X 1 (0) = 0

(3.43)

From Eqs. (3.42) and (3.43) we obtain EC3 α6 C 3 [cos(ω + 1)τ − cos ω0 τ] (cos ω0 τ − cos τ) + 2 ω20 − 1 ω0 − (ω + 1)2 α7 C 3 α8 C 3 + 2 [cos(ω − 1)τ − cos ω0 τ] + 2 [cos(3ω + 1)τ ω0 − (ω − 1)2 ω0 − (3ω + 1)2 α9 C 3 − cos ω0 τ] + 2 [cos(3ω − 1)τ − cos ω0 τ] ω0 − (3ω − 1)2 α10 C3 + 2 [cos(5ω + 1)τ − cos ω0 τ] ω0 − (5ω + 1)2 α11 C3 + 2 [cos(5ω − 1)τ − cos ω0 τ] (3.44) ω0 − (5ω − 1)2

X 1 (τ, C3 ) =

The first order approximate solution of Eq. (3.10) is obtained from Eqs. (2.3), (3.20), (3.44) and (3.40) as:

3 EC3 C3 L A3 (ω20 − 1) X (τ) = α cos ω0 τ + 2 (cos ω0 τ − cos τ) + 2 ω0 − 1 ω0 − (ω + 1)2 8    1 3 1 1 1 L A3 ω [cos(ω + 1)τ − cos ω0 τ] − ω2t A − A3 − L Aω − L A3 ω2 − 2 8 2 8 16

  1 3 3 3 1 1 2 C3 3 2 L A ω A A − + L Aω − L A3 ω2 (ω − 1) − + 2 0 2 t 8 2 8 ω0 − (ω − 1)2 8

 1 1 C3 L A3 (ω20 − 1) + L A3 ω [cos(ω − 1)τ − cos ω0 τ] + 2 16 ω0 − (3ω + 1)2 8  1 1 1 + A3 ω2t + L A3 ω2 + L A3 ω [cos(3ω + 1)τ − cos ω0 τ] 16 8 16

1 1 3 2 1 C3 L A3 (ω20 − 1) + A ωt + L A 3 ω2 + 2 16 8 ω0 − (3ω − 1)2 8  1 + L A3 ω [cos(3ω − 1)τ − cos ω0 τ] (3.45) 16

By substituting Eqs. (3.21) and (3.22) into Eq. (3.24), yields N (Y0 ) = −E sin τ + α12 sin(ω + 1)τ + α13 sin(ω − 1)τ + α14 sin(3ω + 1)τ

3.1 Dynamics of an Angular Misaligned Multirotor System

29

+ α15 sin(3ω − 1)τ + α16 sin(5ω + 1)τ + α17 sin(5ω − 1)τ

(3.46)

The optimal auxiliary function F1Y (τ) for the first approximation Y1 is chosen in the form F1Y (τ) = N (Y0 )

(3.47)

This function may be used also in the form F1Y (τ) = N (Y0 ) F2Y (τ) = 2(cos 2ωτ)N (Y0 ) F3Y (τ) = 2(cos 6ωτ)N (Y0 )

(3.48)

and so on. The equation of the first approximation Y1 is obtained from Eq. (2.13) and (3.47): Y1 + ω20 Y1 = C4 [−E sin τ + α12 sin(ω + 1)τ + α13 sin(ω − 1)τ + α14 sin(3ω + 1)τ + α15 sin(3ω − 1)τ + α16 sin(5ω + 1)τ + α17 sin(5ω − 1)τ, 

Y1 (0) = Y1 (0) = 0

(3.49)

where     1 3 1 5 3 3 1 5 2 A + L(ω0 − 1) A − A α12 = Lω − A + 8 96 8 64    1 1 3 3 1 3 5 5 A− A + A + Lω A− A + − Lω2t 2 16 384 2 16     1 3 1 5 3 3 1 5 A − A − L(ω20 − 1) A − A α13 = Lω2 8 96 8 64    1 1 3 3 1 3 5 5 A− A + A + Lω A− A + + Lω2t 2 16 384 2 16     1 5 1 3 3 5 2 1 3 2 A − A + L(ω0 − 1) A − A α14 = Lω 8 192 8 16     1 3 1 3 5 5 17 5 2 A − A + Lω A − A + Lωt 16 768 16 256     1 1 5 1 3 3 5 A − L(ω20 − 1) A − A α15 = Lω2 − A3 + 8 192 8 16     1 1 5 17 A3 − A5 + Lω A3 − A5 − Lω2t 16 768 16 256 2

1 5 A 256

1 5 A 256





30

3 The First Alternative of the Optimal Auxiliary Functions Method

1 Lω2 A5 − 192 1 =− Lω2 A5 + 192

α16 = − α17

19 A5 1 2 (ω0 − 1)L A5 − Lω2t − LωA5 64 768 768 A5 1 2 19 (ω0 − 1)L A5 + Lω2t + LωA5 64 768 768

(3.50)

The Eq. (3.49) has the solution   EC4 sin ω0 τ α12 − sin τ + 2 [sin(ω + 1)τ 2 ω0 ω0 − 1 ω0 − (ω + 1)2

  ω−1 ω+1 α13 sin(ω − 1)τ − − sin ω0 τ + 2 sin ω τ 0 ω0 ω0 ω − (ω − 1)2

0  3ω + 1 α14 sin(3ω + 1)τ − sin ω0 τ + 2 ω0 ω0 − (3ω + 1)2

 3ω − 1 α15 sin(3ω − 1)τ − sin ω0 τ + 2 ω0 ω0 − (3ω − 1)2

 5ω + 1 α16 sin(5ω + 1)τ − sin ω0 τ + 2 ω0 ω0 − (5ω + 1)2

 5ω − 1 α17 sin(5ω − 1)τ − sin ω0 τ (3.51) + 2 ω0 ω0 − (5ω − 1)2

Y1 (τ, C4 ) =

The first-order approximate solution of Eq. (3.11) will be obtained from Eqs. (2.3), (3.21), (3.50) and (3.49):   (τ) = β cos ω0 τ − EC4 sin τ − sin ω0 τ Y ω0 ω20 − 1 2 3   3L(1−ω0 )A Lω2 A3 1 2 − 8 − 2 Lωt A − 38 A3 − 8 + ω20 − (ω + 1)2 ω+1 − sin ω0 τ] ω0 3

+ − + +

− LωA 16 −

LωA 2

Lω2 A3 8 ω20 − (ω − 1)2

− 38 L(1 − ω20 )A3 +

LωA3 16

+

− 21 LωA

Lω2t  A 2

C4 [sin(ω + 1)τ

− 38 A3

 C4 [sin(ω − 1)τ

ω−1 sin ω0 τ] ω0 3 L(1−ω20 )A3 Lω2 A3 + 16t − ωL16A 8 ω20 − (3ω + 1)2 L(1−ω20 )A3 Lω2 A3 ωL A3 − 16t 16 − 8 ω20 − (3ω − 1)2

C4 [sin(3ω + 1)τ −

3ω + 1 sin ω0 τ] ω0

C4 [sin(3ω − 1)τ −

3ω − 1 sin ω0 τ] ω0

(3.52)

3.1 Dynamics of an Angular Misaligned Multirotor System

31

3.1.3 Numerical Examples In order to show the validity of the OAFM, Eqs. (3.10), (3.11) and (3.12) have been analysed for the following cases: Case 3.1.3.a For the first case we consider A = 0.06; α = 0.07; β = 0.08; ω0 = 1.2; ωt = 0.02; L = 20; E = 0.05. In this case, using the procedure of minimizing the residual error [18], the optimal values of the convergence-control parameters Ci and the approximate frequency are C1 = −1.002544407454696, C2 = −0.004553544428084325, C3 = −1.097600636101952, C4 = −1.092309032033477,  O AF M = 0.662446

(3.53)

It is important to mention that the value of the frequency obtained through numerical integration is ωNUM = 0.662478, and should be emphasized that the OAFM lead to high precision results. Figure 3.12 shows the comparison between the present solution (3.38) and the numerical integration results obtained using a fourth order Runge–Kutta scheme. Figure 3.3 shows the residual (2.30) corresponding to the first approximate solution θ (Figs. 3.2, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, and 3.11. Case 3.1.3.b In the second case it is considered A = 0.06; α = 0.07; β = 0.08; ω0 = 1.2; ωt = 0.1; L = 20; E = 0.05. In this case, following the same procedure, the optimal values of the convergence-control parameters and the frequency are C1 = −1.0405329739831501; C2 = −0.0023584330911787105;

Fig. 3.2 Comparison between the approximate solution given by (3.38) and numerical integration results for Eq. (3.12) _____ numerical _ _ _ _approximate solution

32

3 The First Alternative of the Optimal Auxiliary Functions Method

Fig. 3.3 The residual generated by the approximate solution (3.38)

Fig. 3.4 Phase plane for the approximate results given by (3.38) _____ numerical _ _ _ _approximate solution

Fig. 3.5 Comparison between the approximate solution given by (3.45) and numerical integration results for Eq. (3.10) _____ numerical _ _ _ _approximate solution

3.1 Dynamics of an Angular Misaligned Multirotor System

33

Fig. 3.6 Comparison between the approximate solution given by (3.52) and numerical integration results for Eq. (3.11) _____ numerical _ _ _ _approximate solution

Fig. 3.7 Phase plane for the approximate results given by (3.45) _____ numerical _ _ _ _approximate solution

Fig. 3.8 Phase plane for the approximate results given by (3.52) _____ numerical _ _ _ _approximate solution

34

3 The First Alternative of the Optimal Auxiliary Functions Method

Fig. 3.9 Comparison between the approximate solution given by (3.38) and numerical integration results for Eq. (3.12) _____ numerical _ _ _ _approximate solution

Fig. 3.10 The residual generated by the approximate solution (3.38)

Fig. 3.11 Phase plane for the approximate results given by (3.38) _____ numerical _ _ _ _approximate solution

3.1 Dynamics of an Angular Misaligned Multirotor System

35

Fig. 3.12 Comparison between the approximate solution given by (3.45) and numerical integration results for Eq. (3.10) _____ numerical _ _ _ _approximate solution

C3 = −1.091819177088501; C4 = −1.0845829959029816;  O AF M = 0.676006

(3.54)

In this case the value obtained through numerical integration for the frequency is ωNUM = 0.676796, which means that a very good approximation of the frequency is obtained through OAFM. Figure 3.9 shows the comparison between the present solution (3.38) and the numerical integration results obtained using a fourth order Runge–Kutta scheme while Fig. 3.10 shows the residual corresponding to the first approximate solution θ in the second case. It is observed in this as well that case we obtain an excellent approximation of the solution X (τ). Also, the results plotted in the phase plane emphasize the excellent accuracy (Fig. 3.11, 3.14 and 3.15).

Fig. 3.13 Comparison between the approximate solution given by (3.52) and numerical integration results for Eq. (3.11) _____ numerical _ _ _ _approximate solution (3.38)

36

3 The First Alternative of the Optimal Auxiliary Functions Method

Fig. 3.14 Phase plane for the approximate results given by (3.45) _____ numerical _ _ _ _approximate solution

Fig. 3.15 Phase plane for the approximate results given by (3.52) _____ numerical _ _ _ _approximate solution

Case 3.1.3.c In the last case, the values of the parameters are A = 0.06; α = 0.07; β = 0.08; ω0 = 1.2; ωt = 0.02; L = 20; E = 0.1. The optimal values obtained in this last case are: C1 = −1.005514256667141; C2 = −0.004514545111240005; C3 = −1.1741116543118197; C4 = −1.1853763382434614;  O AF M = 0.662429

(3.55)

Comparison between our approximate solutions and numerical integration results for three different cases is presented in Figs. 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15, 3.16, 3.17, 3.18, 3.19, 3.20, 3.21 and 3.22.

3.1 Dynamics of an Angular Misaligned Multirotor System

37

Fig. 3.16 Comparison between the approximate solution given by (3.38) and numerical integration results for Eq. (3.12) _____ numerical _ _ _ _approximate solution

Fig. 3.17 The residual generated by the approximate solution (3.38)

Fig. 3.18 Phase plane for the approximate results given by (3.38) _____ numerical _ _ _ _approximate solution

38

3 The First Alternative of the Optimal Auxiliary Functions Method

Fig. 3.19 Comparison between the approximate solution given by (3.45) and numerical integration results for Eq. (3.10) _____ numerical _ _ _ _approximate solution

Fig. 3.20 Comparison between the approximate solution given by (3.52) and numerical integration results for Eq. (3.11) _____ numerical _ _ _ _approximate solution

Fig. 3.21 Phase plane for the approximate results given by (3.45) _____ numerical _ _ _ _approximate solution

3.1 Dynamics of an Angular Misaligned Multirotor System

39

Fig. 3.22 Phase plane for the approximate results given by (3.52) _____ numerical _ _ _ _approximate solution

From these figures it can be seen that the results obtained using the present procedure are almost identical with those obtained through numerical integration, which validate the analytical approach. The errors between the analytical and numerical solutions are remarkable good. Also, one can observe that the errors between the analytical and numerical results for the frequencies are very good. We remark that the angular frequency ω is increased by increasing the non-dimensional angular frequency ωt (cases 2 and 3) but the frequency is insensitive to the increasing of the nondimensional mass eccentricity (cases 2 and 3). It is observed that the variation of the angle θ between the two misaligned rotors is harmonic having the period T = 2π/ω. The proposed technique, OAFM, accelerates the convergence of the approximate analytical solutions of the vibration of multirotor system with a flexible coupling taking into consideration the effect of the dynamic angular misalignment.

References 1. V. Marinca, N. Herisanu, Optimal auxiliary functions method for a pendulum wrapping on two cylinders. Mathematics 8, 1364 (2020) 2. M.L. Adams, Rotating Machinery Vibration (From analysis to trouble shooting (CRC Press, Boca Raton, 2010) 3. M. Xu, R.D. Marangoni, Vibration analysis of a motor-flexible-coupling-rotor system subject to misalignment and unbalance, Part I: theoretical model and analysis. J. Sound Vib. 176, 663–679 (1994) 4. K.M. Al-Hussain, Dynamic stability of two rigid rotors connected by a flexible coupling with angular misalignment. J. Sound Vib. 266, 217–234 (2003) 5. K.M. Al-Hussain, I. Redmond, Dynamic response of two rotors connected by rigid mechanical coupling with parallel misalignment. J. Sound Vib. 249, 483–498 (2002) 6. A.S. Sekhar, B.S. Prabhu, Effects of coupling misalignment on vibrations of rotating machinery. J. Sound Vib. 185, 655–671 (1995) 7. P.A. Meroño Pérez, F.C. Gómez de León, L. Zaghar, Characterization of parallel misalignment in rotating machines by means of the modulated signal of incremental encoders. J. Sound Vib. 333, 5229–5243 (2014)

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8. J.K. Sinha, A.W. Lees, M.I. Friswell, Estimating unbalance and misalignment of a flexible rotating machine from a single run-down. J. Sound Vib. 272, 967–989 (2004) 9. T.H. Patel, A.K. Darpe, Experimental investigations on vibration response of misaligned rotors. Mechanical Syst. Signal Process. 23, 2236–2252 (2009) 10. J. Warminski, M.P. Cartmell, A. Mitura, M. Bochenski, Active vibration control of a nonlinear beam with self- and external excitations. Shock and Vib. 20, 1033–1047 (2013) 11. I. Kovacic, L. Cveticanin, M. Zukovic, A. Rakaric, Jacobi elliptic functions: a review of nonlinear oscillatory application problems. J. Sound Vib. 380, 1–36 (2016) 12. S. Momani, Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method. Appl. Math. Comput. 177, 488–494 (2006) 13. Z.M. Odibat, A study on the convergence of variational iteration method. Math. Comp. Model. 51, 1181–1192 (2010) 14. S. Abbasbandy, Homotopy analysis method for heat radiation equations. Int. Commun. Heat Mass Transf. 34, 380–387 (2007) 15. L. Cveticanin, Homotopy-perturbation method for pure nonlinear differential equation. Chaos Solitons Fractals 30, 1221–1230 (2006) 16. N. Herisanu, V. Marinca, Gh. Madescu, An analytical approach to non-linear dynamical model of a permanent magnet synchronous generator. Wind Energy 18, 1657–1670 (2015) 17. N. Herisanu, V. Marinca, Approximate analytical solutions to Jerk equations. Springer Proc. Math. Statist. 182, 169–176 (2016) 18. V. Marinca, N. Herisanu, The Optimal Homotopy Asymptotic Method. Engineering Applications (Springer, Cham, 2015) 19. V. Marinca, N. Herisanu, An optimal iteration method with application to the Thomas-Fermi equation. Cent. Eur. J. Phys. 9, 891–895 (2011) 20. M.O. Kaya, O. Ozdemir, Flexural-torsional coupled vibration analysis of a thin-walled closed section composite Timoshenko beam by using the differential transform method. Springer Proc. Phys. 111, 279–284 (2007) 21. M. Li, Nonlinear vibration of a multirotor system connected by a flexible coupling subjected to the holonomic constraint of dynamic angular misalignment. Math. Probl. Eng. Article ID 243758 (2012) 22. N. Herisanu, V. Marinca, An efficient analytical approach to investigate the dynamics of misalignment multirotor system. Mathematics 8(7), Art.1083 (2020)

Chapter 4

Oscillations of a Pendulum Wrapping on Two Cylinders

The study of simple pendulum has a long history. Thus, Leonardo da Vinci during the Renaissance made many drawings of the motion of pendulum, without ralizing its value for timekeeping. Beginning around 1602, Galileo Galilei was the first to study the properties of pendulum: isochronisms and found that the period of the pendulum is approximately independent of the amplitude or with the swing. Also found that the period is independent on the mass of the bob and proportional to the square root of the length of the pendulum. In 1641 Galileo conceived and dictated to his son a design for a pendulum clock. The pendulum was the first harmonic oscillator used by human being. In 1673, Huygens discovered that a pendulum has the same period when hung from its centre of oscillation as when hung from its pivot [1]. In 1818 British Captain Henry Kater invented the so-named reversible Kater’s pendulum making possible very accurate measurements of gravity. In 1851 Foucault showed that the Earth’s rotation didn’t depend on celestial observations and a “pendulum mania” broke out, as Foucault pendulum [2]. Around 1900 low-thermal-expansion materials began to be used for pendulum rods in the highest precision clocks and other instruments, first nickel steel alloy, and later fused quartz, which made temperature compensation trivial [3]. The accuracy of the best pendulum clocks topped out at around a second per year. In 1921 was invented the quartz crystal oscillator and in 1927, quartz clocks, replaced pendulum clocks as the world’s best timekeepers [4]. Pendulum gravimeters were superseded by “free fall” [5] gravimeters in the 1950’s, but pendulum instruments continued to be used into the 1970’s. In 1721, Graham [6] invented mercury pendulum: the pendulum’s weight is a container of mercury and with a temperature rise, the pendulum rod gets longer, but mercury also expands and its surface level rises slightly in the container, moving its centre of mass closer to the pendulum pivot. In 1726, J. Harrison invented the gridiron pendulum which consists of alternating rods of two different metals, one with lower thermal expansion (steel) and one with high thermal expansion (zinc or brass) [3]. In 1896, C.E. Guillaume invented the nickel-steel alloy [4]. Invar pendulum was first used in the Riefler regulator clock which achieved accuracy of 15 miliseconds per day. In 1826 G. Airy proved that the disturbing effect of a drive force on the period © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_4

41

42

4 Oscillations of a Pendulum Wrapping on Two Cylinders

of a pendulum is smallest if given as a short type of pendulum such as RepsoldBessel pendulum [7], Van Sterneck and Mendelhall gravimeters, double pendulum gravimeters, Gulf gravimeter [8], and so on. Dynamic mechanical systems possessing the pendulum arise in many domains of activity and many researchers have paid attention to obtaining the governing equation of pendulums. These early studies were followed by investigations of other types of pendulum with different conditions along their dynamic behaviour. Hamouda and Pierce [9] analysed the blades of a helicopter rotor (like a simple pendulum) to suppress the root reactions. The general nonlinear equations of motion are linearized. They consider the frequency response for a hingeless rotor blade excited by a harmonic variation of span wise air load distribution. Simple flap and lead-lag pendulum are treated individually. The results include the effect of pendulum tuning on the minimization of hub reactions. The pendulum mass effectiveness was also investigated. A comprehensive discussion of the corrections needed to accurately measure the acceleration of gravity using a plane pendulum as provided by Nelson and Olson [10]. A simple laboratory experiment was described in which g was determined to four significant figures of accuracy. The effect of Coriolis force acting on the bob during station is evaluated adapting a spring-pendulum system analysis to the nearly stiff limit. In their study the linear and quadratic damped were used and perturbation expansion of the small dimensionless parameter was developed. Ge and Ku [11] extended Melnikov approach (which is traditionally restricted to study problems with weakly non-linear phenomena including sufficient small harmonic excitation) to pendulum suspended on a rotating arm described by twodimensional differential equations. These equations possess strongly odd nonlinear function of the displacement and are subjected to large harmonic excitation. Nester et al. [12] presented an experimental investigation to dynamic response of rotor systems fitted with centrifugal pendulum vibration absorbers. Two types of absorbers are considered which exhibit different types of nonlinear behaviour. The behaviour of a spatial double pendulum, comprising two pendulums that swing in different planes was investigated in [13] by Bendersky and Sandler. Mathlab computer programs were used for solving the nonlinear differential equations by Runge–Kutta method. The frequency spectra were obtained using Fourier transformation. Solutions of free vibrations of the pendulums and graphic descriptions of changes in the frequency spectra were used for the dynamic investigation for different initial conditions of motion. The dynamics of the harmonically excited parametric pendulum when it exhibits rotational orbits was considered by Xu et al. [14]. Assuming no damping and small angle oscillations, this system was simplified to the Mathieu equation in which stability was important in investigating the rotational behavior. Analytical and numerical analysis techniques are employed to explore the dynamic responses to different parameters and initial conditions. The dynamic interactions between a parametric pendulum and an electro-dynamical shaker aiming to explain a weak correlation between the theoretical predictions and the experimental results for pendulum rig driven by the shaker was investigated in [15].

4 Oscillations of a Pendulum Wrapping on Two Cylinders

43

A small ellipticity of the driving, perturbing the classical parametric pendulum was studied by Horton et al. [16]. The resonance tongues, which are the most characteristic feature of the classical bifurcation scenario of a parametrically driven pendulum, merge into a simple region of instability. Warminski and Kecik analyzed the vibrations of a nonlinear oscillator with an attached pendulum, excited by moment of its point of suspension. The pendulum and the oscillator are strongly coupled by inertial terms, leading to the so-called autoparametric vibrations [17]. Kecik and Warminski [18] improved dynamics and control motions, a new suspension composed of a semiactive magnetorheological damper and a nonlinear spring. In this way the unstable areas and chaotic or rotating motion of the pendulum are reduced. A variation of the simple pendulum in which the two point masses are replaced by square plates was investigated by Rafat et al. [19]. The equilibrium configurations and normal modes of oscillations are obtained. The equations of motion were solved numerically to produce Poincare sections. The accurate analytic solution of the nonlinear pendulum differential equation is obtained using homotopy analysis technique by Turkyilmazoglu [20]. The obtained explicit analytical expressions for the frequency, period and displacement are compared with numerical ones. The dynamics of a parametric pendulum system operating in rotational regime has been investigated with a view of energy harvesting in [21]. A control strategy aiming to initiate and maintain the desired rotational responses has been developed and verified numerically and experimentally. Elias and Tsuchida [22] analyzed a weakly nonlinear electromechanical pendulum by averaging technique. The stability of the stationary state was studied by means of the Routh-Hurwitz criterion and taking the motor frequency and damper coefficient as control parameters. Awrejcewicz [23] studied the mathematical pendulum motion oscillating in a plane rotating with angular velocity. Energy balance method is applied in [24] to calculate approximations in order to achieve the nonlinear frequency for pendulum attached to rolling wheels that is restrained by a spring. The nonlinear oscillations of a pendulum wrapping and unwrapping on two cylindrical bases were studied by Mazaheri et al. [25]. To obtain an analytical solution the multiple scale method is used and there are analyzed effects of amplitude and radius of cylinder. Boubaker proposed in [26] to enhance the wealth on the pendulum benchmark and attempt to provide an overall picture of historical, current and trend developments in nonlinear control theory based on its simple structure and its rich nonlinear model. The simple structure allows real and virtual experiments to be carried out. The richness of the model has shown on its usefulness in illustrating all emerging ideas in nonlinear control theory. Nadecka et al. [27] studied the synchronization of rotational motion in the system of two parametric pendulums subject to common harmonic excitation. The energy can be harvested from the rotational motion, which is a strongly advantageous alternative

44

4 Oscillations of a Pendulum Wrapping on Two Cylinders

to using energy oscillations. The influence of the parameter mismatch on the response of the system and its synchronization was analyzed. Synchronization of two pendulums mounted on a mutual elastic single degreeof-freedom base is examined by Alevras et al. [28]. The response of the pendulum was considered when their base was externally excited by a random phase sinusoidal force, that leading to stochastic parametric excitation of the pendulums. The dynamic response of the pendulum was numerically investigated with respect to establishment of rotations as well as identification of synchronization with the pendulum characteristics spanning along non-identical parameters. The influence of an external harmonic excitation on the intrinsic modes of a chain of nonlinear pendulum was numerically investigated by Jallouli et al. [29]. The existence and stability domains of solitons are modified when the coupled pendulums are subjected to external parametric excitations. This stabilization mechanism opens the way towards the control of the energy localization phenomena in damped nonlinear periodic lattices for efficient energy transport applications. In this section we study the nonlinear oscillations of a simple pendulum bounded by two cylinders at the point of suspension. The length of this pendulum varies due to wrapping around the cylinders.

4.1 Equation of Motion In what follows we present the equation of motion of a simple pendulum wrapping around two cylinders at the point of suspension [25]. The length of the pendulum is L while the radius of cylinders is r (Fig. 4.1). The motion of the system is described by the generalised coordinate θ, but the string length is changing. The kinetic energy can be expressed in the form T =

Fig. 4.1 Simple pendulum wrapping around the cylinders

1 m(L − r |θ|2 θ˙ 2 2

(4.1)

4.1 Equation of Motion

45

where m is the mass of pendulum and the dot denotes differentiation with respect to time. The potential energy becomes U = mg[L − (L − r |θ|) cos θ − r sin θ]

(4.2)

From the Lagrange’s equation one can put ˙ m(L − r |θ|)2 θ¨ − 2mr (L − r |θ|)(sgnθ)θ˙ 2 + mr (L − r |θ|)θ(sgnθ)+ + mg(L − r |θ|) sin θ − mgr cos θ(sgnθ) + mgr cos θ(sgnθ) = 0

(4.3)

After some manipulation, one obtains: (L − r |θ|)θ¨ + g sin θ − r θ˙ 2 (sgnθ) = 0

(4.4)

¨ + g sin θ − a θ˙ 2 (sgnθ) = 0 θ¨ − a θ|θ| L

(4.5)

where a = r/L. The initial conditions for Eq. (4.5) are ˙ θ(0) = A, θ(0) =0

(4.6)

4.2 Application of OAFM to a Pendulum Wrapping on Two Cylinders If we introduce the independent variable τ = t and the dependent variable ϕ = θA−1 , then Eqs. (4.5) and (4.6) become, respectively 2 ϕ − a2 Aϕ |ϕ| +

g sin Aϕ − a2 Aϕ2 (sgnϕ) = 0 AL

ϕ(0) = 1, ϕ (0) = 0

(4.7) (4.8)

where  is the frequency of the system and prime denotes differentiation with respect to τ. For Eq. (4.5), the linear operator can be written in the form L[ϕ(τ)] = 2 (ϕ + ϕ) with g(τ) = 0, while the corresponding nonlinear operator is

(4.9)

46

4 Oscillations of a Pendulum Wrapping on Two Cylinders

N [ϕ(τ, )] = −2 ϕ − a2 Aϕ |ϕ| +

g sin Aϕ − a2 Aϕ2 (sgnϕ) AL

(4.10)

The Eq. (2.5) becomes ϕ0 + ϕ0 = 0, ϕ0 (0) = 1, ϕ0 (0) = 0

(4.11)

ϕ0 (τ) = cos τ

(4.12)

and has the solution

Substituting Eq. (4.12) into Eq. (4.11), we obtain N [ϕ0 (τ, )] = −2 cos τ + a2 A cos τ|cos τ|+ g + sin(A cos τ) − a2 A sin2 τ(sgn(cos τ)) AL

(4.13)

Having in view that [30, 31] cos τ|cos τ| = cos2 τ(sgn(cos τ) cos2 τ(sgn(cos τ)) − sin2 τ(sgn(cos τ)) = cos 2τ(sgn(cos τ))

(4.14) (4.15)

  1 1 1 1 4 cos τ − cos 3τ + cos 5τ − cos 7τ + cos 9τ + . . . (sgn(cos τ)) = π 2 5 7 9 (4.16)   A5 A7 A9 A3 + − + + . . . cos τ sin(A cos τ) = A − 8 192 9216 737280    5 A3 A5 A7 A9 A A7 + − + − + − . . . cos 3τ + − 24 384 15360 1105920 1920 46080    9 7 9 A A A + . . . cos 5τ + − + + . . . cos 7τ + 2580480 322560 10321920   A9 + . . . cos 9τ (4.17) + 92897280 and substituting Eqs. (4.14)–(4.17) into Eq. (4.13), one can get  gA A2 A4 A6 (4a A − 3π)2 + 1− + − + 3π L 8 192 9216    A8 12a A2 g A A2 A4 A6 + + . . . cos τ + − − + 737280 5π L 4 640 15360 

N [ϕ0 (τ, )] =

4.2 Application of OAFM to a Pendulum Wrapping on Two Cylinders

47

  20a A2 g A A4 A6 cos 3τ + − + − 21π L 1920 46080    28a A2 gA A6 A8 + . . . cos 5τ + − + 2580480 45π L 322560    8 2 A 36a A gA A8 − + . . . cos 7τ + − + + . . . cos 9τ + . . . 10321920 77π L 92897280

−

A8 + ... 1105920



(4.18) Taking into account Eqs. (2.13) and (4.18) we can choose the auxiliary functions F1 , F2 , F3 and F4 in the form F1 (τ) = G(τ) , F2 (τ) = 2(cos 2τ)G(τ), F3 (τ) = 2(cos 4τ)G(τ) , F4 (τ) = 2(cos 6τ)G(τ) G(τ) = α cos τ + β cos 3τ + γ cos 5τ

(4.19)

where α, β, γ are obtained from Eq. (4.18):  2 (4a A − 3π ) g A 1− + 3π L  g A A2 A4 122 a A − − β= 5π L 32 384  4 2 gA A 20 a A + − γ =− 21π L 1920

α=

A2 A4 A6 A8 + − + 8 192 9216 737280  6 8 A A + − 15360 1105920  A6 A8 + 46080 2580480



(4.20)

We also may choose the auxiliary functions F1 , F2 , F3 and F4 as follows F1 (τ) = 2(cos 4τ)G(τ) , F2 (τ) = 2(cos 6τ)G(τ), A F3 (τ) = 2(cos 8τ)(G(τ) + δ cos 7τ) , δ = − 36a77π + 2

gA L

+

F4 (τ) = 2(cos 4τ)(G(τ) + δ cos 9τ)

A8 92897280

(4.21)

or F1 (τ) = 1 , F2 (τ) = G(τ) , F3 (τ) = 2(cos 2τ)G(τ) G(τ) = α cos τ + γ cos 5τ and so on. Substituting Eq. (4.19) into Eq. (2.13), the result is ϕ1 + ϕ1 =

α(C1 + C2 ) + β(C2 + C3 ) + γ (C3 + C4 ) cos τ 2

(4.22)

48

4 Oscillations of a Pendulum Wrapping on Two Cylinders

α(C2 + C3 ) + β(C1 + C4 ) + γ C2 α(C3 + C4 ) + βC2 + γ C1 cos 3τ + cos 5τ 2  2 αC4 + βC3 + γ C2 γ C4 + cos 9τ + 2 cos 11τ (4.23) 2 

+

The solution of Eq. (4.23) is chosen so that it contains no secular terms, which lead to the condition α(C1 + C2 ) + β(C2 + C3 ) + γ (C3 + C4 ) = 0

(4.24)

From Eqs. (4.20) and (4.24) one retrieves 2 = app

M 3π−4a A (C1 3

+ C2 ) −

12a A(C2 +C3 ) 5

+

20a A(C3 +C4 ) 21

(4.25)

where    A4 A6 A8 A2 πg A (C1 + C2 ) 1 − + − + M= L 8 192 9216 737280   2 4 6 8 A A A A − (C2 + C3 ) − + − 32 384 15360 1105920   4 A A6 A8 + (C3 + C4 ) − + 1920 46080 2580480 The solution (4.23) is given by α(C2 + C3 ) + β(C1 + C4 ) + γ C2 (cos τ − cos 3τ) 82 α(C3 + C4 ) + βC2 + γ C1 + (cos τ − cos 5τ) 242 αC4 + βC3 + γ C2 βC1 + γ C3 + (cos τ − cos 7τ) + (cos τ − cos 9τ) 2 48 802 γ C4 (cos τ − cos 11τ) (4.26) + 1202

ϕ1 (τ) =

From Eqs. (4.12), (2.26), (2.3) and from the transformations τ = t and ϕ = θA−1 , we can get the first-order approximate solution of Eq. (4.5) in the form ˜ = A cos t + A[α(C2 + C3 ) + β(C1 + C4 ) + γ C2 ] (cos t − cos 3t) θ(t) 82 A[α(C3 + C4 ) + βC2 + γ C1 ] + (cos t − cos 5t) 242 A[αC4 + βC3 + γ C2 ] + (cos t − cos 7t) 482

4.2 Application of OAFM to a Pendulum Wrapping on Two Cylinders

+

49

A(βC4 + γ C3 ] Aγ C4 ] (cos t − cos 9t) + (cos t − cos 11t) 2 80 1202 (4.27)

where the coefficients α, β and γ are given in Eq. (4.20) and  in Eq. (4.25).

4.3 Numerical Examples To illustrate the accuracy of our approach, we consider different values of the coefficients a, A and L. We represent the behaviour of the solution θ˜ and we compare the results obtained through OAFM with numerical integration results. Also, we represent a graphical comparison of the phase plane and a comparison between the frequencies  given by analytical developments (4.25) and numerical integration results, respectively. Case 4.2.3a First, we consider A = 0.1, a = 0.2, L = 0.6 and g = 9.8. Using the proposed procedure, by minimizing the residual function, the optimal values of the convergencecontrol parameters Ci and the frequency (4.25) are C1 = −0.02612929050874707; C2 = 0.026479977489142312; C3 = −0.008435730447475517; C4 = 0.0022006784301049727; app = 4.056213309077129 The solution given by (4.27) can be written as follows: ˜ = 0.0998307842607 cos t + 0.000186028849 cos 3t θ(t) − 0.000018350481 cos 5t + 1.058064902746 · 10−6 cos 7t + 5.890466227595 · 10−7 cos 9t − 1.097402505723 · 10−7 cos 11t (4.28) Case 4.2.3b For A = 0.1, a = 0.4, L = 0.6 we obtain C1 = −0.06468511377430505; C2 = 0.0663565183850488; C3 = −0.024463908434927746; C4 = 0.007981996714313724; app = 4.0735339712576275 The solution given by (4.27) in this case can be written as follows:

50

4 Oscillations of a Pendulum Wrapping on Two Cylinders

˜ = 0.099649003276 cos t + 0.000385656055 cos 3t θ(t) − 0.000036941545 cos 5t − 2.769256927293 · 10−7 cos 7t + 3.355209133644 · 10−6 cos 9t − 7.960693115042 · 10−7 cos 11t (4.29) Case 4.2.3c For A = 0.1, a = 0.6, L = 0.6 one can get C1 = −0.1261219305903601; C2 = 0.13076543964819184; C3 = −0.05556636277721333; C4 = 0.021026507904137733; app = 4.091781570513826 ˜ = 0.099462706082 cos t + 0.000591204026 cos 3t θ(t) − 0.00005847843 cos 5t − 5.181840137296 · 10−6 cos 7t + 0.000011265133 cos 9t − 3.145558315768 · 10−6 cos 11t

(4.30)

Case 4.2.3d For A = 0.2, a = 0.2, L = 0.6, it holds that C1 = −0.08486110531011988; C2 = 0.08719346899295516; C3 = −0.03657154198248718; C4 = 0.011850205290770313; app = 4.0659463540564085 ˜ = 0.1993020640806 cos t + 0.000789686896 cos 3t θ(t) − 0.000099399722 cos 5t − 3.173485847984 · 10−8 cos 7t + 0.000010044198 cos 9t − 2.363718668036 · 10−6 cos 11t Case 4.2.3e For A = 0.2, a = 0.4, L = 0.6, it holds that C1 = −0.05478213446823842; C2 = 0.05853717115551574; C3 = −0.004627180732353522; C4 = 0.02029235593370952; app = 4.1008614227769575 ˜ = 0.199076124329 cos t + 0.000743701792 cos 3t θ(t) + 0.000180456966 cos 5t + 8.131462054754 · 10−6 cos 7t

(4.31)

4.3 Numerical Examples

51

− 3.192663647494 · 10−7 cos 9t − 8.095283729049 · 10−6 cos 11t (4.32) Case 4.2.3f For A = 0.2, a = 0.6, L = 0.6, we obtain C1 = 0.030085026110128053; C2 = −0.035503685583418404; C3 = 0.02566501404190474; C4 = 0.06269091626063286; app = 4.137450803506345 ˜ = 0.199417 cos t − 0.000080132 cos 3t θ(t) + 0.000496034 cos 5t + 0.000241647 cos 7t − 0.0000373673 cos 9t − 0.0000375142 cos 11t

(4.33)

Case 4.2.3 g In this case, for A = 0.3, a = 0.2, L = 0.6, yields C1 = −0.12598689204928884; C2 = 0.13190482427905972; C3 = −0.05322525168176718; C4 = −0.025963691839690852; app = 4.071192550653585 ˜ = 0.298733 cos t + 0.00129371 cos 3t θ(t) − 0.0000239776 cos 5t − 0.0000223193 cos 7t − 0.0000314489 cos 9t − 0.0000116526 cos 11t

(4.34)

Case 4.2.3 h Considering A = 0.3, a = 0.4, L = 0.6, it follows that C1 = 0.4719373259762131; C2 = −0.5063076905748477; C3 = 0.34756120529541146; C4 = −0.11539436618220579; app = 4.125254940744536 ˜ = 0.29784 cos t + 0.001650441 cos 3t θ(t) + 0.000376774 cos 5t + 0.000455463 cos 7t − 0.000428294 cos 9t + 0.000104476 cos 11t

(4.35)

52

4 Oscillations of a Pendulum Wrapping on Two Cylinders

Case 4.2.3i In this case we consider A = 0.3, a = 0.6, L = 0.6, such that C1 = 0.4150420146527756; C2 = −0.4611401944978064; C3 = 0.32420957736633876; C4 = −0.12122441329678995; app = 4.180777463105995 ˜ = 0.295522 cos t + 0.00448745 cos 3t θ(t) − 0.000365785 cos 5t + 0.000785393 cos 7t − 0.000592619 cos 9t + 0.000163216 cos 11t

(4.36)

Case 4.2.3j Classical simple pendulum is obtained from Eq. (4.5) in the case when no cylinder exist. Therefore, for a = r/L = 0 we obtain from (4.25) the approximate frequency   A2 A4 A6 A8 C 2 + C 3 A2 A4 gA 1− + − + − − + = L 8 192 9216 737280 C1 + C2 32 384    A6 A4 A8 C3 + C4 A6 A8 − + − + (4.37) 15360 1105920 C1 + C2 1920 46080 2580480

2 app

The approximate solution for the simple pendulum is obtained from Eq. (4.27) with the following coefficients given by Eq. (4.20) for this particular case:   A2 A4 A6 A8 gA 1− + − + α = − + L 8 192 9216 737280  2  4 6 8 A A A gA A − + − β=− L 32 384 15360 1105920   A6 A8 g A A4 − + γ = L 1920 46080 2580480 2

(4.38)

The optimal values of the control parameters and the approximate frequency in this case are, respectively C1 = −0.006576676302841165; C2 = 0.006579872704687396; C3 = −0.007543367939537256; C4 = 0.0013865133762133064; app = 3.9941012459580407 ˜ = 0.4000837624301 cos t − 0.000029363344 cos 3t θ(t) − 0.000061202307 cos 5t + 6.789035888081 · 10−6 cos 7t

4.3 Numerical Examples

53

+ 1.421189152905 · 10−8 cos 9t − 2.506926263728 · 10−11 cos 11t (4.39) In Fig. 4.20 we compared the results obtained through OAFM with numerical integration results for the particular care of simple pendulum for A = 0.4, L = 0.6 while in Fig. 4.21 is presented a comparison between the phase planes in this case. Analyzing the comparison between the approximate and numerical integration results presented in Figs. 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10, 4.11, 4.12, 4.13, 4.14, 4.15, 4.16, 4.17, 4.18, 4.19, 4.20 and 4.21 for the cases 4.2.3e-4.2.3j, it can be seen that the results obtained using our procedure are almost identical with those obtained through numerical integration. Moreover, from Table 4.1 one can be observed that the accuracy of the approximate frequency is remarkable good when comparing to numerical results.

Fig. 4.2 Comparison between the approximate solution (4.28) and numerical integration results for A = 0.1, a = 0.2, L = 0.6 numerical approximate solution

Fig. 4.3 Phase plane for A = 0.1, a = 0.2, L = 0.6 (4.28)

numerical

approximate solution

54

4 Oscillations of a Pendulum Wrapping on Two Cylinders

Fig. 4.4 Comparison between the approximate solution (4.29) and numerical integration results for A = 0.1, a = 0.4, L = 0.6 numerical approximate solution

Fig. 4.5 Phase plane for A = 0.1, a = 0.4, L = 0.6 __ (4.29)

merical

approximate solution

Fig. 4.6 Comparison between the approximate solution (4.30) and numerical integration results for A = 0.1, a = 0.6, L = 0.6 numerical approximate solution

4.3 Numerical Examples

Fig. 4.7 Phase plane for A = 0.1, a = 0.6, L = 0.6 (4.30)

55

numerical

approximate solution

Fig. 4.8 Comparison between the approximate solution (4.31) and numerical integration results for A = 0.2, a = 0.2, L = 0.6 numerical approximate solution

Fig. 4.9 Phase plane for A = 0.2, a = 0.2, L = 0.6 (4.31)

numerical

approximate solution

56

4 Oscillations of a Pendulum Wrapping on Two Cylinders

Fig. 4.10 Comparison between the approximate solution (4.32) and numerical integration results for A = 0.2, a = 0.4, L = 0.6 numerical approximate solution

Fig. 4.11 Phase plane for A = 0.2, a = 0.4, L = 0.6 (4.32)

numerical

approximate solution

Fig. 4.12 Comparison between the approximate solution (4.33) and numerical integration results for A = 0.2, a = 0.6, L = 0.6 numerical approximate solution

4.3 Numerical Examples

Fig. 4.13 Phase plane for A = 0.2, a = 0.6, L = 0.6 (4.33)

57

numerical

approximate solution

Fig. 4.14 Comparison between the approximate solution (4.34) and numerical integration results for A = 0.3, a = 0.2, L = 0.6 numerical approximate solution

Fig. 4.15 Phase plane for A = 0.3, a = 0.2, L = 0.6 (4.34)

numerical

approximate solution

58

4 Oscillations of a Pendulum Wrapping on Two Cylinders

Fig. 4.16 Comparison between the approximate solution (4.35) and numerical integration results for A = 0.3, a = 0.4, L = 0.6 numerical approximate solution

Fig. 4.17 Phase plane for A = 0.3, a = 0.4, L = 0.6 (4.35)

numerical

approximate solution

Fig. 4.18 Comparison between the approximate solution (4.36) and numerical integration results for A = 0.3, a = 0.6, L = 0.6 numerical approximate solution

4.3 Numerical Examples

59

Fig. 4.19 Phase plane for A = 0.3, a = 0.6, L = 0.6 (4.36)

numerical

approximate solution

Fig. 4.20 Comparison between the approximate solution (4.39) and numerical integration results for A = 0.4, L = 0.6 numerical approximate solution

Fig. 4.21 Phase plane for A = 0.3, L = 0.6

numerical

approximate solution (4.39)

60 Table 4.1 Comparison between the numerical solution of the frequency and the approximate frequency given by (4.25)

4 Oscillations of a Pendulum Wrapping on Two Cylinders Case no

num

app

3.2.3a

4.056165704763733

4.056213309077129

3.2.3b

4.0735936668241015

4.0735339712576275

3.2.3c

4.091213341156173

4.091781570513826

3.2.3d

4.065980106247986

4.0659463540564085

3.2.3e

4.10137202740024

4.1008614227769575

3.2.3f

4.137539732217073

4.137450803506345

3.2.3 g

4.070864763571452

4.071192550653585

3.2.3 h

4.124733651749398

4.125254940744536

3.2.3i

4.180380645932648

4.180777463105995

From Figs. 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10, 4.11, 4.12, 4.13, 4.14, 4.15, 4.16, 4.17, 4.18 and 4.19 it can be seen that the errors of the approximate solutions comparing to numerical ones increase with respect to increasing values of the parameters a and A. Also, for the particular case of classical simple pendulum the results obtained through our procedure are in very good agreement with numerical integration results. From the cases 4.2.3a–c, d–f and g–i respectively, we deduce that the frequency of the system is increased by increasing the radius of cylinders (parameter a). Also, from the cases 4.2.3a,d,g it can be seen that the frequency of the system is increased by increasing the amplitude A. The same conclusion is obtained from the cases 4.2.3a, e and h or 3.2.3c, f and i, respectively. The sources of nonlinear oscillations of the pendulum wrapping on two cylinders are given by the radius of cylinders (parameters a), the amplitude A and the length L of pendulum.

References 1. R.M. Matthews, Time for Science Education: How Teaching the History and Philosophy of Pendulum Motion Can Contribute to Science Literacy (Springer, New York, 2000). 2. A. Aczel, Leon Foucault: His life, times and achievements, in M.R. Matthews, C.F. Gauld, A. Stinner, The pendulum: Scientific, historical, educational and philosophical perspectives (Springer, 2005, 171–184) 3. Encyclopaedia Britannica, 11-th Edition 4. W. Marrison, The evolution of the quartz crystal clock. Bell Syst. Tech. J. 27, 510–588 (1948) 5. C. Audoin, B. Guinot, S. Lyle, The measurement of time, frequency and the atomic clock (Cambridge University Press, 2001) 6. M. Willis, Time and timekeepers (MacMillan, 1945) 7. G.B. Airy, On the disturbances of pendulum and balances and on the theory of escapements. Trans. Cambridge Philosoph. Soc. III, 105–128 (1830) 8. V.F. Lenzen, R.P. Multauf, Development of gravity pendulums in the 19-th century. United States National Museum Bulletin, vol. 240 (1964) 9. M.N.H. Hamouda, G.A. Pierce, Helicopter vibration suppression using simple pendulum absorbers on the rotor blade. J. Amer. Helicopter Soc. 23, 19–29 (1984) 10. R.A. Nelson, M.G. Olsson, The pendulum-Rich physics from a simple system. Am. J. Phys. 54, 112–121 (1986)

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11. Z.M. Ge, F.N. Ku, Subharmonic Melnikov functions for strongly odd nonlinear oscillators with large perturbations. J. Sound Vibr. 236, 554–560 (2000) 12. T.M. Nester, P.M. Schnitz, A.G. Haddow, S.W. Shaw, Experimental observations of centrifugal pendulum vibration absorber, The 10-th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu Hawaii (2004) 13. S. Bendersky, B. Sandler, Investigation of a spatial double pendulum: an engineering approach, Discrete Dynamics in Nature and Society, Article ID 25193 (2006) 14. X. Xu, M. Wiercigroch, M.P. Cartmell, Rotating orbits of a parametrically-excited pendulum. ChaosSolitons Fractals 23, 1537–1548 (2005) 15. X. Xu, E. Pavlovskaia, M. Wiercigroch, E. Romero, S. Lenci, Dynamic interaction between parametric pendulum and electro-dynamical shaker. Z. Angew. Math. Mech. 87, 172–186 (2007) 16. B. Norton, J. Sieber, J.M.T. Thompson, M. Wiercigroch, Dynamics of the nearly parametric pendulum. Int. J. Nonlin. Mech 46, 436–442 (2011) 17. J. Warminski, K. Kecik, Autoparametric vibrations of a nonlinear system with pendulum, Math. Probl. Eng. Article ID 80705 (2006) 18. K. Kecik, J. Warminski, Dynamics of an autoparametric pendulum-like system with a nonlinear semiactive suspension. Math. Probl. Eng. Article ID 451047 (2011) 19. M.Z. Rafat, M.S. Wheatland, T.R. Bedding, Dynamics of a double pendulum with distributed mass. Am. J. Phys. 77, 216–222 (2009) 20. M. Turkilmazoglu, Accurate analytic approximation to the nonlinear pendulum problem. Phys. Scr. 84, Art. ID 015005 (2011) 21. M. Wiercigroch, A. Najdecka, V. Vaziri, Nonlinear dynamics of pendulum system for energy harvesting. Springer Proc. Phys. 139, 35–422 (2011) 22. L.J. Elias, M. Tsuchida, Study of a electromechanical pendulum with average method. Anais do I CMAC Sudeste 1, 57–60 (2011) 23. J. Awrejcewicz, Classical Mechanics: Dynamics. Advances in Mechanics and Mathematics (Springer, 2012) 24. M. Bayat, I. Pakar, M. Bayat, Application of He’s energy balance method for pendulum attached to rolling wheels that are restrained by a spring. Int. J. Phys Sci. 7, 913–921 (2012) 25. H. Mazaheri, A. Hosseinzadeh, M.T. Ahmadian, Nonlinear oscillation analysis of a pendulum wrapping on a cylinder. Scientia Iranica B 119, 335–340 (2012) 26. O. Boubaker, The inverted pendulum benchmark in nonlinear control theory: a survey. Int. J. Adv. Robotic Syst. 10, 1–5 (2013) 27. A. Najdecka, T. Kapitaniak, M. Wiercigroch, Syncronous rotational motion of parametric pendulums. Int. J. Nonlinear Mech. 70, 84–94 (2015) 28. P. Alevras, D. Yurchenko, A. Naess, Stochastic synchronization of rotating parametric pendulums. Meccanica 49, 1945–1951 (2014) 29. A. Jallouli, N. Kacem, N. Bouhaddi, Stabilization of solitons in coupled nonlinear pendulums with simultaneous external and parametric excitation. Commun. in Nonlin. Sci. Numer. Simul. 42, 1–11 (2017) 30. V. Marinca, N. Herisanu, An optimal iteration method for strongly nonlinear oscillators. J. Appl. Math. Art. ID 9063411 (2012) 31. V. Marinca, N. Herisanu, Nonlinear Dynamical Systems in Engineering (Some approximate Approaches (Springer, Heidelberg, 2011).

Chapter 5

Free Oscillations of Euler–Bernoulli Beams on Nonlinear Winkler-Pasternak Foundation

The use of beams of an elastic foundation has recently become widespread in engineering. Several research papers have appeared in literature on this topic. Horibe and Asano proposed a boundary integral equation method for calculating the large deflection of beams on an elastic foundation of the Pasternak type [1]. The governing equation is transformed into a more convenient form such that the iterative scheme can be applied to large deflection problems. Rao [2] studied large amplitude vibration of slender, uniform beams on elastic foundation with edges immovable axially using a direct numerical integration technique. The nonlinear response of a finite beam on nonlinear viscoelastic foundation subjected to a moving load is studied by Ansari et al. [3]. A comprehensive parametric sensitivity analysis is carried out to investigate the effects of different parameters. The response of the nonlinear system is studied by the method of multiple scales. Senalp et al. [4] considered dynamic response of a simply-supported, finite length Euler–Bernoulli beam with uniform cross-resting on a linear and nonlinear viscoelastic foundation acted upon a moving concentrated force. Tsiatas [5] investigated the nonlinear response of beams with variable properties resting on a nonlinear elastic foundation. The variable cross-sectional properties of the beam result in governing differential equations and the solution was achieved using the analogue equation method of Katsikadelis. Bhattiprolu and Davies [6] considered a pinned beam resting on a nonlinear viscoelastic and unilateral foundation. The static and dynamic responses of the beam are studied with Galerkin’s method. The equations involve the coordinates of the unknown lift-off-points which are determined as a part of the solution for structural response. Ding et al. [7] studied the convergence of the Galerkin method for the dynamic response of an elastic beam resting on a nonlinear foundation with viscous damping subjected to a moving concentrated load. Younesian et al. [8] applied the Variational Iteration Method for the dynamic response of an elastic beam rested on a nonlinear foundation. Koziol and Hryniewicz [9] presented a new semi-analytical solution of the Timoshenko beam subject to a moving load in case of a nonlinear medium underneath. The solution is obtained using wavelet filters of Coiflet type and Adomian Decomposition Method combined with the Fourier transform. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_5

63

64

5 Free Oscillations of Euler–Bernoulli Beams …

nonlinear model of microbeam accounts for the slightly curved beam is studied by Sari and Pakdemirli [10] by means of the method of multiple scales. Abdelghany et al. [11] investigated the dynamic response of a non-uniform Euler–Bernouli simply supported beam subjected to moving load and rested on a nonlinear viscoelastic foundation. Galerkin with Runge–Kutta method are also employed.

5.1 Nonlinear Euler–Bernoulli Beam Model The system under investigation is a simply-supported beam resting on a Winkler and Pasternak foundation. The case of a uniform finite beam is governed by the equation [7, 8, 11, 12]: EI

∂ 4w ∂ 2w ∂ 2w + ρA 2 + k1 w + k2 w3 − k3 2 = 0 4 ∂t ∂x ∂x

(5.1)

in which E is the Young modulus, I the moment of inertia of the cross-sectional area, A the area of the cross section, ρ the material mass density, k1 , k2 and k3 are parts of the Winkler and Pasternak foundation stiffness. The approximate solution is constructed for a simply supported beam based on the Galerkin approximation: w(x, t) =

n 

X i (x)Ti (t)

(5.2)

i=1

where n = 2 and Xi (x) = sin(i π x/ L), L being the length of the beam. Using dimensionless quantities one obtains: x=

x w π2 k 3 k2 i 4 π4 E I k1 , w = , t = tωi , α = , ωi2 = + + , i = 1, 2 (5.3) 4 L L ρA ρAL ρA ρAL 2

By means of Galerkin procedure, one can arrive to the nonlinear equations: 3α 3 (T + 2T1 T22 ) = 0, T¨1 + ω21 T1 + 4 1 3α 3 (T + 2T12 T2 ) = 0, T¨2 + ω22 T2 + 4 2

(5.4)

with the initial conditions T1 (0) = A1 , T˙1 (0) = 0, T2 (0) = A2 , T˙2 (0) = 0

(5.5)

5.1 Nonlinear Euler–Bernoulli Beam Model

65

where dot denotes derivative with respect to time.

5.2 OAFM for Free Oscillations of Euler–Bernoulli Beam If 1 and 2 are the frequencies corresponding to (5.4), then the linear operators are L 1 [T1 (t)] = T¨1 + 21 T1 ,

L 2 [T2 (t)] = T¨2 + 22 T2

(5.6)

Equations (5.4) and (5.5) have the solutions T1 (t) = T10 (t) + T11 (t), T2 (t) = T20 (t) + T21 (t)

(5.7)

where T10 and T20 are determined from the linear equations T¨10 + 21 T10 = 0, T10 (0) = A1 , T˙10 (0) = 0, T¨20 + 22 T20 = 0, T20 (0) = A2 , T˙20 (0) = 0

(5.8)

T10 (t) = A1 cos 1 t, T20 (t) = A2 cos 2 t

(5.9)

It is clear that

The nonlinear operators for Eqs. (5.4) are respectively 3α 3 (T + 2T1 T22 ), 4 1 3α N2 (T1 , T2 ) = (ω22 − 22 )T2 + (2T12 T2 + T23 ). 4 N1 (T1 , T2 ) = (ω21 − 21 )T1 +

(5.10)

Substituting Eqs. (5.9) into Eq. (5.10) we have 9 3 αA3 + αA1 A22 ] cos 1 t 16 1 4 3 3 + αA31 cos 31 t + αA1 A22 [cos 22 + 1 )t + cos(22 − 1 )t] 16 8 9 3 2 2 N2 (T10 , T20 ) = [(ω2 − 2 )A2 + αA32 + αA21 A2 ] cos 2 t 16 4 3 3 2 3 + αA2 cos 32 t + αA1 A2 [cos 21 + 2 )t + cos(21 − 2 )t] 16 8 (5.11) N1 (T10 , T20 ) = [(ω21 − 21 )A1 +

The first-order approximate solutions can be found from the linear equations:

66

5 Free Oscillations of Euler–Bernoulli Beams …

T¨11 + 21 T11 = (C1 + 2C2 cos 21 t + 2C3 cos 22 t){[A1 (ω21 − 21 ) 3 3 + α(3A31 + 4 A1 A22 )] cos 1 t + αA31 cos 31 t} 16 16 T¨21 + 22 T21 = (C4 + 2C5 cos 22 t + 2C6 cos 21 t){[A2 (ω22 − 22 ) 3 3 + α(3A32 + 4 A21 A2 )] cos 2 t + αA32 cos 32 t} 16 16

(5.12)

with the initial conditions T11 (0) = T˙11 (0) = 0, T21 (0) = T˙21 (0) = 0

(5.13)

Avoiding the secular terms, we obtain, respectively 3α (3A21 + 4 A22 ) + 16 3α 22 = ω22 + (3A22 + 4 A21 ) + 16 21 = ω21 +

3αA21 C2 16 C1 + C2 3αA22 C5 16 C4 + C5

(5.14)

From Eqs. (5.12) and (5.13), the first-order approximate solutions can be written as   1 3αA1 2 2 2 2 A1 (ω1 − 1 ) + (3A1 + 4 A2 ) C2 (cos 1 t − cos 31 t) T11 (t) = 16 821 αA31 C2 (cos 1 t − cos 51 t) 12821   A1 (ω21 − 21 ) + 3α A (3A21 + 4 A22 ) C3 16 1 [cos(1 + 22 )t − cos 1 t] + 42 (1 + 2 )   A1 (ω21 − 21 ) + 3α A (3A21 + 4 A22 ) C3 16 1 [cos(1 − 22 )t − cos 1 t] + 42 (2 − 1 ) 3αA31 C3 + [cos(31 + 22 )t − cos 1 t] 4(221 + 31 2 + 22 ) +

3αA31 C3 [cos(31 − 22 )t − cos 1 t] (5.15) − 31 2 + 22 )   1 3αA2 2 2 2 2 A2 (ω2 − 2 ) + (3A2 + 4 A1 ) C5 (cos 2 t − cos 32 t) T21 (t) = 16 822 +

4(221

αA32 C5 (cos 2 t − cos 52 t) 12822   A2 (ω22 − 22 ) + 3α A (3A22 + 4 A21 ) C6 16 2 [cos(2 + 21 )t − cos 2 t] + 41 (1 + 2 )

+

5.2 OAFM for Free Oscillations of Euler–Bernoulli Beam

67



 A2 (ω22 − 22 ) + 3α A (3A22 + 4 A21 ) C6 16 2 [cos(2 − 21 )t − cos 2 t] + 41 (1 − 2 ) 3αA32 C6 + [cos(32 + 21 )t − cos 2 t] 4(222 + 31 2 + 21 ) +

3αA32 C6 [cos(32 − 21 )t − cos 2 t] 4(222 − 31 2 + 21 )

(5.16)

From Eqs. (5.9), (5.15) and (5.16), the first approximations of Eqs. (5.4) are given by Eqs. (5.7).

5.3 Numerical Example For ω1 = 98.86, ω2 = 157.8, α = 11,600, A1 = 0.08, A2 = 0.1, the optimal values of the convergence-control parameters and then the frequencies could be obtained using the procedures described in Sect. 2 as: C1 = 114.951, C2 = 25.4744, C3 = 0.211777, C4 = 0.140021, C5 = 0.514075, C6 = −0.0254753, 1ap = 99.5218, 2ap = 158.237, which lead to the approximate solutions: T1 = 0.08043928 cos 1 t + 0.000064948vos31 t − 0.000119339 cos 51 t − 2.622 · 10−7 cos(1 + 22 )t − 1.151 · 10−6 cos(1 − 22 )t + 0.0000102434 cos(31 + 22 )t − 0.0003937187 · 10−7 cos(31 − 22 )t (5.17) T2 = 0.100018 cos 2 t + 4.3869 · 10−6 cos 32 t − 1.8606 · 10−6 cos 51 t + 4.2439 · 10−7 cos(2 + 21 )t − 1.863 · 10−6 cos(2 − 21 )t − 2.06698 · 10−6 cos(32 + 21 )t − 0.000173991 cos(32 − 21 )t (5.18)

Fig. 5.1 Comparison between the approximate solution (5.17) and numerical results: numerical; approximate

68

5 Free Oscillations of Euler–Bernoulli Beams …

Fig. 5.2 Comparison between the approximate solution (5.18) and numerical numerical; results: approximate

To prove the accuracy of the obtained approximate solutions, in Figs. 5.1 and 5.2 we compare the analytical results (5.17) and (5.18) with numerical integration results. It is observed that the solutions obtained through the proposed procedure are nearly identical with numerical solutions obtained using a fourth-order Runge Kuta method.

References 1. T. Horibe, N. Asano, Large deflection analysis of beams on two-parameter elastic foundation using the boundary integral equation method. JSME Int. J. Series A 44, 231–236 (2001) 2. V. Rao, Large amplitude vibration of slender, uniform beams on elastic foundation. Indian J. Eng. Materials Sci. 10, 87–91 (2003) 3. M. Ansari, E. Esmailzadeh, D. Younesian, Internal resonance of finite beams on nonlinear foundations traversed by a moving load, ASME Proceedings of IMECE 2008, Paper no. IMECE2008–68188, pp. 321–2329 (2016) 4. A.D. Senalp, A. Arikoglu, I. Ozkol, V.Z. Dogan, Dynamic response of a finite length Euler Bernoulli beam on linear and nonlinear viscoelastic foundation to a concentrated moving force. J. Mech. Sci. and Tech. 24, 19579961 (2010) 5. G.C. Tsiatas, Nonlinear analysis of non-uniform beams on nonlinear elastic foundation. Acta Mech. 209, 141–152 (2010) 6. U. Bhattiprolu, P. Davies, Response of a beam on nonlinear viscoelastic unilateral foundation and numerical challenges, ASME Proceed. IDETC/CIE, Paper no. DETC2011–48776, 813– 819 (2011) 7. H. Ding, L.-Q. Chen, S.-P. Yang, Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundation under a moving load. J. Sound Vibr. 331, 2426–2442 (2012) 8. D. Younesian, B. Saadatnia, H. Askari, Analytical solutions for free oscillations of beams on nonlinear elastic foundation using the variational iteration method. J of Theor. Appl. Mech. 50, 639–692 (2012) 9. P. Koziol, Z. Hrynicwicz, Dynamic response of a beam resting on a nonlinear foundation to a moving load: Coiflet-based solution. Shock and Vibr. 19, 995–1007 (2012) 10. G. Sari, M. Pakdemirli, Vibration of a slightly curved microbeam resting on an elastic foundation with nonideal boundary conditions, Math. Probl. Eng., ID736148 (2013)

References

69

11. S.M. Abdelghany, K.M. Ewis, A.A. Mahmoud, M.M. Nassar, Dynamic response of non uniform beam subjected to moving load and resting on a nonlinear viscoelastic foundation. J. Basic and Appl. Sci. 4, 192–199 (2015) 12. N. Herisanu, V. Marinca, Free oscillations of Euler–Bernoulli beams on nonlinear WinklerPasternak foundation. Springer Proc. Phys. 198, 41–48 (2018)

Chapter 6

Nonlinear Vibrations of Doubly Clamped Nanobeam Incorporating the Casimir Force

Micro-electromechanical systems (MEMS), micro-sensors and actuators are an extension of microelectronics and integrated circuits technology. Because of many commercial systems including metal alloy, accelerometers for airbag deployment in automobiles, single crystal silicon switches, filters, resonators, sensors, ink jet printer heads, atomic force microscopes, and so on. Nonlinearity usually arises from sources such as electrostatic actuation itself, squeeze-film damping, geometric nonlinearities and intermolecular forces i.e. Casimir or Van der Waals. Numerous analytical, numerical or experimental investigations have been conducted on the electrostatic and electrodynamic behaviors of the micro-beams. Lin and Zhao [1] studied the bifurcation behavior of nanoscale electrostatic actuators taking into consideration the presence of the Casimir force. Stability analysis showed that one equilibrium point is Hopf point and other is unstable saddle point when there are two equilibrium points. Hu [2] derived the total system energy expressions based on Euler–Bernoulli beam and by neglecting the fringing field capacitances. The closed form solution based on the full-order model is obtained by means of the minimum energy and the assumed mode methods. Using the reduced-order models, he showed that the fourth- and third-order models are not as accurate as the fullorder one. Rhoads et al. [3] regarded a microbeam device which couples the inherent benefits of a resonator with purely-parametric excitation with the simple geometry of a microbeam. An approximate analytical solution to the pull-in voltage of a microbridge considering the elastic boundary effect, fringing field capacitance, residual stresses and the distributed flexibility of the bridge is proposed by Hu et al. [4]. The accuracy of the obtained results is verified by comparison with FEM packages, other solutions and with experimental measured data. The static pull-in instability of electrostatically-actuated microbridges and microcantilevers is investigated by Mojahedi et al. [5] using the homotopy perturbation method. Soroush et al. [6] introduced Adomian Decomposition Method in the study of the pull-in behavior and the interval stress resultants of the nano-actuator using a distributed parameter model. Also the effects of the van der Waals and Casimir forces are taken into account.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_6

71

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6 Nonlinear Vibrations of Doubly Clamped Nanobeam Incorporating …

Yin et al. [7] established a new non-classical Bernoulli–Euler beam invoking size effect for electrostatically actuated microbeams by using the modified couple stress theory. Nonlinear terms associated with the mid-plane stretching and the electrostaticalforce are considered. Askari and Tahani [8] used the homotopy analysis method and Galerkin decomposition procedure to determine analytical approximate solutions for oscillatory behavior of a nanobeam under the effect of the Casimir force. Numerical integration is utilized to find critical value of the Casimir parameter to describe the pull-in instability. Kong [9] get the pull-in instability model for Bernoulli–Euler microbeams based on a modified couple stress theory and presented the approximate analytical solutions to the pull-in voltage and pull-in displacement on the electrostatically actuated microbeam. Caruntu et al. [10] proposed the reduced order model method (ROM) to investigate the nonlinear parametric dynamics of electrostatically actuated MEMS cantilever resonators. Fringe effect and damping forces are included,and the method of multiple scales and ROM are compared in this study. Younis [11] presented an exact analytical solution of the electrostatically actuated initially deformed cantilever beam problem. Simple analytical expressions are derived for two configurations: the curled and tilted configurations for beams of tip deflection of few microns and for largely deformed beams. The pseudo-spectral method is adopted by Maida and Bianchi [12] to numerically solve the problem of pull-in instability in a cantilever microbeam. They showed that poor approximation leads to very unphysical oscillatory attraction/repulsion forces along the cantilever. A novel procedure based on the Sturm’s theorem for real-valued polynomials is developed by Omarov et al. [13] to predict and identify periodic and non-periodic solutions for a grapheme-based MEMS lumped parameter model with general initial conditions. The procedure is supplemented numerically by using Phython codes. Skrzypacz et al. [14] presented bifurcation analysis of dynamic pull-in for a lumped model. The restoring force of the spring is derived based on the nonlinear constitutive stress–strain law and the driving force of the mass attached to the spring is based on the electrostatic Coulomb force, respectively.

6.1 Nonlinear Equation for Nanobeam We consider a clamped–clamped narrow nanobeam of length l, width b, thickness h and density ρ under the action of the Casimir force, as shown in Fig. 6.1. The distance between the beam and the stationary electrode is d0 . In Fig. 6.1, x is the coordinate along the thickness and W is deflection in the z-direction. The Casimir force per unit length of the beam is [15] Fc =

π 2 cW 240(d0 − W )4

(6.1)

6.1 Nonlinear Equation for Nanobeam

73

Fig. 6.1 A schematic of nanobeam with clamped–clamped boundary conditions

where  = 1.099:10−34 Js is Planck’s constant divided by 2π and c = 2.990:108 m/s is the speed of light in vacuum. If ν is Poisson’s ratio, I is the moment of inertia of cross-section about y-axis, E is the effective Young’s modulus and incorporating the von Karaman nonlinearity for mid plane stretching, the equation of motion that governs the transverse deflection W(x,t) of nanobeam, subjected to the Casimir force is as follow [8, 16]:      ∂ W (x, t) ∂ 4 W (x, t) Ebh l ∂ W (x, t) 2 ∂ 2 W (x, t) ρbh + E I − N + d x − Fc = 0 dt 2 ∂t 4 2l 0 ∂x ∂x2

(6.2) where N is the axial loading. The kinematic boundary conditions for the nanobeam deflection of the double-clamped case are: W (0, t) = 0,

∂ W (0, t) = 0, W (l, t) = 0, ∂x

∂ W (l, t) =0 ∂x

(6.3)

and the initial conditions are W (x, 0) = 0,

∂ W (x, 0) =0 ∂t

(6.4)

The following dimensionless variables are utilized:  2 x EI ˆ = W Nˆ = 12l N , xˆ = , tˆ = t , W l ρbkl 4 d0 Ebk 3  2 12l 4 π2 c d0 , λ4 = α=6 k 240Ek 3 d05

(6.5)

Using Eq. (6.2), the dimensionless equation of motion, dropping the hats, can be expressed as

74

6 Nonlinear Vibrations of Doubly Clamped Nanobeam Incorporating …

  (I V ) ¨ − α W +W

1

2



W d x + N W  −

0

λ4 =0 (1 − W )4

(6.6)

The boundary conditions (6.2) become W (0, t) = 0, W  (0, t) = 0, W (1, t) = 0, W  (1, t) = 0

(6.7)

where dot and prime denote derivative with respect to t and x respectively. By multiplying both sides of Eq. (6.6) by (1 − W)4 , we obtain   4 (I V ) 4 ¨ − (1 − W ) α (1 − W ) W + (1 − W ) W

1

4

2



W d x + N W  − λ4 = 0

0

(6.8)

6.2 Galerkin Formulation Separating the dependence of the deflection on W(x, t) into temporal and spatial by means of functions u(t) and X(x) respectively, we apply the Galerkin procedure for Eq. (6.8): W (x, t) = u(t)

X (x) X (0.5)

(6.9)

Within Eq. (6.9), X(x) is a trial function satisfying the kinematic boundary conditions and u(t) is the midpoint deflection of the microbeam and therefore the eigenfunction for the clamped–clamped microbeam is known as X (x) = cosh βx − cos βx −

cosh β − cos β (sinh βx − sin βx) sinh β − sin β

(6.10)

where β = 4.730040745. Substituting Eq. (6.9) into Eq. (6.8), and then multiplying Eq. (6.8) by X(x)/X(0.5), and integrating on the domain [0, 1], we obtain the nonlinear differential equation [8] (a0 + a1 u + a2 u 2 + a3 u 3 + a4 u 4 )u¨ + b0 + b1 u + b2 u 2 + b3 u 3 + b4 u 4 + b5 u 5 + b6 u 6 + b7 u 7 = 0

(6.11)

with the initial conditions u(0) = 0, u  (0) = 0

(6.12)

6.2 Galerkin Formulation

75

The expression of the coefficients ai and bi are given at the end of Sect. 6.4.

6.3 Application of OAFM to Eqs. (6.11) and (6.12) If  is the frequency of the system given by Eq. (6.1), then making the transformation τ = t, Eq. (6.11) can be rewritten as 2 (a0 + a1 u + a2 u 2 + a3 u 3 + a4 u 4 )u  + b0 + b1 u + b2 u 2 + b3 u 3 + b4 u 4 + b5 u 5 + b6 u 6 + b7 u 7 = 0

(6.13)

where u  = du/dτ. The linear operator, the function g(τ) and the nonlinear operator are respectively L[u(τ)] = 2 (u  + u), g(τ) = b0 , N [u(τ)] = 2 (a0 − 1 + a1 u + a2 u 2 + a3 u 3 + a4 u 4 ) + (b1 − 2 )u + b2 u 2 + b3 u 3 + b4 u 4 + b5 u 5 + b6 u 6 + b7 u 7

(6.14) The approximate solutions of Eq. (6.13) can be expressed as u(τ) ˜ = u 0 (τ) + u 1 (τ)

(6.15)

such that the initial approximation θ0 (τ) can be obtained from the linear equation. 20 (u 0 + u 0 ) + b0 = 0, u 0 (0) = u 0 (0) = 0

(6.16)

The Eq. (6.16) has the solution u 0 (τ) = −

b0 (1 − cos τ) 2

(6.17)

The nonlinear operator corresponding to Eq. (6.17) is N [u 0 (τ)] = A0 + A1 cos τ + A2 cos 2τ + A3 cos 3τ + A4 cos 4τ + A5 cos 5τ + A6 cos 6τ + A7 cos 7τ

(6.18)

where Ai , i = 0,1,2,…,7 are the coefficients of cos iτ obtained from the substitution of Eq. (6.17) into Eq. (6.14). The first approximation u1 (τ) can be obtained from equation 2 (u 1 + u 1 ) = (C1 + 2C2 cos τ + 2C3 cos 2τ + 2C4 cos 3τ)(A1 cos τ + A2 cos 2τ)

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6 Nonlinear Vibrations of Doubly Clamped Nanobeam Incorporating …

u 1 (0) = u 1 (0) = 0

(6.19)

Avoiding the secular term in Eq. (6.19), we obtain the condition A1 (C1 + C3 ) + A2 (C2 + C4 ) = 0

(6.20)

The solution of Eq. (6.19) is given by 1 u 1 (τ) = A1 C1∗ + A2 C3∗ )(1 − cos τ) + (A2 C1∗ + A1 C2∗ 3 1 ∗ + A1 C4 )(cos τ − cos 2τ) + (A2 C2∗ + A1 C3∗ )(cos τ − cos 3τ) 8 ∗ C A 1 2 4 + (A2 C3∗ + A1 C4∗ ) + (cos τ − cos 5τ) (6.21) 15 24 where Ci∗ = Ci /2 . From Eqs. (6.17), (6.21) and (6.15), the approximate solution is known.

6.4 Numerical Example Considering α = 6, N = 20, λ4 = 10, by minimizing the residual of the initial Eq. (6.13) for u˜ given by (6.15), the optimal values of the convergence-control parameters are obtained as: C1 = 0.000180703067, C2 = −0.000033253377, C3 = −0.000162423645, C4 = 0.000108804336, and the approximate frequency  = 24.149754208745. In this case, the approximate solution of Eqs. (6.11) and (6.12) is well determined. In Fig. 6.2 the obtained analytical solution is compared with numerical integration results obtained by means of a fourth-order Runge–Kutta method. The values of the coefficients ai and bi which appear into Eq. (6.11) are: Fig. 6.2 Comparison between the approximate solution (6.15) and numerical numerical; results: approximate

6.4 Numerical Example

77

1

1

a0 =

1

X (x)d x, a1 = −4

X (x)d x, a2 = 6

2

0

0

1 a3 = −4

0

1 X (x)d x, a4 =

1 X (x)d x, b0 = −λ4

5

0

b1 =

X (x)X 2

(I V )

0

1 (x)d x − N

0

X (x)X  (x)d x

0

1 b2 = 4N

X 2 (x)X  (x)d x − 4

0

b3 = 6

⎛

X 3 (x)X (I V ) (x)d x − α⎝

0

1

X 2 (x)X (I V ) (x)d x,

0

1

⎞⎛

X 2 (x)d x ⎠⎝

0

1 − 6N

X (x)d x,

6

0

1

1

X 4 (x)d x,

3

1

⎞ X (x)X  (x)d x ⎠

0

X 3 (x)X  (x)d x,

0

1 b4 = −4

X (x)X 4

(I V )

1 (x)d x + 4α

0

⎛

2

1



X (x)X (x)d x + 4N 0

X  (x)d x

0

⎞⎛ 1 ⎞ 1 1  b5 = X 5 (x)X (I V ) (x)d x − 6α⎝ X 2 (x)d x ⎠⎝ X 3 (x)X  (x)d x ⎠ 0

0

1 −N 0

0

X 5 (x)X  (x)d x ⎛

b6 = 4α⎝

1 0

⎛ b7 = −α⎝

1 0

⎞⎛ 1 ⎞  X 2 (x)d x ⎠⎝ X 4 (x)X  (x)d x ⎠, 0

⎞⎛ X 2 (x)d x ⎠⎝

1 0

⎞ X 5 (x)X  (x)d x ⎠

(6.22)

78

6 Nonlinear Vibrations of Doubly Clamped Nanobeam Incorporating …

References 1. W.H. Lin, Y.P. Zhao, Nonlinear behavior for nanoscale electrostatic actuators with Casimir force. Chaos Solitons Fractals 23, 1777–1785 (2005) 2. Y.C. Hu, Closed form solutions for the pull-in voltage of micro curved beams subjected to electrostatic loads. J. Micromech. Microeng. 16, 648–655 (2006) 3. J.F. Rhoads, S.W. Shaw, K.L. Turner, The nonlinear response of resonant microbeam systems with purely parametric electrostatic actuation. J. Micromech. Microeng. 16, 890–899 (2006) 4. Y.C. Hu, P.Z. Chang, W.C. Chuang, An approximate analytical solution to the pull-in voltage of a micro bridge with an elastic boundary. J. Micromech. Microeng. 17, 1870–1876 (2007) 5. M. Mojahedi, M.M. Zand, M.T. Ahmadian, Static pull-in analysis of electrostatically actuated microbeams with homotopy perturbation method. Appl. Mathem. Model. 34, 1032–1041 (2010) 6. R. Soroush, A. Koochi, A.S. Kazemi, A. Noghrehabadi, H. Haddadpour, Investigating the effect of Casimir and van der Waals attraction on the electrostatic pull-in instability of nano-actuators. Physica Scripta 82, Art.ID 045801 (2010) 7. L. Yin, Q. Qian, L. Wang, Size effect on the static behavior of electrostatically actuated microbeams. Acta. Mech. Sin. 27, 445–451 (2011) 8. A.R. Askari, M. Tahani, An analytical approximation to nonlinear vibration of a clamped nanobeam in presence of the Casimir force. Int. J. Aerosp. Lightweight Struct. 2, 317–334 (2012) 9. S. Kong, Size effect on pull-in behavior of electrostatically actuated microbeams based on a modified couple stress theory. Appl. Mathem. Model. 37, 7481–7488 (2013) 10. D. Caruntu, I. Martinez, K.N. Taylor, Voltage amplitude response of alternating current near half natural frequency electrostatically actuated MEMS resonators. Mech. Res. Commun. 52, 25–31 (2003) 11. M.I. Younis, Analytical expressions for the electrostatically actuated curled beam problem. Microsyst. Technol. 21, 1709–1717 (2015) 12. P.D. Maida, G. Bianchi, Numerical investigation of pull-in instability in a micro-switch MEMS device through the pseudo-spectral method. Modeling and Simulation in Engineering ID 8543616 (2016) 13. D. Omarov, D. Nurakhmetov, D. Wei, P. Skrzypacz, On the application of Sturm’s theorem to analyses of dynamic pull-in for a grapheme-based MEMS model. Appl. Comput. Mech. 12, 59–72 (2018) 14. P. Skrzypacz, S. Kadyrov, D. Nurakhmetov, D. Wei, Analysis of dynamic pull-in voltage of a grapheme MEMS model. Nonlinear Anal. Real World Appl. 45, 581–589 (2019) 15. S.K. Lomoreaux, Resource letter of Casimir force. Am. J. Phys. 67, 850–861 (1999) 16. V. Marinca, N. Herisanu, Optimal Auxiliary Functions Method for nonlinear vibration of doubly clamped nanobeam incorporating the Casimir force. Springer Proc. Phys. 251, 51–58 (2021)

Chapter 7

Transversal Oscillations of a Beam with Quintic Nonlinearities

The nonlinear oscillations of the beams received considerable attention over many years. Structures like airplane wings, bridges, buildings, helicopter rotor blades, robot arms or drill strings can be modeled as a beam-like member. It is very important to give an accurate analysis of the nonlinear vibration of these structures. Nonlinear dynamic problems have fascinated engineers, mathematicians and physicists for a long time. Geometrically nonlinear vibration of beams with different boundary conditions has long history. Many analytical and numerical studies were reported in the open literature. Pakdemirli [1] showed that, first discretizing the partial differential equation and then applying perturbation methods lead to incorrect results. An accurate solution is obtained by perturbation methods directly to the partial differential equations. Approximate method for analyzing the vibrations of an Euler–Bernoulli beam resting on a nonlinear elastic foundation is discussed by Nayfeh and Lacarbonara [2] in the cases of primary and subharmonic resonance. They proved that the approximate solutions based on discretization via the Galerkin method are contrasted with direct application of the multiple scales method. The nonlinear characteristics of the parametric resonance of simply supported elastic beams is investigated by Son et al. [3] considering the instability in the lowest mode, whereas Ghayesh and Balar [4] considered the motion of the parametric vibrations of an axially moving Timoshenko beam taking into consideration two nonlinear models. Abe [5] proposed an accuracy improvement to the multiple scales method for nonlinear vibration analyses of continuous systems with quadratic and cubic nonlinearities. Based on differential quadrature method, Peng et al. [6] proposed a new semianalytic method for the geometrically nonlinear vibration of Euler–Bernoulli beams with different boundary conditions. The supercritical equilibrium solutions of an axially moving beam supported by sleeves with torsion springs are analyzed by Ding et al. [7]. The supercritical transport speed ranger, equilibria and critical speeds of axially moving beams with hybrid © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_7

79

80

7 Transversal Oscillations of a Beam with Quintic Nonlinearities

boundary conditions are calculated from a nonlinear integro-partial-differential equation. Huang et al. [8] studied the fundamental and subharmonic resonances of an axially moving beam subject to periodic lateral force excitation. The incremental harmonic balance method was used to evaluate the nonlinear dynamic behavior. Local bifurcations and codimension-3 degenerate bifurcations of a nonlinear beam under parametric excitation is studied by Zhang et al. [9], and Sedighi et al. [10] presented an analytical solution for vibrations of quintic non-linear beam. Bayat and Pakar [11] implemented the variational approach for the Duffing equation with constant coefficients and for a restrained uniform beam carrying an intermediate lumped mass. Al-Qaisia and Hamdan [12] investigated the effect of an initial geometric imperfection on the in-plane nonlinear natural frequencies of an elastic Euler–Bernoulli beam resting on a Winkler elastic foundation. Bayat et al. [13] studied the nonlinear free vibration of a cylindrical shell by introducing the extended version of the Hamiltonian approach. Poorjamshidian et al. [14] presented the nonlinear vibration of a simple-supported flexible beam with constant velocity carrying a moving mass. The response time of the beam is obtained by means of a combination of the homotopy method and traditional perturbation. Wang et al. [15] considered the buckling of Euler–Bernoulli columns coupled with Eringen’s nonlocal theory under a tip load and uniformly distributed axial load. Analytical solutions for this type of clamped-free beam were derived.

7.1 The Governing Equation In what follows, we consider a flexible Euler–Bernoulli beam of length l subjected by a constant axial force P = P0 (Fig. 7.1). The Cartesian coordinate system is adopted in the symmetric plane of the beam and w is displacement of a point in the middle plane of the beam in y-direction. In Fig. 7.2 is considered an infinitesimal length of the Euler–Bernoulli beam with the ends F and G. Applying the Hamilton’s principle [9, 10] we obtain Fig. 7.1 Flexible Euler–Bernoulli beam

7.1 The Governing Equation

81

Fig. 7.2 A segment of an infinitesimal length

t2 δ

t2 Ldt +

t1

δW dt = 0

(7.1)

t1

with L the Lagrange function and W the virtual work. The kinetic energy of the beam l is T = m2 0 (u 2t + vt2 )d x, m being the mass per unit length of the beam and (u,v) are longitudinal and transversal displacement respectively  2 of end F from Fig. 7.2. 1 The strain potential energy of the beam is U = 2E v σ dv, where we consider the linear relation between the stress and strain σ = Eε. The total strain of the point F is ε = ε0 + ε1where the strain caused by the axial displacement of the beam is x = (1 + u x )2 + vx2 − 1, and the strain of the point F located at the ε0 = ds−d dx distance z from the middle plane, caused by the rotation of the cross-sectional plane = z[Wx x (1 + Wx2 )−3/2 ]. By means of Taylor series, the strains ε0 θ(x, t) is ε1 = z dθ ds and ε1 can be written in the form 1 1 1 ε0 = u x + Wx2 − Wx4 − u x Wx2 2 8 2   3 ε1 = z Wx x − Wx2 Wx x 2

(7.2) (7.3)

where ux = ∂u/∂x. It follows that the strain potential energy of the beam is EA U= 2

l  0

1 1 1 u x + Wx2 − Wx4 − u x Wx2 2 8 2

2

EI dx + 2

l 

3 Wx2 − Wx2 Wx x 2

2 dx

0

(7.4) l where A is the area of the beam of length dx and I = 0 z 2 d A. The virtual work is given by the applied force P0 and by the damping, such that

82

7 Transversal Oscillations of a Beam with Quintic Nonlinearities

t2

t2 δW dt = −

t1

t2  l P0 δu(l, t)dt −

t1

cWt δW d xdt t1

(7.5)

0

where c is the damping coefficient. If it is considered that no interaction occurs between transverse and longitudinal vibrations and therefore the longitudinal inertia can be neglected, then the governing equation of hinged-hinged beam subjected to axial constant force is  27 2 3 9 4 3 2 mWtt + cWt + E I W W − 3Wx x − 3Wx Wx x x x + Wx Wx x x x + Wx x x x 2 x xx 4   3 2 (7.6) + P0 Wx x + Wx Wx x = 0 2 

Supposing the transverse deflection w in the form w(x, t) = X (x)T (t)

(7.7)

where T is the amplitude of the fundamental transverse mode and X is the first eigenmode of the hinged-hinged Euler–Bernoulli beam, of the form X(x) = sinπx/l, and applying the Galerkin method, from Eq. (7.6) we obtain m T¨ + c T˙ +



   P0 π4 E I π4 P0 π2 7E I π6 27E I π8 5 T − T3 − − + T =0 4 2 4 6 l l 2l 8l 20l 8 (7.8)

Making the transformations  E I 1/2 lT cl 2 P0 l 2 π2 2 4 t= , T = , μ = , ω = π − , n ml 4 r2 (m E I )1/2 2E I l 4   7π2 r 4 π4 27 r π 8 P0 + 2 , δ = β= 2E I 8l 2E I l 4 20 l 

(7.9)

where r is the radius of gyration of the beam cross-section, then introducing the damping coefficient as 2

μ(Y ) = μ − αY + γY

4

(7.10)

and finally omitting the bar, we obtain the following nondimensional governing equation with constant excitation and quintic nonlinear term [16]: T¨ + (μ − αT 2 + γT 4 )T˙ + ω2n T − βT 3 − δT 5 = 0

(7.11)

7.1 The Governing Equation

83

The Eq. (7.11) describes the transversal vibrations of a hinged-hinged flexible beam subjected to a constant axial force. For the nonlinear differential equation with variable coefficients we will use the Optimal Auxiliary Functions Method.

7.2 OAFM for Nonlinear Differential Eq. (7.11) The initial conditions for Eq. (7.11) are T (0) = a, T˙ (0) = 0

(7.12)

where the amplitude a is unknown at this moment. The frequency of the system (7.11) is , such that making the transformations τ = t, T (τ) = ay(τ)

(7.13)

The original Eq. (7.11) can be rewritten in the form y  +

1 ω2 β δ (μ − αa 2 y 2 + γa 4 y 4 )y  + n2 y − 2 a 2 y 3 − 2 a 4 y 5 = 0    

(7.14)

and the initial conditions (7.12) become y(0) = 1, y  (0) = 0

(7.15)

where primes denote differentiation with respect to τ. The linear and nonlinear operators for Eq. (7.11) are, respectively L(y) = y  + y  N (y) =

(7.16)

 ω2n 1 β − 1 y + (μ − αa 2 y 2 + γa 4 y 4 )y  − 2 a 4 y 5 2   

(7.17)

Assuming that the approximate analytical solution for Eqs.(7.14) and (7.15) is y(τ) = y0 (τ) + y1 (Ci , τ), i = 1, 3, . . . n

(7.18)

then the initial approximation y0 (τ) is obtained from equation L(y0 (τ)) = 0,

y0 (0) = 1,

y0 (0) = 0

(7.19)

84

7 Transversal Oscillations of a Beam with Quintic Nonlinearities

The solution of Eq. (7.19) is y0 (τ) = cos τ

(7.20)

Substituting Eq. (7.20) into Eq. (7.17), we obtain N (y0 (τ)) = A1 cos τ + A2 sin τ + A3 cos 3τ + A4 sin 3τ + A5 cos 5τ + A6 sin 5τ

(7.21)

where   3 2 3 4 ω2n 10δa 4 1 5δa 4 , A3 = − μ − αa γa − 1 − , A = − + 2 2 2    4 8 163 5δa 4 αa 2 δa 4 γa 4 − , A5 = − (7.22) A4 = , A = − 6 4 16 163 16 A1 =

The function given by Eq. (7.20) and (7.21) are “source” for the auxiliary functions, such that the first approximation y1 (Ci , τ) is obtained from the following equation: y1 + y1 = (C1 + 2C2 cos 2τ + 2C2 sin 2τ + 2C3 cos 4τ+ + 2C5 sin 4τ)(A1 cos τ + A3 cos 3τ) + A6 sin 5τ),

y1 (0) = y1 (0) = 0 (7.23)

Form Eq. (7.23) y1 is immediately obtained, such that the approximate solution (7.18) becomes a (A3 C1 + A1 C2 + A6 C3 + A1 C4 )(cos τ − cos 3τ) 8 a a + (A6 C2 + A1 C3 + A1 C5 )(3 sin τ − sin 3τ) + (A3 C2 + A1 C4 )(cos τ 8 24 a a − cos 5τ) + (A6 C1 + A3 C3 + A1 C5 )(5 sin τ − sin 5τ) + (A3 C4 24 48 a − A6 C3 )(cos τ − cos 7τ) + (A6 C2 + A3 C5 )(7 sin τ − sin 7τ) 48 a a A6 C5 (cos τ − cos 9τ) + A6 C4 (9 sin τ − sin 9τ) (7.24) − 80 80

y(τ, Ci ) = a cos τ +

where the values of the convergence-control parameters Ci , i = 1,2,…,5 are optimally determined.

7.3 Numerical Example

85

Fig. 7.3 Comparison between the approximate solution (7.24) and numerical solution for γ = 2.5; α = 2.5; δ = 0.8; β = 3.5; μ = −0.05; ωn = 9.83; numerical analytical

7.3 Numerical Example We illustrate the accuracy of our procedure, considering the case corresponding to the following values of the physical parameters involved in Eq. (7.11): γ = 2.5; α = 2.5; δ = 0.8; β = 3.5; μ = −0.05; ωn = 9.83. By means of the procedure described in [17], we obtain C1 = 0.145927735933; C2 = −0.128431942959; C3 = −0.00431474801; C4 = 0.070184012124; C5 = −0.229972288472; a = 1.464442738889;  = 9.420966459723

(7.25)

As depicted in Fig. 7.3, the analytical results and numerical integration results obtained using a fourth-order Runge–Kutta method are almost identical.

References 1. M. Pakdemirli, A comparison of two perturbation methods for vibrations of systems with quadratic and cubic nonlinearities. Mech. Res. Commun. 21, 203–208 (1994) 2. A. Nayfeh, W. Lacarbonara, On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities. Nonlinear Dyn. 13, 203–220 (1997) 3. I.S. Son, Y. Uchiyama, W. Lacarbonara, H. Yabuno, Simply supported elastic beams under parametric excitation. Nonlinear Dyn. 53, 129–138 (2008) 4. M.H. Ghayesh, S. Balar, Non-linear parametric and stability analysis of two dynamic models of axially moving Timoshenko beams. Appl. Mathem. Model. 34, 2850–2859 (2010) 5. A. Abe, Accuracy improvement of the method of multiple scales for nonlinear vibration analyses of continuous systems with quadratic and cubic nonlinearities. Math. Problems Eng. Art. ID 890813 (2010) 6. J.S. Peng, Y. Lui, J. Yang, A semianalytical method for nonlinear vibration of Euler-Bernoulli beams with general boundary conditions. Mathem. Problems Eng. Art. ID 591786 (2010) 7. H. Ding, G.C. Zhang, L.Q. Chen, Supercritical equilibrium solutions of axially moving beams with hybrid boundary conditions. Mech. Res. Commun. 38, 52–56 (2011)

86

7 Transversal Oscillations of a Beam with Quintic Nonlinearities

8. J.L. Huang, R.K.L. Su, W.H. Li, S.H. Chen, Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances. J. Sound Vib. 330, 471–485 (2011) 9. W. Zhang, X.W. Feng, W.Z. Jean, Local bifurcations and codimension-3 degenerate bifurcations of quintic nonlinear beam under parametric excitation. Chaos Solitons Fractals 24, 977–998 (2005) 10. H.M. Sedighi, K.H. Shirazi, J. Zare, An analytical solution of transversal oscillation of quintic non-linear beam with homotopy analysis method. Int. J. Non-Linear Mech. 47, 777–784 (2012) 11. M. Bayat, I. Pakar, On the approximate analytical solution to non-linear oscillation systems. Shock. Vib. 20, 43–52 (2013) 12. A.A. Al-Qaisia, M.H. Hamdan, On nonlinear frequency veering and mode localizations of a beam with geometric imperfection resting on elastic foundation. J. Sound Vib. 332, 4641–4655 (2013) 13. M. Bayat, I. Pokar, L. Cveticanin, Nonlinear vibration of stringer shell by means of extended Hamiltonian approach. Arch. Appl. Mech. 84, 43–50 (2014) 14. M. Poorjamshidian, J. Sheiki, S.M. Moghadas, M. Nakhaie, Nonlinear vibrations analysis of the beam carrying a moving mass using modified homotopy. J. Solid Mech. 6, 389–396 (2014) 15. C.M. Wang, H. Zhang, N. Challamel, Y. Xiong, Buckling of nonlocal columns with allowance for selfweight. J. Eng. Mech. 142, 04016037 (2016) 16. N. Herisanu, V. Marinca, C., Opritescu, An approximate analytical solution of transversal oscillations with quintic nonlinearities. Springer Proc. Phys. 251, 41–49 (2021) 17. N. Herisanu, V. Marinca, Gh. Madescu, An analytical approach to non-linear dynamical model of a permanent magnet synchronous generator. Wind Energy 18, 1657–1670 (2015)

Chapter 8

Approximate Analytical Solutions to Jerk Equations

The nonlinear jerk equations involving the third temporal derivative of displacement have been originally of interest in the field of mechanics, but these kind of equations have found applications in a variety of physical situations. Jerk appears in some structures exhibiting rotating and translating motions, such as robots and machine-tools structures. From a practical perspective, excessive jerk arising at some machine-tools leads to excitation of vibrations in components in the machine assembly, accelerated wear in the transmission and bearing elements, noisy operations and large contouring errors at discontinuities (such as corners) in the machining path. Also in the case of robots, limiting jerk (defined as the time derivative of the acceleration of the manipulator joints) is very important because high jerk values can wear out of the robot structure, and heavily excite its resonance frequencies. Vibrations induced by non-smooth trajectories can damage the robot actuators, and introduce large errors while the robot is performing tasks such as trajectory tracking. Moreover, low-jerk trajectories can be executed more rapidly and accurately. Recently, there has been some interest in investigating different types of jerk equa... tions. Gottlieb [1, 2], has found simple forms of the jerk function u + f (u, u, ˙ u) ¨ =0 and has explored the flexibility of applying the method of harmonic balance to achieve analytical approximations of periodic solutions to nonlinear jerk equations. Consequent restrictions on the jerk equations amenable to harmonic balance solution are that only problems which have zero initial acceleration and parity and time-reversal invariant (all terms have the same space-parity of reflective behaviour under the transformation u → −u and time-parity of reflective behaviour under the transformation t → −t) can be considered. Wu et al. [3] proposed an improved harmonic balance method for determining accurate expressions of the periodic solutions of nonlinear jerk equations. Ma et al. [4] applied a homotopy approach to the jerk equations. A mixture of methodologies has been employed to study various aspects of nonlinear jerk equations in [5–8].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_8

87

88

8 Approximate Analytical Solutions to Jerk Equations

8.1 OAFM for Jerk Equations The general form of the Jerk equations is ... u + α u˙ 3 + βu 2 u˙ + γ u u˙ u¨ + δ u˙ u¨ 2 + λu˙ = 0

(8.1)

with the initial conditions u(0) = 0, u(0) ˙ = A, u(0) ¨ =0

(8.2)

where α, β, γ , δ, λ and A are known constants and the dot denotes derivative with respect to time. Into Eq. (8.1), at least one of α, β, γ and δ should be non-zero. If δ = 0, we require γ = −2α such that the jerk equations are simply not the time-derivative of an acceleration equation. If we introduce the independent variable τ = ωt and the dependent variable u = ωA x, then Eqs. (8.1) and (8.2) become respectively: 2 (x  + δ A2 x  x 2 ) + (α A2 x 3 + γ A2 x x  x  + λx  ) + β A2 x 2 x  = 0 x(0) = 0, x  (0) = 1, x  (0) = 0

(8.3) (8.4)

where prime denotes differentiation with respect to τ and = ω2 . For Eq. (8.3), the linear operator can be written in the form L[x(τ )] = 2 (x  + x)

(8.5)

with g(τ ) = 0, and the corresponding nonlinear operator is N [x(τ, )] = 2 (δ A2 x  x 2 − x) + (α A2 x 3 + γ A2 x x  x  + λx  ) + β A2 x 2 x  (8.6) The Eq. (2.5) becomes x0 + x0 = 0, x0 (0) = x0 (0) = 0, x0 (0) = 1

(8.7)

and has the solution x0 (τ ) = sin τ Substituting Eq. (8.2) into Eq. (8.6) we obtain

(8.8)

8.1 OAFM for Jerk Equations

89

N [x0 (τ, )] = M1 cos τ + M2 cos 3τ

(8.9)

with    1 1 2 3α − γ 2 2 δA − 1 + λ + A + β A2 M1 = 4 4 4 1 1 1 M2 = − δ A2 2 + (α + γ )A2 − β A2 4 4 4 

(8.10)

The Eq. (2.13) can be written as 2 (x1 + x1 ) + C1 F1 (τ ) + C2 F2 (τ ) + C3 F3 (τ ) + C4 F4 (τ ) = 0, x1 (0) = x1 (0) = x1 (0) = 0

(8.11)

where the optimal auxiliary functions Fi can be chosen in the forms: F1 (τ ) = − 2 sin 2τ (M1 cos τ + M2 cos 3τ ), F3 (τ ) = − 2 sin 3τ,

F2 (τ ) = − 2 sin τ

F4 (τ ) = − 2 sin 5τ

(8.12)

where Ck , k = 1,2,3,4 are unknown parameters at this moment. The solution of Eq. (8.11) is chosen so that it contains no secular terms, which lead to the condition: C1 (M1 − M2 ) + C2 = 0

(8.13)

The solution of Eq. (8.11) is given by 

 3 5 (M1 C1 + C3 ) + (M2 C1 + C4 ) sin τ x1 (τ, Ci ) = 26 124 1 1 − (M1 C1 + C3 ) sin 3τ − (M2 C1 + C4 ) sin 5τ 26 124 From Eqs.(8.8) and (8.14) and taking into account that u(t) ˜ = the first order approximate solution of Eq. (8.1) in the form

A x(τ), ˜ ω

(8.14) we can get

  √ 3M1 C1 A 5M2 C1 3C3 5C4 1+ sin t u(t, ˜ Ci ) = √ + + + 26 124 26 124  √ √ M1 C 1 + C 3 M2 C 1 + C 4 sin 3 t − sin 5 t − (8.15) 26 124

90

8 Approximate Analytical Solutions to Jerk Equations

8.2 Numerical Examples Considering the jerk equations containing velocity-cubed and velocity times displacements squared: α = β = 1, δ = γ = λ = 0, Eq. (8.1) becomes: ... u + u( ˙ u˙ 2 + u 2 ) = 0

(8.16)

u(0) = 0, u(0) ˙ = A, u(0) ¨ =0

(8.17)

with the initial conditions

8.2.1 Case 1 For A = 2, the parameters Ci and are: C1 = −0.000117346, C2 = 0.000219373, C3 = −0.165811985, C4 = −0.02975498, = 3.206684124

The first-order approximate periodic solution of Eq. (8.16) is: √ √ t + 0.007124397 sin 3 t √ + 2.7033546 · 10−4 sin 5 t

u(t) ˜ = 1.094143278 sin

(8.18)

8.2.2 Case 2 In this case for A = 10, we have: C1 = 1.102734811 · 10−8 , C2 = 1.705419027 · 10−5 , C3 = −0.367599608, C4 = −0.040821072, = 72.13317249 and therefore, the first-order approximate periodic solution of Eq. (8.16) becomes (Table 8.2) √ √ t + 0.016646815 sin 3 t √ + 0.000387423 sin 5 t

u(t) ˜ = 1.1255453485 sin

(8.19)

8.2 Numerical Examples

91

To verify the accuracy of the obtained solutions, we have compared the approximate analytical results obtained through OAFM and the numerical integration results. In Figs. 8.1 and 8.2 there are compared the present solutions (8.18) and (8.19) respectively and numerical results obtained by a fourth-order Runge–Kutta method while Tables 8.1 and 8.2 present detailed error analysis.

Fig. 8.1 Comparison between the approximate solution (8.18) and numerical integration results: numerical; approximate

Fig. 8.2 Comparison between the approximate solution (8.19) and numerical integration results: numerical; approximate

Table 8.1 Error analysis in case 1

t

uR−K

ε = abs(uR−K −uapp )

uapprox

T/10

0.647397

0.649905

0.00250

T/9

0.706095

0.709387

0.00329

T/8

0.774124

0.778531

0.00440

T/7

0.852270

0.858271

0.00600

T/6

0.939091

0.947329

0.00823

T/5

1.025365

1.036410

0.01104

T/4

1.074351

1.087289

0.01293

T/3

0.939091

0.947307

0.00821

T/2

−1.85 ×

10–8

−4.84 ×

10–5

0.00004

92 Table 8.2 Error analysis in case 2

8 Approximate Analytical Solutions to Jerk Equations t

uR−K

uapprox

ε = abs(uR−K −uapp )

T/10

0.677218

0.677384

0.00016

T/9

0.737552

0.737743

0.00019

T/8

0.807137

0.807349

0.00021

T/7

0.886578

0.886803

0.00022

T/6

0.974166

0.974389

0.00022

T/5

1.060446

1.060654

0.00020

T/4

1.109077

1.109285

0.00020

T/3

0.974166

0.974466

0.00029

0.000175

0.00017

T/2

1.4 ×

10−8

References 1. H.P.W. Gottlieb, Harmonic balance approach to periodic solutions of nonlinear jerk equations. J. Sound Vibr. 271, 671–683 (2004) 2. H.P.W. Gottlieb, Harmonic balance approach to limit cycles of nonlinear jerk equations. J. Sound Vibr. 297, 243–250 (2006) 3. B.S. Wu, C.W. Lim, W.P. Sun, Improved harmonic balance approach to periodic solutions of non-linear equations. Phys. Lett. A 354, 95–100 (2006) 4. X. Ma, L. Wei, Z. Guo, He’s homotopy perturbation method to periodic solutions of nonlinear jerk equations. J. Sound Vibr. 314, 217–227 (2008) 5. J.I. Ramos, Approximate methods based on order reduction for the periodicsolutions of nonlinear third-order ordinary differential equations. Appl. Math. Comput. 215, 4304–4319 (2010) 6. J.I. Ramos, Analytical and approximate solutions to autonomous, nonlinear, third-order ordinary differential equations. Nonlinear Anal. Real World Appl. 11, 1613–1626 (2010) 7. A.Y.T. Leung Z. Guo, Residue harmonic balance approach to limit cycles of non-linear jerk equations. Int. J. Non-Linear Mech. 46, 898–906 (2011) 8. P. Ju, X. Xue Global residue harmonic balance method to periodic solutionsof a class of strongly nonlinear oscillators. Appl. Math. Model. 38, 6144–6152 (2014)

Chapter 9

Vibration of Nonlinear Nonlocal Elastic Column with Initial Imperfection

The nonlocal elasticity theory pioneered by Eringen [1] is a modification of classical elasticity theory and has been widely accepted and attracted a growing attention in the last years. This theory is often applied to analyze the vibration behavior of nanostructures, and has been used in many areas including light and high toughness fibers, statistical mechanics, ocean engineering, mocro and nano electromechanical systems, and so on. For a detailed literature readers are referred to Gosh and Ray [2] and Askari et al. [3]. The study of Challamel [4] is focused on the geometrically exact elastic stability of two kinematically constrained flexible columns, modeled by the Euler–Bernoulli beam theory. The validity of parallel and translational beam assumptions is discussed. Using nonlocal elasticity theory, Hashemi et al. [5] considered surface effects including surface elasticity, surface stress and surface density on the free vibration analysis of Euler–Bernoulli and Timoshenko nanobeams. The governing equations are obtained and solved for silicon and aluminum nanobeams with three different boundary conditions. Xu et al. [6] presented a semianalytical treatment for calculating large elastic deformation of an initially imperfect nonlocal elastic column which is considered to be prismatic and inextensible. The constitutive equation corresponds to a differential type of Eringen’s nonlocal elasticity theory and Euler–Bernoulli assumption is used. The size effect and the initial imperfection are also discussed. Togun [7] analyzed nonlinear vibration of ananobeam resting on an elastic foundation. The nonlinear equation of motion includes stretching of the neutral axis (that introduces cubic nonlinearity into the equation), forcing and damping. The buckling and postbuckling behavior of a nonlinear discrete repetitive system is studied by Challamel et al. [8]. Ebrahimi and Nasirzadeh [9] used the Timoshenko beam theory for free vibration of nonlocal beam via Hamilton’s principle. The solution is obtained by means of differential transformation method. A discrete column model on the basis of the central finite difference formulation and the equivalent Hencky barchain model are presented by Wang et al. [10] for the elastic buckling of Eringen’s column with allowance for self weight.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_9

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9 Vibration of Nonlinear Nonlocal Elastic Column with Initial Imperfection

In this Chapter, the nonlocal elastic column is considered and the thickness to length ratio is assumed to be small such that the effect of transverse shear deformation is neglected.

9.1 Equation of Motion In what follows, we consider the vibration of a nonlocal elastic column with a slight geometrical curvature as an imperfection (Fig. 9.1). This inextensible, simply supported column of uniform cross-section A and length L is subjected to a conservative force P at its right movable end. The Cartesian coordinate system is chosen such that the abscissa x coincides with the line connecting the two hinged ends, the origin is located at its left end. If θ is the angle of inclination of the arc length reckoned along the column, then the general governing equation can be written in the form [6]:   2  d 2θ ds d 2θ d 2θ 2 − tan θ + EI 2 E I 2 + P sin θ = P(e0 a) cos θ 2 ds ds dθ ds

(9.1)

where EI is the flexural rigidity, e0 a is the parameter that allows for the size effect, s is the arc length, and θ is the initial imperfection. If a0 is the column midspan initial size, then [11]   d sin πs/L a0 θ (s) = sin ds ∗

−1

(9.2)

Using the dimensionless parameters s e0 a x = , μ= , d= L L



a0 π P L2 P , β= , λ2 = EI L EI

(9.3)

and the following result obtained from Eq. (9.2):   2



d 2 θ∗ 3 2 3 2 9 2 2 cos πx + β π β = π β β − 1 1 + β − 1 cos 3πx ds 2 4 4 Fig. 9.1 Initially imperfect nonlocal elastic column—geometry and coordinate system

(9.4)

9.1 Equation of Motion

95

The Eq. (9.1) can be rewritten as

θ + λ2 sin θ − μ2 λ2 θ cos θ − θ2 sin θ   2



3 9 2 2 − π β β − 1 1 + β cos πx − β3 π2 β2 − 1 cos 3πx = 0 4 4

(9.5)

where prime denotes derivative with respect to x. If α is the angle of the rotation at the left end of the column, then the initial conditions for Eq. (9.5) are θ(0) = α, θ (0) = 0

(9.6)

9.2 Application of OAFM to Vibration of Nonlinear Nonlocal Elastic Column with Initial Imperfection In the following we apply our procedure to obtain analytic approximate solution of Eqs. (9.5) and (9.6). For this purpose we choose the linear operator of the form L[θ(x)] = θ + ω2 θ

(9.7)

The initial approximation can be obtained from Eq. (2.5): θ0 + ω2 θ0 = 0, θ0 (0) = α, θ (0) = 0

(9.8)

where ω is the unknown frequency of the system. Equation (9.8) has the solution: θ0 (x) = α cos ωx

(9.9)

The nonlinear operator corresponding to Eq. (9.5) is defined by

N [θ(x)] = −ω2 θ + λ2 sin θ − μ2 λ2 θ cos θ − θ2 sin θ   2



3 9 2 2 − π β β − 1 1 + β cos πx − β3 π2 β2 − 1 cos 3πx 4 4

(9.10)

By substituting Eq. (9.9) into Eq. (9.10) it holds that N [θ0 (x)] = M1 cos ωx + M2 cos 3ωx + M3 cos 5ωx + N1 cos πx + N2 cos πx where

(9.11)

96

9 Vibration of Nonlinear Nonlocal Elastic Column with Initial Imperfection

  3 3 α5 α5 − ω2 α + μ2 λ2 ω2 α − α8 + 192 M1 = λ2 α − α8 + 192  5



α α3 3 5 + μ2 λ2 ω2 − 38 α3 + 128 − 24 α , N2 = − 43 β3 β2 − 1 M2 = λ2 384



2 5 5α5 α + μ2 λ2 ω2 384 , N1 = −π2 β β3 − 1 1 + 49 β2 M3 = λ1920

(9.12)

Within the expressions (9.12) we used the expansions   3  α α5 α3 α5 α5 cos ωx + − + cos 3ωx + sin θ = α − + cos 5ωx 8 192 24 384 1920 (9.13) 

The Eq. (2.13) can be written as θ1 + ω2 θ1 + C1 F1 (x) + C2 F2 (x) + C3 F3 (x) = 0

(9.14)

where the optimal auxiliary functions F1, F2 and F3 can be chosen in the form F1 (x) = −P(x), F2 (x) = −2(cos 2ωx)P(x) F3 = −2(cos 4ωx)P(x), P(x) = M1 cos ωx + M2 cos 3ωx

(9.15)

Considering Eq. (9.15), the Eq. (9.14) can be rewritten as θ1 + ω2 θ1 = [M1 C1 + (M1 + M2 )C2 + M2 C3 ] cos ωx + M2 C1 + M1 C2 + M1 C3 cos 3ωx + (M2 C2 + M1 C3 ) cos 5ωx + M2 C3 cos 7ωx θ1 (0) = θ1 (0) = 0

(9.16)

Avoiding secular terms in Eq. (9.16), we obtain the condition

ω2 =

λ2

μ2 λ2

 α−

 α3 8

  3 α5 (C2 + C3 ) − α − α8 + 192 (C1 + C2 )   3 α5 3α5 + 192 (C1 + C2 ) − 3α8 + 128 (C2 + C3 ) − α(C1 + C2 )

α3 24

−

α5 384



(9.17) The solution of Eq. (9.16) is given by is given by M2 C1 + M1 (C2 + C3 ) (cos ωx − cos 3ωx) 8ω2 M2 C 2 + M1 C 3 M2 C 3 + (cos ωx − cos 5ωx) + (cos ωx − cos 7ωx) 2 24ω 48ω2

θ1 (x, C1 , C2 , C3 ) =

(9.18)

9.2 Application of OAFM to Vibration of Nonlinear Nonlocal …

97

Fig. 9.2 Comparison between the approximate solution (9.20) and numerical integration results: _____numerical; _ _ _approximate solution

From (9.9) and (9.18) and (2.3), the first-order approximate solution becomes ˜ C1 , C2 , C3 ) = α cos ωx + θ1 (x, C1 , C2 , C3 ) θ(x,

(9.19)

9.3 Numerical Example Considering α = π/6, λ = 1, β = 0.1 and β = 0.01, by minimizing the residual of the initial equation, the optimal values of the convergence-control parameters are obtained as C1 = 3.45524, C2 = −2.12519, C3 = 2.69221, and the approximate frequency is ω = 0.985034. In this case, the first-order approximate solution given by (9.19) is (Fig. 9.2) 1 θ˜ (x) = π cos ω − 0.0026422(cos ωx − cos 3ωx) 6 + 0.000898379(cos ω − cos 5ωx) − 0.0003694909(cos ωx − cos 7ωx)

(9.20)

From Eq. (9.20) we can observe that the approximate solution (9.19) obtained through OAFM is highly accurate.

References 1. A.C. Eringen, Nonlocal Continuum Fields Theory (Springer, NY, 2002). 2. S. Gosh, D. Roy, Numeric-analytic form of the Adomian decomposition method for two-point boundary value problems in nonlinear mechanics. J. Eng. Mech. 133, 1124–1133 (2007) 3. H. Askari, E. Esmailzadeh, D. Zhang, Nonlinear vibration analysis of nonlocal nanowires. Composites B 67, 60–71 (2014)

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4. N. Challamel, On geometrically exact post-buckling of composite columns with interlayer slip—The partially composite elastic. Int. J. Nonlin. Mech. 47, 7–17 (2007) 5. S.H. Hashemi, M.R. Fakher, Naremnezhad, Surface effects on free vibration analyses of nanotubes using nonlocal elasticity: a comparison between Euler-Bernoulli and Timoshenko. J. Solid Mech. 5, 290–304 (2013) 6. S.P. Xu, M.R. Xu, C.M. Wang, Stability analysis of nonlocal elastic columns with initial imperfection. Math. Prob. Eng. Article ID 341232 (2013) 7. N. Togun, Nonlocal beam theory for nonlinear vibration of a nanobeam resting on elastic foundation. Boundary Value Problems 2016, 57 (2016) 8. N. Challamel, A. Kocsis, C.M. Wang, Higher-order gradient elasticity models applied to geometrically nonlinear discrete systems. Theor. Appl. Mech. 42, 223–248 (2015) 9. F. Ebrahimi, P. Nasirezadeh, A nonlocal Timoshenko beam theory for the vibration analysis of thick nanobeams using differential transform method. J. Theor. Appl. Mech. 53, 1041–1052 (2015) 10. C.M. Wang, H. Zhang, N. Challamel, Y. Xiang, Buckling of nonlocal columns with allowance for selfweight. J. Eng. Mech. 142, Art. 04016037 (2016) 11. V. Marinca, N. Herisanu, Vibration of nonlinear nonlocal elastic column with initial imperfection. Springer Proc. Phys. 198, 49–56 (2018)

Chapter 10

Nonlinear Vibration of Bernoulli–Euler Beam on a Winkler Elastic Foundation

The nonlinear dynamic response of an imperfect beam resting on elastic foundation has received a great deal of attention in the last decade due to their promising functionality and features. The free and forced vibration of such a beam model with various boundary conditions, types of elastic foundation, initial imperfections, have been the subject of many theoretical, numerical and experimental studies. The nonlinear non-planar dynamic responses of a near-square slightly curved, cantilevered geometrically imperfect beam under harmonic primary resonant base excitation with an one-to-one interval resonance is investigated by Aghababei et al. [1]. Considering two different geometric imperfection shapes, the sensitivity of the perfect beam model predicted limit-cycles to small geometric imperfections is analyzed. The influence of initial geometric imperfections on the nonlinear dynamic behavior of slender flexible cantilever beams is analyzed by Carvalho et al. [2]. By using several nonlinear dynamic tools, a complex dynamic behavior of the beam is observed at the 1:1 interval resonance region. The post-buckling and nonlinear vibration analysis of geometrically imperfect beams made of functionally graded materials resting on nonlinear elastic foundation subjected to axial force is investigated by Yaghobi and Torabi [3]. The effect of an initial geometric imperfection wavelength, amplitude and degree of localization of an elastic Bernoulli–Euler beam resting on a Winkler elastic foundation is considered by Al-Qaisia and Hamdan [4]. Eshragi et al. [5] carried out imperfection sensitivity of large amplitude vibration of curved single-walled carbon nanotubes modeled as a Timoshenko nano-beam. Geometric nonlinearities of von Karman type and nonlocal elasticity theory of Eringen are employed to derive governing equations of motion. The stability of three-layer beams with alumina foam core is studied by numerical FE analysis by Wstawska [6]. The calculations were made on a family of beams with different mechanical properties. Emam et al. [7] investigated the post buckling and free vibration response of geometrically imperfect multilayer nanobeams. The beam is assumed to be subjected to a pre-stress compressive load due to the manufacturing. Ghayesh and Balar [8] considered the nonlinear resonant dynamics of axially functionally graded tapered © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_10

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10 Nonlinear Vibration of Bernoulli–Euler Beam on a Winkler …

beam subjected to initial geometric imperfections, based on the Timoshenko beam theory. A rigorous coupled axial-transverse-rotational nonlinear model is developed taking into account the geometric nonlinearities due to large deformations coupled with an initial imperfection along the length of the beam. Liu et al. [9] investigated the nonlinear dynamic response of a thermally loaded thin composite plate subjected to harmonic excitation. A theoretical formulation is derived in terms of assumed modes and Airy stress function which incorporates an initial global geometric imperfection. The effects of the temperature, equivalent inplane boundary stiffness and initial geometric imperfection on the dynamic behavior are made through a detailed parametric study. Stability problems related to Euler– Bernoulli beams are presented in Krysko et al. [10].

10.1 System Description The beam under study is a pined-pined slender beam resting on a linear Winkler type elastic foundation of stiffness k, length L, mass per unit length m, Young’s modulus E, cross-sectional A and area moment of inertia I, with the imperfection W 0 assumed to be sinusoidal (Fig. 10.1). The equation of motion is considered in the form [4]: m

∂4W ∂2W + E I ∂t 2 ∂x4     2 L   1 ∂ W 2 ∂ W dW0 ∂ W d 2 W0 E A − + + d x + kW = 0 ∂x2 dx2 L 2 ∂x ∂x dx

(10.1)

0

Introducing the dimensionless parameters and variables W , y= r

W0 y0 = , t =t r

Fig. 10.1 Schematic view of beam resting on Winkler foundation with half-sine imperfection over the whole length L



EI x K L4 , k , ξ = = f m L4 L EI

(10.2)

10.1 System Description

101

where r is the radius of gyration of the beam cross-sectional area, Eq. (10.1) becomes ∂ y ∂ y + 4 − ∂t 2 ∂ξ 2

4



⎡ ⎤   1  2 ∂ y d y0 ⎣ 1 ∂ y ∂ y dy0 ⎦ + + dξ + k f y = 0 ∂ξ2 dξ2 2 ∂ξ ∂ξ dξ 2

2

(10.3)

0

The imperfection y0 is assumed to be of the form y0 = R sin πξ

(10.4)

in which R is dimensionless amplitude.

10.2 Discretization and Free Vibration of the Beam Under Study Applying Galerkin-Bubnov procedure, we assume that y(ξ, t) = φ(ξ )T (t)

(10.5)

where φ(ξ ) and T (t) are the eigenfunctions associated to linear mode shape of the beam and generalized coordinates, respectively. To determine the eigenfunction φ it is natural to consider the linearized form of the nonlinear partial differential Eq. (10.3) in the form ⎞ ⎛ 1  ∂4 y d 2 y0 ⎝ ∂ y dy0 ⎠ ∂2 y dξ + k f y = 0 + 4 − ∂t 2 ∂ξ dξ2 ∂ξ dξ

(10.6)

0

If ω is unknown frequency corresponding to linear system (10.6) and assuming that y(ξ, t) = φ(ξ )A sin ωt, then from Eqs.(10.4) and (10.6), one can put ∂ 4φ − (ω2 − k f )φ = π 3 R 2 sin π ξ ∂ξ 4

1

φ  (ξ ) cos π ξ dξ

(10.7)

0

The general solution of Eq. (10.7) is given by φ(ξ ) = A1 sin βξ + A2 cos βξ + A3 sinh βξ + A4 cosh βξ + P sin π ξ

(10.8)

where β4 = ω2 −k f , and the first four terms correspond to the homogeneous solution of Eq. (10.7) and the last term corresponds to the particular solution of (10.7). The

102

10 Nonlinear Vibration of Bernoulli–Euler Beam on a Winkler …

amplitude P of the particular solution is determined from the condition that the function φ p = P sin π ξ verifies Eq. (10.7). We have P=

π 3 R2 β4 − π 4

1

φ  (ξ ) cos π ξ dξ

(10.9)

0

Substituting Eq. (10.9) into Eq. (10.8), after simple manipulations, yields:   π3 R 2 β 1 π4 R 2 = (k1 A1 + k2 A2 + k3 A3 + k4 A4 ) P = 1− 2 β 2 − π4 β 4 − π4

(10.10)

where 1 k1 =

1 cos βξ cos πξdξ; k2 =

0

sin βξ cos πξdξ 0

1 k3 =

1 cosh βξ cos πξdξ; k4 =

0

sinh βξ cos πξdξ

(10.11)

0

Substituting Eq. (10.11) into Eq. (10.10) we can determine the constant P and we find P=

π3 R 2 β2  [A1 sin β  (π2 − β2 ) β4 − π4 − 21 π4 R 2

− A2 (1 + cos β) − A3 sinh β − A4 (1 + cosh β)]

(10.12)

The boundary conditions for the pined-pined beam are φ(0) = φ  (0) = φ(1) = φ  (1) = 0

(10.13)

which lead to the system of equation in which the determinant role of the coefficient matrix is zero. In this way, we can obtain the equation in β:    sinh β − 1 − cosh β sin β 1 + cos β π3 R 2 β 2 sin β + + 2 β4 − π4 − 0.5π4 R 2 π2 + β2 π − β2 π2 − β2     1 + cosh β sinh β 1 + cos β + sin β (cosh β − cos β) + sinh β − 2 − π + β2 π2 + β2 π2 − β2 β (10.14) + (sinh β − sin β) = 0 π

10.2 Discretization and Free Vibration of the Beam Under Study

103

The eigenfunction φ(ξ ) is determined from Eq. (10.8), (10.12) and (10.13). It follows that  (β + π α 3 )(cos β − cosh β) + π(α4 − α2 ) sinh β (sin βξ + α1 sin π ξ ) φ(ξ ) = k (β + π α 1 ) sinh β − (β + π α 3 ) sin β (β + π α 1 )(cos β − cosh β) + π(α4 − α2 ) sin β + cos βξ + α2 sin π ξ + (β + π α 3 ) sin β − (β + π α 1 ) sinh β (10.15) (sin βξ + α3 sin π ξ ) − cos βξ − α4 sin π ξ ] where π3 R 2 β2 sin β π3 R 2 β2 (1 + cos β) , α = − 2 (π2 − β2 )(β4 − π4 − 0.5π4 R 2 ) (π2 − β2 )(β4 − π4 − 0.5π4 R 2 ) 3 2 2 π R β sinh β π3 R 2 β2 (1 + cosh β) , α α3 = − 2 = − 4 (π + β2 )(β4 − π4 − 0.5π4 R 2 ) (π2 + β2 )(β4 − π4 − 0.5π4 R 2 ) (10.16)

α1 =

Now, to determine the generalized coordinate T (t), we substitute Eq. (10.5) into Eq. (10.3), then multiplying by φ(ξ ) and integrating on the domain [0,1], we obtain M1 T¨ + M2 T + M3 q 2 + M4 q 3 = 0

(10.17)

where the dot denotes derivative with respect to time t and 1 M1 =

1 φ (x)d x, 2

0

M2 =

φ(x)φ(x)(I V ) d x

0

⎞⎛ 1 ⎞ ⎛ 1   1 2 3⎝ +R π φ(x) cos π xd x ⎠⎝ φ(x) sin π xd x ⎠ + K f φ 2 (x)d x 0

0

⎞⎛ 1 ⎞ ⎛ 1   1 M3 = Rπ 2 ⎝ φ 2 (x)d x ⎠⎝ φ(x) sin π xd x ⎠ 2 0 0 ⎛ 1 ⎞⎛ 1 ⎞   − Rπ ⎝ φ  (x) sin π xd x ⎠⎝ φ(x)φ  (x)d x ⎠, 0

0

⎞⎛ 1 ⎞ ⎛ 1   1⎝ M4 = − φ 2 (x)d x ⎠⎝ φ(x)φ  (x)d x ⎠ 2 0

0

0

The Eq. (10.17) can be rewritten as:

(10.18)

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10 Nonlinear Vibration of Bernoulli–Euler Beam on a Winkler …

T¨ (t) + ωn2 T (t) + λT 2 (t) + δT 3 (t) = 0

(10.19)

where M2 M3 M4 , λ= , δ= M1 M1 M1

ω2n =

(10.20)

The initial conditions for Eq. (10.19) are T (0) = a, T˙ (0) = 0

(10.21)

It should be emphasized that the nonlinear differential Eq. (10.19) contains quadratic and cubic nonlinearity. If is unknown non-dimensional nonlinear natural frequency, and using the transformations τ = t and T (t) = aq(τ), Eq. (10.19) becomes q  (τ) +

ω2n aλ 2 δa 2 3 q(τ) + q (τ) + q (τ) = 0 2 2 2

(10.22)

and the initial conditions (10.21) become q(0) = 1, q  (0) = 0

(10.23)

where the prime denotes derivative with respect to τ. For the nonlinear differential Eq. (10.22), the linear and nonlinear operators are respectively: L[q(τ)] = q¨ + q

(10.24)

 aλ 2 δa 2 3 ω2n − 1 q + q + 2q 2 2

(10.25)

 N [q(τ)] =

and the function g(τ) = C1 , C 1 being an unknown parameter. In our technique, the approximate analytical solution of Eqs. (10.22) and (10.23) is of the form q(τ) ˜ = q0 (τ, C1 ) + q1 (τ, C2 , C3 , C4 )

(10.26)

where C1–4 are unknown parameters at this moment. The initial approximation q0 (τ, C1 ) is obtained from the linear differential equation

10.2 Discretization and Free Vibration of the Beam Under Study

q0 + q0 = C1 ; q0 (0) = 1, q  (0) = 0

105

(10.27)

The above equation has the solution q0 (τ, C1 ) = C1 + (1 − C1 ) cos τ

(10.28)

Substituting Eq. (10.28) into Eq. (10.25), we obtain   ω2n aλ δa 2 5 3 3 2 2 C C + (3C − 2C + 1) + C + + 3C + 1 1 1 1 1 2 2 2 2 2 1 2  2  ωn 2aλ + − 1 (1 − C1 ) + 2 (C1 − C12 ) 2  2 2 3C1 + 2C1 − 1 3δa + cos τ (1 − C ) 1 2 4   δa 2 (1 − C1 )3 3δa 2 aλ 2 2 cos 2τ + (1 − C ) + C (1 − C ) cos 3τ + 1 1 1 2 2 2 4 2

N [q0 (t, C1 )] =

(10.29) Introducing the auxiliary functions f 1 (τ) = C2 cos τ; f 2 (τ) = 2C3 cos 2τ; f 3 (τ) = 2C4 cos 3τ

(10.30)

the first approximation q1 (τ, C2 , C3 , C4 ) is determined from the equation q1



 ω2n 2aλ + q1 = (C2 cos τ + 2C3 cos 2τ + 2C4 cos 3τ) + 2 cos τ + δC1 cos 2τ 2  (10.31) q1 (0) = 0, q1 (0) = 0

It is important to mention that the auxiliary functions fi given by (10.30) are not unique, but we have the freedom to choose these auxiliary functions in alternative forms, such as: f 1 (τ) = C2 cos τ;

f 2 (τ) = 2C3 cos 2τ;

f 3 (τ) = 2C4 cos 3τ

(10.32)

f 2 (τ) = C3 cos 3τ;

f 3 (τ) = C4 cos 5τ

(10.33)

or f 1 (τ) = C2 cos τ; or yet f 1 (τ) = C2 2 cos τ; f 4 (τ) = C5 cos 6τ;

f 2 (τ) = C3 cos 2τ; f 5 (τ) = C6 cos 7τ

f 3 (τ) = C4 cos 4τ (10.34)

106

10 Nonlinear Vibration of Bernoulli–Euler Beam on a Winkler …

and so on. Another important remark is that the functions which appear in the last parenthesis of Eq. (10.31) are only some terms from Eq. (10.29). It is clear that these terms, also are not unique. Equation (10.31) can be rewritten as q1

  2 aλC2 ωn C 2 2aλC1 δC1 C2 + δC1 C4 cos τ + q1 = + δC1 C3 + + + 2 2 2 2   2 2ωn C3 aλC2 2aλC4 cos 2τ + + + 2 2 2   2 2ωn 2aλC3 δC1 C2 cos 3τ + C + + 4 2 2 2   2aλ cos 4τ + δC1 C4 cos 5τ + C + δC C (10.35) 4 1 3 2

Avoiding the presence of secular term into the last equation, we obtain the dimensionless natural frequency as 2 = −

2(C2 ω2n + 2λC3 ) δC1 (C2 + 2C4 )

(10.36)

From Eq. (10.35) it holds that   1 2ω2n aλC2 aλC2 q1 (τ) = + δC1 C3 (1 − cos τ) + C3 + 2 3 2 2   2  1 2ωn 2aλC4 2aλC3 δC1 C2 (cos τ − cos 2τ) + + C4 + + 2 8 2 2 2   1 2aλ (cos τ − cos 3τ) + C4 + δC1 C3 (cos τ − cos 4τ) 15 2 1 + δC1 C4 (cos τ − cos 5τ) (10.37) 24 

The first-order approximate solution of the Eq. (10.23) is obtained from Eqs. (10.22), (10.26), (10.28) and (10.37) as   aλC2 + δC C T˜ (t) = a[C1 + (1 − C1 ) cos t] + a 1 3 (1 − cos t) 2   a 2ω2n aλC2 2aλC4 (cos t − cos 2 t) + C3 + + 3 2 2 2  2  a 2ωn 2aλC3 δC1 C2 + (cos t − cos 3 t) C4 + + 2 2 8 2

10.2 Discretization and Free Vibration of the Beam Under Study

  a 2aλC4 + δC C 1 3 (cos t − cos 4 t) 15 2 a + δC1 C4 (cos t − cos 5 t) 24

107

+

(10.38)

where is given by Eq. (10.36).

10.3 Numerical Example Following the above described procedure and minimizing the residual of the original equation, for ω2n = 1.2, λ = 0.41, δ = 0.12, R = 0.7, k f = 0.5 and β = 2.06597523009635 × 10–8 obtained from Eq. (10.14), one can get C1 = −0.0169029829, C2 = 0.0057349377, C3 = −0.0083844923, C4 = −0.0001240869, = 1.09249 (10.39) such that the approximate solution given by Eq. (10.38) will be: (t) = − 0.0016690252 + 0.1011071737 cos[1.09606t] T + 0.0005520965 cos[2.19212t] + 0.0000103251 cos[3.28818t] − 5.691331803665242 × 10−7 cos[4.38424t] − 1.0487201519 × 10−9 cos[5.480304t]

(10.40)

In Fig. 10.2 is compared the approximate solution obtained by OAFM and the numerical integration results obtained for Eqs. (10.19) and (10.21) using a fourthorder Runge–Kutta method. We can see that the results obtained by means of OAFM are in very good agreement with numerical integration results. On the other hand, we Fig. 10.2 Comparison between the approximate solution (10.40) and numerical integration results for Eqs.(10.19) and (10.21) in case ω2n = 1.2, λ = 0.41, δ = 0.12 numerical, analytical

108

10 Nonlinear Vibration of Bernoulli–Euler Beam on a Winkler …

remark that the approximate frequency obtained from Eq. (10.36), ap = 1.09249 is nearly identical with that obtained after numerical integration, num = 1.09532.

References 1. O. Aghababaei, H. Nahoi, S.Z. Rad, Sensitivity analysis on dynamic responses of geometrically imperfect base excited cantilever beams. J. Vibroeng. 11, 52–56 (2011) 2. E.C. Carvalho, P.B. Goncalves, Z.G.N. Del Prado, G. Rega, Nonplanar vibration of an imperfect slender beam, in Proceedings of XXXIV Iberian Latin-Americal Congress on Computational Methods in Engineering, Brasil (2013) 3. H. Yaghoobi, M. Torabi, Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation. Appl. Math. Model. 37, 8324–8340 (2013) 4. A.A. Al-Qaisia, M.H. Hamdan, On nonlinear frequency veering and mode localizations of a beam with geometric imperfection resting on elastic foundation. J. Sound Vib. 332, 4641–4655 (2013) 5. I. Eshraghi, S.K. Jalali, N.M. Pugno, Imperfection sensitivity of nonlinear vibration of curved single-walled carbon nanotubes based on a nonlocal Timoshenko beam theory. Materials 9, 78 (2016) 6. I. Wstawska, The influence of geometric imperfections on the stability of three layer beams with foam core. Arch. Mech. Technol. Mater. 37, 65–69 (2017) 7. S.A. Emam, M.A. Eltaher, M.E. Khatter, W.S. Abdalla, Postbuckling and free vibration of multilayer imperfect nanobeams under a pre-stress load. Appl. Sci. 8, 223 (2018) 8. M.H. Ghayesh, S. Balar, Non-linear parametric and stability analysis of two dynamic models of axially moving Timoshenko beams. Appl. Math. Model. 34, 2850–2859 (2010) 9. L. Liu, J.-M. Li, G.A. Kardomateas, Nonlinear vibration of a composite plate to harmonic excitation with initial geometric imperfection in thermal environments. Compos. Struct. 209, 401–442 (2019) 10. A.V. Krysko, J. Awrejcewicz, I.E. Kutepov, V.A. Krysko, Stability of curvilinear EulerBernoulli beams in temperature fields. Int. J. Nonlinear Mech. 94, 207–215 (2017)

Chapter 11

The Nonlinear Thermomechanical Vibration of a Functionally Graded Beam (FGB) on Winkler-Pasternak Foundation

The concept of functionally graded materials (FGM) appeared in 1984 in Sendai area of Japan. These are heterogeneous, anisotropic materials and are made from a mixture of ceramics and metals. The mechanical properties of them varies smoothly and continuously from the surface to other surface of the material. Initially FGM were designed as thermal barrier materials for aerospace application and fusion reactors. These materials had found application in various fields of engineering like automotive, semiconductor industry, manufacturing industry, biomedical science, aerospace, defense industry and general structural element in thermal environments. Many researchers have investigated different aspects of FGM. The effect of damage on free and forced vibration of a FG cantilever beam is studied by Birman and Byrd [1]. The modes of damage include a region with degraded stiffness adjacent to the root of the beam, a single delamination crack and a single crack at the root cross section of the beam propagating in the thickness direction. Dokmeci presented a system of 1-D equations so that to analyze the thermoviscoelastic behavior of an axially FGB of rectangular cross section at high-frequency vibration [2]. The system of 1-D equation governs the extensional, flexural, torsional and thickness shear and also the coupled vibrations of the beam at high frequency. Khorrambadi [3] analyzed the free vibration of FGB with piezoelectric actuators subjected to axial compressive loads. The elasticity modulus of beam is assumed to vary as a power form of the thickness coordinate variable. The effect of the applied voltages, axial compressive loads and FG index of the vibration frequency are discussed. The static Green’s functions for FG Euler–Bernoulli and Timoshenko beams are presented by Carl et al. [4]. All material properties are arbitrary functions along the beam thickness direction. For symmetrical material properties along the beam thickness directions and symmetric cross-sections, the resulting stress distribution is also symmetric. Alshabatat and Naghshineh presented in [5] a design method to optimize the material distribution of FGM with respect to some vibration and acoustic properties. Two novel volume fraction laws are used to describe the material volume distributions through the length of the FGB. Ke et al. [6] discussed the effects of material property distribution and end supports on the nonlinear dynamic behavior of FGB. The direct numerical © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_11

109

110

11 The Nonlinear Thermomechanical Vibration of a Functionally …

integration method and Runge–Kutta method are employed to find the nonlinear vibration of FGB with different end supports. Yaghoobi and Torabi [7] investigated an analytic solution using variational iteration method for nonlinear vibration and postbuckling of beams made of FGM resting on a nonlinear elastic foundation, subject to an axial force. Mohammadi [8] considered the non-linear terms in von-Karaman’s strain–displacement relation with the help of Hamilton’s principle. The equation of potential and kinetic energy of the beam are derived and as a result, the nonlinear motion equation could be reached. Fu et al. [9] used the finite difference method or dynamic equations of FGB with piezoelectric patches. The Eshelby-Mori–Tanaka approach based on an equivalent fiber is used by Thomas et al. [10] to investigate free vibration of functionally graded nanocomposite beam reinforced by randomly oriented straight single-walled carbon nanotubes. The first five normalized mode shapes for this type of beam with different boundary conditions and different carbon nanotubes orientation are presented. The large-amplitude free vibration of clamped immovable thin beams made of FGM is investigated by Elmaguiri et al. [11] using the energy method and a multimode approach. By means of harmonic balance method, the equations of motion are converted into a nonlinear algebraic form and are solved by an iterative numerical method. The linear and nonlinear vibration behavior of monomorph and bimorph beam made from a mixture of PZT4 and PZT-5H with material composition are investigated by Yang et al. [12]. Maganti and Nalluri [13] considered the deformation variables to determine flapwise bending of rotating functionally graded double tapered beam attached to a rigid hub. The equations of motions are derived using hybrid deformation variables employing Lagrange’s approach and Rayliegh-Ritz method is used to evaluate the frequencies of the beam. The effect of temperature field on the natural frequencies of FGB with different conditions is studied by Kashyzadel and Asforjani [14]. Modal analysis has been performed for a FGB with clamped–clamped and clamped-free supports. The vibration of Euler–Bernoulli beam with FGM which is modeled by fourth-order partial differential equations with variable coefficients are examined by Yigit et al. [15] by using the Adomian Decomposition Method. Su et al. [16] presented a unified solution for free and transient analyses of a functionally graded piezoelectric curved beam with general boundary conditions within the framework of Timoshenko beam theory. The formulation is derived by means of the variational principle in conjunction with a modified Fourier series. Fundamental frequency of sandwich beams with functionally graded face sheet and homogenous core is studied by Mhu and Zhao [17]. The classical plate theory is used to analyze the face sheet and a higher-order theory is used to analyze the core of sandwich beams in which both the transverse normal and shear strains of the core are considered. Shwartsman and Majak [18] studied free vibration of axially functionally graded Euler–Bernoulli beams with elastically restrained ends. The method of initial parameters in differential form is treated for the numerical solution of the problem. Numerical method proposed has fourth order of accuracy and the Richardson extrapolation of results with different step sizes gives solutions of the sixth order of accuracy.

11.1 The Governing Equations

111

11.1 The Governing Equations In what follows, we consider a FGB of length L, width b, and thickness h resting on an elastic foundation of Winkler-Pasternak type and subjected to an axial force P (Fig. 11.1). The mechanical properties of the FGB can be varied as a function along thickness, based on the rule of mixtures. Taking into account the rule of mixtures, we have   2z + h k + E1 E(z) = (E 2 − E 1 ) 2h   2z + h k + ρ1 ρ(z) = (ρ2 − ρ1 ) 2h   2z + h k + v1 v(z) = (v2 − v1 ) 2h

(11.1)

(11.2)

(11.3)

where subscript 1 and 2 denote the top surface (z = −h/2) and bottom surface (z = h/2) respectively. The constant k characterizes the distributions of material properties. The case k = 0 corresponds to an isotropic homogenous beam. For a small strain, moderate deformation and rotation, the axial strain of the midplane of the beam accounting for the midplane stretching is given by [19, 20]:  2 ∂2W ∂U 1 ∂W +z 2 + εx = ∂x 2 ∂x ∂x

Fig. 11.1 Schematic of the FGB with nonlinear foundation

(11.4)

112

11 The Nonlinear Thermomechanical Vibration of a Functionally …

where, based on Euler–Bernoulli beam theory, the displacement of an arbitrary point along the x and z axes are U (x, y, z) and W (x, y, z) respectively. If U (x, t) and W (x, t) are displacement components in the midplane, then we have U (x, z, t) = U (x, t) + z

∂W ∂x

W (x, z, t) = W (x, t)

(11.5) (11.6)

where t is the time. The normal stress for the von Karaman type of geometry is given by the law σx x

⎡  2 ⎤ ∂2W E(z) ⎣ ∂U 1 ∂W ⎦ +z 2 + = 1 − v2 (z) ∂ x 2 ∂x ∂x

(11.7)

The curvature of the beam is given by 1 ∂2W = R ∂x2

(11.8)

The axial, coupling and bending stiffness are defined by

h/2 A11 = −h/2

h/2 B11 = −h/2

h/2 D11 = −h/2

E(z) dz 1 − v2 (z)

(11.9)

E(z) zdz 1 − v2 (z)

(11.10)

E(z) 2 z dz 1 − v2 (z)

(11.11)

Assuming that the initial temperature of the beam is zero, resultant force and thermal momentum are defined as

h/2 NT x = − −h/2

E(z) α(z)T dz 1 − v2 (z)

(11.12)

11.1 The Governing Equations

113

h/2 MT x = − −h/2

E(z) α(z)T zdz 1 − v2 (z)

(11.13)

The total induced axial force N x and bending moment M x are related to the stress resultants as ⎡  2 ⎤ 2 ∂U 1 W ∂ ⎦ + B11 ∂ W + N T x + N x = A11 ⎣ (11.14) ∂x 2 ∂x ∂x2 ⎡

 2 ⎤ 2 ∂U 1 W ∂ ⎦ + D11 ∂ W + MT x Mx = B11 ⎣ + ∂x 2 ∂x ∂x2

(11.15)

The equations of motion for axial and transverse vibration of FGB based on Euler–Bernoulli beam theory and von Karaman geometric nonlinearity are ∂ Nx ∂U − =0 ∂t ∂x   ∂ 2 Mx ∂W ∂2W ∂ − = Fw N + I x 1 ∂x ∂x ∂t 2 ∂x2 I1

(11.16)

(11.17)

where

h/2 I1 =

ρdz

(11.18)

−h/2

and F w is reaction of the elastic Winkler-Pasternak foundation. Now, if the axial inertia is neglected, then Eq. (11.14) gives ⎡

N x = N x0

 2 ⎤ 2 ∂U 1 ∂ W ⎦ + B11 ∂ W + N T x + = A11 ⎣ ∂x 2 ∂x ∂x2

(11.19)

and therefore N x is independent of x. From Eq. (11.17) we obtain  A−1 11

∂2W N x − B11 2 − N T x ∂x



 2 ∂U 1 ∂W = − 2 ∂x ∂x

Supposing that the beam has immovable ends,

(11.20)

114

11 The Nonlinear Thermomechanical Vibration of a Functionally …

U (0, t) = U (L , t) = 0

(11.21)

and integrating Eq. (11.20) with respect to x and having in attention that N x is independent of x, we obtain A11 Nx = 2L

L

2

W dx +

B11    W (L , t) − W (0, t) + N T x − P L

(11.22)

0

From Eqs. (11.14) and (11.22) it results ∂U 1 2  = A−1 11 (N x − N T x ) − W − B11 W ∂x 2

(11.23)

or 1 ∂U = ∂x 2L

L

2

W dx +

1 2 B B11     11 W (L , t) − W (0, t) − W − W (11.24) A11 L 2 A11

0

Differentiating Eq. (11.24) with respect to x yields ∂ 2U B11    = −W W − W 2 A11 ∂x

(11.25)

where prime denotes derivative with respect to x. Substituting Eq. (11.24) into Eq. (11.15), we obtain B11 Mx = 2L

L

2

W dx +

B 2  B11    11 W (L , t) − W (0, t) − W + D11 W (11.26) A11 L A11

0

Furthermore, using Eqs. (11.15) and (11.26) we have ∂2 M = ∂x2

 D11 −

2  B11 (I V ) W A11 L

(11.27)

By replacing the Eqs. (11.22) and (11.27) into Eq. (11.17), the governing nonlinear thermomechanical vibration equation of FGM is obtained as follows ⎡

L 2  B11 (I V )  A11 2 ⎣ I W + D11 − W −W W dx A11 L 2L 



0

11.1 The Governing Equations

+

115

  B11   W (L , t) − W (0, t) + N T x − P = Fw L

(11.28)

where the dot denotes derivative with respect to time. Due to the nonlinear elastic Winkler–Pasternak foundation, the second side of the Eq. (11.8) is defined as follows: 3

Fw = −k L W − k N L W + ks

∂2W ∂x2

(11.29)

where k L and k NL are the linear and nonlinear coefficients respectively and k s is the shear coefficient of elastic foundation. In order to derive general results independent of dimensions and specific sizes, we introduce the dimensionless parameters: x W , r= x= , W = L h NT = k2 =

NT x L 2 D11 − kN Lr 2 L D11 −

2 B11

,

D11 −

A11 4

2 B11 A11

A11 r 2 I1 t , 2α = 2 ,t = B A L D11 − A1111 P L2

P=

, k3 =



2 B11

A11

ks L 2 D11 −

2 B11 A11

, λ=

B11r 2 D11 −

2 B11

A11



 2  1 B11 D11 − D11 A11 L

, k1 =

kL L 4 D11 −

+ α(P − N T )

2 B11 A11

(11.30)

By replacing Eq. (11.30) into (11.28) it results W  + W (I V ) + k1 W + k2 W 3 − k3 W  ⎡ 1

   ⎣ − αW W 2 d x + λ W  (1, t) − W  (0, t) = 0

(11.31)

0

The Eq. (11.31) is a partial differential equation in two dimension, displacement x and time t. Using the Galerkin method, Eq. (11.31) becomes an ordinary differential equation. Assuming that the transverse displacement is expressed as W (x, t) = X (x)T (t)

(11.32)

where X(x) is the linear fundamental vibration mode and T (t) is the time dependent function to be determined. Substituting Eq. (11.32) into Eq. (11.31) and applying Galerkin’s method, in which the orthogonality property of the mode shapes is used, yields: T¨ + αT + bT 2 + cT 3 = 0

(11.33)

116

11 The Nonlinear Thermomechanical Vibration of a Functionally …

where

1 X (x)X

a=

(I V )

1 (x)d x + k1

0

1 X (x)d x − k3 2

0

  b = −αλ X  (1) − X  (0)

1

X (x)X  (x)d x

0

X (x)X  (x)d x

0

1 c = k2

⎛ 1 ⎞⎛ 1 ⎞



(I V ) 2  X (x)d x − α ⎝ X (x)d x ⎠⎝ X (x)X (x)d x ⎠

0

0

(11.34)

0

For the case of simply supported beam, the fundamental vibration mode is X (x) =

√ 2 sin π x

(11.35)

The Eq. (11.33) contains a quadratic nonlinear term due to the presence of bending-extension coupling effect in FGB and cubic nonlinear term due to the Winkler-Pasternak foundation. We remark that in Eq. (11.33) there exists no small or large parameter. The beam centroid is subjected to the following initial conditions T (0) = A, T˙ (0) = 0

(11.36)

11.2 Application of OAFM to Eqs. (11.33) and (11.36) If we introduce the independent variable τ = ωt and the dependent variable T = Ay, then Eqs. (11.33) and (11.36) become, respectively y  +

a b A 2 c A2 3 y + y + 2 y =0 ω2 ω2 ω y(0) = 1,

y  (0) = 0

(11.37) (11.38)

where prime denotes derivative with respect to τ and ω is the frequency of the system. For Eq. (11.37) the linear operator may be written L[y(τ )] = y  + y and the corresponding nonlinear operator becomes

(11.39)

11.2 Application of OAFM to Eqs. (11.33) and (11.36)

N [y(τ )] =

117

a  b A 2 c A2 3 − 1 y + y + 2 y ω2 ω2 ω

(11.40)

The Eqs. (2.5) and (2.6) become y0 + y0 = 0,

y0 (0) = 1,

y  (0) = 0

(11.41)

and has the solution y0 (τ ) = cos τ

(11.42)

Substituting Eqs. (11.42) into (11.40), we obtain     bA bA a 3c A2 c A2 cos τ + 2 cos 2τ + + − 1 + cos 3τ N y0 τ (] = 2 2 2 2ω ω 4ω ω 4ω2 (11.43) The Eq. (2.13) can be written as y1 + y1 + C1 F1 (τ ) + C2 F2 (τ ) + C3 F3 (τ ) + C4 F4 (τ ) + C5 F5 (τ ) y1 (0) = y1 (0) = 0

(11.44)

where F1 (τ ) = −P(τ ); F2 (τ ) = −2 cos τ P(τ ); F4 (τ ) = − cos 3τ ; F5 (τ ) = − cos 4τ

F3 (τ ) = −2 cos 2τ P(τ )

and P(τ ) =

bA + 2ω2



a 3c A2 − 1 + ω2 4ω2

 cos τ +

bA cos 2τ ω2

Alternatively, we may choose these functions on the forms F1 (τ) = −P(τ),

F2 (τ) = −2 cos τP(τ),

F3 (τ) = −2 cos 5τP(τ)

(11.45)

and P(τ) = or yet

bA + 2ω2



a 3c A2 −1+ 2 ω 4ω2

 cos τ

(11.46)

118

11 The Nonlinear Thermomechanical Vibration of a Functionally …

F1 (τ) = −P(τ), F2 (τ) = −2 cos τP(τ), F3 (τ) = −2 cos 3τP(τ) F4 (τ) = −P(τ) cos 4τ, F5 (τ) = −2 cos 5τP(τ)

(11.47)

and P(τ) =

bA bA cA + 2 cos 2τ + cos 3τ 2 2ω ω 4ω2

(11.48)

and so on. Using only Eqs. (11.44) and (11.45), the Eq. (11.43) becomes    bA 4a + 3c A2 b AC3 3b A C + − 1 C + + C2 1 2 2 2 2 2ω 4ω 2ω 2ω2       bA 4a + 3c A2 1 cos τ + C C + − 1 + C + ) (C 1 3 3 1 4ω2 ω2 2      4a + 3c A2 4a + 3c A2 + − 1 C2 cos 2τ + − 1 C3 2 4ω 4ω2    bA bA  + 2 C2 + C4 cos 3τ + C + C 3 5 cos 4τ, y1 (0) = y1 (0) = 0 ω ω2 (11.49)

y 1 + y1 =

The solution of Eq. (11.49) is chosen so that to contain no secular terms, which lead to the condition C2 3 3 ω2 = a + c A 2 + b A 4 2 C1 + C3

(11.50)

The solution of Eq. (11.49) is given by   bA bA 4a + 3c A2 + 2C − 1 C2 + C3 + (C ) 1 3 2 2 2ω 4ω 2ω2     1 bA 4a + 3c A2 + + 2C − 1 C2 (cos τ − cos 2τ) + (C ) 1 3 3 2ω2 4ω2     1 bA 4a + 3c A2 + C + − 1 C + C 2 3 4 (cos τ − cos 3τ) 8 2ω2 4ω2   1 bA (11.51) + C + C 3 4 (cos τ − cos 4τ) 15 2ω2

y1 (τ, Ci ) =

From Eqs. (11.49), (11.51) and (2.3) and taking into account that τ = ωt and T = Ay, we can get the first-order approximate solution of Eqs. (11.33) and (11.36) in the form

11.2 Application of OAFM to Eqs. (11.33) and (11.36)

119

   b A2 4a + 3c A2 b A2 T (t) = A cos ωt + C1 + A − 1 C2 + C3 (1 − cos ωt) 2ω2 4ω2 2ω2   2   1 b A (C1 + 2C3 ) 4a A + 3c A3 + + − A C2 (cos ωt − cos 2ωt) 3 2ω2 4ω2   2   1 b A C2 4a A + 3c A2 + + − A C3 + AC4 (cos ωt − cos 3ωt) 8 2ω2 4ω2   1 b A2 C 3 + (cos ωt − cos 4ωt) (11.52) + AC 5 15 2ω2 

where ω is given by Eq. (11.50).

11.3 Numerical Examples We consider the following three cases, in which the parameters a, b and c are obtained from Eq. (11.34): √ 3k2 a = π4 + k1 + π2 k3 , b = −2 2π3 αλ, c = + απ 4 2

(11.53)

Case 11.3.a For k 1 = k 3 = 50 and k 3 = 25, α = 1, λ = 0.1, A = 0.6, the parameters C i and ω are obtained using the described procedure as C1 = −0.8035880485, C2 = −0.2868078311 C3 = 0.1650363864 C4 = −0.0241556085, C5 = 0.0000545129, ω = 25.9571

(11.54)

The first-order approximate solution of Eqs. (11.33) and (11.36) is T˜ (t) = 0.000590623 + 0.598237 cos ωt − 0.0000679857 cos 2ωt + 0.00121644 cos 3ωt + 0.0000235972 cos 4ωt (11.55) To emphasize the accuracy of the obtained solution, we compare the approximate result obtained through OAFM with the numerical integration results. Figure 11.2 presents a comparison of the present solution (11.55) and numerical results obtained by a fourth-order Runge–Kutta method. Case 11.3.b In the second case, for k 1 = 50, k 2 = 40 and k 3 = 20, we have

120

11 The Nonlinear Thermomechanical Vibration of a Functionally …

Fig. 11.2 Comparison between the approximate solution (11.55) and numerical integration results _____numerical, _ _ _analytical

C1 = −0.8086487353, C2 = −0.2991968209, C3 = 0.1707921303 C4 = −0.027121251, C5 = 0.00005828, ω = 23.9352

(11.56)

and T˜ (t) = 0.00059746 + 0.59808 cos ωt − 0.0000422829 cos 2ωt + 0.00133548 cos 3ωt + 0.0000290429 cos 4ωt

(11.57)

Case 11.3.c In the last considered case, k 1 = 40, k 2 = 30 and k 3 = 50, such that C1 = −0.865250275, C2 = −0.3458418138 C3 = 0.1808644585 C4 = −0.0266366276, C5 = 0.0000323662, ω = 25.7944 (11.58) and therefore T˜ (t) = 0.000379825 + 0.598285 cos ωt + 0.0000164287 cos 2ωt + 0.00129126 cos 3ωt + 0.0000273128 cos 4ωt

(11.59)

In Figs. 11.3 and 11.4 are compared the solutions (11.57) and (11.59) with the corresponding numerical integration results. From Figs. 11.2, 11.3 and 11.4 one can be observed that the first-order approximate analytical results obtained by means of OAFM are almost identical with the numerical simulation results in all considered cases for various values of the parameters k i , i = 1,2,3.

11.3 Numerical Examples

121

Fig. 11.3 Comparison between the approximate solution (11.57) and numerical integration results _____ numerical, _ _ _ _analytical

Fig. 11.4 Comparison between the approximate solution (11.59) and numerical integration results _____ numerical, _ _ _ _analytical

References 1. V. Birman, L.W. Byrd, Vibrations of damaged cantilever beams manufactured from functionally graded materials. AIAA J. 45, 2747 (2007) 2. M.C Dokmeci, C. Altay, High-frequency thermoviscoelastic vibration of functionally graded thin beams. J. Acoust. Soc. America 119, 3336 (2006) 3. M.K. Khorambadi, Free vibration of functionally graded beams with piezoelectric layers subjected to axial load. J. Solid Mech. 1, 22–28 (2009) 4. O. Carl, P. Villamil, C. Zhang, Stress and free vibration analysis of functionally graded beams using static Greens’ functions. Proc. Appl. Math. Mech. 11, 199–200 (2011) 5. N.T. Alshabatat, K. Naghshineh, Optimization of natural frequencies and sound power of beams using functionally graded material. Adv. Acoust. Vibr. ID 752361 (2014) 6. L.-L. Ke, J. Yong, S. Kitipornchai, An analytical study on the nonlinear vibration of functionally graded beams. Meccanica 45, 743–752 (2010) 7. H. Yaghoobi, M. Torabi, An analytical approach to large amplitude vibration and post-buckling of functionally graded beams rest on non-linear elastic foundation. J. Theor. Appl. Mech. 51, 39–52 (2013) 8. N. Mohammadi, Nonlinear vibration analysis of functionally graded beam on WinklerPasternak foundation under mechanical and thermal loading via perturbation analysis method. Int. J. Eng. and Tech. Sci. 3, 144–158 (2015) 9. Y. Fu, J. Wang, Y Mao, Nonlinear vibration and active control of functionally graded beams with piezoelectric sensors and actuators. J. Intell. Mater. Syst. Struct. 22, 2093–2102 (2011)

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10. B. Thomas, P. Inamdar, T. Roy, B.K. Nanda, Finite element modeling and free vibration analysis of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotubes. Int. J. Theor. Appl. Res. Mech. Eng. 2, 97 (2013) 11. M.N. Elmaguri, M. Haterbouch, A. Bouayad, O. Oussouaddi, Geometrically nonlinear free vibration of functionally graded beams. J. Mater. Environ. Sci. 6, 3620–3633 (2015) 12. J. Yang, S. Kitipornchai, C. Feng, Nonlinear vibration of PZT4/PZT-5H monomorph and bimorph beams with graded microstructures. Int. J. Struct. Stab. Dyn. 15, 1540015 (2015) 13. N.V.R. Maganti, M.R. Nalluri, Flapwise bending vibration analysis of functionally graded rotating double-tapered beams. Int. J. Mech. Mater. Eng. 10, Art.21 (2015) 14. K.R. Kashizadeh, A.A. Asfarjani, Finite element study on the vibration of functionally graded beam with different temperature conditions. Adv. Mater. 5, 57–65 (2016) 15. G. Yigit, A. Sahim, M. Bayram, Modeling of vibration for functionally graded beams. Open Math. 14, 661–672 (2016) 16. Z. Su, G. Jin, T. Ye, Vibration analysis and transient response of a functionally graded piezoelectric curved beam with general boundary conditions. Smart Mater. Struct. 25, 065003 (2016) 17. L. Mu, G. Zhao, Fundamental frequency analysis of sandwich beams with functionally graded face and metallic foam core. Shock Vibr. ID 3287645 (2016) 18. B.S. Shvartsman, J. Majak, Free vibration analysis of axially functionally graded beams using method of initial parameters in differential form. Adv. Theor. Appl. Mech. 9, 31–42 (2016) 19. S. Emam, A.H. Nayfeh, Postbuckling and free vibrations of composite beams. Compos. Struct. 88, 636–642 (2009) 20. V. Marinca, N. Herisanu, The nonlinear thermomechanical vibration of a functionally graded beam on Winkler-Pasternak foundation. MATEC Web Conf. 148, 13004 (2018)

Chapter 12

Nonlinear Free Vibration of Microtubes

Microtubes have been widely used and studied in different applications such as capacitive switches, signal filtering, biology, information technology and semiconductor technology, cancer therapy, resonant sensors and so on. In the last years, there has been a great deal of interest in dynamic behaviour of micro-pipes or microtubes. Younis and Nayfeh [1] investigated the response of a resonant microbeam to an electric actuation. A nonlinear model is used to account for the mid-plane stretching a DC electrostatic force and an AC harmonic force. Also, Zand et al. [2] studied nonlinear oscillations of microbeams actuated by suddenly applied electrostatic force. Effects of electrostatic actuation, residual stress, midplane stretching and fringing fields are considered in modelling. Large amplitude flexural vibration behavior is presented by Shen [3] for microtubes embedded in an elastic matrix of cytoplasm. The mocrotube is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effect. Formulation are based on shell theory with a von Karman-Donnelltype of kinematic nonlinearity. Zeverdejani and Beni [4] analysed the free vibration of protein microtubes embedded in the cytoplasm by using Euler–Bernoulli model based on modified strain gradient theory. The protein microtube is modeled as a simply suported or clamped–clamped beam and the elastic medium is modeled with Pasternak support foundation. The microfluid-induced nonlinear free vibration of microtubes is studied by Yang et al. [5]. Based on Hamilton’s principle and a modifies couple stress theory and taking into consideration the geometric nonlinearity, a derivation of the nonlinear equation of motion is obtained. A mathematical formulation is proposed by Semnani et al. [6] to investigate the nonlinear flowinduced dynamic characteristics of a cantilevered pipe conveying fluid from macro to micro scale. Hosseini and Bahaadini [7] investigated the size dependent stability of cantilever nocro pipes. They used the modified strain gradient theory in conjuction with the Bernoulli–Euler beam model when two or three length scale parameters or all of them are zero. The micro Coriolis flow meters are extensively used by Ghazani et al. [8] in fluidic micro-circuits. The out-of-plane vibration and stability of curved microtubes are investigated to study the dynamic behaviour of curved Coriolis flow

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_12

123

124

12 Nonlinear Free Vibration of Microtubes

meters. The structural analysis of microtube vibration modes calculated by an atomistic approach is reported by Havelka et al. [9]. Molecular dynamics was applied to refine the atomic structure of a microtube including its anisotropy. A size-dependent Timoshenko beam model is used by Bahaadini et al. [10] for the study of free vibration and instability analysis of a nanotube conveying nanoflow. To capture the size effects, nonlocal strain gradient theory and Knudsen number are applied.

12.1 Problem Formulation The system under consideration is a pined-pined straight and slender microtube with length L, flexural rigidity EI, and mass per unit length m, as shown in Fig. 12.1. The internal flow in the microtube is due to an incompressible fluid of mass per unit length M, and flowing with velocity Vf . The displacement of the Bernoulli–Euler beam can be assumed to be: U = −Z ψ(x, t), V = 0, W = W (x, t)

(12.1)

where U, V and W are the displacement components in the x-, y-, and z-directions, respectively, ψ is the rotation angle of the central axis of the microtube given by ψ(x, t) =

∂ W (x, t) ∂x

Based on the Hamilton’s principle [5]

Fig. 12.1 Schematic view of a fluid-conveying microtube

(12.2)

12.1 Problem Formulation

125

t2 δ



 T p + T f − Q − Us dt = 0

(12.3)

t1

and on a modified couple stress elasticity theory, we have: m Tp = 2

L  0

M Tf = 2

L  0

L Q=



∂W ∂x

2 dx

∂W ∂W + Vf ∂t ∂x

E I + G Al 2

Us = − 0



2 +

V f2

   ∂2W 2

0

L

(12.4)



∂W P ∂x

∂x2

dx

dx

(12.5)

(12.6)

2 dx

(12.7)

where I is the area moment of inertia, A is cross-sectional area and P is the additional axial force L Us = − 0



∂W P ∂x

2 dx

(12.8)

Applying the variational technique, the dynamic equation of motion of this microtube can be derived as 

E I + G Al 2

 ∂4W ∂x4

∂2W ∂2W + (M + m) 2 + 2M V f ∂t ∂ x∂t ⎡ ⎤ L 2  ∂ W ∂2W ⎣ E A MV 2 − + dx⎦ = 0 ∂x2 2L ∂x

(12.9)

0

The boundary conditions for the pinned–pinned microtube are ∂ 2 W (L , t) ∂ 2 W (0, t) = W (0, t) = = W (L , t) = 0 2 ∂x ∂x2 Defining the following dimensionless variables and expressions:

(12.10)

126

12 Nonlinear Free Vibration of Microtubes

W M EI M x t ∗ , t = 2 , v = LVf , β= , x = , y= L L L m+M EI m+M E AL 2 G Al l d k2 = , ξ= , η= , h= , EI EI D D π D2 π D4 L (1 − h 2 ), I = (1 − h 4 ) (12.11) δ= , A= D 4 64 ∗

and omitting the star, Eq. (12.9) may be written in dimensionless form  ∂2 y ∂4 y ∂2 y ∂2 y 1 ∂2 y (1 + ξ) 4 + 2 + 2 βv + v2 2 = k 2 2 ∂x ∂x ∂ x∂t ∂x 2 ∂x

1  0

∂y ∂x

2 dx

(12.12)

Then the boundary conditions (12.10) are: y(0, t) =

∂ 2 y(1, t) ∂ 2 y(0, t) = y(1, t) = =0 ∂x2 ∂x2

(12.13)

12.2 Free Vibration of the Microtube In what follows, the free vibration of a pined-pined straight and slender microtube will be analyzed. Galerkin-Bubnov decomposition method is used to convert the nonlinear partial differential equation of motion (12.12) into a nonlinear ordinary differential equation. Assuming that the displacement expression is of the form y(x, t) = X (x)T (t)

(12.14)

where X(x) is the eigenfunction of the free undamped vibrations of a beam which satisfies the boundary conditions (12.13). In this section, we consider X (x) = sin πx, and T (t) is the generalized coordinate of the discretized system. Multiplying Eq. (12.9) with X(x) and then integrating this new equation on the domain x ∈ [0, 1][0, 1] and taking into account the identities 1 0

1 0

∂ 4 y(x, t) 1 X (x)d x = π 4 T (t), 4 ∂x 2 ∂2 y ∂x2

1  0

∂y ∂x

2

1 0

∂ 2 y(x, t) π2 T (t), X (x)d x = − ∂x2 2

π4 X (x)d x = − T 3 (t), 4

1 0

∂2 y X (x)dt = 0, ∂ x∂t

12.2 Free Vibration of the Microtube

1 0

127

∂2 y 1 X (x)dt = T¨ 2 ∂t 2

(12.15)

where the dot defines the derivative with respect to t, we obtain the nonlinear ordinary differential equation T¨ (t) + ω2n T (t) + αT 3 (t) = 0

(12.16)

where ω2n = (1 + ξ)π4 − π2 v 2 , α =

1 4 2 π k 4

(12.17)

The corresponding initial conditions for Eq. (12.16) are T (0) = a, T˙ (0) = 0

(12.18)

We mention that Eq. (12.16) is a well-known Duffing nonlinear differential equation, with strongly or weakly nonlinear cases, depending on the value of the parameter α.

12.3 OAFM for Eqs. (12.16) and (12.18) If is the frequency of the system (12.16) and making the transformation τ = t into (12.16), we obtain T  (τ) +

ωn α T (τ) + 2 T 3 (τ) = 0 2

(12.19)

where prime denotes derivative with respect to τ. For (12.19) we choose linear and nonlinear operators in the forms L[T (τ)] = T  (τ) + T (τ)  N [T (τ)] =

 ω2n α − 1 T (τ) + 2 3 (τ) 2 T

(12.20)

(12.21)

Assuming that the approximate analytical solution T˜ (τ) for Eqs. (12.18) and (12.19) is T˜ (τ) = T0 (τ) + T1 (τ, C1 , C2 , C3 , C4 )

(12.22)

128

12 Nonlinear Free Vibration of Microtubes

in which the initial approximation T0 (τ) will be defined as follows: T0 (τ) + T0 (τ) = 0, T0 (0) = a, T0 (0) = 0

(12.23)

The solution of Eq. (12.23) is T0 (τ) = a cos τ

(12.24)

Substituting Eq. (12.24) into Eq. (12.21), one can get: N [T0 (τ)] = A1 cos τ + A2 cos 3τ

(12.25)

where   a 3αa 2 2 2 , A 1 = 2 ωn − + 4

A2 =

α 2 a . 4

(12.26)

Taking into consideration the form of Eqs. (12.24) and (12.25), we define the auxiliary functions as F1 (τ) = −N [T0 (τ)], F2 = −2(cos 2τ)N [T0 (τ)], F3 (τ) = −2(cos 4τ)N [T0 (τ)], F4 (τ) = −(cos 6τ)N [T0 (τ)],

(12.27)

such as the first approximation T1 (τ, C1 , C2 , C3 , C4 ) is obtained from the equation T1 (τ) + T1 (τ) = (C1 + C3 cos 2τ + 2C3 cos 4τ + 2C4 cos 6τ)(A1 cos τ + A2 cos 3τ) T1 (0) = T1 (0) = 0, (12.28) where C 1 , C 2 , C 3 and C 4 are unknown parameters at this moment. The Eq. (12.28) can be rewritten as T1 + T1 = [A1 (C1 + C2 ) + A2 C2 + A3 C3 + A2 C4 ] cos τ + [A1 (C2 + C3 ) + A2 C1 ] cos 3τ + A1 (C3 + C4 ) + A2 C2 ] cos 5τ + A1 C4 cos 7τ + A2 C3 cos 9τ + A2 C4 cos 11τ,

(12.29)

Avoiding the presence of secular a term in Eq. (12.29) needs 2 = ω2n +

3α 2 αa 2 C2 + C3 + C4 a + . 4 4 C1 + C2

(12.30)

Taking into account Eq. (12.30), from Eq. (12.29) we find the following solution:

12.3 OAFM for Eqs. (12.16) and (12.18)

129

A1 (C2 + C3 ) + A2 C1 (cos τ − cos 3τ) 8 A1 (C3 + C4 ) + A2 C2 A1 C 4 + (cos τ − cos 5τ) + (cos τ − cos 7τ) 24 48 A2 C 4 A2 C 3 (cos τ − cos 9τ) + (cos τ − cos 11τ). (12.31) + 80 120

T1 (τ, Ci ) =

From (12.22), (12.24), (12.31) and τ = t, one can get the first-order approximate solution A1 (C2 + C3 ) + A2 C1 T˜1 (t, C1 , C2 , C3 , C4 ) = a cos t + (cos t − cos 3 t) 8 A1 (C3 + C4 ) + A2 C2 + (cos t − cos 5 t) 24 A2 C 3 A1 C 4 (cos t − cos 7 t) + (cos t − cos 9 t) + 48 80 A2 C 4 + (cos t − cos 11 t). (12.32) 120 where is given by Eq. (12.30) and the optimal values of the convergence-control parameters will be determined by minimizing the residual of the initial differential equation.

12.4 Numerical Example To show the validity of our technique, we consider a particular case, when a = 0.03, η = 0.1, h = 0.8, δ = 20, v = 3, μ = 0.38,

(12.33)

the optimal values of the convergence-control parameters and the frequency are: C1 = −0.0037510868, C2 = −0.0005173432, C3 = −0.0018105005, C4 = −0.0012962683, = 3.933145,

(12.34)

It is easy to verify the accuracy of the obtained results by plotting the approximate analytical solution given by Eqs. (12.24) and (12.31). Figure 12.2 show the comparison between the present solution and the numerical integration results obtained using a fourth-order Runge–Kutta method. From Fig. 12.2 it can be seen that the solution obtained by OAFM is very accurate being nearly identical with the numerical integration results.

130

12 Nonlinear Free Vibration of Microtubes

Fig. 12.2 Comparison between the approximate solution (12.32) with the parameters given by (12.34) and numerical integration results of Eqs. (12.16) and (12.18): numerical; analytical

References 1. M.I. Younis, A.H. Nayfeh, A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dyn. 31, 91–11 (2003) 2. M.M. Zand, M.T. Ahmadian, R. Rashidian, Semi-analytic solutions to nonlinear vibrations of microbeams under suddenly applied voltages. J. Sound Vib. 325, 389–439 (2009) 3. H.S. Shen, Nonlinear vibration of microtubes in living cells. Curr. Appl. Phys. 11, 812–882 (2011) 4. M.K. Zeverdejani, Y.T. Beni The nano scale vibration of protein microtubes based on modified strain gradient theory. Curr. Appl. Phys. 13, 1566–157 (2013) 5. T.Z. Yang, S. Ji, X-D. Yang B. Fang, Microfluid induced nonlinear free vibration of microtubes. Int. J. Eng. Sci. 76, 47–5 (2014) 6. A.M.D. Semnani, H. Zafari, E. Dehdashri, A parametric study on nonlinear flow-induced dynamics of a fluid-conveying cantilever pipe in a post flutter region from macro to micro scale. Int. J. Nonlinear Mech. 85, 207–225 (2016) 7. M. Hosseini, R. Bahaadni, Size-dependent stability analysis of cantilever micro-pipes conveying fluid based on modified strain gradient theory. Int. J. Eng. Sci. 101, 1–13 (2016) 8. M.R. Ghazani, H. Molki, A.A. Beigloo, Nonlinear vibration and stability analysis of the curved microtube conveying fluid as a model of the micro Coriolis flow meters based on strain gradient theory. Appl. Math. Model. 45, 1020–1030 (2017) 9. D. Havelka, M. Deriu, M. Cifra, D. Kucera, Deformation pattern in vibrating microtube: structurate mechanics study based on atomistic approach. Sci. Rep. 7, 4227 (2017) 10. R. Bahaadini, A.R. Saidi, M. Hosseini, On dynamics of nanotubes conveying nanoflow. Int. J. Eng. Sci. 123, 181–196 (2018)

Chapter 13

Nonlinear Free Vibration of Elastically Actuated Microtubes

In the last years microtubes received an increased importance in a large variety of applications related to capacitive switches, signal filtering, semiconductor technology or resonant sensors. Electrically actuated microtubes are studied by the MEMS community and there are many potential applications in optical, aerospace and biomedical engineering. There are known several actuation methods for MEMS devices, but electrosatatic actuation is the most well established actuation method because of its simplicity and high efficiency. Younis and Nayfeh [1] presented the response of resonant nanobeams to an electric actuation. They used a nonlinear model to account for the mid-plane stretching, a DC electrostatic force and an AC harmonic force including in the model design parameters by lumping them into nondimensional parameters. The influence of Casimir force on the nonlinear behavior of nanoscale electrostatic actuators is studied by Lin and Zhao [2]. Stability analysis showed that one equilibrium point is Hopf point and the other is unstable saddle point when there are two equilibrium points. Nayfeh et al. [3] studied the pull-in instability in MEMS resonators and find that characteristics of the pull-in phenomenon in the presence of AC loads differ from these under purely DC loads. The frequency or the amplitude of the AC loading can be adjusted to reduce the driving voltage and switching time. Chao et al. [4] discussed the prediction of the DC dynamic pull-in voltages of a double clamped microbeam based on a continuous model. Considering lower-order modes and approximating the beam deflection by a different order series, bifurcation based on phase portraits is considered to derive static and dynamic pull-in voltages. Zand et al. [5] considered vibrations of microbeams subjected to suddenly applied step voltages. Different sources of nonlinearity such as electrostatic force and midplane stretching are studied in modeling. Younis et al. [6] investigated clamped–clamped micromachined arches which are made curved intentionally through the induced residual stress from fabrication. They developed a reduced-order model and utilized it to investigate the static and dynamic behavior of the arch for various loading ranges including softening behavior, dynamic snap-through and dynamic pull-in. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_13

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13 Nonlinear Free Vibration of Elastically Actuated Microtubes

Wang et al. [7] presented a size-dependent model for electrostatically actuated microbeam based MEMS using strain gradient elasticity theory. The normalized pullin voltage is shown to increase nonlinearity with the decrease of the beam height, and the size effect becomes prominent when the beam dimension is comparable to the material length scale parameter. Askari and Tahani [8] studied the EulerBernouli beam with mid-plane stretching and applied axial loading to describe the deformations of a nanobeam under the van der Waals force. The governing equations are nonlinear due to inherent nonlinearity in the expression of the van der Waals force and the geometrical nonlinearity due to consideration of mid-plane stretching of the elastic beam. Shen [9] considered large amplitude flexural vibration of microtubes embedded in an elastic matrix of cytoplasm, by modelling the microtube as a nonlocal shear deformable cylindrical shell. Developments are based on shell theory with a von Karaman-Donnell-type kinematic nonlinearity. Zeverdejani and Beni [10] used the Euler–Bernoulli model based on a modified strain gradient theory to investigate free vibration of microtubes embedded in the cytoplasm. They considered the protein microtube as clamped–clamped or simply supported beam, using a Pasternak support foundation for modelling the elastic medium. The electromechanical coupling characteristics of carbon nanotubes reinforced cantilever nanoactuator are investigated by Yang et al. [11] by considering surface effect, nonlocal scale effect, containing the long-range forces among atoms, van der Waals force as molecular interaction and Casimir force as macro effect of quantum field fluctuation. The extremely nonlinear governing equation is derived by utilizing energy methods based on Eringen’s nonlocal elasticity theory and Young–Laplace’s surface effect model. Yang et al. [12] studied some microfluid induced nonlinear free vibration analyzing the nonlinear equation of motion derived by using Hamilton’s principle and a modified couple stress theory by considering the geometric nonlinearity. Askari and Tahani [13] studied size dependent stability analysis studied sizedependent stability of a fully clamped microelectromechanical beam under the effect of shock acceleration pulse. The size-dependent Euler–Bernoulli beam model based on the modified couple stress theory with von Karman-type geometric nonlinearity is utilized in theoretical formulations. The model’s predictions based on the classical theory are compared with those obtained using the finite element method and six modes Galerkin approximations. The micro Coriolis flow meters are extensively used by Ghazani et al. [14] in fluidic micro-circuits. The out-of-plane vibration and stability of curved microtubes are investigated to study the dynamic behavior of curved Coriolis flow meters. Singh et al. [15] discussed the mass sensitivity with respect to linear and nonlinear response of nonuniform cantilever beam with linear and quadratic variation in width. The nonlinear Bernoulli–Euler beam equation with harmonic forcing is used. The mode-shape corresponding to linear undamped free vibration case for different types of beams with a tip mass at the end is proposed. Havelka et al. [16] presented an atomistic approach in calculating microtube vibration modes in a structural analysis. The atomic structure of microtube, including its anisotropy was refined employing molecular dynamics. Bahaadini et al. [17] utilized

13 Nonlinear Free Vibration of Elastically Actuated Microtubes

133

a size-dependent Timoshenko beam model for the study of free vibration and instability analysis of a nanotube conveying nanoflow. To capture the size effects, nonlocal strain gradient theory and Knudsen number are applied.

13.1 Problem Formulations The system under consideration is a pined-pined electrically actuated straight and slender microtube with length L, flexural rigidity EI, under the axial force Fr (Fig. 13.1). Based on the modified couple stress theory of Yang et al. [18], the strain tensor and curvature tensor are included in the strain energy density. The strain energy U in a deformed isotropic linear elastic material occupying the domain D is given by U=

1 2



  σi j εi j + m i j χi j d D

(13.1)

(D)

where σ, ε, m and χ are the Cauchy stress tensor, strain tensor, deviatoric part of the couple stress tensor and symmetric curvature tensor respectively, and are defined as

Fig. 13.1 Schematic view of an electrostatically actuated microtube

134

13 Nonlinear Free Vibration of Elastically Actuated Microtubes

σi j = λεkk δi j + 2μεi j

(13.2)

 1 Ui, j + U j,i 2

(13.3)

εi j =

m i j = ql 2 μχi j χi j =

 1 θi, j + θ j,i 2

(13.4) (13.5)

where λ and μ are Lame’s constants (μ is also known as shear modulus), l is a material length scale parameter, U is the displacement vector, θ is the rotation vector known as θ=

1 curlU 2

(13.6)

and δ ij is the Kronecker symbol: δ ij = 1 for i = j and δ ij = 0 if i = j. In Fig. 13.1, x, y and z are the coordinates along the length, width and thickness respectively, W is deflection of the microtube, t is the time, Fr is axial force, G0 is the initial gap between the non-actuated tube and the stationary electrode and V is DC voltage. If we neglect the effect of fringing field, then the electrostatic excitation by the polarized voltage V is Fes =

ε0 DV 2 2(G 0 − W )2

(13.7)

where ε = 8.854 × 10–12 [C2 N−1 m−2 ] is the permittivity of vacuum within the gap. According to the basic hypothesis of the Euler–Bernoulli beams, the displacement (u, ˜ W˜ ) of an arbitrary point on the microtube can be written as u˜ = u(x, t) − z

∂ W˜ , W˜ = W (x, t) ∂x

(13.8)

where (u, W ) are the axial and transverse displacement of a point on the mid-plane of microtube (i.e. z = 0). Using the von Karman nonlinear strain, the strain components associated with the displacement field presented in Eq. (13.8) are εx =

  ∂2W ∂u 1 ∂W 2 −z 2 + ; ε y = εz = εx y = εx z = ε yz = 0 ∂x ∂x 2 ∂x

Using Eqs. (13.5), (13.6) and (13.9) it results

(13.9)

13.1 Problem Formulations

χx y = −

135

1 ∂2W , χx z = χ yz = χx = χ y = χz = 0 2 ∂x2

(13.10)

The Cauchy stress tensor is 

   ∂2W ∂U 1 ∂W 2 σx = E −z 2 + ∂x ∂x 2 ∂x

(13.11)

The deviatoric part of the couple stress tensor is m x y = −μl 2

∂2W ∂x2

(13.12)

where E is the Young modulus of the microtube. Inserting Eqs. (13.9)–(13.12) into Eq. (13.1), one has 1 U= 2

L

⎡



⎣ E I + μAl

   ∂2W 2 2 ∂x2

0

⎤   2 1 ∂W 2 ⎦ ∂U + dx + EA ∂x 2 ∂x 

(13.13)

where I is the moment of inertia of the cross-sectional area about the y-axis and A is the cross-sectional area of the microtube. The kinetic energy of the microtube is 1 T = ρA 2

L  0

∂U ∂t

2

 +

∂W ∂t

2  dx

(13.14)

The virtual work is given by the electrostatic force and axial force L

L Fes δW d x +

δW = 0

FδU d x

(13.15)

0

The Hamiltonian principle for an elastic body can be written as t2 δ(T − U + W )dt = 0 t1

Substituting Eqs. (13.13)–(13.15) into (13.16) it holds that

(13.16)

136

13 Nonlinear Free Vibration of Elastically Actuated Microtubes

     ∂U 1 ∂W 2 ∂ 2U ∂ + ρA 2 − = Fr EA ∂t ∂x ∂x 2 ∂x      ∂U 1 ∂W 2 ∂W ∂2W ∂ + ρA 2 − EA ∂t ∂x ∂x 2 ∂x ∂x    ∂2W ε0 DV 2 ∂2  = + 2 E I + ρ Al 2 2 ∂x ∂x 2(G 0 − W )2

(13.17)

(13.18)

It is known that in theory of MEMS, the longitudinal oscillation is small in compar2 ison with the transverse vibration, hence the inertia term ρA ∂∂ xu2 in Eq. (13.17) can be neglected. In this case, taking into consideration Eqs. (13.17) and (13.18), we can write ρA

 4  4 ∂2W  ∂2W  2 ∂ W 2 ∂ W + E I + ρ Al ρ A + E I + ρ Al ∂t 2 ∂x4 ∂t 2 ∂x4 ⎤ ⎡ L    ∂W 2 ⎦ ∂2W ⎣ EA ε0 Dv 2 + − dx = F r 2 ∂x 2L ∂x 2(G 0 − W )2

(13.19)

0

The initial and boundary conditions for pinned–pinned microtube are ∂ 2 W (L , t) ∂ 2 W (0, t) = W (0, t) = = W (L , t) = 0 ∂x2 ∂x2

(13.20)

Defining the following dimensionless variables and parameters  x x∗ = ; L

W t y= , t∗ = 2 G0 L L

EI ; ρA

6ε0 L 4 V 2 μAL 2 d Fr L 2 ; ξ= ; h= ; N= 3 EI EI D E G0   π D4  E AL 2 π D2  1 − h2 ; I = 1 − h4 ; K 2 = A= 4 64 EI

α=

(13.21)

and omitting the star, Eq. (13.19) may be written in dimensionless form ⎤ ⎡ 1  2 ∂ y ∂2 y ∂2 y ⎣ α ∂4 y dx⎦ = (1 + ξ) 4 + 2 + 2 N − K 2 ∂x ∂t ∂x ∂x (1 − y)2 0

The corresponding initial and boundary conditions (13.20) are

(13.22)

13.1 Problem Formulations

y(0, t) =

137

∂ 2 y(0, t) ∂ 2 y(1, t) = y(1, t) = =0 ∂x2 ∂x2

(13.23)

We expand the function y(x, t) in a Taylor series around y = 0: 1 = 1 + 2y + 3y 2 + 4y 3 + 5y 4 + 6y 5 + · · · (1 − y)2

(13.24)

such that Eq. (13.22) can be rewritten in the form ⎤ ⎡ 1  2 ∂ y ∂2 y ∂2 y ⎣ ∂4 y dx⎦ (1 + ξ) 4 + 2 + 2 N − K 2 ∂x ∂t ∂x ∂x 0   − α 1 + 2y + 3y 2 + 4y 3 + 5y 4 + 6y 5 = 0

(13.25)

13.2 Free Vibration of the Microtube Free vibration of the pined-pined straight and slender microtube will be investigated in this section starting by using the Galerkin-Bubnov decomposition method to convert the nonlinear partial differential equation of motion (13.25) into a nonlinear ordinary differential equation. We assume the displacement expression in the form y(x, t) = X (x)T (t)

(13.26)

where X(x) represents the eigenfunction of free undamped vibration of the beam, satisfying the boundary conditions (13.23), and T (t) is the generalized coordinate of the discretized system. By considering X (x) = sin πx, after multiplication of Eq. (13.25) with this expression and then integrating the resulting equation on the domain x ∈ [0, 1][0, 1] and taking into account the expressions 1 0

1 0

∂ 4 y(x, t) 1 X (x)d x = π4 T (t), 4 ∂x 2 ∂2 y ∂x2

1  0

∂y ∂x

2

1 0

∂ 2 y(x, t) π2 T (t) X (x)d x = − ∂x2 2

π4 X (x)d x = − T 3 (t), 4

1 0

(13.27) ∂2 y 1 X (x)dt = T¨ 2 ∂t 2

we obtain the nonlinear ordinary differential equation of the form [19] T¨ + AT − BT 2 + DT 3 − E T 4 − F T 5 = 0

(13.28)

138

13 Nonlinear Free Vibration of Elastically Actuated Microtubes

where the dot represents the derivative with respect to time t, and A = π4 (1 + ξ) − π2 N − 2α, D=

k π − 3πα, 4 2

4

E=

B = 8πα,

32α , 3π

F=

15α 4

(13.29)

The corresponding initial conditions for Eq. (13.28) are T (0) = a, T˙ (0) = 0

(13.30)

13.3 Application of OAFM to Elastically Actuated Microtube Making the transformation τ = t into the governing Eq. (13.28), we have T  (τ) +

A B D E F T (τ) − 2 T 2 (τ) + 2 T 3 (τ) − 2 T 4 (τ) − 2 T 5 (τ) = 0 (13.31) 2     

where  is the frequency of the system and prime denotes derivative with respect to τ. The linear operator corresponding to (13.31) may be identified as L[T (τ)] = T  (τ) + T (τ)

(13.32)

in which case the nonlinear operator will be  N [T (τ)] =

 A B D E F − 1 T (τ) − 2 T 2 (τ) + 2 T 3 (τ) − 2 T 4 (τ) − 2 T 5 (τ) 2      (13.33)

Now, according to OAFM, we consider the approximate analytical solution T˜ (τ) of the form T˜ (τ) = T0 (τ) + T1 (τ, C1 , C2 , C3 , C4 )

(13.34)

where the initial approximation T0 (τ) will be obtained from the linear differential Eq. (2.3) and therefore T0 (τ) + T0 (τ) = 0, T0 (0) = a, T0 (0) = 0

(13.35)

13.3 Application of OAFM to Elastically Actuated Microtube

139

whose solution will be T0 (τ) = a cos τ

(13.36)

Substituting this expression in (13.33) we obtain N [T0 (τ)] = Q 0 + Q 1 cos τ + Q 2 cos 2τ + Q 3 cos 3τ + Q 4 cos 4τ + Q 5 cos 5τ (13.37) where Q0 = −

 a  4B + 3a 2 E , 2 8

Q1 =

A 3a 2 D a2 F + − − 1, 2 2  4 162

Q2 =

a (B − a E) 22

(13.38)

Considering the form of Eqs. (13.36) and (13.37), we may define the auxiliary function of the form F1 (τ) = N (τ), F2 (τ) = −2 cos τN (τ), F3 (τ) = −2 cos 2τN (τ) F4 (τ) = −2 cos 3τN (τ), N (τ) = Q 0 + Q 1 cos τ + Q 2 cos 2τ

(13.39)

and in these conditions, the first approximation T1 may be obtained from the equation T 1 (τ) + T1 (τ) = (C1 + 2C2 cos τ + 2C3 cos 2τ + 2C4 cos 3τ)(Q 0 + Q 1 cos τ + Q 2 cos 2τ) T1 (0) = T1 (0) = 0

(13.40)

in which C 1 , C 2 , C 3 and C 4 are the convergence-control parameters unknown at this moment. After simple manipulations the Eq. (13.40) may be rewritten under the form T 1 + T1 = Q 0 C1 + Q 1 C2 + Q 2 C3 + [2Q 0 C2 + Q 1 (C1 + C3 ) + Q 2 (C2 + C4 )] cos τ + [2Q 0 C3 + Q 1 (C2 + C4 ) + Q 2 C1 ] cos 2τ + [2Q 0 C4 + Q 1 C3 + Q 2 C2 ] cos 3τ + (Q 1 C4 + Q 2 C3 ) cos 4τ + Q 2 C4 cos 5τ

(13.41)

where the condition to avoid secular term may be identified as 2 = A +

  3 C2 + C4 a C2 3a 2 D a B + a2 E + (B − a E) − 4 C1 + C3 2 C1 + C3 4

In these conditions, from Eq. (13.41) we obtain the solution

(13.42)

140

13 Nonlinear Free Vibration of Elastically Actuated Microtubes

T1 (τ) = a(Q 0 C1 + Q 1 C2 + Q 2 C3 )(1 − cos τ) +

a [2Q 0 C3 + (C2 3

a [2Q 0 C4 + Q 1 C3 + Q 2 C2 ](cos τ 8 a Q 2 C4 a (cos τ − cos 5τ) − cos 3τ) + [Q 1 C4 + Q 2 C3 ](cos τ − cos 4τ) + 15 24 (13.43) + C4 )Q 1 + Q 2 C1 ](cos τ − cos 2τ) +

At this stage, the first-order approximate solution of the considered problem is obtained taking into account (13.35), (13.36) and (13.43), where τ = t: T (t, C1 , C2 , C3 , C4 ) = a cos t + a(Q 0 C1 + Q 1 C2 + Q 2 C3 )(1 − cos t) a a + [2Q 0 C3 + (C2 + C4 )Q 1 + Q 2 C1 ](cos t − cos 2t) + [2Q 0 C4 3 8 a + Q 1 C3 + Q 2 C2 ](cos t − cos 3t) + [Q 1 C4 15 a Q 2 C4 (cos t − cos 5t) + Q 2 C3 ](cos t − cos 4t) + (13.44) 24 in which  is given by Eq. (13.42). Now, in order to obtain the final explicit analytical solution, the optimal values of the convergence-control parameters should be determined by minimizing the residual of the initial differential equation.

13.4 Numerical Examples In what follows we will test the validity of the proposed procedure considering several particular cases [19]. Case 13.4a. As a first numerical example we consider the following parameters: a = 0.03, ξ = 0.5, N = 1.5, α = 0.2, k = 2. In this case, following the proposed procedure, we obtain the optimal values for the convergence-control parameters C1 = 0.6213539559, C2 = 1.2316488142 C3 = −2.3695907339, C4 = 2.7499506532

(13.45)

while the obtained frequency is ap = 11.4416. In this case, the explicit approximate analytical solution will be T˜ (t) = − 0.0000333319(1 − cos[t]) + 0.03 cos[t] + 0.0000504558(cos[t]

13.4 Numerical Examples

141

− cos[2t]) − 0.0000136034(cos[t] − cos[3t])− − 1.20031663 × 10−8 (cos[t] − cos[4t]) + 1.97179608 × 10−6 (cos[t] − cos[5t])

(13.46)

which, for comparison purposes, is plotted in Fig. 13.2 along with the numerical integration solution. Moreover, also for the purposes to emphasize the accuracy of the analytical solution, Fig. 13.3 presents a comparison between the phase portraits obtained for both analytical and numerical solutions. Case 13.4b. In the second case we consider a = 0.05, ξ = 0.11, N = 0.5, α = 0.1, k = 5, in which case the optimal values of the convergence-control parameters are C1 = 2.8988302952, C2 = −4.0290046797, C3 = 1.460489958, C4 = 0.3112246531 Fig. 13.2 Comparison between the analytical approximate solution (13.46) and numerical integration results in case 13.4a: numerical solution; analytical solution

Fig. 13.3 Phase portraits in case 13.4a: numerical integration results; analytical results (13.46)

(13.47)

142

13 Nonlinear Free Vibration of Elastically Actuated Microtubes

Fig. 13.4 Comparison between the analytical approximate solution (13.48) and numerical integration results in case 13.4b: numerical solution; analytical solution

Fig. 13.5 Phase portraits in case 13.4b: numerical integration results; analytical results (13.48)

and the corresponding frequency ap = 10.2075. Therefore, the approximate analytical solution becomes in this case (Figs. 13.4 and 13.5). T˜ (t) = 0.00007921(1 − cos[t]) + 0.05 cos[t] + 0.00003737(cos[t] − cos[2t]) − 0.00002299806(cos[t] − cos[3t]) + 2.283024 × 10−6 (cos[t] − cos[4t]) + 3.883582 × 10−7 (cos[t] − cos[5t])

(13.48)

Case 13.4c. The third case is developed for the following values of the physical parameters: a = 0.07, ξ = 0.11, N = 0.5, α = 0.1, k = 10. Using the same procedure, the optimal values of the convergence-control parameters are in this case C1 = 2.8988302952, C2 = −4.0290046797 C3 = 1.460489958, C4 = 0.3112246531

(13.49)

13.4 Numerical Examples

143

and ap = 0.5836. Consequently, the approximate analytical solution will be T˜ (t) = 0.000485164(1 − cos[t]) + 0.07 cos[t] + 0.000258764(cos[t] − cos[2t]) − 0.000214699(cos[t] − cos[3t]) + 0.0000232923(cos[t] − cos[4t]) + 5.36416 × 10−6 (cos[t] − cos[5t])

(13.50)

The corresponding graphical results are presented in Figs. 13.6 and 13.7. Case 13.4d. As a last case, the considered values for the parameters are: a = 0.09, ξ = 0.5, N = 1.5, α = 0.2, k = 2. In this case we obtain the optimal values: C1 = −0.0876954594, C2 = −0.1514756484, C3 = −0.043917181007, C4 = −0.0116343324

(13.51)

which lead to the frequency ap = 11.456. The OAFM analytical solution is Fig. 13.6 Comparison between the analytical approximate solution (13.50) and numerical integration results in case 13.4c: numerical solution; analytical solution

Fig. 13.7 Phase portraits in case 13.4c: numerical integration results; analytical results (13.50)

144

13 Nonlinear Free Vibration of Elastically Actuated Microtubes

Fig. 13.8 Comparison between the analytical approximate solution (13.52) and numerical integration results in case 13.4d: numerical solution; analytical solution

Fig. 13.9 Phase portraits in case 13.4d: numerical integration results; analytical results (13.52)

T˜ (t) = − 0.0000188884(1 − cos[t]) + 0.09 cos[t] + 9.18482 × 10−6 (cos[t] − cos[2t]) − 3.38383 × 10−6 (cos[t] − cos[3t]) + 5.80597 × 10−7 (cos[t] − cos[4t]) + 7.42801 × 10−8 (cos[t] − cos[5t])

(13.52)

For this last case, graphical results are presented in Figs. 13.8 and 13.9. In Figs. 13.2, 13.3, 13.4, 13.5, 13.6, 13.7, 13.8, 13.9, in all considered cases, as one can be seen, a very good accuracy is achieved comparing OAFM results with numerical integration results.

References 1. M.I. Younis, A.H. Nayfeh, A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dyn. 31, 91–11 (2003) 2. W.H. Lin, Y.P. Zhao, Nonlinear behavior for nanoscale electrostatic actuators with Casimir force. Chaos Solitons Fractals 23, 1777–1785 (2005)

References

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3. A.H. Nayfeh, M.I. Younis, E.M.A. Rahman, Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dyn 48, 153–163 (2007) 4. P. Chao, C.W. Chiu, T.H. Liu, DC dynamic pull-in predicition for a generalized clampedclamped micro-beams based on a continuation model and bifurcation analysis. J. Micromech. Microeng. 18, 115008 (2008) 5. M.M. Zand, M.T. Ahmadian, R. Rashidian, Semi-analytic solutions to nonlinear vibrations of microbeams under suddenly applied voltages. J. Sound Vibr. 325, 389–439 (2009) 6. M.I. Younis, F.M. Alsaleem, R. Miles, W. Cui, Nonlinear dynamics of MEMS arches under harmonic electrostatic actuation. J Microelectromech. Syst. 19, 647–656 (2010) 7. B. Wang, S. Zhou, J. Zhao, X. Chen, Size-dependent pull-in instability of electrostatically actuated microbeam-based MEMS. J. Micromech. Microeng. 21, 027001 (2011) 8. A.R. Askari, M. Tahani, Investigating nonlinear vibration of a fully clamped nanobeam in the presence of the van der Waals attraction. Appl. Mech. Mater. 226, 181–185 (2012) 9. H.S. Shen, Nonlinear vibration of microtubes in living cells. Curr. Appl. Phys. 11, 812–882 (2011) 10. M.K. Zeverdejani, Y.T. Beni, The nano scale vibration of protein microtubes based on modified strain gradient theory. Curr. Appl. Phys. 13, 1566–157 (2013) 11. W.D. Yang, X. Wang, C.Q. Fang, G. Lu, Electromechanical coupling characteristics of carbon nanotube reinforced cantilever nano-actuator. Sensor Actuators A: Phys. 220, 178–187 (2014) 12. T.Z. Yang, S. Ji, X.D. Yang, B. Fang, Microfluid-induced nonlinear free vibration of microtubes. Int. J. Eng. Sci. 76, 47–55 (2014) 13. A.R. Askari, M. Tahani, Size-dependent dynamic pull-in analysis of beam-type MEMS under mechanical shock based on the modified couple stress theory. Appl. Math. Model. 39, 934–946 (2015) 14. M.R. Ghazani, H. Molki, A.A. Beigloo, Nonlinear vibration and stability analysis of the curved microtube conveying fluid as a model of the micro Coriolis flow meters based on strain gradient theory. Appl. Math. Model. 45, 1020–1030 (2017) 15. S.S. Singh, P. Pal, A.K. Pandey, Mass sensitivity of non-uniform microcantilever beams. J. Vibr. Acoust. 138, 064502 (2016) 16. D. Havelka, M. Deriu, M. Cifra, D. Kucera, Deformation pattern in vibrating microtube: structurate mechanics study based on atomistic approach. Sci. Rep. 7, 4227 (2017) 17. R. Bahaadini, A.R. Saidi, M. Hosseini, On dynamics of nanotubes conveying nanoflow. Int. J. Eng. Sci. 123, 181–196 (2018) 18. F. Yang, Chong, P. Tong, Couple stress theory based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002) 19. N. Herisanu, V. Marinca, An effective analytical approach to nonlinear free vibration of elastically actuated microtubes. Meccanica, 56, 813–823 (2021)

Chapter 14

Analytical Investigation to Duffing Harmonic Oscillator

In the papers [1–7] are presented some conservative nonlinear oscillatory systems which can often be modelled by potentials having a rational form. Such models also lead to differential equations for which the usual expansion in a small parameter perturbation procedure does not apply [2]. Mickens [1, 2] applied two non-standard finite difference scheme to numerically integrate the equation of Duffing harmonic oscillator. Tiwari at all [3] presented and approximate frequency-amplitude relation close to the exact one assuming a single term solution and following the Ritz procedure. Also they applied a rational harmonic balance approximation for Duffing harmonic oscillator. Hu and Tang [4] used the first-order harmonic balance via First Fourier coefficient to construct an approximate frequency-amplitude relation. Herisanu and Marinca [6] proposed and optimization procedure, namely the optimal variational iteration method for analytically solving this type of equations. This procedure provides a good approximation for both frequency and periodic solution. Ozis and Yildirim [5] applied the energy balance method obtaining the frequency in the form ω2 = 1 −

2 2(1 + A2 ) ln A2 2 + A2

(14.1)

where A is the initial position of the Duffing harmonic oscillator. An example of the system which has a rational form is the Duffing harmonic oscillator presented in [2] d2 y ay 3 + =0 dt 2 b + cy 2

(14.2)

where y is the displacement, t is the time, a, b and c are nonnegative parameter. Defining

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_14

147

148

14 Analytical Investigation to Duffing Harmonic Oscillator

x=y



b , a

t =t



c a

(14.3)

and dropping the bar on t, gives the following non-dimensional equation x¨ +

x3 =0 1 + x2

(14.4)

where dot denotes the derivative with respect to the time t. For small or large x, respectively, the Eq. (14.4) can be written as x¨ + x 3 ≈ 0 for smal x

(14.5)

x¨ + x ≈ 0 for large x

(14.6)

We remark that for small x, the equation of motion (14.5) is that of a Duffing type nonlinear oscillator, while for large x the equation of motion approximates that of a linear harmonic oscillator. Hence, Eq. (14.4) is referred to as the Duffing harmonic nature [2]. The restoring force in Eq. (14.4) is the same for both negative and positive amplitudes. The initial condition for Eq. (14.4) are x(0) = A, x(0) ˙ =0

(14.7)

14.1 OAFM for Duffing Harmonic Oscillator Using a new independent variable and a new function τ = t, x(t) = Au(τ)

(14.8)

the Eq. (14.4) can be written in the following form 2 u  + A2 2 u 2 u  + A2 u 3 = 0

(14.9)

with the corresponding initial conditions u(0) = 1, u  (0) = 0

(14.10)

where prime denotes the derivative with respect to the new variable τ. The linear and nonlinear operators for Eq. (14.9) are respectively (g(τ) = 0)

14.1 OAFM for Duffing Harmonic Oscillator

149

L[u(τ)] = 2 (u  + u)

(14.11)

N [u(τ)] = 2 (A2 u 2 u  − u) + A2 u 3

(14.12)

The approximate analytical solution is of the form u(τ) ˜ = u 0 (τ) + u 1 (τ)

(14.13)

The initial approximation is obtained from the equation L[u 0 (τ)] = 0, u 0 (0) = 1, u  (0) = 0

(14.14)

leading to the solution u 0 (τ) = cos τ

(14.15)

Introducing Eq. (14.15) into Eq. (14.12) it follows that N [u 0 (τ)] = α cos τ + β cos 3τ

(14.16)

where α=

1 3 2 A (1 − 2 ) − 2 , β = A2 (1 − 2 ) 4 4

(14.17)

The auxiliary functions Fi (τ) are F1 (τ) = −N (τ),

F2 (τ) = −2(cos 2τ)N (τ),

F3 (τ) = −2(cos 4τ)N (τ) (14.18)

where N (τ) = α cos τ + β cos 3τ + C4 cos 5τ

(14.19)

It follows that the first approximation can be determined from the equation L[u 1 (τ.Ci )] = (C1 + 2C2 cos 2τ + 2C3 cos 4τ)(α cos τ + β cos 3τ + C4 cos 5τ) or 2 (u 1 + u 1 ) = [α(C1 + C2 ) + β(C2 + C3 ) + C3 C4 ] cos τ

(14.20)

150

14 Analytical Investigation to Duffing Harmonic Oscillator

+ (αC2 + βC1 + C2 C4 ) cos 3τ + (αC3 + βC2 + C1 C4 ) cos 5τ + (βC3 + C2 C4 ) + C3 C4 cos 4τ

(14.21)

Avoiding the secular term in Eq. (14.21) we can determine the frequency 2 =

(3C1 + 4C2 + C3 )A2 + 4C3 C4 A2 (3C1 + 4C2 + C3 ) + 4C1 + 4C2

(14.22)

From Eqs. (14.21), (14.15), (14.8) and (14.13) we obtain the first-order approximate analytical periodic solution for Duffing harmonic oscillator (14.4)  3A2 3A2 2 A2 − − 2 C2 + (1 − 2 )C1 4 4 4  2  3A 3A2 2 A −  C2 C4 ](cos t − cos 3t) + − 1 C3 242 4 4  A2 2 (1 −  )C2 + C1 C4 (cos t − cos 5t) 4  2  A A 2 (1 −  )C3 + C2 C4 (cos t − cos 7t) 482 4 C3 C4 (cos t − cos 9t) (14.23) 802

x(t) = A cos t + + + + +

A 42



14.2 Numerical Examples The accuracy of our procedure is proved considering two different cases for the initial amplitudes A: Case 14.2a For the initial amplitude A = 3, the obtained approximate frequency of the system is  = 0.9231 and the first-order approximate solution (14.23) in this case is plotted in Fig. 14.1. Case 14.2b In the second case, for A = 5, the obtained approximate frequency of the system is  = 0.9708 and the first-order approximate solution (14.23) in this case is plotted in Fig. 14.2. Figures 14.1 and 14.2 show the approximate solutions (14.23) and the corresponding numerical solutions in two considered cases, for different initial amplitudes of the considered Duffing harmonic oscillator.

14.2 Numerical Examples

151

Fig. 14.1 The approximate solution in the first case, for the initial amplitude A = 3: _______ numerical integration results; _ _ _ _ analytical results

Fig. 14.2 The approximate solution in the first case, for the initial amplitude A = 5: _______ numerical integration results; _ _ _ _ analytical results

References 1. R.E. Mickens, Semi-classical quantization using the method of harmonic balance. Il Nuova Cimento 101, 359–366 (1989) 2. R.E. Mickens, Mathematical and numerical study of the Duffing-harmonic oscillator. J. Sound Vibr. 244, 563–567 (2001) 3. S.B. Tiwari, S. Nageswara Rao, N.S. Swang, K.S. Sai, H.R. Nataraje, Analytical study on a Duffing-harmonic oscillator. J. Sound Vibr. 295, 1217–1222 (2005) 4. H. Hu, J.H. Tang, Solution of a Duffing-harmonic oscillator by the method of harmonic balance. J. Sound Vibr. 294, 637–639 (2006) 5. T. Ozis, A. Yildirim, Determination of the frequency-amplitude relation for a Duffing-harmonic oscillator by the energy balance method. Comp. Math. Appl. 54, 1184–1187 (2007) 6. N. Herisanu, V. Marinca, A modified variational iteration method for strongly nonlinear problems. Nonlinear Sci. Lett. A 1, 183–192 (2010) 7. N. Herisanu, V. Marinca, Analytical investigation to Duffing harmonic oscillator, in The 26th International Conference Noise and Vibration, Nis 177–179 (2018)

Chapter 15

Free Vibration of Tapered Beams

Tapered beams can model engineering structures which require a variable stiffness along the length, such as moving arms and turbine blades [1–3], or can be modeled as a slender, flexible cantilever beam carrying a lumped mass with rotary inertia at an intermediate point along its span hence it exhibits large-amplitude vibrations [4, 5]. Linearization techniques are employed in order to approximate such nonlinear problems. One may question the accuracy of using a linear mode method, which are a frequently used method in the analyses of nonlinear continuous systems to approximate the large amplitude nonlinear behavior [6]. In dimensionless form, the governing differential equation corresponding to fundamental mode of tapered beams is given by   2  d 2u 2 du d 2u +u+a u + + bu 3 = 0 dt 2 dt 2 dt

(15.1)

where a and b are modal constants which result from the discretization procedure [6, 7], describe the large-amplitude free vibration of the considered slender inextensible cantilever beam, which is assumed undergoing planar flexural vibrations. The terms from the brackets in Eq. (15.1) represent inertia-type cubic nonlinearity arising from the in extensibility assumption. The last term is a static-type cubic nonlinearity associated with the potential energy stored in bending. For Eq. (15.1), the initial conditions are u(0) = A,

du (0) = 0 dt

(15.2)

Under the transformations τ = t, u(τ) = Ax(t)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_15

(15.3)

153

154

15 Free Vibration of Tapered Beams

where  is the natural frequency, Eqs. (15.1) and (15.2) can be rewritten in the form 2 x  + x + a A2 2 x 2 x  + a A2 2 x x 2 + λA2 x 3 = 0

(15.4)

x(0) = 1, x  (0) = 0

(15.5)

where the prime denotes differentiation with respect to τ.

15.1 OAFM for Free Vibration of Tapered Beams The Eqs. (15.4) and (15.5) can be written in the following form x b + a A2 (x 2 x  + x x 2 ) + 2 A2 x 3 = 0 2   x(0) = 1, x  (0) = 0

x  +

(15.6)

The linear and nonlinear operators are respectively L[x(τ)] = x  + x N [x(τ)] =

(15.7)

  x b − x + a A2 x 2 x  + x x 2 + 2 A2 x 3 2  

(15.8)

From the equation x0 + x0 = 0, x0 (0) = 1, x0 (0) = 0

(15.9)

x0 (τ) = cos τ

(15.10)

it follows that

The nonlinear operator for the solution (15.10) becomes  N [x0 (τ)] =

1 1 2 3b A2 aA + − 1 − 2 2 42



 cos τ +

b A2 a A2 − 42 2

 cos 3τ (15.11)

The auxiliary functions are F1 (τ) = −N (τ),

F2 (τ) = −2 cos 2τN (τ),

F3 (τ) = −2 cos 4τN (τ)

(15.12)

15.1 OAFM for Free Vibration of Tapered Beams

155

where N (τ) = α cos τ + β cos 3τ + C4 cos 5τ

(15.13)

The first approximation is obtained from the following equation: x1 + x1 = (C1 + 2C2 cos 2τ + 2C3 cos 4τ)(α cos τ + β cos 3τ + C4 cos 5τ) x1 (0) = x1 (0) = 0

(15.14)

where α and β are obtained from Eq. (15.11): α=

1 1 2 3b A2 b A2 a A2 a A − 1 − + ; β = − 2 2 42 42 2

(15.15)

and Ci , i = 1, 2, 3, 4 are unknown parameters at this moment. After simple manipulations, Eq. (15.14) can be rewritten in the form x1 + x1 = [α(C1 + C2 ) + β(C2 + C3 ) + C3 C4 ] cos τ + (αC2 + βC1 + C2 C4 ) cos 3τ + (αC3 + βC2 + C1 C4 ) cos 5τ + (βC3 + C2 C4 ) cos 7τ + C3 C4 cos 9τ (15.16) x1 (0) = x1 (0) = 0 Avoiding the secular term into Eq. (15.16), we obtain α(C1 + C2 ) + β(C2 + C3 ) + C3 C4 = 0

(15.17)

From Eqs. (15.15) and (15.17) we can determine the frequency 2 =

(4 + 3b A2 )(C1 + C2 ) + b A2 (C2 + C3 ) 2a A2 (C1 + 2C2 + C3 ) + 4C1 + 4C2 − 4C3 C4

(15.18)

Taking into account the solution of Eq. (15.16), and Eqs. (15.10), (15.3) and (2.32.3) one can obtain the first-order approximate periodic solution of Eqs. (15.1) and (15.2):    2 1 1 2 3b A2 bA a A2 a A C1 C − 1 − + + − 2 2 2 42 42 2    2 1 A 1 2 3b A2 bA C3 + + C2 C4 ](cos t − cos 3t) + − 1 − aA + 24 2 2 42 42  a A2 C3 C4 − C1 + C2 C4 ](cos t − cos 7t) + (cos t − cos 9t) 2 80 (15.19)

u(t) ˜ = A cos t +

A 8



156

15 Free Vibration of Tapered Beams

15.2 Numerical Examples In what follows we consider two cases in order to prove the accuracy of OAFM. Case 15.2a For a = 2, b = 1, A = 0.5, the obtained approximate frequency of the system is  = 0.7502 and the first-order approximate solution (15.19) in this case is plotted in Fig. 15.1. Case 15.2b In the case a = 1, b = 3, A = 0.5, the obtained approximate frequency of the system is  = 1.4591 and the first-order approximate solution (15.19) in this case is plotted in Fig. 15.2. Figures 15.1 and 15.2 show the approximate solutions (15.19) and the corresponding numerical solutions in two considered cases, for different physical parameters and initial amplitudes. Fig. 15.1 The approximate solution in the first case, for a = 2, b = 1, A = 0.5: _______ numerical integration results; _ _ _ _ analytical results

Fig. 15.2 The approximate solution in the second case, for a = 1, b = 3, A = 1: _______ numerical integration results; _ _ _ _ analytical results

References

157

References 1. D.J. Goorman, Free Vibrations of Beams and Shafts (Wiley, New York, 1975) 2. S.P. Timoshenko, J.N. Goodier, Theory of Elasticity (McGraw Hill, Tokyo, 1993) 3. S.H. Hoseini, T. Pirbodaghi, M.T. Ahmadian, G.H. Farahi, On large amplitude free vibrations of tapered beams: an analytical approach. Mech. Res. Commun. 361, 892–897 (2009) 4. N. Herisanu, V. Marinca, Explicit analytical approximations to large-amplitude nonlinear oscillations of an uniform cantilever beam carrying an intermediate lumped mass and rotary inertia. Meccanica 45, 847–855 (2010) 5. M.N. Hamdan, N.N. Shabaneh, On the large amplitude free vibration of a restrained uniform beam carrying an intermediate lumped mass. J. Sound Vibr. 199, 711–726 (1997) 6. M.N. Hamdan, M.A.F. Dado, Large amplitude free vibration of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia. J. Sound Vibr. 206, 151–168 (1997) 7. V. Marinca, N. Herisanu, Free vibration of tapered beams, in The 26th International Conference Noise and Vibration, Nis 181–183 (2018)

Chapter 16

Dynamic Analysis of a Rotating Electrical Machine Rotor-Bearing System

Rotating electrical machines are widely used in engineering and industry applications due to their reliability. Having in view that there is a need to continuously increase the performance of this kind of machines, they are intensively studied from both mechanical and electrical point of view, since these complex engineering systems combine electrical and mechanical concepts. Reliable analytical and numerical tools should be developed and implemented in order to predict and analyze possible problems related to dynamic behavior of these systems, which could be described by nonlinear and strongly nonlinear differential equations. The most often encountered problems could be generated by unbalanced forces of the rotor, electrical unbalances, coupling or driven equipment, shaft misalignment and nonlinearity of the bearing stiffness, bad bearings and mechanical looseness, resonance, critical speeds, etc. Specific for this type of systems is the interaction of mechanical and electrical phenomena, which should be understood and taken under control in order to avoid problems occurrence and making the machine to run smoothly and reliably to higher speeds and loads. Many scientists are concerned in investigating the nonlinear behavior of rotating electrical machines using various techniques. Raja et al. [1] exploited bio-inspired computational intelligence to analyze the nonlinear vibrational dynamics of rotating electrical machine model by applying artificial neural networks, genetic algorithms and active-set methods. In [2] it is introduced an optimal variational method to investigate the nonlinear behavior of a rotating electrical machine modeled as an oscillator with cubic elastic restoring force and time variable coefficients. An analytical approach for expeditiously understanding and solving specific problems encountered by rotating electrical machines is presented in [3]. Martinez et al. [4] developed a Finite Element Model in order to study the vibrations in induction motors under steady-state introducing a model which utilizes a weak coupling strategy between both magnetic and elastodynamic fields on the structure. Xu et al. [5] investigated the vibration characteristics of an inclined rotor with both static and displacement eccentricity and the static angle eccentricity in the three-dimensional space. Kirschneck et al. [6] introduced an approach for a multiphysical modal analysis that makes it © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_16

159

160

16 Dynamic Analysis of a Rotating Electrical Machine …

possible to predict the dynamics of the strongly coupled magnetomechanical system proposing a method applied to a single-bearing design direct-drive wind turbine generator rotor to calculate the changes of the structural dynamics caused by the electromagnetomechanical coupling. The machines understudy in this section are subjected to a parametric excitation caused by an initial trust and a forcing excitation caused by an unbalanced force of the rotor while the entire system is being supported by nonlinear bearings with nonlinear stiffness characteristics and damping properties. The nonlinear suspension makes the analytical study very difficult, leading to strong nonlinear differential equations, which are hard to be solved through classical methods. Supplementary problems could arise in case of some horizontal rotating machines, when the gravity effect is not negligible for certain stiffness conditions. For the gravity deflection the shaft centre leaves the bearing centerline, which leads to vibration occurrence. Also, the misalignment could occur in the electrical machine after some amount of running. The dynamical behavior of the investigated electrical machine will be governed by the following second-order strongly nonlinear differential equation with variable coefficients [7] m¨x + k1 (1 − q sin ω2 t)x + k2 x3 = f sin ω1 t x(0) = A, x˙ (0) = 0

(16.1)

This equation may be written in the more convenient way x¨ + ω2 x − a sin ω2 tx + bx3 − c sin ω1 t = 0 x(0) = A, x˙ (0) = 0

(16.2)

where ω2 = km1 , a = km1 q , b = km2 , c = mf and the dot denotes derivative with respect to time and A is the amplitude of the oscillations. It should be emphasized that it is unnecessary to suppose the existence of any small or large parameters in Eq. (16.2).

16.1 Application of OAFM to the Investigation of Nonlinear Vibration of the Considered Electrical Machine The validity of our procedure is illustrated on the electrical machine whose dynamic behavior is governed by Eq. (16.2). The Eq. (16.2) describes a system of oscillating = t. Under the with an unknown period T. We switch to a scalar time τ = 2πt T transformations τ = t, x(t) = Ay(τ) the original Eq. (16.2) becomes

(16.3)

16.1 Application of OAFM to the Investigation of Nonlinear Vibration …

2 y + ω2 y − ay sin

161

ω2 c ω1 τ + bA2 y3 − sin τ = 0, y(0) = 1, y (0) = 0  A  (16.4)

where the prime denotes derivative with respect to τ. We define the linear and nonlinear operators, respectively (g(τ) = 0):   L[y(τ )] = 2 y + y   ω1 ω2 c τ + bA2 y3 − sin τ N[y(τ)] = ω2 − 2 y − ay sin  A 

(16.5) (16.6)

For Eq. (16.4) we demand an approximate analytical solution y˜ (τ) in two components y˜ (τ) = y0 (τ) + y1 (τ, Ci ), i = 1, 2, . . . , n

(16.7)

where C i are unknown parameters at this moment. The initial approximation y0 (τ) is determined from the linear equation   L y0 (τ) = 0, y0 (0) = 1, y0 (0) = 0

(16.8)

The solution of Eq. (16.8) is y0 (τ) = cos τ

(16.9)

The nonlinear operator (16.6) for the initial approximation (16.9) becomes     1 3 2 2 2 N y0 (τ) = ω −  + bA cos τ + bA2 cos 3τ 4 4

ω

 c 1  ω2 ω2 2 − a sin + 1 τ + sin − 1 τ − sin τ 2   A 

(16.10)

The first approximation y1 (τ, Ci ) can be determined from the equation   L y1 (τ, Ci ) + C1 F1 (τ) + C2 F2 (τ) + · · · + C6 F6 (τ) = 0, y1 (0) = 0, y1 (0) = 0

(16.11)

where, taking into consideration (16.10), we have the freedom to choose the functions F1 (τ) = −N (τ), F2 (τ) = −2(cos 2τ)N (τ),   b A2 3 N (τ) = ω2 − 2 + b A2 cos τ + cos 3τ 4 4

162

16 Dynamic Analysis of a Rotating Electrical Machine …

  b A2 3 2 2 2 cos 3τ, F4 (τ) = − cos 5τ F3 (τ) = ω −  + b A cos τ + 4 4 ω ω



ω1 2 2 F4 (τ) = − sin + 1 τ, F5 (τ) = − sin − 1 τ, F6 (τ) = sin τ (16.12)    However, the choices (16.12) are not unique. We can choose another alternative F1 (τ) = −N (τ),

F2 (τ) = −2(cos 2τ)N (τ),

ω 2 F3 (τ) = −2(cos 4τ)N (τ), F4 (τ) = − sin +1 τ    ω1 3 2 2 2 τ, N (τ) = ω −  + b A cos τ F5 (τ) = sin  4

(16.13)

or F1 (τ) = −1,

F2 (τ) = −N (τ),

F3 (τ) = −2(cos 2τ)N (τ)

ω ω1 2 + 1 τ, F6 (τ) = sin τ F4 (τ) = −2 cos 4τN (τ), F5 (τ) = sin     ω

1 3 2 +1 τ (16.14) N (τ) = ω2 − 2 + b A2 cos τ − a sin 4 2  and so on. Having a view Eqs. (16.12) and (16.5), Eq. (16.11) becomes 

2



y1

+ y1



 

1 2 3 2 2 2 = (C1 + C2 ) ω −  + bA + bA C2 cos τ 4 4 

 1 3 2 2 2 2 + C2 ω −  + bA + bA C1 cos 3τ 4 4  

ω 1 2 +1 τ + C3 + b A2 C2 cos 3τ + C4 sin 4 

ω ω1 2 + C5 sin − 1 τ + C6 sin τ,   y1 (0) = y1 (0) = 0 (16.15)

Avoiding the presence of secular terms in Eq. (16.15) we have   1 3 (C1 + C2 ) ω2 − 2 + bA2 + bA2 = 0 4 4

(16.16)

From Eq. (16.16) we obtain the frequency of the system in the form C2 3 1 2 = ω2 + bA2 + bA2 4 4 C1 + C2

(16.17)

16.1 Application of OAFM to the Investigation of Nonlinear Vibration …

163

The solution of Eq. (16.15) is  

1 2 1 1 3 2 2 2 C2 ω −  + bA + bA C (cos τ − cos 3τ) y1 (τ) = 82 4 4   1 1 C3 + bA2 C2 (cos τ − cos 5τ) + 242 4

ω

 2 C4  ω2 2 + 2 + 1 sin τ − sin +1 τ  ω2 + 2ω2  



  2  C5 ω2 ω2 + 2 11 sin τ − sin −1 τ  ω2 − 2ω2  

2  C6 ω1 ω1 + 2 τ− sin τ (16.18) sin   2 − ω21 Substituting Eqs. (16.9), (16.18) and (16.3) into Eq. (16.9) we obtain the solution of Eq. (16.1) in the first order of approximation using OAFM in the form:  

A 1 2 3 2 2 2 x(t) = A cos t + C ω + bA bA −  + C 2 1 (cos t − cos 3t) 82 4 4   A 1 2 C bA + + C 3 2 (cos t − cos 5t) 242 4

 2 AC4  ω2 + 1 sin t − sin(ω2 + )t + 2 ω2 − 2ω2  

 2 AC5  ω2 − 1 sin t − sin(ω2 − )t + 2 ω2 − 2ω2  

2 AC6 ω1 sin t (16.19) + sin ω t − 1   − ω21

16.2 Numerical Example In order to develop a numerical application, we consider a particular case characterized by the following parameters: ω1 = 1.21, ω2 = 1.51, ω = 1.63796, b = 2.1951, c = 0.195122, A = 1

a = 2.95122, (16.20)

Following the described procedure, the optimal values of the convergence-control parameters are obtained by minimizing the residual generated by the approximate solution (16.19) after substitution in the governing equation:

164

16 Dynamic Analysis of a Rotating Electrical Machine …

Fig. 16.1 Comparison between the approximate solution (16.22) and corresponding numerical results: numerical integration results; analytical results

C1 = −0.12949743551566287, C2 = −8.6126665038499, C3 = 25.324695766371054, C4 = 0.09686844412991548 C5 = 0.07534250937939393,

C6 = 0.01

(16.21)

Finally the approximate analytical solution of the first-order of approximation (16.19) becomes x(t) = cos[3.38829t] − 0.0370021(cos[3.38829t] − cos[10.1649t]) − 0.00338774(cos[3.38829t] − cos[16.9415t]) − 0.108767(sin[1.87829t] − 0.554348 sin[3.38829t]) + 0.0114617(sin[1.21t] − 0.35711 sin[3.3882t]) + 0.0888774(1.44565 sin[3.38829t] − sin[4.89829t])

(16.22)

Figure 16.1 shows the comparison between our approximate solution (16.22) and numerical integration results obtained by means of a fourth-order Runge–Kutta method. It can be seen that the solution obtained using our technique is nearly identical with that obtained through numerical integration method.

References 1. M.A.Z. Raja, S.A. Niazi, S.A. Butt, An intelligent computing technique to analyze the vibrational dynamics of rotating electrical machine. Neurocomputing 219, 280–299 (2017) 2. N. Herisanu, V. Marinca, An optimal approach to study the nonlinear behavior of a rotating electrical machine. J. Appl. Math. 465023 (2012) 3. W.R. Finley, M.M. Hodowanec, W.G. Holter, An analytical approach to solving motor vibration problems. IEEE Trans. Ind. Appl. 36, 1467–1480 (2000) 4. J. Martinez, A. Belahcen, J.G. Detoni, A 2D magnetic and 3D mechanical coupled finite element model for the study of the dynamic vibrations in the stator of induction motors. Mech. Syst. Signal Process. 66–67, 640–656 (2016) 5. X.P. Xu, Q.K. Han, F.L. Chu, A four degrees-of-freedom model for a misalignment electrical rotor. J. Sound Vib. 358, 356–374 (2015)

References

165

6. M. Kirschneck, D.J. Rixen, H. Polinder, R.A.J. van Ostayen, Electromagnetomechanical coupled vibration analysis of a direct-drive off-shore wind turbine generator. J. Comput. Nonlinear Dyn. 10, 041011 (2015) 7. N. Herisanu, V. Marinca, Analysis of nonlinear dynamic behavior of a rotating electrical machine rotor-bearing system using Optimal Auxiliary Functions Method, Polonia. Springer Proc. Math. Stat. 249, 159–168 (2018)

Chapter 17

Investigation of a Permanent Magnet Synchronous Generator

The permanent magnet synchronous generators (PMSG) are rotating electrical machines having a classic three-phase stator like that of an induction motor, and the rotor has surface-mounted permanent magnets. They are widely used to convert the mechanical wind energy into electrical energy, which is a hot topic nowadays [1–6]. That is why various aspects of design and functioning of permanent magnet synchronous machines received an increased interest from scientists. Small signal stability of permanent magnet synchronous generator (PMSG)-based wind turbines connected to the power grid is properly studied in [7] in order to facilitate damping strategy design. Song et al. [8] applied the Taguchi method to optimal design of permanent magnet synchronous motors to optimize the thrust and thrust ripple, and using finite-element analysis, the relative importance of each design parameter was estimated in detail. Based on a linearized model, the relation between the PMSG electromagnetic torque and boost converter current is extracted, and then system’s control-loops are developed by Rahimi in [9]. Dynamic control for PMSG is investigated in [10]. In this Chapter, the behavior of the PMSG is predicted using the classical dq equivalent circuit models. Starting from the equations of the smooth-air-gap synchronous machine, the D-Q axis equations of PMSG in the rotor reference frame, lead to a system of three nonlinear differential equations with unknowns instantaneous values of stator current components and electrical angular speed. Analytical approximate solutions which are obtained are of considerable importance for practical analysis of electrical power system dynamic behavior with problems caused by possible perturbations generated by some short circuits, sudden change of loads, disconnection of load and other switching transients in power station or stability problem of such systems. The proposed approach has been applied on a low-power generator and this work should be continued with the case of high-power electrical generators, connected directly into a large electrical power system.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_17

167

168

17 Investigation of a Permanent Magnet Synchronous Generator

17.1 Governing Equations of PMSG Using the classical D-Q equivalent circuit models, the equation of the smooth-air-gap synchronous machine in the rotor reference frame are of the form [11]: dψ D − ωE ψQ dt dψ Q u Q = RG i Q + + ωE ψD dt ψ D = LG i D + ψPM ; ψ Q = LG i Q

u D = RG i D +

(17.1)

where the instantaneous values of D and Q axis stator voltage components are uD and uQ ; the stator components are iD , iQ ; RG and LG are electrical resistance and synchronous inductance of the generator phase windings; ψD and ψQ are instantaneous values of D and Q axis stator flux components; ψPM is the permanent magnet flux and ωE is the electrical angular such that M = ωE /P1 , where P1 is the number of pole pairs of the generator and M is the mechanical angular speed of the turbinegenerator system. If RL is the electrical resistance of the external load connected to the output of the generator, the PMSG output voltages are: u D = −R L i D u Q = −R L i Q

(17.2)

The motion equation of the generator is described by [1]: 3 JM dω E = T M + P1 ψPM P1 dt 2

(17.3)

where JM is total axial moment of inertia and TM is the mechanical torque of the wind turbine: TM =

1 r ωE ρπr 3 v2 Ct (λr ), λr = 2 P1 v

(17.4)

Within Eq. (17.4) ρ is the air density, r is the turbine radius, v is the wind speed, λr is the tip-speed ratio and Ct is the torque coefficient provided by the turbine manufacturer: Ct (λr ) = 0.125 + 0.2092λr − 0.1209

(17.5)

17.1 Governing Equations of PMSG

169

For the values ρ = 1.225 kg/m3 , P1 = 16, r = 2.5 m, the torque becomes: TM = 3.758252931v2 + 0.982783141vω E −

0.035079416 2.5 ωE √ v

(17.6)

Concerning the wind speed, different from other works, we consider the analytical model of the speed as [6]: v(t) = vm + A sin

π 3π t + B sin t TG TG

(17.7)

where vm is the mean wind speed of the base wind velocity that is a constant. The base wind velocity vm is considered only in the case in which the generator is active and A and B are two different amplitudes and TG is the gust period. Considering a practical case of a real wind turbine PMSG, the √ characteristics of the steady-state regime are: RG = 0.9, LG = 0.03H, ψ P M = 2W b, P1 = 16 pole pairs, JM = 4.75 Kgm2 . The nominal speed of rotation is nN = 70 rpm, which lead to N = 7.330352856 rad/s or ωN = 117.2856457 rad/s, where N denotes the nominal values (or rated values). Corresponding to these values, from Eq. (17.6) one retrieves TMN = 684.192163461 Nm and for the steady-state regime one can get iQN = −20.158204693. It follows from (17.1) and (17.2) that iDN = −11.118492391 √ V, uQN = 110.4525 V, 2I N = A, DN = 60.9213984503   RLN = 5.479285888, u√ i Q2 N + i D2 N = 23.0211 A, 2u N = u 2Q N + u 2D N = 126.1395 V. In the nominal point of working, the wind turbine develops the mechanical power P1N = TM N ≈ 5015 W and the electric generator develops the electrical power P2N = 3U N I N ≈ 435 W. It is often convenient to express the generator’s parameters, variables and the governing equation in dimensionless quantities. For this aim these terms are divided by base quantities. Usually, the following set of base quantities √ is widely used: the base voltage UB (peak stator phase nominal√voltage U B = 2U N , the base current IB (peak stator phase nominal current I B = 2I N ), the base power SB (nominal apparent power SB = 3UN IN ), the base angular speed ωB (nominal electrical angular speed ωB = ωN . The additional quantities are the base torque TB = P1 SB /ωB , the base flux linkage ψB = UB /ωB , the base impedance ZB = UB /IB ; the base time tB = 1ω/B . By means of the following transformations UD UQ ID iQ D Q ; uQ = ; iD = ; iQ = ; ψD = ; ψQ = ; UB UB IB iB B B P M ωE RL RG ωB L G ψPM = ;ω = ; rl = ; rg = ; xg = ; B ωB ZB ZB ZB TM J M ω2 t Tm = ;k = 2 B ;τ = = ωB t (17.8) TB tB 3P1 U N I N

uD =

170

17 Investigation of a Permanent Magnet Synchronous Generator

the governing Eqs. (17.1) can be written in dimensionless form as rl + r g did id = 0 − ωi q + dτ xE diq rl + r g ψPM − ωi d + iq + ω =0 dτ xE xE dω k − Tm − ψPM i q = 0 dτ

(17.9)

The initial conditions for Eqs. (17.9) are obtained considering the steady-state regime characterized by a constant angular speed at constant speed of the wind vm = 10 m/s and external electrical load rl = 0.4528. One gets the initial conditions: id (0) = −0.438786995; iq (0) = −0.843879596; ω(0) = 0.49923991

(17.10)

The governing Eqs. (17.9) can be retrieved in the form: did + 0.9601432255id − ωi q = 0 dτ diqd + 0.960143255iq + ωi d + 2.061756973ω = 0 dτ dω − 0.168386689iq − 0.00080884v2 − 0.02480742vω dτ 1.124718044 2.5 ω =0 + √ v

(17.11)

where v is given by Eq. (17.7), considering A = 10, B = 4, vm = 10 m/s and TG = 20.5: v(τ ) = 10 + 10 sin

π π τ + 4 sin 3 τ λ λ

(17.12)

where λ = 2407. The dynamical system (17.11) with the initial conditions (17.10) and with the wind speed (17.12) will be investigated in what follows using the Optimal Auxiliary Functions Method.

17.2 Approximate Solution of Eqs. (17.11) and (17.10)

171

17.2 Approximate Solution of Eqs. (17.11) and (17.10) In order to apply our procedure to obtain an approximate solution of Eqs.(17.11) and (17.10), we consider the linear operators for the system (17.11) in the following form: did + 0.9601432255id dτ   diq + 0.960143255iq L 2 i q (τ ) = dτ dω L 3 [ω(τ )] = dτ

L 1 [i d (τ )] =

(17.13)

and the nonlinear operators   N1 i d (τ ), iq (τ ), ω(τ ) = −ωi q   N2 i d (τ ), iq (τ ), ω(τ ) = ωi d + 2.061756973ω   N3 i d (τ ), iq (τ ), ω(τ ) = −0.168386689iq − 0.000808844v2 − 1.124718044 2.5 − 0.024807429vω + ω √ v

(17.14)

where the wind velocity is given by Eq. (17.12). The initial approximations id0 , iq0 , ω0 are determined from Eqs. (17.17), which become did0 + 0.9601432255?id0 (τ ) = 0 id0 (0) = −438786995 dτ

(17.15)

diq0 + 0.960143255?iq0 (τ ) = 0 iq0 (0) = −0.843879596 dτ

(17.16)

dω0 (τ ) = 0 ω0 (0) = 0.499239911 dτ

(17.17)

The solutions of Eqs. (17.15)–(17.17) are id0 (τ ) = −0.438786995 exp(−0.9601432255τ )

(17.18)

iq0 (τ ) = −0.843879596 exp(−0.960143255τ )

(17.19)

ω0 (τ ) = 0.499239911

(17.20)

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17 Investigation of a Permanent Magnet Synchronous Generator

The nonlinear operators (17.14) for the initial approximations (17.18)–(17.20) are:   N1 id0 (τ ), iq0 (τ ), ω0 (τ ) = −0.421298371 exp(−0.9601432255τ )

(17.21)

  N2 id0 (τ ), iq0 (τ ), ω0 (τ ) = 1.029311368 − 0.21905998 exp(−0.960143255τ ) (17.22)   N3 id0 (τ ), iq0 (τ ), ω0 (τ ) = 0.14209809 exp(−0.9601432255τ )− 3π 2 π τ) − − 0.000808844(10 + 10 sin τ + 4 sin λ λ   π 3π 0.1980091808 − 0.01238478 10 + 10 sin τ + 4 sin τ + λ λ 10 + 10 sin π τ + 4 sin 3π τ λ

λ

(17.23) Taking into account the expressions (17.21)–(17.23) and (2.13), in the following we consider F1id (τ ) = 1 π (2i − 1)π (2i − 1)π id − 0.960143255 sin , i = 1, 2, . . . , 5 = − (2i − 1) cos Fi+1 λ λ λ iq F1 (τ ) = 1 π (2i − 1)π (2i − 1)π iq − 0.960143255 sin , i = 1, 2, . . . , 9 Fi+1 = − (2i − 1) cos λ λ λ π (2i − 1)π ω Fi+1 , i = 1, 2, 3, 4 (17.24) (τ ) = (2i − 1) cos λ λ The linear differential equations for the first approximations are did1 (τ ) π π 3π + 0.9601432255id1 (τ ) + C1 − (C2 cos τ + 3C3 cos τ+ dτ λ λ λ 5π 7π 9π + 5C4 cos τ + 7C5 cos τ + 9C6 cos τ ) − 0.960143255 λ λ λ π 3π 5π 7π 9π (C2 sin τ + C3 sin τ + C4 sin τ + C5 sin τ + C6 sin τ ) = 0, λ λ λ λ λ (17.25) id1 (0) = 0

diq1 (τ ) π πτ 3π τ 5π τ + 0.96014iq (τ ) + C7 − (C8 cos + 3C9 cos + 5C10 cos + dτ λ λ λ λ 7π 9π 11π 13π τ + 9C12 cos τ + 11C13 cos τ + 13C1 cos τ+ + 7C11 cos λ λ λ λ

17.2 Approximate Solution of Eqs. (17.11) and (17.10)

173

15π 17π π τ + 17C16 cos τ ) − 0.960143255(C8 sin τ + λ λ λ 3π 5π 7π 9π + C9 sin τ + C10 sin τ + C11 sin τ + C12 sin τ+ λ λ λ λ 11π 13π 15π 17π τ + C1 sin τ + C15 sin τ + C16 sin τ ), + C13 sin λ λ λ λ iq1 (0) = 0 (17.26) + 15C15 cos

π 3π 5π dω1 (τ ) π + (C17 cos τ + 3C18 cos τ + 5C19 cos τ+ dτ λ λ λ λ 7π τ ) = 0, ω1 (0) = 0 + 7C20 cos λ

(17.27)

and the solutions are   π id1 (τ ) = 0.438786995 exp(−0.9601432255τ ) − 1 + C2 sin τ + λ 3π 5π 7π 9π + C3 sin τ + C4 sin τ + C5 sin τ + C6 sin τ (17.28) λ λ λ λ   π iq1 (τ ) = 0.84387959 exp(−0.96014325τ ) − 1 + C8 sin τ + λ 3π 5π 7π 9π + C9 sin τ + C10 sin τ + C11 sin τ + C12 sin τ+ λ λ λ λ 11π 13π 15π 17π + C13 sin τ + C14 sin τ + C15 sin τ + C16 sin τ λ λ λ λ (17.29) ω1 (τ ) = −C17 sin

π 3π 5π 7π τ − C18 sin τ − C19 sin τ − C20 sin τ λ λ λ λ

(17.30)

The approximate solutions of Eqs. (17.11) and (17.10) are id(τ ) = id0 (τ ) + id1 (τ )

(17.31)

iq(τ ) = iq0 (τ ) + iq1 (τ )

(17.32)

ω(τ ) = ω0 (τ ) + ω1 (τ )

(17.33)

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17 Investigation of a Permanent Magnet Synchronous Generator

The optimal values of the convergence-control parameters Ci are obtained by means of a collocation approach as: C2 = −1.3089571472, C3 = −0.4503649177, C4 = −0.0961880721, C5 = −0.0529407913, C6 = −0.0001149325, C8 = −0.1013914279, C9 = −0.0111456049, C10 = −0.0738099321, C11 = −0.0471348271, C12 = −0.0324127981, C13 = −0.0222245798, C14 = −0.0131280042, C15 = −0.0077014973, C16 = −0.0034482411, C17 = −1.3204012731, C18 = −0.5072910784, C19 = −0.0225673217, C20 = −0.0141314789 (17.34) Finally, the approximate solution of Eq. (17.11) and (17.10) can be written as: 3π π τ − 0.4503649177 sin τ− 2407 2407 5π 7π − 0.096188072 sin τ − 0.052940791 sin τ− 2407 2407 9π − 0.000114932 sin τ (17.35) 2407

id(τ ) = −0.438786995 − 1.3089571472 sin

3π π τ − 0.0111456049 sin τ− 2407 2407 7π 5π τ − 0.047134827 sin τ− − 0.0738099321 sin 2407 2407 9π 11π − 0.032412798 sin τ − 0.022224579 sin τ− 2407 2407 13π 15π 17π − 0.013128004 sin τ − 0.0077014973 sin τ − 0.0034482411 sin τ 2407 2407 2407

iq(τ ) = −0.843879596 − 0.1013914279 sin

(17.36)

π 3π ω(τ ) = 0.499239911 + 1.3204012731 sin τ + 0.5072910784 sin τ+ 2407 2407 7π 5π τ + 0.0141314789 sin τ (17.37) + 0.0225673217 sin 2407 2407

Figures 17.1, 17.2 and 17.3 show the obtained approximate solutions of Eqs. (17.11) and (17.10), which, for validation purposes, are compared with numerical solutions obtained using a fourth-order Runge–Kutta method.

17.2 Approximate Solution of Eqs. (17.11) and (17.10) Fig. 17.1 Comparison between the analytical and numerical results for id. _______numerical integration results; _ _ _ _ analytical results

Fig. 17.2 Comparison between the analytical and numerical results for iq. _______numerical integration results; _ _ _ _ analytical results

Fig. 17.3 Comparison between the analytical and numerical results for ω. _______numerical integration results; _ _ _ _ analytical results

175

176

17 Investigation of a Permanent Magnet Synchronous Generator

References 1. B. Basu, Z. Zhang, S.K.R. Nielson, Damping of edgewise vibration in wind turbine blades by means of circular liquid dampers. Wind Energy 19, 213–226 (2016) 2. I. Boldea, Variable Speed Generators (CRC Press, Boca Raton, 2006) 3. P. Vasss, Electrical Machines and Drives. A Space-Vector Theory Approach (Claredon Press, Oxford, 1996) 4. A. Binder, Elektrische Maschinen Und Antriebe (Springer, Berlin-Heidelberg, 2012) 5. P.C. Krause, O. Wasynczuk, S.D. Sudhoff, Analysis of Electrical Machinery (IEEE Press, New York, 1995) 6. N. Herisanu, V. Marinca, G. Madescu, Nonlinear dynamics of a wind turbine permanent magnet generator system in different wind profile conditions, in AIP Conference Proceedings, vol. 1863 (2017), p. 460002 7. D. Xie, Y.P. Lu, J.B. Sun, C.H. Gu, Small signal stability analysis for different types of PMSGs connected to the grid. Renew. Energy 106, 149–164 (2017) 8. J.C. Song, F. Dong, J.W. Zhao, S.L. Lu S.K. Dou, H. Wang, Optimal design of permanent magnet linear synchronous motors based on Taguchi method. IET Electr. Power Appl. 11, 41–48 (2017) 9. M. Rahimi, Modeling, control and stability analysis of grid connected PMSG based wind turbine assisted with diode rectifier and boost converter. Int. J. Electr. Power Energy Syst. 93, 84–96 (2017) 10. C.H. Lin, Dynamic control for permanent magnet synchronous generator system using novel modified recurrent wavelet neural network. Nonlinear Dyn. 77, 1261–1284 (2014) 11. N. Herisanu, V. Marinca, Gh. Madescu, An analytical approach to non-linear dynamical model of a permanent magnet synchronous generator. Wind Energy 18, 1657–1670 (2015)

Chapter 18

Dynamic Response of a Permanent Magnet Synchronous Generator to a Wind Gust

Wind turbine dynamics recently becomes a subject of great interest for scientists [1, 2], once with increasing the importance of renewable, green energy harvesting technologies and their installations. It is a known reality that both mechanical and electrical loads are greatly influenced by the action of the wind speed and its variations, which simultaneously affects the tower/nacelle and blade system, and also the electrical parameters or the mechanical behavior of the stator/rotor system. It is shown that random wind fluctuation can excite significant variations in wind turbine torque as well as in generated electric power [3, 4]. Other works reveal the dynamic response of permanent magnet synchronous generators (PMSG) to specific wind speed profiles in order to assess both the electrical and mechanical stress of different components of the wind power station [5] or propose some modeling procedures for wind speed simulation, which are needed in the investigation of wind power systems [6]. The jumps in wind speed, represented by wind gusts, are often present in real operating conditions and could produce mechanical and electrical shocks, which could lead to damages from both mechanical and electrical point of view. From mechanical point of view, the wind gust generates torque pulsations in the drive train, and consequently an additional mechanical stress occurs. Therefore, it is very important to know the way in which such wind variations affect the mechanical and electrical performance of the wind turbine in order to develop an efficient tool to evaluate and control the whole phenomenon. The influence of the wind gust on various wind power systems is a topic of increased interest for engineers. Giaourakis and Safacas [7] presented a quantitative and qualitative behavior analysis of a wind energy conversion system under a wind gust and converter faults, which has been carried out via simulation. Borowy and Salameh [8] studied the dynamic response of a stand-alone wind energy conversion system with battery energy storage to a wind gust. Bystryk and Sullivan [9] analyzed control strategies for a small scale wind turbine in intermittent wind gusts using a computer model. Generally, such developments need some reliable models which must obey the theory of electrical machines [10–13]. Once such a complex model is achieved, the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_18

177

178

18 Dynamic Response of a Permanent Magnet Synchronous Generator …

solving process can be managed by various analytical and/or numerical approaches. Usually, perturbation methods or asymptotic approaches are used to solve this kind of non-linear problems [14, 15]. In this Chapter, the electrical and mechanical reaction of a low-power PMSG in the presence of a wind gust is investigated using a new approach, namely the Optimal Auxiliary Functions Method. Explicit analytical solutions are developed and analyzed in order to emphasize the effect of the wind gust from both mechanical and electrical point of view. Generally, such analytical approach is very useful to further develop an efficient tool for protection issues and risk assessment concerning the PMSG. A wind power system consists on different subsystems which should separately modeled in view of the simulation of the whole system. Such subsystems are presented in Chapter 17. The model of the wind speed is described by a variable function with respect to time, which simulates the manner in which the “wind profile” is altered. In this Chapter it is considered that the wind profile contains two components: v(t) = vm + vG (t)

(18.1)

where vm is the mean wind speed, or the base wind velocity, that is a constant, and vG is the gust wind component. The base wind velocity vm is considered only in the case in which the generator is active and the gust wind velocity component could be considered as the usual (1-cosine) gust [16]. Therefore, one can consider: vm = const.,

  2π A 1 − cos vG (t) = t 2 TG

(18.2)

The governing equations are di d − ωi q + 0.960143255 i d = 0 dτ di q + ωi d + 0.960143255 i q + 2.061756973ω = 0 dτ dω − 0.168386689i q − 0.000808844v2 dτ 1.124718044 2.5 ω =0 − 0.024807429vω + √ v

(18.3)

with the initial conditions: i d (0) = −0.438786995 i q (0) = −0.843879596 ω(0) = 0.499239911

(18.4)

18 Dynamic Response of a Permanent Magnet Synchronous Generator …

179

By means of the change of the variable τ = ω B t and considering A = 6, vm = 10 m·s−1 and T G = 12 into Eq. (18.2), the Eq. (18.1) can be expressed as:   2π τ v(τ ) = 10 + 3 1 − cos λ

(18.5)

18.1 Approximate Solution of the Dynamic Model of the Wind-Power System In the following, it is applied the above described procedure to obtain an approximate solution of Eqs. (18.3) and (18.4). The linear operators for the system (3.16.4) are L 1 (i d (τ )) =

  di q (τ ) di d (τ ) dω(τ ) ; L 2 i q (τ ) = ; L 3 (ω(τ )) = dτ dτ dτ

(18.6)

These operators are constructed in a completely different way as in Sect. 17.1. The approximate solutions given by Eq. (17.16) in our case can be written as i˜d (τ ) = i d0 (τ ) + i d1 (τ, Ci ), i = 1, 2, ... i˜q (τ ) = i q0 (τ ) + i q1 (τ, D j ), j = 1, 2, ... ω(τ ˜ ) = ω0 (τ ) + ω1 (τ, E k ), k = 1, 2, ...

(18.7)

The initial approximations id0 , iq0 , and ω0 are determined from the Eqs. (2.3) and (18.4), which in this case read: di d0 (τ ) = 0 i d0 (0) = −0.438786995 dτ di q0 (τ ) = 0 i q0 (0) = −0.843879596 dτ dω0 (τ ) = 0 ω0 (0) = 0.499239911 dτ

(18.8)

The solution of the system (18.8) is i d0 (τ ) = −0.438786995 i q0 (τ ) = −0.843879596 ω0 (τ ) = 0.499239911 The nonlinear operators for the system (18.3) are N1 (i d , i q , ω) = −ωi q + 0.960143255i d

(18.9)

180

18 Dynamic Response of a Permanent Magnet Synchronous Generator …

N2 (i d , i q , ω) = ωi d + 0.960143255i q + 2.061756973ω N3 (i d , i q , ω) = −0.168386689i q − 0.000808844v2 1.124718044 2.5 − 0.024807429vω + ω √ v

(18.10)

where the wind velocity v is given by Eq. (18.5). By substituting Eqs. (18.9) into Eqs. (18.10) it holds that N1 (i d0 , i q0 , ω0 ) = 0 N2 (i d0 , i q0 , ω0 ) = 0 N3 (i d0 , i q0 , ω0 ) = −0.159239505 + 0.1002444079 cos − 0.003639798 cos

0.1980691808 4π τ + λ 13 − 3 cos 2πτ λ

2π τ λ (18.11)

The linear equations for the first approximation given by Eqs. (2.13), in this case are di d1 (τ ) + C1 F1 (τ ) + C2 F2 (τ ) + C3 F3 (τ ) + C4 F4 (τ ) = 0, i d1 (0) = 0 dτ di q1 (τ ) + D1 G 1 (τ ) + D2 G 2 (τ ) + · · · + D5 G 5 (τ ) = 0, i q1 (0) = 0 dτ dω1 (τ ) (18.12) + E 1 H1 (τ ) + E 2 H2 (τ ) + E 3 H3 (τ ) = 0, ω1 (0) = 0 dτ Taking into account the expression (18.11) and the initial condition from (18.12), in what follows it is considered (2i − 1)π τ , i = 1, 2, 3, 4 λ (2i − 1)π G i (τ ) = cos τ , i = 1, 2, ..., 5 λ (2i − 1)π Hi (τ ) = cos τ , i = 1, 2, 3 λ Fi (τ ) = cos

(18.13)

The first approximations are obtained from (18.12) and (18.13) under the form:   1 1 1 λ πτ 3π τ 5π τ 7π τ C1 sin + C2 sin + C3 sin + C4 sin π λ 3 λ 5 λ 7 λ  1 1 λ πτ 3π τ 5π τ D1 sin + D2 sin + D3 sin i q1 (τ ) = π λ 3 λ 5 λ  1 7π τ 9π τ 1 + D4 sin + D5 sin 7 λ 9 λ i d1 (τ ) =

18.1 Approximate Solution of the Dynamic Model …

ω1 (τ ) =

  1 1 λ πτ 3π τ 5π τ E 1 sin + E 2 sin + E 3 sin π λ 3 λ 5 λ

181

(18.14)

The approximate solutions of Eqs. (18.3) and (18.4) are obtained from Eqs. (18.9), (18.14) and (18.7). The optimal values of C i , Dj and E k are determined by the collocation method as follows C1 = −0.001880098936, C2 = 0.000249576329, C3 = 0.0001099044306, C4 = 0.000047008063,D1 = −0.000429218279, D2 = −0.0001028685602, D3 = 0.000044633562, D4 = 0.000029287755, D5 = 0.000009332777, E 1 = 0.001592362394, E 2 = −0.000310689633, E 3 = −0.000065075109 The approximate solution (18.7) for Eqs. (18.3) and (18.4) can be written as follows: 3π π i d (τ ) = −0.438786995 − 0.842240332sin τ + 0.111804356099sin τ λ λ 5π 7π + 0.049234613317sin τ + 0.021058512367sin τ (18.15) λ λ 3π π i q (τ ) = −0.843879596 − 0.192279746121sin τ −0.046082708017sin τ λ λ 5π 7π 9π + 0.01999479165sin τ + 0.0131202292sin τ + 0.00418086576sin τ λ λ λ (18.16) π ω(τ ) = 0.499239911 + 0.713341093867sin τ λ 3π 5π −0.139181686295sin τ −0.029152126573sin τ λ λ

(18.17)

where λ = 1407.36. Figures 18.1,18.2 and 18.3 show the approximate solution of Fig. 18.1 Comparison between the analytical and numerical results for id . _______numerical integration results; _ _ _ _ analytical results

182

18 Dynamic Response of a Permanent Magnet Synchronous Generator …

Fig. 18.2 Comparison between the analytical and numerical results for iq . _______numerical integration results; _ _ _ _ analytical results

Fig. 18.3 Comparison between the analytical and numerical results for ω. _______numerical integration results; _ _ _ _ analytical results

Eqs. (18.3) and (18.4) which, for validation purposed, are compared with numerical solutions obtained using a fourth-order Runge–Kutta method. The changes occurring in the wind profile at the moment τ = 0 generate a nonstationary regime in the system characterized by a variation of the angular speed ω and also a significant variation of the currents of the generator id and iq . The angular speed increases from the value ω = 0.5 to ω = 1.35, which means 2.7 times. Such increasing in angular speed under the action of the wind gust could lead to exceeding the limit of overload and should be treated with special attention. From Fig. 18.3 one can be seen that the angular speed of the turbine exceeded with 35% the nominal angular speed (ω = 1). These dynamic changes in the operating conditions generated by the wind gust, affects the whole mechanical system of the turbine generator, which exhibits mechanical loads emphasized by the mechanical torque, which are illustrated in Fig. 18.4. From Fig. 18.4 it is observed a sudden increasing of the torque with almost 20%, which reach a maximum around τ = 350. The increasing gradient of the torque is higher than that of the wind speed, which produces a substantial mechanical stress in the turbine under the action of the wind gust. One can be seen from Fig. 18.5 that under the action of the wind gust, the elec-

18.1 Approximate Solution of the Dynamic Model …

183

Fig. 18.4 Variation of mechanical torque of the wind turbine during the wind gust

Fig. 18.5 Variation of electrical power during the wind gust

trical power at the generator has a similar variation with the angular speed since the electrical voltage is proportional with ω. In the present section, an effective analytic solution is obtained to the governing equations and, as far as we are aware, there are no analytical solutions available in the literature for the particular problem that it is solved. Comparison with numerical integration results obtained using a fourth-order Runge–Kutta method reveals that the proposed analytical approach is very accurate. The proposed dynamical model describing the generator wind turbine system under a simple wind gust allows analyzing both mechanical and electrical phenomena and determining the performances of the dynamic regime produced by wind turbulence. It was proved that the wind speed can be considered in the system of equations describing the dynamic model to predict the system response to specific changes in speed. For this purpose, it is necessary to know the wind profile as a function of time.

184

18 Dynamic Response of a Permanent Magnet Synchronous Generator …

References 1. P.W. Carlin, A.S. Laxson, E.B. Muljadi, The history and state of the art of variable-speed wind turbine technology. Wind Energy 6, 129–159 (2003) 2. Z. Zhang, J. Li, S.R.K. Nielson, B. Basu, Mitigation of edgewise vibration in wind turbine blades by means of roller damper. J. Sound Vibr. 333, 5283–5298 (2014) 3. O. Wasynczuk, D.T. Man, J.P. Sullivan, Dynamic behavior of a class of wind turbine generators during random wind fluctuation. IEEE Trans. Power App. Syst. 100, 2837–2845 (1981) 4. S.A. Papathanassiou, M.P. Papadopulos, Dynamic behaviour of variable speed wind turbines under stochastic wind. IEEE Trans. Energy Conv. 14, 1617–1623 (1999) 5. H. Shariatpanah, R. Fadeinedjad, M. Rashidinejad, A new model for PMSG-based wind turbine with yaw control. IEEE Trans. Energy Conv. 28, 929–937 (2013) 6. C. Nichita, D. Luca, B. Dakyo, E. Ceanga, Large band simulation of the wind speed for real time wind turbine simulators. Trans. Energy Conv. 17, 523–529 (2002) 7. D.G. Giaourakis, A.N. Safacas, Quantitative and qualitative behavior analysis of a DFIG wind energy conversion system by a wind gust and converter faults. Wind Energy 19, 527–546 (2016) 8. B.S. Borowy, Z.M. Salameh, Dynamic response of a stand-alone wind energy conversion system with battery energy storage to a wind gust. IEEE Trans. on Energy Conv. 12, 73–78 (1997) 9. J. Bystryk, P.E. Sullivan, Small wind turbine power control in intermittent wind gusts. J. Wind Eng. Ind. Aerodyn. 99, 624–637 (2011) 10. I. Boldea, Variable Speed Generators (CRC Press, Boca Raton, 2006) 11. P. Vas, Electrical Machines and Drives. A Space-Vector Theory Approach (Claredon Press, Oxford, 1996) 12. A. Binder, Elektrische Maschinen Und Antriebe (Springer, Berlin-Heidelberg, 2012) 13. C.M. Ong, Dynamic Simulation of Electric Machinery (Prentice Hall, New Jersey, 1998) 14. A.H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1993) 15. J. Awrejcewicz, V.A. Krysko, Introduction to Asymptotic Methods (Chapman and Hall/CRC Press, Boca Raton, 2006) 16. P.M. Anderson, A. Bose, Stability simulation of wind turbine systems. IEEE Trans on Power App. Syst. 102, 3791–3795 (1983)

Chapter 19

Axisymmetric Flow and Heat Transfer on a Moving Cylinder

Investigation of axisymmetric flow is significance in engineering and industrial processes. For instance Gorla [1] is concerned with the boundary layer flow on laminar incompressible fluid in the vicinity of an axisymmetric stagnation point for a variation with time of the free stream on a circular cylinder. The unsteady viscous flow in the vicinity of an axisymmetric stagnation point of an infinite cylinder is investigated by Takhar et al. [2] when the free stream velocity and the velocity of the cylinder very arbitrarily with time. Also the self-similar solution is obtained when the velocity of the cylinder and the free stream velocity vary inversely as a linear function of time. Laminar stagnation flow, axisymmetrically and obliquely impinging on a generator of a circular cylinder is formulated by Widman and Putkaradze [3] as an exact solution of the Navier–Stokes equations. The outer stream is composed of a rotational axial-flow superposed onto irrotational radial stagnation flow normal to a cylinder. Resnic et al. [4] studied the steady axisymmetric stagnation flow and heat transfer on a thin infinite cylinder in the cases of constant wall temperature and constant wall heat flux. An analytic solution for the flow of a second-grade fluid over of a radially stretching sheet is presented by Ahmad et al. [5] in the form of a series. The stagnation flow in the annular domain between two cylinders is studied by Hong and Wang [6]. Fluid is injected in wand radially from fixed outer cylinder towards and axially translating and rotating inner cylinder like the convective cooling of a moving rod. Hayat and Nawaz [7] considered the MHD flow analysis of a viscous fluid between two radially stretching sheets. The fluid is electrically conducting, the sheets are not conducting and an incompressible fluid saturates the porous medium. A system of two singular equations is presented by Doo et al. [8] and the equivalence between the 3D axisymmetric inviscid stagnation flow related to Navier–Stokes equations is established. Nadeem et al. [9] discussed the axisymmetric stagnation flow of a micro-polar nanofluid in a moving cylinder with heat transfer. Shahzad et al. [10] reported an exact solution for the axisymmetric two-dimensional flow and heat transfer of an elastically conducting viscous fluid over a nonlinear radially stretching porous sheet within a porous medium. Hayat et al. [11] investigated the heat transfer effects in the axisymmetric flow of an incompressible third grade fluid © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_19

185

186

19 Axisymmetric Flow and Heat Transfer on a Moving Cylinder

between the stretching surfaces. That steady laminar incompressible flow and heat transfer of a viscous fluid between two circular cylinders for two different types of thermal boundary conditions are studied by Mastroberardino [12]. Dual solutions are found for the study axisymmetric stagnation point flow and heat transfer of a viscous and incompressible fluid due to a permeable moving flat plate with partial sleep by Rosca et al. [13]. The flow of an electrically conducting fluid in the vicinity of an axisymmetric stagnation point on a moving cylinder under the influence of magnetic field is started by Hazarika and Sarmah [14] and the normal impingement of the rotational stagnation point on a liquid layer is studied by Weidman [15]. Khan et al. [16] analysed the two dimensional axisymmetric flow and heat transfer of a modified second grade fluid over an isothermal nonlinear radially stretching sheet. The unsteady MHD axisymmetric flow Carreau nanofluid over a radially stretching sheet is investigated by Azam et al. [17]. Nadeem et al. [18] considered steady MHD flow of nanofluids between two concentric circular cylinders with the consideration of a heat generation /absorption effect. The floor is assessed with respect to constant surface temperature and constant heat flux thermal boundary conditions. Mahapatra and Sidui [19] studied heat transfer in non-axisymmetric Homann stagnation point flows towards a stretching sheet. They showed that in under certain conditions there is a new family of axisymmetric viscous stagnation-point flow depending on the ratios of share to strain rate at the plate.

19.1 Equations of Motion We consider steady laminar incompressible flow between two cylinders [5, 9, 20]. The vertical inner cylinder of radius R is rotating with angular velocity and is moving with velocity W in the axial z direction. The inner cylinder is enclosed by an outer cylinder of radius bR. The flow is axisymmetric about z-axis and the fluid is injected radially with velocity U from the outer cylinder towards the inner cylinder. In Fig. 19.1 is shown the geometry of the problem. The cylindrical coordinates are (r, θ, z) and the corresponding velocity components are given by (u, v, w). Supposing that and effects can be ignored, the constant property continuity equation, the constant property Navier–Stokes equations with r, θ and z components of momentum and energy conservation equations are as follows: ∂u ∂w u + + =0 (19.1) r ∂r ∂z  2  ∂u ∂u v2 1 ∂p ∂ u u 1 ∂u ∂ 2u ϕ σB02 u +w − =− +ν − u + + − u − ∂r ∂z r ρ ∂r ∂r 2 r ∂r r2 ∂z 2 k0 ρ (19.2)   2 ∂v ∂v uv ∂ v 1 ∂v ∂ 2 v v u (19.3) +w + =ν + 2 − 2 + 2 ∂r ∂z r ∂r r ∂r ∂z r

19.1 Equations of Motion

187

Fig. 19.1 Geometry of the problem

u

 2  ∂w 1 ∂ρ ∂ w 1 ∂w ∂ 2 w ∂w ϕ +w =− +ν + + − w ∂r ∂r ρ ∂z ∂r 2 r ∂r ∂z 2 k0 

  σB02 2 ∂T 1 ∂T ∂T λ ∂2T ∂2T cp u + +w = + u + ∂r ∂z ρ ∂r 2 r ∂r ρ ∂z 2       2       ∂u 2 u 2 ∂w 2 ∂v ∂u v 2 ∂w 2 ∂v +ν 2 + 2 + + + − + + ∂r ∂z ∂r r ∂z ∂r ∂r r

(19.4)

(19.5)

where ρ is the density, p is the pressure, ν is the kinematic viscosity, δ is the electrical conductivity, B0 is constant magnetic field applied in the z-direction, ϕ is the porosity, k0 is the permeability of the porous medium, cp is the specific heat, T is the temperature and λ is the thermal diffusivity. Let us define the following similarity transformations and non-dimension variables u = −U f (η)η− 2 , v = Rh(η), w = 2U f  (η)ξ + W g(η)

(19.6)

η = r 2 R −2 , ξ = z R −1

(19.7)

1

Using the above transformations, Eq. (19.1) is automatically satisfied. Supposing that the temperature of the outer cylinder is held constant at Tb and that of the inner cylinder at Ts and considering the temperature T of the form

188

19 Axisymmetric Flow and Heat Transfer on a Moving Cylinder

 T (η) = Tb + (Ts − Tb ) ξ2 K (η) + ξq(η) + s(η)

(19.8)

then eliminating pressure between Eqs. (19.2) and (19.4), six similarity equations are derived from Eqs. (19.2)–(19.5) to (19.8):  η f (I V ) (η) + 2 f (I I I ) (η) + Re f (η) f (I I I ) (η) − f  (η) f  (η) − (M + ) f  (η) = 0 (19.9)  ηg  (η) + g  (η) + Re f (η)g  (η) − f  (η)g(η) − f (η) = 0 (19.10)   h(η) 2 f (η)h(η) =0 + Re 4 f (η)h  (η) + η η  ηK  (η) + K  (η) + Re Pr f (η)K  (η) − 2 f  η)K (η)

4ηh  (η) + 4h  (η) −

+ 4 Pr Ecη f  2 (η) + Re Pr EcM f 2 (η) = 0

(19.11)

(19.12)

 ηq  (η) + q  (η) + Re Pr f (η)q  (η) − αg(η)K (η) + 4α Pr Ecη f  η)g  (η) + Re Pr EcM f 2 (η) = 0

(19.13)

 2ηs  + 2s  + K (η) + Re Pr 2 f (η)s  (η) − αg(η)q(η)

2 4 f (η) f  (η) 2 f (η) + 2α2 ηg 2 (η) + 8 f 2 (η) − + Pr Ec 2 η η   h 2 (η) 2 2  + β 2ηh (η) − 2h(η)h (η) + + Re Pr EcM f 2 (η) = 0 (19.14) 2η σB 2

where Re = 21 U Rν−1 is the cross flow Reynolds number, M = ρR0 is Hartman νϕ is the porosity parameter, Pr = νρ p λ−1 is Prandl number, E c = number,  = ρλR

is Eckert number, α = W , β = R are velocity ratios. U U The boundary conditions are no slip on the inner cylinder, and uniform injection on the outer cylinder: U2 c p (Ts −Tb )

f (1) = 0, f  (1) = 0, f (b) =

√ b, f  (b) = 0

(19.15)

g(1) = 1 , g(b) = 0 , h(1) = 1 , h(b) = 0

(19.16)

g(1) = 1 , g(b) = 0 , h(1) = 1 , h(b) = 0

(19.17)

For the heat problem, the boundary conditions are: K (1) = 0 , K (b) = 0 , q(1) = 1 , q(b) = 0 , s(1) = 1 , s(b) = 0

(19.18)

19.2 Optimal Auxiliary Functions Method …

189

19.2 Optimal Auxiliary Functions Method for Solving the System (3.17.9)–( 3.17.18) We remark that Eq. (19.9) is decoupled from the other equations, and the Eqs. (19.9)– (19.11) are decoupled from the (19.12)–(19.14). At first we will apply OAFM for Eqs. (19.9)–(19.11). For these equations, the approximate analytic solutions are of the form f˜(η) = f 0 (η) + f 1 (η, Ci )

(19.19)

g(η) ˜ = g0 (η) + g1 (η, C j )

(19.20)

˜ h(η) = h 0 (η) + h 1 (η, Ci )

(19.21)

At this moment we should emphasize that the linear operator L and the initial approximation f0 given from Eq. (2.5) are not unique. In the following, we present only three possibilities to choose the linear operators and the initial known function Ff (η) for Eqs. (19.9)–(19.11). Here Ff (η) plays the role of the function g from Eq. (2.5). Case 19.2 a L( f ) = η f (I V ) + 2 f  , F f f (η) = 0

(19.22)

L(g) = ηg  + g  , F f g (η) = 0

(19.23)

L(h) = ηh  + h  , F f h (η) = 0

(19.24)

The Eqs. (2.5) and (2.6) can be rewritten as L( f 0 ) = 0, f 0 (1) = f 0 (1) = f 0 (b) = 0, f 0 (b) =

√ b

(19.25)

L(g0 ) = 0 , g0 (1) = 1 , g0 (b) = 0

(19.26)

L(h 0 ) = 0 , h 0 (1) = 1 , h 0 (b) = 0

(19.27)

From Eqs. (19.25), (19.26) and (19.27) it is obtained √ b[(η − 1)2 ln b + (2b − 2)(η − 1 − η ln η)] f 0 (η) = (b − 1)[2b − 2 − (b + 1) ln b]

(19.28)

190

19 Axisymmetric Flow and Heat Transfer on a Moving Cylinder

g0 (η) = 1 −

ln η ln b

(19.29)

h 0 (η) = 1 −

ln η ln b

(19.30)

Case 19.2 b L( f ) = η f (I V ) , F f f (η) = 0

(19.31)

L(g) = g  , F f g (η) = 0

(19.32)

L(h) = h  , F f h (η) = 0

(19.33)

In this case the initial approximations are √ b f 0 (η) = (η − 1)2 (3b − 1 − 2η) (b − 1)3

(19.34)

g0 (η) =

η−b 1−b

(19.35)

h 0 (η) =

η−b 1−b

(19.36)

Case 19.2 c L( f ) = f

(I V )

√ 2(b−η) 2 8 b  f , F f f (η) = − + (2η − 3b + 1)e b−1 5 b−1 (b − 1)

(19.37)

L(g) = g  −

2(b−η) 4 4b g, F f g (η) = − 2 e b−1 2 3 (b − 1) e (b − 1)

(19.38)

L(h) = h  −

2(b−η) 4 4b h, F f h (η) = − 2 e b−1 (b − 1)2 e (b − 1)3

(19.39)

The initial approximations are obtained in the form f 0 (η) =

√

 b

g0 (η) =

η−1 b−1

2 e

b−η 2(b−η) e b−1 b−1

2(b−η) b−1

(19.40) (19.41)

19.2 Optimal Auxiliary Functions Method …

h 0 (η) =

191

b−η 2(b−η) e b−1 b−1

(19.42)

In what follows, we consider only the case 19.2a. The nonlinear operators are obtained from Eqs. (19.9)–(19.11), (19.22)–(19.24) and are respectively  N ( f ) = Re f f (I I I ) − f  f  − (M + ) f 

(19.43)

 N (g) = Re f g  − f  − f 

(19.44)

  h(η) 2fh  N( f ) = − + Re f h + 4η 2η

(19.45)

Inserting the initial approximations given by Eqs. (19.28)–(19.30) into Eqs. (19.43)–(19.45) it follows that 2bRe 2 ln b + 2(b − 1)2 [(3b − 1) ln b − (b − 1)2 [2b − 2(b + 1) ln b]2 η ln η + 2 − 2b 4(b − 1) ln η + (2b − 2) ln b + 2b − 2 + − η2 η √ Re(M + ) b[2η ln b − 2b − 2(b − 1) ln η] 2 (19.46) − 2 ln bη] − (b − 1)[2b − 2 − (b + 1) ln b] √ ln b + 2 − 2b bRe [ − 2(η − 1) ln η N (g0 ) = (b − 1)[2b − 2 − (b + 1) ln b] η ln b ln2 η 2b − 2 + 2( + 1) ln b − 2(b − 1) + η − 2( + 1) ln bη + ln b ln b (2b − 2) ln η ] (19.47) + 2( + 1)(b − 1) ln η − ln b

N ( f0 ) =

√ 1 ln η − + bRe[(η − 1)2 ln b 4η ln b 4η   1 ln η − + 2(b − 1)(η − 1 − η ln η)] 1 − ln b η ln b

N (h 0 ) =

(19.48)

It follows that the auxiliary functions corresponding to the first approximation f1 (η) from Eq. (19.19) are obtained from Eqs. (19.29) and (19.46): Fi (η) ∈

1 1 ln η , , , ln η, η2 , η, η ln η, . . . η η2 η

(19.49)

In this way, the first approximation is defined by Eq. (2.13) which becomes

192

19 Axisymmetric Flow and Heat Transfer on a Moving Cylinder

η f 1(I V ) + 2 f 1 + C1 + C2 η + C3 η2 + C4 η ln η = 0 f (1) = f 1 = f 1 (b) = f 1 (b) = 0

(19.50)

In what concerns the Eq. (19.50), we can also write: η f 1(I V ) + 2 f 1 + C1 +

C2 + C3 ln η + C4 η + C5 η2 = 0 η

(19.51)

C4 C3 C5 ln η + 2 + =0 η η η

(19.52)

or η f 1(I V ) + 2 f 1 + C1 η + C2 η2 +

and so on. Using only Eq. (19.50) then yields the solution 1 1 f 1 (η) = αη(1 − ln η) + βη2 + γη + δ − C1 η3 − C2 η4 12 72   1 17 4 1 C3 η5 + η − η4 ln η C4 − 240 864 72

(19.53)

where

b−1 b3 − b2 − b + 1 (b − 1)2 − C1 − C2 2(b − 1) − (b + 1) ln b 12 36    −b4 + 2b3 3b4 − 2b3 − 2b2 − 2b + 3 11b3 − 17b2 − 17b + 11 C3 + + ln b C4 − 240 432 36(b − 1) 

3 ln b b2 + b + 1 b3 + b2 + b + 1 b ln b b+1 7(b2 + b + 1) C4 β= α+ C1 + C2 + C3 + − 2(b − 1) 8 36 96 36(b − 1) 216 

2 3 2 3 2 b 7(b + b) ln b b +b b +b +b b ln b α − C1 − C2 − C3 + − + C4 γ=− b−1 4 18 48 18(b − 1) 108 α=

3b − 1 ln b − 2b + 2 2b2 + 2b − 1 α+ C1 + C2 2(b − 1) 24 72 

28b2 + 28b − 11 5b3 + 5b2 + 5b − 3 b3 ln b C3 + − C4 + 480 36(b − 1) 864

δ=

(19.54)

The auxiliary functions F(η)i to the first approximations g1 from Eq. (19.20) are the following

1 F(η)i ∈ η, ln η, , η ln η, ln2 η, . . . η

(19.55)

The first approximation g1 is obtained from Eq. (2.13): g1 + g1 + C5 η + C6 η2 + C7 η ln η + C8 η ln η = 0, g(1) = g1 (b) = 0

(19.56)

19.2 Optimal Auxiliary Functions Method …

193

such that     C5 C6 b2 − 1 b3 − 1 ln η + 1 − η3 + ln η g1 (η) = 1 − η2 + ln b 4 ln b 9   b ln b + 2 − 2b ln η + 2η − 2 − η ln η C7 + ln b  2  b ln η + 1 − b2 C8 ln η + η2 − 1 − η2 ln η (19.57) + ln b 4 Using the methods given above, the first approximation h1 can be obtained from ηh 1 + h 1 + C9 η + C10 η2 + C11 ln η + C12 η ln η = 0, h(1) = h 1 (b) = 0 (19.58) with the solution     C9 C10 b2 − 1 b3 − 1 2 3 ln η + 1−η + ln η h 1 (η) = 1 − η + ln b 4 ln b 9   b ln b + 2 − 2b ln η + 2η − 2 − η ln η C11 + ln b  2  b ln η + 1 − b2 C12 2 2 ln η + η − 1 − η ln η (19.59) + ln b 4 The approximate analytical solutions of Eqs. (19.9)–(19.11), (19.15) and (19.16) are obtained from Eqs. (19.19), (19.28), (19.53), (19.20), (19.29), (19.57) and (19.21), (19.30), (19.59), respectively: 1 1 f˜(η) = αη(1 − ln η) + βη2 + γη + δ − C1 η3 − C2 η4 12 72   1 17 1 C3 η5 + η4 − η4 ln η C4 + f 0 (η) − 240 864 72

(19.60)

where α, β, γ, δ are given by Eq. (19.54).     C5 C6 b2 − 1 b3 − 1 ln η + 1 − η2 + ln η + 1 − η3 + ln η g(η) ˜ =1− ln b ln b 4 ln b 9   b ln b + 2 − 2b ln η + 2η − 2 − η ln ηη C7 + ln b  2  b ln η + 1 − b2 C8 2 2 ln η + η − 1 − η ln η (19.61) + ln b 4     ln η C9 C10 b2 − 1 b3 − 1 ˜ h(η) =1− + 1 − η2 + ln η + 1 − η3 + ln η ln b ln b 4 ln b 9

194

19 Axisymmetric Flow and Heat Transfer on a Moving Cylinder

 b ln b + 2 − 2b ln η + 2η − 2 − η ln η C11 ln b  2  b ln η + 1 − b2 C12 ln η + η2 − 1 − η2 ln η + ln b 4 

+

(19.62)

For the Eqs. (19.12)–(19.14) the linear operator is the same L(K ) = L(q) = L(s) = L(x) = ηx  + x 

(19.63)

and the nonlinear operators are respectively  N (K ) = Re Pr f K  − 2 f  K + 4 Pr Ecη f  2 + Re Pr EcM f 2

(19.64)

 N (q) = Re Pr f q − f q − αgk + M Ec f 2 + 4α Pr Ecη f  g 

(19.65)



1 1 1 K + Re Pr f q  − f  q + αgq + EcM f 2 2 2 2  

2  f 2ff h2 2 2 2 2 2  + α ηg + β ηh − hh + (19.66) + Pr Ec 2 + 4 f − η η 4η

N (s) =

The approximate analytical solutions of Eqs. (19.12)–(19.14) are: K˜ (η) = K 0 (η) + K 1 (η, Ci )

(19.67)

q(η) ˜ = q0 (η) + q1 (η, C j )

(19.68)

s˜ (η) = s0 (η) + s1 (η, Ci )

(19.69)

where the initial approximations K0 , q0 and s0 can be obtained from the equations ηK 0 + K 0 = 0, K 0 (1) = K 0 (b) = 0

(19.70)

ηq0 + q0 = 0, q0 (1) = q0 (b) = 0

(19.71)

ηs0 + s0 = 0, s0 (1) = 1, s0 (b) = 0

(19.72)

Solving Eqs. (19.70)–(19.72) then yields the solution K 0 (η) = q0 (η) = 0, s0 (η) = 1 −

ln η ln b

(19.73)

Substituting Eqs. (19.73) into Eqs. (19.64)–(19.66), we gain successively

19.2 Optimal Auxiliary Functions Method …

 b Pr Ec MRe[(η − 1)2 ln b 2 (b − − 2 − (b + 1) ln b]

16 2 2 + (2b − 2)(η − 1−η ln η)] + (η ln b + 1 − b) η

N (K 0 ) =

195

1)2 [2b

 λ Pr Ec MRe[(η − 1)2 ln b (b − 1)2 [2b − 2 − (b + 1) ln b]2

8α + (2b − 2)(η − 1−η ln η)]2 − √ (b − 1)[2b − 2 − (b + 1) ln b](η ln b + 1 − b) b

(19.74)

N (q0 ) =

(19.75)

√

(η − 1)2 Re Pr b ln b + (2b N (K 0 ) = (b − 1)[2b − 2 − (b + 1) ln b] ln b η

  (η − 1)2 Pr Ecb 1 ln b − 2) 1 − − ln η + 2 2 η (b − 1) [2b − 2 − (b + 1) ln b] η  2 1 + (2b − 2) 1 − − ln η + 16[(η − 1) ln b − 16(b − 1) ln η]2 η  

1 (η − 1)2 ln b + (2b − 2) 1 − − ln η −4 η η [η − 1 − (b − 1) ln η]}     2η ln η 4 − 4 ln η + ln2 η ln2 η 2 − ln η 1 +η + β2 + + + α2 η − ln b ln b 4η 2η ln b 4η ln2 b   2 2 Re Pr M Ecb (η − 1) ln b + (2b − 2)(η − 1 − η ln η)] + (19.76) 2(b − 1)2 [2b − 2 − (b + 1) ln b]2 From Eq. (19.74) and (19.75), the auxiliary functions Fi can be written as 

1 Fi ∈ η4 , η3 , η2 , η, η ln η, η2 ln η, η2 ln2 η, , ln η, . . . η

(19.77)

and from Eq. (19.76) these auxiliary functions are 

1 1 ln η ,... Fi ∈ η, , 2 , η ln η, ln2 η, η ln2 η, η η η

(19.78)

The first approximations K1 , q1 , s1 are obtained from the following equations ηK 1 + K 1 + C13 η + C14 η2 + C15 ln η + C16 η ln η = 0, K 1 (1) = K 1 (b) = 0 (19.79) ηq1 + q1 + C17 η + C18 η2 + C19 ln η + C20 η ln η = 0, q1 (1) = q1 (b) = 0 (19.80) ηs1 + s1 + C21

1 + C22 η ln2 η + C23 η + C24 = 0, s1 (1) = s1 (b) = 0 η2

(19.81)

196

19 Axisymmetric Flow and Heat Transfer on a Moving Cylinder

From the above equations it results in     C13 b2 − 1 b3 − 1 C14 2 3 1−η + 1−η + K 1 (η) = ln η + ln η 4 ln b 9 ln b   b ln b + 2 − 2b ln η + 2η − 2 − η ln η + C15 ln b  2  C16 b ln b − b2 + 1 2 2 ln ηη + η − 1 − η ln η + (19.82) 4 ln b     C18 C17 b2 − 1 b3 − 1 1 − η2 + ln η + 1 − η3 + ln η q1 (η) = 4 ln b 9 ln b   b ln b + 2 − 2b ln η + 2η − 2 − η ln η + C19 ln b   C20 b2 ln b − b2 + 1 2 2 ln η + η − 1 − η ln η + (19.83) 4 ln b 

b−1 1 ln η C21 s1 (η) = 1 − − η b ln b   3 2b2 ln2 b − 4b2 ln b + 3b2 − 3 2η2 ln2 η − 4η2 ln η + 3η2 + ln η − C22 + 8 8 ln b 8     1 − η2 b2 − 1 b−1 + ln η C23 + 1 − η + ln ηη C24 + (19.84) 4 4 ln b ln b

The approximate analytical solutions of Eqs. (19.12)–(19.14) are, respectively:     C14 b2 − 1 b3 − 1 C13 2 3 ˜ 1−η + ln η + 1−η + ln η K (η) = 4 ln b 9 ln b   b ln b + 2 − 2b ln η + 2η − 2 − η ln η + C15 ln b  2  C16 b ln b − b2 + 1 2 2 ln η + η − 1 − η ln η (19.85) + 4 ln b     C18 C17 b2 − 1 b3 − 1 1 − η2 + ln η + 1 − η3 + ln η q(η) ˜ = 4 ln b 9 ln b   b ln b + 2 − 2b ln η + 2η − 2 − η ln η + C19 ln b   C20 b2 ln b − b2 + 1 ln η + η2 − 1 − η2 ln η + (19.86) 4 ln b  

1 b−1 ln η + 1− − ln η C21 s˜ (η) = 1 − ln b η b ln b   3 2b2 ln2 b − 4b2 ln b + 3b2 − 3 2η2 ln2 η − 4η2 ln η + 3η2 + ln η − C22 + 8 8 ln b 8

19.2 Optimal Auxiliary Functions Method …  +

   1 − η2 b2 − 1 b−1 + ln η C23 + 1 − η + ln η C24 4 4 ln b ln b

197

(19.87)

19.3 Numerical Results In order to emphasize the effectiveness of the method and to prove the accuracy of the obtained results, we will consider a specific set of numerical values for the physical parameters involved in the governing equations, which are: b = 2, Ec = 0.1, M = 1,  = 1, α = 1, β = 1, Pr = 0.7. Moreover, in order to investigate in influence of Re number we will consider 3 for different Re numbers, which are 1, 4 and 7. In this situation, following the procedure described above for determining the optimal values of the convergence-control parameters, which ensure the convergence of the solution, for Re = 1 we obtain the following optimal values: C1 = 1403.5606580130336, C2 = −804.2283347147892, C3 = −599.9275548052959, C4 = 1871.8001233771774, C5 = 325.6753295253647, C6 = −67.77144388196257, C7 = −543.8963537244399, C8 = 89.43063711778753 C9 = −161.23249058301138, C10 = 34.84776686951173, C11 = 259.5686396395336, C12 = −43.73424584471891 C13 = 384.08384057024324, C14 = −347.15205596692687, C15 = −744.397862208022, C16 = 832.622899026811 C17 = −22.878035371196603, C18 = 15.424368137434392, C19 = 67.67949004152135, C20 = −44.871875430381564 C21 = −18.35411789181781, C22 = 8.203667057969469, C23 = −22.283213322660515, C24 40.41496454921898

(19.88)

Taking into account the above values of the convergence-control parameters, the ˜ and also the approximate approximate solutions of the similarity functions f˜, g˜ and h,    ˜ ˜ solutions of the velocity profiles f , g˜ and h are graphically presented in Figs. 19.2, 19.3, 19.4, 19.5, 19.6, and 19.7 in comparison with corresponding numerical integration results obtained using a fourth-order Runge Kutta method combined with a shooting approach. Moreover, in Figs. 19.8, 19.9, and 19.10 we graphically present the variation of K, q and s, corresponding to the heat transfer problem. Figures 19.2, 19.3, 19.4, 19.5, 19.6, 19.7, 19.8, 19.9, and 19.10 emphasize a very good accuracy of the proposed approximate results, comparing to the numerical integration ones obtained through a fourth-order Runge Kutta method in combination with a shooting approach. In order to analyze the influence of Re number, in Figs. 19.11, 19.12, and 19.13 we graphically present the variation of f, g and h for three different values of Re: Re = 1, Re = 4 and Re = 7.

198

19 Axisymmetric Flow and Heat Transfer on a Moving Cylinder

Fig. 19.2 Comparison between the approximate solution and numerical integration results for f: ______ numerical, _ _ _ analytical solution (19.60)

Fig. 19.3 Comparison between the approximate and numerical integration results for the velocity profile f’: ______ numerical, _ _ _ analytical solution obtained from (19.60)

Fig. 19.4 Comparison between the approximate solution and numerical integration results for g: ______ numerical, _ _ _ analytical solution (19.61)

19.3 Numerical Results

199

Fig. 19.5 Comparison between the approximate and numerical integration results for the velocity profile g’: ______ numerical, _ _ _ analytical solution obtained from (19.61)

Fig. 19.6 Comparison between the approximate solution and numerical integration results for h: ______ numerical, _ _ _ analytical solution (19.62)

Fig. 19.7 Comparison between the approximate and numerical integration results for the velocity profile h’: ______ numerical, _ _ _ analytical solution obtained from (19.62)

200

19 Axisymmetric Flow and Heat Transfer on a Moving Cylinder

Fig. 19.8 Comparison between the approximate solution and numerical integration results for K: ______ numerical, _ _ _ analytical solution (19.85)

Fig. 19.9 Comparison between the approximate and numerical integration results for the velocity profile q: ______ numerical, _ _ _ analytical solution obtained from (19.86)

Fig. 19.10 Comparison between the approximate solution and numerical integration results for s: ______ numerical, _ _ _ analytical solution (19.87)

19.3 Numerical Results

201

Fig. 19.11 The influence of Re number on f

Fig. 19.12 The influence of Re number on g

Fig. 19.13 The influence of Re number on h

It was found that an increase in the velocity profile f can be observed with increase in Re (Fig. 19.11), while a decrease in the velocity profiles g and h can be observed with increase Re (Figs. 19.12 and 19.13).

202

19 Axisymmetric Flow and Heat Transfer on a Moving Cylinder

A monotonically increasing behaviour of the similarity function f can be observed in Fig. 19.2, and a monotonically decreasing behaviour of the functions g, h and s can be observed in Figs. 19.4, 19.6 and 19.10, while the functions K and q have an arbitrary behaviour (Figs. 19.8 and 19.9).

References 1. R.S.R. Gorla, The final approach to steady state in an axisymmetric stagnation flow following a change in free stream velocity. Appl. Sci. Res. 40, 247–251 (1983) 2. H.S. Takhar, A.J. Chamkhe G. Neth, Unsteady axisymmetric stagnation-point floor of a viscous fluid on a cylinder, Int. J. Eng. Sci. 37, 1143–1157 (1999) 3. P.D. Weidman, V. Putkaradze, Axisymmetric stagnation flow obliquely impinging on a circular cylinder. Eur. J. Mech-B Fluids 22(2), 123–131 (2003) 4. C. Resnic T. Grosan I. Pop, Heat transfer in axisymmetric stagnation flow on a thin cylinder, Studia Univ. Babes Bolyai Math. LIII(2), 119-125 (2008) 5. I. Ahmed, M. Sajid, T. Hayat, M. Ayub, Unsteady axisymmetric flow of a second-grade fluid over a radially stretching sheet. Comp. Math. Appl. 56, 1351–1357 (2008) 6. L. Hong, C.Y. Wang, Annular axisymmetric stagnation flow on a moving cylinder. Int. J. Eng. Sci. 47, 141–152 (2009) 7. T. Hayat, M. Nawaz, Effect of heat transfer on magnetohydrodynamic axisymmetric flow between two stretching sheets, Z. Naturforsch. 65a, 961–968 (2010) 8. C. Doo, S.J. Huang, M.J. Zhang, On 3D axisymmetric invested stagnation point flow related to Navier-Stokes equations. Nonlinear Anal. Forum 16, 67–75 (2011) 9. S. Nadeem, A. Rehman, K. Vajravelu, J. Lee, C. Lee, Axisymmetric stagnation flow of a micropolarnanofluid in a moving cylinder, Math. Problems Eng. 2012, 378259 (2012) 10. ShahzadA, Ali R, Khan M. On the exact solution for axisymmetric flow and heat transfer over a nonlinear radially stretching sheet, Chim. Phys. Lett. 29(8), 084705 (2012) 11. T. Hayatt, A. Shafiq, A. Alsaedi, M. Awais, MHD axisymmetric flow of third grade fluid between stretching sheets with heat transfer. Comput. Fluids 36, 103–108 (2013) 12. A. Mastroberardino, Series solutions of annular axisymmetric stagnation flow and heat transfer on moving cylinder. Appl. Math. Mech. 34(9), 1043–1054 (2013) 13. A.V. Rosca, N.C. Rosca, I. Pop, Axisymmetric stagnation point flow and heat transfer towards a permeable moving flat plate with surface slip condition. Appl. Math. Comput. 233, 139–151 (2014) 14. G.C. Hazarika, A. Sarmah, Effect of magnetic field on flow near an axisymmetric stagnation point on a moving cylinder. Int. J. Modern Eng. Tech. 1(5), 46–54 (2014) 15. P. Weidman, Axisymmetric rotational stagnation point flow impinging on a flat liquid surface, Eur. J. Mech.-B/Fluids 56, 188–191 (2016) 16. M. Khan, M. Rahman, M. Manzur, Axisymmetric flow and heat transfer to a modified second grade fluid over a radially stretching sheet. Results Phys. 7, 878–889 (2017) 17. M. Azam, M. Khan, A.S. Alshomrani, Effects of magnetic field and partial slip on unsteady axisymmetric flow of Caureau nanofluid over a radially stretching surface. Results Phys. 7, 2671–2682 (2017) 18. A. Nadeem, A. Mahmood, J.I. Siddique, L. Zhao, Axisymmetric magnetohydrodynamic flow of nanofluidunder heat generation/absorption effects. Appl. Math. Sci. 11, 2059–2087 (2017) 19. T.R. Mahapatra, S. Sidui, Heat transfer in non-axisymmetric Homan stagnation point flows towards a stretching sheet. Eur. J. Mech. B/Fluids 65, 522–529 (2017) 20. V. Marinca, N. Herisanu, Construction of analytic solution to axysymmetric flow and heat transfer on a moving cylinder. Symmetry 12(8), 1335 (2020)

Chapter 20

Blasius Problem

The flow of the non-Newtonian fluids is very important for engineers, because of its several applications in various fields of science and engineering. In fluid mechanics a Blasius boundary layer (named after Paul Richard Heinrich Blasius), describes the steady two dimensional laminar boundary layer that forms a semi-infinite plate which is held parallel to a constant unidirectional flow. In the last few decades, these fluids have attracted considerable attention from researchers in many branches of nonlinear dynamical systems. Many powerful methods have been presented. Perturbation methods have been widely applied to solve nonlinear problems [1] but unfortunately they are based on such assumption that a small parameter exist. Until now, lots of other analytical procedures were proposed to solve Blasius equation. Belhachmi et al. [2] studied in detail the concave solutions of initial value problems involving the Blasius equation, Fazio [3] introduced a numerical parameter and require to an extended scaling group involving this parameter, Ganji et al. [4] and Towsyfyan et al. [5] employed the homotopy perturbation method to solve Blasius nonlinear differential equation, Aminikhah [6] applied Laplace transform and a new homotopy perturbation method. Variational iteration method is used by Liu and Kurra [7], optimized artificial neural networks approximation with sequential quadratic programing algorithm and hybrid AST-INP techniques is employed by Ahmad and Bilal [8] and an integrated neural network and gravitational search algorithm (HNN GSA) is applied by Biglari et al. [9]. Peker et al. [10] studied the Blasius problem by means of Padé approximation with differential transformation method, Adenhounme and Codo [11] applied the Adomian Decomposition Method to solve the Blasius problem. Lal and Neeraj [12] employed the Taylor series with higher radius of convergence and parameters of asymptotic variation and Robin [13] developed a new uniform approximation from existing rational approximations. He applied a perturbation procedure [14] and OHAM is used by Marinca and Herisanu [15]. Wazwaz obtained an explicit analytic solution and initial slope with high accuracy by means of Variational Iteration Method [16]. Other important works on Blasius problem were done by Boyd [17, 18], and Costin and Tanveer [19]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_20

203

204

20 Blasius Problem

20.1 The Governing Equation For the steady, incompressible, two-dimensional boundary layer equations for continuity and momentum are, respectively:

u

∂u ∂v + =0 ∂x ∂y

(20.1)

∂v ∂ 2u ∂u +v =ν 2 ∂x ∂y ∂y

(20.2)

where ν is kinematic viscosity. If δ is the boundary-layer thickness and L is natural length scale and U is a constant velocity of the mainstream at infinity, then balancing between viscosity and convective inertia it results the scaling argument U U2 ≈ν 2 L δ

(20.3)

From the scaling argument it is apparent that the boundary layer grows with the downstream coordinate x, e.g.: δ(x) ≈

 νx 1/2 U

(20.4)

such that we introduce the following similarity variable η=

 1/2 U y =y δ(x) νx

(20.5)

the stream function  = (νxU )1/2 f (η)

(20.6)

u = U f  (η)

(20.7)

and the function

Now, differentiating to find the velocities, after simple manipulations we obtain Blasius equation [20]: f  (η) +

1 f (η) f  (η) = 0 2

(20.8)

20.1 The Governing Equation

205

subject to the boundary conditions f  (0) = 0,

f (0) = 0,

f  (∞) = 1

(20.9)

where prime denotes the derivative with respect to η.

20.2 Approximate Solution of the Blasius Problem In the following, we apply our procedure to obtain an approximate solution of Eqs. (20.8) and (20.9). The initial approximation f 0 (η) which verify the boundary conditions (20.9) can be chosen as f 0 (η) = η +

e−kη − 1 k

(20.10)

where k is an unknown parameter at this moment. Taking into consideration Eq. (20.10), we define the linear operator and the function g in the form: L[ f (η)] = f  (η) + k f  (η), g(η) = 0

(20.11)

L[ f (η)] = f  (η) + k f  (η), g(η) = −k

(20.12)

L[ f (η)] = f  (η) − k 2 f  (η), g(η) = k 2

(20.13)

or

or

If we consider only the linear operator given through Eq. (20.11), we can obtain the nonlinear operator as N [ f (η)] =

1 f (η) f  (η) − k f  (η) 2

(20.14)

It is clear that the Eq. (2.5) is defined by L[ f 0 (η)] = 0 whose solution is given by Eq. (20.10).

(20.15)

206

20 Blasius Problem

Now, substituting Eq. (20.10) into Eq. (20.14), it holds that  N [ f 0 (η)] =

 1 1 1 2 kη − − k e−kη + e−2kη 2 2 2

(20.16)

We have the freedom to choose the auxiliary functions Fi (η) in the following forms: F1 (η) = η3 e−kη ;

F2 (η) = η2 e−kη ;

F4 (η) = ηe−2kη ;

F5 (η) = e−2kη

F3 (η) = ηe−kη (20.17)

or F1 (η) = η2 e−kη ;

F2 (η) = ηe−kη ;

F3 (η) = η2 e−2kη ;

F4 (η) = ηe−2kη (20.18)

or F1 (η) = η2 e−kη ; F4 (η) = e−2kη ;

F2 (η) = e−kη ;

F3 (η) = ηe−2kη

F5 (η) = ηe−3kη ;

F6 (η) = e−3kη

(20.19)

and so on. Hence, we consider the auxiliary functions Fi given by Eqs. (20.17). In this case, Eq. (2.13) may be written as   f 1 (η) + k f 1 (η) = C1 η3 + C2 η2 + C3 η + C4 e−kη + (C5 η + C6 )e−2kη ,

f 1 (0) = f 1 (0) = f 1 (∞) = 0

(20.20)

whose solution is      C1 4 2C1 C2 9C1 2C2 C3 3 f 1 (η) = η + + 2 η + + 3 + 2 η2 4k 2 k3 3k k4 k 2k   24C1 24C1 6C2 2C3 C4 6C2 2C3 C4 + + 4 + 3 + 2 η+ 6 + 5 + 4 + 3 5 k k k k k k k k   3C5 C6 −kη C5 C5 C6 −2kη 24C1 + 2 + 3 e − η+ 4 + 3 e − 6 4k 2k 4k 3 2k 4k k 6C2 2C3 C4 C5 C6 − 4 − 3 − 4− 3 (20.21) k5 k k 4k 4k The approximate solution (2.3) is obtained from Eqs. (20.10) and (20.21). The parameters k, C1 , C2 , …, C6 are determined by collocation method, as follows:

20.2 Approximate Solution of the Blasius Problem

207

k = 0.7213456124, C1 = −0.0200901684, C2 = 0.6332041491, C3 = −5.7019089005, C4 = 15.6635426754, C5 = −5.4440609736, C6 = −15.4081506377 The approximate solution (2.3) for Eqs. (20.8) and (20.9) can be written as follows: f (η) = (−0.0096524331η4 + 0.2985860773η3 − 2.7728487526η2 + 11.2835892031η − 18.5769471635)e−0.7213456128η + (3.6260417956η + 20.3162186219)e−1.4426912248η + η − 1.73927146 (20.22)

20.3 Discussion To solve Eqs. (20.8) and (20.9), Blasius [20] provides in 1908 a power series solution f (η) =

 ∞ 

1 k Ak δk+1 3k+2 − η 2 (3k + 2)! k=1

(20.23)

where A0 = A1 = 1,

Ak =

 k−1 

3k − 1 r =0

3r

Ar Ak−r −1 k ≥ 2



 m m! . But the expression (20.23) is not closed, because σ = f  (0) with = n!(m−n)! n is unknown. Using two different approximations, Blasius obtained the numerical result σ = 0.332. In 1938, Howarth [21] gained a more accurate value σ = 0.33206, the expression of f (η) given by Eq. (20.23) is valid in a small region 0 ≤ η ≤ 5.69. Blasius’ power series (20.23) is fundamentally an analytic- numerical solution because the value of σ is obtained by numerical techniques. Asaithambi [22] found this number correct to nine decimal precision as σ = 0.332057336. Using our procedure, from Eq. (20.22) we obtain 

σ = f (0) = 0.3320542804

(20.24)

Obviously, our first-order approximate result (20.22) ensures an error ε = 0.0009%, which is remarkable good.

208

20 Blasius Problem

Table 20.1 Comparison of analytical and numerical results 

 f num [22]

η

f

0

0

0

0

0

0.4

0.02658

0.0265

0.13276

0.1327

0.8

0.10601

0.1061

0.26415

0.2647

1

0.16535

0.1655

0.32918

0.3297

1.4

0.32262

0.322

0.45626

0.4562

2

0.64995

0.6560

0.63043

0.6297

2.4

0.92238

0.9222

0.72898

0.7289

2.8

1.23084

1.2309

0.81037

0.8115

3

1.3964

1.3961

0.84445

0.8460

3.6

1.92807

1.9295

0.92187

0.9233

4

2.30397

2.3057

0.95551

0.9555

4.4

2.69098

2.6923

0.97786

0.9758

5

3.2839

3.2832

0.99600

0.9915

5.4

3.68348

3.6803

1.0012

0.9974

6

4.28503

4.2796

1.00311

0.9989

6.4

4.68614

4.6793

1.00228

0.9996

7

5.28684

5.2792

0.99992

0.9999

7.4

5.68647

5.6792

0.99827

1

8

6.2848

6.2792

0.99628

1

8.4

6.68313

6.6792

0.99539

1

8.8

7.08116

7.0792

0.99484

1

9

7.28012

7.2797

0.99469

1

f num [22]

f

From Table 20.1 we can conclude that the first-order approximate solution (20.22) obtained by means of OAFM is highly accurate.

References 1. V. Marinca, N. Herisanu, Construction of analytic solution to axysymmetric flow and heat transfer on a moving cylinder. Symmetry 12(8), 1335 (2020) 2. Z. Belhachmi, B. Brighi, K. Taous, On the concave solutions of the Blasius equation. Acta Math. Univ. Comenianae 2, 199–214 (2000) 3. R. Fazio, Transformation methods for the Blasius problem and its recent variants, in Proceedings of the World Congress on Engineering, London, vol. II (2008) 4. D.D. Ganji, H. Babazadeh, T. Noori, M.M. Pirouz, M. Janipour, An application of homotopy perturbation method for non-linear Blasius equations to boundary layer flow over a flat plate. Int. J. Nonl. Sci. 7, 399–404 (2009) 5. H. Towsyfyan, A.A. Salehi, G. Davoudi, The application of homotopy perturbation method to Blasius equations. Res. J. Math. Stat. 5, 1–4 (2013)

References

209

6. H. Aminikhah, Analytical approximation to the solution of nonlinear Blasius viscous flow equation by LTNHPM. ISRN Math. Anal., ID 957473 (2012) 7. Y. Liu, S.N. Kurra, Solution of Blasius equation by variational iteration. Appl. Math. 1, 24–27 (2011) 8. I. Ahmad, M. Bilal, Numerical solution of Blasius equation through neural networks algorithms. Amer. J. Comput. Math. 4, 223–232 (2014) 9. M. Biglari, E. Assareh, I. Poultagani, M. Nedaei, Solving Blasius differential equation by using hybrid neural network and gravitational search algorithm (HNN GSA). Glob. J. Sci. Eng. Techn. 11, 29–36 (2013) 10. M.A. Peker, O. Karaoglu, G. Oturanc, The differential transformation method and Pade approximant for a form of Blasius equation. Math. Comut. Appl. 16, 507–513 (2011) 11. V. Adenhounme, F.P. Codo, Solving Blasius problem by Adomian decomposition method. Int. J. Sci. Eng. Res. 3, 1–4 (2012) 12. S.A. Lal, P.M. Neeraj, An accurate Taylor series solution with high radius of convergence for Blasius function and parameters of asymptotic solution. J. Appl. Fluid Mech. 7, 557–564 (2014) 13. W. Robin, Some new uniform approximate analytical representations of the Blasius function. Glob. J. Math. 2, 150–155 (2015) 14. J.H. He, A simple perturbation approach to Blasius equation. Appl. Math. Comput. 140, 217– 222 (2003) 15. V. Marinca, N. Herisanu, The optimal homotopy asymptotic method for solving Blasius equation. Appl. Math. Comput. 231, 134–138 (2014) 16. A.M. Wazwaz, The variational iteration method for solving two forms of Blasius equation on a half infinite domain. Appl. Math. Comput. 188, 485–499 (2007) 17. J.P. Boyd, The Blasius function in the complex plane. Experimental Math. 8, 381–394 (1999) 18. J.P. Boyd, The Blasius function: Computations before computers, the value of tricks, undergraduate projects, and open research problems. SIAM Rev. 50, 791–804 (2008) 19. O. Costin S. Tanveer, Analytical approximation of Blasius similarity solution with rigorous error bounds. SIAM J. Math. Anal. 46(6), 3782–3813 (2014) 20. H. Blasius, Grenzschichten in Flussigkeiten mit kleiner Reibung. Z. Math. Phys. 56, 1–37 (1908) 21. L. Howarth, On the solution of the laminar boundary layer equations. Proc. Lond. Math. Soc. A 164, 547–579 (1938) 22. A. Asaithambi, Solution of the Falkner-Skan equation by recursive evaluation of Taylor coefficients. J. Comput. Appl. Math. 176, 203–214 (2005)

Chapter 21

Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder

It is well known that the subject of non-Newtonian fluids is very popular and is an area of active research especially in industry and engineering problems. Examples of non-Newtonian fluids include microchip production, performance of lubricants, food processing, movements of biological fluids, wire and fibre coating, paper production, gaseous diffusion and so on [1, 2]. These fluids are described by a nonlinear relationship between stress and the rate of deformation tensors and therefore several models have been proposed. It is very difficult to suggest a single-model which exhibits all properties of non-Newtonian fluids. As a consequence several fluid models have been proposed to predict the non-Newtonian behaviour of various types of materials. Fourth grade fluid is one of the important fluids and its equation is based on strong theoretical foundations, where relation between stress and strain is not linear. Some experiments may be well described by the fluids on the order four. Because the exact solutions of these equations are difficult to achieve, approximate analytical and numerical methods are widely used to solve nonlinear differential equations modelling such physical phenomena. There exists some analytical approaches such as the Lindstedt-Poincare method, the KBM method, the Adomian Decomposition Method, the elliptic perturbation method, the harmonic balance method [3–5], or some iteration procedures [6, 7]. Most of the perturbation methods unfortunately require the inclusion of a small parameter in the equation, but this parameter is not already specified in the equation and it is artificially introduced and finally equated to unity to obtain the solution of the original problem. Considerable efforts have been made to study strongly nonlinear non-Newtonian fluids for various geometrical configurations via analytical techniques. Some developments in this direction are discussed by different investigators. Among them Khalid and Vafai [8] discussed hydrodynamic squeezed flow and heat transfer over a sensor surface. Miladinova et al. [9] studied this film flow over a power law liquid falling from an inclined plate where it was observed that saturation of nonlinear interaction occur in a permanent finite amplitude wave. Sajid et al. [2] investigated the sleep effects of thin film flow grade fluid down a vertical cylinder using the homotopy analysis method. Analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder is presented © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_21

211

212

21 Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder

by Hayat and Sajid [10]. Gul et al. [11] investigated effects of sleep condition on thin film flow of third grade fluids for lifting and drainage problem under the condition of constant viscosity. The homotopy perturbation method and the traditional perturbation method are applied by Siddiqui et al. [12] to the nonlinear equations modelling thin film flow of a fourth grade fluid falling in the outer surface of an infinitely long vertical cylinder. For other studies see [13–15].

21.1 Governing Equations of Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder In what follows we consider a non-Newtonian fluid of fourth grade falling on the outside surface of an infinitely long vertical cylinder of radius R. The flow is considered in thin, uniform axisymmetric film with thickness δ, in contact with stationary air. The velocity field is of the form [10, 12–16]: v = [0, 0, u(r )]

(21.1)

In cylindrical coordinates we have        4   d 1 4 1 d du 2 du ∂p = (2α1 + α2 ) γ 3 + γ4 + γ5 + γ6 + r r ∂r r dr dr r 2 dr dr (21.2)      3 ∂p μ d du 2 d du = r + (β2 + β3 ) + ρg (21.3) r ∂z r dr dr r dr dr ∂p =0 ∂θ

(21.4)

where p = p(z) is the pressure. From Eq. (21.3) it is clear that     3  d 2u 2(β2 + β3 ) du du ρg ∂u 2 d 2 u r 2 + + r =0 + + 3r 2 dr dr μ ∂r dr dr μ

(21.5)

The corresponding boundary conditions are u(R) = 0, Defining

du (R + δ) = 0 dr

(21.6)

21.1 Governing Equations of Thin Film Flow of a Fourth …

η=

r , R

f =

213

R g R3 μ(β2 + β3 ) δ u, k = 2 , β = , d =1+ , ν ν R4ρ2 R

(21.7)

the Eq. (21.5) and (21.6) reduces to d2 f df η 2 + + kη + 2β dη dη



df dη

f (1) = 0,

3



df + 3η dη

2

d2 f dη2

 =0

f  (d) = 0

(21.8) (21.9)

21.2 Approximate Solution of the Eqs. (21.8) and (21.9) To apply our procedure to obtain an approximate solution of Eqs. (21.8) and (21.9) we consider the linear operator for the Eq. (21.8) in two alternatives. For the first alternative we consider L[ f (η)] = η f  (η) + f  (η) + kη

(21.10)

where f  (η) = d f /dη and the nonlinear operator   N [ f (η)] = 2β f 3 (η) + 3η f 2 (η) f  (η)

(21.11)

The initial approximation f 0 (η) is determined from Eq. (2.5) which becomes η f 0 + f 0 + kη = 0,

f 0 (1) = 0,

f 0 (d) = 0

(21.12)

The solution of Eq. (21.12) is f 0 (η) =

  k d2 −η 2 η

(21.13)

The nonlinear operator (21.11) for the initial approximation (21.13) becomes N





f 0 (η)

  1 3 d6 4 2 2 4 = βk − 3d + 3d η − η 4 η2

(21.14)

or   N f 0 (η) = 2β(η f 03 )

(21.15)

214

21 Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder

Equation (2.13) can be written in the form η f 1 (η) + f 1 (η) + kη + C1 F1 (η) + C2 F2 (η) + · · · + C6 F6 (η) = 0 f 1 (1) = 0, f 1 (d) = 0 (21.16) The Eq. (21.6) can be rewritten as    η f1 +



1 2 kη 2



+ [C1 F1 (η) + C2 F2 (η) + · · · + C6 F6 (η)] = 0

(21.17)

where  2  2 2 3 d d d2 F1 (η) = − η; F2 (η) = − η ; F3 (η) = −η η η η 4 5  2  2   d d 1 − η ; F4 (η) = − η ; F6 (η) = − [2β f 03 + kη F4 (η) = η η C6 (21.18) By integrating Eq. (21.17) and considering f 1 (0) = 0, we obtain  2  2  2  3 3 d2 d d βk − η + C2 −η + + C3 −η η η 4 η  2  2 4 5 d d − η + C5 −η + C4 (21.19) η η

f  (η) = C1



Finally from Eqs. (21.1), (21.19) and (2.3) we can obtain the differentiation of the approximate solution of Eqs. (21.8) and (21.9) in the form  2  2  2  2  3 3 d d d k βk + C1 − η + C2 −η + + C3 −η 2 η η 4 η  2  2 4 5 d d − η + C5 −η + C4 (21.20) η η

f˜ (η) =



From Eq. (21.20) it is easy to obtain the approximate solution of Eqs. (21.8) and (21.9) by integration and having in view that f˜(1) = f (1) = 0.

21.3 Numerical Example for the First Alternative

215

21.3 Numerical Example for the First Alternative For the case when k = 1, β = 8, d = 1.5, we obtain C1 = 0.0446457909 117; C2 = −0.5830431778 7714; C3 = −1.5722449276 332089; C4 = −0.1607374508 74938; C5 = 0.0214437762 867209

(21.21)

In [10] β ≥ 0.3 is considered a parameter corresponding to strong non-linearity. Therefore the explicit analytic expression given by Eq. (21.20) for the convergencecontrol parameters given by (21.21), the first order approximate solution becomes   2.25 −η f  (η) = 0.5446457909 117669 η 2 3   2.25 2.25 − η + 0.4277550723 667911 −η − 0.5830431778 77 η η 4 5   2.25 2.25 − η + 0.0214437762 867209 −η − 0.1607374508 74938 η η

(21.22) In Table 21.1 is presented a comparison between the present solution obtained from formula (21.22) and the numerical solution of Eqs. (21.8) and (21.9). It can be seen that the solution obtained through OAFM is near identical with that given by the numerical results, demonstrating a very good accuracy. For the second alternative, the linear and nonlinear operators, and the function g are respectively [17]: Table 21.1 Comparison of analytical and numerical results   f˜ (η), Eq. (21.22) η f (η), numerical





ε = f  (η) − f˜ (η)

1.00

0.2782771936982422

0.27868677728446

4.09*10–4

1.05

0.2613391820243754

0.2619592020457328

6.20*10–4

1.10

0.243037748916253

0.243037749162644

2.46*10–10

1.15

0.2237293973885208

0.223832705241852

1.03*10–4

1.20

0.2031782753168658

0.203219171567154

4.08*10–5

1.25

0.180723124261417

0.180659142650312

6.39*10–5

1.30

0.1554016332342926

0.155372275453134

2.93*10–5

1.35

0.126040063073303

0.126185641229246

1.45*10–4

1.40

0.091316354263098

0.091367589247079

5.12*10–5

1.45

0.0498036954997723

0.048981783573852

8.21*10–4

1.50

0

0

0

216

21 Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder

L[ f (η)] = η

d2 f df + dη2 dη

(21.23)

g(η) = kη  N [ f (η)] = 2b

df dη

3

(21.24) 

df + 3η dη

2

d2 f dη2

 (21.25)

The initial approximation f0 is obtained from Eq. (2.5): η

d f0 d 2 f0 + kη = 0, + dη2 dη

f 0 (1) = 0,

d f0 (d) = 0 dη

(21.26)

The solution of Eq. (21.6) is f0 (η) =

k 1 + d2 ln η2 − η2 4

(21.27)

Substituting Eq. (21.27) into Eq. (21.25), the nonlinear operator becomes  2  2  d 1 3 d2 −η + 2η N[f0 (η)] = − bk 2 η η

(21.28)

The first approximation f1 is given by Eq. (2.13): η

d f1 d 2 f1 + + C1 F1 (η) + C2 F2 (η) + · · · + C p F p (η) = 0 dη2 dη f 1 (1) = 0,

d f1 (d) = 0 dη

(21.29) (21.30)

Taking into account that   d 2 f1 d d f1 d f1 η 2 + = η dη dη dη dη   2 2  2  3   2 d d d d − −η + 2η = −η η η η dη η We have freedom to choose the auxiliary function Fi as F1 (η) = −

   2  2 d d 2 d η dη − η . . . .. η − η , F2 (η) = − dη dη η

(21.31)

(21.32)

21.3 Numerical Example for the First Alternative

  2 p d d η −η . . . .F p (η) = − dη η

217

(21.33)

where p is an arbitrary integer positive number. Using only the expression (21.33) of the auxiliary functions, Eq. (21.29) can be written as     2 3  2  2  2 d d f1 d d d d η − − η + C2 η − η + C3 η −η C1 η dη dη dη η η η p  2 d d f1 = 0, f 1 (1) = −η (d) = 0 (21.34) + · · · + C pη η dη From Eq. (21.34) we find the following solution: 2 3 p    d2 d2 d2 d f1 d f1 = C1 − η + C2 − η + · · · + Cp − η , f (1) = (d) = 0 dη η η η dη

(21.35) Solving Eq. (21.35), and then substituting this solution and Eq. (21.27) into Eq. (2.3), we obtain approximate solution of Eqs. (21.8) and (21.9) by OAFM in the second alternative in the form    4   d 1 1 2 1 3 k 2 2 + C1 d ln η + − η − C2 + 2d η − η f (η, Ci ) = 2 2 2 η 3  6  d 3 1 − C3 + 3d 4 ln η − d 2 η2 + η4 2η2 2 4  8  d 4d 6 4 1 5 4 2 3 − 6d d η − C4 − η + η − 3η3 η 3 5  10  8 d 5d 5 2 4 1 7 6 4 2 + ... d η − C5 − − 10d ln η + 5d η − η + 4η4 2η2 4 7 (21.36)

21.4 Numerical Results by OAFM (The Second Alternative) We will show that the error of the solutions decreases when the number of terms in the auxiliary functions Fi increases. 21.4.a Case 1 First, we consider k = 2, b = 2, d = 1.2, p = 4. One can get

218

21 Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder

C1 = 0.02010809 507891; C2 = −0.44321206 1022469; C3 = −1.46971845 415762; C4 = 2.05824705 104109

(21.37)

The approximate velocity with four constants in this case by using OAFM is    2 ∂ f (η) 1.44 1.44 = 1.02010809507891 − η − 0.443212061022469 −η ∂η η η   3 4 1.44 1.44 − 1.46971845415762 − η + 2.05824705104109 −η η η (21.38) 21.4.b Case 2 In this case for same values: k = 2, b = 2, d = 1.2 but p = 5, it holds that C1 = 0.00493140 714889; C2 = −0.08652252 98529617; C3 = −4.13589639 137958; C4 = 9.72574923 625005 C5 = −7.4369884 0812736

(21.39)

and velocity becomes    2 1.44 1.44 ∂ f (η) = 1.00493140714889 − η − 0.0865225298529617 −η ∂η η η 3 4   1.44 1.44 − η + 9.72574923625005 −η − 4.13589639137958 η η 5  1.44 −η − 7.43698840812736 (21.40) η It is easy to verify the accuracy of the obtained solutions if we compare these analytical results with numerical ones. From Tables 21.2 and 21.3 it can be seen that the analytical solutions of our problem obtained by OAFM are very accurate. The examples presented in this section lead to the very important conclusion that the accuracy of the obtained results is growing along with increasing the number of constants in the auxiliary functions. Our approach does not depend upon small parameters. 21.4.c Case 3 For k = 1, b = 2, d = 1.2, p = 4 we obtain C1 = 0.00022854 018451; C2 = −0.00216198 649445; C3 = −0.53701336 395192; C4 = 0.47615841 198521

(21.41)

21.4 Numerical Results by OAFM (The Second Alternative)

219

Table 21.2 Comparison between the OAFM solutions (21.38) and (21.40) with numerical solutions for k = 2, b = 2, d = 1.2, p = 4 (Eq. (21.38)) and p = 5 (Eq. (21.40)) η

Numerical solution

f (η) Eq. (21.38)

f (η) Eq. (21.40)

Error of Eq. (21.38)

Error of Eq. (21.40)

1

0.3149892726

0.3149902864

0.3149892725

1.01*10–6

3.66*10–7

1.02

0.2920872677

0.2917634189

0.2922025741

3.25*10–4

1.15*10–4

0.2677958741

6.89*10–6

9.48*10–9 3.81*10–5

1.04

0.2677958645

0.2677841234

1.06

0.2418828814

0.2422461104

0.2418448028

3.63*10–4

1.08

0.2140851463

0.2145652735

0.2140852072

4.80*10–4

6.12*10–8

0.1841492536

2.94*10–4

2.63*10–5 6.4*10–8

1.1

0.1841229205

0.1844175114

1.12

0.1517392282

0.1517400487

0.1517391641

8.20*10–7

1.14

0.1167864570

0.1166398160

0.1167536690

1.46*10–4

3.27*10–5 3.14*10–8

1.16

0.0793786596

0.0793792588

0.0793786281

5.96*10–7

1.18

0.0400814166

0.0403378899

0.0401505928

2.56*10–4

6.91*10–5

1.2

0

0

0

0

0

Table 21.3 Comparison between the OAFM solution (21.42) and (21.44) with numerical solutions for k = 1, b = 2, d = 1.2, p = 4 (Eq. (21.42)) and p = 5 (Eq. (21.44)) η

Numerical solution

f (η) Eq. (21.42)

f (η) Eq. (21.44)

Error of Eq. (21.42)

Error of Eq. (21.44)

1

0.1917839251

0.1917878034

0.1917843781

9.41*10–5

4.53*10–7

1.02

0.1745937891

0.1745669533

0.1745912673

2.68*10–5

2.52*10–6

0.156867684

5.64*10–6

3.55*10–7 1.11*10–6

1.04

0.1568673294

0.1568678463

1.06

0.1385961651

0.138557774

0.1385972549

3.83*10–5

1.08

0.1197907602

0.1198159933

0.1197910245

2.52*10–5

2.64*10–7

0.1004859761

1.30*10–5

7.88*10–7 7.67*10–7

1.1

0.10048675874

0.1004998341

1.12

0.0807509285

0.0807519952

0.0807510956

1.07*10–6

1.14

0.0606850155

0.0606806934

0.0606866534

4.30*10–6

1.66*10–6

0.0404254824

3*10–7

8.20*10–8 5.43*10–6 0

1.16

0.0404254003

0.0404257002

1.18

0.0201368303

0.0201799364

0.0201313916

4.31*10–5

1.2

0

0

0

0

   2 1.44 1.44 ∂ f (η) = 0.50022854018451 − η − 0.00216198649445 −η ∂η η η 3 4   1.44 1.44 − η + 0.476158411985218 −η − 0.537013363951926 η η

(21.42)

220

21 Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder

21.4.d. Case 4 In the last case, for k = 1, b = 2, d = 1.2 but p = 5 we have C1 = −0.0005652 4385115; C2 = 0.01651354 436441 C3 = −0.6765478 04124; C4 = 0.87717256 349806 C5 = −0.3886916 4165841

(21.43)

   2 ∂ f (η) 1.44 1.44 = 0.49943475061489 − η + 0.01651354436441 −η ∂η η η   3 4 1.44 1.44 − 0.676547804124 − η + 0.87717256379806 −η η η  5 1.44 − 0.38869164165841 (21.44) −η η

References 1. T. Gul, S. Islam, R.A. Shah, I. Khan, S. Shafie, Thin film flow in MHD third grade fluid on a vertical belt with temperature dependent viscosity. Plos One, 9, e.97552 (2014) 2. M. Sajid, M. Awais, S. Nadeem, T. Hayat, The influence of slip condition on thin film flow of a fourth grade fluid by the homotopy analysis method. Comp. Math. Appl. 56, 2019–2026 (2008) 3. A.H. Nayfeh, Perturbation Methods (Wiley, New York, 2000) 4. P. Hagedorn, Nonlinear Oscillations (Claredon Press, Oxford, 1981) 5. J.D. Cole, Perturbation Methods in Applied Mathematics (Blaisdell Publishing Company, Waltham, 1968) 6. R.E. Mickens, Iteration procedure for determining approximate solution to nonlinear oscillations. J. Sound Vibr. 16, 185–188 (1987) 7. V. Marinca, N. Herisanu, A modified iteration perturbation method for some nonlinear oscillation problems. Acta Mech. 184, 231–242 (2006) 8. A.R.A. Khalid, K. Vafai, Hydrodynamic squeezed flow and heat transfer over the sensor surface. Int. J. Eng. Sci. 42, 509–519 (2004) 9. S. Miladinova, G. Lebon, E. Toshev, Thin film flow of a power law liquid falling down an inclined plane. J. Non-Newton. Fluid Mech. 22, 69–70 (2004) 10. T. Hayat, M. Sajid, On analytic solution of thin film flow of a fourth grade fluid down a vertical cylinder. Phys. Lett. A 361, 316–322 (2007) 11. T. Gul, R.A. Shah, S. Islam, M. Arif, MHD thin film flows of a third grade fluid on a vertical belt with slip boundary conditions. J. Appl. Math, ID 70728614 (2013) 12. A.M. Siddiqui, R. Mahmood, Q.K. Ghori, Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder. Phys. Lett. A 352, 404–410 (2006) 13. V. Marinca, N. Herisanu, I. Nemes, Optimal homotopy asymptotic method with application to thin film flow. Centr. Eur. J. Phys. 6, 649–653 (2008) 14. V. Marinca, N. Herisanu, An optimal homotopy perturbation approach to thin film flow of a fourth grade fluid, in AIP Conference Proceedings, vol. 1479 (2012), pp. 2383–2386

References

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15. V. Marinca, N. Herisanu, C. Bota, B. Marinca, An optimal homotopy asymptotic method applied to the steady flow or a fourth grade fluid past a porous plate. Appl. Math. Lett. 22, 245–251 (2009) 16. V. Marinca, N. Herisanu, Optimal auxiliary functions method for thin film flow of a fourth grade fluid down a vertical cylinder. Ro. J. Techn. Sci. Appl. Mech. 62(2), 181–189 (2017) 17. B. Marinca, V. Marinca, Approximate analytical solution for thin film flow of a fourth grade fluid down a vertical cylinder. Proc. Roman. Acad. Ser. A 19, 69–76 (2018)

Chapter 22

Viscous Flow Due to a Stretching Surface with Partial Slip

The study of boundary layer viscous flow due to a stretching surface is very important because of its several engineering applications as: crystal growing, drawing of elastic films, the heat treated materials traveling between a feed roll and the wind-up roll or a conveyor belt poses the features of a moving continuous surface. There exist situations where there may be a partial slip between the fluid and the boundary. In the last decades, the nonlinear problems are widely used as models to describe complex physical phenomena in various field of science. The boundary layer flow over a continuously stretching surface was first studied by Sakiadis [1, 2]. Crane [3] found a closed form exponential solution for the planar viscous flow of a linear stretching case. Sparrow et al. [4, 5] studied a number of flow problems taking the velocity slip into account. In Ackroyd [6] is given an algorithm for the moving boundaries and for the flow of a fluid past a rotating disk. Rajagopal et al. [7] obtained numerical solution for the flow of viscoelastic second order fluid past a stretching sheet. Ariel [8] and Anderson [9] have reported the analytical closed form solutions of the second grade fluid and of the fourth order nonlinear differential equations arising due to the MHD flow. Wang [10] studied an axisymmetric cases for the viscous flow. Mirogolbabei et al. [11] considered a number of boundary layer flows induced by the axisymmetric stretching of a sheet. The unsteady axisymmetric flow and heat transfer of a viscous fluid is treated by Sajid et al. [12]. Also, the analytic solution for these problems has been considered by Sajid et al. in [13], Miklavˇciˇc and Wang [14] first analyzed properties of the flow to a shrinking sheet with suction. Many other researchers denoted themselves on investigating these problems [15–19]. In general in fluid mechanics of viscous fluids, by means of similarity transformations, the set of partial differential equations are reduced to that of ordinary differential equations. Much attention of the researchers was focused in obtaining of the analytical solutions for the nonlinear problems. But, in general, the nonlinear problems cannot be solved analytically by means the traditional methods. For the weakly nonlinear problems, many methods exist for approximating the solutions. Most widely applied method by researches is perturbation technique [20] which is based on the existence of small parameters. Others methods for approximating the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_22

223

224

22 Viscous Flow Due to a Stretching Surface with Partial Slip

solutions of nonlinear problems are the modified Lindstedt-Poincare method [21], the Adomian decomposition method [22], the parameter-expansion method [23], the optimal homotopy perturbation method [24–26], the optimal homotopy asymptotic method [27–29]. Recently, several methods has been used for solving different nonlinear differential equations such as: traveling-wave transformation method [28], Cole-Hopf transformation method [29].

22.1 The Governing Equations Consider the three dimensional flow of a viscous fluid bounded by a stretching surface. If (u,v,w) are the velocity components in the Cartesian directions (x,y,z) respectively, then the continuity and steady constant property Navier–Stokes equations for viscous fluid flow are [7, 10–13]: u x + v y + wz = 0

(22.1)

uu x + vu y + wu z = −

px + v∇ 2 u ρ

(22.2)

uvx + vv y + wvz = −

py + v∇ 2 v ρ

(22.3)

pz + v∇ 2 w ρ

(22.4)

uwx + vw y + wwz = −

where p is the pressure, ρ is the density and υ is the kinematic viscosity. If a is the stretching constant, a > 0, W is the suction velocity and m is a parameter describing the type of stretching, then the velocity components on the stretching surface are: u = ax, v = a(m − 1)y, w = −W.

(22.5)

It is known that for m = 1, we have planar stretching case, while for m = 2 we have axisymmetric stretching case. In order to simplify the governing equations we use the similarity transform [10]:  √ a , (22.6) u = ax (η), v = a(m − 1)y (η), w = −m av(η), η = z v 



where prime denotes the differentiation with respect to η. Equation (22.1) is identically satisfied and Eq. (22.4) becomes

22.1 The Governing Equations

225

1 p = vρwz − ρW 2 + C, 2

(22.7)

with C a constant. From the Eqs. (22.2) and (22.3) we deduce that:  (η) + m(η) (η) − 2 (η) = 0

(22.8)

as there is no lateral pressure gradient at infinity. If N denotes a slip constant, then on the surface of the stretching sheet, the velocity slip is assumed to be proportional to the local sheer stress [30]: u − ax = ρvN u z < 0, v − a(m − 1)y = ρvN vz < 0.

(22.9)

From the similarity transform (22.6) we obtain  (0) = 1 + λ (0),

(22.10)

√ with λ = ρN av > 0 non-dimensional parameter indicating the relative importance of partial slip. If λ = 0 there is no slip. Given a suction velocity of –W on the stretching surface, we have the boundary condition (0) = α,

(22.11)

W > 0 is a non-dimensional constant which determines the transpiwhere α = m √ av ration rate at the surface and α < 0 if injection from the surface occurs, α > 0 for suction and α = 0 for an impermeable sheet. Also, since there is no lateral velocity at infinity, we have yet the condition

 (∞) = 0.

(22.12)

22.2 Application of OAFM to Viscous Fluid Given by Eqs. (22.8), (22.10), (22.11) and (22.12) We introduce the basic ideas of the OAFM by considering Eq. (22.8) with the initial / boundary conditions given by Eqs. (22.10), (22.11) and (22.12). We choose the linear operator L by the form: L((η)) =  + K  .

(22.13)

226

22 Viscous Flow Due to a Stretching Surface with Partial Slip

We can also choose the linear operator in the form L((η)) =  + K 2  , and so on, where K > 0 is an unknown parameter. Equation (2.5) becomes (g(η) = 0):   0 (η) + K 0 (η) = 0, 0 (0) = α, 0 (0) = 1 + λ0 (0), 0 (∞) = 0,

(22.14)

with the solution 0 (η) = α +

1 − e−K η K (1 + λK )

(22.15)

The nonlinear operator N ((η)) is obtained from Eqs. (22.8) and (22.13): N ((η)) = −K  (η) + m(η) (η) − 2 (η)

(22.16)

By means of the Eqs. (22.15) and (22.16) it is obtained N (0 (η)) =

βe−K η + (m − 1)e−2K η (1 + λK )2

(22.17)

where β = K (K − mα)(1 + λK ) − m

(22.18)

Having in view Eqs. (2.13), (22.15) and (22.17), the first approximation is obtained from the equation:   −kη  + (η − 1)e−2K η  1 + K 1 − (C 1 η + C 2 ) βe     − (C3 + C4 η) e−2K η + K 1 e−3K η − (C5 + C6 η) e−3K 1 + K 2 e−4K η = 0, (22.19) where Ci , i = 1…6, K and Ki , i = 1,2 are unknown parameters. The initial / boundary conditions are 1 (0) = 0, 1 (0) = λ1 (0), 1 (∞) = 0.

(22.20)

The Eq. (22.19) can be written as:  −K η  1 + K 1 = (βC 1 + βC 2 η)e

+ [(mC2 − C2 + C4 )η + (m − 1)C1 + C3 ]e−2K η + [(K 1 C4 + C6 )η + K 1 C3 + C5 ]e−3K η + [K 2 C6 η + K 2 C5 ]e−4K η (22.21)

22.2 Application of OAFM to Viscous Fluid Given …

227

with the initial / boundary conditions given in Eq. (22.20), whose solution is: βC2 2 βC1 2βC2 η +( 2 + )η + M2 ]e−K η 2K 2 K K3 (1 − m)C2 − C4 1−m 1−m C3 C4 −2K η +[ η+ C1 + C2 − − ]e 4K 3 4K 3 2K 4 4K 3 2K 4 K 1 C4 + C6 K 1 C3 + C5 7(K 1 C4 + C6 ) −3K η −[ η+ + ]e 3 3 18K 18K 108K 4 K2 K2 5K 2 −[ C6 η + C5 + C6 ]e−4K η , (22.22) 3 3 48K 48K 288K 4

1 (η, Ci ) = M1 + [

where 1 − m − 4β + λK (3 − 3m − 8β) 1 − m − 8β + λK (2 − 2m − 12β C1 + C2 4K 3 (1 + λK ) 4K 4 (1 + λK ) 9 + 4K 1 + λK (16K 1 + 27) 8K 1 + 27 + λK (2K 1 − 18) − C3 − C4 3 36K (1 + λK ) 108K 4 (1 + λK ) 9K 2 + 16 + λK (45K 2 + 64) 27K 2 + 112 + λK (81K 2 + 208) − C5 − C6 144K 3 (1 + λK ) 864K 4 (1 + λK ) (2β + m − 1)(1 + 2λK ) 8β + 3m − 3 + 4λK (m − 1 + 3β) M2 = C1 + C2 2K 3 (1 + λK ) 4K 4 (1 + λK ) K 1 + 3 + 3λK (K 1 + 2) 27 + 5K 1 + 9λK (K 1 + 4) + C3 + C4 6K 3 (1 + λK ) 36K 4 (1 + λK ) 2 + K 2 + 2λK (3 + 2K 2 ) 20 + 7K 2 + 4vK (9 + 4K 2 ) + (22.23) C5 + C6 12K 3 (1 + λK ) 144K 4 (1 + λK ) M1 =

If we consider m = 1 and β = 0 (the planar stretching case for impermeable sheet) into Eq. (22.17) then we have N (0 (η)) = 0 and therefore we can obtain the exact solution of the equation:  +  − 2 = 0, (0) = 0,  (0) = 1 + λ (0),  (∞) = 0

(22.24)

which is: (η) = α +

1 − e−K η , K (1 + λK )

(22.25)

where K is obtained from the equation: λK 3 + (1 − αλ)K 2 − αK − 1 = 0.

(22.26)

228

22 Viscous Flow Due to a Stretching Surface with Partial Slip

Beyond of this remarkable case, the approximate analytic solution of the Eqs. (22.8), (22.10), (22.11) and (22.12) can be obtained from Eqs. (2.3), (22.15) and (22.23).

22.3 Numerical Examples We illustrate the accuracy of OAFM by comparing obtained approximate solutions with the numerical integration results obtained by means of a fourth-order Runge–Kutta method in combination with the shooting method. In all cases, the unknown parameters are optimally identified via Galerkin method. For this, we use the following eleven weighted functions f i given by [30]: f 1 (η) = γe−3K η + η2 e−2K η + δηe−K η , f 2 (η) = ηe−K η , f 3 (η) = 2ηe−K η , f 4 (η) = 1 + γηe−K η , f 5 (η) = e−2K η , f 6 (η) = e−K η , f 7 (η) = ηe−4K η , f 8 (η) = e−4K η , f 9 (η) = δηe−2K η + η3 e−2K η f 10 (η) = e−K η + K 1 e−2K η + K 3 ηe−4K η , f 11 (η) = ηe−K η + K 2 e−3K η + K 4 e−4K η

(22.27)

where γ, δ, K 1 , K 2 , K 3 and K 4 are unknown parameters. The parameters K 1 , K 2 , K 3 , K 4 , γ, δ, C1 , . . . , C6 are determined from equations: ∞ J j = 0 R(m, λ, α, η) f j (η)dη = 0, j = 1, . . . , 11

(22.28)

where the residual R(m, α, λ, η) is given by Eq. (2.30): 



2

R(m, λ, α, η) =  (η) + m(η) (η) −  (η)

(22.29)

and (η) is given by Eq. (2.3) with the initial / boundary conditions (22.10)-(22.12). Example 1 Consider planar stretching case with impermeable sheet, m = 1, α = 0 and λ = 1. In this case, from Eqs. (22.28) and (22.29) we obtain K = 0.4631238249, δ = −0.1052314539, K 3 = 2.13 · 10−6 , C1 = −0.0119658357, C3 = 0.0642359221, C5 = 0.0530417312,

γ = 0.6858214854, K1 = 0, K 2 = 0, K 4 = 0.21 · 10−6 , C2 = 0.0007396024, C4 = 0.0326512044 C6 = 0.0102396495

22.3 Numerical Examples

229

The first-order approximate solution given by Eq. (2.3) can be written in the form (η) = 0.7549045180 + (−0.0011830813η2 + 0.0280632559η − 0.1942637382)e−0.4631238249η − (0.0821765397η + 0.5165462911)e−0.9262476499η − 0.0440894886e−1.3893714748η

(22.30)

Example 2 In the second case for the same planar stretching case with impermeable sheet m = 1, α = 0 but λ = 5, from Eqs. (22.18) the values of the parameters are K = 0.5252370049, γ = 4.8796805607, δ = −1.2890550305, K 1 = 0, K 2 = 0, K 3 = 1.24 · 10−6 , K 4 = 1.02 · 10−6 C1 = −0.0279185716, C2 = −1.9918945423 · 10−6 , C3 = 6.2493581461 · 10−12 , C4 = 3.90211197551 · 10−12 C5 = −9.621041312 · 10−12 , C6 = 0 The approximate solution (2.3) becomes: ¯ (η) = 0.5251657155 + (−1.3352013705 · 10−9 η2 − − 0.0000374387η−0.5251657155)e−0.5252370049η + + (6.7324778915 · 10 - 12 η + 1.4853702952 · 10 - 11 )e−1.0504740099η + + (9.1284670601 · 10 - 13 η + 5.7165228355 · 10 - 12 )e−1.5757110148η (22.31) Example 3 In the third case for the planar stretching case with impermeable sheet m = 1, α = 0 and with sleep parameter λ = 10, from Eqs. (22.28) we obtain K = 0.4331821853, γ = 3.0639340595, δ = −0.5722383481, K 1 = 0, K 2 = 0, K 3 = 1.03 · 10−6 , K 4 = 2.11 · 10−6 C1 = −0.0125089730, C2 = −9.6275057188 · 10−7 , C3 = 6.6010897191 · 10−12 , C4 = −3.360346796 · 10−12 C5 = −1.0095931161 · 10−11 , C6 = −2.0561054881 · 10−12 The first-order approximate solution (2.3) is (η) = 0.4331053011 + (−1.2809679965 · 10−9 η2 − 0.0000332989η − 0.4331053011)e−0.433182185η   + 1.0336892501 · 10−11 η + 2.7423124117 · 10−11 e−0.8663643706η

230

22 Viscous Flow Due to a Stretching Surface with Partial Slip

Table 22.1 Values of  (0) and (∞) in the case of planar stretching case with impermeable sheet m = 1, α = 0 λ

1

5

10

num (0) O AF M (0)

−0.4301597090

−0.1448401942

−0.0812419798

−0.4301595579

−0.1448401939

−0.0812419808

num (∞)

0.7548776662

0.5251657154

0.4331053011

 O AF M (∞)

0.7549045179

0.5251657158

0.4331053009

  + 0.932245039 · 10−12 η + 1.065716162 · 10−11 e−1.2995465559η (22.32) The values of  (0) and (∞) are given in Table 22.1 for the same planar stretching case with impermeable sheet, calculated by means OAFM, and by numerical integration. Example 4 In this case, we consider planar stretching case but suction sheet, m = 1, α = 3 and with slip parameter λ = 1. The parameters obtained by means of Eqs. (22.28) are: K = 1.0265261151, γ = 3.011278740, δ = −0.9721507438, K 1 = 0, K 2 = 0, K 3 = 1.002 · 10−6 , K 4 = 0.52 · 10−6 , C1 = 2.6781920761 · 10−11 , C2 = −4.2525109956 · 10−12 , C3 = 4.655928821 · 10−9 , C4 = 1.6964107211 · 10−9 C5 = −6.9684341491 · 10−10 , C6 = 0.0001204043 and the first-order approximate solution (2.3) becomes (η) = 3.0795956234 + (−2.3525169361 · 10−12 η2

 +2.0134806302 · 10−11 η − 4.9211079655 · 10−11 e−1.0265261151η   + 8.7180902556 · 10−11 η − 6.9432001142 · 10−11 e−2.0530522302η   + 1.3753249143 · 10−6 η − 0.0795955933 e−3.0795783454η (22.33)

Example 5 For planar stretching case and suction sheet, m = 1, α = 3 and slip parameter λ = 5, we have K = 1.0016112565, γ = 2.9079023283, δ = −0.9038557714, K 1 = 0, K 2 = 0, K 3 = 1.55 · 10−6 , K 4 = 0.62 · 10−6 , C1 = 1.3658235079 · 10−6 , C2 = −2.1653508962 · 10−7 , C3 = −0.0005801616156, C4 = 0.0002087511, C5 = −0.9806268698, C6 = 0.0158334825

22.3 Numerical Examples

231

and therefore, the first-order approximate solution (2.3) has the form (η) = 3.0205594824 + (−5.4255801019 · 10−7 η2 + 4.6777709040 · 10−6 η − 0.0000118225)e−1.0016112565η + (0.0000200454η − 0.0000156837)e−2.0032225131η + (0.0003400695η − 0.0205318961)e−3.0048337696η

(22.34)

Example 6 The planar stretching case and suction sheet, m = 1, α = 3 but slip parameter λ = 10 we obtain from Eq. (22.28): K = 1.1395297367, γ = 2.9650196892, δ = −0.9588570845 K 1 = 0, K 2 = 0, K 3 = 1.002 · 10−6 , K 4 = 1.32 · 10−6 , C1 = 0.0000646091, C2 = −0.0000113762, C3 = −0.0863263871, C4 = 0.02944192315, C5 = −0.2391591505, C6 = −0.06102791273 and the first-order approximate solution (2.3) can be written as (η) = 3.0106792212 + (−0.0000661253η2 + 0.0005189796η − 0.0011722736)e−1.1395297367η + (0.0027526798η − 0.0032398505)e−2.2790594735η + (−0.0012678992η − 0.0062671170)e−3.4185892103η

(22.35)

In Table 22.2, we present the values of of  (0) and (∞) in the case of planar stretching case with suction sheet. Example 7 Now, we consider the axisymmetric flow with impermeable sheet, m = 2, α = 0 and with slip parameter λ = 1. From Eq. (22.28) we obtain the following set of parameters Table 22.2 Values of  (0) and (∞) in the case of planar stretching case with suction sheet m = 1, α = 3 λ

1

5

10

num (0) O AF M (0)

−0.7548776662

−0.1875797976

−0.0967852665

−0.7548776659

−0.1875797939

−0.0967852020

num (∞)

3.0795956235

3.0205594824

3.0106792212

 O AF M (∞)

3.0795956233

3.0205594411

3.0106777729

232

22 Viscous Flow Due to a Stretching Surface with Partial Slip

K = 1.2284861855, γ = −1.2107679860, δ = 5.9462560050, K 1 = K 2 = 0.7335772195, K 3 = 1.11 · 10−6 , K 4 = 1.06 · 10−6 , C1 = −0.0272717347, C2 = −0.0098884303, C3 = −0.0130978383, C4 = −0.1950919615, C5 = −0.06294893217, C6 = 0.0018821549 In this case, the first-order approximate solution (2.3) is (η) = 0.5509446955 + (−0.0077420177η2 − 0.0679124177η − 0.6421304231)e−1.2284861855η + (0.0469289624η + 0.0831404603)e−2.4569723711η + (0.0041898146η + 0.0075368933)e−3.6854585567η + + (0.0000154947η + 0.0005083699)e−4.9139447423η

(22.36)

Example 8 For the case of the axisymmetric flow with impermeable sheet, m = 2, α = 0 and with slip parameter λ = 5, from Eq. (22.28) we have. K = 0.8408529662, δ = 12.6879512489, K 3 = 2.03 · 10−6 , C1 = −0.0053548861, C3 = −0.00821412956, C5 = −0.01091079627,

γ = −2.5936764722, K 1 = K 2 = 0.5953831575, K 4 = 1.01 · 10−6 , C2 = −0.0012448162, C4 = −0.03031382709, C6 = 0.00176755942

and therefore, the first-order approximate solution (2.3) becomes (η) = 0.3780173693 + (−0.0023588678η2 − 0.0315158198η − 0.4396463038)e−0.8408529662η + (0.0208604015η + 0.0573797361)e−1.6817059324η + (0.001423055078η + 0.0040581045)e−2.5225588986η + (−0.0000368811η + 0.0001911237)e−3.3634118648η

(22.37)

Example 9 In this case we consider the axisymmetric flow with impermeable sheet, m = 2, α = 0 and with slip parameter λ = 10. From Eq. (22.28) yields

22.3 Numerical Examples

233

Table 22.3 Values of  (0) and (∞) in the case of stretching flow with impermeable sheet m = 2, α = 0 λ

1

5

10

num (0) O AF M (0)

−0.4625096500

−0.1493933452

−0.0829116367

−0.4625096441

−0.1493933439

−0.0829118109

num (∞)

0.5509502913

0.37801840311

0.31065126857

 O AF M (∞)

0.5509446952

0.3780173713

0.31063735490

K = 0.5239724222, δ = −1.8794890289, K 3 = 0.43 · 10−6 , C1 = 0.0112870692, C3 = 0.0366309932, C5 = −0.0310033957,

γ = 1.6648535124, K 1 = K 2 = −3.4855129921, K 4 = 0.31 · 10−6 , C2 = −0.0004699006, C4 = −0.0059152635, C6 = −0.00815982977

The first-order approximate solution can be written in the form (η) = 0.3106373548 + (−0.000610251η2 + 0.0246579539η − 0.317017534)e−0.5239724222η + (−0.0247341994η + 0.0442009306)e−1.0479448445η + (−0.0157948648η − 0.0156201635)e−1.5719172668η + (−0.0041191012η + 0.0222006138)e−2.095889689η

(22.38)

In Table 22.3 we present the values of  (0) and (∞) for the stretching flow with impermeable sheet. Example 10 For the axisymmetric flow with suction sheet m = 2, α = 3 and slip parameter λ = 1 we obtain the following values of the parameters: K = 1.5574274146, γ = −0.7860056759, δ = 3.6690514687, K 1 = K 2 = −0.0507748232, K 3 = 1.003 · 10−6 , K 4 = 0.32 · 10−6 , C1 = 0.01071333, C2 = −0.0024557002, C3 = −9.2340545611, C4 = 4.93780472001, C5 = −2.72171778619, C6 = 23.18498897742928 with the first-order approximate solution obtained from (2.3): (η) = 3.0235364880 + (0.0026339287η2 + 0.0162169332η − 0.0279371463)e−1.5574274146η

234

22 Viscous Flow Due to a Stretching Surface with Partial Slip

+ (0.0864947892η − 0.0510827083)e−3.1148548292η + (0.0937682076η − 0.0527720186)e−4.6722822438η + (0.0064922112η + 0.0027118480)e−6.2297096585η

(22.39)

Example 11 For the case of the axisymmetric flow with suction sheet m = 2, α = 3 and slip parameter λ = 5, the parameters are K = 1.7780169491, γ = −0.5957717503, δ = 2.8222930255, K 1 = K 2 = −0.0131160498, K 3 = 1.013 · 10−6 , K 4 = 1.22 · 10−6 , C1 = 0.00008624, C2 = −0.0000222658, C3 = −0.3512786202, C4 = 0.1994379874, C5 = −1.6374819712, C6 = 1.6614059805 and therefore, the first-order approximate solution (2.3) has the form (η) = 3.0053612258 + (−0.0001065840η2 + 0.0005858601η − 0.0008994531)e−1.7780169491η + (0.0035222493η − 0.0022439786)e−3.55603389829η + (0.0065444365η − 0.0021759449)e−5.3340508474η + (0.0000797662η − 0.0000397491)e−7.1120677965η

(22.40)

Example 12 In the last case we consider the axisymmetric flow with suction sheet m = 2, α = 3 and slip parameter λ = 10, such that the parameters are K = 1.8412120648, γ = −0.554362126, δ = 2.6330245292, K 1 = K 2 = −0.0066382243, K 3 = 1.004 · 10−6 , K 4 = 0.52 · 10−6 , C1 = 9.5831454966 · 10−6 , C2 = −2.5525898638 · 10−6 , C3 = −0.0821658733, C4 = 0.04718459612, C5 = −0.8534171847, C6 = 0.5288479009 The first-order approximate solution (2.3) can be written as (η) = 3.0027282856 + (−0.0000243991η2 + 0.0001301957η − 0.0001939768)e−1.8412120648η + (0.0008131418η − 0.0005329181)e−3.6824241297η + (0.0020278091η − 0.0019878153)e−5.3340508474η + (0.0000116973η − 0.0000135853)e−7.3648482595η

(22.41)

22.3 Numerical Examples

235

In Table 22.4 are presented the values of  (0) and (∞) for stretching flow with suction sheet. From Tables 22.1, 22.2, 22.3, 22.4 we deduced that there exist an excellent agreement between the numerical results and the results obtained by means of OAFM. On the other hand, considering the effect of slip parameter on the velocity (η) in both flows, Figs. 22.1, 22.2, 22.3, 22.4 have been displayed. It is shown in Fig. 22.1 the variation of  for planar flow and impermeable sheet. In Fig. 22.2 is plotted the variation of  for planar flow and suction sheet. In Fig. 22.3 shows the variation of  for axisymmetric flow and impermeable sheet. Figure 22.4 has been plotted the variation of  for axisymmetric flow and suction sheet. It is clear that the velocity components decreases with an increase in the slip parameter for all cases. In Figs. 22.5, 22.6 and 22.7 have been plotted the planar cases for every value of slip parameter λ and in Figs. 22.8, 22.9 and 22.10 have been plotted the stretching cases for different values of λ. It is evident that the velocity is less for the axisymmetric flow when compared with the planar case. Finally, the residual functions obtained for the approximate analytic solutions given by Eqs. (22.30)-(22.41) are plotted in Figs. 22.11, 22.12, 22.13, 22.14, 22.15, 22.16, 22.17, 22.18, 22.19, 22.20, 22.21 and 22.22.

Table 22.4 Values of  (0) and (∞) in the case of stretching flow with suction sheet m = 2, α =3 λ

1

5

10

num (0) O AF M (0)

−0.8578597370

−0.1935567447

0.0163824516

−0.8578597339

−0.1935567450

0.0163824517

num (∞)

3.0235285642

3.005309966

3.0027282376

 O AF M (∞)

3.0235359880

3.0053592258

3.0027282799

Fig. 22.1 Variation of  by increasing the slip parameter λ for planar flow and impermeable sheet (m = 1, α = 0) — numerical solution; …… OAFM solution

236 Fig. 22.2 Variation of  by increasing the slip parameter λ for planar flow and suction sheet (m = 1, α = 3):

Fig. 22.3 Variation of  by increasing the slip parameter λ for axisymmetric flow and impermeable sheet (m = 2, α = 0): — numerical solution; …… OAFM solution

Fig. 22.4 Variation of  by increasing the slip parameter λ for axisymmetric flow and suction sheet (m = 2, α = 3): — numerical solution; …… OAFM solution

Fig. 22.5 Variation of  by increasing the coefficient α for planar flow (m = 1, λ = 1): — numerical solution; …… OAFM solution

22 Viscous Flow Due to a Stretching Surface with Partial Slip

22.3 Numerical Examples Fig. 22.6 Variation of  by increasing the coefficient α for planar flow (m = 1, λ = 5) — numerical solution; …… OAFM solution

Fig. 22.7 Variation of  by increasing the coefficient α for planar flow (m = 1, λ = 10): — numerical solution; …… OAFM solution

Fig. 22.8 Variation of  by increasing the coefficient α for axisymmetric flow (m = 2, λ = 1): — numerical solution; …… OAFM solution

237

238 Fig. 22.9 Variation of  by increasing the coefficient α for planar flow (m = 2, λ = 5) — numerical solution; …… OAFM solution

Fig. 22.10 Variation of  by increasing the coefficient α for planar flow (m = 2, λ = 10) — numerical solution; …… OAFM solution

Fig. 22.11 The residual R(1,1,0,η) for Eq. (22.30) obtained by OAFM

22 Viscous Flow Due to a Stretching Surface with Partial Slip

22.3 Numerical Examples Fig. 22.12 The residual R(1,5,0,η) for Eq. (22.31) obtained by OAFM

Fig. 22.13 The residual R(1,10,0,η) for Eq. (22.32) obtained by OAFM

Fig. 22.14 The residual R(1,1,3,η) for Eq. (22.33) obtained by OAFM

Fig. 22.15 The residual R(1,5,3,η) for Eq. (22.34) obtained by OAFM

239

240 Fig. 22.16 The residual R(1,10,3,η) for Eq. (22.35) obtained by OAFM

Fig. 22.17 The residual R(2,1,0,η) for Eq. (22.36) obtained by OAFM

Fig. 22.18 The residual R(2,5,0,η) for Eq. (22.37) obtained by OAFM

22 Viscous Flow Due to a Stretching Surface with Partial Slip

22.3 Numerical Examples Fig. 22.19 The residual R(2,10,0,η) for Eq. (22.38) obtained by OAFM

Fig. 22.20 The residual R(2,1,3,η) for Eq. (22.39) obtained by OAFM

Fig. 22.21 The residual R(2,5,3,η) for Eq. (22.40) obtained by OAFM

Fig. 22.22 The residual R(2,10,3,η) for Eq. (22.41) obtained by OAFM

241

242

22 Viscous Flow Due to a Stretching Surface with Partial Slip

References 1. B.C. Sakiadis, Boundary layer behavior on continuous solid surfaces. I. Boundary layer equations for two dimensional and axisymmetric flow. AIChE J. 7, 26–28 (1961) 2. B.C. Sakiadis, Boundary layer behavior on continuous solid surfaces. II. Boundary layer on a continuous flat surface. AIChE J. 7, 221–225 (1961) 3. L.J. Crane, Flow past a stretching plate. Z. Angew Math. Phys. 21, 645–647 (1970) 4. E.M. Sparrow, G.S. Beavers, L.Y. Hung, Flow about a porous-surfaced rotating disk. Int. J. Heat Mass Trans. 14, 993–996 (1971) 5. E.M. Sparrow, G.S. Beavers, L.Y. Hung, Channel and tube flows with surface mass transfer and velocity slip. Phys. Fluids 14, 1312–1319 (1971) 6. J.A.D. Ackroyd, A series method for the solution of laminar boundary layers on moving surfaces. Z. Angew Math. Phys. 29, 729–741 (1978) 7. K.R. Rajagopal, T.Y. Na, A.S. Gupta, Flow of a viscoelastic fluid over a stretching sheet. Rheol. Acta 23(2), 213–215 (1984) 8. P.D. Ariel, MHD flow of a viscoelastic fluid past a stretching sheet with suction. Acta Mech. 105, 49–56 (1994) 9. H.I. Anderson, Slip flow of a viscoelastic fluid past a stretching surface. Acta Mech. 95, 227–230 (1992) 10. C.Y. Wang, Analysis of viscous flow due to a stretching sheet with surface slip and suction. Nonlinear Anal. Real World Appl. 10(1), 375–380 (2009) 11. H. Mirogolbabei, D.D. Ganji, M.M. Etghani, A. Sobati, Adapted varational iteration method and axisymmetric flow over a stretching sheet. World J. Modell. Simul. 5(4), 307–314 (2009) 12. M. Sajid, I. Ahmed, T. Hayat, M. Ayub, Series solution for unsteady axisymmetric flow and heat transfer over a radially stretching sheet. Commun. Nonlinear Sci. Numer. Simul. 13(10), 2193–2202 (2008) 13. M. Sajid, T. Hayat, S. Asghar, K. Vajravelu, Analytic solution for axisymmetric flow over a nonlinearity stretching sheet. Arch. Appl. Mech. 78(2), 127–134 (2007) 14. M. Miklavcic, C.Y. Wang, Viscous flow due to a shrinking sheet. Quart. Appl. Math. 64, 283–290 (2006) 15. K.V. Prasad, K. Vajravelu, P.S. Dutt, The effects of variable fluid properties on the hydromagnetic flow and heat transfer over a non-linearity stretching sheet. Int. J. Therm. Sci. 40, 603–610 (2010) 16. M. Turkyilmazoglu, Multiple solutions of heat and mass transfer of MHD slip flow for the viscoelastic fluid over a stretching sheet. Int. J. Therm. Sci. 50, 2264–2276 (2011) 17. F. Labropulu, D. Li, I. Pop, Non-orthogonal stagnation-point flow over a stretching surface in a non-Newtonian fluid with heat transfer. Int. J. Therm. Sci. 49, 1042–1050 (2010) 18. C.Y. Wang, C.O. Ng, Slip flow due to a stretching cylinder. Internat. J. Non-Linear Mech. 46, 1191–1194 (2011) 19. M. Sajid, K. Mahmood, Z. Abbas, Axisymmetric stagnation-point flow with a general slip boundary condition over a lubricated surface. Chin. Phys. Lett. 29(2), 307–310 (2012) 20. A. Nayfeh, Problems in Perturbation (Wiley, New York, 1985) 21. Y.K. Cheung, S.H. Chen, S.L. Lau, A modified Lindstedt-Poincare method for certain strongly nonlinear oscillations. Int. J. Nonlinear Mech. 26, 367–378 (1991) 22. G. Adomian, A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 135, 501–544 (1998) 23. F.O. Zengin, M.D. Kaya, S.A. Demirbag, Application of parameter-expantion method to nonlinear oscillators with discontinuities. Int. J. Nonlinear Sci. Numer. Simul. 9, 267–270 (2008) 24. N. Heri¸sanu, V. Marinca, Optimal homotopy perturbation method for non-conservative dynamical system of a rotating electrical machine. Z. Naturforsch A. 670, 509–516 (2012) 25. V. Marinca, N. Heri¸sanu, I. Neme¸s, Optimal homotopy asymptotic method with application to thin film flow. Cent. Eur. J. Phys. 6, 648–653 (2008)

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26. V. Marinca, R.D. Ene, B. Marinca, Analytic approximate solution for Falkner-Skan equation. Sci. World J. Article ID 617453 (2014) 27. H. Jafari, M. Ghorbani, E. Ebadattalab, R. Moallem, D. Baleanu, Optimal homotopy asymptotic method-a tool for solving fuzzy differential equations. J. Comput. Complex. Appl. 2(4), 112– 123 (2016) 28. X.-J. Yang, J.A. Tenreiro Machado, D. Baleanu, C. Cattani, On exact traveling-wave solutions for local fractional Korteweg-de Vries equation. Chaos 26, Article ID 084312 (2016) 29. X.J. Yang, J.A. Tenreiro Machado, J. Hristov, Nonlinear dynamics for local fractional Burgers’ equation arising in frctal flow. Nonlinear Dynam. 84, 3–7 (2016) 30. V. Marinca, R.D. Ene, B. Marinca, Optimal Auxiliary Functions Method for viscous plate due to a stretching surface with partial slip. Open Eng. 8, 261–274 (2018)

Chapter 23

Axisymmetric MHD Flow and Heat Transfer to Modified Second Grade Fluid

The flow of the non-Newtonian fluids and heat transfer to modified second grade fluid, has very important role for physicians, applied mathematicians and engineers, because of its several applications in different fields of science and engineering, such as cooling of polymer films, chemical reactions, solar energy collections, electric devices, drawing of cooper, hot rolling and so on. Interest in the flows of nonNewtonian axisymmetric flow with heat transfer has grown considerably in the past few decades. Crane [1] made pioneering research of exact analytic solution for two dimensional boundary layer flow due to stretching sheet. Afterwards, this problem has been extensively studied in light of different aspects of stretching velocities, non-Newtonian fluids, heat and mass transfer, MHD heat generation/absorption, suction/blowing, etc. Dunn and Fosdick [2] calculated relation between the stress, various non-Newtonian parameters and temperature profiles for an incompressible, homogeneous fluid of second grade/order: T = −pI + μA1 + α1 A2 + α2 A21

(23.1)

where T is the Cauchy stress tensor, p the pressure, I the identity tensor, μ the viscosity, α1 and α2 the normal stress coefficients, while A1 and A2 are the first and second Rivlin-Ericksen tensors, respectively defined as: A1 = L + LT ; A2 =

dA1 + A1 L + LT A1 dt

(23.2)

The normal stress coefficients α1 and α2 satisfy the restrictions α1 ≥ 0, α1 + α2 = 0

(23.3)

In the study of glacier flows, Man and Sun [3] used a generalization of Eq. (23.1) in two models, called “modified second order luids” and “the power law fluid of grade 2”. In this section, the first model is named modified/generalised second grade/order © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_23

245

246

23 Axisymmetric MHD Flow and Heat Transfer …

and is written in the form: T = −pI + μ
m/2 A1 + α1 A2 + α2 A21

(23.4)

where
= tr A21 and m is a real number. Man and Sun consider
= 21 tr A21 in which μ includes 21 . Follows that the model given by Eq. (23.4) is a combination of the classical power law viscoelastic fluid and a fluid of second grade/order: if α1 = α2 = 0, Eq. (23.4) reduces to the constitutive relation of a power law fluid, and if m = 0 it describes a second grade/order fluid. When m = 0 and α1 = α2 = 0 the fluid is Newtonian. If μ
m/2 is viewed as the „viscosity”, then Eq. (23.4) clearly exhibits shear thickening when m >0, i.e. the viscosity increases if increases velocity shear; when m 0 and s is a real power-law exponent parameter, then the stretching sheet is assumed to have velocity U = cr s with radial direction in its own plane. Also the steady two-dimensional axisymmetric flow is developed due to radially stretching sheet coinciding with the half space z > 0 of the vertical axis. If u and w are the radical and axial velocity components, then using the cylindrical coordinate system, we can write: V = [u(r, z), 0, w(r, z)]

(23.5)

The governing equations for the steady two-dimensional axisymmetric low and < 0, are [4, 15, heat transfer in the absence of pressure gradient and considering du dt 16]: ∂u u ∂w + + =0 ∂r r ∂z   ∂u m ∂ 2 u ∂u ∂u u +w = ν(m + 1) − ∂r ∂z ∂z ∂z2   ∂ 3u δB20 α1 ∂ 3 u ∂u ∂ 2 u ∂u ∂ 2 u u − + + w + − u ρ ∂r∂z2 ∂z3 ∂r ∂z2 ∂z ∂r∂z ρ

(23.6)

(23.7)

248

23 Axisymmetric MHD Flow and Heat Transfer …

    ∂T ∂T ∂T ∂ ∂q +w ρcp u = k1 (T) − r + Q0 (T − T∞ ) ∂r ∂z ∂z ∂z ∂z

(23.8)

in which ν is the kinematic viscosity, δ is the electrical conductivity, B0 is the strength of the magnetic field, ρ is the density, cp is the specific heat of field, k1 (T) is the thermal variable conductivity, qr is the radiation heat flux, Q0 is the uniform volumetric heat generation (Q0 > 0) or heat absorption (Q0 < 0) and T∞ is the ambient temperature. Rosseland approximation for radiation applied to the radiation heat flux lead to the relation: qr = −

4δ1 ∂T4 3k∗ ∂z

(23.9)

where δ1 and k∗ are Stefan-Boltzmann constant and the Rosseland mean absorption coefficient, respectively. It is know that: T4 ≈ 4T3∞ T − 3T4∞

(23.10)

such that from Eqs. (23.9) and (23.10) follows that: 16δ1 T3∞ ∂ 2 T ∂qr =− · 2 ∂z 3k∗ ∂z

(23.11)

The boundary conditions for the problem are: u(r, z) = U = crs , w(r, z) = 0 at z = 0 ∂u(r, z) → 0 as z → ∞ u(r, z) → 0, ∂z T = Tw at z = 0 T → T∞ as z → ∞

(23.12)

where s is a real number. In order to non-dimensionalize the problem, let us introduce the similarity transformations: z T − T∞ 1 1 η = Re m+2 , m = −2, ψ = −r2 URe− m+2 f(η), θ(η) = r Tw − T∞

(23.13)

where f(η) is the dimensionless stream function and Re is the local Reynolds number given by: Re =

ρ m+1 1−m r U u

(23.14)

23.1 The Governing Equations

249

The thermal conductivity of the fluid is of the form.  k1 (T) = k∞

T − T∞ 1+a Tw − T∞

 (23.15)

where a is thermal variable conductivity parameter and k∞ is the thermal conductivity of the fluid far away the surface sheet. From Eqs. (23.13) and (23.15) we gain: k1 (T) = k∞ (1 + aθ)

(23.16)

The stream function ψ is defined as: u=−

1 ∂ψ 1 ∂ψ ,w = r ∂z r ∂r

(23.17)

so that the continuity Eq. (23.6) is automatically satisfied. From Eqs. (23.13) and (23.17), the present problem can be expressed as: m    [m + 3 + s(2m + 1)]ff + m2 + 3m + 2 f −f − M2 f   − s(m + 2)f2 + k(3s − 1) 2f f − f2 − k[m + 3 + s(2m + 1)]ffIV (23.18) θ (1 + R + a θ) + a θ2 + Pr

m + 3 + s(2m + 1)  f θ + Q θ = 0, m = −2 m+2 (23.19)

where the dimensionless variabless k, M2 , R, Pr and Q represent respectively, generalized second order parameter, the generalized magnetic field parameter, the radiation variable, the generalized Prandtl number and heat generation/absorption parameter and the primes denotes the differentiation with respect to η. The above parameters are defined as: 2

16δ1 T3∞ − 2 δB20 r α1 Re m+2 , R= , M2 = Re m+2 , 2 ρr ρU 3k∗ k∞ ρCp Ur − 2 Q U2 2 Re m+2 , Q = 0 r Re− m+2 Pr = k∞ ρCp k∞

k=

(23.20)

The boundary conditions (23.12), become: f(0) = 0, f (0) = 1, f (∞) = 0, f (∞) = 0

(23.21)

θ(0) = 1, θ(∞) = 0

(23.22)

250

23 Axisymmetric MHD Flow and Heat Transfer …

23.2 OAFM for Solving the System (23.18), (23.19), (23.21), (23.22) We remark that Eq. (23.18) is decoupled from Eq. (23.19). At first we apply OAFM for Eq. (23.18) and (23.21). For Eqs. (23.18) and (23.19) the approximate analytical solutions are of the forms: f(η) = f0 (η) + f1 (η, Ci ), i = 1, 2, . . . , q

(23.23)

  θ(η) = θ0 (η) + θ1 η, Cj , j = q + 1, q + 2, . . . , p

(23.24)

It is known that linear operators Lf and Lθ and the initial approximations f0 (η) and θ0 (η) are not unique. In what follows we present only three possibilities to choose the linear operators and the initial approximations for Eq. (23.18). 23.2.a Case 1 f0 (η) =

1 − e−λη λ

(23.25)

where λ is an unknown parameters at this moment, with λ > 0. For the initial approximation given by Eq. (23.25), the initial and boundary conditions (23.21) are fulfilled. In this case, the linear operator is written as: L(f) = fIV (η) + λf (η)

(23.26)

From Eqs. (23.18) and (23.26), the nonlinear operator N(f) is given by (Ff = 0):  2  m m + 3m + 2 −f − λ f   + M2 f − s(m + 2)f2 + k(3s − 1) 2f f − f2

N(f) = [m + 3 + s(2m + 1)]ff +

− {1 + kf[m + 3 + s(2m + 1)]}fIV

(23.27)

23.2.b Case 2 For the initial approximation: f0 (η) = ln(1 + η)

(23.28)

the conditions (23.21) are also fulfilled, and the linear operator becomes: L(f) = fIV +

3  f η+1

(23.29)

23.2 OAFM for Solving the System (23.18), (23.19), (23.21), (23.22)

251

23.2.c Case 3 In the last case, we consider initial approximation in the form: f0 (η) = 2



1+η−1

(23.30)

and the linear operator as: L(f) = fIV +

5 f 2(η + 1)

(23.31)

From these three possibilities to choose the initial approximation and the linear operator, in what follows we will consider only the case 23.2.a and therefore initial approximation f0 is given by Eq. (23.25) and linear operator is given by Eq. (23.26). For Eqs. (23.19) and (23.22) the initial approximation and linear operator, are respectively: θ0 (η) = e−λη

(23.32)

L(θ) = θ + λθ

(23.33)

The corresponding nonlinear operator for Eq. (23.19) is: N(θ) = θ (R + a θ) − λθ + a θ2 + Pr

m + 3 + s(2m + 1)  fθ +Qθ m+2

(23.34)

In order to choose the auxiliary functions ai we will determine N(f0 ) and N(θ0 ) by substituting Eq. (23.25) into Eq. (23.27) and then substituting Eq. (23.32) into Eq. (23.34), which means that:  

 2 kλ − 1 [m + 3 + s(2m + 1)] − M2 e−λη 

  + 1 − kλ2 [m + 3 + s(2m + 1)] + kλ2 (3s − 1) − s(m + 2) e−2λη   + m2 + 3m + 2 λm+2 e−(m+1)λη (23.35)

N(f0 ) =

  m + 3 + s(2m + 1) −λη 2 N(θ0 ) = (1 + R)λ + Q − Pr e m+2   m + 3 + s(2m + 1) −2λη e + 2aλ2 + Pr m+2

(23.36)

Taking into account the considerations presented in Chapter 2 and expressions of f0 , θ0 , N(f0 ) and N(θ0 ) given in Eqs. (23.25), (23.32), (23.35) and (23.36) respectively, let us now consider the auxiliary functions as:

252

23 Axisymmetric MHD Flow and Heat Transfer …

Fi (η) ∈ α1 ηe−λη , α2 ηe−1.5λη , α3 ηe−2λη , α4 ηe−λη , α5 e−1.5λη, α6 e−2λη , (α7 η + α8 )e−λη , (α9 η + α10 )e−1.5λη , . . . where αi are real numbers i = 1, 2, . . .. It is clear that we can consider different possibilities to choose these auxiliary functions, as follows. 23.2.1. Case 1 In the first case, we choose four parameters Ci for the first approximation f1 (η,  Ci ),  i = 1, 2, 3, 4 and only two parameters for the first approximation θ1 η, Cj , j = 5, 6, such that Eq. (2.13) can be written for the f1 and θ1 in the forms:  −λη − C2 e−λη − C3 e−1.5λη − C4 e−2λη = 0, fIV 1 + λf1 − C1 ηe f1 (0) = f1 (0) = f1 (∞) = f1 (∞) = 0

θ1 + λθ1 − C5 ηe−λη − C6 ηe−1,5λη = 0, θ1 (0) = θ1 (∞) = 0

(23.37) (23.38)

Again we emphasize that the expression given by Eqs. (23.37) and (23.38) are not unique. Solving Eqs. (23.37) and (23.38), then yields the solutions: C2 8C3 C4 3C1 + 4 + + 4 5 4 λ λ 27λ 8λ  3C1 C1 2 3C1 C2 C2 8C3 C4 −λη η + e − η + + + + + 2λ3 λ4 λ3 λ5 λ4 9λ4 4λ4 16C3 1.5λη C4 + e + 4 e−2λη (23.39) 4 27λ 8λ     32C6 −λη 4C6 32C6 −1.5λη 3C5 2 C5 e e η + 2η+ θ1 (η) = − + η + (23.40) 2λ λ 9λ3 3λ2 9λ3 f1 (η) =

The approximate analytic solution of Eqs.(23.18), (23.19), (23.21) and (23.22) is obtained from Eqs. (23.25), (23.39), (23.23), (23.32), (23.40) and (23.24): 3C1 1 C2 8C3 C4 + + 4 + + 4 4 λ λ5 λ 27λ 8λ    3C1 C1 2 3C1 C2 C2 8C3 C4 1 −λη η + e − η + + + + + + 2λ3 λ4 λ3 λ5 λ4 9λ4 4λ4 λ 16C3 −1.5λη C4 + e + 4 e−2λη (23.41) 4 27λ 8λ

f(η) =

23.2 OAFM for Solving the System (23.18), (23.19), (23.21), (23.22)

253

    3C5 2 C5 32C6 −λη 4C6 32C6 −1.5λη e e η − 2η− θ(η) = 1 − + η + (23.42) 2λ λ 9λ3 3λ2 9λ3 23.2.2. Case 2 In the second case, we consider five parameters Ci for the first approximation  f1 (η,  Ci ), i = 1, 2, . . . , 5 and three parameters for the first approximation θ1 η, Cj , j = 6, 7, 8: −λη f1I V + λf − C2 e−λη − C3 ηe−1.5λη − C4 e−2λη − C5 e−2λη = 0, 1 − C 1 ηe f1 (0) = f1 (0) = f1 (∞) = f1 (∞) = 0 (23.43)

θ1 + λθ1 − C6 ηe−λη − C7 e−λη − C8 e−1.5λη = 0, θ1 (0) = θ1 (∞) = 0

(23.44)

In this case the approximate analytic solution of Eqs. (23.18), (23.19), (23.21) and (23.22) can be written as: 3C1 1 C2 16C3 8C4 C5 + 5 + 4 + + + 4 4 4 λ λ λ 27λ  27λ 8λ   3C1 1 C1 2 3C1 C2 C2 80C3 8C4 C5 −λη + η + e − η + + + + + + 2λ3 λ4 λ3 λ λ4 27λ4 9λ4 4λ4 λ5   16C3 64C3 16C4 −1.5λη C5 e + η + + + 4 e−2λη (23.45) 4 4 4 27λ 27λ 27λ 8λ     4C8 −λη 4C8 −1.5λη C7 2 C7 C6 e η − η − θ(η) = 1 − + + 2e (23.46) 2λ λ2 λ 3λ2 3λ

f(η) =

23.2.3. Case 3 In this last case, we consider five parameters for f1 and four parameters for θ1 . The approximate analytic solution of Eq. (23.18) and (23.21) is given by Eq. (23.45). The first approximate θ1 is obtained from equation: θ1 + λθ1 − C6 ηe−λη − C7 e−λη − C8 ηe−1.5λη − C9 e−1.5λη = 0, θ1 (a) = θ1 (∞) = 0

(23.47)

The approximate solution of Eqs.(23.19) and (23.22) is:     32C8 C7 2 C7 C6 4C9 −λη e η − η − θ(η) = 1 − + − 2λ λ2 λ 9λ2 3λ2   4C8 32C8 4C9 −1.5λη e + + + 3λ2 9λ3 3λ2

(23.48)

254

23 Axisymmetric MHD Flow and Heat Transfer …

23.3 Numerical Examples We illustrate the accuracy of our technique for different values of the parameters which are involved in Eqs. (23.18) and (23.19) and for every auxiliary functions which appear in the case 23.2.1–23.2.3. Also we will show that the error of the solutions decreases when the number of terms in the auxiliary functions increases. The results obtained using OAFM are compared with the exact solutions given in [4]. The optimal convergence control parameters are determined by collocation method.

23.3.1 Example 1 First, we consider: M2 = 1, m = 1, k = 1, R = 0.5, Pr = 1, a = 2.61735467356482, s = 7.9619625998 and Q = 8.05319889015817

(23.49)

The values of λ and optimal convergence control parameters, corresponding to the 23.2.1, are: λ = 0.9751236487, C1 = 9.29451604192024 · 10−4 , C2 = −5.57358300359673 · 10−3 , C3 = 9.03173290247 · 10−3 , C4 = 3.31828999833 · 10−3 , C5 = −1.11532022264632 · 10−4 , C6 = −1.5759670385173 · 10−4

(23.50)

The approximate analytical solution for the problem (23.18), (23.19), (23.21) and (23.22) yields: f(η) = 1.025926647162

− 5.010459062912 · 10−4 η2 − 2.928151667764 · 10−3 η + 1.0323049503196 e−0.9751236487η + 5.9195480180038 · 10−3 e−1.46268547305η + 4.587543854078 · 10−4 e−1.9502472974η

(23.51)

θ(η) = 5.7188656235197 · 10−5 η2 + 1.172951887926 · 10−4 η + 1.0006043321088 e−0.9751236487η

− 2.20986949425986 · 10−4 η + 3.833451594 · 10−4 e−1.46268547305η (23.52)

In Table 23.1 we present a comparison between the approximate solution given by Eqs. (23.51) and (23.52) and exact solution presented in [4] for some values of η and the corresponding relative errors. The values of λ and optimal convergence control parameters corresponding to the case 23.2.2 are:

f(η) Equation (23.51)

0.638890069538

0.879923046334

0.970827901642

1.005120477735

1.018063893479

1.022952769939

1.024800861830

1.025500086721

1.025768764140

1.025865256751

η

1

2

3

4

5

6

7

8

9

10

1.026073220831

1.025974062574

1.025711300391

1.025034990687

1.023169853383

1.018280349664

1.005323519884

0.970988861723

0.880045109030

0.638902588027

fex (η) [4]

2.1E-04

2.1E-04

2.1E-04

2.3E-04

2.2E-04

2.2E-04

2.0E-04

1.6E-06

8.1E-05

1.3E-05

Error    f ex − f 

0.020279820517 7.652972161·10–3 2.887993159·10–3 1.089838603·10–3 4.11271197·10–4 1.552009018·10–4 5.856798259·10–5

7.656116071·10–3 2.890079877·10–3 1.091085646·10–3 4.119611299·10–4 1.55561561·10–4 5.874853106·10–5

0.053740051723

0.142407234741

0.377368831169

θex (η) [4]

0.020289976707

0.053745250647

0.142417457522

0.377410984151

θ(η) Equation (23.52)

4.8E-07

7.6E-07

6.9E-07

1.2E-06

2.1E-06

3.1E-06

4.2E-06

5.2E-06

1.0E-05

4.2E-05

Error   θex − θ

Table 23.1 Comparison between the exact solution [4] and approximate solution (23.51) and (23.52) for M2 = 1, m = 1, k = 1.5, R = 0.5, Pr = 1, a = 2.61735467356482, s = 7.9619625998, Q = 8.05319889015817, corresponding of the Case 23.2.1

23.3 Numerical Examples 255

256

23 Axisymmetric MHD Flow and Heat Transfer …

λ = 0.9748412451, C1 = 0.011080109805234, C2 = −0.089041390275063, C3 = 0.0933108796403789, C4 = −0.07589610096100960289, C5 = 0.168676097352772, C6 = −3.0113732789279 · 10−4 , C7 = −9.3182942157916 · 10−8 C8 = −1.5239950961041 · 10−10

(23.53)

such that the approximate solution of Eq. (23.18), (23.19), (23.21) and (23.22) can be written as: f(η) = 1.0262243829

− 0.0051323439785135η2 − 0.05930785333023η + 1.25100433028015 e−0.9748412451η + (0.0612284010985751η + 0.201433114511259)e−1.462261868η + 0.02334683287e−1.9496824902η (23.54)

θ(η) = 4.779390625207 · 10−8 η2 + 3.0905096923 · 10−4 η + 1.00000000021382 e−0.9748412451η − 2.138230474 · 10−8 e−1.94968249024η

(23.55)

In Table 23.2, we present a comparison between the approximate solution given by Eqs. (23.54) and (23.55) and exact solution presented in [4], for some values of η and the corresponding relative errors. In the second example we choose: M2 = 1, m = 1, k = 1, R = 0.5, Pr = 1, a = 1.56404819650345, s = 1.747990665689 and Q = 1.9731453936482 (23.56) The approximate solution for the problem (23.18), (23.19), (23.21) and (23.22) in the case 23.2.1 is: f(η) = 1.1636043277 + (1.71132200250175 · 10−3 η2 − 0.012597780424491η − 1.12702714860682)e−0.86003571798η − 0.045580780936037e−1.2900527897η − 0.0187602726814719e−1.7200703596η

(23.57)

θ(η) = (6.538227701010977 · 10−5 η2 + 8.318377788029 · 10−5 η + 1.00083097916049)e−0.860051798η − (2.67941065609811 · 10−4 η + 8.3079416049 · 10−4 )e−1.2900527697η

(23.58)

The approximate solution for the problem (23.18), (23.19), (23.21) and (23.22) in the case 23.2.2 is written as:

f(η) Equation (23.54)

0.638902448711

0.888000045728

0.970988813632

1.005323611734

1.018291133021

1.023199581431

1.02506454638

1.025776793688

1.026050485252

1.026156365151

η

1

2

3

4

5

6

7

8

9

10

1.026073220831

1.025974062574

1.025711300391

1.025034990687

1.023169853383

1.018280349664

1.005323519884

0.970988861723

0.8800451090306

0.638902588027

fex (η) [4]

8.3E-04

7.6E-05

6.6E-05

5.0E-05

3.0E-05

1.1E-05

9.2E-08

4.8E-08

6.2E-08

1.4E-07

Error    f ex − f 

0.020279820517 7.652972161·10–3 2.887993159·10–3 1.089838603·10–3 4.11271197·10–4 1.552009018·10–4 5.856798259·10–5

7.652973836·10–3 2.8879939229·10–3 1.0898389365·10–3 4.1127126349·10–4 1.5520096269·10–4 5.8568008506·10–5

0.053740051723

0.142407234741

0.377368831169

θex (η) [4]

0.020279824058

0.053740058781

0.142407247281

0.37736884772

θ(η) Equation (23.55)

2.6E-11

6.1E-11

1.4E-10

3.3E-10

7.6E-10

1.7E-10

3.6E-09

7.1E-09

1.2E-08

1.6E-08

Error   θex − θ

Table 23.2 Comparison between the exact solution [4] and approximate solution (23.54) and (23.55) for the values of constants like in Table 23.1 corresponding to the case 23.2.2

23.3 Numerical Examples 257

258

23 Axisymmetric MHD Flow and Heat Transfer …

Table 23.3 Comparison between exact solution [4] and approximate solution (23.57) (with four parameters) and approximate solution (23.59) (with five parameters) for M2 = 1, m = 1, k = 1.5, R = 0.5, Pr = 1, a = −1.56404819650345, s = 1.747990665689, Q = 1.9731453936482 η

fex (η) [4]

1

0.670878108664 0.671272538762 3.9E-04

0.670813158321 6.5E-05

2

0.954902751037 0.955355614753 4.5E-04

0.954899531371 3.2E-06

3

1.075148137544 1.075018561856 4.7E-04

1.075171346413 2.3E-05

4

1.126055526619 1.126492148966 4.3E-04

1.126080766420 2.5E-05

5

1.147607806817 1.147970400898 3.6E-04

1.147636746112 2.9E-05

6

1.156732234334 1.157034867689 3.0E-04

1.156770296123 3.8E-05

7

1.160595174903 1.16085077276

2.6E-04

1.160643781402 4.8E-05

8

1.162230599131 1.162459400749 2.2E-04

1.162288084712 5.7E-05

9

1.162922976444 1.163124710042 2.0E-04

1.162986751123 6.4E-05

10 1.163216103043 1.163405105182 2.0E-04

1.163283876512 6.8E-05

f(η) Error f(η)   Equation (23.57) fex (η) − f(η) for Equation (23.59) Eq. (23.57)

Error   fex (η) − f(η) for Eq. (23.59)

f(η) = 1.1635042378 − −(3.123154708587 · 10−4 η2 + 2.355924618695 · 10−3 η +1.1388080057752)e−0.8599135721η − (0.0160931189312η + 0.007964939814)e−1.28987035815η − 0.016731292232e−1.7198271442η (23.59) θ(η) = (7.50313104217 · 10

−8 2

η + 3.87110804841345 · 10 −0.8599135721η

+ 1.0000000005132268)e

−4

η

− 5.1322614321 · 10−11 e−1.28987035815η

(23.60) In Table 23.3 we compare the approximate solutions (23.57) and (23.59) with exact solution given in [4] for some values of η and corresponding errors and in Table 23.4 we compare the approximate solutions (23.58) and (23.60) with exact solution given in [4].

23.3.2 Example 2 In the second example we consider: M2 = 1, m = 1, k = 2, R = 0.5, Pr = 1, a = −1.0368823909487, s = 0.326771423799 and Q = 0.75951185089871 (23.61) The approximate solution for the problem in study in the case 23.2.3 is defined as:

23.3 Numerical Examples

259

Table 23.4 Comparison between exact solution [4] and approximate solution (23.58) (with two parameters) and approximate solution (23.60) (with three parameters) for some value of the constants like in Table 23.3 η

θex (η) [4]

θ(η) Equation (23.58)

Error θ(η)   θex − θ Equation (23.60) for Eq. (23.58)

Error   θex − θ for Eq. (23.60)

1

0.42336251355

0.42336401198

2.1E-06

0.42336251365

9.5E-11

2

0.179235817884

0.179235100443

7.1E-06

0.1792358179

1.6E-11

3

0.0758817163788

0.075819355896

6.2E-05

0.075881726382

3.0E-12

4

0.0321253784127

0.032129053512

3.6E-06

0.032125478413

9.7E-12

5

0.0136007232899

0.013604860877

4.4E-06

0.0136007231899

0

6

5.758036398·10–3

5.7615535051·10–3

3.5E-06

5.758036397475·10–3

0

7

2.4377367627·10–3 2.440269938·10–3

2.5E-06

2.437736762·10–3

0

8

1.0320463632·10–3 1.03369054917·10–3 1.6E-06

1.032046362·10–3

0

9

4.369297424·10–4

6.8E-13

10 1.84979674·10–4

4.3792357967·10–4

9.9E-07

4.369297417654·10–4

1.855272471·10–4

5.5E-07

1.849796735308·10–4 4.8E-13

f(η) = 1.2906172743 − (1.966068103341 · 10−3 η2 + 0.026509752606η +1.414767473335179)e−0.7752201456η + (0.0252574737597η + 0.1134232599204)e−1.128302184η + 0.01072693913138e−1.5504402912η (23.62) θ(η) = (5.996659364 · 10−8 η2 + 3.459729463454 · 10−4 η + 0.9999999904822)e−0.7752201456η + (4.693096867 · 10−9 η + 9.517761409 · 10−9 )e−1.128302184η

(23.63)

In Table 23.5 we compare the approximate solution (23.62) and (23.63) with exact given in [4], for some values of η and corresponding errors.

23.3.3 Example 3 In the last case, we consider: 4 M2 = 0, s = − , k = 3, a = 0 3

(23.64)

It is interesting that we obtain two solutions in the case 4.21.2.3 for f and these are:

fex (η) [4]

0.695904322677

1.016550473487

1.164291981447

1.232305629359

1.263731368101

1.278183503355

1.284842495875

1.287910705177

1.289324049536

1.289975615637

η

1

2

3

4

5

6

7

8

9

10

1.290042014981

1.289380884161

1.287954694438

1.284872022072

1.278198999886

1.263736683567

1.232364736798

1.164291712551

1.016550488327

0.695904284142

f(η) Equation (23.62)

6.6E-05

5.7E-05

4.4E-05

2.9E-05

1.5E-05

5.3E-06

8.9E-07

8.5E-07

1.8E-08

4.2E-08

Error    f ex − f 

0.020767399241 9.568823293·10–3 4.4089472789·10–3 2.031474527·10–3 9.36025763·10–4 4.312848716·10–4

9.568833104·10–3 4.408953074·10–3 2.031477297·10–3 9.36025763·10–4 4.312856084·10–5

0.045071881723

0.097820362389

0.212301393661

0.460761756993

θ(η) Equation (23.63)

0.020767416989

0.045071912502

0.097820412510

0.212301466087

0.460761832281

θex (η) [4]

7.4E-10

1.4E-09

2.8E-09

5.5E-09

9.8E-09

1.8E-08

3.1E-08

5.0E-08

7.2E-08

7.6E-08

Error   θex − θ

Table 23.5 Comparison between exact solution [4] and approximate solution (23.62) and (23.63) for the case 23.2.3 for M2 = 1, m = 1, k = 2, R = 0.5, Pr = 1, a = −1.0368823909487, s = 0.326771423799, Q = 0.75951185089811

260 23 Axisymmetric MHD Flow and Heat Transfer …

23.3 Numerical Examples

261

f(η) = 1.6923505011 − (0.00206679968564η2 − 0.036863583142η +1.914368455858)e−0.5910065471η + (0.0339322164859η + 0.0203815389827)e−0.88650982065η + 0.018202564931e−1.182013094η (23.65) f (η) = 0.4197300603066 − (0.0201956307010988η2 − 0.109074977019532η +0.596986257385764)e−2.3826374712η + 0.022423423871865e−4.7652749424η (23.66) For Q = −0.5238185317005 and Q = −8.0514877494588, the approximate solutions θ(η) are, respectively:

θ(η) = 1.046627495766 · 10−6 η2 + 4.7282721371291 · 10−5 η+1.00009548180792

e−0.5910065471η − 1.09246686714 · 10−5 η + 9.548180792034 · 10−5 e−0.88650982η + 0.018202564931e−1.182013094η (23.67)

θ(η) = 8.058122032 · 10−7 η2 + 7.2571191548738 · 10−5 η+1.00001275929671

e−2.3826374712η − 1.080644888896231 · 10−5 η + 1.275929671 · 10−5 e−3.573956118η

(23.68)   In Tables 23.6 and 23.7 we compare the first approximate solution

f, θ given by

Eqs. (23.65) and (23.67) and the second approximate solution f, θ given by Eqs. (23.66) and (23.68) with exact solutions given in [4]: Form Tables 23.1, 23.2, 23.3, 23.4, 23.5, 23.6 and 23.7 we can conclude that the analytical solutions obtained by our approach are very accurate comparing to exact solutions. If we compare the results presented in Tables 23.1 and 23.2 and then the results presented in Table 23.3 and Table 23.4, we deduce that the analytical results obtained by OAFM are more accurate along with an increased number of terms in the auxiliary functions. In Table 23.8 we present a comparison between the values of f(0) and θ(0) obtained using our procedure from Eqs. (23.54), (23.55), (23.59), (23.60), (23.65), (23.66), (23.67), (23.68) and exact results. Are carried out that the results obtained using OAFM are nearly identical with those obtained from exact solutions.

fex (η) [4]

0.755056607087

1.173211031881

1.404787310239

1.533035346390

1.604039856923

1.643393580591

1.665176861337

1.677240541244

1.683921472232

1.687621407812

η

1

2

3

4

5

6

7

8

9

10

k = 3, a = 0, Q = −0.5238185317005

1.687674784738

1.683955039191

1.677255412884

1.665177094833

1.643385908671

1.604051738312

1.533031577193

1.404787189412

1.173211076693

0.755056616124

f(η) Equation (23.65)

5.3E-05

3.4E-05

1.5E-05

2.3E-07

7.7E-06

8.1E-06

3.7E-06

2.0E-08

4.5E-08

9.1E-09

Error    f ex − f 

0.015977242414 8.848308358·10–3 4.900260761·10–3 2.713832923·10–3

8.84825184·10–3 4.90020963·10–3 2.713762543·10–3

0.028849868396

0.052093818641

0.094065160099

0.169852347214

0.306700488533

0.553803397964

θ(η) Equation (23.67)

0.015977185984

0.028849819896

0.052093786033

0.094065146784

0.169852347339

0.306700418625

0.553805397793

θex (η) [4]

7.0E-08

5.1E-08

5.7E-08

5.6E-08

4.8E-08

3.3E-08

1.4E-08

1.3E-10

8.7E-11

1.7E-10

Error   θex − θ

  Table 23.6 Comparison between exact solution [18] and approximate solution f, θ given by Eqs. (23.65) and (23.67) in the case 4.3 for M2 = 0, s = − 43 ,

262 23 Axisymmetric MHD Flow and Heat Transfer …

fex (η) [4]

0.380971090898

0.416140090332

0.419386684498

0.419644249197

0.419714058005

0.419716612079

0.419716847874

0.419716869612

0.419716871339

0.419716871826

η

1

2

3

4

5

6

7

8

9

10

Q = −8.0514877494588

0.419730080238

0.419730059697

0.419730054947

0.419730013517

0.419729645801

0.419726344811

0.419695319833

0.419386884587

0.416139774958

0.380972176771

f(η) Equation (23.66)

1.3E-05

1.3E-05

1.3E-05

1.3E-05

1.3E-05

1.2E-05

5.1E-07

2.0E-07

3.1E-07

1.1E-06

Error    f ex − f 

0.092314089719 8.521891099·10–3 7.866902395·10–4 7.262259384·10–5 6.704096191·10–6 6.18884008·10–7 5.71319556·10–8 5.274114263·10–9 4.868785852·10–10 4.494615061·10–11

8.521891145·10–3 7.866906231·10–4 7.262262869·10–5 6.704091855·10–6 6.188821364·10–7 5.713154102·10–8 5.274040199·10–9 4.868785852·10–10 4.494484760·10–11

θ(η) Equation (23.68)

0.092314089637

θex (η) [4]

1.3E-15

9.8E-15

6.8E-14

4.1E-13

2.0E-12

4.3E-12

3.5E-11

3.8E-10

4.6E-11

8.2E-11

Error   θex − θ

Table 23.7 Comparison between exact solution [4] and approximate solution f, θ given by Eqs.(23.66) and (23.68) for M2 = 0,s = − 43 ,k = 3, a = 0,

23.3 Numerical Examples 263

264

23 Axisymmetric MHD Flow and Heat Transfer … 



Table 23.8 Comparison between f (0) and θ (0) with exact results fex (0) and θex (0) fex (0)



f (0)

θex (0)



θ (0)

Equations (23.54) −0.97453223774 −0.9744551528 −0.974532237 and (23.55)

−0.97453219

Equations (23.59) −0.8595264609 and (23.60)

−0.8595264609

−0.85932646

Equations (23.62) −0.77487400228 −0.774841042 and (23.63)

−0.77487400228

−0.774874165

Equations (23.65) −1.59094192146 −0.590926417 and (23.67)

−0.59094192146

−0.59093988

Equations (23.66) −2.38255849865 −2.38257982 and (23.68)

−2.38255849865–6 −2.382580505

−0.858943269

References 1. L.J. Crane, Flow past a stretching plate. Z. Angew Math. Phys. 21, 645–647 (1970) 2. J.E. Dunn, R.L. Fosdick, Thermodynamics, stability and boundness of fluids of complexity 2 and fluids of second grade. Arch. Rational Mec. Anal. 56, 191–252 (1974) 3. C.S. Man, Q.K. Sun, On the significance of normal stress effects in the flow of glacier. J. Glaciol. 33, 268–273 (1987) 4. B. Marinca, V. Marinca, Some exact solution for MHD flow and heat transfer to modified second grade fluid with variable thermal conductivity in the presence of thermal radiation and heat generation/absorption. Comput. Math Appl. 76, 1515–1524 (2018) 5. G. Gupta, M. Massoudi, Flow of a generalized second grade fluid between heated plates. Acta Mech. 99(1–4), 21–33 (1993) 6. P.D. Ariel, Axisymmetric flow of a second grade fluid past a stretching sheet. Int. J. Eng Sci. 39(5), 529–553 (2001) 7. M. Massoudi, C.E. Maneschy, Numerical solution to the flow of a second grade fluid over the stretching sheet using the method of quasi-linearization. Appl. Math. Comput. 149, 165–173 (2004) 8. F. Carapan, Axisymmetric motion of a generalized Rivlin-Ericksen fluids with shear-depent normal stress coefficients. Math. Models Methods Appl. Sci. 2(2), 168–176 (2008) 9. M. Massoudi, A. Vaidya, R. Wulandana, Natural convection flow of a generalized second grade fluid between two vertical walls. Nonlinear Anal. Real World Appl. 9, 80–93 (2008) 10. A. Keҫeba¸s, M. Yürüsoy, Numerical solutions of unsteady boundary layer equations for a generaliyed second grade fluid. J. Theor Appl. Mech. 49, 71–82 (2011) 11. M. Khan, A. Shahzad, On axisymmetric flow of Sisko fluid over a radially stretching sheet. Int. J. Nonlinear Mech. 47(9), 999–1007 (2012) 12. M. Nawaz, A. Alsaedi, T. Hayat, T. Alhothauli, Dufour and Soret effects in an axisymmetric stagnation flow of second grade fluid with newtonian heating. J. Mech. 29(1), 27–34 (2013) 13. H. Zaman, A. Sohail, A. Ali, T. Abbas, Effects of Hall current and flow of unsteady MHD axisymmetric second grade fluid with suction and blowing over an exponentially stretching sheet. Open J. Modell Simul. 2, 23–33 (2014) 14. J. Ahmed, A. Begun, A. Shahzad, R. Ali, MHD axisymmetric flow of power-law fluid over an unsteady stretching sheet with convective boundary conditions. Results in Phys 6, 973–981 (2016) 15. M. Awais, M.Y. Malik, S. Bilal, T. Salahudin, A. Hussain, Magnetohydrodynamic (MHD) flow od Sisko fluid near the axisymmetric stagnation point towards a stretching cylinder. Results Phys 7, 49–56 (2017)

References

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16. M. Khan, M. Rahman, M. Manzur, Axisymmetric flow and heat transfer to modified second grade fluid over a radially stretching sheet. Results Phys 7, 878–889 (2017) 17. D. Lu, M. Ramzan, N. Huda, J.D. Chung, U. Farooq, Nonlinear radiation effect on MHD Carreau nanofluid flow over a radially stretching surface with zero mass flux at the surface. Sci. Rep. 8, Art. 3709 (2018) 18. K.M. Al-Hussain, Dynamic stability of two rigid rotors connected by a flexible coupling with angular misalignment. J. Sound Vib. 266, 217–234 (2003)

Chapter 24

Thin Film Flow of an Eyring Powel Fluid on a Vertical Moving Belt

The problem of the thin film flow on an Eyring-Powell fluid is difficult to analyze mathematically because of the nonlinear character of the governing equation of motion. Since the pioneering study of the Crane [1], who studied the exact solution for the steady two dimensional flows due to a stretching surface in a quiescent fluid, many authors have considered some aspects of this problem. The analytical study of an infinite, lubricated slider bearing consisting of connected surface with Eyring-Powell fluid as lubricant is considered by Siddiqui et al. [2]. They showed that only viscous and non-Newtonian terms have effects and the variation of pressure and from that the load carrying capacity of the bearing is presented for a range of fluid and bearing parameters. Siddiqui et al. [3] applied the variational iteration method and Adomian decomposition method to find an approximate solution of a nonlinear differential equation that arises in the thin film flow of an Eyring-Powell fluid on a vertically moving belt. The problem of steady, laminar, two-dimensional boundary layer flow and heat transfer on an incompressible, viscous, non-Newtonian fluid over a nonisothermal stretching sheet in the presence of viscous dissipation and interval heat generation/absorption is investigated by Prasad et al. [4]. The non-Newtonian behavior of the fluid is characterized by the constitutive equation due to Powell and Eyring. Panigrahi et al. [5] founded sufficient conditions for existence of similar solutions of the mixed convection flow of a Powell-Eyring fluid over a nonlinear stretching permeable surface in the presence of magnetic field. The effects of different material parameters are discussed. Siddiqui et al. [6] studied the flow of an incompressible, isothermal Eyring-Powell fluid in a helical screw rheometer. The complicated geometry of the helical screw rheometer is simplified by “unwrapping or flattening” the channel bands and the outside rotating barrel, assuming the width of the channel is larger as compared to the depth.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_24

267

268

24 Thin Film Flow of an Eyring Powel Fluid on a Vertical Moving Belt

Rehman et al. [7] examined the mixed convection heat transfer effects over the steady incompressible flow of Eyring-Powell fluids over a vertical circular cylinder. The nonlinear coupled system of ordinary differential equations is solved by homotopy analysis method. Hina [8] explored the combined influence of slip and magneto-hydrodynamics in the peristaltic motion of Eyring-Powell fluid in a complaint wall channel with heat/mass transfer. Babu et al. [9] analyzed the mass transfer flow of Powell-Eyring fluid due to the porous stretching sheet with magnetic field. A second-order approximation of the Eyring-Powell fluid model is used to obtain the flow equations and is solved by shooting method in combinations with Runge–Kutta fourth order scheme. Animasaun et al. [10] presented the analysis of the motion within the thin layer formed in a horizontal object which is neither perfect horizontal nor vertical and neither an inclined surface nor a cone/wedge. The governing nonlinear equations are solved numerically through Runge–Kutta integration procedure along with shooting technique. Reddy and Reddy [11] considered thermal radiation and chemical reaction impacts on MHD peristaltic motion of the Eyring-Powell fluid through a porous medium in a channel with compliant walls under slip conditions for velocity temperature and concentration. Reddy et al. [12] studied numerically the steady laminar incompressible viscous magneto-hydrodynamics boundary layer flow of an Eyring- Powell fluid over a nonlinear stretching flat surface in nanofluid with slip condition and heat transfer through melting effect. Parand et al. [13] discussed a spectral method based on the rational Legendre functions to approximate the solution of the boundary layer flow as an Eyring- Powell non-Newtonian fluid over a stretching sheet. By applying the quasiliniarization method on the nonlinear differential equation at each iteration, the equations convert to a system of linear algebraic equations.

24.1 The Governing Equation of Motion The governing equations of motion for an incompressible fluid, neglecting the thermal effects are given by:

ρ

divV = 0

(24.1)

dV = −∇ P + div S + ρ f dt

(24.2)

where ρ is the constant density of the fluid, V is the velocity vector, P is the dynamic pressure, S is the specific stress force, dtd is the material derivative and f is the specific body force. The constitutive equation for the Eyring-Powell fluid is given by [14]

24.1 The Governing Equation of Motion

  1 1 S = μ divV + sinh−1 divV β c

269

(24.3)

where μ is the coefficient of shear viscosity and β, c are the material constants   of the Eyring-Powell fluid model. Because the 1c divV is a small quantity,  1c divV 1

(25.8) (25.9)

The equations which govern the flow, are the continuity equation and momentum equation

ρ

divV = 0

(25.10)

dV = divS + r − ∇p dt

(25.11)

where ρ is the fluid density, r is the Darcy resistance and S = S1 +S2 +S3 +S4 . For the flow of a fourth grade fluid past a porous plate, we consider that the plate is at y = 0 and fluid occupies the porous space y > 0. We use spatial coordinate such that the x-axis is parallel to the plate and the y-axis to be normal to it. Also, we suppose that the flow is independent of variable x, and therefore from Eq. (25.10) it holds that ∂v =0 ∂y

(25.12)

The velocity field is V =(u = u)(y), v, w), which together with Eq. (25.12) yields

288

25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium

v = −V0 , w = 0

(25.13)

where u, v and w are the components of the velocity field. V0 > 0 corresponds to uniform suction velocity and V0 < 0 corresponds to the uniform blowing (injection) velocity. From Eqs. (25.1 – 25.9), the components of the tensor T are  Sxx = α2

 4 3 du d2 u d2 u 2 du d u + 2γ V − γ V + 2γ (25.14) 2 0 3 0 6 dy dy3 dy dy2 dy2  2 du du d2 u = (2α1 + α2 ) − 2(3β1 + β2 )V0 dy dy dy2  2 3 d2 u du d3 u 2 du d u + 6γ1 V20 + + γ + 2(γ · )V 1 2 0 dy dy3 dy dy3 dy2  4  2 2 du 4 d u + γ3 V0 + 2(2γ3 + 2γ4 + 2γ5 + γ6 ) (25.15) 2 dy dy

du dy

Syy

2

− 2β2 Vo

Szz = 0 Sxy

(25.16)

 3 3 du d2 u d4 u du 2d u = μ − α1 V0 2 + β1 V0 3 + 2(β2 + β3 ) − γ1 V30 4 dy dy dy dy dy  2 2 du d u − 2(3γ2 + γ3 + γ4 + γ5 + 3γ7 + γ8 )V0 (25.17) dy dy2 Sxz = Syz = 0, Sxy = Syx , Sxz = Szx , Syz = Szy

(25.18)

The Eq. (25.11) of the momentum can be written in the forum ∂Sxy ∂Sxx ∂Sxz du = + + + rx − (∇p)x dy ∂x ∂y ∂z

(25.19)

0=

∂Syy ∂Syz ∂Sxy + + + ry − (∇p)y ∂x ∂y ∂z

(25.20)

0=

∂ρyz ∂ρzz ∂ρxz + + + rz − (∇p)z ∂x ∂y ∂z

(25.21)

ρV0

in which the subscripts x, y and z indicate the components in the x−, y− and z− directions. In an unbounded porous medium, Darcy’s law holds for viscous fluid flows having low velocity. This law relates the pressure drop induced by the frictional drag and velocity ignores the boundary effects on the flow. According to this law, Neil and

25.1 The Governing Equations

289

Bejan [17] have been suggested the equation ∇p = −

μϕ V k

(25.22)

in which k is the permeability and ϕ is the porosity of the porous space. In accordance with Tan and Masuoka [18], Hayat et al. [15] proposed the following relationship between the pressure drop and velocity in steady flow for a fourth grade fluid over a porous boundary, as  2 2 du du 2d u + β1 V0 2 + 2(β1 + β2 )u (∇p)x = −uμ − α1 V0 dy dy dy

− γ1 V30

 d3 u du d2 u ϕ (25.23) − 2V + γ + γ + γ + 3γ + γ u (3γ ) 0 2 3 4 5 7 8 dy dy2 k dy3

The pressure gradient in the Eq. (25.23) can be interpreted as a measure of the resistance to fluid in the bulk of the porous space, such that we can write (∇p)x = rx

(25.24)

Follows that from Eqs. (25.10), (25.11), (25.23) and (25.24), we can get −ρV0

 2 2 du d u d2 u du d3 u d4 u = μ 2 − α1 V0 3 + β1 V20 4 + 6(β2 + β3 ) dy dy dy2 dy dy dy  2  5 du d2 u 3d u − γ1 V0 5 − 2V0 (3γ2 + γ3 + γ4 + γ5 + 3γ7 + γ8 ) 2 dy dy2 dy

 2 3 2 du du d u ∂p ϕ 2d u − μu − α + β + V V − 1 0 1 0 dy dy3 ∂x k dy dy2  2 d3 u du + 2(β2 + β3 )u − γ0 V30 3 dy dy  du d4 u (25.25) − 2V0 (3γ2 + γ3 + γ4 + γ5 + 3γ7 + γ8 )u dy dy4  2

∂p du ϕ du = V0 μ + α1 + 2(β2 + β3 ) ∂y k dy dy  du d2 u (25.26) + 2(3γ2 + γ3 + γ4 + γ5 + 3γ7 + γ8 )u dy dy2 ∂p =0 ∂z

(25.27)

290

25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium

where the modified pressure p is given by  2 du du d2 u p = p − Syy = p − (2α1 + α2 ) + 2V0 (3β1 + β2 ) dy dy dy2    2 2 2 3 d2 u du d3 u 2 2 du d u 2 d u − 6γ1 V0 + − γ3 V0 − 2(γ1 + γ2 )V0 dy dy3 dy dy3 dy2 dy2  4 du − 2(2γ3 + 2γ4 + 2γ5 + γ6 ) (25.28) dy Equation (25.27) indicates that p is independent of the variable z. Eliminating the pressure gradient between Eqs. (25.26) and (25.28), by cross differentiation, we obtain equation μ

2 

5 d2 u d4 u du d2 u d3 u 2d u + ρV − α V + β V + 6(β + β ) 2 0 1 0 1 0 2 3 dy dy2 dy3 dy2 dy4 dy5  3  2 3 6 du d u d2 u 3d u + V − 2(3γ + γ + γ + γ + 3γ + γ )V − γ 2 1 0 2 3 4 5 7 8 0 dy dy3 dy6 dy2

 2 4 3 du d2 u du d2 u d3 u du d u ϕ 2d u μ − α +6 + V + β V − 1 0 1 0 dy dy2 dy3 dy dy4 k dy dy2 dy3   du 3 d4 u du d2 u + 2(β2 + β3 ) + 2u − γ1 V30 4 − 2(3γ2 + γ3 + γ4 + γ5 2 dy dy dy dy      2 d2 u d2 u du 2 du d2 u =0 (25.29) + 3γ7 + γ8 )V0 + 2 + u dy dy2 dy2 dy dy The relevant boundary conditions for Eq. (25.29) are u(0) = 0, u(∞) = U0 ,

dk u = 0, k = 1, 2, 3, 4 dyk

(25.30)

where U0 is the reference velocity. Introducing dimensionless variables and parameters u U0 y V0 α1 U20 β1 U40 γ1 U60 , V0 = , y= , α1 = , β = , γ = , 1 1 U0 v U0 ρv2 ρv3 ρv4 2(β2 + β3 )U40 ϕv2 2(3γ2 + γ3 + γ4 + γ5 + 3γ7 + γ8 )U60 β= , ϕ = , γ = ρv3 ρv4 kU20 (25.31) u=

25.1 The Governing Equations

291

The Eq. (25.29) can be rewritten in the dimensionless form omitting the bars in the form   (VI) (V) (IV) − γ1 V03 u +β1 V02 u −α1 V03 u +V0 u + V0 u + 3β 2u u2 + u2 u   IV − γV0 2u3 + 6u u u + u2 u − ϕ[u − α1 V0 u + β1 V02 u     (IV) + β u3 + 2uu u − γ1 V03 u −γV0 u u2 + uu + uu u ] = 0

(25.32)

The corresponding boundary conditions (25.30) are u(0) = 0, u(∞) = 1, u (∞) = u (∞) = u (∞) = u(I V ) (∞) = 0

(25.33)

where the prime denotes the differentiation with respect to y. The sixth order nonlinear differential Eqs. (25.32) and (25.33) will be solved using OAFM in the next section.

25.2 OAFM for the Steady Flow of a Fourth Grade Fluid in a Porous Medium In what follows we will apply our technique to obtain an approximate analytical solution of Eq. (25.32) and (25.33). The initial approximation u0 (y) which verify the boundary conditions (25.32) can be choose as u0 (y) = 1 − e−ky

(25.34)

in which k is an unknown positive parameter at this moment. Taking into consideration Eq. (25.34), we define linear operator and the function g(y) in the forms (the linear operator would not be unique)     L u(y) = V0 u + k2 u ; g(y) = V0 k2

(25.35)

    L u(y) = V0 u + ku ; g(y) = 0

(25.36)

    L u(y) = V0 u + k2 u ; g(y) = 0

(25.37)

or

or yet

and so on.

292

25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium

If we consider only the linear operator and the function g(y) through Eq. (25.35), we can obtain the nonlinear operator. Substituting Eq. (25.35) into Eq. (25.32), after some manipulation, one retrieves   (VI) (V) (IV) N u(y) = −γ1 V03 u +β1 V02 u −α1 V0 u +u − V0 k 2 u      2 2  3    2 IV + 3β 2u u + u u − γV0 2u + 6u u u + u u    − ϕ u − α1 V0 u + β1 V02 u + β u3 + 2uu u   (IV) − γ1 V03 u −γV0 u2 u + uu2 + uu u (25.38) It is evident that the Eq. (2.5) is defined by   L uo (y) + V0 k2 = 0

(25.39)

whose solution is given by Eq. (25.34). Now, inserting Eq. (25.34) into Eq. (25.38), it holds that   N u0 (y) = Me−ky + Ne−2ky + Pe−3ky

(25.40)

where M = k3 − V0 k2 + α1 V0 k4 + β1 V20 k5 + γ1 V30 k6   − ϕ k + α1 V0 k2 + β1 V20 k3 + γ1 V30 k4 N = −2βk3 − 2γV0 k4

  P = 9βk5 + 9γV0 k6 − ϕ 3βk3 + 3γV0 k4

(25.41)

Having in view the considerations from Chapter 2 and Eqs. (25.46) and (25.52), the auxiliary functions are chosen in the forms f1 (y) = −e−kη ; f2 (y) = −ye−ky ; f3 (y) = −e−2ky ; f4 (y) = −e−3ky

(25.42)

or f1 (y) = −e−kη ; f2 (y) = −ye−ky ; f3 (y) = −y2 e−ky ; f4 (y) = −e−2ky ; f5 (y) = −ye−2ky

(25.43)

or yet f1 (y) = −e−ky ; f2 (y) = −ye−ky ; f3 (y) = −e−2ky ; f4 (y) = −ye−2ky ; f5 (y) = −e−3ky ; f6 (y) = −ye−3ky and so on.

(25.44)

25.2 OAFM for the Steady Flow of a Fourth Grade Fluid in a Porous Medium

293

If we consider only the auxiliary functions given by Eq. (25.42), then the first approximation u1 y, Cj is obtained from Eq. (2.13), which can be rewritten as   V0 u1 + k2 u1 − (C1 + C2 y)e−ky − C3 e−2ky − C4 e−3ky = 0, u1 (0) = u1 (∞) = 0

(25.45)

whose solution is

  C2 C1 1 C2 2 y y + + V0 4k2 2k 4k2  C3 C4 C3 C4 + 2 − 2 e−ky + 2 e−2ky + 2 e−3ky 3k 8k 3k 8k

u1 (y, C1 , C2 , C3 , C4 ) = −

(25.46)

The approximate analytical solution of Eqs. (25.32) and (25.33) is obtained from (25.34), (25.46) and (2.3)

  1 C2 2 C2 C1 y u(y) = 1 − e − y + + V0 4k2 2k 4k2  C3 C4 C3 C4 + 2 − 2 e−ky + 3 e−2ky + 2 e−3ky 3k 8k 3k 8k −ky

(25.47)

In the following, in order to prove the accuracy of the obtained results, we will determine the values of the convergence-control parameters k and Ci which appear in Eq. (25.47) by means of the minimization of square residual error (2.30) and (2.31). Then we will study the influence of some material parameters on the velocity profile.

25.2.1 Case 1 For V0 = 0.432; γ1 = 0.123; γ = −0.234; β1 = 0.111; α1 = 0.222; β = 0.081 and ρ = 0.333 the values of the convergence-control parameters are C1 = −4.1469367478 · 10−5 ; C2 = −6.025883482 · 10−9 ; C3 = −8.2401451636224 · 10−8 ; C4 = −2.4136673327441 · 10−4 ; k = 0.8045793 (25.48) such that, the approximate analytical solution (25.47) can be expressed in the form

294

25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium

u(y) = 1 + (4.3341919054393 · 10−8 y2 + 5.96505067680313 · 10−5 y − 0.999892015563873)e−0.8045793y − 9.8218240794894 · 10−8 e−1.6091586y − 1.07886217948446 · 10−4 e−2.4137379y

(25.49)

25.2.2 Case 2 For V0 = 0.432; γ1 = 0.123; γ = −0.434; β1 = 0.111; α1 = 0.222; β = 0.081 and ϕ = 0.333, one can get C1 = 5.5114336433963 · 10−4 ; C2 = 8.183548206 · 10−7 ; C3 = 1.82845864 · 10−8 ; C4 = −1.027060125 · 10−9 ; k = 0.4335017

(25.50)

u(y) = 1 − (1.0924639179115 · 10−6 y2 + 1.4740193091457 · 10−3 y + 1.00000007349)e−0.4335017y + 7.507555120971 · 10−8 e−0.8670034y − 1.58139559481 · 10−9 e−1.3005351y

(25.51)

25.2.3 Case 3 For V0 = 0.432; γ1 = 0.123; γ = −0.634; β1 = 0.111; α1 = 0.222; β = 0.081 and ϕ = 0.333, we have C1 = −1.965319959974 · 10−4 ; C2 = 1.599239420811 · 10−7 ; C3 = 6.508375841294 · 10−9 ; C4 = 1.452846613312 · 10−9 ; k = 0.2949712

(25.52)

u(y) = 1 + (2.980594306043 · 10−7 y2 + 7.701413847068 · 10−4 y − 1.00000006259)e−0.2949712y + 5.771761710911 · 10−8 e−0.5899424y + 4.831533596 · 10−9 e−0.8849136y

(25.53)

25.2 OAFM for the Steady Flow of a Fourth Grade Fluid in a Porous Medium

295

25.2.4 Case 4 For V0 = 0.532; γ1 = 0.123; γ = −0.234; β1 = 0.111; α1 = 0.222; β = 0.081 and ϕ = 0.333, it holds that C1 = 7.164651467787 · 10−5 ; C2 = 2.8382635338476 · 10−5 ; C3 = 2.925562150482 · 10−4 ; C4 = −1.8262558811867 · 10−5 ; k = 0.6510347

(25.54)

u(y) = 1 − (2.0486933449735 · 10−5 y2 + 1.348989283176 · 10−4 y + 1.0004223581397)e−0.6510347y + 4.324821303369 · 10−4 e−1.3020694y − 1.0123990622902 · 10−5 e−1.9531041y

(25.55)

25.2.5 Case 5 For V0 = 0.632; γ1 = 0.123; γ = −0.234; β1 = 0.111; α1 = 0.222; β = 0.081 and ϕ = 0.333, we obtain C1 = −3.989642482022 · 10−4 ; C2 = 2.30316711358 · 10−7 ; C3 = 4.841742428044 · 10−9 ; C4 = −2.1501451936361 · 10−9 ; k = 0.5471352

(25.56)

u(y) = 1 + (−1.665151392689 · 10−7 y2 + 5.76584648859 · 10−4 y − 1.00000000711)e−0.5471352y + 8.53049115335 · 10−9 e−1.0942704y − 1.420598690973 · 10−9 e−1.6414056y

(25.57)

25.2.6 Case 6 For V0 = 0.432; γ1 = 0.123; γ = −0.234; β1 = 0.111; α1 = 0.222; β = 0.081 and ϕ = 0.333, one has C1 = −1.15807499133 · 10−4 ; C2 = −1.08013632799 · 10−5 ; C3 = −4.06399813489 · 10−5 ; C4 = 3.345069216428 · 10−5 ; k = 1.7904321

(25.58)

296

25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium

u(y) = 1 + (6.51545084094 · 10−7 y2 + 7.681256126816 · 10−5 y − 0.999993237249809)e−1.7904321y − 9.782115519558 · 10−6 e−3.5808642y + 3.020267959 · 10−6 e−5.3712963y

(25.59)

25.2.7 Case 7 For V0 = 0.432; γ1 = 0.123; γ = −0.234; β1 = 0.111; α1 = 0.222; β = 0.041 and ϕ = 0.333, yields C1 = 6.05366161536 · 10−4 ; C2 = 1.0566964037134 · 10−6 ; C3 = 7.42689620461 · 10−8 ; C4 = −2.402115824318 · 10−8 ; k = 0.4073112

(25.60)

u(y) = 1 − (1.501343744308 · 10−6 y2 + 1.723882434999 · 10−3 y + 1.000000303526)e−0.4073112y + 3.45421724803 · 10−7 e−0.8146224y − 4.189551235358 · 10−8 e−1.2219336y

(25.61)

25.2.8 Case 8 For V0 = −0.543; γ1 = 0.123; γ = 0.123; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333, we have C1 = 6.598771114254 · 10−4 ; C2 = +6.73408917526 · 10−8 ; C3 = +6.6440204244036 · 10−6 ; C4 = −9.594719260992 · 10−6 ; k = 1.6623165 (25.62) u(y) = 1 − (1.865113761883 · 10−7 y2 + 3.656392515643 · 10−4 y + 0.99999932323)e−1.6623165y − 1.475986696824 · 10−6 e−3.324633y + 7.99397906415 · 10−7 e−4.9869495y

(25.63)

25.2 OAFM for the Steady Flow of a Fourth Grade Fluid in a Porous Medium

297

25.2.9 Case 9 For V0 = −0.543; γ1 = 0.123; γ = 0.234; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333, we obtain C1 = −3.884556620875 · 10−4 ; C2 = +3.18499116405 · 10−8 ; C3 = −9.480745514073 · 10−6 ; C4 = +1.1919461858315 · 10−5 ; k = 0.8731781

(25.64)

u(y) = 1 + (−1.6793668507 · 10−8 y2 + 4.096268402444 · 10−4 y − 0.9999959653077)e−0.87317819y − 7.633520864628 · 10−6 e−1.7463562y + 3.598828583608 · 10−6 e−2.61953435y

(25.65)

25.2.10 Case 10 For V0 = −0.543; γ1 = 0.123; γ = 0.345; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333, on can get C1 = +1.21662867601 · 10−4 ; C2 = +5.519585482452 · 10−8 ; C3 = +3.631112466734 · 10−7 ; C4 = −2.364782173515 · 10−7 ; k = 0.5927107

(25.66)

u(y) = 1 − (4.28749745511 · 10−8 y2 + 1.89082630327 · 10−4 y + 1.00000047954)e−0.59271079y + 6.34502128769 · 10−7 e−1.1854214y − 1.549586582588 · 10−7 e−1.17781321y

(25.67)

25.2.11 Case 11 For V0 = −0.654; γ1 = 0.123; γ = −0.123; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333, we obtain C1 = −9.581751353935 · 10−4 ; C2 = 7.874399784686 · 10−7 ; C3 = 2.276975662499 · 10−6 ; C4 = −1.96734646961 · 10−7 ; k = 1.3793456

(25.68)

298

25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium

u(y) = 1 + (−2.1822606753162 · 10−7 y2 + 5.30926818893 · 10−4 y − 1.00000041234)e−1.3793456y + 6.099766322438 · 10−7 e−2.7586912y − 1.976363957499 · 10−7 e−4.1380368y

(25.69)

25.2.12 Case 12 For V0 = −0.765; γ1 = 0.123; γ = 0.123; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333, one has C1 = 7.116857425385 · 10−4 ; C2 = 2.89211555174 · 10−7 ; C3 = −6.48549590267 · 10−8 ; C4 = 1.212592439226 · 10−7 ; k = 1.1800531

(25.70)

u(y) = 1 + (−8.009265015112 · 10−8 y2 − 3.94248506681 · 10−4 y − 0.999999965478)e−1.1899531y − 2.02935207146 · 10−8 e−2.3601062y − 1.4228539816 · 10−8 e−3.5405593y

(25.71)

25.2.13 Case 13 For V0 = −0.543; γ1 = 0.123; γ = 0.345; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333, yields C1 = −7.1884612963096 · 10−5 ; C2 = 6.524034643826 · 10−8 ; C3 = 3.383347335115 · 10−7 ; C4 = −1.506038530634 · 10−7 ; k = 1.1849871

(25.72)

u(y) = 1 + (−2.53479484382 · 10−8 y2 + 5.583752347732 · 10−5 y − 1.000000012322)e−1.18498719y + 1.47910244646 · 10−7 e−2.3699742y − 2.468988239 · 10−8 e−3.5549613y

(25.73)

25.2 OAFM for the Steady Flow of a Fourth Grade Fluid in a Porous Medium

299

25.2.14 Case 14 For V0 = −0.543; γ1 = 0.123; γ = 0.345; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333, we have C1 = 8.433025436648 · 10−4 ; C2 = 5.031884670002 · 10−7 ; C3 = 2.68208195679 · 10−6 ; C4 = −4.657839283656 · 10−6 ; k = 1.7780012

(25.74)

u(y) = 1 − (1.3029883175248 · 10−7 y2 + 4.368118410197 · 10−4 y − 1.0000001816388)e−1.7780012y + 5.2081904946 · 10−7 e−3.5560024y − 3.39180271294 · 10−7 e−5.3340036y

(25.75)

In Tables 25.1, 25.2, 25.3, 25.4, 25.5, 25.6, 25.7, 25.8, 25.9, 25.10, 25.11, 25.12, 25.13 and 25.14 we present a comparison between approximate solutions from previously cases with numerical results and corresponding relative errors. If we compare the results presented in Table 25.1, 25.2, 25.3, 25.4, 25.5, 25.6, 25.7, 25.8, 25.9, 25.10, 25.11, 25.12, 25.13 and 25.14, we can arrive at conclusion that the analytical approximate results obtained by OAFM are nearly identical with those obtained through numerical integration. In Figs. 25.1, 25.2, 25.3, 25.4, 25.5 and 25.6 the influence of the parameters V0 , β and γ on the steady velocity is discussed in two cases: suction and injection (blowing) velocity. In these figures are plotted the steady velocity against the horizontal distance y. Figure 25.1 shows the variations of steady velocity for different values of the uniform suction and for fixed values γ = −0.234; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.081, ϕ = 0.333. Table 25.1 Comparison between approximate solution (25.49) obtained by OAFM and numerical results by V0 = 0.432; γ = −0.234; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.081 and ϕ = 0.333

Y

u(y) Numerical

0

0

u(y), Error Equation (25.49) ε = |u(y) − u(y)| 0

0

0.2 0.148646461854

0.1486717443

2.52 E-05

0.4 0.275197153086

0.27523428483

3.71 E-05

0.6 0.382936531822

0.3829777737

4.14 E-05

0.8 0.474660833106

0.4747018913

4.11 E-05

1

0.5527506415378 0.5527892881

3.86 E-05

2

0.7999680113553 0.7999887704

2.07 E-05

3

0.9105358214068 0.9105454321

9.61 E-06

4

0.959987203511

0.9599915408

4.33 E-06

5

0.982104302

0.9821062514

1.99 E-06

6

0.99199616075

0.9919970364

8.75 E-07

300

25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium

Table 25.2 Comparison between approximate solution (25.51) obtained by OAFM and numerical results by V0 = 0.432; γ = −0.434; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.081 and ϕ = 0.333 Y

u(y) Numerical

u(y), Equation (25.51)

Error ε = |u(y) − u(y)|

0

0

0

0

0.2

0.0827778052723

0.082777438

6.14 E-08

0.4

0.158703445498

0.1587033937

5.17 E-08

0.6

0.2283441278637

0.2283441282

4.00 E-10

0.8

0.2922201073846

0.2922201077

4.00 E-10

1

0.350808573511

0.3508085738

3.00 E-10

2

0.5785504917734

0.5785504919

2.00 E-10

3

0.7263985925614

0.7263985926

1.00 E-10

4

0.8223803120156

0.8223803123

3.00 E-10

5

0.8846908213849

0.8846908211

2.00 E-10

6

09,251,422,698,476

0.9251422706

8.00 E-10

Table 25.3 Comparison between approximate solution (25.53) obtained by OAFM and numerical results by V0 = 0.432; γ = −0.634; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.081 and ϕ = 0.333

y

u(y) Numerical

u(y), Error Equation (25.53) ε = |u(y) − u(y)|

0

0

0

0

0.2 0.0574329907125 0.05743299082

1.16 E-10

0.4 0.1115674330028 0.1114674328

1.60 E-10

0.6 0.1625927723718 0.1625927721

2.49 E-10

0.8 0.2106875738988 0.2106875733

2.97 E-10

1

0.2560201471363 0.2560201467

3.75 E-10

2

0.4464939785329 0.4464939783

1.97 E-10

3

0.5882026715898 0.5882026713

2.73 E-10

4

0.6936310841997 0.6936310832

9.32 E-10

5

0.7720676991009 0.7720676971

2.05 E-09

6

0.8304229603142 0.8304229571

3.16 E-09

It is observed that the steady velocity decreases with the increase of the suction velocity V0 . In Fig. 25.2 we present the variation of the steady velocity for different values of the uniform injection for γ = 0.123; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.111, ϕ = 0.333. From Fig. 25.2 we deduce that the steady velocity u(y) increases with increasing injection velocity V0 . In Figs. 25.3 and 25.4 are presented graphically the variation of the steady velocity for different values of the parameters β for V0 = 0.432, γ = −0.234; γ1 = 0.123;

25.2 OAFM for the Steady Flow of a Fourth Grade Fluid in a Porous Medium

301

Table 25.4 Comparison between approximate solution (25.55) obtained by OAFM and numerical results by V0 = 0.532; γ = −0.234; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.081 and ϕ = 0.333 y

u(y) Numerical

u(y), Equation (25.55)

Error ε = |u(y) − u(y)|

0

0

0

0

0.2

0.1220213698323

0.1220175419

3.82 E-06

0.4

0.2291535249689

0.2291501027

3.42 E-06

0.6

0.3232132677826

0.3232126557

6.12 E-06

0.8

0.4057957119321

0.4057990681

3.35 E-16

1

0.4783013331224

0.4783089934

7.66 E-06

2

0.7278305009781

0.7278544867

2.39 E-06

3

0.858009535196

0.8580320925

2.29 E-05

4

0.92592376380219

0.9259401537

9.64 E-05

5

0.961354263283

0.9613644487

9.92 E-06

6

0.97983870790462

0.9798439247

5.21 E-06

Table 25.5 Comparison between approximate solution (25.57) obtained by OAFM and numerical results by V0 = 0.632; γ = −0.234; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.081 and ϕ = 0.333 y

u(y) Numerical

u(y), Equation (25.57)

Error ε = |u(y) − u(y)|

0

0

0

0

0.2

0.1037557980839

0.1037537981

1.00 E-10

0.4

0.1967461305318

0.19674633055

1.82 E-10

0.6

0.2800885560713

0.2800885561

9.28 E-11

0.8

0.3547835424859

0.3547835425

1.15 E-11

1

0.42172849097

0.4217284909

7.20 E-11

2

0.6656020618467

0,665,602,069

9.00 E-11

3

0.8066271996883

0.8066271993

2.11 E-10

4

0.8881780189588

0.8881780187

2.50 E-10

5

0.93533653428082

0.9353365340

2.80 E-10

6

0.9626069600996

0.9626069598

1.74 E-10

β1 = 0.111; α1 = 0.222; ϕ = 0.333 and V0 = -0.543, γ = 0.345; γ1 = 0.123; β1 = 0.111; α1 = 0.222; ϕ = 0.333, respectively. In Figs. 25.5 and 25.6 we present the variation of the steady velocity for different values of the parameter γ for suction (V0 = 0.432); γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.081; ϕ = 0.333 and for injection (V0 = −0.543);γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.111; ϕ = 0.333, respectively.

302

25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium

Table 25.6 Comparison between approximate solution (25.59) obtained by OAFM and numerical results by V0 = 0.432; γ = −0.234; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333 y

u(y) Numerical

u(y), Equation (25.59)

Error ε = |u(y) − u(y)|

0

0

0

0

0.2

0.300995061225

0.3009990722

4.33 E-07

0.4

0.5113983095591

0.5113952367

3.07 E-06

0.6

0.6584671770724

0.6584664485

7.28 E-07

0.8

0.7612683880983

0.7612676308

7.57 E-07

1

0.8331264853765

0.8331257563

7.29 E-07

2

0.97215323011721

0.9721529135

3.16 E-07

3

0.99535111638718

0.9953530275

8.41 E-08

4

0.9992245574071

0.9992245390

1.84 E-08

5

0.9998705991691

0.9998705955

3.56 E-09

6

0.99997840642856

0.9999784058

6.42 E-10

Table 25.7 Comparison between approximate solution (25.61) obtained by OAFM and numerical results by V0 = 0.432; γ = −0.234; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333 y

u(y) Numerical

u(y), Equation (25.61)

Error ε = |u(y) − u(y)|

0

0

0

0

0.2

0.0779146066703

0.0779146071

4.52 E-10

0.4

0.149758527408

0.1497590833

5.56 E-07

0.6

0.2160047573197

0.2160047576

3.30 E-10

0.8

0.2770894382845

0.2770899213

5.39 E-07

1

0.3334147303584

0.3334147306

2.91 E-10

2

0.5556640782968

0.5556640785

4.79 E-10

3

0.70381221982

0.7038122199

6.60 E-11

4

0.8025655886842

0.8025655886

4.63 E-11

5

08,683,931,296,965

0.8683931296

9.60 E-11

6

0.91227279887205

0.9122727988

1.70 E-10

From the Figs. 25.3 and 25.4 it is clear that the steady velocity u(y) in the cases of the suction and injection velocity increases with increasing parameter β. On the other frond, from Figs. 25.5 and 25.6 we deduce that in the case of suction velocity the steady velocity u(y) increases with increasing parameter β, but in the case of injection velocity, the steady velocity u(y) decreases with increasing parameter β.

25.2 OAFM for the Steady Flow of a Fourth Grade Fluid in a Porous Medium

303

Table 25.8 Comparison between approximate solution (25.63) obtained by OAFM and numerical results by V0 = −0.543; γ = 0.123; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333 y

u(y) Numerical

u(y), Equation (25.63)

Error ε = |u(y) − u(y)|

0

0

0

0

0.2

0.2827924886889

0.2827925836

5.51 E-08

0.4

0.4856133857182

0.4856135533

1.68 E-07

0.6

0.6310780565189

0.6310779054

1.52 E-07

0.8

0.7354064110477

0.7354065959

1.85 E-07

1

0.8102316905585

0.8102316509

1.60 E-07

2

0.96398791282435

0.9639879569

1.41 E-08

3

0.9831660398948

0.9831660482

8.41 E-10

4

0.998703129577254

0.99870313077

1.27 E-09

5

0.999753894832937

0.9997538947

1.29 E-10

6

0.9999532969828

0.9999532969

7.14 E-11

Table 25.9 Comparison between approximate solution (25.65) obtained by OAFM and numerical results by V0 = −0.543; γ1 = 0.123; γ = 0.234; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333 y

u(y) Numerical

u(y), Equation (25.65)

Error ε = |u(y) − u(y)|

0

0

0

0

0.2

0.1603060965923

0.1603059757

1.21 E-07

0.4

0.29491414858

0.2949140317

1.16 E-07

0.6

0.4079437091836

0.4079436345

7.16 E-08

0.8

0.5028539421273

0.5028540298

8.77 E-08

1

0.5825494861011

0.5825494851

9.57 E-10

2

0.8257350684456

0.8257350675

8.68 E-10

3

0.9272530147681

0.9272529764

3.83 E-08

4

0.96963173363039

0.9696316944

3.92 E-08

5

0.9873227515977

0.9873227521

2.65 E-08

6

0.99470787613966

0.994707861

1.48 E-08

In this section we consider only n = 11 control points (see Tables 25.1, 25.2, 25.3, 25.4, 25.5, 25.6, 25.7, 25.8, 25.9, 25.10, 25.11, 25.12, 25.13 and 25.14), but for statistical tests, are necessary to sustain some asymptotic properties. Two statistical tests are more important: test of homoscedasticity and test of autocorrelations Bartlett and Durbin Watson test, respectively.

304

25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium

Table 25.10 Comparison between approximate solution (25.67) obtained by OAFM and numerical results by V0 = −0.543; γ = 0.345; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333 y

u(y) Numerical

u(y), Equation (25.67)

Error ε = |u(y) − u(y)|

0

0

0

0

0.2

0.1117519893926

0.1117519903

1.01 E-09

0.4

0.2110154716521

0.2110154721

1.95 E-10

0.6

0.299186062295

0.2991860621

5.61 E-08

0.8

0.3775034140276

03,775,034,133

7.39 E-10

1

0.4470686459001

0.4470686448

1.09 E-09

2

0.6942669176533

0.6924669169

6.36 E-10

3

0.8309505927849

0.8309505919

7.83 E-10

4

0.9065272823587

0.9065272799

2.38 E-09

5

0.9483160036633

0.9483159988

4.77 E-09

6

0.9714222979202

0.9714222926

5.29 E-09

Table 25.11 Comparison between approximate solution (25.69) obtained by OAFM and numerical results by V0 = −0.654; γ = 0.123; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333 y

u(y) Numerical

u(y), Equation (25.69)

Error ε = |u(y) − u(y)|

0

0

0

0

0.2

0.2411682783697

0.2411682803

2.00 E-09

0.4

0.4241744182476

0.42417441744

8.24 E-10

0.6

0.5630452824401

0.5630452801

2.29 E-09

0.8

0.6684248993996

0.668424897

2.13 E-09

1

0.7483902955616

0.7483902941

1.36 E-09

2

0.93669255663244

0.9366925567

2.35 E-11

3

0.98407123288554

0.9840712309

1.86 E-09

4

0.995992167614263

0.995921665

1.11 E-09

5

0.998891590477986

0.9989915898

5.67 E-10

6

0.999746274398214

0.9997462741

2.14 E-10

For Bartlett test [n/ 2] 2 ε B = ni=1 i 2 i=[n/ 2] εi

(25.76)

25.2 OAFM for the Steady Flow of a Fourth Grade Fluid in a Porous Medium

305

Table 25.12 Comparison between approximate solution (25.71) obtained by OAFM and numerical results by V0 = −0.765; γ = −0.123; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.111 and ϕ = 0.333 y

u(y) Numerical

u(y), Equation (25.71)

Error ε = |u(y) − u(y)|

0

0

0

0

0.2

0.2101654382833

0.2101654504

1.22 E-08

0.4

0.3761613651177

0.3761613902

2.51 E-08

0.6

0.5072706852358

0.5072707226

3.74 E-08

0.8

0.6108253576282

0.6108254054

4.78E-08

1

0.692616416911

0.6926164722

5.53 E-08

2

0.90551533284738

0.9055153949

6.21 E-08

3

0.9709569446366

0.9709570063

4.20 E-08

4

0.99107264767306

0.9910726704

2.28 E-08

5

0.997255878454247

0.9972558893

1.09 E-08

6

0.9991565020868

0.9991565069

4.88 E-09

Table 25.13 Comparison between approximate solution (25.73) obtained by OAFM and numerical results by V0 = −0.543; γ = 0.345; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.222 and ϕ = 0.333 y

u(y) Numerical

u(y), Equation (25.73)

Error ε = |u(y) − u(y)|

0

0

0

0

0.2

0.2110154715574

0.2110154817

1.02 E-08

0.4

0.3775034138781

0.3775034113

2.45 E-09

0.6

0.508859824542

0.5088598223

2.19 E-09

0.8

0.61249800026667

0.6124979986

1.56 E-09

1

0.6942669174699

0.6942669164

9.56 E-10

2

0.90652728224662

0.9065272806

1.55 E-09

3

0.97142229786879

0.9714222968

9.48 E-10

4

0.9912628510358

0.9912628502

8.52 E-10

5

0.99732876451465

0.9973287639

5.61 E-10

6

0.9991833149409

0.9991833146

2.67 E-10

the representative values of limits are Binf = 0.000162; Fsup = 8.6831168141     for confidence level 0.99%. In Eq. (25.76) we use the notation εi = u yi − u yi , i = 1, 2, . . . , n. First test assure that errors have the constant variance and possible to control the analytic approximate solutions for any real interval. In the considered case this interval is [0, 6]. For Durbin Watson test we have

306

25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium

Table 25.14 Comparison between approximate solution (25.75) obtained by OAFM and numerical results by V0 = −0.543; γ = 0.345; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.333 and ϕ = 0.333 y

u(y) Numerical

u(y), Equation (25.75)

Error ε = |u(y) − u(y)|

0

0

0

0

0.2

0.2991860622389

0.2991860919

2.98 E-08

0.4

0.5088598246398

0.5088598444

1.98 E-08

0.6

0.6558021197131

0.6558021283

8.55 E-09

0.8

0.7587813281471

0.7587813302

2.05 E-09

1

0.8309505927173

0.8309505918

8.86 E-10

2

0.97142229789737

0.9714222963

1.50 E-09

3

0.99516895639805

0.9951689559

3.73 E-10

4

0.9991833149425

0.9991833147

1.28 E-10

5

0.9998619398751

0.9998619398

7.50 E-10

6

0.99947666101772

0.9999766610

1.75 E-09

Fig. 25.1 Variation of a steady velocity with the increasing suction velocity V0 for γ = −0.234; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.081 and ϕ = 0.333. V0 = 0.632 (Eq. 25.57); V0 = 0.532 (Eq. 25.55); V0 = 0.432 (Eq. 25.49)

2n DW =

(εi − εi−1 )2 n 2 i=1 εi

i=1

(25.77)

The representative limits for Durbin Watson test are DWinf = 1.36 and DWsup = 2.92. For confidence level 0.99%is necessary that DWinf < DW < DWsup . This test explain no correlation between errors and this means that no exist an analytic terms in the errors that is, we obtain the best analytic approximate solution. For all 14 approximate solutions (25.49), (25.51), (25.53), (25.55), (25.57), (25.59), (25.61), (25.63), (25.65), (25.67), (25.69), (25.71), (25.73), (25.75) and for n = 24 points, all errors pass both tests are for Bartlett test

25.2 OAFM for the Steady Flow of a Fourth Grade Fluid in a Porous Medium

307

Fig. 25.2 Variation of a steady velocity with the increasing function velocity V0 for γ = 0.123; γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.111; ϕ = 0.333. V0 = −0.765 V0 = -0.765 (Eq. 25.71); V0 = −0.654 (Eq. 25.69); V0 = −0.543 (Eq. 25.63)

Fig. 25.3 Variation of a steady velocity with the increasing parameters β for the case of suction velocity V0 =0.432 and γ=−0.234; γ1 =0.123; β1 =0.111; α1 =0.222; ϕ=0.333. β = 0.041 (Eq. 25.61); β=0.081 (Eq. 25.49); β=0.181 (Eq. 25.59)

Fig. 25.4 Variation of a steady velocity with the increasing parameters β for the case of injection velocity Vo = −0.543; γ = 0.345; γ1 = 0.123; β1 = 0.111; α1 = 0.222; ρ = 0.333. β = 0.111 (Eq. 25.67); β = 0.222 (Eq. 25.73); β = 0.333 (Eq. 25.75)

308

25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium

1.35721; 1.42038; 2.10375; 2.46131; 1.45721; 1.99237; 1.83041 (25.78) and for Durbin-Watson 1.38999; 1.81372; 1.83045; 2.31473; 1.93145; 2.04973; 2.11138 (25.79) It is clear that by means of OAFM we obtain the best analytic approximate solution for our problem, given by Eqs. (25.32) and (25.33).

Fig. 25.5 Variation of a steady velocity with the increasing parameters γ for the case of suction velocity V0 = 0.432 and γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.081; ϕ = 0.333. γ = −0.634 (Eq. 25.53); γ = −0.434 (Eq. 25.51); γ= −0.234 (Eq. 25.49)

Fig. 25.6 Variation of a steady velocity with the increasing parameters γ for the case of injection velocity V0 = −0.543 and γ1 = 0.123; β1 = 0.111; α1 = 0.222; β = 0.111; ϕ = 0.333. γ = 0.123 Eq. (25.63); γ = 0.234 (Eq. 25.65); γ = 0.345 (Eq. 25.67)

References

309

References 1. T. Hayat, Y. Wang, K. Hutter, Flow of a fourth grade fluid, mathematical models and methods in applied. Science 12(6), 757–811 (2002) 2. T. Hayat, A.H. Kara, E. Momoniat, The unsteady flow of a fourth-grade fluid past a porous plate. Math. Comput. Modell. 41, 1347–1353 (2005) 3. Y. Wang, W. Wu, Unsteady flow of a fourth-grade fluid due to an oscillating plate. Int. J. Non-Linear Mech. 42(3), 436–441 (2007) 4. M. Sheikholeslami, H.R. Ashorynejad, D. Domairry, I. Hashim, Investigation of the laminar viscous flow in a semi-porous channel in the presence of uniform magnetic field using optimal homotopy asymptotic method. Sayns Malaysiana 41(10), 1281–1285 (2010) 5. T. Hayat, S. Noreen, M. Sajid, Steady flow of a fourth grade fluid in a porous medium. J. Porous Media 13(11), 67–79 (2010) 6. S. Islam, Z. Bano, A.M. Siddiqui, The optimal solution for the flow of a fourth grade fluid with partial slip. Comput. Math. Appl. 61, 1507–1516 (2011) 7. S.V.H.N. Kumari, P.Y.V.K.R. Kumar, M.V.R. Murthy, S. Sreenadh, Peristaltic motion of a fourth grade fluid through a porous medium under effect of magnetic field in an inclined channel. J. Basic Appl. Sci. Res 19, 1052–1064 (2011) 8. T. Aziz, A. Fatima, F. M. Mahomed, Shock wave solution for a nonlinear partial deferential equation arising in the study of a non-Newtonian fourth-grade fluid model, Math. Problems in Engineering Art.ID 573170 (2013) 9. T. Aziz, F.M. Mahomed, Reductions and solutions for the unsteady flow of a fourth grade fluid on a porous plate. Appl. Math. Comput. 219, 9187–9195 (2003) 10. B. Sahoo, Blasius flow and heat transfer of a fourth grade fluid with slip. Appl. Math. Mech. 34(12), 1465–1480 (2013) 11. M.S. Reddy, M.S. Reddy, Peristaltic pumping of a fourth grade fluid through a porous medium under the effect of magnetic field in a symmetric channel. Int. J. Concept. Comput. Inform. Techn. 2(1), 58–67 (2014) 12. H. Zaman, T. Abbas, A. Sohail, A. Ali, Couette flow problem for an unsteady MHD fourth-order grade fluid with Hall currents. J. Appl. Math. Phys. 2, 1–10 (2014) 13. M. Yurusoy, New analytical solutions fot the flow of a fourth grade fluid past a porous plate. Math. Comput. Sci. 1(2), 29–35 (2016) 14. P.G. Moakher, M. Abbasi, M. Khaki, Fully developed flow of fourth grade fluid through the channel with slip condition in the presence of a magnetic field. J. Appl. Fluid Mech. 9(5), 2239–2245 (2016) 15. A.A. Khan, F. Masood, R. Ellahi, M.M. Bhati, Mass transport on chemicalized fourth-grade fluid propagating peristaltically through a curved channel with magnetic effects. J. Mol. Liq. 258, 186–195 (2018) 16. D.A. Neild, A. Bejan, Convection in Porous Media, 2nd edn. (Springer, Berlin, 1988) 17. W.C. Tan, T. Masuoka, Stokes first problem for an Oldroyd-B fluid in a porous space. Phys. Fluids 17, 123101–123107 (2005) 18. N. Heri¸sanu, V. Marinca, Gh. Madescu, Application of the optimal auxiliary functions method to a permanent magnet synchronous generator, Int. J. Nonlinear Sci. Numer. Simul. 20, 399–406 (2019)

Chapter 26

Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

In general, the constitutive equations that describe non-Newtonian fluids are nonlinear and highly complicated. There is not a single constitutive equation available in the literature by which one can study the flow behavior of non-Newtonian fluids. For example Truesdell and Noll [1] have studied the fluids of different types of grade n, such as the second grade fluid, the third grade fluid, fourth grade fluid, Maxwell fluid, Oldroyd B fluid, power-law fluids, etc. Thien and Walsh [2] considered the motion of an Oldroyd—fluid being squeezed between two parallel disks on infinite extent. The fluid inertia is neglected and the squeezing velocity varies exponentially with time. Khan [3] developed a generalized approximation method to obtain a solution of steady unidirectional flows of an Oldroyd-8constant magneto-hydrodynamic fluid in bounded domain. The fluid is electrically conducting in the presence of an uniform magnetic fluid, Siddiqui et al. [4] analyzed the thin film flow problem of an Oldroyd six constant fluid falling in the outer surface of an infinitely long vertical cylinder. The thermophysical properties of the fluid are assumed to the independent of temperature and surface tension. The non-Newtonian nature of the fluid is responsible for introducing nonlinearities in the resulting differential equation. The thin film flow of an Oldroyd 6-constant fluid on a vertical moving belt is investigated by Hammed and Ellahi [5], analytically and numerically by means of higher-order Chebyshev spectral methods. The governing equations for the flow field are derived for a steady one-dimensional flow. Xiong et al. [6] used a simple Oldroyd-B constitutive model to study the role of the viscoelasticity of dilute polymer solution in two-dimensional flows past a bluff body by means of numerical simulation. This investigation is motivated by the numerous experimental results obtained in quasi two-dimensional systems, such a soap film channels. Zhang et al. [7] studied the modal and non-modal linear instability of inertia—dominated channel flow of viscoelastic fluids modeled by Oldroyd B and FENE-P closures. The effects of polymer viscosity and relaxation time are considered for both fluids with the additional parameter of the maximum possible extension for the FENE-P fluids. The work of Hayat et al. [8] is concentrated on the mathematical modeling for three-dimensional flow of an incompressible Oldroyd-B fluid over a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_26

311

312

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

bidirectional stretching surface, including the effects of internal heat source/sink. Two cases of heat transfer namely the prescribed surface temperature (PST) and prescribed surface magneto—hydrodynamic thin film flow, Gul et al. [9] studied the unsteady magnetohydrodynamic thin film flow of an incompressible Oldroyd-B fluid over an oscillating inclined belt, making a certain angle with the horizontal. This problem is solved using these efficient techniques, namely optimal homotopy asymptotic method and Adomian decomposition method. The same procedure of optimal homotopy asymptotic method is applied by Shah et al. [10] in the study of the wire coating in a pressure type die with the both of Oldroyd 8-constant fluid with pressure gradient. Ullah et al. [11] computed and inspected the rotational of Oldroyd-B fluid between two co-axial circular cylinders. The rotation of both cylinders produces motion of fluid. The velocity field and tangential stress corresponding to the motion of an Oldroyd-B fluid with fractional derivative has been obtained using Laplace and Hankel integral transformations. Ene et al. [12] considered the flow of the Oldroyd 6-constant fluid over a moving belt with constant velocity. The time-dependent thin film flow problem of viscoelastic fluid consisting of nano-sized particles through an inclined belt in presence of transverse magnetic field, has studies by Dey and Khound [13]. The constitutive equation of fluid flow is characterized by Oldroyd-B fluid model bearing rheological parameters. The lower surface of the belt is oscillatory about a non-zero constant mean velocity. Krishna and Qadri [14] deals with the unsteady hydrodynamic flow of an incompressible Oldroyd-B fluid in a parallel plate channel, initially induced by a constant pressure gradient. The pressure gradient is suddenly withdrawn and the upper plate moves with a uniform velocity, while the lower plate continued to be a rest. The polymer flow during wire coating dragged from both of viscoelastic incompressible and laminar fluid inside pressure type die is carried out numerically by means of Runge–Kutta fourth order with shooting technique by Khan et al. [15]. In wire coating the flow depends on the velocity of the wire, geometry of the die and viscosity of the fluid. Tiwana et al. [16] examined the unsteady magnetohydrodynamic convective fluid flow of Oldroyd-B model by means of ramped wall temperature and velocity simultaneously. The fluid flow is closed to an infinite vertical flat plate immersed through porous medium. The Laplace transformation is used to find solutions of momentum and energy equations. Ahmad et al. [17] studied a mixed convective peristaltic flow of an Oldroyd-4 constant fluid in a two dimensional channel with flexible walls. The equations are solved numerically by shooting method and implicit finite difference scheme. By the influence of buoyancy forces the symmetric of the velocity profile is disturbed about the central line of the channel and also the size of the trapped bolus increase in the left half of the channel.

26.1 Governing Equations The fundamental equations governing the motion of an incompressible fluid, neglecting the thermal effects and body forces, are given by:

26.1 Governing Equations

313

L = grad V

(26.1)

A1 = L + LT

(26.2)

where V is the velocity vector.

where A1 is the first Rivlin-Ericksen tensor. If μ and λi , i = 1, 2, . . . , 7 are material constants, then the extra tensor S for Oldroyd -6 constant fluid is defined by DS λ3 λ5 λ5 S + λ1 + (SA1 + A1 S) + (trS)A1 + [tr(SA1 )]I Dt 2 2  2  DA1 λ 7 + λ4 A21 + (bL A1 )2 I = μ A1 + λ2 Dt 2 The contravariant derivative

DS Dt

in terms of material derivative

d dt

(26.3)

is defined as

DS dS = − LS − SLT Dt dt

(26.4)

The Cauchy tensor T is defined by: T = −ρI + S

(26.5)

where ρ is the density, I is the unit tensor and t is the time. It should be noted that the model (26.3) includes the following special cases: – λi = 0, i = 1, 2, . . . , 7 the model reduces to the classical model of a Newtonian fluid. – λ1 = λ3 = λ5 = λ6 = λ7 = 0 the second-grade fluid model. – λ2 = λ3 = λ4 = λ5 = λ6 = λ7 = 0 the Maxwell model. – λ3 = λ4 = λ5 = λ6 = λ7 = 0 the Oldroyd-B model. – λ5 = λ6 = λ7 = 0 the Johnson-Segalman model. – λ6 = λ1 = 0 the Oldroyd 6-constant model. The extra tensor of Oldroyd-6 constant fluid and velocity are respectively ⎛

⎞ ⎛ ⎞ Sxx Sxy Sxz 0 S(x) = ⎝ Syx Syy Syz ⎠, V(x) = ⎝ v(x) ⎠ 0 Szx Szy Szz

(26.6)

The equation of continuity and momentum equation are respectively divV = 0

(26.7)

314

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

ρ

 ∂V + ρ V · gradV = divT ∂t

(26.8)

From Eq. (26.6), the continuity Eq. (26.7) is satisfied identically, and the momentum Eq. (26.8) can be written as dp =

dSxy dx

(26.9)

where Sxy

 3 dv 1 dv = μ + μα1 M dx dr  M = 1 + α2

dv dx

(26.10)

2 (26.11)

α1 = λ1 λ4 − (λ3 + λ5 )(λ4 − λ2 )

(26.12)

α2 = λ1 λ3 − (λ3 + λ5 )(λ3 − λ1 )

(26.13)

and the modified pressure p becomes:

 2  4 dv dv 1 p = p1 − + μ(λ4 α2 − λ3 α1 ) μ(λ4 − λ3 ) M dx dx

(26.14)

From Eqs. (26.9) and (26.10), after nondimensionalization, we obtain  2 2  4  2 dv d v dx d v d2 v + (3α1 − α2 ) + α1 α2 2 2 2 dx dx dx dx dx

 2 2 dv − m 1 + α2 =0 dx

(26.15)

where α1 and α2 are given in terms of nondimensional material constants and m is a nondimensional gravity parameter. The boundary conditions are

v(0) = 1,

 3 dv dv + α1 dx dx

x=1

=0

(26.16)

26.2 OAFM for Eqs. (26.15) and (26.16)

315

26.2 OAFM for Eqs. (26.15) and (26.16) In what follows we apply our procedure to obtain the approximate solutions of Eq. (26.15) and (26.16), and the exact solution in some particular cases. For α2 = 0, Eq. (26.15) is replaced with the expression 

 2 2



mα2 1 + α2 v



d α1 v (α2 − α1 )v   −1 =0 + dx mα2 mα2 1 + α2 v2

(26.17)

or will the equivalent form

d α1 v  (α2 − α1 )v   −1=0 + dx mα2 mα2 1 + α2 v2

(26.18)

Hence, by integrating last equation, we can get α1 v  (α2 − α1 )v   =x+k + mα2 mα2 1 + α2 v2

(26.19)

where k is constant of integration. For x = 1 in the Eq. (26.19), it holds that mα2 (α1 − α2 )(1 + k) = 0 α1

(26.20)

For α1 = α2 , follows that k = -1. Substituting last result into Eq. (26.19), after some manipulation, one retrieves α1 v3 (x) + mα2 (1 − x)v2 (x) + v (x) + m(1 − x) = 0, α2 = 0, α1 = α2

(26.21)

Generally speaking, it is possible to distinguish the following cases from Eq. (26.21) with the boundary conditions (26.16). 26.2.1 Case 1 For α2 = 0 within Eq. (26.15) we obtain the expression v (x) + 3α1 v2 (x)v (x) − m = 0

(26.22)

v (x) + α1 v3 (x) − mx = k

(26.23)

or by integration

316

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

where k is constant of integration. For x = 1 in this last equation and making use Eq. (26.16), we obtain k = −m such that the Eq. (26.36) becomes α1 v3 (x) + v (x) + m(1 − x) = 0, v(0) = 1

(26.24)

To solve nonlinear differential Eq. (26.24) we will apply OAFM. For this purpose, the linear operator, nonlinear operator and function g(x) are respectively   L v (x) = v (x); N v (x) = α1 v3 (x); g(x) = m(1 − x)

(26.25)

The linear operator and the function g(x), are not unique. Also we can choose   L v (x) = v (x); N v (x) = α1 v3 (x) + m(1 − x); g(x) = 0

(26.26)

  L v (x) = v (x) + m(1 − x); N v (x) = α1 v3 (x); g(x) = 0

(26.27)

or

and so on. Let as consider only the Eq. (26.25). The approximate solution of Eq. (26.24) be consider as v (x) = v0 (x) + v1 (x)

(26.28)

where v0 (x) can be obtained from Eqs. (2.5) and (26.25) v0 (x) + m(1 − x) = 0

(26.29)

v0 (x) = −m(1 − x)

(26.30)

It holds that

Substituting Eq. (26.30) into Eq. (26.25), we obtain  N v0 (x) = −α1 m3 (1 − x)3

(26.31)

As it was considered in Chapter 2 and taking into consideration Eqs. (26.30) and (26.31), the auxiliary functions are f1 (x) = −1; f2 (x) = −(1 − x); f3 (x) = −(1 − x)2 , . . . , fp−1 (x) = −(1 − x)p (26.32) where p is an arbitrary positive number at this moment.

26.2 OAFM for Eqs. (26.15) and (26.16)

317

  The first approximation v1 x, Cj is obtained from Eqs. (2.13) and (26.32) v1 (x) − C1 − C2 (1 − x) − C3 (1 − x)2 − . . . − Cp+1 (1 − x)p = 0

(26.33)

The approximate solution of the Eq. (26.37) is obtained from Eqs. (26.28), (26.30) and (26.33) v (x) = C1 + (C2 − m)(1 − x) + C3 (1 − x)2 + . . . + Cp+1 (1 − x)p

(26.34)

where the constants Cj , j = 1, 2, 3, . . . , p + 1 are determined considering the Eqs. (2.31), where    R x1 , C1 , C2 , . . . , Cp−1 = α1 v3 1 + v (x) + m(1 − x)

(26.35)

The algebraic system (2.31) can has a unique solution if α1 > 0 or three solutions for α1 < 0, as follows. 26.2.1.1 Subcase 1 We consider α1 = 0.1, m = 1 and will show that the accuracy of the result obtained by means of OAFM is growing along with increasing the number p of the parameters Cj which appear in Eq. (26.34). For p = 7 into Eq. (26.34) and using Eq. (2.31) we obtain a unique solution because Eq. (26.16) lead to v (1) = 0 C1 = 0; C2 = −0.00002236287365; C3 = 3.879902736 · 10−4 ; C4 = 0.09728242783; C5 = 0.0101436325225; C6 = −0.0520184879006 C7 = 0.027681371076; C8 = −0.00515332082 (26.36) The approximate solution of Eq. (26.15), in this subcase with 8 parameters is obtained from Eq. (26.34) v (x) = −1.00002236287365(1 − x) + 3.879902736 · 10−4 (1 − x)2 + 0.09728242783(1 − x)3 + 0.0101436325225(1 − x)4 − 0.0520184879006(1 − x)5 + 0.027681371076(1 − x)6 − 0.00515332082(1 − x)7

(26.37)

The velocity is obtained by integrating Eq. (26.37) with condition obtained from Eq. (26.16)1 :v(0) = 1  v (x) = 1 + 0.5000001118143682 (1 − x)2 − 1   − 1.293300912 · 10−4 (1 − x)3 − 1 − 0.0243206069575 (1 − x)4 − 1   − 0.002028726504 (1 − x)5 − 1 + 0.008669747983 (1 − x)6 − 1

318

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

  − 0.00395448158143 (1 − x)7 − 1 + 6.441651025 (1 − x)8 − 1 (26.38) Now, if we consider p = 8 into Eq. (26.34), one can get C1 = 0; C2 = −0.00002373299519; C3 = 3.962291091 · 10−4 ; C4 = 0.097277570627987; C5 = 0.010077184054; C6 = −0.0518468476021 C7 = 0.027523109361; C8 = −0.00510254167; C9 = 2.9971342 · 10−8 (26.39) and the approximate solution of Eq. (26.15) with 9 parameters is v (x) = −1.00002373299519(1 − x) + 3.962291091 · 10−4 (1 − x)2 + 0.097277570627987(1 − x)3 + 0.010077184054(1 − x)4 − 0.0518468476021(1 − x)5 + 0.02752310936(1 − x)6 − 0.00510254167(1 − x)7 + 2.9971342 · 10−8 (1 − x)8

(26.40)

The corresponding velocity for this subcase becomes  v(x) = 1 + 0.5000001186649759 (1 − x)2 − 1   − 1.320763697 · 10−4 (1 − x)3 − 1 − 0.024319392655 (1 − x)4 − 1   − 0.002015436811 (1 − x)5 − 1 + 0.00864114126667 (1 − x)6 − 1   − 0.003931872765714 (1 − x)7 − 1 + 6.378177088 · 10−4 (1 − x)8 − 1  − 3.33014444 · 10−9 (1 − x)9 − 1 (26.41) The Table 26.1 are given the values of v (x) from Eqs. (26.37) and (26.40) in comparison with the numerical results. It can be seen that the obtained approximate solution are nearly identical with numerical results and the results obtained using 9 parameters are more accurate than the results obtained using 8 parameters. In Figs. 26.1 and 26.2 are graphically presented the approximate solution (26.40) for v (x) and approximate solution (26.41) for v(x) respectively, and numerical solutions. 26.2.1.2 Subcase 2 For α1 = −0.1 and m = 1. From Eq. (26.16)2 , one can get three solutions for the derivative of the velocity: b1 ) v (1) = 0. The values of the convergence-control parameters Cj , j = 1, 2, . . . , 9 which appear in the approximate solution (26.34) in this subcase, are C1 = 0; C2 = 0.00603307647341; C3 = −0.09673865302428; C4 = 0.53033209040411; C5 = −2.18139193291359; C6 = 4.3375162871814

26.2 OAFM for Eqs. (26.15) and (26.16)

319

Table 26.1 The results of v (x) obtained from Eqs. (26.37) and (26.40) and numerical solution for α1 = 0.1, α2 = 0, m = 1 x

v (x), v (x), Equation (26.37) Equation (26.40)

v’(x) Numerical

0

−0.9216987491

−0.921698998

−0.9216989941 4.5 E-07

4.1 E-09

0.1 −0.8406019253

−0.8406020769

−0.8406020638 4.4 E-07

1.3 E-08

0.2 −0.7566757833

−0.75667590581

−0.7566758866 1.1 E-07

1.9 E-08

0.3 −0.669932639

−0.6699327712

−0.669932755

1.2 E-07

1.6 E-08

0.4 −0.5804437771

−0.580443969603 −0.5804439612 1.8 E-07

8.4 E-09

0.5 −0.4883531695

−0.4883533174

−0.4883533129 1.4 E-07

4.5 E-09

0.6 −0.3938888436

−0.3938888928

0.393888875

3.1 E-08

1.7 E-08

0.7 −o.2973703527

−0.297370378801 −0.2973703791 2.6 E-08

2.9 E-10

0.8 −0.1992094038

−0.199209448

1.2 E-09

0.9 −0.099905525

−0.099905964

1.0 0

ε 1 =  v (x) − v (x) Equation (26.37)

−0.1992094492 4.5 E-07

ε 1 =  v (x) − v (x) Equation (26.40)

−0.099905982

4.6 E-08

1.8 E-08

0

0

0

Fig. 26.1 Comparison between the approximate solution (26.40) and numerical sol. for α1 = 0.1,α2 = 0, m = 1 Eq. (26.40) Numerical

C7 = −5.101846548802; C8 = 3.2378586626; C9 = −0.8852402875

(26.42)

The first approximate solution of Eq. (26.15) is v1 (x) = −0.9939669235265(1 − x) − 0.0967386583024(1 − x)2 + 0.530332090411(1 − x)3 − 2.1813913291353(1 − x)4 + 4.3375162871814(1 − x)5 − 5.101846548802(1 − x)6 + 3.23785866296(1 − x)7 − 0.88524028275(1 − x)8

(26.43)

320

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

Fig. 26.2 Comparison between the approximate solution (26.41) and numerical sol. for α1 = 0.1, α2 = 0, m = 1 Eq. (26.41) Numerical

A comparison between the values of the first approximate solution (26.43) and numerical results, for α1 = −0.1, α2 = 0 and m = 1 is present in Table 26.2 The velocity v1 (x) corresponding to Eq. (26.43) is given by  v1 (x) = 1 + 0.49698346175 (1 − x)2 − 1   + 0.03224621943333 (1 − x)3 − 1 − 0.1325830226 (1 − x)4 − 1   + 0.4362783864 (1 − x)5 − 1 − 0.72291938116667 (1 − x)6 − 1   + 0.7288352211428 (1 − x)7 − 1 − 0.40473233275 (1 − x)8 − 1  − 0.098360031411 (1 − x)9 − 1 (26.44) Table 26.2 The results for the first approximate solution obtained from Eq. (26.43) for α1 = −0.1, α2 = 0, m = 1, and numerical solutions   x v 1 (x) v1 (x) ε = v1 (x) − v 1 (x) Equation (26.43) Numerical Equation (26.43) 0

−1.1534681051

−1.15346730514

0.1

−1.000007290062

−1.0

7.3 E-06

0.2

−0.864646007204

−0.864640887

5.1 E-06

0.3

−0.740628624114

−0.7406251939

3.4 E-06

0.4

−0.624338537201

−0.624336377

2.2 E-06

0.5

−0.51354477817

−0.513543528

1.5 E-06

0.6

−0.406723574523

−0.406728427

4.8 E-06

0.7

−0.302775906987

−0.302775641125

2.6 E-07

0.8

−0.20080983599

−0.200809756

8.0 E-08

0.9

−0.100013297786

−0.10001003012

3.2 E-06

1.0

0

0

0

8 E-07

26.2 OAFM for Eqs. (26.15) and (26.16)

321

Fig. 26.3 Comparison between the approximate solution (26.43) and numerical solution for α1 = −0.1, α2 = 0, m = 1 Eq. (26.43) Numerical

Fig. 26.4 Comparison between the approximate solution (26.44) and numerical sol. for α1 = −0.1, α2 = 0, m = 1 Eq. (26.44) Numerical

In Figs. 26.3 and 26.4 are graphically presented the first approximate solution (26.43) for v1 (x) and the first approximate velocity (26.44) for v1 (x) respectively and the numerical solution. b2) v (1) = √10.1 . The values of the convergence-control parameters Ci , i = 1,2,…,9 from the approximate solution (26.34) in this subcase are

C1 = 3.16227876101; C2 = 1.4994174962062; C3 = −0.109772938017; C4 = −0.003278766675; C5 = 0.14221324114903; C6 = −0.2870101064523 C7 = 0.302333670005; C8 = −0.16722870723; C9 = 0.0381378959375 (26.45) The second approximate solution of Eq. (26.15) for α1 = −0.1, α2 = 0, m = 1 and v (1) = √10.1 has the form

322

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

v2 (x) = 3.16227876101 + 0.4994174962062(1 − x) − 0.109772938017(1 − x)2 − 0.003278766675(1 − x)3 + 0.14221324114903(1 − x)4 − 0.2870101064523(1 − x)5 + 0.302333670005(1 − x)6 − 0.16722870723(1 − x)7 − 0.0381378959375(1 − x)8

(26.46)

The velocity v2 (x) corresponding to Eq. (26.46) is written as  v2 (x) = 1 + 3.16227876101x − 0.249708748103 (1 − x)2 − 1   + 0.036590979333 (1 − x)3 − 1 + 8.1969166875 · 10−4 (1 − x)4 − 1   − 0.02844264822 (1 − x)5 − 1 + 0.047835017733 (1 − x)6 − 1   − 0.0431905242857 (1 − x)7 − 1 + 0.0209035884 (1 − x)8 − 1  − 0.004237543992 (1 − x)9 − 1 (26.47) In Table 26.3, a comparison between the values of the second approximate solution (26.46) of Eq. (26.15) for α1 = −0.1, α2 = 0, m = 1 and numerical results is given. In Figs. 26.5 and 26.6 are depicted to second approximate solution (26.46) for v2 (x) and the second approximate velocity (26.47) for v2 (x) respectively, and the numerical solution. b3) v (1) = − √10.1 . The values of the convergence-control parameters Ci , i = 1,2,…,9 for the Eq. (26.34), are

Table 26.3 The results for v2 (x) obtained from Eq. (26.46) and numerical results for α1 = −0.1, α2 = 0, m = 1   x v2 (x), v’2 ε = v2 (x) − v2 (x) Equation (26.46) Numerical Equation (26.46) 0

3.5770905449

3.57708944514

1.1 E-06

0.1

3.5413823649

3.54138126515

1.1 E-06

0.2

3.5046652764

3.5046643531

9.2 E-07

0.3

3.4668610859

3.46685998672

1.1 E-06

0.4

3.4278805911

3.42787953312

1.0 E-06

0.5

3.3876201549

3.38761905599

1.1 E-06

0.6

3.3459632881

3.3459632975915

9.0 E-09

0.7

3.3027767409

3.302775641125

1.1 E-06

0.8

3.2578981158

3.25789701647

1.1 E-06

0.9

3.2111311387

3.211139353322

8.2 E-06

1

3.162278760168

3.162277660168

1.1 E-06

26.2 OAFM for Eqs. (26.15) and (26.16)

323

Fig. 26.5 Comparison between the second approximate solution v2 (x) given by (26.46) and numerical solution for α1 = −0.1, α2 = 0, m = 1 Eq. (26.46) Numerical

Fig. 26.6 Comparison between the approximate solution v2 (x) given by (26.47) and numerical sol. for α1 = −0.1, α2 = 0, m=1 Eq. (26.47) Numerical

C1 = 3.162277660168; C2 = 1.49339067787234; C3 = 0.2240342039305; C4 = −0.6329341661717; C5 = 2.37298743597; C6 = −4.648761097271 C7 = 5.41399825632; C8 = −3.40774987836; C9 = 0.92372008825 (26.48) The third approximate solution of Eq. (26.15) for α1 = −0.1, α2 = 0, m = 1 becomes v3 (x) = 3.162277660168 + 0.49339067787234(1 − x) + 0.22403422039305(1 − x)2 − 0.6329341661717(1 − x)3 + 2.37295743597(1 − x)4 − 4.648761097271(1 − x)5 + 5.413998256532(1 − x)6 − 3.40774987836(1 − x)7 + 0.92372008825(1 − x)8

(26.49)

324

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

The velocity v3 (x) obtained from Eq. (26.49) by integration is  v3 (x) = 1 − 3.162277660168x − 0.24669533893617 (1 − x)2 − 1   − 0.0746780679667 (1 − x)3 − 1 + 0.158233541525 (1 − x)4 − 1   − 0.47459148715 (1 − x)5 − 1 + 0.7747935161667 (1 − x)6 − 1   − 0.773428322286 (1 − x)7 − 1 + 0.42596873475 (1 − x)8 − 1  − 0.1026355653555 (1 − x)9 − 1 (26.50) In Table 26.4 is presented a comparison between some values of v3 (x) obtained from Eq. (26.49) and numerical results. In Figs. 26.7 and 26.8 are illustrated approximate solution (26.49) and approximate solution (26.50) respectively, and numerical results. 26.2.2 Case 2 For α1 = α2 into Eq. (26.18) imply that  d v −1=0 dx m

(26.51)

Integrating twice the last equation, we obtain v(x) =

m 2 ˜ x + kx + k˜˜ = 0 2

(26.52)

Table 26.4 Results for v3 (x) obtained from Eq. (26.49) and numerical results for α1 = −0.1, α2 = 0, m = 1   x v3 (x), v’3 (x) ε = v3 (x) − v3 (x) Equation (26.49) Numerical Equation (26.49) 0

−2.4236130402

−2.42362214

9.1 E-06

0.1

−2.54137216524

−2.54138126515

9.1 E-06

0.2

−2.64001736516

−2.6400234655

9.1 E-06

0.3

−2.726225692106

−2.72623479223

9.1 E-06

0.4

−2.803532733077

−2.803541833

9.1 E-06

0.5

−2.874066427058

−2.874075527

9.1 E-06

0.6

−2.93923215753

−2.9392348705

2.7 E-06

0.7

−2.99999089993

−3

9.1 E-06

0.8

−3.057078159854

−3.05708726

9.1 E-06

0.9

−3.1111263937541

−3.1111320514

5.6 E-06

1

−3.16226856016

−3.162277660168

9.1 E-06

26.2 OAFM for Eqs. (26.15) and (26.16)

325

Fig. 26.7 Comparison between the second approximate solution v3 (x) given by (26.49) and numerical solution for α1 = 0.1, α2 = 0, m = 1 Eq. (26.49) Numerical

Fig. 26.8 Comparison between the approximate solution v3 (x) given by (26.50) and numerical sol. for α1 = 0.1, α2 = 0, m = 1 Eq. (26.50) Numerical

where the constants k˜ and k˜˜ can be identified from the boundary conditions (26.16). The number of real solutions of Eq. (26.16)2 depends of the sign of α1 . 26.2.2.1 Subcase 1 If α1 = α2 > 0, then Eq. (26.16)2 , has a unique solution. v (1) = 0. In this subcase follows that k˜ = −m, k˜˜ = 1. The exact solution of Eq. (26.15), for α1 = α2 > 0 is v(x) =

m 2 x − mx + 1 2

(26.53)

26.2.2.2 Subcase 2 If α1 = α2 < 0, then Eq. (26.16)1 has three distinct solutions 1 1 ; v3 (1) = − √ v1 (1) = 0; v2 (1) = √ −α1 −α1

(26.54)

326

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

In this subcase the exact solution of Eq. (26.15) are obtained from Eqs. (26.52), (26.54) and (26.16)1 , and are respectively m 2 x − mx + 1 2  1 m v2 (x) = x2 + √ −m x+1 2 −α1  1 m 2 v3 (x) = x − √ +m x+1 2 −α1 v1 (x) =

(26.55) (26.56) (26.57)

26.2.3 Case 1 For α1 = α2 , α2 = 0 and α1 > 0 we study the effects of the non-Newtonian parameters, α1 , α2 and m on the velocity field and on the derivative of the velocity. In all subcases the approximate solution of Eq. (26.35) and (26.16) is unique. To solve nonlinear differential Eq. (26.35) we will apply OAFM with linear operator, nonlinear operator and the function g(x) of the form   L v (x) = v (x); N v (x) = α1 v3 (x) + mα2 (1 − x)v2 (x); g(x) = m(1 − x)

(26.58)

Similar to Case 3.24.2.1., the initial approximation v0 of the approximate solution v(x) = v0 (x) + v1 (x), is obtained from the linear equation v0 + m(1 − x) = 0, v0 (1) = 0

(26.59)

v0 (x) = −m(1 − x)

(26.60)

Follows that

Substituting Eq. (26.60) into Eq. (26.58)2 , it holds that  N v0 (x) = (α2 − α1 )m3 (1 − x)3

(26.61)

From Eqs. (26.59) and (26.61), follows that auxiliary functions can be chosen of the form f1 (x) = −1, f2 (x) = −(1 − x), f3 (x) = −(1 − x)2 , . . . , f9 (x) = −(1 − x)8 (26.62) The first approximation v1 (x) is obtained from the equation v1 (x) − C1 − C2 (1 − x) − C3 (1 − x)2 + . . . + C9 (1 − x)8 = 0

(26.63)

26.2 OAFM for Eqs. (26.15) and (26.16)

327

such that approximate solution of Eq. (26.15) becomes v1 = C1 − (C2 − m)(1 − x) + C3 (1 − x)2 + . . . + C9 (1 − x)8

(26.64)

In what follows, we consider 7 subcase. 2.3a Subcase For α1 = 21 , α2 = 43 , m = 1 the values of the convergence –control parameters Cj are C1 = 0; C2 = −0.001143402102; C3 = 0.0180138108175; C4 = −0.36403707256735; C5 = 0, 37609902980414; C6 = −0.69246912033634; C7 = 0.69091581731622; C8 = −0, 2644761734; C9 = 0.025438124375

(26.65)

such that the approximate solution (26.64) becomes v (x) = 1.001143402102(1 − x) + 0.0180138108175(1 − x)2 − 0.36403707256735(1 − x)3 + 0.37609902980414(1 − x)4 − 0.69246912033634(1 − x)5 + 0.6909091581731622(1 − x)6 − 0.2644761734(1 − x)7 + 0.025438124375(1 − x)8

(26.66)

The corresponding approximate velocity is  v(x) = 1 + 0.500571701051 (1 − x)2 − 1   − 0.006004603603 (1 − x)3 − 1 − 0.091009268125 (1 − x)4 − 1   − 0.07521980596 (1 − x)5 − 1 + 0.11541152005 (1 − x)6 − 1   − 0.0987022596143 (1 − x)7 − 1 + 0.033059521675 (1 − x)8 − 1  − 0.0282645826333 (1 − x)9 − 1 (26.67) In Table 26.5 are given some values of v (x) obtained from Eq. (26.66) in comparison with the numerical results. 26.2.3.2 Subcase 2 For α1 = 1, α2 = 43 , m = 1, the values of the convergence-control are C1 = 0; C2 = 0.00063930225404; C3 = 0.00980623018; C4 = 0.3073530341498; C5 = −0.1482852389282; C6 = −0.2995770483424; C7 = 0.473507526392; C8 = −0, 27280991348; C9 = 0.0594166002

(26.68)

328

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

Table 26.5 Results for v (x) obtained from Eq. (26.66) and numerical results for α1 = 21 , α2 = 43 , m=1 v’(x) Numerical

  ε = v (x) − v (x) Equation (26.66)

−1.21165898592

−1,2,116,591,730

1.9 E-07

−1.0623253618677

−1.062315451

9.0 E-08

0.2

−0.9187084116

−0.9187085631

1.5 E-07

0.3

−0.7819489758

0.7819491092

1.3 E-07

0.4

−0.6526780521

−0.6526781649

1.1 E-07

0.5

−0.5308776458

−0.5308777412

9.9 E-08

0.6

−0.41592277785

−0.4159219291

8.5 E-08

0.7

−0.3067395707

−0.3067316279

5.7 E-08

0.8

−0.2019993578

−0.2019993959

3.8 E-08

0.9

−0.10026689921

0.1002699951

3.1 E-06

1.0

0

0

0

x

v (x), Equation (26.66)

0 0.1

In this subcase the approximate solutions v (x) and v(x) are respectively v (x) = −0.99936069774592(1 − x) − 0.00980623018(1 − x)2 + 0.3073530341498(1 − x)3 − 0.1482852389282(1 − x)4 − 0.2995770483424(1 − x)5 + 0.473507526392(1 − x)6 − 0.27280991348(1 − x)7 + 0.0594166002(1 − x)8

(26.69)

 v(x) = 1 + 0.49968034885 (1 − x)2 − 1   − 0.003268743393 (1 − x)3 − 1 − 0.076838258525 (1 − x)4 − 1   + 0.02965704778 (1 − x)5 − 1 + 0.04992950805 (1 − x)6 − 1   − 0.06764393232 (1 − x)7 − 1 + 0.034101239175 (1 − x)8 − 1  − 0.006601844467 (1 − x)9 − 1 (26.70) In Table 26.6, the numerical results are compared with the approximate results obtained from Eq. (26.69) 2.3c Subcase For α1 = 15, α2 = 43 , m = 1 one can get C1 = 0; C2 = 0.00529341557; C3 = −0.094315275574; C4 = 1.43664244695315; C5 = −2.57494213220618; C6 = 2.506423537639; C7 = −1.437159787325;

26.2 OAFM for Eqs. (26.15) and (26.16) Table 26.6 Results of v (x) obtained from Eq. (26.69) and numerical results for α1 = 1, α2 =

3 4

329

x

v (x), Equation (26.69)

v’(x) Numerical

ε =  v (x) − v (x) Equation (26.69)

0

−0.8895619719

−0.8895618639

1.1 E-07

0.1

−0.8107604338

−0.8107603399

8.3 E-08

0.2

−0.7304196102

−0.7304195271

8.9 E-08

0.3

−0.6482227845

−0.6482227118

7.3 E-08

0.4

−0.5638182786

−0.5638182157

6.3 E-08

0.5

−0.4768430636

−0.4768430116

5.2 E-08

0.6

−0.3869749988

−0.3869754307

4.3 E-07

0.7

−0.2940318986

−0.2940318678

3.1 E-08

0.8

−0.1981117209

−0.19811170018

3.1 E-08

0.9

−0.0997441568

0.0997449982

8.4 E-07

1.0

0

0

0

and m = 1

C8 = 0.4507823808; C9 = −0.05868459325

(26.71)

v (x) = −0.99470658443(1 − x) − 0.094315275574(1 − x)2 + 1.43664244695315(1 − x)3 − 2.57494213220618(1 − x)4 + 2.506423537634(1 − x)5 − 1.437159784325(1 − x)6 + 0.4507823808(1 − x)7 − 0.05868459325(1 − x)8

(26.72)

 v(x) = 1 + 0.4973532922 (1 − x)2 − 1   + 0.03143842519 (1 − x)3 − 1 − 0.3591606115 (1 − x)4 − 1   + 0.5149884264 (1 − x)5 − 1 − 0.41773725616667 (1 − x)6 − 1   + 0.2055308541 (1 − x)7 − 1 − 0.0563477976 (1 − x)8 − 1  + 0.00652051036 (1 − x)9 − 1 (26.73) In Table 26.7, a comparison between the values of the approximate solution (26.72) and numerical results is presented. Figures 26.9 and 26.10 have been displayed the variations of v (x) and v(x) respectively, with increasing parameter α1 for α2 = 34 and m = 1. From these figures it is evident that v (x) and v(x) increase with the increase of α1 . 2.3d Subcase For α1 = 1, α2 = 0.5, m = 1, we obtain C1 = 0; C2 = 0.002328044595; C3 = −0.0389247788458;

330

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

Table 26.7 Result of v (x) obtained from Eq. (26.72) and numerical results for α1 = 1.5, α2 = 43 , m=1

  ε = v (x) − v (x) Equation (26.72)

x

v (x), Equation (26.72)

0

−0.7659540084

−0.7659535804

4.3 E-07

0.1

−0.7071346666

−0.7071342848

3.8 E-07

0.2

−0.6460052832

−0.64600494392

3.4 E-07

0.3

−0.58220651319

−0.5820648347

2.9 E-07

0.4

−0.5146908688

−0.5146906144

2.5 E-07

0.5

−0.44312007407

−0.443119862156

2.1 E-07

0.6

−0.3664666471

−0.3664660658

5.8 E-07

0.7

−0.2838285798

−0.28382845266

1.3 E-07

0.8

−0.1946237987

−0.19462371389

8.5 E-08

0.9

−0.0992103913

−0.09921065211

2.6 E-07

1.0

0

0

0

v’(x) Numerical

Fig. 26.9 Variation of v (x) , α1 = 0.5 Eq. (26.67)blue line , α1 = 1 Eq. (26.70)red line , α1 = 1.5 Eq. (26.73)-green line.

C4 = 0.75873887101689; C5 = −0.84751381806795; C6 = 0.2205912345775; C7 = 0.316797530938; C8 = −0.2920555813; C9 = 0.07627066085375

(26.74)

v (x) = −0.9947671550496(1 − x) − 0.03892477884(1 − x)2 + 0.75873887101689(1 − x)3 − 0.84751381806795(1 − x)4 + 0.2205912345775(1 − x)5 + 0.316797530938(1 − x)6 − 0.2920555813(1 − x)7 + 0.0762766085375(1 − x)8

(26.75)

26.2 OAFM for Eqs. (26.15) and (26.16)

331

Fig. 26.10 Variation of v(x) α1 = 0.5 Eq. (26.67)blue line, α1 = 1 Eq. (26.70)red line, α1 = 1 Eq. (26.73)green line α1 = 1.5

 v(x) = 1 + 0.4988357752 (1 − x)2 − 1   + 0.01297492628 (1 − x)3 − 1 − 0.18968474775 (1 − x)4 − 1   + 0.1695027636 (1 − x)5 − 1 − 0.03676520575 (1 − x)6 − 1   − 0.04525679028571 (1 − x)7 − 1 − 0.0365069476625 (1 − x)8 − 1  + 0.0084751787255 (1 − x)9 − 1 (26.76) In Table 26.8, a comparison between the values of the approximate solution (26.75) and numerical results is presented. Table 26.8 Result of v (x) obtained from Eq. (26.75) and numerical results for α1 = 1, α2 = 43 , m=1 x

v (x), Equation (26.75)

v’(x) Numerical

  ε = v (x) − v (x) Equation (26.75)

0

−0.8037614752

−0.80376088332

5.9 E-07

0.1

−0.7406052308

−0.7406046907

5.4 E-07

0.2

−0.6748379523

−0.6748374722

4.8 E-07

0.3

−0.6059930186

−0.6059925986

4.2 E-07

0.4

−0.5335268983

−0.5335265383

3.6 E-07

0.5

−0.4568345185

−0.4568342159

3.0 E-07

0.6

−0.3753061189

−0.375307056669

9.4 E-07

0.7

−0.2884754936

−0.2884762136

7.2 E-07

0.8

−0.1962900912

−0.1962899712

1.2 E-07

0.9

−0.0994798611

−0.0994797451104

1.1 E-07

1.0

0

0

0

332

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

26.2.3.5 Subcase 5 For α1 = 1, α2 = 0.25, m = 1 we have C1 = 0; C2 = 0.00480400033812; C3 = −0.082578859666378; C4 = 1.324999217386; C5 = −2.0361176356477; C6 = 1.5875116501345; C7 = −0.616722474615; C8 = 0.0643599987; C9 = 0.01741772175

(26.77)

v (x) = −0.995195941684(1 − x) − 0.082578859666378(1 − x)2 + 1.324999217386(1 − x)3 − 2.0361176356477(1 − x)4 + 1.5875116501345(1 − x)5 − 0.616722474615(1 − x)6 + 0.064359987(1 − x)7 + 0.01741772175(1 − x)8

(26.78)

 v(x) = 1 + 0.4975979708 (1 − x)2 − 1   + 0.02752628655333 (1 − x)3 − 1 − 0.33124980425 (1 − x)4 − 1   + 0.4072235271289 (1 − x)5 − 1 − 0.2645852750224 (1 − x)6 − 1   + 0.08810321065714 (1 − x)7 − 1 − 0.00804499838 (1 − x)8 − 1  + 0.00193530245666 (1 − x)9 − 1 (26.79) In Table 26.9 is presented a comparison between the approximate solution (26.78) and numerical results. Table 26.9 Result of v (x) obtained from Eq. (26.78) and numerical results for α1 = 1, α2 = 0.25, m=1   x v (x), v’(x) ε = v (x) − v (x) Equation (26.78) Numerical Equation (26.78) 0

−0.7363263814

−0.7363258783

5.1 E-07

0.1

−0.6845987356

−0.6845986828

1.5 E-07

0.2

−0.6296562542

−0.62965585178

4.0 E-07

0.3

−0.5709372334

−0.5709368813948

3.5 E-07

0.4

−0.5077616828

−0.5077613810842

3.0 E-07

0.5

−0.4393308639

−0.439330612417

2.5 E-07

0.6

−0.3647687926

−0.3647683449

4.5 E-07

0.7

−0.2832851903

−0.28328503942

1.5 E-07

0.8

−0.1945307469

−0.194530646359

1.0 E-07

0.9

−0.09920873609

−0.0992081493

5.8 E-07

1

0

0

0

26.2 OAFM for Eqs. (26.15) and (26.16)

333

Figures 26.11 and 26.12 have been displayed the variations of v (x) and v(x) respectively, with increasing parameter α2 for α1 = 1,m = 1. From these figures it is clear that v (x) and v(x) decrease with the increase of α2 .Eq. (26.76)-blue line 26.2.3.6 Subcase 6 For α1 = 1, α2 = 43 , m = 1, 5, we obtain C1 = 0; C2 = 0.0065336777154; C3 = −0.117864430573377; C4 = 1.71657870359306; C5 = −3.34404109955317; C6 = 3.5150602572346; C7 = −2.198597317828 C8 = −0.7702799085; C9 = −0.116650570375 v (x) = −1.4934663222846(1 − x) − 0.117864430573577(1 − x)2 Fig. 26.11 Variation of v (x) Eq. (26.70)green line, α2 = 0.25 Eq. (26.76)red line, α2 = 0.5 Eq. (26.78)blue line, α2 = 0.75

Fig. 26.12 Variation of v(x) , α2 = 0.25 Eq. (26.70)green line, α2 = 0.5 Eq. (26.76)red line, α2 = 0.75

(26.80)

334

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

+ 1.71657870359306(1 − x)3 − 3.3440410995317(1 − x)4 + 3.515060257234(1 − x)5 − 2.198597317828(1 − x)6 + 0.7702799085(1 − x)7 − 0.116650570375(1 − x)8

(26.81)

 v(x) = 1 + 0.74673316111423 (1 − x)2 − 1   + 0.0392881091192 (1 − x)3 − 1 − 0.42914467575 (1 − x)4 − 1   + 0.6688082199106 (1 − x)5 − 1 − 0.5858433761667 (1 − x)6 − 1   + 0.314085331075 (1 − x)7 − 1 − 0.0962849885625 (1 − x)8 − 1  + 0.0129611744778 (1 − x)9 − 1 (26.82) In Table 26.10 is presented a comparison between the approximate solution (26.81) and numerical results. 26.2.3.7 Subcase 7 For α1 = 1, α2 = 34 , m = 2, the values of the convergence-control parameters Cj are C1 = 0; C2 = 0.0062660397944; C3 = −0.16939866644895; C4 = 3.805250532482; C5 = −9.662358550631; C6 = −13.035363243779; C7 = −10.35186684227 C8 = 4.565122602; C9 = −0.8643778195

(26.83)

In this last subcase, the approximate solution v (x) and v(x) are respectively Table 26.10 Result of v (x) obtained from Eq. (26.81) and numerical results for α1 = 1, α2 = 43 , m = 1.5 x

v (x), Equation (26.81)

v’(x) Numerical

  ε = v (x) − v (x) Equation (26.81)

0

−1.2687008703

−1.12687001763

6.9 E-07

0.1

−1.1568379478

−1.15683732002

6.3 E-07

0.2

−1.0436026945

−1.0436021358

5.6 E-07

0.3

−0.9284731780

−0.9284726881

4.9 E-07

0.4

−0.8107607595

−0.8107603397

4.2 E-07

0.5

−0.6895748039

−0.6895744549

3.5E-07

0.6

−0.5638169119

−0.56381771591

8.1 E-07

0.7

−0.4322871775

−0.4322869676

2.1 E-07

0.8

−0.2940320077

−0.2940318678

1.4 E-07

0.9

−0.1491100741

−0.149110878734

8.0 E-07

1

0

0

0

26.2 OAFM for Eqs. (26.15) and (26.16)

335

v (x) = −1.99373396020564(1 − x) − 0.16939866644895(1 − x)2 + 3.805250532482(1 − x)3 − 9.662358550631(1 − x)4 + 13.05363243775(1 − x)5 − 10.35186684227(1 − x)6 + 4.565122602(1 − x)7 − 0.8643778195(1 − x)8

(26.84)

 v(x) = 1 + 0.996866980102 (1 − x)2 − 1   + 0.0564662221333 (1 − x)3 − 1 − 0.951312633121 (1 − x)4 − 1   + 1.932471710126 (1 − x)5 − 1 − 2.17256054063 (1 − x)6 − 1   + 1.47883812032 (1 − x)7 − 1 − 0.57064032525 (1 − x)8 − 1  + 0.0960419799444 (1 − x)9 − 1 (26.85) In Table 26.11 is presented a comparison between the approximate solution (26.84) and numerical results. In Figs. 26.13 and 26.14 are depicted the variation of v (x) and v(x) respectively with increasing parameter m for α1 = 1, α2 = 0.75. From these figures it is clear that v (x) and v(x) decrease with the increase of m. 26.2.4 Case 1 For α1 = α2 , α2 = 0 and α1 < 0 we will show that Eqs. (35) and (16) have a single solution on the domain [0,1], but on a subdomain included in [0,1] there are still two solutions for v (x). On the other hand, there is a single solution for velocity on domain [0,1]. For this purpose, we consider Table 26.11 Results of v (x) obtained from Eq. (26.84) and numerical results for α1 = 1, α2 = 43 , m=2 x

v (x), Equation (26.84)

v’(x) Numerical

  ε = v (x) − v (x) Equation (26.84)

0

−1.635994619

−1.6359991617

3.1 E-07

0.1

−1.4897764361

−1.4897761665

2.7 E-07

0.2

−1.34271125836

−1.3427107537797

5.0 E-07

0.3

−1.1942559409

−1.19425573148544

2.1 E-07

0.4

−1.0436023149

−1.0436021358

1.8 E-07

0.5

−0.88956201314

−0.88956186328

1.5 E-07

0.6

−0.7304235842

−0.7304241402

5.5 E-07

0.7

−0.5638183056

−0.5638182157646

9.0 E-08

0.8

−0.3869754905

−0.3869754306

6.0 E-08

0.9

−0.1981078583

−0.19810700169

8.6 E-08

1

0

0

0

336

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

Fig. 26.13 Variation of v (x) Eq. (26.84)green line, m = 1 Eq. (26.89)red line, m = 1.5 Eq. (26.73)blue line, m=2

Fig. 26.14 Variation of v(x) Eq. (26.84)green line, m = 1 Eq. (26.89)red line, m = 1.5 Eq. (26.73)blue line m=2

1 1 α1 = − , α2 = , m = 1 3 3

(26.86)

From the boundary conditions (26.16) one retrieves three subcase: 26.2.4.1 Subcase 1 √ For v (1) = 3, the convergence-control parameters Cj are √ C1 = 3; C2 = 1.99985577301647; C3 = −0.286038430687; C4 = 0.312571693849; C5 = −0.3142201122083; C6 = 0.2684760111435; C7 = −0.1668815886552 C8 = 0.06400834387; C9 = −0.011147991075 The approximate solution v (x) and v(x) are respectively

(26.87)

26.2 OAFM for Eqs. (26.15) and (26.16)

v (x) =

√

337

3 + 0.99985619301674(1 − x) − 0.286038430687(1 − x)2

+ 0.31251693849(1 − x)3 − 0.3142201122083(1 − x)4 + 0.268470111435(1 − x)5 − 0.1668815886552(1 − x)6 + 0.06400834387(1 − x)7 − 0.011147991075(1 − x)8

(26.88)

√

 3x − 0.4999280965 (1 − x)2 − 1   + 0.0953461435333 (1 − x)3 − 1 − 0.07814292345 (1 − x)4 − 1   + 0.06284402244 (1 − x)5 − 1 − 0.04474600185 (1 − x)6 − 1   + 0.023840226943 (1 − x)7 − 1 − 0.0088001042984 (1 − x)8 − 1  + 0.001238665674 (1 − x)9 − 1 (26.89)

v(x) = 1 +

In Table 26.12 is presented a comparison between the approximate solution (26.88) and numerical results. 26.2.4.2 Subcase 2 For v (1) = 0, it is very interesting to remark that the function v (x) is not continuous in the critic point xc = 0.479894418625 and the convergence-control parameters Cj are well determined only on subdomain [xc , 1]. These parameters are C1 = 0; C2 = 2.6309957225; C3 = −59.99867574852; C4 = 490.882847891; C5 = −1849.05441536; C6 = 3235.9311998; C7 = −2132.431791958872 Table 26.12 Results of v (x) obtained from Eq. (26.88) and numerical results for α1 = − 13 , √ α2 = 13 , m = 1, v (1) = 3   x v (x), v’(x) ε = v (x) − v (x) Equation (26.88) Numerical Equation (26.88) 0

2.5986749269

2.59867451285033

4.1 E-07

0.1

2.5175960309

2.517595657265

3.7 E-07

0.2

2.4359837279

2.43598339555877

3.3 E-07

0.3

2.35367716377

2.35367687645523

2.9 E-07

0.4

2.2704784954

2.2704782458471

2.5 E-07

0.5

2.18614087

2.186140661607

2.1 E-07

0.6

2.1003509021

2.100350792203

1.1 E-07

0.7

2.0127024667

2.012702341002

4.3 E-07

0.8

1.9226543229

1.922054261014

8.4 E-07

0.9

1.829459716

1.829460341003

6.2 E-07

1

1.73205080757

1.73205080757

0

338

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

Table 26.13 Results of v (x) obtained from Eq. (26.91) and numerical results for α1 = − 13 , α2 = 13 , m = 1, v(1) = 0, x ∈ [xc , 1] x

v (x), Equation (26.91)

v’(x) Numerical

  ε = v (x) − v (x) Equation (26.91)

0.479894418625

−0.8415485509

−0.8415485596536

8.7 E-09

0.5

−0.6861406515

−0.686140661834

1.1 E-08

0.6

−0.460990252

−0.460990259768

7.7 E-09

0.7

−0.321395731

−0.321395736172

5.0 E-09

0.8

−0.2057237111

−0.2057237131

2.2 E-09

0.9

−0.1006780314

−0.1006780296

1.8 E-09

1

0

0

0

C8 = 64.01783024162; C9 = −153.642792576

(26.90)

The approximate solution v (x) defined on the subdomain [xc , 1], can be written as v (x) = 1.63099572225(1 − x) − 59.99867574852(1 − x)2 + 490.882847891(1 − x)3 − 1849.05441536(1 − x)4 + 3235.9311998(1 − x)5 − 2132.431791958872(1 − x)6 + 64.01783024162(1 − x)7 − 153.642792576(1 − x)8 , x ∈ [xc , 1], xc = 0.479894418625

(26.91)

In this subcase, the velocity v(x) does not exist because the condition v(0) = 1 is violated (0 ∈ / [xc , 1]). In Table 26.13 is presented a comparison between the approximate solution (26.91) and numerical results. 26.2.4.3 Subcase 3 √ nFor v (1) = − 3 , similar to previous subcase, the convergence-control parameters Cj can be determined only on subdomain [xc , 1], xc = 0.479894418625. In consequence, we obtain √ C1 = − 3; C2 = −0.78452432252; C3 = 63.9926458022; C4 = −524.3776926001; C5 = 1991.42803806; C6 = −3535.1278766; C7 = 2379.63883862 C8 = 4.92505495912 · 10−5 ; C9 = −9.450109917 · 10−5

(26.92)

The approximate solution v (x) is defined on the subdomain [xc , 1] and has the form.

26.2 OAFM for Eqs. (26.15) and (26.16)

339

√ v (x) = − 3 − 1.78452432252(1 − x) + 63.9926458022(1 − x)2 − 524.3776926001(1 − x)3 + 1991.42803806(1 − x)4 − 3535.1278766(1 − x)5 + 2379.638838625209(1 − x)6 + 4.92505495912 · 10−5 (1 − x)7 − 9.450109917 · 10−5 (1 − x)8 (26.93) The velocity v(x) as we specified previous does not exist. In Table 26.14 is presented a comparison between the approximate solution (26.93) and numerical results. From the Tables 26.1, 26.2, 26.3, 26.4, 26.5, 26.6, 26.7, 26.8, 26.9, 26.10, 26.11, 26.12, 26.13 and 26.14 and from Figs. 26.1, 26.2, 26.3, 26.4, 26.5, 26.6, 26.7, 26.8, 26.9, 26.10, 26.11, 26.12, 26.13, 26.14, 26.15, 26.16, 26.17 and 26.18 can we observe that the results obtained by means of OAFM are of a very high accuracy in comparison with the numerical results. Table 26.14 Results of v (x) obtained from Eq. (26.93) and numerical results for α1 = − 13 , √ α2 = 13 , m = 1, v (1) = − 3, x ∈ [xc , 1]   x v (x), v’(x) ε = v (x) − v (x) Equation (26.93) Numerical Equation (26.93) 0.479894418

−0.841548568171

−0.8415485596536

0.5

−0.9999999978

−1

2.2 E-09

0.6

−1.2393605136

−1.2393605322

1.3 E-08

0.7

−1.391304599543

−1.3913046045

5.1 E-09

0.8

−1.51693055396

−1.516930555957

2.1 E-09

0.9

−1.62878331049 √ − 3

−1.628783311424 √ − 3

1.2 E-09

1

Fig. 26.15 Comparison between the second approximate solution v (x) given by Eq. (101) and numerical solution for α1 = − 13 , α2 = 13 , m = 1, √ v (1) = 3 Eq. (26.88) Numerical

8.5 E-09

0

340

26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt

Fig. 26.16 Comparison between the second approximate solution v (x) given by Eq. (101) and numerical solution for α1 = − 13 , α2 = 13 , m = 1, √ v (1) = 3 Eq. (26.89) Numerical

Fig. 26.17 Comparison between the approximate solution v (x) given by Eq. (26.91) and numerical solution for α1 = − 13 , α2 = 13 , m = 1, v (1) = 0 and x ∈ [xc , 1] Eq. (26.91) Numerical

Fig. 26.18 Comparison between the approximate solution v (x) given by Eq. (26.93) and numerical solution for α1 = − 13 , α2 = 13 , m = 1, v (1) = 0 and x ∈ [xc , 1] Eq. (26.93) Numerical

26.2 OAFM for Eqs. (26.15) and (26.16)

341

It is noteworthy what in the some situations, exist only a single solution or three solutions for Eqs. (26.15) and (26.16). An interesting question arises when the derivate of the velocity is well-defined on a subdomain, but the velocity does not exist on that subdomain.

References 1. C. Truesdell, W. Noll, The Nonlinear Field Theories of Mechanics, Handbook Der Physics, 3rd edn. (Springer, Berlin, 1965) 2. N. Phan-Thien, W. Walsh, Squeeze-film flow of an Oldroyd-B fluid. Similarity solution and limiting Weissenberg number. Zeitschrift für angewandte Mathematik und Physik 36(6), 747– 759 (1984) 3. R.A. Khan, Couette flows an Oldroyd 8-constant fluid with magnetic field. Neural, Parallel and Scientific Computation 17, 433–444 (2009) 4. A.M. Siddiqui, F. Iskhlaq, A. Ashra, Q.A. Azim, The film flow of Oldroyd six constant fluid down a vertical cylinder. Math. Eng. Sci. Aerosp. 1, 139–148 (2010) 5. M. Hammed, R. Ellahi, Thin film flow of non-Newtonian magnetohydrodynamic fluid on vertically moving belt. Numer. Methods Fluids 66(1), 1409–1419 (2011) 6. Y.L. Xiong, C.H. Bruneau, H. Kellog, A numerical study of the dimensional flow past a bluff body for dilute polymer solution. J. Nonnewton. Fluid Mech. 116, 8–26 (2013) 7. M. Zhang, I. Lashgari, T.A. Zaki, L. Brandt, Linear stability analysis of channel flow of viscoelastic Oldrod-B and FENE-P fluids. J. Fluid Mech. 737, 249–279 (2013) 8. T. Hayat, S.A. Shehzad, S.A. Mezel, A. Alsaedi, Three dimensional flow of an Oldroyd-B fluid over a bidirectional stretching surface with prescribed surface temperature and prescribed surface heat flux. J. Hydrol. Hydromec. 62, 117–125 (2014) 9. T. Gul, S. Islam, R. A. Shak, A. Khalid, J. Khan, S. Shafie, Unsteady magnetohydrodynamic film flow of an Oldroyd-B fluid over an oscillating inclined belt. Plos One 0126698 (2015) 10. R.A. Shah, S. Islam, A.M. Siddiqui, T. Haroon, Wire coating analysis with Oldroyd 8-constant fluid by Optimal homotopy asymptotic method. Comput. Math. Appl. 63, 695–707 (2012) 11. S. Ullah, N.A. Khan, S. Bajwa, N.A. Khan, M. Tanvar, K. Liaqat, Some exact solutions for the rotational flow of Oldroyd-B fluid between two circular cylinders. Adv. Mech. Eng. 9(8), 1–15 (2017) 12. R.D. Ene, V. Marinca, B. Marinca, Thin film flow of an Oldroyd 6-constant fluid over a moving belt: an analytic approximate solution. Open Phys. 14, 44–64 (2016) 13. D. Dey, A.S. Khound, Analysis of thin film flow of Oldroyd-B nanofluid in a oscillating inclined belt with convective boundary conditions. Int. J. Math. Arch. 9(7), 142–150 (2018) 14. M.V. Krishna, R.Y. Qadri, Run-up flow of Oldroyd-B fluid through a parallel plate channel. IOSR-J. Math. 12(5), 1–8 (2016) 15. Z. Khan, H. U. Rasheed, J. Tlili, I. Khan, T. Abbas, Runge-Kutta 4th oder method analysis for viscoelastic Oldroyd 8-constant fluid used as coating material for wire with temperature dependent viscosity. Scientific Raports 8, Art. 14504 (2018) 16. M.H. Tiwana, A.B. Mann, M. Rizwan, K. Maqbool, S. Javeed, S. Raza, M.S. Khan, Unsteady magnetohydrodynamic convective fluid flow Oldroyd-B model considering ramped wall-temperature and ramped wall velocity. Mathematics 7(8), 676–689 (2019) 17. I. Ahmad, A. Abbasi, W. Abbasi, W. Farooq, Mixed convective peristaltic flow of an Oldroyd 4 constant fluid in a planar channel. Int. J. Thermofluid Sci. Tehnol. 4(6), 19060302 (2019)

Chapter 27

Cylindrical Liouville-Bratu-Gelfand Problem

This Chapter is devoted to finding an approximate solution to the cylindrical Liouville-Bratu-Gelfand problem, a particular boundary value problem related to the classical nonlinear Bratu problem. The Bratu problem has a rich history and served as a test for different approximate and numerical methods. The so-called Liouville-Bratu-Gelfand problem, deals with finding of positive solutions for the equation: u + λeu = 0, x ∈  u = 0, x ∈ ∂

(27.1)

where  is the Laplace operator, λ is a positive parameter,  is a bounded domain in RN and N is a discrete parameter allowed to vary continuously. Equation (27.1) is known as the solid fuel ignition model. But for a general domain  it is difficult to solve Eq. (27.1) and if  is the unit ball in R N ( = B1 (0)) then Gidas et al. [1] showed that all solutions of Eq. (27.1) are radially symmetric and therefore Eq. (27.1) is equivalent to the ordinary differential equation’s boundary value problem [2]: u  +

N −1  u + λeu = 0, r ∈ [0, 1], N = 1, 2, 3, . . . , u  (0) = u(1) = 0 r (27.2)

where u(r ) = u(|x|) and prime denotes differentiation with respect to r. If we assume that N = 1, Eq. (27.1) is replaced with: u  + λeu = 0, r ∈ (0, 1]u  (0) = u(1) = 0

(27.3)

which was first studied by Liouville in 1853 [3]. The exact solution of Eq. (27.3) is defined as: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_27

343

344

27 Cylindrical Liouville-Bratu-Gelfand Problem

 u(r ) = −2 ln

cosh 0.5(r − 0.5)θ cosh 0.25θ

 (27.4)

where θ is obtained from the equation: θ=

√

2λ cosh 0.25θ

(27.5)

Bratu’s problem (27.3), [4] has one solution when λ = λc = 3.513830719 and u  (0) = 3, 999754001 [5],two solutions for λ < λc ,and no solution for λ < λc . For N = 2, there exist two solutions if 0 < λ < λc = 2 and a unique solution for λ = 2 [6]. For N = 3, Frank-Kamenetsky [7] obtained λc = 3.32. Gelfand [8] proved the existence of a value λc for which Eq. (27.2) has infinitely many nontrivial solutions. Joseph and Lundgren [9] completely characterized the solution structure of (27.2) versus N. For N < 10, Eq. (27.2) has a continuum solution that oscillates around the line λ = 2(N − 2) and u(0) → ∞. For N ≥ 10, Eq. (27.2) has unique solution for each λ ∈ (0, 2(N − 2)) and no solution for λ ≥ 2(N − 2). Moreover λ → 2(N − 2) for u∞ → ∞ . For N = 3, Chandrasekhar used Eq. (27.2) as a model for the temperature distribution of an isothermal gas sphere in expansion of the universe [10]. Having in view (27.2), there exist many applications of the equation such as: questions in geometry and relativity concerning the Chandrasekhar model, the fuel ignition model of the thermal combustion theory, the model of the thermal reaction process, the radioactive heat transfer, nanotechnology, chemical reaction theory, non deformable material of constant density during the ignition period or electro-spinning process for the fabrication of nano-fibers, electrostatics and plasma physics ([10–14]). Equation (27.2) is a nonlinear differential equation and have no small parameters. It is still very difficult to find approximate analytic solutions. In general many authors try to solve Eq. (27.2) by analytical and numerical methods. For example Boyd [15] studied numerical and analytical solutions of the two- and three-dimensional Bratu’s problem. The analytic solution is based on the collocation method with one basis function at one interpolation for two dimensional problem, and collocation method with three functions of interpolation. Jacobsen and Schmitt [2] determined the existence and multiplicity results for radial solutions of the Liouville-Bratu-Gelfand problem, associated with a class of quasilinear radial operators, which included perturbations of k-Hessian and p-Laplace operators. Buckmire [6] obtained numerical solutions to Eq. (27.2) at the critical value λc = 2 using nonstandard finite difference schemes known as Mickens finite differences. The nonlinear Bratu problem in 2-dimensions is solved by Odejide and Aregbesola [16] by means of various procedures including the finite difference method, weighted residual method, and continuation-conjugate gradient method. Abbasbandy et al. [17] transform the Bratu problem into a nonlinear second order boundary value problem and then solved it by the Lie-group shooting method. Caglar et al. [11] examined the dynamics exhibited by the solution of Bratu’s equation calculating the corresponding Lyapunov exponent, power spectra and cobweb diagrams.

27 Cylindrical Liouville-Bratu-Gelfand Problem

345

Using the iteration formulas derived using different perturbation-iteration algorithms, Aksoy and Pakdermili [18] obtained new solutions of Bratu type equations. A numerical solution of the Bratu’s problem using non-polynomial spline functions is obtained by Jalilian [12]. Ananthaswamy and Rajendran [19] applied a homotopy perturbation method to obtain analytical solutions of Bratu’s problem. Sinccollocation method is developed and analysed by Rashidinia and Taher [20] in order to obtain numerical solution of Bratu’s problem (27.2). The nonlinear Bratu equation is solved numerically in one, two and three dimensions by Karkovski [21]. The results are obtained by means of three different numerical methods namely pseudospectral method, finite difference method and radial basis functions method which lead to same results. Green’s functions, different iterative procedures and nonlinear conjugate-gradient method are used by Mohsen [22] to obtain the integral solution of the one-dimensional Bratu problem. Marinca and Herisanu [23] obtained two approximate solutions for λ = 1 and an unique approximate solution for λ = 2 by the optimal homotopy asymptotic method. A one-dimensional nonlinear Bratu boundary value problem is solved by Kafri and Khuri [13] combining Green’s function and fixed point iterative schemes such as Picard’s and Krasnoselskii-Mann. Hajipour et al. [24] investigated an accurate discretization method to solve the one-, two-, and three-dimensional Bratu-type problems by discretization of the nonlinear equation via a fourth order nonstandard compact finite difference formula. In this Chapter we will obtain an analytical approximate solution for cylindrical Liouvile-Bratu –Gelfand in the case N = 2.

27.1 OAFM for Cylindrical Liouville-Bratu-Gelfand Problem Henceforth we consider N = 2 into Eq. (27.2) which means that: 1 u  (r ) + u  (r ) + λeu(r ) = 0, r ∈ (0, 1] r

(27.6)

u  (0) = u(1) = 0

(27.7)

The Eq. (27.6) contains the terms r1 and λeu(r ) and this leads to a singularity for r = 0 and to very strong nonlinearity. To avoid this singularity, we try to find an initial approximation u 0 (r ) of the form: u 0 (r ) = r h(r )

(27.8)

where h(r) does not contain the terms of the form r1t , with t ≥ 2. Moreover after taking into account (27.7), we have freedom to choose initial approximations u0 in many forms as:

346

27 Cylindrical Liouville-Bratu-Gelfand Problem

u 0 (r ) = (1 − r )er

(27.9)

u 0 (r ) = 1 − r k ,

(27.10)

1 + rk , k>1 2  p u 0 (r ) = 1 − r k , k > 2, p > 1 u 0 (r ) = ln

(27.11) (27.12)

and so on. Also, the linear operator in all cases (27.9)-( 27.12) is: 1 L[u(r )] = u  (r ) + u  (r ) r

(27.13)

and the function g(r) which appears in Eq. (2.5) for each case (27.9–27.12) is, respectively: k−2

g(r ) = (r + 2)er ; g(r ) = k 2 r k−2 ; g(r ) = − 2kr k 2 (1+r )   g(r ) = kp · r k−2 1 − r k k − k 2 + (1 − kp)r k Finally, we consider the following two combinations for the initial approximation:       u 0 (r, C1 , C2 , C3 ) = C1 1 − r 2 + C2 1 − r 3 + C3 1 − r 4

(27.14)

      u 0 (r, C1 , C2 , C3 , C4 ) = C1 1 − r 2 + C2 1 − r 3 + C3 1 − r 4 + C4 (1 − r )er (27.15)

27.1.1 Case A For the initial approximation u 0 (r ) given by Eq. (27.14), the linear operator and the function g(r) are respectively: 1 L[u(r )] = u  (r ) + u  (r ); g(r ) = 4C1 + 9C2 r + 16C3r 2 r

(27.16)

The corresponding nonlinear operator in this case becomes: N [u(r )] = λeu − 4C1 − 9C2 r − 16C3r 2

(27.17)

27.1 OAFM for Cylindrical Liouville-Bratu-Gelfand Problem

347

The Eqs. (2.5) and (2.6) can be written in the form: 1 u 0 + u 0 + 4C1 + 9C2 r + 16C3r 2 = 0 r

(27.18)

u 0 (0) = u 0 (1) = 0

(27.19)

The solution of Eq. (27.18) and (27.19) is given by (27.14). Substituting (27.14) into Eq. (27.17), we obtain: 2 3 4 N [u 0 (r )] = λeC1 (1−r )+C2 (1−r )+C3 (1−r ) − 4C1 − 9C2 r − 16C3r 2

(27.20)

The function given by (27.14) and (27.20) are “source” for the auxiliary functions f k which appear in Eq. (2.13). The auxiliary functions F i are chosen in the form: F1 (r ) = 25r 3 ,

F2 (r ) = 36r 4 ,

F3 (r ) = 49r 5 ,

F4 (r ) = 64r 6

(27.21)

The Eq. (2.13) becomes: u 1 + r1 u 1 + 25C4 r 3 + 36C5 r 4 + 49C6r 5 + +64C7 r 6 = 0, u 1 (0) = u 1 (1) = 0

(27.22)

and it has the solution:         u 1 (r, C4 , C5 , . . . , C7 ) = C4 1 − r 5 + C5 1 − r 6 + C6 1 − r 7 + C7 1 − r 8 (27.23) From Eqs. (27.14) and (27.23), the approximate solution (which contains 7 parameters) of Eqs. (27.6) and (27.7) can be written as:         u(r ) = u 0 (r ) + u 1 (r ) = C1 1 − r 2 + C2 1 − r 3 + C3 1 − r 4 + C4 1 − r 5       + C5 1 − r 6 + C6 1 − r 7 + C7 1 − r 8 (27.24)

27.1.2 Case B For the initial approximation u 0 (r ) given by Eq. (27.15), following the same way, the linear operator is given by Eq. (27.16), but the function g and the nonlinear operator are, respectively: g(r ) = 4C1 + 9C2 r + 16C3 r 2 + C4 (r + 2)er

(27.25)

348

27 Cylindrical Liouville-Bratu-Gelfand Problem

N [u(r )] = λeu − 4C1 − 9C2 r − 16C3 r 2 − C4 (r + 2)er

(27.26)

Substituting Eq. (27.15) into Eq. (27.26), we have: 2 3 4 r N[u0 (r)] = λeC1 (1−r )+C2 (1−r )+C3 (1−r )+C4 (1−r)e −4C1 − 9C2 r − 16C3 r2 − C4 (r + 2)er

(27.27)

The auxiliary functions F i are chosen as: F1 (r ) = 25r 3 , F2 (r ) = 36r 4 , F3 (r ) = 49r 5 , F5 (r ) = 81r 7 , F6 = 100r 8 , F7 = 121r 9

F4 (r ) = 64r 6 ,

(27.28)

The Eq. (2.13) can be written as: u1 + 1r u1 + 25C5 r3 + 36C6 r4 + 49C7 r5 + 64C8 r6 +81C9 r7 + 100C10 r8 + 121C11 r9 = 0 u 1 (0) = u 1 (1) = 0

(27.29)

Equation (27.29) has the solution:       5 6 − r7 u1 (r,C5 , C6, . . . , C11 ) = C 5 1 − r + C6 1 − r  + C7 1   +C8 1 − r8 + C9 1 − r9 + C10 1 − r10 + C11 1 − r11

(27.30)

such that the approximate solution (with 11 parameters) of Eqs. (27.18) and (27.19) is:       u(r ) = u 0 + u 1 (r) = C1 1 − r 2 + C2 1 − r 3 + C3 1 − r 4 (27.31) +C4 (1 − r )er +C5 1 − r 5 + C6 1 − r 6 +C7 1 − r 7  +C8 1 − r 8 + C9 1 − r 9 + C10 1 − r 10 + C11 1 − r 11 Remark 1 As was considered earlier, the choice of the auxiliary functions F i is not unique. Also we can choose the auxiliary functions in the forms: F1 (r) = r3 , F2 (r) = r5 , F3 (r) = r7

(27.32)

F1 (r) = r4 , F2 (r) = r5 , F3 (r) = r7 , F4 (r) = r8

(27.33)

F1 (r) = 0, F2 (r) = r6 , F3 (r) = r9 , F4 (r) = r9 , F5 (r) = r10

(27.34)

F1 (r) = (1 − r)er , F2 (r) = r3 , F3 (r) = 0, F4 (r) = r5 , F5 (r) = r7

(27.35)

27.1 OAFM for Cylindrical Liouville-Bratu-Gelfand Problem

349

and so on. Reamrk 2 In the sequel we will show that the error of the solutions decreases when the number of terms in the approximate solutions increases.

27.2 Numerical Examples To illustrate the accuracy of OAFM for different values of the parameter λ and for the cases 27.1.1 (7 parameters) and 27.1.2 (11 parameters), we consider the cases:

27.2.1 Case A For λ = λc = 2 it is known that there exists a unique solution. Applying the collocation method in the case 27.1.1, we obtain: C1 = 1.99909956543098; C2 = 5.25326566883928 × 10−3 ; C3 = −0.999854657245149; C4 = −0.09329411426; C5 = 1.0593731926; C6 = 0.770171682; C7 = 0.18588546453334. The analytical approximate solution of Eqs. (27.6) and (27.7) in the subcase 27.1.1 yields: u(r ) = −0.18588546453334r 8 + 0.770171682r 7 −1.0593731926r 6 + 0.09329411426r 5 +0.999854657245149r 4 − 0.0052532666883928r 3 −1.99909956543098r 2 + 1.38629103472806

(27.36)

27.2.2 Case B For λ = λc = 2 in the case 27.1.2, we have: C1 = −1.77843291313257; C2 = −2.5186715446098; C3 = −1.94841597357164; C4 = 7.55684878963775; C5 = −0.222560550112507; C6 = 0.472400274318908; C7 = 0.441914524249252; C8 = −1.4516409346361; C9 = 1.27747819306747; C10 = −0.530613129110163; C11 = 0.0899876252654083

The analytical approximate solution of the Eqs. (27.18) and (27.19) in the subcase 27.1.2 becomes: u(r ) = 7.55684878963775(1 − r )er − 0.0899876252654083r 11

350

27 Cylindrical Liouville-Bratu-Gelfand Problem

+ 0.530613129110163r 10 − 1.27747819306747r 9 + 1.4516409346361r 8 − 0.441914524249252r 7 − 0.472400274318908r 6 + 0.222560550112507r 5 + 1.94841597357164r 4 + 2.51867154467098r 3 + 1.77843291313257r 2 − 6.17055442833295

(27.37)

In Table 27.1 we present the exact solution of the Eqs. (27.6) and (27.7) for λ = λc = 2 which is compared with the approximate solution u(r ) obtained from Eq. (27.36) (the case with 7 parameters) and from Eq. (27.37) (the case with 11 parameters). From this Table results that there exist an excellent agreement between the exact solution and the results obtained by means of OAFM. In addition it is Table 27.1 The results of the first-order approximate solution u(r ) given by Eq. (27.36) (7 parameters), u(r ) given by Eq. (27.37) (11 parameters) and exact solution [6] for Eqs. (27.6) and (27.7) in the case λ = λc = 2 r

uex (r) [6]

u(r ) Eq. (27.36)

u(r ) ε= Equation (27.37) u ex (r ) − u(r ) for Eq. (27.36)

ε= u ex (r ) − u(r ) for Eq. (27.37)

1.38629436111989 1.3862910347280 3.32 × E 06

1.3862943613048

−1.06 × E - 10

0.1 1.36639369941355 1.3663947234712 −1.02 × E - 06

1.3663936998969

2.38 × E 11

0.2 1.30785293481332 1.3078562300198 −3.29 × E - 06

1.30785293473881 7.51 × E 11

0.3 1.21393896863778 1.2139397206316 −7.52 × E - 07

1.21393896865131 −1.43 × E - 11

0.4 1.08945435088334 1.0894513410005 3.01 × E 06

1.08945435064901 2.34 × E 10

0.5 0.94000725849147 0.9400039873455 3.27 × E 06

0.94000725820532 2.86 × E 10

0.6 0.77132496162396 0.7713278000115 −2.84 × E - 06

0.77132496163986 −1.59 × E - 11

0.7 0.58874212120515 0.5887522484271 −1.01 × E - 05

0.58874212125496 −4.98 × E - 11

0.8 0.39690187744767 0.3969107000247 −8.90 × E - 06

0.39690187750477 −5.70 × E - 11

0.1996434200268 −2.74 × E - 06

0.19964067063868 −7.43 × E - 11

0.9 0.1996406705644 0

0

0

−5.67 × E - 15

−5.67 × E - 15

27.2 Numerical Examples

351

clear that the error of the approximate solutions decreases when the number of the unknown parameters C i increases.

27.2.3 Case C For λ = 1 < λc , there exist two solutions. For the first approximate solution given by Eq. (27.31), we obtain: C1 = −0.3202908219065; C2 = −8.1001752803416; C3 = −33.6746094477689; C4 = 23.954579823642; C5 = −39.6111565432065; C6 = 377.39406373464; C7 = −889.807965835214; C8 = 1132.41065472403; C9 = −848.500228883599; C10 = 353.585902315362; C11 = −63.4885848726215.

and therefore the first approximate solution is: u(r ) = 23.954579823642(1 − r )er + 63.4885848726215r 11 − 353.585902315362r 10 + 848.500228883599r 9 − 1132.41065472403r 8 + 899.807965835214r 7 − 377.39406373464r 6 + 39.6111565432065r 5 + 33.6746094477689r 4 + 8.1001752803416r 3 + 0.3202908219065r 2 − 20.112390910642

(27.38)

Now, for λ = 1 , for the second approximate solution given by Eq. (27.31), we have: C1 = 0.287350186125982, C2 = −0.0373034441696372; C3 = −0.0421689251971278; C4 = 0.111598137074538; C5 = −0.0111623859011855; C6 = 0.0292801231921856; C7 = −0.0593151868521791; C8 = 0.0816998992010747; C9 = −0.0694900000812; C10 = 0.032804685273703; C11 = −6.598717436121 × 10−3

The second approximate solution of Eqs. (27.6) and (27.7) is: u(r ) = 0.1115981370745382(1 − r )er + 6.598717436121 × 10−3 r 11 − 0.032804685273703r 10 + 0.0694900000812r 9 − 0.0816998992010747r 8 + 0.0593151868521791r 7 − 0.0292801231921856r 6 + 0.0111623859011855r 5 + 0.0421689251971278r 4 + 0.0373034441696372r 3

352

27 Cylindrical Liouville-Bratu-Gelfand Problem

− 0.287350186125982r 2 + 0.20509623415546

(27.39)

The exact solutions of Eqs. (27.6) and (27.7) for λ = 1 , are compared with u(r ) given by (27.38) and (27.39) in Tables 27.2 and 27.3, respectively. Figures 27.1, 27.2 and 27.3 present a comparison between the approximate solutions (27.37), (27.38) and (27.39) respectively obtained by OAFM and exact solution of Eqs. (27.6) and (27.7). It is easy to emphasize the accuracy of the obtained results for (27.6) and (27.7) since with these graphical representation the analytical results are nearly identical with exact ones, which proves the validity of the approximate results.

Table 27.2 The results of the first approximate solution u(r ) given by Eq. (27.38) and first exact solution [6] for Eq. (27.6) and (27.7) in the case λ = 1 r

u ex (r ) [6]

u(r ) Eq. (27.38)

ε = u ex (r ) − u(r )

0 0.1

3.84218871571889

3.8421889130001

−1.97 × E - 7

3.72889074626776

3.7288913040883

−5.57 × E - 7

0.2

3.42306591963373

3.4230658391926

8.04 × E - 8

0.3

2.99877907292626

2.9987787078762

3.65 × E - 7

0.4

2.52450968496725

2.524509983819

−2.99 × E - 7

0.5

2.04421961055683

2.0442200099081

−3.99 × E - 7

0.6

1.58052432398946

1.580524296074

2.79 × E - 8

0.7

1.14296463728601

1.142964686834

−4.95 × E - 8

0.8

0.73425655346587

0.7342565251829

2.82 × E - 8

0.9

0.35389240596932

0.3538927360043

−3.30 × E - 7

1

0

0

0

Table 27.3 The results of the second approximate solution u(r ) given by Eq. (27.39) and the second exact solution [6] for Eq. (27.6) and (27.7) in the case λ = 1 u ex (r )[6]

u(r ) Eq. (27.39)

ε = u ex (r ) − u(r )

0.31669436764075

0.31669437123

−3.59 × E - 9

0.1

0.31326585049806

0.313265854200984

−3.70 × E - 9

0.2

0.303015422832294

0.303015422959866

−1.27 × E - 10

0.3

0.28604726530485

0.28604726549978

−1.95 × E - 10

0.4

0.262531127456037

0.262531127485849

−2.98 × E - 11

0.5

0.23269678387383

0.23269677831415

−5.56 × E - 9

0.6

0.19682680569215

0.196826805653958

−3.91 × E - 11

0.7

0.1552481066827

0.155248106714071

−3.13 × E - 11

0.8

0.1083227634444

0.1108321763423151

−2.12 × E - 11

0.9

0.05643860246923

0.056438610933986

−8.46 × E - 9

0

0

0

r

27.2 Numerical Examples

353

Fig. 27.1 Comparison between approximate solution (27.37) and exact solution of Eqs. (27.6) and (27.7) λ = 2______(exact); _ _ _ _(approx.)

Fig. 27.2 Comparison between approximate solution (27.38) and exact solution of Eqs. (27.6) and (27.7) for λ = 1______(exact); _ _ _ _(approx.)

Fig. 27.3 Comparison between approximate solution (27.39) and exact solution of Eqs. (27.6) and (27.7) for λ = 1______(exact); _ _ _ _(approx.)

References 1. B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209–243 (1979) 2. J. Jacobsen. K. Schmitt, The Liouville-Bratu-Gelfand problem for radial operators. J. Diff. Eqs. 184, 289–298 (2002) 3. J. Liouville, Sur l’équation aux derivées partielles. J. de Math. 18, 71–72 (1853)

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27 Cylindrical Liouville-Bratu-Gelfand Problem

4. G. Bratu, Sur les équations intégrales non-linéares. Bull. Math. Soc. So. France 42, 113–142 (1914) 5. V. Marinca, N. Herisanu, The optimal homotopy asymptotic method. Engineering applications (Springer, Cham, 2015) 6. R. Buckmire, Application of a Mickens finite-difference scheme to the cylindrical BratuGelfand problem. Numer. Methods Partial Differ. Equ. 19, 327–337 (2003) 7. D.A. Frank-Kamenetsky, Diffusion and Heat Exchange in Chemical Kinetics (Princeton University Press, Princeton N. J., 1955). 8. I.M. Gelfand, Some problems in the theory of quasi-linear equations. Amer. Math. Soc. Trans. Ser. 2(29), 295–381 (1963) 9. D.D. Joseph, T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources. Arch. Rat. Mech. Anal. 49, 241–269 (1973) 10. S. Chandrasekhar, An Introduction to the Study of Stellar Structure (Dover, 1967) 11. H. Caglar, N. Caglar, M. Özer, A. Valaristos, A.N. Miliou, A.N. Anagnostopoulos, Dynamics of the solution of Bratu’s equation. Nonlinear Anal. 71, 672–678 (2009) 12. R. Jalilian, Non-polynomial spline method for solving Bratu’s problem. Comp. Phys. Comm. 181, 1868–1872 (2010) 13. H.Q. Kafri, S.A. Khuri, Bratu’s problem: a novel approach using fined—point iterations and Green’s functions. Comp. Phys. Comm. 198, 97–104 (2016) 14. S. Hichar, A. Guerfi, S. Donis, M.T. Meftah, Application of nonlinear Bratu’s equation in two and three dimensions to electrostatics. Report Math. Phys. 76(3), 283–295 (2015) 15. J.P. Boyd, An analytical and numerical study of the two—dimensional Bratu equation. J. Sci. Comp. 1(2), 183–206 (1986) 16. S.A. Odejide, Y.A.S. Aregbesola, A note on two dimensional Bratu problem. Kragujevac J. Math. 29, 49–56 (2006) 17. S. Abbasbandy, M.S. Hashemi, C.S. Liu, The Lie-group shooting method for solving the Bratu equation. Commun. Nonl. Sci. Numer Simulat. 16, 4238–4249 (2010) 18. Y. Aksoy, M. Pakdemirli, New perturbation iteration solutions for Bratu type equations. Comput. Math. Appl. 59, 2802–2808 (2010) 19. V. Ananthaswamy, L. Rajendran, Analytical solutions of some two-point nonlinear elliptic boundary value problem. Appl. Math. 3, 1044–1058 (2012) 20. J. Rashidinia, N. Taher, Application of the Sinc approximation to the solution of Bratu’s problem. Int. J. Math. Modell. Comput. 2(3), 239–246 (2012) 21. J. Karkovski, Numerical experiments with the Bratu equation in one, two and three dimensions. Comput. Appl. Math. 32, 231–244 (2013) 22. A. Mohsen, On the integral solution of the one-dimensional Bratu problem. J. Comput. Appl. Math. 251, 61–66 (2013) 23. A.M. Wazwaz, The variational iteration method for solving two forms of Blasius equation on a half infinite domain. Appl. Math. Comput. 188, 485–499 (2007)

Chapter 28

The Polytrophic Spheres of the Nonlinear Lane—Emden—Type Equation Arising in Astrophysics

Several studies and numerical methods have been focused on the study of polytrophic spheres—nonlinear Lane—Emden equation. Horedt [1] presented series expansion of Lane—Emden functions near an interior point of a polytrope for N = 1, 2 or 3. The same author [2] presented seven digit numerical solutions of the Lane—Emden equation for the plane—parallel, cylindrical and spherical case for n = −10, −5, − 4, −3, −2, −1.5, −1.01, −0.9, −0.5, 0, 0.5, 1, 1.5, 2, 3, 4, 5, 6, 10, 20. For N = n = 3, Beech [3] suggested   a good approximation of the polytrophic model in the form y(x) = sec h √x3 . Roxburgh and Stockman [4] studied polytrophic equation by means of power series, calculating 5000 coefficients. The same equation is solved by Hunter by the same method [5] but longer than 60—term series are needed for the outer regions of n > 3. Nouh [6] improved the convergence radii of the series, optimized the model developed by Hunter [5], using the combination of two accelerating techniques: Euler—Abel transformation and Padé approximation. Traditional power series approach and Homotopy Analysis Method are used by Van Gorder and Vajravelu [7]. Tabrizidooz et al. [8] presented a numerical algorithm based on hybrid functions and Lagrange interpolating polynomials. Approximate solutions of the Lane—Emden type are obtained by Adibi and Rismani [9] by an improved Legendre—spectral method. The Legendre—Gauss points used as collocation nodes and Legendre interpolation is employed in the Volterra term. Boyd [10] used Newton—Kantorovich iteration and Chebyshev pseudospectral method to the three—dimensional spherical polytrophic Lane—Emden problem in the domain [0, ζ ] where y(ζ ) = 0. Differential transformation method is proposed by Mukherjee et al. [11] to obtain a relatively new exact series method and a shifted Jacobi—Gauss collocation spectral method is proposed by Bhrawy and Alofi [12]. Yin et al. [13] presented a coupled method of Laplace transform and Legendre wavelets to obtain exact solutions of Lane—Emden—type equations. Baranwal et al. [14] proposed an iterative method which is a hybrid of variational iteration method and Adomian decomposition method and further refined by introducing a new correction functional. Pandey and Kumar [15] proposed a numerical © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_28

355

356

28 The Polytrophic Spheres of the Nonlinear Lane …

method using Bernstein operational matrix of differentiation derived from Bernstein polynomials. Yang and Hou [16] initiated a numerical method using hybrid of block—pulse functions, Chebyshev polynomials and collocation method. Boubaker and Van Gorder [17] applied the Boubaker Polynomials Expansion Scheme to obtain analytical—numerical solutions by polytrophic and isothermal gas spheres. Khan et al. [18] used Optimal Homotopy Asymptotic Method and genetic algorithm to determine an approximate solution of Lane—Emden—type equation. Hosseini and Abbasbandy [19] proposed the spectral Adomian decomposition method based on a combination between spectral method and Adomian decomposition method to solve Lane—Emden equations. Homotopy perturbation method with Laplace transform is used to solve numerically the Lane—Emden type equation by Tripathi and Mishra [20]. Al-Hayaniet al. [21] presented homotopy analysis method with genetic algorithm in the study of Lane—Emden type equation. In this Chapter we have obtained an approximate analytical solutions for the polytrophic sphere of Lane—Emden—type equation in the case N = 3, n = 3 and n = 5. Since analytical method seems to offer generally much deeper insight into the nature of the problem, as compared to numerical integrations, it will be of importance to have available approximate analytical solutions of the Lane—Emden equation [22].

28.1 The Nonlinear Lane—Emden Equation In that follow, we consider the equation of hydrostatic equilibrium [1, 23]. ∇P = ρ∇φ

(28.1)

in which the gradient act in N-dimensional space. P denotes the pressure, ρ the density and φ the gravitational potential. We shall support that the polytrophic equation is: P = kn ρ1+1/n

(28.2)

where kn and n are constants (n is the polytrophic index). For example n → 0 represent a homogeneous liquid; n = 23 a monoatomic gas in adiabatic equilibrium; n → ∞ an isothermal gas. Now, substituting Eq. (28.2) into Eq. (28.1), one finds:   1 1+ kn ρ1/n ∇ρ = −ρ∇φ n and integrating in the conditions: n = −1, ±∞, we obtain:

(28.3)

28.1 The Nonlinear Lane—Emden Equation

  1 n (n + 1)kn ρ0/ − ρ1/ n = φ − φ0

357

(28.4)

If the initial value of the gravitational potential φ0 ≡ 0, and then inserting the gravitational potential φ from Eq. (28.4) into Poisson’s equation: φ = 4πGρ

(28.5)

the last equation can be rewritten in the form: (n + 1)kρ1/n = −4πGρ n = −1, ±∞

(28.6)

ρ = ρ0 yn (r), P = P0 yn+1 (r)

(28.7)

If we put:

and introducing an additional dimensionless variable x: r = cx

(28.8)

with: c2 =

(n + 1)kn 1− 1n ρ0 4πG

(28.9)

we obtain the well-known of the Lane—Emden equation in N-dimensional polar coordinates:   d dy xN−1 + yn = 0, N = 1, 2, . . . , n = −1, ±∞ (28.10) x1−N dx dx or y +

N−1  y + yn = 0 x

(28.11)

where prime denotes differentiation in respect to x. The initial conditions are: y(0) = 1, y (0) = 0

(28.12)

The Eqs. (28.11) and (28.12) admits solutions of practical interest when N = 1 (polytrophic slab), N = 2 (polytrophic cylinder) and N = 3 (polytrophic sphere). Also, the values of parameter n are in the interval [0, 5]. The main difficulty of Eqs. (28.11) and (28.12) is the singularity behavior occurring al x = 0. It is known that

358

28 The Polytrophic Spheres of the Nonlinear Lane …

the solutions of Eqs. (28.11) and (28.12) could be exact only for n = 0, n = 1 and n = 5. These solutions are, respectively: − 21  sin x 1 1 y(x) = 1 − x2 ; y(x) = ; y(x) = 1 + x2 6 x 3

(28.13)

For the other values of n, the Lane—Emden equation is to be integrated numerically. In the present paper we have obtain an effective approximate analytical solutions for polytrophic sphere (N = 3) in the cases n = 3 and n = 5.

28.2 OAFM for the Polytrophic Spheres of the Lane—Emden Equation Henceforth, we consider N = 3 (polytrophic sphere) into Eqs. (28.11) and (28.12), which means that: 2 y (x) + y (x) + yn (x) = 0, x ∈ [0, 10) x

(28.14)

y(0) = 1, y (0) = 0

(28.15)

After taking into account the form of Eqs. (28.14) and (28.15), let us note that: y(−x) = y(x)

(28.16)

The linear operator and the function g(x) are respectively: L[y(x)] = y (x), g(x) = −2

(28.17)

The Eqs. (2.5) and (2.6) can be written as: y0 (x) − 2 = 0, y0 (0) = 1, y0 (0) = 0

(28.18)

The solution of Eqs. (28.18) is: y0 (x) = 1 + x2

(28.19)

The corresponding nonlinear operator becomes: N[y(x)] =

2  y (x) − yn (x) x

Substituting Eq. (28.19) into Eq. (28.20), we obtain:

(28.20)

28.2 OAFM for the Polytrophic Spheres of the Lane—Emden Equation

  n(n − 1) 4 n(n − 1)(n − 2) 6 x − x + ... N y0 (x) = 3 − nx2 − 2! 3!

359

(28.21)

The function given by Eq. (28.19) and (28.21) are „source” for the auxiliary functions F i , such that these functions are chosen in the form: Fi (x) = −x 2i , i = 0, 1, 2, . . .

(28.22)

The Eq. (2.13) becomes: y1 (x) − C1 − C2 x2 − C3 x4 − C4 x6 − C5 x8 = 0, y1 (0) = y1 (0) = 0

(28.23)

We remark that into Eqs. (2.24) and (28.23) we put p = 5 and the expression (28.23) is not unique. For example, we can consider instead of Eq. (28.23), the expressions: y1 − C1 − C2 x2 − C3 x4 = 0, y1 (0) = y1 (0) = 0

(28.24)

or y1 − C1 − C2 x2 − C3 x4 − C4 x6 − C5 x8 − C6 x10 = 0 y1 (0) = y1 (0) = 0

(28.25)

and so on. Considering only the expression (28.23), follow that: y1 (x, Ci ) =

C1 2 C2 4 C3 6 C4 8 C5 10 x + x + x + x + x 2 12 30 56 90

(28.26)

such that the approximate analytical solution of the first order (2.3) becomes: y(x) = y0 (x) + y1 (x, Ci )   C1 2 C2 4 C3 6 C4 8 C5 10 x + x + x + x + x =1+ 1+ 2 12 30 56 90

(28.27)

The unknown parameters Ci are determined by minimizing the square residual error: 

1

y(C1 , C2 , C3 , C4 , C5 ) = 0

We consider two cases for n.



2 2   n y (x) + y (x) + y (x) dx x

(28.28)

360

28 The Polytrophic Spheres of the Nonlinear Lane …

28.2.1 Case 1 For n = 3, the optimal convergence–control parameters, determined from Eqs. (28.28) and (2.31) are: C1 = -2.3333333334; C2 = 0.3; C3 = -0.11330952381; C4 = 0.0318415638;C5 = -0.0044665404. Therefore in this first case, the approximate solution of Eqs. (28.14) and (28.15) is: y(x) = 1 − 0.166666667x2 + 0.025x4 − 0.0037698413x6 + 5.685993533 × 10−5 x8 + 4, 9622822671 × 10−5 x10 , x ∈ [0, 1] (28.29)

28.2.2 Case 2 For n = 5, we obtain: y(x) = 1 − 0.166666667x2 + 0.0416578921x4 − 0.00115741143x6 + 0.0033758014x8 − 0.0012731831x10

(28.30)

The numerical solution [2] of Eqs. (28.14) and (28.15) for n = 3 and exact solution (28.13) 3 for n = 5 are compared with the approximate analytical solutions of the first order y(x) given by Eqs. (28.29) and (28.30) in Tables 28.1 and 28.2, respectively: From Tables 28.1 and 28.2 follows that an excellent agreement between the numerical (exact) solutions and the results obtained by means of OAFM on domain [0, 1]. Table 28.1 Results of the approximate analytical solution y(x) given by Eq. (28.29) and numerical results [2] for n = 3 and x ∈ [0, 1]

x

y(x)Num. [2]

y(x) Eq. (28.29)

ε = |y(x)Num − y(x)|

0

1

1

0

0.1

0.9983358

0.998335281

5.2E − 07

0.2

0.9933731

0.993371035

2.1E − 06

0.3

0.9851998

0.985195645ssss

3.4E − 05

0.4

0.9739583

0.973952054

6.2E − 06

0.5

0.9598391

0.959831585

7.5E − 06

sss0.6

0.9430732

0.943066005

7.2E − 06

0.7

0.9239238

0.923916106

7.7E − 06

0.8

0.9026721

0.902673553

1.4E − 06

0.9

0.8796172

0.879628273

1.1E − 05

1

0.8550576

0.855082462

2.5E − 05

28.2 OAFM for the Polytrophic Spheres of the Lane—Emden Equation Table 28.2 The results of the approximate analytical solution y(x) given by Eq. (28.30) and the exact solution (28.13) 3 for n = 5 and x ∈ [0, 1]

361

x

y(x)ex Eq.(28)3

y(x) Eq. (28.30)

ε = |y(x)ex − y(x)|

0

1

1

0

0.1

0.99833749

0.998337487

2.4E − 09

0.2

0.99339927

0.993399255

1.4E − 08

0.3

0.98532928

0.985329205

7.5E − 08

0.4

0.97435470

0.974354446

2.5E − 07

0.5

0.96076892

0.960768049

8.7E − 07

0.6

0.94491118

0.944907862

3.3E − 06

0.7

0.92714553

0.927132354

1.3E − 05

0.8

0.90784130

0.907791979

4.9E − 05

0.9

0.88735651

0.887190024

1.7E − 04

1

0.86602540

0.885519729

5.1E − 04

The analytical solutions given by Eq. (28.29) and (28.30) converge for any value of x ≤ 1 but a slight difficulty occurs for x > 1. For example if x = 1,2 and n = 3 the numerical and approximate solutions are respectively: yNum (1, 2) = 0.8025919; y(1, 2) = 0.800646111

(28.31)

and for n = 5 the exact and approximate solutions are respectively: yex (1, 2) = 0.82199493; y(1, 2) = 0.818453798

(28.32)

To accelerate the convergence of the approximate solutions of Eq. (28.11) and (28.12), instead of approximate solutions (28.29) and (28.30), we consider the quotient of two polynomials y(x) =

1 + k1 x2 + k2 x4 1 + k3 x2 + k4 x4

(28.33)

which verify the initial conditions (28.12). In expression (28.33),k1 , k2 , k3 and k4 are unknown parameters and will be optimally determined. These parameters are not related will be coefficients of Eqs. (28.29) and (28.30). In the case 28.2.1, for n = 3 we obtain: k1 = 9.25925929925 × 10−3 ; k2 = −2.4250440917 × 10−4 k3 = 0.1574074074074; k4 = 9.9206349206244 × 10−3

362

28 The Polytrophic Spheres of the Nonlinear Lane …

Table 28.3 The results of approximate solution y(x) given by Eq. (28.34) and the numerical results [2] for n = 3 and x ∈ [0, 6.92]:

x

y(x)app Eq. (28.34)

y(x) Numerical

ε = |y(x)Num − y(x)|

0

1

1

0

0.2

0.993373225

0.9933731

1.2E − 07

0.4

0.973958255

0.9739583

4.4E − 08

0.6

0.943073172

0.9430732

2.7E − 08

0.8

0.902672089

0.9026721

1.1E − 08

1

0.855017569

0.8550576

3.0E − 08

2

0.582851018

0.5828505

5.1E − 07

3

0.359237187

0.3592265

4.4E − 05

4

0.209350163

0.2092816

6.8E − 05

5

0.111058197

0.1108198

2.4E − 04

6

0.044311377

0.04373798

5.7E − 04

6.92

0

10-4

1E − 04

and the approximate analytical solution of the first order (28.33) becomes: y(x) =

1 − 0.259259299 · 10−3 x 2 − 2.4250440917 · 10−4 x 4 1 + 0.1574074074x 2 + 9.9206349206244 · 10−3 x 4

(28.34)

In the case 28.2.2.for n = 5, occur: k1 = 0.103327411135187; k2 = 5.0409685088165 × 10−4 ; k3 = 0.266695273475284; k4 = 6.7664846355483 × 10−3 . with the approximate solution: y(x) =

1 + 0.103327411135187x 2 + 5.0409685088265 · 10−4 x 4 1 + 0.266695273475284x 2 + 6.7642846305483 · 10−3 x 4

(28.35)

In Tables 28.3 and 28.4 are compared numerical solution of Eqs. (28.14) and (28.15) for n = 3 and exact solution of Eqs. (28.14) and (28.15) for n = 5, respectively with approximate analytical solutions given by Eqs. (28.34) and (28.35), respectively on the domain [0, L), L > 6.9 and not on the domain [0,1]. It can be seen that the solutions obtained by OAFM are in very good agreement with that given by numerical or exact solution on the domain [0, L] ⊃ [0, 1].

28.2 OAFM for the Polytrophic Spheres of the Lane—Emden Equation Table 28.4 The results of approximate solution y(x) given by Eq. (28.35) and the exact results for n = 5 and x ∈ [0, 9]

363

x

y(x)app Eq. (28.35)

y(x)exact [2]

ε = |y(x)exact − y(x)|

0

1

1

0

0.2

0.993524416

0.993399267

1.3E − 04

0.4

0.974781307

0.974354703

4.3E − 04

0.6

0.945642517

0.944911182

7.3E − 04

0.8

0.907508055

0.907841209

3.3E − 04

1

0.866795998

0.866025403

7.7E − 04

3

0.499141274

0.5

8.5E − 04

5

0.32768110

0.327326835

3.5E − 04

7

0.23997259

0.24019223

2.6E − 04

9

0.189216342

0.188982236

2.3E − 04

References 1. G.P. Horedt, Exact solutions of the Lane—Emden equation in N-dimensional space. Astron Astrophis. 160, 148–156 (1986) 2. G.P. Horedt, Seven-digit tables on Lane—Emden functions. Astrophy. Space Sci. 126, 357–408 (1986) 3. M. Beech, An approximate solution for the polytrope n = 3. Astrophis. Space Sci. 132, 393–396 (1987) 4. I.W. Roxburgh, L.M. Stockman, Power series solutions of the polytrope equations. Mon. Not. R. Astrom Soc. 303, 466–470 (1999) 5. C. Hunter, Series solution for polytropes and the isothermal sphere. Mon. Not. R. Astrom Soc. 328, 839–847 (2001) 6. M.I. Nouh, Accelerated power series solution of polytrophic and isothermal gas sphere. New Astron. 9(6), 467–473 (2004) 7. R. Van Gorder, K. Vajravelu, Analytic and numerical solutions to the Lane—Emden equation. Phys. Lett. A 372(39), 6060–6065 (2008) 8. N.R. Tabrizidooz, N.R. Marzban, M. Razzaghi, Solution of the generalized Emden—Fowler equations by the hybrid functions method. Phys. Scr. 80, 025001 (2009) 9. H. Adibi, A.M. Rismani, On using a modified Legendre—spectral method for solving singular IVP’s of Lane—Emden type. Comp. Math. with Appl. 60, 2126–2130 (2010) 10. J. Boyd, Chebyshev spectral methods and the Lane—Emden problem. Num. Math. Theor. Meth. Appl. 4(2), 142–157 (2011) 11. S. Mukherjee, B. Roy, P.K. Chaterjee, Solution of Lane—Emden equation by differential transform method. Int. J. Nonl. Sci. 12(4), 478–484 (2011) 12. A.H. Bahrawy, A.S. Alofi, Jacobi-Gauss collocation method for solving nonlinear Lane— Emden type equations. Commun. Nonlinear Sci. Numer. Simulat. 17, 62–70 (2012) 13. F. Yin, J. Song, F. Lu, H. Leng, A coupled method of Laplace transform and Legendre wavelets for Lane—Emden—type differential equations. J. Appl. Math. Art. ID 163821 (2012) 14. S. Baranwal, V.K. Pandey, M.P. Tripathi, O.P. Singh, An analytic algorithm of Lane—Emden— type equations arising in astrophysics—a hybrid approach. J. Theor. Appl. Phys. 6, 22–28 (2012) 15. R.K. Pandey, N. Kumar, Solution of Lane—Emden type equations using Bernstein operational matrix of differentiation. New Astron. 17(3), 303–308 (2012) 16. C. Yang, J. Hou, A numerical method for Lane—Emden equations using hybrid functions and the collocation method. J. Appl. Math. Art.ID 316534 (2012)

364

28 The Polytrophic Spheres of the Nonlinear Lane …

17. K. Boubaker, R.A. Van Gorder, Application of BPES to Lane—Emden equations governing polytrophic and isothermal gas sphere. New Astron. 17(6), 565–569 (2012) 18. J.A. Khan, M.A.Z. Raja, I.M. Qureshi, An application of evolutionary computational technique to non-linear singular system arising in polytrophic and isothermal sphere. Global J. Res. in Eng. Numer. Methods 125(1), 1–8 (2012) 19. S. Hosseini, S. Abbasbandy, Solution of Lane—Emden type equations by combination of the spectral method and Adomian decomposition method. Math. Problems Eng. Art. ID 534754 (2015) 20. R. Tripathi, H.K. Mishra, Homotopy perturbation method with Laplace transform for solving Lane—Emden type differential equations. Springer Plus 5, 1859 (2016) 21. W. Al-Hayani, L. Alzubaidy, A. Entesar, Solutions of singular IVP’s of Lane—Emden type by homotophy analysis method with genetic algorithm. Appl. Math. Inf. Sci. 2, 407–416 (2017) 22. G.P. Horedt, Approximate analytical solutions of the Lane—Emden equation in N-dimensional space. Astron. Astrophys. 172, 359–367 (1987) 23. S. Chandrasekhar, An Introduction to the Study of Stellar Structure (Dover, 1967)

Part III

Some Variants and Modifications of the Basic Optimal Auxiliary Functions Method

Chapter 29

The Second Alternative to the Optimal Auxiliary Functions Method

In this Chapter, we present an alternative to the OAFM. For the nonlinear differential equation L[u(x)] + N[u(x)] + g(x) = 0, x ∈ 

(29.1)

with the boundary/initial conditions   du(x) =0 B u(x), dx

(29.2)

The approximate solution is of the form u˜ (x) = u0 (x) + u1 (x, Ci ), i = 1, 2, . . . , p

(29.3)

where the initial approximation u0 (x) can be determined from the linear equation L[u 0 (x)] + g(x) = 0,

  du 0 (x) B u 0 (x), dx

(29.4)

The first approximation u1 (x,Ci ) is determined from the linear equation L[u1 (x, Ci )] +

p 

Ck Fk (fi , gi ) = 0

(29.5)

k=1

where the auxiliary functions Fk are dependent on the functions fi and gi . The functions fi depend on the initial approximation u0 (x), and gj depend on the nonlinear operator N calculated in u0 (x): N[u0 (x)].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_29

367

368

29 The Second Alternative to the Optimal Auxiliary Functions Method

Based on results known from the theory of differential equations (see Ref. [1] from Chap. 2), in the present Chapter we have the freedom to choose the first approximation in the form u 1 (x, Ci ) =

p 

Fi (C j , f k , gr )

(29.6)

i=1

where Fi are auxiliary functions depending on the p unknown parameters Cj , and also on the functions fk defined in Eq. (29.4) and gr which appear in the composition of N[u0 (x)]. In consequence, the first approximation u1 (x,Ci ) is determined from Eq. (29.6).

29.1 Dynamic Response of a Permanent Magnet Synchronous Generator to a Wind Gust The governing equations of the smooth-air-gap synchronous generator in the rotor reference frame [2] may be written using the classical D-Q equivalent circuit models as (see also Chapters 17 and 18): D − ωE ψQ u D = RG i D + dψ dt dψ Q u Q = RG i Q + dt + ω E ψ D ψD = L G i D + ψP M ; ψQ = L G i Q

(29.7)

where uD and uQ are the instantaneous values of D and Q axis stator voltage components, iD , iQ are the stator current components, RG and LG are electrical resistance and synchronous inductance of the generator phase windings, ψD and ψQ are instantaneous values of D and Q axis stator flux components, ψPM is the permanent magnet flux and ωE is the electrical angular speed, such that the mechanical angular speed of the turbine-generator system will be M = ωE /P1, where P1 is the number of pole pairs of the generator. The output voltages of the PMSG can be expressed taking into account the electrical resistance RL of the external load connected to the output of the generator, and the corresponding iD and iQ : u D = −R L i D u Q = −R L i Q

(29.8)

The motion equation of the considered generator is given by [2]: 3 JM dω E = TM + P1 ψ P M i Q P1 dt 2

(29.9)

29.1 Dynamic Response of a Permanent Magnet Synchronous …

369

in which JM represents the total axial moment of inertia and TM represents the mechanical torque of the wind turbine, given by: TM =

1 r ωE ρπr 3 v2 Ct (λr ) , λr = 2 P1 v

(29.10)

in which r is the turbine radius, v is the wind speed, ρ is the air density, λr is the tip-speed ratio and Ct is the torque coefficient provided by the turbine manufacturer as: Ct (λr ) = 0.125 + 0.2092λr − 0.1209λr2.5

(29.11)

Taking into account the characteristics of the device under study, P1 = 16, r = 2.5 mm and considering ρ = 1.225 kg/m3 , the expression of the torque can be written as: TM = 3.758252931v2 + 0.982783141vω E −

0.035079416 2.5 ωE √ v

(29.12)

An analytical representation of the wind profile, which influences the response of the generator is considered as described in [2] under the form v(t) = vm + Asin

π 3π t + Bsin t TG TG

(29.13)

in which vm represents the mean wind speed of the base wind velocity, which is the constant component, A and B are the amplitudes of the variable components, while TG is the gust period. We implement the following transformations UD UQ ID IQ D ; uQ = ; iD = ; iQ = ; ψD = ; UB UB IB IB B Q P M ωE RL RB ψQ = ; ψP M = ;ω = ; rl = ; rg = B B ωB zB zB ωB L G TM JM ω2B t xg = ; Tm = ;k = ;τ = = ωB t zB TB tB 3P12 U N I N

uD =

(29.14)

and therefore the original Eq. (29.7) can be re-written in dimensionless form rl + r g di d − ωi q + id = 0 dτ xE di q rl + r g ψP M + ωi d + iq + ω =0 dτ xE xE

370

29 The Second Alternative to the Optimal Auxiliary Functions Method

k

dω − Tm − ψ P M i q = 0 dτ

(29.15)

The associated initial conditions for these equations will be obtained considering the steady-state regime, which is characterized by a constant angular speed at a constant speed of the wind vm = 10 m/s and external electrical load rl = 0.4528. In this situation, we obtain the initial conditions as i d (0) = −0.438786995; i q (0) = −0.843879596; ω(0) = 0.499239911

(29.16)

After some simple manipulations, the governing Eq. (29.15) may be written in the form: di d + 0.9601432255i d − ωi q = 0 dτ di q + 0.9601432255i q + ωi d + 2.061756973ω = 0 dτ dω − 0.168386689i q − 0.000808844v2 dτ 1.124718044 2.5 ω =0 − 0.024807429vω + √ v

(29.17)

where v is given by (29.13), considering A = 10, B = 4, vm = 10 and TG = 20.5, so that 3π π v(τ ) = 10 + 10sin τ + 4sin τ λ λ

(29.18)

where λ = 2407.

29.1.1 Application of an Alternative of OAFM to the Considered Problem (29.16) and (29.17) The application of the proposed solution procedure always starts with identifying the linear operators of the considered equations, so that in the considered case we may establish three linear operators of the following form: L 1 [i d (τ )] =   L 2 i q (τ ) = L 3 [ω(τ )] =

di d dτ di q dτ dω dτ

(29.19)

29.1 Dynamic Response of a Permanent Magnet Synchronous …

371

and consequently the corresponding nonlinear operators remain to be   r +r N1 i d (τ ), i q (τ ), ω(τ ) = l x E q i d − ωi q   r +r N2 i d (τ ), i q (τ ), ω(τ ) = l x E q i q + ωi d + ω ψxPEM   N3 i d (τ ), i q (τ ), ω(τ ) = −ψ P M i q − TM

(29.20)

As a first step of the iteration procedure, the initial approximations of the considered system will be determined from Eq. (29.4), which become did0 (τ ) dτ diq0 (τ ) dτ dω0 (τ ) dτ

= 0; =0 =0

(29.21)

with the initial conditions given by (29.16). In these conditions, the initial approximations are identified as id0 (τ ) = −0.438786995 iq0 (τ ) = −0.843879596 ω0 (τ ) = 0.499239911

(29.22)

The corresponding nonlinear operators (29.20) for the initial approximations will be: N1 [id0 (τ ), iq0 (τ ), ω0 (τ )] = 0

(29.23)

N2 [id0 (τ ), iq0 (τ ), ω0 (τ )] = 0

(29.24)

N3 [id0 (τ ), iq0 (τ ), ω0 (τ )] = 0.14209809  2 −0.000808844 10 + 10sin πλ τ + 4sin 3π τ λ −0.012384789 10 + 10sin πλ τ + 4sin 3π τ λ + √ 0.1980091808 π 3π

(29.25)

10+10sin λ τ +4sin

λ

τ

Now, considering these expressions (29.23)–(29.25) and taking account of (29.6), in what follows we can consider Fi (Cj , fk , gr ) = Cij sin

(2j − 1)π λ

(29.26)

and following the procedure described in this Chapter, we can get the solutions of the first approximations as

372

29 The Second Alternative to the Optimal Auxiliary Functions Method

π 3π id1 (τ ) = C11 sin τ + C12 sin τ + · · · λ λ

(29.27)

π 3π iq1 (τ ) = C21 sin τ + C22 sin τ + · · · λ λ

(29.28)

π 3π ω1 (τ ) = C31 sin τ + C32 sin τ + · · · λ λ

(29.29)

Finally, the approximate solutions of Eqs. (29.17) and (29.16) will be id(τ ) = id0 (τ ) + id1 (τ )

(29.30)

iq(τ ) = iq0 (τ ) + iq1 (τ )

(29.31)

ω(τ ) = ω0 (τ ) + ω1 (τ )

(29.32)

According to the described procedure, the next step will be the identification of the optimal values of the convergence-control parameters, and after that the explicit solution will be available. However, in what follows, we will show that the number of convergence-control parameters involved in the auxiliary functions could greatly influence the convergence of solutions.

29.1.2 Numerical Examples In the following examples we will employ an increased number of convergencecontrol parameters in the approximate solution, in order to evaluate the influence of this number on the convergence of solutions idapp , iqapp and ωapp . In what follows, the number of parameters is denoted by x, y and z, respectively for the convergencecontrol parameters of idapp , iqapp and ωapp . For a better evaluation, in each case the approximate analytical solutions are graphically represented in comparison with numerical integration results.

29.1.2.1

Case 1: X = 2, Y = 5, Z = 2

In this first case the optimal values of the convergence-control parameters Cij are obtained by means of a collocation approach as:

29.1 Dynamic Response of a Permanent Magnet Synchronous …

C11 = −0.721178281655, C12 = −0.775467568786 C21 = −0.09272729779, C22 = −0.034880157984, C23 = −0.081061912708, C24 = −0.051727806797, C25 = −0.048680696911, C31 = 1.143270540022, C32 = 0.630062415013

373

(29.33)

Figures 29.1, 29.2 and 29.3 show the obtained approximate solutions of Eqs. (29.16) and (29.17), which, for ensuring a way to estimate their accuracy, are compared with numerical solutions obtained using a fourth-order Runge–Kutta method. Fig. 29.1 Comparison between the analytical and numerical results for id in Case-1

Fig. 29.2 Comparison between the analytical and numerical results for iq in Case-1

Fig. 29.3 Comparison between the analytical and numerical results for ω in Case-1

374

29 The Second Alternative to the Optimal Auxiliary Functions Method

Fig. 29.4 Comparison between the analytical and numerical results for id in Case-2

Fig. 29.5 Comparison between the analytical and numerical results for iq in Case-2

29.1.2.2

Case 2: X = 4, Y = 7, Z = 3

In the second case the optimal values of the convergence-control parameters Ci are: C11 = −1.2876493766, C12 = −0.471433409076 C13 = −0.1027742455, C14 = −0.07531102628, C21 = −0.09906695036, C22 = −0.00511906590, C23 = −0.07189171775, C24 = −0.05332650921 C25 = −0.03816510200, C26 = −0.02305774489, C27 = −0.00816264558, C31 = 1.363785231067, C32 = 0.478132815441, C33 = 0.053852277381

(29.34)

Figures 29.4, 29.5 and 29.6 show the comparison between the obtained approximate solutions of Eqs. (29.16) and (29.17) in case 2 and numerical integration results.

29.1.2.3

Case 3: X = 6, Y = 9, Z = 4

In the third case we obtain the following optimal values of the convergence-control parameters:

29.1 Dynamic Response of a Permanent Magnet Synchronous …

375

Fig. 29.6 Comparison between the analytical and numerical results for ω in Case-2

Fig. 29.7 Comparison between the analytical and numerical results for id in Case-3

C11 = −1.3121425222, C12 = −0.46172281501 C13 = −0.108400382, C14 = −0.06232646941, C15 = −0.026619404, C16 = −0.00972490599 C21 = −0.10206627291, C22 = −0.0111020519, C23 = −0.07350272084, C24 = −0.0482234299 C25 = −0.0320759468, C26 = −0.02155488364, C27 = −0.0124749316, C28 = −0.00569735143, C29 = −0.00224172165, C31 = 1.32012272028, C32 = 0.505408846362, C33 = 0.021852060919 C34 = 0.010749981796

(29.35)

Figures 29.7, 29.8 and 29.9 provide the comparison between the obtained approximate analytical solutions and numerical integration results in the third case.

29.2 Lambert W Function with Application in Electronics and Seismic Waves The Lambert W function is defined to be the solution of the algebraic equation [1]

376

29 The Second Alternative to the Optimal Auxiliary Functions Method

Fig. 29.8 Comparison between the analytical and numerical results for iq in Case-3

Fig. 29.9 Comparison between the analytical and numerical results for ω in Case-3

W(x)exp(W(x)) = x

(29.36)

where x can be any complex number or when restricted to real values, then W(x) is defined only for x ≥ − 1e . We note that for x < − 1e , Eq. (29.36) has no real solution (Fig. 29.10). Solid curve is the principal branch for − 1e ≤ x < ∞ and dashed curves is branch for − 1e ≤ x ≤ 0. The Lambert W function is a multivalued function, or multibranched function. In Fig. 29.10 are plotted the branches considering only real valued of x where two branches: W(x) and W−1 (x) are identified. W(x) is the principal branch (also named the upper branch) defined in the condition W(x) ≥ −1 (solid curve in Fig. 29.10), while the branch W−1 (x) is defined in the condition W−1 (x) ≤ −1 (dashed curve in Fig. 29.10). On the other hand the principal branch can be divided into two subbranches: W+ (x) defined in the condition W+ (x) ≥ 0 and subbranch W− (x) defined in the conditions −1 ≤ W− (x) < 0. This means that the subbranch W+ (x) is defined for all arguments values x ≥ 0 and the subbranch W− (x) is defined only for the values − 1e ≤ x < 0 where as the branch W−1 (x) is defined for the values   − 1e ≤ x < 0. But branches are jointed in the point P − 1e , −1 . Follows that on the domain − 1e ≤ x < 0 there are two possible values for W(x): the one for W− (x) and the second for W−1 (x) while there is only one possibility for W(x) on the domain x ≥ 0.

29.2 Lambert W Function with Application in Electronics and Seismic Waves

377

Fig. 29.10 Two real branches of the Lambert function

The origin of Eq. 29.36 begins in 1758 when J.H.Lambert solved the trinomial equation y = a + yp by giving a series for y in powers of a. The Eq. 29.36 was first described by Polya and Szego in 1925 [3]. The number of works dealing with this remarkable function from 1758 along the years is enormous. In the following we mention only a smaller fraction of the past contributions to this topic, quoting only the publications deal exclusively with the Lambert W function. Corless et al. [4] presented a discussion of the complex branches of W function, an asymptotic expression valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision and a method for the symbolic integration of expressions containing the W function. They presented a some applications of Lambert W function like as jet fuel problem, model a combustion problem, an enzyme kinetics problem, a linear constant-coefficient delay equations, a anomaly in the calculation of exchange forces between two nuclei within the hydrogen molecular ion, the Ricker’s equation, Volterra equation for population growth, asymptotic root of trinomial, epidemic and competitive, pedagogical application and so on. Also they proposed the following approximate solutions for Eq. 29.36: W(x) = L1 − L2 +

L2 L2 (L2 − 2) L2 (2L22 − 9L2 + 6) + + L1 2L21 6L31

378

29 The Second Alternative to the Optimal Auxiliary Functions Method

L2 (2L32 − 22L22 + 36L2 − 12) + +O 12L41



L2 L1

5 (29.37)

where L1 = Log x, L2 = Log Log x, and 11 1 W(x) = −1 + p − p2 + p3 + · · · 3 72

(29.38)

√ √ for p = 2(1 + ex) and |p| < 2. Conde et al. [5] constructed an exact closed form solutions based on the Lambert W function for the forward current–voltage characteristics of non-ideal singleexponential divides containing all possible combinations of series and shunt parasitic resistances. Barry et al. [6] provided three simple analytical functions for W(x) for three different domains: ⎛ ⎞ 

1.2x 2x ⎝ ⎠  − 0.45869ln  (29.39) W+ (x) ≈ 1.45869ln 2.4x ln(1 + 2x) ln ln(1+2.4x)

for x ≥ 0 √ 2(1 + ex)   W− (x) ≈ −1 +  √ √ 1 + N1 (x) · 2(1 + ex)/ N2 (x) + 2(1 + ex)

(29.40)

where − 1e ≤ x < 0 and 

√ 1 N1 (x) = 1 − √ (N2 (x) + 2) (29.41) 2 √ √ √ (2237 + 1457 2)e − 4108 2 − 5764  N2 (x) = 3 2 + 6 − · 2(2 + ex) √ √ (215 + 199 2)e − 430 2 − 786 (29.42) W−1 (x) = ln(−x) − 5.9586 · (1 −

√

1 + 0.0042(1 + ln(−x))e−0.00201 −(1+ln(−x)) √ √ 0.3361 −0.5(1 + ln(−x)) + 1 + 0.0042(1 + ln(−x))e−0.0021 −(1+ln(−x))



(29.43) where − 1e ≤ x < 0. Blondeau and Monir [7] constructed a simple and fast evaluation algorithm with prescribed accuracy by means of Lambert W function for Monte Carlo simulation requiring large numbers of realizations of the generalized Gaussian noise. The

29.2 Lambert W Function with Application in Electronics and Seismic Waves

379

outflow speed and the mass loss rate if the solar wind of plasma particles ejected by the Sun and the radial dependence of the flow speed applied to Parker’s isothermal solar wind equation and Bondi’s equation are solved by Cranmer [8] by use of the Lambert W function. The problem of equilibrium electric charge of a solitary body plasma, nonlinear ion-acustic waves, the dynamics of shaping virtual cathodes in a medium with viscous friction and the study of the dispersion equation for electron plasma waves in the framework of generalized Boltzmann kinetics are discussed by Dubinov and Dubinova [9] by applying Lambert W function. Valluri et al. [10] presented some applications of the Lambert W function to the formalism of quantum statistics, Maxwell–Boltzmann systems and black body radiation. Wang [11] showed that the stable fixed point-asymptotic values of the population in different classes can be expressed in terms of the Lambert W functions. Vial [12] presented an approximate solution for the downward time of travel in the case of a mass falling with a linear drag force. A quasi-analytical solution implying the Lambert function is equivalent to the search for Wien’s displacement law. Fukushima [13] used two series expansion formulas: that around the branch point W = −1 and that around the zero W = 0. As result, the new method require so call of transcendental functions like the exponential function itself and the logarithm. The analytical method applied to photovoltaic curves by means of approximation that convert equations into explicit form, e.g. the Lambert W function is used by Lun et al. [14]. The Ricker wavelet is the second derivative of a Gaussian function and its spectrum is a single-valued smooth curve, numerical evaluation of the Lambert W function is implemented by Wang [15] by a stable interpolation procedure, followed by a recursive computation for high precision. The Lambert W function is used Fathabadi [16] to perform the equations which express current–voltage(I-V), powervoltage(P–V) and power-current(P-I). Roberts and Valluri [17] presented a solution of the quantum mechanics problem of the allowable energy levels of a bound particle in a one-dimensional finite square well by the images of simple geometric shapes lines circles and by the Lambert W function. The Lambert W function in ecological and evolutionary models, including EulerLotka equation, the marginal value theorem, the Lotka-Voltera predator–prey model and SIR epidemic model is applied by Lehtonen [18]. Lambert W function based framework has been proposed by Biswas et al. [19] to model individual speed in a mixed traffic stream. The predicted speeds by the model developed have been utilised in estimation of the Passenger Car Unit(PCU) for individual vehicle type. A virtual analog model of the Lockhard and Serge Wavefolders in digital domain namely aliasing suppression is studied in [20] by evaluation of the Lambert W function. The construction of an accurate approximations for the principal branch of the Lambert W function is given by Iacono and Boyd [21]. In particular, a simple, global analytic approximation is derived that covers the whole branch with a maximun relative error smaller then 5×10−3 for x > 0. Analytic bounds for W are also constructed for x > 0 which are much tighter than these currently available. The construction of accurate approximation to W based on Chebyshev spectral theory is discussed. Li and Liu[22] presented a simple and accurate procedure for explicitly calculating the friction factors by using Lambert W function. Calasan and Nedic [23] studied

380

29 The Second Alternative to the Optimal Auxiliary Functions Method

the exact analytical closed form solutions for inductor air gap length inductance and other constructional parameter of inductor lead to nonlinear equations which is solved by using Lagrange inversion theorem and the Lambert W function. The roots of the trinomial are given by Belkiˇc [24] in terms of the Lambert and Euler series for the asymmetric and symmetric cases, the complete Bell polynomials, the partial Bell polynomials, the Pochhammer symbols, logarithmic function on the confluent Fox–Wright function. The analytical solutions for the trinomial roots are numerically illustrated in the genome multiplicity corrections for survival of synchronous cell populations after irradiation. Vazquez-Leal al. [25] proposed two accurate piece-wise approximate solutions, one for the lower branch and another one for the upper branch by means of the power series extended method (PSEM) in combination with asymptotic solutions. The approximations are validated for a problem of economy and acoustic waves of nonlinear ions.

29.2.1 Evaluation of the Lambert W Function by OAFM To find an approximate analytic solutions of Eq. 29.36, we assume that the approximate solutions can be written in the form: W(x) = W(C1 f1 (x), C2 f2 (x), · · · , Cn fn (x))

(29.44)

where C1 , C2 , . . . , Cn are n unknown parameters at this moment, f1 , f2 , . . . , fn are n known auxiliary functions. In what follows we present four alternatives to determine the Lambert W function starting from Eq. (29.36) which can be rewritten for x > 0 in the form W(x) + lnW(x) = ln x

29.2.1.1

(29.45)

First Alternative for the Lambert W Function

Taking into account Eqs. (29.44) and (29.45), in this first case we consider W (x) = ln[1 + C1 f 1 (x) + C2 f 2 (x) + · · · + Cn f n (x)]

(29.46)

  with fi (x) = xi , i = 1, 2, · · · , n and x ∈ − 1e , 0 . For the time being we consider only the subbranch W− (x) which is defined as −1 ≤ W− (x) < 0 (see Introduction). We will prove that the first approximate solution W− (x) is more accurate along with an increased number n of the convergence-control parameters into Eq. (29.46).

29.2 Lambert W Function with Application in Electronics and Seismic Waves

381

Subcase 29.2.1.1.1 In this first subcase we consider n = 5 in Eq. (29.46):   1 (29.47) W − (x) = ln(1 + C1 x + C2 x + C3 x + C4 x + C5 x ), x ∈ − , 0 e 2

3

4

5

  Using collocation method on the domain − 1e , 0 , one can get C1 = 1.93849179067, C2 = 21.2681930077, C3 = 171.03378669875 C4 = 562.943563185, C5 = 679.9257743992027 such that the approximate solution for Lambert function becomes: W− (x) = ln(1 + 1.93849179067x + 21.2681930077x2 + 171.03378660875x3 + 562.943563185x4 + 679.9257743992027x5 ) (29.48) Subcase 29.2.1.1.2 In the second subcase we consider n = 6 in Eq. (29.46) and therefore we have: W− (x) = ln(1 + 0.37874191719552x − 18.6077122570364x2 − 201.501944313274x3 − 1089.030988959008x4 − 2883.0081799408773x5 − 2884.549838666x6 )

(29.49)

Subcase 29.2.1.1.3 In the last subcase for n = 7 in Eq. (29.46) we obtain: W− (x) = ln(1 + 0.9357624081875x + 4.09398113173125x2 + 85.5013827833875x3 + 770.00368403151x4 + 3696.07175732704x5 + 8937.716852810666x6 + 8642.096610590476x7

(29.50)

In Table 29.1 we present a comparison between the first approximate solutions given by Eqs. (29.48), (29.49) and (29.50) respectively and numerical solution of Eq. (29.45) more precisely for the subbranch W− (x) defined by −1 ≤ W− (x) ≤ 0. From this table we can conclude that a large number of optimal convergence -control parameters ci in the Eq. (29.46) lead to a better accuracy of the results. In Table 29.2 we compare the first approximate solution W− (x) given by Eq. (29.50) with the first approximate solutions given by Barry et al. [6], Iacono and Boyd [21], Winitzki [26], Luo et al. [27] and with numerical results. The second solution of Lambert W function  defined by W−1 (x) ≤ −1 is  W−1 (x) obtained from Eq. (29.46) for n = 7 and x ∈ − 1e , 0 :

−0.1083716413

−0.169745141

−0.2511669103

0.3463788754

−0.4743316627

−0.9694225405

−0.15

−0.2

−0.25

−0.3

−e−1

−0.0638213481

−0.05

−0.1

W− (x) Eq. (29.48)

x

  Table 29.1 The relative error ε = 1 −   x ∈ − 1e , 0



3.06

3.08

3.08

3.01

3.43

3.09

21

ε%

−0.0027854474

−0.4902714098

−0.3578944909

−0.2594697714

−0.1796805941

−0.1194541164

−0.4733779418

W− (x) Eq. (29.49)

0.278

0.178

0.137

0.115

0.105

0.101

10.17

ε%

−0.9999996717

−0.4894023788

−0.3574020952

−0.2591711861

−0.1794912841

−0.1183256603

−0.0527060089

W− (x) Eq. (29.50)

−0.1118325602 −0.1794912699 −0.2591711192 −0.3574029585 −0.4894022204

5.41 × 10−6 7.85 × 10−6 2.57 × 10−5 2.41 × 10−4 3.24 × 10−5

−1

−0.0527059849

4.59 × 10−5

3.29 × 10−5

W Numerical

ε%

W− (x)  WNum (x) ·100 between the first approximate solutions (29.48), (29.49) and (29.50) and numerical solution of Eq. (29.45),

382 29 The Second Alternative to the Optimal Auxiliary Functions Method

29.2 Lambert W Function with Application in Electronics and Seismic Waves

383

Table 29.2 Comparisons between the first approximate relation given by (29.50), the first approximate solution given by [6, 21, 26, 27] with numerical solution of Eq. (29.14) for x ∈  −1  −e , 0 ε%

W− (x) [6]

ε%

−0.05 −0.0527060089

4.55 × 10−5

−0.0527088635

0.005 −0.0527697486 0.121

−0.1

5.41 × 10−6

−0.1118433225

0.009 −0.1119555703 0.110

−0.15 −0.1794912841

7.85 × 10−6

−0.17951329941 0.012 −0.1796664311 0.097

−0.2

2.57 × 10−5 −0.2592033541

0.013 −0.2593869904 0.083

−0.25 −0.3574020952

2.41 × 10−4

−0.3574456765

0.012 −0.3576401412 0.066

−0.3

−0.4894023788

3.24 × 10−5

−0.4894428633

0.008 −0.4896238778 0.045

−e−1

−0.9999996717

3.29 × 10−5

−1

0

x

W− (x) Eq. (29.50) −0.11183256603 −0.2591711861

W− (x) [21]

ε%

−1

0

x

W− (x)[26]

ε%

W− (x)[27]

ε%

W− (x) Numerical

−0.05

−0.0526792795

0.051

−0.0527059829

3.8 × 10−6

−0.0527059849

−0.1

−0.1115518018

0.251

−0.1118324843

6.7 × 10−5

−0.1118325602 −0.1794912699

−0.15

−0.1782055170

0.716

−0.1794896616

8.9 × 10−4

−0.2

−0.2548533042

1.666

−0.2591522713

7.2 × 10−3

−0.2591711192

−0.25

−0.3446753785

3.561

−0.3572287643

0.049

−0.3574029585

−0.3

−0.452414465953

7.558

−0.4877684288

0.334

−0.4894022204

−e−1

−0.6413136819

35.87

−0.8075278304

19.24

−1

W −1 (x) = ln(1 + 51.4759378396253x + 1032.5956045293753x 2 + 10542.2565599212501x 3 + 60360.5252815200141x 4 + 195475.7431282117998x 5 + 333867.01718933333333x 6 + 233126.37264285709989x 7 )

(29.51)

In Table 29.3 we compare the second solution of Lambert W function given by Eq. (29.51) with the approximate solution given by Barry et al. [6] and with numerical solution of Eq. (29.45). It can be seen that the second approximate solution (29.51) is very accurate in comparisons with presented in [6] and with the numerical solution on   the procedure the domain x ∈ −e−1 , 0 . Now for the principal branch W+ (see Introduction) defined by W+ ≥ 0 we propose the following approximate solution of Eq. (29.45) for n = 8 and x ≥ 0 W + (x) =

1 + C1 x + C2 x 2 + C3 x 3 · ln(1 + C8 x) C4 + C5 x + C6 x 2 + C7 x 3

(29.52)

Making use the collocation method previously mentioned, the coefficients are found to be:

384

29 The Second Alternative to the Optimal Auxiliary Functions Method

Table 29.3 Comparisons between the second approximate solution given by Eq. (29.51), the second   approximate solution given in [6] and numerical solution of Eq. (29.45) for x ∈ −e−1 , 0 x

W−1 (x) Eq. (29.51)

ε%

W−1 (x) [6]

ε%

W−1 (x) Numerical

−0.05

−4.4997562042

4.55 × 10−5

−4.5006064735

0.019

−4.4997552805

−0.1

−3.5771526282

5.41 × 10−6

−3.5776128663

0.013

−3.5771520612

−0.15

−2.9935951775

7.85 × 10−6

−2.9948046157

0.040

−2.9935949755

−0.2

−2.5426411692

2.57 × 10−5

−2.5426704444

0.001

−2.5426413511

−0.25

−2.1532917908

2.41 × 10−4

−2.153210122171

0.004

−2.1532923012

−0.3

−1.7813361372

3.24 × 10−5

−1.7812104236

0.007

−1.7813370201

−e−1

−0.9999992420

3.29 × 10−5

−1

0

−1

C1 = 1.563718862524682, C2 = 0.4736732053105695, C3 = −6.794794493225987 · 10−3 , C4 = 0.78322008017523, C5 = 1.684226134859049, C6 = 0.6267334672285997 C7 = −8.97161260368 · 10−3 , C8 = 0.7813550427411

(29.53)

The approximate solution of the principal branch of Lambert function becomes: W+ (x) 1 + 1.5637188625x + 0.4736732053x2 − 6.7947944932 · 10−3 x3 0.78322008 + 1.68422613x + 0.62673346x2 − 8.9716126 · 10−3 x3 · ln(1 + 0.78135504274118x) (29.54)

=

The domain of interest for the expression (29.54) is [0,20]. On the domain [20,18000] the approximate solution of the principal branch of Lambert function is W + (x) 1 + 0.7935401612x + 0.0107087151x 2 + 1.485074877 · 10−6 x 3 −1.25242177 + 10.86949741x + 0.01253676x 2 + 1.723056 · 10−6 x 3 · ln(0.454353251340102x) (29.55) =

In this way we can extend the domain of interest for the approximate solution of the principal branch of Lambert function for example to [18000, 1000000]. In Table 29.4 we compare some values of the principal branch W+ (x) given Eq. (29.54) on domain [0, 20] and of Eq. (29.55) on domain [20,18000] with results obtained by Barry et al. [6], Iacono and Boyd [21], Winitzki [26], Luo et al. [27] and with numerical results. In the following alternatives we present only the principal branch W+ (x).

29.2 Lambert W Function with Application in Electronics and Seismic Waves

385

Table 29.4 Comparisons between some values of W+ (x) obtained in present paper into [6, 21, 26, 27] and numerical solutions of Eq. (29.45) for x ∈ [0, 18000] x

W+ (x) Eqs. (29.54) and (29.55)

ε%

W+ (x) [6]

0

0

0

0

0

0

0

1

0.5671543193

0.002

0.5676614511

0.091

0.5684742943

0.234

2

0.8526196735

0.001

0.8528634814

0.030

0.8550026221

0.281

4

1.2021843849

0.001

1.2017233619

0.037

1.2061339442

0.330

5

1.3267449335

0.001

1.3259689847

0.059

1.3313006656

0.345

10

1.7455462059

0.001

1.7436238653

0.109

1.7522646089

0.386

20

2.2050221441

8.5 ×10−4

2.2017846251

0.145

2.2141478825

0.415

40

2.6982511322

0.053

2.6922286636

0.170

2.7083639972

0.428

50

2.8623728701

0.051

2.8558729167

0.175

2.87317781951

0.429

100

3.3872599446

0.048

3.3792900348

0.187

3.3999442021

0.423

1000

5.2509856870

0.026

5.2393789777

0.195

5.2661283181

0.315

10,000

7.2306534447

0.016

7.2184918761

0.184

7.2422225909

0.143

18,000

7.74425018065

0.079

7.7363268332

0.181

7.7580021753

0.098

W+ (x) [27]

W+ (x) [21]

ε%

ε%

x

W+ (x) [26]

0

0

0

0

0

0

1

0.5576166358

1.680

0.5672248048

0.014

0.5671432904

2

0.8357929395

1.972

0.8527498047

0.017

0.8526055020

4

1.1817624653574

1.617

1.2013151151

0.071

1.2021678732

5

1.30661547514

1.520

1.3248840987

0.139

1.3267246659

10

1.7309852410

0.833

1.736468843332

0.519

1.7455280031

20

2.201171154305

0.174

2.1792582948

1.167

2.2050032750

40

2.7058499328

0.335

2.6418632753

2.012

2.6968098988

50

2.8741326317

0.465

2.7936111726

2.351

2.8608901781

100

3.4113299118

0.759

3.2707566654

3.393

3.3856301409

1000

5.3050401902

1.056

4.8848812446

6.948

5.2496028535

10,000

7.3015379891

0.964

6.5107756765

9.971

7.2318460384

18,000

7.8221512519

0.126

6.9261649456

10.634

7.7503845728

29.2.1.2

ε%

ε%

W+ (x) Numerical

The Second Alternative for the Lambert W Function

We have freedom to choose the principal branch of Lambert W function in the form  W + (x) = C1 ln 1 +

C2 x C3 x C4 x + + ln(e + x) ln(e + 10x) ln(e + 100x)

386

29 The Second Alternative to the Optimal Auxiliary Functions Method +

C5 x C6 x C7 x C8 x + + + ln(e + 1000x) ln(e + 10000x) ln(e + 100000x) ln(e + 1000000x)



(29.56) To obtain a very good accuracy on the domain [0, 10000] it holds that: C1 = 4.96751584122101, C2 = 0.05645663601338, C3 = 0.4436205350713, C4 = −106.61451940759011, C5 = 1008.00478003014702, C6 = −3025.48657462962403, C7 = 3611.52492766021901, C8 = −1494.28405244096111 The approximate upper branch of the Lambert function on [0, 10000] becomes:  0.05645663601338x W+ (x) = 4.96751584122101 ln 1 + ln(e + x) 0.44362053507130x 106.61451940759011x − + ln(e + 10x) ln(e + 100x) 1008.00478003014702x −3025.48657462962403x + + ln(e + 1000x) ln(e + 10000x)  3611.52492766021901x 1494.28405244096111x − + ln(e + 100000x) ln(e + 1000000x)

(29.57)

In Table 29.5, we compare the upper branch of the Lambert function given by Eq. (29.57) with numerical solution for x ∈ [0, 10000]. Table 29.5 Some values of the upper branch W+ (x) given by Eq. (29.57) and comparison with numerical solution on the region [0, 10000]

x

W+ (x) Eq. (29.57)

W+ (x) Numerical

ε%

0

0

0

0

1

0.5668434939

0.5671432904

0.053

2

0.8521754528

0.8526055020

0.050

3

1.2016211622

1.2021678732

0.045

5

1.3258970493

1.3267246659

0.062

10

1.7441158992

1.7455280931

0.076

20

2.2051563176

2.2050032750

0.069

40

2.6990626001

2.6968098988

0.083

50

2.8631669696

2.8608901781

0.079

100

3.38330466899

3.3856301409

0.076

1000

5.2456030308

5.2496028535

0.076

10,000

7.2263230007

7.2318460384

0.076

29.2 Lambert W Function with Application in Electronics and Seismic Waves

29.2.1.3

387

The Third Alternative for the Lambert W Function

Another expression for the upper branch of the Lambert function is defined by W+ (x) = C1

ln(1 + x) + C2 ln(1 + 1.5x) + C3 ln(1 + 2x) + C4 ln(1 + 2.5x) 1 + C5 [ln(ln(e + x)) + C6 ln(ln(e + 1.5x)) + C7 ln(ln(e + 2x))] (29.58)

such that by means of collocation method in the region [0, 500000] one obtain W+ (x) = −191.48958013356011 ·

A(x) , x ∈ [0, 500000] B(x)

(29.59)

where A(x) = ln(1 + x) − 3.00338242072745 ln(1 + 1.5x) + 2.991374129427148 ln(1 + 2x) − 0.99770434972371 ln(1 + 2.5x) B(x) = 1 + 20.66458142161533[ln(ln(e + x)) −6.38518946584951 ln(ln(e + 1.5x)) + 5.3962970666356ln(ln(e + 2x))]

29.2.1.4

The Fourth Alternative for the Upper Branch of Lambert W Function

In the last case we choose the upper branch for the Lambert function in the form  W+ (x) = C1 ln

 1 + 10x 1 + 20x 1 + 70x + C2 ln + · · · + C7ln ln(e + 10x 2 ) ln(e + 20x 2 ) ln(e + 70x 2 ) (29.60)

It can be shown that the W+ given by Eq. (29.58) can be rewritten as 

1 + 10x ln(e + 10x 2 ) 1 + 30x 1 + 20x + 136.998032137134 ln − 27.063136181231 ln ln(e + 20x 2 ) ln(e + 30x 2 ) 1 + 50x 1 + 40x − 3.462976073015 ln − 206.056924512383 ln 2 ln(e + 40x ) ln(e + 50x 2 )  1 + 70x 1 + 60x − 111.2587354772 ln + 209.84046150358 ln ln(e + 60x 2 ) ln(e + 70x 2 ) (29.61)

W+ (x) = −3.0053749227444360 ln

388

29 The Second Alternative to the Optimal Auxiliary Functions Method

In Table 29.6 we compare the last two upper branch of the Lambert function given by Eq. (29.58) for x ∈ [0, 500000] and for [0, 100000] with numerical solution. From Tables 29.2, 29.3, 29.4, 29.5 and 29.6 it is obvious that all procedures presented in Refs. [6, 21, 26, 27] give a very good accuracy but OAFM is by far the best method delivering faster convergence and better accuracy. From the considerations above it follows that the approximate solution given by Eqs. (29.50), (29.51) or (29.54), (29.55), (29.57), (29.59) and (29.61) are not unique. Let us note that it can also chosen following expressions for Lambert function:   C3ln(1 + ln(1 + x)) C5ln(1 + ln(1 + 2x)) + + ··· W (x) = ln(1 + C1 x) C2 + C4 + ln(1 + x) C6 + ln(1 + 2x) (29.62)   1 + C2 x + C3 x 2 + C4 x 3 W (x) = ln(1 + C1 x) + ··· C5ln(e + x) + C6ln(1 + x 2 ) + C7ln(e + x 3 ) (29.63) W (x) = C1ln W (x) =

1 + C2 x 1 + C5 x + C4 ln + ··· ln(e + C3 x) ln(e + C6 x)

C1ln(1 + C2 x) + C3ln(1 + C4 x 2 ) + C4 ln(1 + C5 x 3 ) 1 + C6ln(1 + C7 x) + C8ln(1 + C9 x 2 ) + C10 ln(1 + C11 x 3 )

(29.64)

(29.65)

and so on. Table 29.6 Some values of the of the upper branch W+ (x) given by Eq. (29.59) for x ∈ [0, 500000] the upper branch W+ (x) given by Eq. (29.61) for x ∈ [0, 1000000] and comparisons with numerical results x

W+ (x) Eq. (29.59)

ε%

W+ (x) Eq. (29.61)

ε%

W+ (x) Numerical

0

0

0

0

0

0

1

0.5672735863

0.023

0.5671300251

0.002

0.5671432904

2

0.8528018758

0.023

0.8527465244

0.016

0.8526055020

4

1.2034445921

0.023

1.2025856720

0.035

1.2021678732

5

1.3269819787

0.019

1.3269826142

0.019

1.3267246659

10

1.7459291573

0.023

1.7451817957

0.019

1.7455280931

20

2.2057721727

0.035

2.2046514488

0.016

2.2050032750

40

2.6978878874

0.040

2.6963683011

0.016

2.6968098988

50

2.8620382397

0.040

2.8606309805

0.009

2.8608900781

100

3.3869317103

0.038

3.3860667358

0.013

3.3856301409

1000

5.2514241487

0.035

5.12520832966

0.047

5.2496028535

10,000

7.2377850410

0.026

7.2323692047

0.007

7.2318460384

100,000

9.2829500406

0.017

9.2780736515

0.070

9.2845714291

500,000

10.7391083961

0.079

–

–

10.7476740199

29.2 Lambert W Function with Application in Electronics and Seismic Waves

389

29.2.2 Application of the Lambert Function in Electronics and Seismic Waves 29.2.2.1

Current–voltage Characteristic of Non-Ideal Diode

We consider a diode modeled by an electronic circuit containing a series resistance Rs a shunt resistance Rp , a junction reverse current I0 , a junction ideality factor η and the terminal voltage VT (Fig. 29.11). This model is described as [25]: 

  V − IRs V − IRs −1 + I = I0 exp ηVT Rp

(29.66)

where I is the terminal current, V is voltage and VT = kT/q with q = 1.60217646 × 10−19 C (electron change), k is the Boltzmann constant k = 1.3806503 × 10−23 J/k and T is the Kelvin temperature of function. The Eq. (29.66) can be rewritten in the form 



1 V V Rs Rs = I+1− I+ + exp − ηVT ηVT I0 Rp Rp

(29.67)

or in equivalent form: exp(aI + b) = cI + d

(29.68)

where variables are defined as follows: a=−

Rs V 1 Rs V ,b = ,c = + ,d = 1 − ηVT ηVT Is Rp Rp

Fig. 29.11 Circuit model of the diode

(29.69)

390

by

29 The Second Alternative to the Optimal Auxiliary Functions Method

 Multiplying Eq. (29.66) by − ac exp −aI −

ab c



then the current I can be determined



 a bc − ab d 1 I = − − W − exp c a c c

(29.70)

Using variables given in Eqs. (29.68) and (29.69) lead to the explicit solution for the current in term of Lambert W function

  V − R p I0 Rs R p I0 V + Rs I0 ηVT exp + I (V ) = W (29.71) Rs ηVT (Rs + R p ) ηVT (1 + G p Rs ) Rs + R p where Gp = R1p is the shunt conductance. It is clear that the arguments of the Lambert function in Eq. (29.70) only contain the corresponding variable and parameters. Similarly the explicit solution giving the voltage in term of the current can be expressed in the form. 

 R p I0 R0 (I + I0 ) ·W + (Rp + Rs )I v(I) = Rp I0 − ηVT W ηVT ηVT

(29.72)

In Fig. 29.12 depicted the characteristic I = I(V) for following values of the parameters I0 = 10−15 , ηVT = 0.003, Rs = 103 , Rp = 106 . Within Eq. (29.71) the Lambert function is given by Eq. (29.61). From the plot it can be seen that the

Fig. 29.12 Characteristic I-V obtained from Eqs. (29.71) and (29.61) ( numerical ( )

) and

29.2 Lambert W Function with Application in Electronics and Seismic Waves

391

results obtained using Eq. (29.61) are nearly identical with the solution obtained by numerical integration.

29.2.2.2

The Frequency Band of the Ricker Wavelet

The Ricker wavelet is known in the study of seismic data and have the scaled form [26, 27]. √





1 22 1 22 πω r(t) = − 1 − ωp t exp − ωp t 2 2 4

(29.73)

where t is the time and ωp is the most energetic frequency. For Eq. (29.73) the Fourier transform is defined by ω2 ω2 R(ω) = − 2 exp − 2 ωp ωp

(29.74)

This the most energetic frequency ωp is the peak frequency and is obtained from ∂R = 0, whose solution is ω = ωp . The peak amplitude is the condition ∂ω R(ωp ) = −

1 e

(29.75)

where e is the Euler’s constant e = 2.71828182849 · · · . The bandwidth of the amplitude spectrum is measured at a half of this peak: R(ω) = −

1 2e

(29.76)

From Eqs. (29.73) and (29.75) yields ω2 ω2 1 − 2 exp − 2 = − ωp ωp 2e

(29.77)

so that from Eq. (29.77) one retrieves 

ω2 1 = −W − ω2p 2e The frequency band [ω1 , ω2 ] can be obtained from Eq. (29.78)

(29.78)

392

29 The Second Alternative to the Optimal Auxiliary Functions Method

 ω1 = ωp





  1 1 ; ω2 = ωp −W1 − −W− − 2e 2e

(29.79)

 1  1 where W− − 2e is obtained from Eq. (29.50) and W1 − 2e is obtained from  1  1 Eq. (29.51). Therefore the approximate values of W− − 2e and W−1 − 2e are respectively:

W−

1 − 2e



 ≈ −0.23196095295; W−1

1 − 2e

 ≈ −2.6783469901;

(29.80)

The frequency band becomes [ω1 , ω2 ] ≈ [0.481623247882, 1.6365656082174]

(29.81)

The frequency band is centred at the frequency ωe =

1 (ω1 + ω2 ) ≈ 1.0590944279 2

(29.82)

For Eqs. (29.81) and (29.82) the error is ε = 1.31 × 10−7 %.

29.3 Nonlinear Blasius and Sakiadis Flows In fluid mechanics, a Blasius (named after Paul Richard Heinrich Blasius) boundary layer describes the steady flow of two dimensional viscous incompressible fluids has been intensely studied. The Blasius equation is the “mother” of all boundary layer theory on a half-infinite interval. Since the pioneering paper of Howarth [28] different aspects of this problem have been studied by many researches. Fazio [29] introduced a numerical parameter and required to an extended scaling group involving this parameter. Momentum and energy laminar boundary layers of an incompressible fluid with thermal radiation about a moving plate in a quiescent ambient fluid (Sakiadis flow), the flow induced over a resting flat-plate by a uniform free stream (Blasius flow) and different effects like suction/blowing etc., are investigated numerically by Cortell [30–32]. Olanrewaju et al. [33] investigated the radiation and viscous dissipation effects of Blasius and Sakiadis flows under a convective surface boundary conditions by using shooting technique along side with the sixth order of Runge–Kutta integration scheme and the the variation of some parameters. Mohammed et al. [34] proposed a new application of the successive linearization method for solving Blasius, Sakiadis, Falkner-Skan and MHD problems. Ramesh et al. [35] studied the behavior of Sakiadis and Blasius flow of Walliamson fluid with

29.3 Nonlinear Blasius and Sakiadis Flows

393

convective boundary conditions with help of fourth and fifth order Runge–KuttaFahlberg method. Naveed et al. [36] discussed the heat transfer characteristic for Blasius and Sakiadis flows over a curved surface coiled in a circle of radius R having constant temperature by means of shooting algorithm. The Blasius and Sakiadis problem are treated by Abdella et al. [37] by a new Sinc-collocation approach based on first derivative interpolation. A predictor corrector two-point block method is proposed by Majid and See [38] to solve Blasius and Sakiadis flow numerically without reducing it to a system of first order equation. Nandeppanavar et al. [39] deal with the effect of external magnetic field and other parameters on the flow and heat transfer in the presence of suspended carbon nanotubes over a flat plate. The derived governing Blasius equations of flow and energy are solved numerically using fourth-order Runge Kutta method in combination with shooting technique. Narsu and Kumar [40] investigated the effects of unsteady MHD chemically reacting radiated flow with variable conductivity about a flat plate in a uniform stream of fluid ( Blasius flow) and about a sheet in a quiescent ambient fluid (Sakiadis flow) both under a convective surface boundary conditions. Gopal and Kishan [41] considered the flow and heat transfer features of Sakiadis and Blasius flow of MHD Carreau fluid with thermal radiation with Cattaneo-Christov heat flux model. Numerical solutions are carried out by Runge–Kutta based on shooting technique. The Sakiadis flow of thixotropic fluid under the effect of Lorentz force and Newtonian heating is analyzed by Ghiasi and Saleh [42]. A theoretical study on the effect of magnetohydrodynamic field in the Blasius and Sakiadis flows of heat transfer characteristics with variable conditions and variable properties are studied by Krishna et al. [43]. The transformed boundary layer equations like momentum and energy equations are solved numerically by using Runge–Kutta method. The optimal homotopy asymptotic method is used by Marinca and Herisanu [44–46]or Marinca and Marinca [47] to determine an approximate solution of Blasius problem. The Blasius equation is presented in Sect. 3.18: f  (η) +

1 f (η) f  (η) = 0, η ∈ [0, ∞) 2

(29.83)

with the boundary conditions f (0) = 0,

f  (0) = 0,

f  (∞) = 1

(29.84)

For the Sakiadis case, the governing Eqs. (29.83) and (29.84) has the boundary conditions f (0) = 0,

f  (0) = 1,

f  (∞) = 0

(29.85)

394

29 The Second Alternative to the Optimal Auxiliary Functions Method

29.3.1 Approximate Solutions for the Blasius and Sakiadis Problems Using the Alternative of the OAFM In what follows, we apply our technique to obtain an approximate solution of Eqs. (29.83)–(29.84) and (29.83)–(29.85) respectively.

29.3.1.1

Approximate Solution to the Blasius Problem

From the consideration made in Chaps. 2 and 3, it follows that the linear operator and initial approximation (f0 ) are not unique. In the case of Eqs. (29.83) and (29.84), the initial approximation f0 which verify the boundary conditions (29.84) can be chosen as f 0 (η) = η +

e−kη − 1 k

(29.86)

where k is an unknown positive parameter. Having in view Eq. (29.86), we can define the linear operator as: L[ f (η)] = f  (η) + k f  (η)

(29.87)

Another posibility to choose the initial approximation is  η f 0 (η) = η − k ln 1 + k

(29.88)

The corresponding linear operator for Eq. (29.88) can be 2 f   L[ f (η)] = f  (η) +  k 1 + ηk

(29.89)

or L[f(η)] = f −

2f 2 +   2 η 2 2 2 k 1+ k k 1 + ηk

(29.90)

If we consider only the linear operator L given by Eq. (29.89) we obtain the nonlinear operator as N[f(η)] =

1 2f (η)  f(η)f (η) −  2 k 1 + ηk

such that Eq. (29.83) is obtained from Eqs. (29.89) and (29.91).

(29.91)

29.3 Nonlinear Blasius and Sakiadis Flows

395

Inserting Eq. (29.88) into Eq. (29.91), it gives  η 1 η η −  +  N[f0 (η)] =  3 η  ln 1 + η 2 2 k 2 1+ k 2k 1 + k k 1 + ηk

(29.92)

The boundary conditions for the first approximation are f 1 (0) = f 1 (0) = f 1 (∞) = 0

(29.93)

Taking into account Eqs. (29.88), (29.92) and (29.93) and the observations maded in this Chapter, we have the freedom to choose the first approximation in the form f 1 (η, Ci ) = ln(1 + C1 η2 + C2 η3 + · · · + C p η p+1 )

(29.94)

where c1 , c2 , · · · , cp are unknown parameters. The analytical approximate solution of Eqs. (29.83) and (29.84) is obtained from Eqs. (29.3), (4.36) and (29.94):  η + ln(1 + C1 η2 + C2 η3 + · · · + C p η p+1 ) f (η, ci ) = f 0 (η) + f 1 (η) = η − k ln 1 + k

(29.95)

The optimal convergence control parameters Ci are determined by using the conditions (2.31), where R(η, ci ) = f  (η, Ci ) +

1 f (η, Ci ) f  (η, Ci ), i = 1, 2, · · · , p 2

(29.96)

and f (η, Ci ) is given in Eq. (29.95). In what follows we will shown that the accuracy of the results obtained by an alternative to the OAFM is growing along with increasing the number p of the parameres ci .

29.3.1.2

In the the Case p = 6 We Obtain

k = 4.136440558322; C1 = 0.0451056233184; C2 = 0.0199680280744 C3 = −0.00335917554; C4 = 0.001665797515; C5 = −0.0002536093067; C6 = 0.00001213043235256

(29.97)

The approximate solution of Eqs. (29.83) and (29.84) in this case is written in the form f(η) = η − 4.136440558322 ln(1 + 0.241753745999η)

396

29 The Second Alternative to the Optimal Auxiliary Functions Method

+ ln(1 + 0.0451056233184η2 + 0.0199680280744η3 − 0.00335917554η4 + 0.00166579751η5 − 0.0002536093067η6 + 0.00001283043235256η7 ) (29.98) In Table 29.7, we present a comparison between the approximate solution given by Eq. (29.98) with numerical solution [28] for some values of variable η and the corresponding errors. Lal and Neeraj [48] showed that lim [η − f(η)] = 1.7207876575205038 such that for η ≥ 8, we can write that

η→∞

f (η) = η − 1.7207876575205058

(29.99)

It follows that the domain D = [0, 8] chosen in our procedure for Blasius problem is sufficient of significant. In the Case p = 8 One Can Get k = 2.5001378439, C1 = −0.0341093397, C2 = 0.0544718640 C3 = −0.0176773550, C4 = 0.0048952234, C5 = −0.0008840941, C6 = 8.98134158 · 10−5 , C7 = −4.5347403 · 10−6 , C8 = 8.3659093 · 10−8 (29.100)

Table 29.7 Comparison between the approximate solution given by Eq. (29.98) and numerical results [28] for Blasius problem η

fNumerical (η) [28]

f(η) Eq. (29.98),

Error = |fNumerical (η) − f(η)|

0

0

0

0

0.2

0.0066412

0.006418582

6.38E-05

0.6

0.0597215

0.059749865

2.83E-05

1

0.1655717

0.165584803

1.31E-05

1.4

0.3229815

0.322975577

5.92E-06

2

0.6500243

0.650065389

4-11E-05

2.4

0.9222901

0.922399411

1.01E-04

3

1.3968082

1.396900384

9.21E-05

3.4

1.7469501

1.746963516

1.34E-05

4

2.3057464

2.30585738

1.11E-04

5

3.28329

3.284382105

1.01E-3

6

4.27964

4.279822606

1.22E-04

7

5.27926

5.27547322

3.78E-03

8

6.27923

6.279480302

2.51E-04

29.3 Nonlinear Blasius and Sakiadis Flows Table 29.8 Comparison between the approximate solution given by Eq. (29.101) and numerical results [28] for Blasius problem

397

η

FNumerical (η), [28]

f(η), Eq. (29.101)

Error

0

0

0

0

0.2

0.0066412

0.0066418

6.58E-07

0.6

0.0597215

0.05975216

3.14E-05

1

0.1655717

0.165571192

5.07E-07

1.4

0.3229815

0.32292591

5.56E-05

2

0.6500243

0.649967725

5.65E-05

2.4

0.9222901

0.922443657

1.53E-04

3

1.3968082

1.397106905

2.98E-04

3.4

1.7469501

1.747172681

2.22E-04

4

2.3057464

2.305742988

3.41E-06

5

3.28329

3.283285682

4.32E-6

6

4.27964

4.279634827

5.17E-06

7

5.27926

5.279254032

5.96E-06

8

6.27923

5.279213301

6.05E-06

The approximate solution of Blasius Eqs. (29.83) and (29.84) is f (η) = η − 2.5013784392 ln(1 + 0.399779463356η + ln(1 − 0.03410933979297η2 + 0.05447186056187η3 − 0.017677355024η4 + 0.004895237662η5 − 0.00088409414η6 + 8.98134158273 · 10−5 η7 − 4.53347403075 · 10−6 η8 + 8.365909375 · 10−8 η9

(29.101)

In Table 29.8, we present a comparison between our approximate solution (29.101) and numerical result [28]. In Table 29.9 we present a comparison between our approximate solution (29.101) with published results [49, 50]. From Tables 29.7, 29.8 and 29.9 it can be observed that the analytical methods applied to solve the Blasius equation are very accurate, but OAFM is by far the best method delivering faster convergence and better accuracy. However the accuracy of the obtained results by OAFM is growing along with increasing the number of parameters in the auxiliary functions.

29.3.1.3

Approximate Solutions of the Sakiadis Problem

For the Sakiadis problem we present two variants of approximate solution for Eq. (29.83), taking into consideration the boundary conditions (29.85).

398

29 The Second Alternative to the Optimal Auxiliary Functions Method

Table 29.9 Comparison between the present results given Eq. (29.101), we with numerical results and with the results published in [49, 50] for Blasius problem η

fNumerical (η) [28],

f(η) Eq. (29.101),

f(η) [49],

f(η) [50],

0

0

0

0

0

0.2

0.0066412

0.0066418

0.0066409

0.0069699

0.6

0.0597215

0.05975216

0.0597345

0.0626959

1

0.1655717

0.165571192

0.1655715

0.1738016

1.4

0.3229815

0.32292591

0.3229812

0.3391217

2

0.6500243

0.649967725

0.6500224

0.6828833

2.4

0.9222901

0.922443657

0.9222734

0.9691873

3

1.3968082

1.397106905

1.3964712

1.4674133

3.4

1.7469501

1.747172681

1.7451217

1.8335195

4

2.3057464

2.305742988

2.2897787

2.4153361

5

3.28329

3.283285682

–

–

6

4.27964

4.279634827

–

–

7

5.27926

5.279254032

–

–

8

6.271923

5.274213301

–

–

29.3.1.4

In the First Variant, the Initial Approximation is Chosen in the Form

f 0 (η) = ln 1 +

 η , f 0 (0) = 0, f 0 (0) = 1, f 0 (∞) = 0 kη + 1

(29.102)

where k is a unknown positive parameter. The linear operator becomes L[f(η)] = f + 2



 k k+1 2k(k + 1) − f − (k + 1)η + 1 kη + 1 (k + 1)2 [(k + 1)η + 1]2 (29.103)

and therefore the nonlinear operator can be written as N[f0 (η)] = 2ff − 2



 k k+1 2k(k + 1) − f − (k + 1)η + 1 kη + 1 (k + 1)2 [(k + 1)η + 1]2 (29.104)

Taking into account Eq. (29.102), the nonlinear operator N from the last equation becomes 

  η 2k3 2k2 2(k + 1)2 ln 1 + + − N[f0 (η)] = (kη + 1)2 ((k + 1)η + 1)2 kη + 1 (kη + 1)3

29.3 Nonlinear Blasius and Sakiadis Flows

−

399

2(k + 1)3 4k2 (k + 1)((k + 1)η + 1) − ((k + 1)η + 1)3 (kη + 1)2 ((k + 1)η + 1)2

(29.105)

From Eqs. (29.102) and (29.105), we have many posibilities to chose the first approximation with boundary conditions f 1 (0) = f 1 (0) = f 1 (0) = 0:

C1 η2 + C2 η2.5 + C3 η3 + C4 η3.5 f 1 (η, Ci ) = ln 1 + 1 + C 5 η + C 6 η2 + C 7 η3 + C 8 η4

 (29.106)

or 

C1 η2 + C2 η2.5 + C3 η3 + C4 η3.5 + C5 η4 f 1 (η, Ci ) = ln 1 + 1 + C 6 η2 + C 6 η2 + C 7 η3 + C 8 η4 + C 9 η6

(29.107)

or yet f1 given by Eq. (29.94), and so on. Considering only expression given by Eq. (29.106) the approximate solution of Sakiadis problem is written as

f (η) = ln 1 +

η kη + 1





C1 η2 + C2 η2.5 + C3 η3 + C4 η3.5 (29.108) + ln 1 + 1 + C 5 η + C 6 η2 + C 7 η3 + C 8 η4

In this subcase, the values of the optimal convergence control parameters are obtained by means of collocation method: k = 0.247911925706, C1 = 0.5387254949333086, C2 = −0.067050002658011, C3 = 0.017429200239331 C4 = 0.0061454656373252, C5 = 0.9898708171573922 C6 = 0.027661226022537, C7 = 0.08739434907052391, C8 = 0.0015152551362013

(29.109)

The approximate solution of Sakiadis problem is

f (η) = ln 1 + +

η 0.247911925706η + 1

 + ln(1

0.538725412η2 − 0.057050002η2.5 − 0.0174292002η3 + 0.00614546η3.5 1 + 0.098987081η − 0.027661226η2 + 0.087394349η3 + 0.001515255η4



(29.110) In Table 29.10, we present a comparison between the approximate solution (29.110) obtained by OAFM and numerical results [32].

400

29 The Second Alternative to the Optimal Auxiliary Functions Method

Table 29.10 Comparison between the approximate solution (29.110) and numerical results for Sakiadis problem

η

fNumerical (η), [32]

f(η), Eq. (29.110)

Error

0

0

0

0

0.2

0.1911395

0.191139621

1.22E-07

0.4

0.3647266

0.364716414

1.02E-05

0.6

0.5212411

0.521241810

7.10E-07

0.8

0.6614207

0.661426569

5.07E-06

1

0.7862015

0.786202925

1.43E-06

1.5

1.0380120

1.0379989241

1.31E-05

2

1.2185520

1.21855053

3.05E-06

3

1.4327300

1.432734351

4.35E-06

4

1.5330800

1.533085362

5.36E-06

5

1.5788440

1.578849932

5.93E-06

10

1.6154630

1.615471182

8.13E-06

In the last variant for Sakiadis problem, the initial approximation is of the form f0 (η) =

1 − e−kη k

(29.111)

where k is a positive unknown parameter. The linear operator can be in the following forms: L[f(η)] = f − kf

(29.112)

L[f(η)] = f − k2 f

(29.113)

L[f(η)] = f + 2kf + k2 f

(29.114)

L[f(η)] = f + 3kf + 2k2 f

(29.115)

or

or

or

and so on.

29.3 Nonlinear Blasius and Sakiadis Flows

401

For linear operator defined in Eq. (29.112), the corresponding nonlinear is N[f(η)] =

1  ff − kf 2

(29.116)

Substituting Eq. (29.111) into Eq. (29.116) one can get

 1 −kη 1 −2kη + e N[f0 (η)] = k2 − e 2 2

(29.117)

The boundary conditions for the first approximation f1 (η, ci ) are f 1 (0) = f 1 (0) = = 0 such that it is natural to choose

f 1 (∞)

f 1 (η, ci ) = (C1 η2 + C2 η2.5 + C3 η3 + C4 η3.5 + C5 η4 )e−kη + (C6 η2 + C7 η2.5 + C8 η3 )e−2kη

(29.118)

Also we can choose the first approximations in the forms:   f 1 (η, Ci ) = C1 η2 + C2 η2.5 + C3 η3 + C4 η3.5 e−kη   + C5 η2 + C6 η2.5 + C7 η3 + C8 η3.5 + C9 η4 e−2kη

(29.119)

or     f 1 (η, Ci ) = C1 η2 + C2 η3 e−kη + C3 η2 + C4 η3 + C5 η4 + C6 η5 e−2kη (29.120) and so on. Considering Eqs. (29.111) and (29.118), the approximate solution for Sakiadis in this last case, is defined as  1 − e−kη  + C1 η2 + C2 η2.5 + C3 η3 + C4 η3.5 + C5 η4 e−kη k   + C6 η2 + C7 η2.5 + C8 η3 e−2kη (29.121)

f (η) =

Using the collocation procedure, the values of the optimal convergence-control parameters are k = 0.6027135448, C1 = 1.406438402462772, C2 = −0.3485157110693, C3 = −0.596379801434892, C4 = 0.3272322341874, C5 = −0.048461835386407, C6 = −1.327737233699063, C7 = 0.359388648157453, C8 = −0.266952533728397

(29.122)

402

29 The Second Alternative to the Optimal Auxiliary Functions Method

Table 29.11 Comparison between the approximate solution of Sakadis problem given by Eq. (29.123) and numerical results [32]

η

fNumerical (η), [32]

f(η), Eq. (29.123)

Error

0

0

0

0

0.2

0.1911395

0.191139585

8.5E-08

0.4

0.3647266

0.364725056

1.54E-06

0.6

0.5212411

0.5214573

4.73E-07

0.8

0.6614207

0.661423127

2.43E-06

1

0.7862015

0.786202312

8.12E-07

1.5

1.0380120

1.038005301

6.59E-06

2

1.2185520

1.218552973

9.73E-07

3

1.4327300

1.432730656

6.56E-07

4

1.5330800

1.533080035

3.5E-08

5

1.5788440

1.578844165

1.65E-07

10

1.6154630

1.615463035

3.50E-08

Therefore the second solution of the Sakiadis problem yields: f(η) = 1.6591629782(1 − e−0.6027135448η ) + (1.406438402462772η2 − 0.348157110649304η2.5 − 0.596379801434892η3 + 0.32723223418745η3.5 − 0.048461835386407η4 )e−0.6027135448η + (−1.327737233699063η2 + 0.359388648157453η2.5 − 0.266952533728397η3 )e−1.2054270896η

(29.123)

In Table 29.11, we present a comparison between the approximate solution (29.123) for Sakiadis problem and numerical result. From Tables 29.10 and 29.11 it can be seen that the obtained approximate solution are nearly identical with numerical results. It is easy to verify the accuracy of the obtained solutions if we graphically compase these analytical approximate solutions with the coresponding numerical ones. Figures 29.13 and 29.14 show the comparison between the present solutions gived by Eqs. (29.101) and (29.123) respectively and and the numerical solutions. In Figs. 29.15 and 29.16 have been plotted the residual given by Eq. (29) for the Blasius and Sakiadis, respectively. From Figs. 29.15 and 29.16 we can observe that the approximate solutions obtained through OAFM are highly accurate.

29.3 Nonlinear Blasius and Sakiadis Flows Fig. 29.13 Comparison between the approximate solution (29.101) and numerical solution [28] for Blasius equation. Numerical solution (blue), Approx. solution (29.101)

Fig. 29.14 Comparison between the approximate solution (29.123) and numerical solution [32] for Blasius equation. Numerical solution (blue), Approx. solution (29.123)

Fig. 29.15 Residual given by Eq. (29.84) obtained using the approximate solution (29.101) for Blasius equation

403

404

29 The Second Alternative to the Optimal Auxiliary Functions Method

Fig. 29.16 Residual obtained from Eq. (29.123) for Sakiadis equation

29.4 Poisson–Boltzman (P.B) Equations The P.B equation has been widely used in electrochemistry (Gouy-Chapman theory), in biophysics (P.B theory), in colloid chemistry (Derjaguin-Landau-VerweyOverbeek theory), in solution chemistry (Debye-Hückel theory), molecular dynamics (Brownian dynamics) and so on [51]. P.B equation constitutes a typical implicit solvent model and provides a simplified continuum description of the discrete particle distributions in solution. For instance the phenomenon of counterion condensation in a solution of high charged rigid polyelectrolyte within the cell model. It is very interesting the following assertion of Muthukumar [52]: “We do not known how life began on our planet. But we do known that polyelectrolytes must have existed before life began, since the life as we known it requires replicating charged polymers thas contains information. The description of polyelectrolytes is perhaps the most challenging subject today among all biological and chemical systems in their liquid and solid state.” On the other hand, the process describing how polyelectrolytes chains migrate in external electric fields (electrophoresis) is an important phenomena with many applications. Most applications of P.B equation appear into two areas such as investigations of the chemical physics of ionic or colloidal solutions and calculations of the electrostatic energy or large biomolecules [53]. The forerunner of known solution to what is now called P.B equation appears to be due to Liouville who, while investigating surface of constant curvature showed that the nonlinear differential equation [54] λ d2 log λ ± 2 = 0. dudv 2a

(29.124)

has the solution λ(u, v) =

4a 2 ex p[ϕ(u) + ϕ(v)] dϕ(u) dψ(v) · · [1 ± ex p(ϕ(u) + ϕ(v))]2 du dv

(29.125)

where ϕ(u) and ψ(v) are two arbitrary functions. In particular, for u = v = x, the variable λ(x) represents the local charge density for a systems of ions of charge q,

29.4 Poisson–Boltzman (P.B) Equations

405

which in turn is related to the local electrostatic potential qϕ(x) through Boltzmann’s equation, that is λ(x) = ex p[qϕ(x)K B T ]

(29.126)

The Eq. (29.124) is reduced to the P.B equations as expressed by Eq. (29.125). There exists a vast literature in the domain of the physical fundamentals, methodologies and applications of P.B electrostatics to molecular structures and dynamics. In what follows we only list some of these. Fuoss et al. [55] presented the exact solution of the problem of the potential of an isolated rodlike molecule in the presence of an electrically equivalent number of counterion, with the distribution of density derived from potential. The derivation depends on explicit use of the Boltzmann distribution function and assumed linear superposition of fields in a case where the Poisson equation is nonlinear. Andrietti et al. [56] solved analytically the unidimensional P.B equation for 1:2 (2:1) electrolyte, but the errors are not small. Barrat and Joanny [57] obtained the exact electrostatic potential for the two cases depending of the charge parameter. A Monte Carlo method is proposed by Deserno [58] for determining the equilibrium in distribution functions belonging to free energies of the P.B type. The phenomenon of counterion condensation in a solution of highly charged rigid polyelectrolytes within the cell model is investigated by Deserno et al. [59]. The proposed method is based on the charge distribution function in connection with Manning parameter. The nonlinear P.B equation is solved by Shestakov et al. [60] by means of Newton– Krylov iterations coupled with pseudo-transient continuation and finite element method. The P.B potential is used to compute the electrostatic energy and evaluate the force on a user specified contour. Chapot et al. [61] showeed that the far field electrostatic potential created by a highly charged finite size cylinder within the P.B theory is remarkable close to the potential created within the linearized P.B approximation. Values of the effective surface potential are proved as a functions of the bare surface charge and Debye length of the ionic solution. Vishniakov et al. [62] considered the P.B problem in spherical symmetry using the distribution of a self-consisted potential around a charged grain in a thermal collisional plasma. It has been demonstrated that for large potentials, it is possible to neglect the curvature of the grain surface and the use the solution of the plane problem. Polat and Polat [63] developed an analytical solution of the one-dimensional nonlinear P.B equation for two plates of arbitrary surface potentials interacting in symmetrical electrolyte solution, within condition that potentials of infinite separation have the same sign. Mallarino et al. [64] studied counterion condensation and its ramifications both numerically using Monte Carlo simulating and analytically with special emphasis on the strong-coupling regime. Huang et al. [65] reviewed the development of the electric double layer model, from the dimensionless Gouy-Chapmann model to the symmetric Bikerman-Freise model and finally toward size-asymmetric mean field theory models. It provided the general derivation within the framework of Helmholtz free energy of the lattice-gas model.

406

29 The Second Alternative to the Optimal Auxiliary Functions Method

Samaj and Trizac [66] considered a set of counterions confined to a domain with curved hard walls carrying a uniform fixed surface charge density. The particle system in thermal equilibrium is assumed to be described by P.B theory. It is showed that the contact density of particles at the charged surface is determined by a first order Abel differential equation of the second kind which is a counterpart of Enig’s equation in the critical theory of gravitation and combustion or explosion. Khan et al. [67] deals with a theoretical study of the dynamics of an electroosmatic flow in cylindrical domain. After solving linearized P.B. equation, the Cauchy momentum equation with electrostatic body force is solved analytically by means of the temporal Fourier and finite Hankel transforms. The pseudo-spectral method with Chebyshev and Legendre polynomials are used by Nikzad et al. [68] in order to compute the electric solution via the P.B equation. Reindl et al. [69] analyzed electrolytes in contact with planar, cylindrical and spherical electrodes. The surface charge density has a strong effect on the capacitance for small curvatures, whereas for large curvatures the behavior becomes independent of charge density. The universal behavior at large curvatures is captured in an analytic expression.

29.4.1 P.B Equation for a Charged Rod in Absence of Added Salt Fuoss et al. [55] considered the interaction between polyelectrolyte chains and their counterion clouds by means of the P.B equation. They assumed that in an infinite rodlike molecules, all are parallel. In this chapter we consider an infinitely long cylinder (Fig. 29.17) of radius a which is coaxially enclosed in a cylindrical cell of radius R and line charged density λ > 0. Adding an appropriate amount of oppositely charged counterions imply global charge neutrality of the system. The local electrostatic potential (r), where r in the radial coordinate, is defined through Poisson equation:

(r) = −

ρ(r) ε

(29.127)

where is the Laplace operator in cylindrical coordinates, ρ(r) is the charge density distribution and ε is the dielectric constant outside the cylinder. The charge density distribution can be written by the Boltzmann equation ρ(r) = en(r)

(29.128)

where e is the positive unit charge and n(r) is the cylindrical counterion density, which is related to the electrostatic potential (r) by the Boltzmann factor: n(r) = n(R) exp[βe(r)]

(29.129)

29.4 Poisson–Boltzman (P.B) Equations

407

Fig. 29.17 Geometry of the cell model

where n(R) is the average counterin concentration, β = (KB T)−1 is the inverse temperature and KB is the Boltzmann constant. The strength of the electrostatic interactions is conveniently expressed by Bjerum length lB = βe2 /4πε

(29.130)

The change parameter also known as the dimensionless Manning parameter which is a measure of the charges on the surfaces of counterions is defined as: ξ = λlB /e

(29.131)

The screening potential k > 0 and reduced electrostatic potential y(r) are defined as:

408

29 The Second Alternative to the Optimal Auxiliary Functions Method

k2 = 8π · lB n(R)

(29.132)

y(r) = −βe(r)

(29.133)

Taking into account Eqs. (29.127), (29.128), (29.129), (29.132) and (29.133), the P.B equation can be written in cylindrical coordinates as y (r) +

k2 y (r) = exp(y(r)), r ∈ [a, R] r 2

(29.134)

where prime denotes derivation with respect to variable r. Equation (29.134) is a nonlinear differential equation with variable coeficient and of the second order. The solution of this equation is obtained using the Gauss law at a and R: y (a) = −

2ξ , y (R) = 0 a

(29.135)

The nonlinear differential P.B Eq. (29.134) with the boundary conditions (29.135) will be investigated in what follows using an analytic approximate solution obtained by OAFM.

29.4.2 OAFM for P.B given by Eqs. (29.134) and (29.135) In the following we apply our technique to obtain an analytic approximate solutions for Eqs. (29.134) and (29.135). Apparently from Eq. (29.134) the linear operator, the function g and the nonlinear operator are respectively. L[y(r)] = y +

k2 y , g(r) = 0, N[y(r)] = − · exp(y(r)) r 2

(29.136)

The linear Eq. (29.4) becomes y0 +

y0 2ξ = 0, y0 (a) = − , y0 (R) = 0 r a

(29.137)

But this last equation lead to the solution y0 (r) = A + B ln r with y0 (r) = the boundary conditions are not fulfilled.

B r

but

29.4 Poisson–Boltzman (P.B) Equations

409

In this case we choose the function g(r) = 0 so that g(r ) = −

C3 C1 − C2 − 3 r r

(29.138)

The function g(r) is not unique. Also we can choose the following expression for g(r) C4 C1 − C2 − C3r − 4 r r

(29.139)

C1 − C2r − C3r 3 − C4r 3 r

(29.140)

g(r ) = − or g(r ) = − or yet

g(r ) = −C1 − C2 r − C3r 2 −

C4 C5 − 4 r3 r

(29.141)

and so. Using only the expression (29.138) Eq. (4.4.) becomes: y0 +

y0 c3 2ξ c1 − − c2 − 3 = 0, y0 (a) = − , y0 (R) = 0 r r r a

(29.142)

where C1 , C2 and c3 are unknown at this moment. The solution of Eq. (29.142) is   2aξ 1 2 2 − (a + R)C1 − a + a R + R C2 ln r y0 (r ) = R−a 2   1 2a Rξ 1 − a RC1 − a R(a + R)C2 + R−a 2 r 1 + C1r + C2r 2 + C3 2 

(29.143)

Taking into consideration the expression (29.138) for g(r) the nonlinear operator N is given by

410

29 The Second Alternative to the Optimal Auxiliary Functions Method

1 C1 C3 + C2 + 3 N [y(r )] = − k 2 ex p[y(r )] + 2 r r

(29.144)

1 C1 C3 N [y0 (r )] = − k 2 ex p[y0 (r )] + + C2 + 3 2 r r

(29.145)

and therefore

Having in view that exp[y0 (r)] = 1 + y0 +

y20 y3 + 0 + ··· 2! 3!

(29.146)

the preceding expresion (29.145) can be regarded as 

α 1 N[y0 (r)] = N ln r, r , β , α ≥ 0, β ≥ 1 r

(29.147)

  where N ln r, rα , r1β means that the nonlinear operator depends on ln r, r α and Now, for the first approximation y1 (r, cj ) the boundary conditions are y1 (a) = y1 (R) = 0

1 . rβ

(29.148)

such that the approximation y1 can be written in the form 

α 1 y1 (r, c j ) = [(r − a)(r − R)] F ln r, r , β , c j , p ≥ 2, β ≥ 4 r p

(29.149)

where F is an arbitrary function which depends on ln r, rα and r1β and is not unique. It is clear that for the expression (29.148) the conditions (29.149) are hold. For the functions F, we can choose the expressions: 

α 1 F ln r, r , β = C4 + C5r + C6r 2 + C7r 3 + C8r 4 r C11 C10 C12 C13 + 2 + 3 + 4 + C9r 5 + r r r r

(29.150)

or 

α 1 F ln r, r , β = C4 + C5ln r + C6r + C7r 2 r C9 C8 C10 C11 C12 C13 C14 + 2 + 3 + 4 + 5 + 6 + 7 + r r r r r r r

(29.151)

29.4 Poisson–Boltzman (P.B) Equations

411

or yet   F ln r, r α , r1β = C4 + C5ln r + C6ln r + C7r + C8r 2 + C9r 3 +C10 r 4 + Cr11 + Cr122 + Cr133 + Cr144 + Cr155

(29.152)

and so on. The first options is expression (29.150). For p = 2 into Eq. (29.149) the first approximation can be rewriten in the form  y1 (r, Ci ) = [r 2 − (a + R)r + a R]2 C4 + C5r + C6r 2 + C7r 3  C11 C10 C12 C13 + 2 + 3 + 4 + C8r 4 + C9r 5 + r r r r

(29.153)

In this way the approximate solution y(r, ci ) is obtained from Eqs. (29.3), (29.143) and (29.153) and depends of the optimal convergence-control parameters ci .

29.4.3 Numerical Examples According to Refs. [55, 57, 59, 68] the consider a = 0.5 [nm], r = 6[nm] and the screening potential k is obtained from the relation [55, 57]:k2 R2 = 4ξ. Also, we consider two cases for charge parameter (Manning parameter):low charge parameter for ξ < 1 and high parameter for ξ > 1.

29.4.3.1

Case 1. √

For ξ = 0.5 the Screening Potential is k = 602 . Using the Collocation Method, the Optimal Convergence-Control Parameters Are c1 = 0.267777310637375; c2 = −3.920347837373 · 10−3 ; c3 = 1.434005189746775; c4 = 3.853620593336786; c5 = −1.4074687443415; c6 = 0.3322080296340778; c7 = −0.0489741490933696; c8 = 4.10019063449295115 · 10−3 ; c9 = −1.48844382550093 · 10−4 ; c10 = −6.801409561893227; c11 = 7.44724946355476; c12 = −4.567748669828187; c13 = 1.192216549637498

412

29 The Second Alternative to the Optimal Auxiliary Functions Method

The approximate solution for Eqs. (29.134) and (29.135) can be written as y(r) = −1.3418961239422 ln r − 0.104983820826511

1 r

+ 0.267777310637375r − 3.920347837373 · 10−3 r2 + 1.434005189746775 + (r2 − 6.5r + 3)2 (3.853620593336786 − 1.4074687443415r + 0.33220808963340778r2 − 0.0489741490933696r3 + 4.10019063449295115 · 10−3 r4 6.801409561893227 − 1.48844382580093 · 10−4 r5 − r  7.44724916355436 4.567748669821187 1.192216549637498 − + + r r3 r4 (29.154)

29.4.3.2

Case 2.

For ξ = 1.25, the Screening Potential is k = This Case is

√

15 12

and the Approximate Solution in

y(r) = −1.3418961239422 ln r − 0.104983820826511

1 r

+ 0.267777310637375r − 3.920347837373 · 10−3 r2 + 1.434005189746775 + (r2 − 6.5r + 3)2 (3.853620593336786 − 1.4074687443415r + 0.33220808963340778r2 − 0.0489741490933696r3 + 4.10019063449295115 · 10−3 r4 6.801409561893227 − 1.48844382580093 · 10−4 r5 − r  7.44724916355436 4.567748669821187 1.192216549637498 − + + r r3 r4 (29.155) It is easy to verify the accuracy of the obtained solutions if we compare these analytical approximate results with the corresponding exact solutions. Within Tables 29.12 and 29.13 we present a such of comparisons. It can be seen that the analytical solution given by Eqs. (29.154) and (29.155) by OAFM are extremely accurate.

29.4 Poisson–Boltzman (P.B) Equations

413

Table 29.12 Comparison between the approximate solution (29.154) of Eqs. (29.134) and (29.135) for a = 0.5 [nm], R = 6 [nm], ξ = 0.5 and k = (p)

√

2 60

    Error = y(r) exact − y(r)

r

yexact (r), [55, 57]

y(r), Eq. (29.154)

0.5

2.287077631511

2.2870776300253

1.0E-09

1

1.6156587138886

1.615658691063

2.2E-08

1.5

1.25190994888564

1.2519099256444

2.3E-08

2

1.0149805623112

1.014980468874

9.4E-08

2.5

0.84839168536812

0.84839238306324

6.9E-07

3

0.72711153759488

0.7271115696892531

3.2E-08

3.5

0.63788843937892

0.63788810251954

3.4E-07

4

0.57290370512582

0.5729038342524

3.4E-07

4.5

0.5270229321816

0.52702314027581

2.1E-07

5

0.49748953124662

0.4974895724615

4.1E-08

5.5

0.48150705334562

0.48150705214621

4.2E-09

6

0.47768414311282

0.47768414189581

1.3E-09

Table 29.13 Comparison between the approximate solution (29.155) of Eqs. (29.134) and (29.135) for a = 0.5 [nm], R = 6 [nm], ξ = 1.25 and k =

√

2 12

r

yexact (r), [55, 57]

y(r), Eq. (29.155)

  Error = yexact (r) − y(r)

0.5

3.835858989195

3.8358589980393

1.0E-08

1

2.2806785769662

2.2806785759738

1.0E-09

1.5

1.52554068661292

1.525540685943123

1.0E-09

2

1.06120226889234

1.061202267170668

1.1E-09

2.5

0.74656054133664

0.7465605391681005

4.5E-09

3

0.52317062099844

0.5231706207691416

2.3E-10

3.5

0.36154109738874

0.3615410981167

8.7E-10

4

0.24491266101634

0.2449126608785

2.3E-10

4.5

0.162993334260972

0.16299333367548

6.1E-10

5

0.109131410535882

0.1091314104825023

5.3E-10

5.5

0.7889484450236

0.078894853940013

9.4E-09

6

0.069300671266342

0.069300680566014

9.3E-09

414

29 The Second Alternative to the Optimal Auxiliary Functions Method

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Chapter 30

Piecewise Optimal Auxiliary Functions Method

In this Chapter, the Piecewise Optimal Auxiliary Functions Method (POAFM) is introduced in order to obtain a globally smooth approximate solution to a nonlinear dynamical system. In Chaps. 3–29, we applied OAFM to find an approximate solution only in a certain domain . In order to obtain an approximate solution on a domain  ⊃ , we modify the OAFM in the following manner: divide the new domain  in some subdomains.  = 1 ∪ 2 ∪ . . . . ∪ n where 1 = [0,a1 ], 2 = [a1 ,a2 ],…. n = [an−1 ,an ], and n is a fixed value. On every subdomain i ,i = 1, n is applied the standard OAFM. The values of the end points a1 , a2 ,…, an can be determined so that. (a) (b)

On every subdomain i , i = 1, n, the error between our approximate solution obtained by OAFM and numerical solution is acceptable The conditions of the continuity and derivability of the approximate solutions at the common point aj , j = 1, n − 1, of two adjacent subintervals j , and j+1 are fulfilled.

30.1 The Lane-Emden Equation of the Second Kind The Lane-Emden equation written in terms of the classical variables: 2 dy d2 y = e−y + dx2 x dx

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_30

(30.1)

417

418

30 Piecewise Optimal Auxiliary Functions Method

is based on the works [1, 2]. Bonnor [28] used a modified form of Boyle’s law for a spherical mass of gas at uniform temperature in equilibrium under its own gravitation and an external pressure. The Eq. (30.1) is known as Bonnor-Ebert [3] gas sphere-isothermal gas sphere embedded in a pressurized medium at the maximum possible mass allowing for hydrostatic equilibrium [4]. Also, Eq. (30.1) is often referred to as the Lane-Emden equation of the second kind [5]. The Lane-Emden equation was used the first to describe the internal structure of a self-gravitating polytropic body. If the temperature effects are also incorporated, like in the study of isothermal gas sphere, the astrophysical phenomenon is described by a Lane-Emden equation. The isothermal self-gravitating sphere is defined as the asymptotic limit of the Lane-Emden equation where polytropic index is taken to be very large. From the Lane-Emden equation follows that as the polytropic index is increased, the radius at which the surface condition is met also increases [6]. The thermodynamics of isothermal, self-gravitating fluid sphere plays an important role in some astrophysical problems in the context of star formation, as they provide a basic model of the dense clumps from which stars form, galactic dynamics, stellar structure or the physical behavior of the emission of electricity from hot bodies. The isothermal selfgravitating sphere is relevant in two main contexts [7]: the equilibria of a molecular cloud cores in star formation regions and the equilibria of N-bodies systems. There is no exact solution to the problem of the isothermal sphere and as a consequence, many papers have been devoted to the derivation of analytic approximations forms that agree with numerical results to different degrees of accuracy. In the framework of the non-relativistic Newtonian mechanics, isothermal spheres have been much investigated in [8] by means of Taylor series expansions around the origin, radius of convergence is limited to x = 3.27. In [9] is presented seven digit numerical solutions of the Lane-Emden equation for polytropes and the isothermal sphere with a very high precision (contained between ε = 10–12 and ε = 10–15 ). In [10] is found a very good approximation for finite polytropic index and for the isothermal case at a level of 1, we consider: L(y) = y   + 2x y  2 p−2 N [y(x)] = −e−y(x) + 2αpx + 1+αx 2 p g(x) =

2 p−2 − 2αpx 1+αx 2 p

−

4αp2 x 2 p−2 (1+αx 2 p )2

(30.24)

4αp2 x 2 p−2 (1+αx 2 p )2

Remark 1 The function g(x) was obtained from Eq. (2.5) after the choice of the initial approximation y0 (x) and the nonlinear operator N [y(x)] was obtained from Eqs. (30.7) and (2.1) after the determination of g(x). In this subcase, the auxiliary functions, from the above consideration are: Fi (x) =

4αi pi2 x 2 pi −2 2αi pi x 2 pi −2 +  2 , i = 1, q, αi > 0, pi > 1 1 + αi x 2 pi 1 + αi x 2 pi

The Eq. (2.13) can be written as: y1 +

2  y − C1 F1 (x) − C2 F2 (x) − . . . − Cq Fq (x) = 0, x ∈ 2 = [1, a2 ] x 1 (30.25)

In this moment it’s worth emphasizing that the initial conditions for Eq. (30.25) are obtained from Eq. (30.22) into end point x = 1: y1 (1) = y 2 (1),

y1 (1) = y 2 (1)

(30.26)

Solving Eq. (30.25), yields the solution:     y1 (x) = y 2 (1) + C1 ln 1 + α1 x 2 p1 + C2 ln 1 + α2 x 2 p2 + . . .   + Cq ln 1 + αq x 2 pq

(30.27)

We remark that the conditions from (30.26) is not used in the expression of y1 (x) given by Eq. (30.27). The approximate solution for this subcase is given by Eq. (2.3). But considering Eq. (30.27), for greater efficiency, we propose the following solution on the domain 2 = [1, a2 ], where a2 is at this moment unknown, for the values 1 1 , α2 = 13 , α3 = 3.5 , p1 = 1.1, p2 = 1.2, p3 = 1.3 with the substitution of α1 = 2.5 2 2 x with x -1. y 2 (x) = 0.1588275667 +

A2 B2

(30.28)

30.1 The Lane-Emden Equation of the Second Kind

425

where:  1.1   2 x −1 x2 + 1 A2 = 0.302900438ln + C1 ln 1 + 2 2.5   1.2  1.3   2  2 x −1 x −1 + C2 ln 1 + + C3 ln 1 + 3 3.5  1.1   2 x −1 B2 = 1 + C4 ln 1 + 2.5   1.2  1.3   2  2 x −1 x −1 + C5 ln 1 + + C6 ln 1 + 3 3.5 Let us emphasize that the expression (30.28) verifies the initial conditions into x = 1: y 2 (1) = y 1 (1) = 0.1588275667 y 2 (1) = y 1 (1) = 0.302900438 These values are obtained from Eq. (30.22) and C1 , C2 , . . . , C6 are unknown parameters at this moment.

30.1.2.4

Subcase B 

In this subcase we consider y0 (x) = αx 2 p coshkx and the operator L(y) = y  + 2x y  . As in the previous subcase we obtain:

g(x) = −α 2 p(2 p − 1)x 2 p−2 coshkx + 2k(2 p − 1)x 2 p−1 sinhkx − αk 2 x 2 p coshkx N (y) = −e−y + 2αp(2 p − 1)x 2 p−2 coshkx + 2αk(2 p − 1)x 2 p−1 sinhkx + αk 2 x 2 p coshkx

(30.29)

so that the auxiliary functions Fi are given by: Fi (x) = 2αi pi (2 pi − 1)x 2 pi −2 coshki x + 2αi ki (2 pi − 1)x 2 pi −1 sinhki x + αi ki2 x 2 pi coshki x From Eq. (2.13), the first approximation y1 (x) will be:

(30.30)

426

30 Piecewise Optimal Auxiliary Functions Method

y1 (x) = y 1 (1) + C1 α1 x 2 p1 coshk1 x + C2 α2 x 2 p2 coshk2 x + . . . On 2 = [1, a2 ] we propose another approximate solution defined as: become: y 2 (x) = 0.1588275667 +

A3 B3

(30.31)

where:  2  1.1 x −1 x 2 − 1 coshx x A3 = 0.302900438 + C1 cosh 2cosh1 2.5 2.5  2  2 1.2 1.3 x −1 x − 1 x x cosh + C3 cosh + C2 3 3 3.5 3.5  2 1.1 x −1 x B3 = 1 + C4 cosh 2.5 2.5  2  2 1.2 1.3 x −1 x −1 x x + C5 cosh + C6 cosh 3 3 3.5 3.5 

30.1.2.5

Subcase C

In the last subcase, we consider y0 (x) = αx 2 p+1 sinhkx and the operator L(y) =  y  + x2 y  . It follows that: g(x) = −αk 2 x 2 p+1 sinhkx − 2αk(4 p + 3)x 2 p coshkx − 2αp(2 p + 1)x 2 p−1 sinhkx N (y) = −e−y + αk 2 x 2 p+1 sinhkx + 2αk(4 p + 3)x 2 p coshkx + 2αp(2 p + 1)x 2 p−1 sinhkx

(30.32)

Fi (x) = αi ki2 x 2 pi +1 sinhki x + 2αi ki (4 pi + 3)x 2 pi coshki x + 2αi pi (2 pi + 1)x 2 pi −1 sinhki x y1 (x) = y 1 (1) + C1 α1 x 2 p1 +1 sinhk1 x + C2 α2 x 2 p2 +1 coshk2 x + . . . and the corresponding approximate solution of the form:

(30.33)

30.1 The Lane-Emden Equation of the Second Kind

427

y 2 (x) = 0.1588275667 +

A4 B4

(30.34)

where:  2  1.1 x −1 x x 3 − x sinhx x + C1 sinh A4 = 0.302900438 sinh1 2.5 2.5  2  2 1.2 1.3 x −1 x x −1 x x x sinh + C3 sinh + C2 3 3 3.5 3.5  2 1.1 x −1 x x sinh B4 = 1 + C4 2.5 2.5  2  2 1.2 1.3 x −1 x x −1 x x x sinh + C6 sinh + C5 3 3 3.5 3.5 

Remark 2 All the approximate solutions given by Eqs. (30.28), (30.31) and (30.34) satisfy the initial conditions in x = 1 and y 2 (−x) = y 2 (x). The parameters Ci , i = 1, 6 are determined by means of collocation on domain 2 = [1, a2 ], and a2 is the end point which is determined from the condition that the error is acceptable. In the same manner, we can determine approximate solutions y 3 , y 4 ,… for 3 = [a2 , a3 ], 4 = [a3 , a4 ], . . . in appropriate conditions. In the following, we will use only the subcase 30.1.1.2.a. If we consider that the error is within ε = 10−4 % from the approximate solution, then a2 = 8. If we consider that the error is within ε = 10−3 % from the approximate solution then a2 = 9. Considering acceptable the case a2 = 8, we obtain: C1 = 1.35470347825534,

C2 = −1.42339449090575

C3 = −0.949812354689086, C5 = 40.28686966192230,

C4 = −29.98041737108601 C6 = −11.82731274266151

The approximate solution on domain 2 = [1, 8] can be expressed by: become: y 2 (x) = 0.1588275667 +

A5 B5

where: A5 = 0.302900438 · ln

x2 + 1 2 

1.1  x2 − 1 + 1.35470347825534 · ln 1 + 2.5 

(30.35)

428

30 Piecewise Optimal Auxiliary Functions Method



1.2  x2 − 1 − 1.42339449090575 · ln 1 + 3  1.3   2 x −1 − 0.949812354689086 · ln 1 + 3.5  1.1   2 x −1 B5 = 1 − 29.98041737108601 · ln 1 + 3.5  1.2   2 x −1 − 40.28686966192230 · ln 1 + 3.5  1.3   2 x −1 − 11.827311274266151 · ln 1 + 3.5 

In Table 30.2 is presented a comparison between the approximate solution (30.35), the approximate solution given in [17], the approximate Padé solution given by Eq. (30.23) and numerical solutions [9]. From Table 30.2 we can conclude that the proposed method POAFM is very efficient and highly accurate as compared to other variants. The approximate Padé solution is efficient only in a small domain: x ∈ [1, 4]. Now, taking into consideration a new region 3 = [8, a3 ], and values of y 2 (8) and y 2 (8) obtained from Eq. (30.35), we search the approximate solution in the form: y 3 (x) = 3.175411001 +

A6 B6

(30.36)

where:  1.1   2   2 x − 82 x − 3 + C7 · ln 1 + A6 = 0.316966777 · ln 16 25     1.3   2  1.2 x − 82 x 2 − 82 + C8 · ln 1 + + C9 · ln 1 + 30 35    1.1 1.2   2  2 x − 82 x − 82 B6 = 1 + C10 · ln 1 + + C11 · ln 1 + 25 30     2 1.3 x − 82 + C12 · ln 1 + 35 

In expression (30.36) we considered that y 2 (8) = 3.175411001 and y 2 (8) = 0.316966777. The first function which appears at the numerator of the Eq. (30.36),

y(x) Numerical [9]

0.1588277

0.559823

1.063335

1.572233

2.044092

2.46721

2.842589

3.175397

x

1

2

3

4

5

6

7

8

3.175411001

2.842579980

2.467221464

2.044097091

1.572232980

1.063336999

0.559822999

0.158827566

y2 (x) Equation (30.35)

4.4E-04

3.2E-04

4.6E-04

2.5E-04

1.3E-06

1.9E-04

1.8E-06

8.4E-05

Error %

3.175508142

2.842720787

2.467266432

2.044066001

1.572182071

1.063309902

0.559819501

0.158827600

y(x) [17]

3.5E-03

4.6E-03

2.3E-03

1.3E-03

3.2E-03

2.3E-03

6.2E-04

6.2E-05

Error %

3.0344567975

2.7566184406

2.4223789749

2.0256639083

1.5670031275

1.0625901373

0.5597961881

0.1588275576

y(x) Equation (30.23)n

4.4

3.0

1.8

3.7

0.3

0.07

4.8E-03

8.9E-05

Error %

Table 30.2 Comparison between the numerical solution [9], the approximate POAFM solution given by Eq. (30.35), the approximate solution given by Eq. (24) from[17], and the approximate Padé solution given Eq. (30.23)

30.1 The Lane-Emden Equation of the Second Kind 429

430

30 Piecewise Optimal Auxiliary Functions Method

 2 x h(x) = 0.316966777ln 16 − 3 , verifies the conditions h(8) = 0 and h  (8) = 0.316966777 so that y 3 verifies the initial conditions in x = 8. The coefficients αi 1 1 1 which appear in Eq. (30.27) and then in Eq. (30.36) are: α1 = 25 , α2 = 30 , α3 = 35 . −3 Searching for an error of order 10 % on the domain 3 = [8, a3 ] we find that a3 = 80 and the approximate solution is: y 3 (x) = 3.175411001 +

A7 B7

(30.37)

where:  A7 = 0.316966777 · ln

 x2 −3 16 

1.1  x 2 − 82 − 0.196586347064206 · ln 1 + 25  1.2   2 x − 82 − 0.609263280864408 · ln 1 + 30  1.3   2 x − 82 + 0.873739357799364 · ln 1 + 35  1.1   2 x − 82 B7 = 1 + 12.2551328628728 · ln 1 + 25  1.2   2 x − 82 − 18.758526636962 · ln 1 + 30  1.3   2 x − 82 + 6.94091694513224 · ln 1 + 35 

In Table 30.3 we compare results obtained form Eq. (30.37) with analytical approximations [12] and with numerical solutions [9]. It is easy to observe from Table 30.3 that the approximation y 3 (x) given by Eq. (30.37) is our best result. It is reasonable to select y 4 (x) on 4 = [80, a4 ] and to work with an error of order ε = 10−3 %. Let us continue in the same manner with: y 4 (x) = 8.184501222 +

A8 B8 

 2 1.1    2 x − 802 x A8 = 0.023087149 · ln − 39 + C13 · ln 1 + 160 2500

(30.38)

30.1 The Lane-Emden Equation of the Second Kind

431

Table 30.3 Results of the approximate solutions given by Eq. (30.37) and [12] in comparison with the numerical results of [9] x

y(x) Numerical [9]

y 3 (x) Equation (30.7)

Error %

y(x) [12]

Error %

8

3.175397

3.175411001

4.4E-04

3.17541912977

0.03

13

4.387994

4.388102299

2.5E-03

4.38796421305

6.8E-04

20

5.403887

5.403902410

2.8E-04

5.40376757901

0.07

30

6.284061

6.284112301

8.2E-04

6.28403901872

3.4E-04

40

6.867771

6.867586588

2.7E-03

6.86788458114

1.6E-03

50

7.302273

7.302243200

4.1E-04

7.30273267599

6.3E-03

60

7.648490

7.648731716

3.2E-03

7.64981689800

0.017

70

7.936828

7.937158173

4.2E-03

7.93978285251

0.037

80

8.184440

8.184501222

7.5E-04

8.18980757136

0.065



 1.2  1.3   2 x 2 − 802 x − 802 + C14 · ln 1 + + C15 · ln 1 + 3000 3500   1.1  1.2   2  2 x − 802 x − 802 B8 = 1 + C16 · ln 1 + + C17 · ln 1 + 2500 3000     2 1.3 x − 802 + C18 · ln 1 + 3500 

where y 3 (8) = 8.184501222, y 3 (8) = 0.023087149, α1 = 2500−1 , α2 = 3000−1 , α3 = 3500−1 and the function y 4 (x) satisfies the initial conditions for x = 80. This means that: y 4 (x) = 8.184501222 +  A9 = 0.023087149 · ln

 x2 − 39 160  

A9 B9

1.1  x 2 − 802 + 1.76626734099279 · ln 1 + 2500  1.2   2 x − 802 − 2.57054103026032 · ln 1 + 3000  1.3   2 x − 802 + 0.952755993415469 · ln 1 + 3500

(30.39)

432

30 Piecewise Optimal Auxiliary Functions Method

Table 30.4 Comparisons between numerical solution [9], approximate POAFM solution given by Eq. (30.39), with the approximate solution given in [12] x

y(x) Numerical[9]

y 4 (x) Equation (30.39)

Error %

y(x) [12]

Error %

80

8.184440

8.184501221

7.9E-04

8.18980757136

0.06

90

8.401877

8.401790313

9E-04

8.4103137221362

0.10

300

10.66923

10.669198700

2.9E-04

10.728700817

0.55

500

11.68443

11.684447961

1E-04

11.741382275611

0.48

700

12.36685

12.366792082

4.7E-04

12.4117491978

0.36

800

12.63944

12.639414351

2E-04

12.678175428231

0.31

1000

13.09606

13.095417792

4.9E-03

13.123710242725

0.21

1200

13.46941

13.469567123

1.2E-03

13.48794301977

0.14

1400

13.78479

13.785014899

1.6E-03

–

–

1600

14.05756

14.057787781

1.6E-03

–

–

1800

14.29770

14.297872162

1.2E-03

–

–

2000

14.51209

14.512134481

3.0E-04

–

–

2200

14.70565

14.705495502

1.0E-03

–

–

2500

14.96423

14.964074412

1.1E-03

–

–





1.1  x 2 − 802 B9 = 1 − 3.88521563997006 · ln 1 + 2500  1.2   2 x − 802 + 9.4412042059993 · ln 1 + 3000   2 1.3  x − 802 − 5.42657852707826 · ln 1 + 3500 In Table 30.4, the numerical solution is compared with the approximate solution obtained from Eq. (30.39) and with approximate solutions obtained by [12]. From Table 30.4 it results that there exists an excellent agreement between the numerical results and the results obtained by means of POAFM. Our procedure converges significantly faster than that presented in [12]. In the same manner, we can obtain the approximate solutions on domains 5 = [2500, a5 ], 6 = [a5 , a6 ], and so on.

References 1. J.H. Lane, On theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining heat and depending of the laws of gases known to terrestrial expanded. Amer. J. Sci. and Arts 50, 5774 (1870)

References

433

2. R. Emden, Gaskugeln-Anwendungen Der Mechanischen Warmtheorie (Druck und Verlag Von B.G. Teubner, Leipzig, Berlin, 1907) 3. R. Ebert, Uber die Verdichtung von H. I. Gebicten, Zeitschrift fuer Astrophysik 37, 217–232 (1955) 4. R.A. Gorder, Analytical solutions to a quasilinear differential equation related to the LaneEmden equation of the second kind. Celest. Mech. Dyn. Astr. 109, 137–145 (2011) 5. J. Binney, S. Tremaine, Galactic Dynamics (Princeton University Press, Princeton, 1987) 6. B.M. Mirza, Approximate analytical solutions of the Lane-Emden equation for a selfgravitating isothermal gas sphere. MNRAS 395, 2288–2291 (2009) 7. A.C. Raga, J.C. Rodriguez-Ramirez, M. Villasante, A. Rodriguez-Gonzalez, V. Lora, A new analytic approximation to the isothermal, self-gravitating sphere. Rev. Mexicana de Astronomia y Astrofisica 49, 63–69 (2013) 8. M.A. Lampert, R.A. Martinelli, Solution of the nonlinear Poisson-Boltman equation in the interior of charged, spherical and cylindrical vesicles. I, The high-charge limit. Chem. Phys. 88, 399–413 (1984) 9. G.P. Horedt, Seven-digit tables of Lane-Emden functions. Astrophys. Space Sci. 126, 357–408 (1986) 10. F.K. Liu, Polytropic gas sphere: an approximate analytic solution of the Lane-Emden equation. MNRAS 281, 1197–1205 (1996) 11. P. Natarajan, D. Lynden-Bell, An analytic approximation to the isothermal sphere. MNRAS 286, 268–270 (1997) 12. C. Hunter, Series solutions for polytropes and the isothermal sphere. MNRAS 328, 839–847 (2001) 13. A.M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type. Appl. Math. Comput. 118, 287–310 (2001) 14. P.H. Chavanis, Gravitational instability of finite isothermal sphere. Astron. Astrophys. 381, 340–356 (2002) 15. R.A. Van Gorder, Exact first integrals for a Lane-Emden equation of the second order modeling a thermal explosion in a rectangular slab. New Astron. 16, 492–497 (2011) 16. C.M. Bender, K.A. Milton, L.M. Simmons Jr., A new perturbation approach to nonlinear problems J. Math. Phys. 30, 1447–1455 (1989) 17. R. Iacono, M. De. Felice, Approximate analytic solutions to isothermal Lane-Emden equation. Celest. Mech. Dyn Astr. 118, 291–298 (1999) 18. A.S. Saad, M.I. Nouh, A.A. Shaker, T.M. Kamel, Approximate analytical solutions to the relativistic isothermal gas sphere. General Relativity and Quantum Cosmology, arXiv: 1704.08947 [gr-qc] (2017) 19. J.H. He, Variational approach to the Lane-Emden equation. Appl. Math. Comput 143, 539–541 (2003) 20. M.I. Nouh, Accelerated power series solution of polytropic and isothermal gas sphere. New Astron. 9(6), 467–473 (2004) 21. C. Harley, E. Momoniat, Steady state solutions for a thermal explosion in a cylindrical vessel. Mod. Phys. Lett. B 21, 831–841 (2007) 22. R.A. Gorder, An elegant perturbation solution for the Lane-Emden equation of the second kind. New Astron. 16, 65–67 (2011) 23. K. Boubaker, R.A. Van Gorder, Application of the BPES to Lane-Emden equations governing polytropic and isothermal gas spheres. New Astron. 17(6), 565–569 (2012) 24. V.K. Baranwal, R.K. Pandey, M.P. Tripathi, O.P. Singh, An analytic algorithm of Lane-Emden type equations arising in astrophysics. JJ. Theor. Appl. Phys. 6, 22–31 (2012) 25. K. Reger, R.A. Van Gorder, Lane-Emden equations of second kind modelling thermal explosion in infinite cylinder and sphere. Appl. Mathe. Mech. (English Edition) 34(12), 1439–1452 (2013) 26. E.H. Doha, A.H. Bhrawy, R.M. Hafez, R.A. Van Gorder, A Jacobi rational pseudospectral method for Lane-Emden initial value problems arising in astrophysics on a semi-infinite interval. Comp. Appl. Math. 33, 607–619 (2014)

434

30 Piecewise Optimal Auxiliary Functions Method

27. E.H. Doha, A.H. Bhrawy, R.M. Hafez, R.A. Van Gorder, A Jacobi rational-Gauss collocation method for Lane-Emden equations of astrophysical significances. Nonl. Anal. Modell. Control 19(14), 537–550 (2014) 28. W.B. Bonnor, Boyle’s law and gravitational instability. MNRAS 116, 351–359 (1956) 29. S. Chandrasekhar, An Introduction to the Study of Stellar Structures (Dover, Toronto, N.Y., 1967)

Chapter 31

Some Exact Solutions for Nonlinear Dynamical Systems by Means of the Optimal Auxiliary Functions Method

In this Chapter, we will present some particular cases in which we can determine exact solutions to nonlinear dynamical systems. More precisely, for nonlinear dynamical systems given by the nonlinear differential equation L[u(x)] + N [u(x)] + g(x) = 0, x ∈ 

(31.1)

with boundary/initial conditions   du(x) =0 B u(x), du

(31.2)

where L is a linear operator, N is a nonlinear operator, g is a known function,  is the domain of interest and B is a boundary operator.In Chap. 2, we considered that the approximate solution has the form u(x) = u 0 (x) + u 1 (x, Ci ), i = 1, 2, . . . , p

(31.3)

The initial approximation u0 (x) is obtained from the linear equation L[u 0 (x)] + g(x) = 0,

  du 0 (x) =0 B u 0 (x), dx

(31.4)

If the nonlinear operator N calculated for u0 (x) has the property that N [u 0 (x)] = 0

(31.5)

then from Eqs. (31.4), (31.5) and (31.1), it is clear that u(x) = u 0 (x) is the exact solution for Eq. (31.1). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Marinca et al., Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems, https://doi.org/10.1007/978-3-030-75653-6_31

435

436

31 Some Exact Solutions for Nonlinear Dynamical Systems …

31.1 Some Exact Solutions for MHD Flow and Heat Transfer to Modified Second Grade Fluid with Variable Thermal Conductivity in the Presence of Thermal Radiation and Heat Generation/Absorption In this first Section, we consider the MHD flow and heat transfer to modified second grade fluid with variable thermal conductivity in the presence of thermal radiation and heat generation/absorption presented in Chapter 23. The governing nonlinear differential equation are [1]    m [m + 3 + s(2m + 1)] f f  + m 2 + 3m + 2 f  − f  −   −M 2 f  − s(m + 2) f  2 + k(3s − 1) 2 f  f  − f  2 − k[m + 3 + s(2m + 1)] f f I V = 0

(31.6) θ  (1 + aθ + R) + aθ  2 + Pr

m + 3 + s(2m + 1)  f θ + Qθ = 0, m = −2 (31.7) m+2

with the boundary conditions: f (0) = 0, f  (0) = 1, f  (∞) = 0, f  (∞) = 0

(31.8)

θ (0) = 1, θ (∞) = 0

(31.9)

In the above equations, the primes denotes the differentiation with respect to η, the dimensionless variables k, M2 , R, Pr and Q represent respectively generalized second order parameter, the generalized magnetic field parameter, the radiation parameter, the generalized Prandl number and heat generation/absorption parameter. They are defined as: 2

k=

16δ1 T∞ − 2 δ B02 r α1 Re m+2 ,R = , M2 = Re m+2 , 2 ρr ρU 3k ∗ k∞

Pr =

ρC p Ur − 2 Q 0 Ur 2 − 2 Re m+2 , Q = Re m+2 k∞ ρC p k∞

31.1 Some Exact Solutions for MHD Flow and Heat Transfer …

437

31.1.1 Some Exact Solutions for Eqs. (31.6)–(31.9) Using OAFM For Eq. (31.6), the linear operator can be written in the form: L[ f (η)] = f I V (η) + λ f  (η)

(31.10)

where λ is an unknown (positive) parameter at this moment, and the corresponding nonlinear operator is: m   N [ f (η)] = [m + 3 + s(2m + 1)] f f  + [ m 2 + 3m + 2 − f    − λ] f  − M 2 f  − s(m + 2) f  2 + k(3s − 1) 2 f  f  − f  2 − {1 − k f [m + 3 + s(2m + 1)]} f I V

(31.11)

For Eq. (31.7) the linear and nonlinear operators are respectively: L[θ (η)] = θ  + λθ  N [θ (η)] = θ  (R + aθ ) − λθ  + aθ  2 + Pr

(31.12)

m + 3 + s(2m + 1)  f θ + Qθ m+2 (31.13)

The Eq. (31.4) with linear operator given by Eq. (31.10), becomes: f 0I V (η) + λ f 0 (η) = 0, f 0 (0) = 0, f 0 (0) = 1, f 0 (∞) = 0, f 0 (∞) = 0 (31.14) and has the solution: f 0 (η) =

1 − e−λη ,λ > 0 λ

(31.15)

Now, Eq. (31.4) with linear operator given by Eq. (31.12) is: θ0 + λθ0 = 0, θ0 (0) = 1, θ0 (∞) = 0

(31.16)

θ0 (η) = e−λη

(31.17)

with the solution:

Substituting Eq. (31.15) into Eq. (31.11) and then substituting Eq. (31.17) into Eq. (31.13), we obtain respectively:

438

31 Some Exact Solutions for Nonlinear Dynamical Systems …

N [ f 0 (η)] = F1 e−λη + F2 e−2λη + F3 e−(m+1)λη

(31.18)

N [θ0 (η)] = G 1 e−λη + G 2 e−2λη

(31.19)

with   F1 = kλ2 − 1 [m + 3 + s(2m + 1)] − M 2   F2 = 1 − kλ2 [m + 3 + s(2m + 1)] + kλ2 (3s − 1) − s(m + 2)   (31.20) F3 = m 2 + 3m + 2 λm+2 m + 3 + s(2m + 1) m+2 m + 3 + s(2m + 1) m = −2 G 2 = 2aλ2 + Pr m+2 G 1 = (1 + R)λ2 + Q − Pr

(31.21)

To obtain exact solutions for Eqs. (31.6)–(31.9) we require: N [ f 0 (η)] = 0

(31.22)

N [θ0 (η)] = 0

(31.23)

For this aim we consider the following possible cases:

31.1.1.1

Case 1

For m = 0 in Eq. (31.18) we require: F1 + F3 = 0 F2 = 0 G1 = 0 G2 = 0

(31.24)

(31.25)

Taking into account Eqs. (31.20) and (31.21), Eqs. (31.24) and (31.25) can be rewritten as:   2 kλ − 1 (s + 3) − M 2 + 2λ2 = 0   1 − kλ2 (s + 3) + kλ2 (3s − 1) − 2s = 0

(31.26)

31.1 Some Exact Solutions for MHD Flow and Heat Transfer …

439

1 (1 + R)λ2 + Q − Pr(s + 3) = 0 2 1 2aλ2 + Pr(s + 3) = 0 2

(31.27)

From Eqs. (31.26) we have:    2 − k 2M 2 + 1 M2 + 1 4 − k   , M 2 = , λ = s= 3k k+2 4 − k 1 + 3M 2

(31.28)

From Eqs. (31.27) we obtain:   k 11M 2 + 4 − 14 ,  a=  2 4 M + 1 4 − k − 3k M 2     14 − k 11M 2 + 4 (1 + R) M 2 + 1 4−k  − , M 2 = Q = Pr  2 k+2 3k 2 4 − k − 3k M

(31.29)

In this case, the exact solutions of Eqs. (31.6)–(31.9) are: 1−e f ex (η) =

−



M 2 +1 k+2 η

M 2 +1 k+2

, θex (η) = e

−



M 2 +1 k+2 η

(31.30)

for λ, s, a and Q given by Eqs. (31.28) and (31.29).

31.1.1.2

Case 2

For m = 1 into Eq. (31.18) we obtain the system: F1 = 0 F2 + F3 = 0

(31.31)

G1 = 0 G2 = 0

(31.32)

 2  kλ − 1 (s + 3) − M 2 = 0   1 − kλ2 (3s + 4) − kλ2 (3s − 1) − 3s + 6λ3 = 0

(31.33)

which become:

440

31 Some Exact Solutions for Nonlinear Dynamical Systems …

1 3s + 4 =0 (1 + R)λ2 + Q − Pr 2 4 3s + 4 2aλ2 + Pr =0 4

(31.34)

The solutions of Eqs. (31.33) and (31.34) are:   k 5M 2 + 15s + 4  , M 2 + 3s + 4 = 0 λ=  2 6 M + 3s + 4  3 2  36 M 2 + 3s + 4 − k 3 (3s + 4) 5M 2 + 15s + 4 = 0  2 9Pr(3s + 4) M 2 + s + 4 a=− 2 ,  2k 2 5M 2 + 15s + 4  2 k 2 5M 2 + 15s + 4 3s + 4 2 − (1 + R) Q = Pr  2 , M + 3s + 4 = 0 2 3 36 M + 3s + 4

(31.35) (31.36)

(31.37)

The solutions of Eqs. (31.6) – (31.9) are: f ex (η) =

1 − e−λη , θex (η) = e−λη λ

(31.38)

where λ is given by Eq. (31.35), with the conditions (31.36) and (31.35). Obviously, into Eq. (31.38) there are many solutions, depending of M, kands. We present only three solutions. 32 ,the conditions (31.35) and (31.36) are accomplished. a) If s = M 2 = 1,k = 3 32 7 7 It results in:



7 3 49 7 3 4 3 16 , a=− , Q = Pr λ= 7 12 2 3 49 and therefore the exact solution becomes: √ 3 4 √ 3 4 1 − e− 7 η f ex (η) = , θex (η) = e−− 7 η 3

4 7

√ 2 , λ = 3 1.3 with the exact solution b). If s = 2, M 2 = 1, k = 11 3 845

(31.39)

31.1 Some Exact Solutions for MHD Flow and Heat Transfer …

441

√ 3

√ 3 1 − e− 1.3η f ex (η) = , θex (η) = e− 1.3η √ 3 1.3

c) In the Case s = 3, M 2 = 1, k =

14 √ , 3 39

1 − e− f ex (η) =

3

9 13

with the exact solution

√ 3 9

13 η

9 13

3

31.1.1.3

λ=

(31.40)

, θex (η) = e−

√ 3 9

13 η

(31.41)

Subcase 2.1

If M 2 + 3s + 4 = 0, we seek M 2 = 0, s = − 43 and the condition λ3 − 5kλ2 + 4 = 0 If k


3

972 , 125

√ 3 4

4 3

3η

, θex (η) = e−

√ 3 4

3η

(31.46)

Eq. (31.41) has three real solutions.λ1 is given by Eq. (31.42) and

442

31 Some Exact Solutions for Nonlinear Dynamical Systems …



√ −1 + i 3 3 125k 3 1 1 5k 125k 3 + − + λ2 = 1− 18 2 5832 3 3 972 

√ −1 − i 3 3 125k 3 125k 3 1 1 + 1− − + + 2 972 3 3 972 

√ −1 − i 3 3 125k 3 1 1 5k 125k 3 + − + 1− λ3 = λ2 = 18 2 5832 3 3 972 

√ 1 1 −1 + i 3 3 125k 3 125k 3 − + + + 1− 2 972 3 3 972

(31.47)

(31.48)

where i 2 = −1. For numerical example k = 1.5 < 3 972 , from Eq. (31.42) we obtain λ1 = 125 −0.6002583190417. For k = 3 > 3 972 , from Eqs. (31.42), (31.46) and (31.42) λ j are defined as 125 follows: λ1 = 0.590941921465 λ2 = 2.382558498658 λ3 = −0.473500419805

(31.49)

Corresponding, the values of the parameters a and Q are: a1 = a2 = 0, Q 1 = −0.349212354467(1 + R) Q 2 = −5.676584996392(1 + R)

(31.50)

and with the exact solutions: 1 − e−0.590941921465η 1 − e−2.382558498658η , f 2ex (η) = 0.590941921465 2.382558498658 −0.590941921465η −2.382558498658η , Q 2ex (η) = e Q 1ex (η) = e f 1ex (η) =

31.1.1.4

(31.51)

Subcase 2.2

For 5M 2 + 15s + 4 = 0 follows that λ = 0 for s = − 83 and λ = − 3 23 if s = − 83 and therefore this subcase is unacceptable.

31.1 Some Exact Solutions for MHD Flow and Heat Transfer …

31.1.1.5

443

Case 3

For m = −1, into Eq. (31.20) we obtain, F3 = 0 and from Eqs. (31.18) and (31.19) follows that: F1 = 0 F2 = 0

(31.52)

G1 = 0 G2 = 0

(31.53)

From Eq. (31.51) we obtain two solutions. The first solution of Eqs. (31.52) is given by: s1 = 

λ1 M

 2

  1 5 − 4M 2 − 9 − 16M 2 + 16M 4 4

√

5 − 8M 2 − 9 − 16M 2 + 16M 4 1

 , M 2 = =  √ 3 11 − 12M 2 − 3 9 − 16M 2 + 16M 4 k

(31.54)

(31.55)

We remark that for M 2 = 13 , Eq. (31.52) is not defined. In this case we define: λ1

    1 = lim λ1 M 2 1 3 M2→ 3

(31.56)

After simple manipulation, we have:   4 lim λ1 M 2 = 1 5 M2→ 3

(31.57)

In consequence, we define: √

5 − 8M 2 − 9 − 16M 2 + 16M 4

 , λ1 M =   √ 11 − 12M 2 − 3 9 − 16M 2 + 16M 4 k       1 4 1 1 f or M 2 ∈ 0, ∪ , ∞ andλ1 = 3 3 3 5 

2



The second solution of Eqs. (31.52) is: s2 =

  1 5 − 4M 2 + 9 − 16M 2 + 16M 4 , 4

(31.58)

444

31 Some Exact Solutions for Nonlinear Dynamical Systems …



λ2 M

2



√

5 − 8M 2 + 9 − 16M 2 + 16M 4

 , M 2 ∈ [0, 1) =  √ 2 2 4 k 11 − 12M + 3 9 − 16M + 16M

(31.59)

Corresponding, from Eqs. (31.52) the parameters a and Q are given by: a1 =

−s1 s2 Pr, a2 = − 2 Pr 2 λ1 λ1

Q 1 = (s1 − 2)Pr, Q 2 = (s2 − 2)Pr

(31.60)

where s1 and s2 are given Eqs. (31.54) and (31.59). Finally, in this case, for M 2 ∈ [0, 1), we obtain two exact solutions: f 1ex (η) =

1 − e−λ1 η 1 − e−λ2 η , f 2ex (η) = λ1 λ2

θ1ex (η) = e−λ1 η , θ2ex (η) = e−λ2 η

(31.61) (31.62)

where λ1 and λ2 are given by Eqs. (31.58) and (31.59), respectively. For M 2 > 1 we obtain only the exact solution in the form: f ex (η) =

1 − e−λ1 η , θex (η) = e−λ1 η λ1

(31.63)

with λ1 given by Eq. (31.58).

31.1.1.6

Case 4

For m = −2 into Eq. (31.20) we obtain F3 = 0. But from Eqs. (31.56) and (31.26) this case is unacceptable.

31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics 31.2.1 Case 1. The Flow of a Fourth Grade Fluid Past a Porous Plate The nonlinear differential Equations in this first case can be written as [2]

31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics (I V )

445

(V )

u  (y) + v0 u  (y) − α1 v0 u  (y) + β1 v02 u (y) − γ1 v03 u (y)   + βu  2 (y)u  (y) − γ v0 2u  (y)u  2 (y) + u  2 (y)u  (y) = 0 θ  (y) + Pr [v0 θ  (y) + E c (u  ( y −α1 v0 u  (y)u  (y) + β1 v02 u  (y) 1 + u  4 (y) − γ v0 u  3 (y)u  (y) − γ1 v03 (y)(I V ) u(y)u  (y))] = 0 3

(31.64)

(31.65)

where α1 , β1 , γ1 , β, γ , Pr , E c are dimensionless material parameters, v0 is a constant which represents the suction (v0 > 0) or injection velocity (v0 < 0), and the prime denotes differentiation with respect to dimensionless variable y. The boundary conditions for Eqs. (31.64) and (31.65) are u(0) = 0, u(∞) = 1; u (k) (∞) = 0 , k = 1, 2, 3 θ (0) = 1, θ (∞) = 0

(31.66) (31.67)

The linear and nonlinear operators for Eq. (31.64) (g(η) = 0) are respectively L[u(y)] = u  (y) + ku  (y) (I V )

(31.68) (V )

N [u(y)] = (v0 − k)u  − α1 v0 u  + β1 v02 u −γ v03 u +βu  u  2   − γ v0 2u  u  2 + u  2 u 

(31.69)

where k > 0 is an unknown parameter at this moment (g(x) = ku  (y)). The initial approximation u 0 (η) can be obtained from Eq. (6d) u 0 + ku 0 = 0

(31.70)

u 0 (0) = 0, u 0 (∞) = 1

(31.71)

with the boundary conditions

From Eqs. (31.70) and (31.71), the initial approximation u 0 (y) becomes u 0 (η) = 1 − e−ky , k > 0

(31.72)

Inserting Eq. (31.72) into Eq. (31.69), one has   N [u 0 (y)] = k v0 − k − k 2 α1 v0 − β1 k 2 v02 − γ1 k 4 v03 e−ky − k 4 (β + 3γ v0 k)e−3ky

(31.73)

446

31 Some Exact Solutions for Nonlinear Dynamical Systems …

From Eqs. (6c) and (31.73), we obtain the conditions β + 3γ v0 k = 0

(31.74)

v0 − k − α1 k 2 v0 − β1 k 2 v02 − γ1 k 4 v03 = 0

(31.75)

such that k=−

β 3γ v0

81v02 γ 4 + 27βγ 3 − 9β 2 α1 γ 2 + 3β1 β 3 γ − γ1 β 4 = 0

(31.76) (31.77)

The exact solutions of the Eq. (31.64) is obtained from Eqs. (31.72) and (31.76): βy

u(y) = u 0 (y) = 1 − e 3γ v0 , βγ v0 < 0

(31.78)

with the condition (31.77). To obtain the exact solution of Eq. (31.65), we will insert Eq. (31.78) into Eq. (31.65) and therefore:   θ  + Pr v0 θ  + Pr E c[ k 2 + α1 k3 v0 + β1 k 4 v02 + γ1 k 5 v03 e−2ky + +

β 3

+ γ1 v0 k 4 e−4ky ] = 0

(31.79)

where k is given by Eq. (31.76). The boundary conditions for Eq. (31.79) are obtained from Eq. (31.67). The linear differential Eq. (31.79) with the boundary conditions (31.67), has the exact solution:    Pr E c k 2 + α1 k 3 v0 + β1 k 4 v02 + γ1 k 5 v03  −v0 Pr y − e−2ky + e θ (y) = e−Pr v0 y + 2 4k − 2kv0 Pr   β Pr E c 3 + γ1 v0   e−v0 Pr y − e−4ky (31.80) + 2 16k − 4kv0 Pr with the conditions (31.77) and v0 > 0 (suction velocity). We consider two numerical subcases.

31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics

31.2.1.1 a.

447

A Subcase

For v0 = 13 , β = −1, γ = 0.2, α1 = 0.1, β1 = 0.2, γ1 = −0.3576, Pr = 3, E c = 1, k = 5, the exact solution of Eqs. (31.64), (31.65), (31.66) and (31.67) is u 1 (y) = 1 − e−5y

(31.81)

    θ1 (y) = e−y + 0.0222 e−y − e−10y − 0.0035726315 e−y − e−20y (31.82)

31.2.1.2 b.

B Subcase

For v0 = 23 , β = −1„α1 = 0.1, β1 = 0.3, γ1 = 1.8, Pr = 2, E c = 1, k = 1, the exact solution of Eqs. (31.64), (31.65), (31.66) and (31.67) becomes u 2 (y) = 1 − e−y   −1.333333y θ2 (y) = e−1.333333y + 1.424637534 − e−2y +  −1.333333y e −4y  −e +0.1625 e

(31.83)

(31.84)

In Figs. 31.1, 31.2, 31.3, 31.4 are presented the exact solutions for Eqs. (31.64) and (31.65).

31.2.2 Case 2. The Flow of a Second Grade Fluid Over a Stretching Sheet with Suction/Injection The boundary value problem governing the flow is given by [3]: Fig. 31.1 Exact solution u 1 (y) given byEq (6.2.20) for Eq. (31.64) with v0 = 13 , β = −1, γ = 0.2, α1 = 0.1, β1 = 0.2,γ1 = −0.3576, Pr = 3, E c = 1,k = 5

448

31 Some Exact Solutions for Nonlinear Dynamical Systems …

Fig. 31.2 Exact solution θ1 (y) given by Eq. (31.82) for Eq. (31.65) with v0 = 13 , β = −1, γ = 0.2, α1 = 0.1, β1 = 0.2, γ1 = −0.3576, Pr = 3, E c = 1, k = 5

Fig. 31.3 Exact solution u 2 (y) given by Eq (31.83) for Eq. (31.64) with v0 = 23 , β = −1, γ = 0.5, α1 = 0.1, β1 = 0.3, γ1 = 1.8, k = 1

Fig. 31.4 Exact solution θ2 (y) given by Eq. (31.84) for Eq. (31.65) with v0 = 23 , β = −1, γ = 0.5, α1 = 0.1, β1 = 0.3, γ1 = 1.8, Pr = 2, E c = 1, k = 1

u  (η) + u(η)u  (η) − u  2 (η)   (I V ) −k u(η) u (η) − 2u  (η)u  (η) + u  2 (η) = 0

(31.85)

u(0) = a; u  (0) = 1, u  (∞) = u  (∞) = 0

(31.86)

where k and a are known dimensionless parameters. The linear operator, nonlinear operator and the function g(η) are respectively L[u(η)] = u  + λu 

(31.87)

31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics

  (I V )    2 u −2u u + u N [u(η)] = −λu + uu − u − k u 



2

g(η) = 0

449

(31.88) (31.89)

where λ is a positive unknown parameter. The initial approximation u 0 (η) is obtained from equations:  u  0 (η) + λu 0 (η) = 0

(31.90)

u 0 (0) = a, u 0 (0) = 1, u 0 (∞) = 0

(31.91)

The above equations have the solution u 0 (η) = a +

1 − e−λη , λ>0 λ

(31.92)

From Eqs. (31.92) and (31.88) one can get N [u 0 (η)] = kaλ3 + (k + 1)λ2 − aλ − 1

(31.93)

The positive solutions of Equation kaλ3 + (k + 1)λ2 − aλ − 1 = 0

(31.94)

for k = 2.42539052971426 and a = −0.729843788128358 are λ1 = 2, λ2 =

1 2

(31.95)

The corresponding exact solutions for this numerical case, become u 1 (η) = −0.229843788128358 − 0.5e−2η

(31.96)

u 2 (η) = 1.270156211871642 − 2e−0.5η

(31.97)

These exact solutions are plotted in Figs. 31.5 and 31.6, respectively.

450

31 Some Exact Solutions for Nonlinear Dynamical Systems …

Fig. 31.5 Exact solution u 1 (y) given by Eq. (31.96) for Eqs. (31.85) and (31.86) for k = 2.42539052971426, a = −0.729843788128358

Fig. 31.6 Exact solution u 2 (y) given by Eq. (31.97) for Eqs. (31.85) and (31.86) for k = 2.42539052971426, a = −0.729843788128358

31.2.3 Case 3. Thin Film of an Oldroyd 6-Constant Fluid Over a Moving Belt The governing equation of thin film flow of an Oldroyd 6-constant fluid over a moving belt [4] can be written as follows: 2  v  (x) + (3α1 − α2 )v  2 (x)v  (x) + α1 α2 − m 1 + α2 v  2 = 0

(31.98)

where α1 and α2 are nondimensional material constants and m is a nondimensional gravity parameter. The boundary conditions are v(0) = 1; v  (1) + α1 v  3 (1) = 0

(31.99)

To solve Eqs. (31.98) and (31.99) we have L[v(x)] = v  (x) 2  N [v(x)] = (3α1 − α2 )v  2 v  + α1 α2 v  4 v  − m 1 + α2 v  2

(31.100) (31.101)

31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics

g(x) = −m

451

(31.102)

In what follows, we consider two subcases.

31.2.3.1 a.

A Subcase

If α1 > 0, then from the boundary condition (31.99)2 one retrieves v  (1) = 0 and in consequence, the initial approximate v0 (x) is obtained from the equations v0 − m = 0

(31.103)

v0 (0) = 1, v  (1) = 0

(31.104)

 1 2 x −x +1 v0 (x) = m 2

(31.105)

It is clear that 

Inserting Eq. (31.105) into Eq. (31.100), lead to N [v0 (x)] = 3(α1 − α2 )m 3 (x − 1)2 + α2 (α1 − α2 )m 5 (x − 1)4

(31.106)

From the Eqs. (6e) and (31.101) it holds that α1 = α2 . In this subcase, the exact solution is defined by v(x) =

31.2.3.2 b.

 m 2 x − 2x + 1 2

(31.107)

Subcase B

If α1 < 0, from the boundary condition (31.99)2 we obtain v  (1) = ± √

1 −α1

(31.108)

Initial approximation v0 is determined from equations v0 (x) − m = 0 v0 (0) = 0, v0 (1) = ± √

(31.109) 1 −α1

(31.110)

452

31 Some Exact Solutions for Nonlinear Dynamical Systems …

and therefore, the exact solution is v(x) = v0 (x) =

1 1 m m (x − 1) − ± √ (x − 1)2 ± √ 2 2 −α1 −α1

(31.111)

In Fig. 31.7 is plotted the parabola given by Eq. (31.107) for m = 2 and in Figs. 31.8 and 31.9 are plotted the parabolas given by Eq. (31.111) for m = 2 and α1 = −1.

31.2.4 Case 4. Viscous Flow Due to a Stretching Surface with Partial Slip The governing equation of viscous flow due to a stretching surface with partial slip is known as [5] φ  (η) + mφ(η)φ  (η) − φ  2 (η) = 0 Fig. 31.7 Exact solution (31.107) of Eq. (31.98) for α1 = α2 and m = 2

Fig. 31.8 First exact solution (31.111) of Eq. (31.98) for α1 = α2 = −1 and m = 2

(31.112)

31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics

453

Fig. 31.9 The second exact solution (31.111) of Eq. (31.98) for α1 = α2 = −1 and m = 2

where m is a parameter describing the type of stretching. For m = 1, we have planar stretching case while m = 2 we have axisymmetric stretching case. The boundary conditions are φ(0) = a; φ  (0) = 1 + λφ  (0); φ  (∞) = 0

(31.113)

where λ is a non-dimensional parameter indicating the partial slip and is a nondimensional constant which determines the transpiration rate at the surface; a < 0 if injection from the surface occurs, a > 0 for suction and a = 0 for an impermeable sheet. In this case we should choose L[φ(η)] = φ  + kφ 

(31.114)

N [φ(η)] = −kφ  + mφφ  − φ  2

(31.115)

g(η) = 0

(31.116)

The initial approximation is obtained from the equations φ0 + kφ0 = 0, φ0 (0) = a, φ0 (0) = 1 + λφ0 (0), φ0 (∞) = 0

(31.117)

which has the following solution φ0 (η) = a +

1 − e−kη k(1 + λk)2

(31.118)

454

31 Some Exact Solutions for Nonlinear Dynamical Systems …

where k > 0 is an unknown parameter. From Eqs. (31.118) and (31.115) one can put N [φ0 (η)] =

βe−kη + (m − 1)e−2kη (1 + λk)2

(31.119)

The exact solutions can be obtained from the condition N [φ0 (η)] = 0, such that β = 0, m − 1 = 0

(31.120)

where β = k(k − ma)(1 + λk) − m. After simple manipulation, we have that m = 1 and λk 3 + (1 − λa)k 2 − ak − 1 = 0

(31.121)

There are many possibilities to determine exact solutions from Eq. (31.121), but we present only three numerical subcases.

31.2.4.1 For a =

Subcase 1 1 2

(suction) and λ = 1 one gets k = 1 and the exact solution φ1 (η) = 1 − 0.5e−η

31.2.4.2

(31.122)

Subcase 2

For a = − 56 (injection) and λ = 1, results in k =

1 2

and the exact solution

φ2 (η) = 0.5 − 1.333333e−0.5η

(31.123)

In Figs. 31.10 and 31.11 are plotted the first solution given by Eq. (31.122) and the second solution given by Eq. (31.123), respectively.

31.2.4.3

Subcase 3

For a = 1.13745860882 and λ = −8.2749172691, Eq. (31.121) has two positive solutions k1 = 21 and k2 = 1. The exact solutions are respectively φ1 (η) = 0.5 + 0.6374586036e−0.5η

(31.124)

31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics

455

Fig. 31.10 The exact solution (31.122) of Eq. (31.112) for a = 21 , λ = 1, k = 1

Fig. 31.11 The exact solution (31.123) of Eq. (31.122)for a = − 56 , λ = 1, k =

φ2 (η) = 1 + 0.1374586882−η

1 2

(31.125)

In Figs. (31.12) and (31.13) are plotted the exact solution (31.124) and (31.125), respectively. Fig. 31.12 The exact solution (31.124) of Eq. (31.112) for a = 1.13745860882; λ = −8.2749172691; k = 0.5

456

31 Some Exact Solutions for Nonlinear Dynamical Systems …

Fig. 31.13 The exact solution (31.125) of Eq. (31.112) for a = 1.13745860882; λ = −8.2749172691; k = 1

31.2.5 Case 5. Thermal Radiation on MHD Flow Over a Stretching Porous Sheet The governing equations of thermal radiation on MHD flow over a stretching porous sheet [6] are f  (η) + f (η) f  (η) − f  2 (η) − M f  (η) = 0

(31.126)

θ  (η) + β f (η)θ  (η) − β S f  (η)θ (η) = 0

(31.127)

where M is the magnetic parameter, β is the radiation parameter and S is the wall temperature parameter. The boundary conditions for Eqs. (31.126) and (31.127) are f (0) = λ;

f  (0) = 1;

f  (∞) = 0

θ (0) = 1; θ (∞) = 0

(31.128) (31.129)

The linear, nonlinear operators and g(η) are L[ f (η)] = f  (η) + γ f  (η) N [ f (η)] = −γ f  (η) + f (η) f  (η) − f  2 (η) − M f  (η)

(31.130) (31.131)

L[θ (η)] = θ  − k 2 θ

(31.132)

  N [θ (η)] = − k 2 + β S f  (η) θ (η) + β f (η)θ  (η)

(31.133)

g(η) = 0

(31.134)

31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics

457

The initial approximation f 0 (η) for Eq. (31.126) is determined from linear equation f 0 (η) + γ f 0 (η) = 0,

f 0 (0) = λ,

f 0 (0) = 1,

f 0 (∞) = 0

(31.135)

and it holds that f 0 (η) = λ +

1 − e−γ η , γ >0 γ

(31.136)

Substituting Eq. (31.136) into Eq. (31.131), lead to equation   N [ f 0 (η)] = γ 2 − γ λ − 1 − M e−γ η

(31.137)

The exact solution of Eqs. (31.126) and (31.128) is obtained from the condition N [ f 0 (η)] = 0

(31.138)

such that we obtain γ =

  1 λ + λ2 + 4M + 4 2

(31.139)

Exact solution of Eqs. (31.126) and (31.128) is f (η) = f 0 (η) = λ + 2

1−e λ+

 √  − 21 λ+ λ2 +4M+4 η

√ λ2 + 4M + 4

(31.140)

Now, for Eq. (31.127) the initial approximate θ0 (η) is obtained from the linear equation θ0 − k 2 θ0 = 0, θ0 (0) = 1, θ0 (∞) = 0

(31.141)

θ0 (η) = e−kη , k > 0

(31.142)

which is

Inserting Eqs. (31.142) and (31.140) into Eq. (31.133), lead to the expression      β βk e−kη + − β S e−(k+γ )η N [θ0 (η)] = k 2 − k βλ + γ λ From the equation N [θ0 (η)] = 0, it holds that

(31.143)

458

31 Some Exact Solutions for Nonlinear Dynamical Systems …

  β =0 k 2 − k βλ + λ

(31.144)

βk − βS = 0 λ

(31.145)

Solving Eqs. (31.144) and (31.145), one retrieves   √ λS λ + λ2 + 4M + 4   ; k = λS β= √ 2 + λ λ + λ2 + 4M + 4

(31.146)

Therefore, the exact solution for Eq. (31.127) and (31.129) is θ (η) = θ0 (η) = e−λSη

if β =

(31.147)

 √  λS λ+ λ2 +4M+4  √ . 2+λ λ+ λ2 +4M+4

For numerical case S = M = λ = 1, yields γ = 2; k = 1; β =

2 3

(31.148)

and the exact solution (31.140) and (31.147) are defined as f (η) = 2.5 − 0.5e−2η

(31.149)

θ (η) = e−η

(31.150)

In Figs. 31.14 and 31.15 are presented the exact solution (31.149) and (31.150), Fig. 31.14 The exact solution (31.149) for Eqs. (31.126) and (31.128) for λ = 1, M = 1, γ = 2

31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics

459

Fig. 31.15 The exact solution (31.150) for Eqs. (31.127) and (31.129) for λ = 1, M = 1, γ = 2, S = 1

respectively.

31.2.6 Case 6. Upper-Convected Maxwell Fluid Over a Porous Stretching Plate In this Section, the governing nonlinear differential equation can be written in the form [7] f  (η) − M 2 f  (η) − f  2 (η) + f (η) f  (η) + β[2 f (η) f  (η) f  (η)− − f 2 (η) f  (η)] = 0

(31.151)

where M and β are material parameters. The boundary conditions are f (0) = R,

f  (0) = 1,

f  (∞) = 0

(31.152)

where R > 0 corresponds to suction velocity, and R < 0 for injection velocity. The linear and nonlinear operator (g(η) = 0) are respectively L[ f (η)] = f  (η) − k 2 f  (η), k > 0   N [ f (η)] = k 2 f  − M 2 f  − f  2 + f f  + β 2 f f  f  − f 2 f 

(31.153) (31.154)

From the linear equation f 0 (η) − k 2 f 0 (η) = 0;

f 0 (0) = R,

we obtain the initial approximation

f 0 (0) = 1,

f 0 (∞) = 0

(31.155)

460

31 Some Exact Solutions for Nonlinear Dynamical Systems …

f 0 (η) = R +

 1 1 − e−kη k

(31.156)

From Eqs. (31.156) and (31.154) yields   N [ f 0 (η)] = k 2 − M 2 − 1 − k R − β(1 + k R)2 e−kη + βe−3kη

(31.157)

The exact solution is obtained using equation N [ f 0 (η)] = 0

(31.158)

The last equation, leads to the conditions β = 0; k 2 − k R − M 2 − 1 = 0

(31.159)

such that the exact solution of Eqs. (31.151) and (31.152) is given by     √ −0.5 R+ R 2 +4M+4 η 2 1−e f (η) = f 0 (η) = R + √ R + R 2 + 4M + 4

(31.160)

We present two numerical subcases for suction and injection velocity.

31.2.6.1 a.

Subcase A

For R = 1 (suction velocity), M = 1 and k = 2 the exact solution is defined by f 1 (η) = 1.5 − 0.5e−2η

31.2.6.2 b.

(31.161)

B Subcase

For R = −2 (injection velocity), M = defined by

1 2

and k =

f 2 (η) = −2e−0.5η

1 , 2

the exact solution is

(31.162)

In Fig. 31.16 and 31.17 are plotted the exact solutions (31.161) and (31.162) respectively.

31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics

461

Fig. 31.16 The exact solution (31.161) for Eqs. (31.151) and (31.152) for M = 1, β = 0,k = 2,R = 1

Fig. 31.17 The exact solution (31.162) for Eqs. (31.151) and (31.152) for M = 21 , β = 0, k = 21 , R = −2

31.2.7 Case 7. Unsteady Viscous Flow Over a Shrinking Cylinder The boundary value problem governing the problem of unsteady viscous flow over a shrinking cylinder is [8]   η f  (η) + f  (η) + f (η) f  (η) − f  2 (η) − S η f  (η) − f  (η) = 0

(31.163)

where S is the unsteady parameter.The boundary conditions are f (1) = γ ,

f  (1) = −1,

f  (∞) = 0

(31.164)

where γ is the dimensionless suction parameter (γ > 0). The linear and nonlinear operators (g(η) = 0) are respectively L[ f (η)] = f  (η) + k f  (η)

(31.165)

  N [ f (η)] = (1 − kη) f  (η) + f (η) f  (η) − f  2 (η) − S η f  (η) − f  (η) (31.166)

462

31 Some Exact Solutions for Nonlinear Dynamical Systems …

where k > 0 is an unknown parameter. The initial approximation is obtained from the linear equations f 0 + k f 0 = 0,

f 0 (1) = γ ,

f  (1) = −1,

f 0 (∞) = 0

(31.167)

whose solution is f 0 (η) = γ +

e−k(η−1) − 1 k

(31.168)

Using Eqs. (31.168) and (31.166) it follows that   N [ f 0 (η)] = −k(k + S)η + k(γ + 1) + S − 1 e−k(η−1)

(31.169)

From equation N [ f 0 (η)] = 0, we may be written as 1 1 S=− ; k= γ γ

(31.170)

and therefore, the exact solution for Eq. (31.163) and (31.164), becomes f (η) = f 0 (η) = γ e

1−η γ

, γ >0

(31.3.171)

In particular case, for γ = 2, the exact solution of Eqs. (31.163) and (31.164) is f (η) = 2e0.5(1−η) and is plotted in Fig. 31.18. Fig. 31.18 The exact solution (31.162) of Eqs. (31.163) and (31.164) for γ = 2, k = −S = 21 , R = −2

(31.172)

31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics

463

31.2.8 Case 8. The Flow of a Viscous Incompressible Fluid Over a Porous Stretching Wall The governing equation of a two-dimensional flow of a viscous and incompressible fluid past a flat sheet is [9] f  (η) + f (η) f  (η) −

2m  2 f (η) = 0 m+1

(31.173)

The boundary conditions are f (0) = S;

f  (0) = 1 + λ f  (0);

f  (∞) = 0

(31.174)

where m > 0 is nonlinear stretching parameter, S is suction parameter if S > 0 or blowing parameter if S < 0, and λ is a slip parameter. The linear and nonlinear operators are respectively L[ f (η)] = f  (η) + k f  (η) N [ f (η)] = [ f (η) − k] f  (η) −

(31.175)

2m  2 f (η) m+1

(31.176)

where k is a positive unknown parameter. The initial approximation is obtained from equation f 0 + k f 0 = 0,

f 0 (0) = S;

f 0 (0) = 1 + λ f 0 (0);

f 0 (∞) = 0

(31.177)

The above equation has the solution f 0 (η) = S +

1 − e−k η k(1 + λ k)

(31.178)

Substituting Eq. (31.181) into Eq. (31.176) one gets N [ f 0 (η)] =

(1 + λ k)(k 2 − k S) − 1 −k η 1−m e + e−2k η 2 (1 + λ k) 1 + m)(1 + λ k)2

(31.179)

If N [ f 0 (η)] = 0, then Eqs. (31.173) and (31.174) are exactly solvable. This special case implies that m = 1; λ k 3 + (1 − λ S)k 2 − Sk − 1 = 0

(31.180)

464

31 Some Exact Solutions for Nonlinear Dynamical Systems …

The last equation has unique solution if D = (2 + 3λ S − 3λ2 S 2 − 2λ3 S 3 − 27λ2 )2 − 4(1 + λ S + λ2 S 2 )3 > 0 and three solutions if D ≤ 0. For example in the case m = λ = S = 1 we obtain D > 0 and k = 1.3247175. The unique solution can be written in the form f (η) = 1.3247175 − 0.3247175e−1.3247175η

(31.181)

In the case m = 1, λ = 0.35, S = 0.75, we obtain D < 0 and three solutions for k but only k = 1.286155675 fulfilled condition k > 0. The exact solution becomes f ex (η) = 1.365849668 − 0.615849668e−1.28615675η On the other hand, if

2m m+1

f  + f f  + f  2 = 0,

(31.182)

= −1 (or m = -1/3), Eq. (31.173) and (31.174) become f (0) = S,

f  (0) = 1 + λ f  (0);

f  (∞) = 0 (31.183)

By integrating Eq. (31.183) one gets f  (η) + f (η) f  (η) = C1 = S + (1 + λ S) f  (0) f  (η) +

1 2 f (η) = [S + (1 + λ S)] f  (0)η + C2 2

(31.184) (31.185)

where the constant C2 is given by C2 = f  (0) + 21 f 2 (0). For η → ∞ into the last S 1 such that f  (0) = 1+λ and therefore equation, it follows that f  (0) = − 1+λ S S

S +S+2 C2 = λ2(1+λ . S) The Eq. (31.185) may be rewritten as 3

f  (η) +

1 2 λ S3 + S + 2 f (η) = 2 2(1 + λ S)

(31.186)

31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics

465

which is a Riccati equation with the exact solution 

λ S3 + S + 2 f ex (η) = 2(1 + λ S)



 21 tanh

λ S3 + S + S 1+λS

 21

 (η − η0 ) , 1 + λ S = 0 (31.187)

where η0 is a constant which can be determined from Eq. (31.174)1 : 

λ S3 + S + 2 tanh η0 = −S 2(1 + λ S)

− 21 (31.188)

31.3 Exact Solutions to Oscillations of Some Nonlinear Dynamical Systems 31.3.1 Oscillations of a Uniform Cantilever Beam Carrying an Intermediate Lumped Mass and Rotary Inertia In this Chapter we consider a clamped beam at the base, free at the tip, which carries a lumped mass and rotary inertia at an arbitrary intermediate point along its spam. The beam is assumed to the inextensible, uniform of constant length and mass per unit length and thickness of this conservative beam is assumed to be small compared to the length so that effects of rotary inertia and shearing deformation will be ignored. In these conditions, we consider the discrete, single-mode of order three nonlinearities beam temporal problem [10] ¨ + αu(t)u  2 (t) + βu 3 = 0 u(t) ¨ + u(t) + αu 2 (t)u(t)

(31.189)

subject to the initial conditions u(0) = A, u(0) ˙ =B

(31.190)

The dot denotes derivative in respect to time t. In this system α and β are modal constants. If  is the unknown frequency of the system (31.189), then the linear and nonlinear operator (g(t) = 0) are respectively L[u(t)] = u(t) ¨ + 2 u(t)   N [u(t)] = 1 − 2 u(t) + αu 2 u¨ + αuu  2 + βu 3

(31.191) (31.192)

466

31 Some Exact Solutions for Nonlinear Dynamical Systems …

The initial approximations u 0 (t) can be obtained from the linear equation u¨ 0 (t) + 2 u 0 (t) = 0, u 0 (0) = A, u˙ 0 (0) = 0

(31.193)

u(0) = Acost

(31.194)

whose solution is

Substituting Eq. (31.194) into Eq. (31.192), it can be shown that   1  3 3 2 2 3 N [u 0 (t)] = A 1 −  − α A + β A cost+ 2 4   1 3 1 3 2 β A − α A  cos3t + 4 2 

(31.195)

From the condition N [u 0 (t)] = 0, we obtain

= α=

1 1 + β A2 2

(31.196)

β 2 + β A2

(31.197)

such that, the exact solution of Eq. (31.189) and (31.190) is

1 u(t) = u 0 (t) = Acos 1 + β A2 t 2 −1  with the condition α = β 2 + β A2 . In particular case, for β = 2 and A = α=

3 4

(31.198)

it is obtained

5 1 ; = 2 4

(31.199)

and the exact solution can be written in the form u(t) = 0.75cos1.25t

(31.200)

In Fig. 31.19 is plotted the exact solution given by Eq. (31.200) for the problem (31.189) and (31.190).

31.3 Exact Solutions to Oscillations of Some Nonlinear Dynamical Systems

467

Fig. 31.19 The exact solution (31.200) of Eqs. (31.189) and (31.190) for α = 21 , β = 2,A = 34 . R = −2

31.3.2 Nonlinear Jerk Equations The most general jerk function with invariance of the time-reversal and space-reversal and which has only cubic nonlinearities may be written as [45] ... u (t) + α u˙ 3 (t) + βu 2 (t)u(t) ˙ + γ u(t)u  (t)u(t) ¨ + δ u(t) ¨ u¨ 2 (t) + λu˙ = 0 (31.201) where the parameters α, β, γ , δ are constants. The corresponding initial conditions are u(0) = 0, u(0) ˙ = A, u(0) ¨ =0

(31.202)

Here, at the least one of α, β, γ , and δ should be non-zero. In addition, if δ = 0, we require γ = 1 − 2α such that the jerk equation is simply not the time-derivative of an acceleration equation. Using the transformations τ = ωt; u =

A x;  = ω2 ω

(31.203)

the Eqs (31.201) and (31.202) can be rewritten in the form     2 x  + δ A2 x  x  2 +  α A2 x  3 + γ A2 x x  x  + λx  + β A2 x 2 x  = 0 (31.204) x(0) = 0, x  (0) = 1, x  (0) = 0

(31.205)

where prime denotes differentiation with respect to τ . The linear and nonlinear operator are defined by   L[x(τ )] = 2 x  + x 

(31.206)

468

31 Some Exact Solutions for Nonlinear Dynamical Systems …

    N [x(τ )] = 2 −x + δ A2 x  x  2 +  α A2 x  2 + γ A2 x x  x  + λx  + + β A2 x 2 x 

(31.207)

From the equation   2 x0 + x0 = 0, x0 (0) = 0, x0 (0) = 1, x0 (0) = 0

(31.208)

one can get x0 (τ ) = sin(τ )

(31.209)

From Eqs. (31.209) and (31.207), we can deduce that     1 2 3α − γ 2 1 2 2 δA − 1  + λ + A  + β A cosτ N [x0 (τ )] = 4 4 4   1 1 1 (31.210) + − δ A2 2 + (α + γ )A2 − β A2 cos3τ 2 2 4 

The exact solution of Eqs. (31.201) and (31.202) can be obtained from the conditions     1 2 1 3α − γ 2 2 δA − 1  + λ + A  + β A2 = 0 (31.211) 4 4 4 1 1 1 − δ A2 2 + (α + γ )A2 − β A2 = 0 2 2 4

(31.212)

The above equations have the solutions      β = λ + α A2 γ + α − δ λ + α A2 ; ω = λ + α A2

(31.213)

The exact solutions of Eqs. (31.201) and (31.202) are obtained from (31.203), (31.209) and (31.213): u(t) = √

A λ + α A2

 sin λ + α A2 t

(31.214)

    with conditions β = λ + α A2 γ + α − δ λ + α A2 and γ = −2α. In particular case for, A = γ = 1, α = 2, the exact solution (31.214) becomes √ u(t) =

3 √ sin 3t 3

(31.215)

31.3 Exact Solutions to Oscillations of Some Nonlinear Dynamical Systems

469

Fig. 31.20 The exact solution (31.215) of Eqs. (31.201) and (31.202) for A = λ = 1, α = 2, β = 3γ − 6δ + 6

In Fig. 31.20 is plotted the exact solution (31.215).

31.4 Exact Solutions to Duffing Equation Problems of nonlinear oscillations in conservative systems have a long history and the well-known nonlinear Duffing and double-well Duffing oscillators are illustrative examples in this field [11, 12]. Many mechanical systems involve nonlinearity of Duffing type [13–19]. The Duffing equation with a double-well potential (with a negative linear stiffness) is also an important model. A physical realization of such a Duffing oscillator model is a mass particle moving in an asymmetric double-well potential [20–22]. This form of the equation also appears in the transverse vibrations of a beam when the transverse and longitudinal deflections are coupled. Although the Duffing equation with a nonnegative linear stiffness is very often used as an example to demonstrate the validity of various methods for constructing analytical approximate solutions to nonlinear oscillators [11, 12], no corresponding report to the Duffing oscillator with a negative linear stiffness appears up to now. One of the most exciting recent advances of nonlinear science and theoretical physics has been the development of some methods to look for exact solutions of nonlinear differential equations. This is important because many mathematical models are described by nonlinear differential equations and exact solutions are preferable instead of approximate ones. In the last years scientists were concerned in finding exact solutions and some efficient procedures were developed. The inverse scattering transform [23], the Hirota linear method [24], the Bäcklund transformation [25], the homogeneous balance method [26, 27], the exp-function method [28–32] the Jacobi elliptic function expansion method [33, 34], the F-expansion method [35], the auxiliary equation method [36], the tanh method [37–39], the simplest equation method [40, 41] are reliable methods for obtaining exact solutions of nonlinear differential equations.

470

31 Some Exact Solutions for Nonlinear Dynamical Systems …

In this Chapter" we propose a new approach to search explicit and exact solutions of nonlinear differential equations. We are taking into account two basic ideas. The first idea is to propose the solutions of a differential equation in the form of a quotient of trigonometric function. The second idea is to apply the simplest nonlinear differential equations that have lesser order than the studied equations. We have two advantages of our method. The first advantage is that our approach generalizes a number of methods that were applied before. The second advantage of the method is its simplicity of realization. We apply our approach to obtain exact solutions of Duffing and double-well Duffing equations. In the following analysis, the nonlinear Duffing oscillator is taken as an example. The oscillation is defined by the equation [42] x¨ + ω20 x + αx 3 = 0

(31.216)

where x is the displacement, ω0 is the circular frequency, α is a negative constant which may not be a small value. In order to solve Eq. (31.216), the following transformation is needed τ = ωt

(31.217)

where ω is called the frequency of the system. Substitution of Eq. (31.217) into Eq. (31.216) lead to ω2 x  + ω20 x + αx 3 = 0

(31.218)

d where  = dτ . For nonlinear differential Eq. (31.218) with the initial conditions

x(0) = A, x  (0) = B

(31.219)

thelinar operator L and nonlinear operator N (g(τ) = 0) are respectively L[x(τ)] = ω2 (x  + x)

(31.220)

N [x(τ)] = (ω20 − ω2 )x + αx 3

(31.221)

The initial approximation x0 is obtained from Eq. (2.5) ω2 (x0 + x0 ) = 0, x0 (0) = A, x0 (0) = B whose solution is given by

(31.222)

31.4 Exact Solutions to Duffing Equation

471

x0 (τ) = A cos τ + B sin τ

(31.223)

Substituting Eq. (31.223) into Eq. (31.221) yields 3αA 2 (A + B 2 )] cos τ + [B(ω20 − ω2 ) 4 3αB 2 αA 2 αB (A + B 2 )] sin τ + (A − 3B 2 ) cos 3τ + (3A2 − B 2 ) sin 3τ + 4 4 4

N (x0 , τ) = [A(ω20 − ω) +

(31.224)

Taking into consideration Eqs. (31.223) and (31.224) it is clear that the approximate solution x(τ) is of the form x(τ) = C1 cos τ + C2 sin τ + C3 cos 3τ + C4 sin 3τ + . . .

(31.225)

where ω is obtained from the condition that the secular terms disappears. Now, instead of Eq. (31.225) and taking into consideration the procedures from Chapter 4, we try to choose the following expressions for the exact solution (not approximate solution) of Eq. (31.216) a0 + x(θ) = b0 +

n  j=1 n 

(a j sin θ + an+ j cos θ) sin j−1 θ (31.226) (b j sin θ + bn+ j cos θ) sin

j−1

θ

j=1

with the remarks: (1)

The function θ satisfies the relation θ = sin θ

(2) (3)

(31.227)

n can be determined by balancing the highest degree nonlinear term and the derivative terms of higher order in Eq. (31.218). In our case n = 1. In Eq. (31.226) appear the coefficients ai and bi (i = 0,2n), but does not appear the convergence-control parameters Ci because the purpose of this Section is to obtain exact solutions and not approximate solutions. Actually, from Eq. (31.227) one has sin θ = ±

1 cosh(τ + ϕ)

(31.228)

where τ is given by Eq. (31.217) and ϕ is constant. Differentiating (31.228) with respect to τ and substituting (31.227) into it, yields

472

31 Some Exact Solutions for Nonlinear Dynamical Systems …

cos θ = − tanh(τ + ϕ)

(31.229)

Here minus sign must be taken, otherwise cos θ = tanh(τ + ϕ) cannot satisfy (31.227) [39]. So, the quotient solution takes the following form x(θ) =

a0 + a1 sin θ + a2 cos θ b0 + b1 sin θ + b2 cos θ

(31.230)

Here stress is needed to lay on the coefficients ai , bi (i = 0,1,2) in order to get nontrivial solutions of Eq. (31.218). From the quotient solution (31.230), the following relation can be easily obtained x  (θ) =

A(sin θ, cos θ) (b0 + b1 sin θ + b2 cos θ)3

(31.231)

where A(sin θ, cos θ) = (a1 b0 + a1 b22 − a0 b0 b1 − a2 b1 b2 ) sin θ + (a0 b12 − a1 b0 b1 + 2a0 b22 − 2a2 b0 b2 ) sin2 θ + (a1 b22 − a2 b1 b2 − 2a1 b02 + 2a0 b0 b1 ) sin3 θ + (2a1 b0 b2 − a2 b0 b1 − a0 b1 b2 ) sin θ cos θ + (a2 b12 − a1 b1 b2 + 2a0 b0 b2 − 2a2 b02 ) sin2 θ cos θ x 3 (θ) =

B(sin θ, cos θ) (b0 + b1 sin θ + b2 cos θ)3

(31.232) (31.233)

where B(sin θ, cos θ) = a03 + 3a0 a22 + 3(a02 a1 + a1 a22 ) sin θ + 3(a0 a12 − a0 a22 ) sin2 θ + (a13 − 3a1 a22 ) sin3 θ + (a23 + 3a02 a2 ) cos θ + (3a12 a2 − a23 ) sin2 θ cos θ

(31.234)

So, substituting Eqs. (31.230), (31.232) and (31.233) into Eq. (31.218) it results in an algebraic equations about expansion coefficients ai and bi . Setting the coefficients of various sin j θ(j = 0,1,2,3) and sin j θ cos θ(j = 0,1,2) as zero, one can obtain a set of algebraic equation about the expansion coefficients aj and bj , i.e. ω20 (a0 b02 + a0 b22 + 2a2 b0 b2 ) + α(a03 + 3a0 a22 ) = 0 ω2 (a1 b02 + a1 b22 − a0 b0 b1 − a2 b1 b2 ) + ω20 (a1 b02 + a1 b22 + 2a0 b0 b1 + 2a2 b1 b2 ) + 3α(a02 a1 + a1 a22 ) = 0

31.4 Exact Solutions to Duffing Equation

473

ω2 (a2 b02 + a2 b22 + 2a0 b0 b2 ) + α(a23 + 3a02 a2 ) = 0 ω2 (a0 b12 − a1 b0 b1 + 2a0 b22 − 2a2 b0 b2 ) + ω20 (a0 b12 − a0 b22 + 2a1 b0 b1 − 2a2 b0 b2 ) + 3α(a0 a12 − a0 a22 ) = 0 ω2 (a1 b22 − a2 b1 b2 − 2a1 b02 + 2a0 b0 b1 ) + ω20 (a1 b12 − a1 b22 − 2a2 b1 b2 ) + α(a13 − 3a1 a22 ) = 0

(31.235)

ω2 (2a1 b0 b2 − a2 b0 b1 − a0 b1 b2 ) + 2ω20 (a0 b1 b2 + a1 b0 b2 + a2 b0 b1 ) = 0 ω2 (a2 b12 − a1 b1 b2 + 2a0 b0 b2 − 2a2 b02 ) + ω20 (a2 b12 − a2 b22 + 2a1 b1 b2 ) + α(3a12 a2 − a23 ) = 0 from which the expansion coefficients ai and bi can be determined under certain conditions satisfied by the parameters ω0 and α < 0. Solving the above nonlinear algebraic Eqs. (31.235) we can obtain: √ Case 31.4.1:a0 = a2 = b0 = b2 = 0, ω = 2ω0 . Case 31.4.2:a0 = −a2 , b0 = −b2 = ±

√ −α a , ω0 0

a1 = b1 = 0.

Case 31.4.3:a0 = a2 = b1 = b2 = 0, a1 = ± − α2 ω0 , ω = iω0 , i 2 = −1.

Case 31.4.4:a0 = a2 , a1 − b1 = 0, b0 = b2 = ±

√

−α a , ω0 0

ω=

√ 2ω0 .

Case 31.4.5: √

−α 2 a2 = b0 = 0, b1 = ± a1 − a02 , ω0 √ √ −α b2 = ± a0 , ω = 2ω0 ,|a1 | > |a0 | ω0 √ −α a , ω ω0 0 √ b2 = ± ω−α a2 , 0 √ b1 = ± ω−α a1 . 0

Case 31.4.6:a1 = a2 = b0 = b1 = 0 , b2 = ±

Case 31.4.8:a2 = b2 = 0, b0 =

√

−α a , ω0 1 √ ± ω−α a0 , 0

Case 31.4.7:a0 = b0 = 0, b1 = ±

Case 31.4.9:a0 = b2 = 0, b1 = ±

√

−α ω0

=

ω0 √ . 2

ω=

√ 2ω0 .

√ √ a12 + a22 , b0 = ± ω−α a2 , ω = 2ω0 . 0

Case 31.4.10: a0 = a1 = b1 = b2 = 0, b0 = ±

√

−α a , ω0 2

ω=

ω0 √ . 2

474

31 Some Exact Solutions for Nonlinear Dynamical Systems …

Case 31.4.11: a1 = b1 = 0, b0 = ±

√

−α a , ω0 0

b2 = ±

√

−α a . ω0 2

Case 31.4.12: a2 = −a0 , a1 = b1 = 0, b2 = −b0 , ω = iω0 . Case 31.4.13: a1 = b1 = 0, b2 = ±

√ −α a , ω0 0

Case 31.4.14: a1 = 0, b0 = ± √ √ ± ω−α a0 , ω = 2ω0 |a2 | > |a0 | 0

b0 = ±

√

−α a , ω0 2

b1

√

−α a , ω0 2

ω0 ω= √ . 2 √ = ± ω−α a22 − a02 ,b2 0

=

In the cases 31.216, 31.217, 31.219, 31.222, 31.223, 31.226 and 31.227 the solutions (31.230) are constants. We find the following types of solutions to Eq. (31.230): In the case 31.218:

1 2 (31.236) x1,2 = ± − ω0 α cos(ω0 t + ϕ) In the case 31.220: x3,4

√ a0 cosh( 2ω0 t + ϕ) ± a1 ω0 , |a1 > |a0 || = ±√ −α ± a 2 − a 2 − a sinh(√2ω t + ϕ) 1

0

0

(31.237)

0

where a0 and a1 are real parameters. In the case 31.221:   ω0 ω0 x5,6, = ± √ coth √ t + ϕ −α 2

(31.238)

In the case 31.224: x7,8

√ ±a1 − a2 sinh( 2ω0 t + ϕ) ω0 = ±√ −α a cosh(√2ω t + ϕ) ± a 2 + a 2 2 0 1 2

(31.239)

In the case 31.225:   ω0 ω0 x9,10 = ± √ tanh √ t + ϕ −α 2 In the case 31.228 we obtain the same solutions given by Eq. (31.238). In the case 31.229:

(31.240)

31.4 Exact Solutions to Duffing Equation

475

√ √ a0 cosh( 2ω0 t + ϕ) − a2 sinh( 2ω0 t + ϕ) ω0 x11,12 = ± √ , |a2 | > |a0 | −α a cosh(√2ω t + ϕ) + a sinh(√2ω t + ϕ) ± a 2 − a 2 2 0 0 0 2 0

(31.241) where a0 and a1 are real parameters.

31.5 Solutions of the Double-Well Duffing Equation Now, we study the unforced and undamped double-well Duffing equation x¨ − ω20 x + βx 3 = 0

(31.242)

where β is a positive constant which may not be a small value. Under the transformation (31.243), Eq. (31.242) becomes: ω2 x  − ω20 x + βx 3 = 0

(31.243)

where ω is the frequency of the system and  = ddτ . Equation (31.242) may have the same quotient trigonometric solution (31.228) and θ satisfies Eq. (31.227). Substituting Eqs. (31.228), (31.229) and (31.231) into Eq. (31.243), it results a set of algebraic equations about the expansion coefficients aj and bj , i.e. −ω20 (a0 b02 + a0 b22 + 2a2 b0 b2 ) + β(a03 + 3a0 a22 ) = 0 ω2 (a1 b02 + a1 b22 − a0 b0 b1 − a2 b1 b2 ) − ω20 (a1 b02 + a1 b22 + 2a0 b0 b1 + 2a2 b1 b2 ) + 3β(a02 a1 + a1 a22 ) = 0 −ω20 (a2 b02 + a2 b22 + 2a0 b0 b2 ) + β(a23 + 3a02 a2 ) = 0 ω2 (a0 b12 − a1 b0 b1 + 2a0 b22 − 2a2 b0 b2 ) − ω20 (a0 b12 − a0 b22 + 2a1 b0 b1 − 2a2 b0 b2 ) + 3β(a0 a12 − a0 a22 ) = 0 −ω20 (a2 b02 + a2 b22 + 2a0 b0 b2 ) + β(a23 + 3a02 a2 ) = 0 ω2 (a0 b12 − a1 b0 b1 + 2a0 b22 − 2a2 b0 b2 ) − ω20 (a0 b12 − a0 b22 + 2a1 b0 b1 − 2a2 b0 b2 ) + 3β(a0 a12 − a0 a22 ) = 0

476

31 Some Exact Solutions for Nonlinear Dynamical Systems …

ω2 (a1 b22 − a2 b1 b2 − 2a1 b02 + 2a0 b0 b1 )− ω20 (a1 b12 − a1 b22 − 2a2 b1 b2 ) + β(a13 − 3a1 a2 ) = 0 ω2 (2a1 b0 b2 − a2 b0 b1 − a0 b1 b2 ) − 2ω20 (a0 b1 b2 + a1 b0 b2 + a2 b0 b1 ) = 0 ω2 (a2 b12 − a1 b1 b2 + 2a0 b0 b2 − 2a2 b02 )− ω20 (a2 b12 − a2 b22 + 2a1 b1 b2 ) + β(3a12 a2 − a23 ) = 0

(31.244)

where the parameters ω0 and β > 0 are known. Solving the nonlinear algebraic Eqs. (31.244) we can obtain. Case 31.5.1: a2 = a0 = 0, a1 = β2 ω0 b02 − b22 , |b0 | > |b2 |, ω = ω0 . Case 31.5.2: a1 = b1 = 0, a2 = −a0 , b2 = −b0 , ω real. √

Case 31.5.3: a1 = a2 = b0 = b1 = 0, b2 = ± ω0β a0 , ω = iω0 .

√ √ Case 31.5.4: a1 = ±a0 , a2 = b1 = 0, b2 = ± ω0β a0 , ω = i ω0 2. √

√

Case 31.5.5: a0 = b0 , b1 = ± ω0β a1 , b2 = ± ω0β a2 , ω real. √

√

Case 31.5.6: a2 = b2 = 0, b0 = ± ω β0 a0 , b1 = ± ω0β a1 , ω real. √

Case 31.5.7: a0 = a1 = b1 = b2 = 0, b0 = ± ω0β a2 , ω = i ω20 .

√ √ √ Case 31.5.8: a1 = b1 = 0, b0 = ± ω0β a0 , b2 = ± ω0β a2 , ω = i 2ω0 . √

√

Case 31.5.9: a2 = −a0 , a1 = b1 = 0, b0 = ± ω0β a0 , b2 = ∓ ω0β a0 . Case 31.5.10: a0 = a2 , b0 = b1 = Case 31.5.11:a0 = a2 , b0 = b2 −

√ β a , ω0 0 √ β a , ω0 0

a1 = b1 = 0, ω real. √ ω = i 2ω0 .

In the cases 31.243, 31.246, 6.5.9 and 6.5.10 it is obtained constant solutions and in the case 31.244, 31.245, 31.248, 6.5.8 and 6.5.11 it is obtained complex solutions. Only in the case 31.242 we can find the following solutions of Eq. (31.242): 1 ω0 , |b0 | > |b2 | x = ± √ 2(b02 − b22 ) b cosh(ω t + ϕ) − b2 sinh(ω0 t + ϕ) β 0 0 (31.245) It is interesting to remark that if we consider the Jacobi elliptic equation, which is widely studied [43]

31.5 Solutions of the Double-Well Duffing Equation

x˙ 2 = a + bx 2 + cx 4

477

(31.246)

where a, b and c are constants, differentiating Eq. (31.246) with respect to t, we have 2 x˙ x¨ = 2bx x˙ + 4cx 3 x˙

(31.247)

x¨ − bx − 2cx 3 = 0

(31.248)

or simplifying

which means that Eqs. (31.246) and (31.248) are equivalent and thus the solutions given above by our method are also solutions for the Jacobi elliptic Eq. (31.246) in case bc < 0.

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