Keller-Box Method and Its Application 9783110271782, 9783110271379
Most of the problems arising in science and engineering are nonlinear. They are inherently difficult to solve. Tradition
130 111 14MB
English Pages 412 [414] Year 2014
Contents
Chapter 0 Introduction
References
Chapter 1 Basics of the Finite Difference Approximations
1.1 Finite difference approximations
1.2 The initial value problem for ODEs
1.3 Some basic numerical methods
1.4 Some basic PDEs
1.5 Numerical solution to partial differential equations
References
Chapter 2 Principles of the Implicit Keller-box Method
2.1 Principles of implicit finite difference methods
2.2 Finite difference methods
2.3 Boundary value problems in ordinary differential equations
References
Chapter 3 Stability and Convergence of the Implicit Keller-box Method
3.1 Convergence of implicit difference methods for parabolic functional differential equations
3.1.1 Introduction
3.1.2 Discretization of mixed problems
3.1.3 Solvability of implicit difference functional problems
3.1.4 Approximate solutions of difference functional problems
3.1.5 Convergence of implicit difference methods
3.1.6 Numerical examples
3.2 Rate of convergence of finite difference scheme on uniform/non-uniform grids
3.2.1 Introduction
3.2.2 Analytical results
3.2.3 Numerical results
3.3 Stability and convergence of Crank-Nicholson method for fractional advection dispersion equation
3.3.1 Introduction
3.3.2 Problem formulation
3.3.3 Numerical formulation of the Crank-Nicholson method
3.3.4 Stability of the Crank-Nicholson method
3.3.5 Convergence
3.3.6 Radial flow problem
3.3.7 Conclusions
References
Chapter 4 Application of the Keller-box Method to Boundary Layer Problems
4.1 Flow of a power-law fluid over a stretching sheet
4.1.1 Introduction
4.1.2 Formulation of the problem
4.1.3 Numerical solution method
4.1.4 Results and discussion
4.1.5 Concluding remarks
4.2 Hydromagnetic flow of a power-law fluid over a stretching sheet
4.2.1 Introduction
4.2.2 Flow analysis
4.2.3 Numerical solution method
4.2.4 Results and discussion
4.3 MHD Power-law fluid flow and heat transfer over a non-isothermal stretching sheet
4.3.1 Introduction
4.3.2 Governing equations and similarity analysis
4.3.3 Heat transfer
4.3.4 Numerical procedure
4.3.5 Results and discussion
4.4 MHD flow and heat transfer of a Maxwell fluid over a non-isothermal stretching sheet
4.4.1 Introduction
4.4.2 Mathematical formulation
4.4.3 Heat transfer analysis
4.4.4 Numerical procedure
4.4.5 Results and discussion
4.4.6 Conclusions
4.5 MHD boundary layer flow of a micropolar fluid past a wedge with constant wall heat flux
4.5.1 Introduction
4.5.2 Flow analysis
4.5.3 Flat plate problem
4.5.4 Results and discussion
4.5.5 Conclusion
References
Chapter 5 Application of the Keller-box Method to Fluid Flow and Heat Transfer Problems
5.1 Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet
5.1.1 Introduction
5.1.2 Mathematical formulation
5.1.3 Solution of the problem
5.1.4 Results and discussion
5.1.5 Conclusions
5.2 Convection flow and heat transfer of a Maxwell fluid over a non-isothermal surface
5.2.1 Introduction
5.2.2 Mathematical formulation
5.2.3 Skin friction
5.2.4 Nusselt number
5.2.5 Results and discussion
5.2.6 Conclusion
5.3 The effects of variable fluid properties on the hydromagnetic flow and heat transfer over a nonlinearly stretching sheet
5.3.1 Introduction
5.3.2 Mathematical formulation
5.3.3 Numerical procedure
5.3.4 Results and discussion
5.3.5 Conclusions
5.4 Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid over a vertical stretching sheet
5.4.1 Introduction
5.4.2 Mathematical formulation
5.4.3 Numerical procedure
5.4.4 Results and discussion
5.5 The effects of linear/nonlinear convection on the non-Darcian flow and heat transfer along a permeable vertical surface
5.5.1 Introduction
5.5.2 Mathematical formulation
5.5.3 Numerical procedure
5.5.4 Results and discussion
5.6 Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid over a stretching surface
5.6.1 Introduction
5.6.2 Mathematical formulation
5.6.3 Numerical procedure
5.6.4 Results and discussion
5.6.5 Conclusions
References
Chapter 6 Application of the Keller-box Method to More Advanced Problems
6.1 Heat transfer phenomena in a moving nanofluid over a horizontal surface
6.1.1 Introduction
6.1.2 Mathematical formulation
6.1.3 Similarity equations
6.1.4 Numerical procedure
6.1.5 Results and discussion
6.1.6 Conclusion
6.2 Hydromagnetic fluid flow and heat transfer at a stretching sheet with fluid-particle suspension and variable fluid properties
6.2.1 Introduction
6.2.2 Mathematical formulation
6.2.3 Solution for special cases
6.2.4 Analytical solution by perturbation
6.2.5 Numerical procedure
6.2.6 Results and discussion
6.2.7 Conclusions
6.3 Radiation effects on mixed convection over a wedge embedded in a porous medium filled with a nanofluid
6.3.1 Introduction
6.3.2 Problem formulation
6.3.3 Numerical method and validation
6.3.4 Results and discussion
6.3.5 Conclusion
6.4 MHD mixed convection flow over a permeable non-isothermal wedge
6.4.1 Introduction
6.4.2 Mathematical formulation
6.4.3 Numerical procedure
6.4.4 Results and discussion
6.4.5 Concluding remarks
6.5 Mixed convection boundary layer flow about a solid sphere with Newtonian heating
6.5.1 Introduction
6.5.2 Mathematical formulation
6.5.3 Solution procedure
6.5.4 Results and discussion
6.5.5 Conclusions
6.6 Flow and heat transfer of a viscoelastic fluid over a flat plate with a magnetic field and a pressure gradient
6.6.1 Introduction
6.6.2 Governing equations
6.6.3 Results and discussion
6.6.4 Conclusions
References
Subject Index
Author Index
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Kuppalapalle Vajravelu, Kerehalli V. Prasad Keller-Box Method and Its Application
De Gruyter Studies in Mathematical Physics
Edited by Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, São Paulo, Brazil Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia
Volume 8
Kuppalapalle Vajravelu, Kerehalli V. Prasad
Keller-Box Method and Its Application
Physics and Astronomy Classification Scheme 2010 2.60.-X, 2.60. lj, 2.70.-c, 47.11.-j, 47.11.Bc, 44.20.+b, 44.25.+f, 44.40.+a, 47.15.cd, 47.15.gm, 47.50.-d, 47.5d.+r, 47.85.-g, 83.80.Rs, 47.10.ad, 44.05.+e Authors Prof. Dr. Kuppalapalle Vajravelu University of Central Florida Department of Mathematics 4000 Central Florida Blvd. P.O.Box 161364 Orlando, FL 32816-1364 USA [email protected] Prof. Dr. Kerehalli V. Prasad Bangalore University Department of Mathematics Central College Campus Bangalore-560056 India
ISBN 978-3-11-027137-9 e-ISBN 978-3-11-027178-2 Set-ISBN 978-3-11-027179-9 ISSN 2194-3532 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2014 Higher Education Press and Walter de Gruyter GmbH, Berlin/Boston Printing and binding: CPI books GmbH, Leck ♾Printed on acid-free paper Printed in Germany www.degruyter.com
Preface
During the past decades there has been an increased interest in solving systems of nonlinear differential equations associated with physical problems. Throughout engineering and technological industries, we are confronted with nonlinear boundary-value problems that cannot be solved by analytical methods. Although remarkable progress has been made in developing new and powerful techniques for solving the nonlinear differential equations, notably in the fields of fluid mechanics, biology, finance, aerospace engineering, chemical, and control engineering, much remains to be done. In the present book, we highlight the development, analysis and application of the finite difference technique, the Keller-box method, for the solution of coupled nonlinear boundary-value problems. We have tried to present an account of what has been accomplished in the field to date. Accordingly, we shape this book to those interested in the Keller-box method as a working tool for solving physical and engineering problems. This book can help the reader develop the toolset needed to apply the method, without sifting through the endless literature on the subject. Issues of finite differences, converting a system to first order differential equations, linearization by Newton’s method, initial approximations, some basic numerical techniques, and obtaining a tridiagonal system by the Thomas algorithm, are discussed heuristically. As mentioned above, there are plenty of applications of the Keller-box method in the literature. In selecting applications and specific problems to work through, we have restricted our attention to fluid flow and heat transfer phenomena. Hence, in order to illustrate various properties and tools useful when applying the Keller-box method, we have selected recent research results in this area. We appreciate the support and motivation of the editor A.C.J. Luo. We also acknowledge the role of Higher Education Press (China) and de Gruyter for making this book a reality. Last but not the least, we thank Prof. Mike Taylor for reading the manuscript and suggesting some needed changes. Orlando, Florida
K. Vajravelu K.V. Prasad
Contents
Chapter 0 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Basics of the Finite Difference Approximations · · · · · · · · · · · · 1.1 Finite difference approximations · · · · · · 1.2 The initial value problem for ODEs· · · · 1.3 Some basic numerical methods · · · · · · · 1.4 Some basic PDEs · · · · · · · · · · · · · · · · · 1.5 Numerical solution to partial differential References · · · · · · · · · · · · · · · · · · · · · · · · · ·
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Chapter 1
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Chapter 2 Principles of the Implicit Keller-box Method · 2.1 Principles of implicit finite difference methods · · · · · · · · · 2.2 Finite difference methods · · · · · · · · · · · · · · · · · · · · · · · · 2.3 Boundary value problems in ordinary differential equations · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Chapter 3
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Stability and Convergence of the Implicit Keller-box Method· · · · · · · · · · · · · · · · · · · · · · · · · · 89 3.1 Convergence of implicit difference methods for parabolic functional differential equations· · · · · · · · · · · · · · · · · · · · · · · 90 3.1.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 90 3.1.2 Discretization of mixed problems· · · · · · · · · · · · · · · · · 91 3.1.3 Solvability of implicit difference functional problems· · · 94 3.1.4 Approximate solutions of difference functional problems · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 96 3.1.5 Convergence of implicit difference methods · · · · · · · · · 99 3.1.6 Numerical examples · · · · · · · · · · · · · · · · · · · · · · · · · · 103
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3.2 Rate of convergence of finite difference scheme on uniform/non-uniform grids · · · · · · · · · · · · · · · · · · · · · · · 3.2.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · 3.2.2 Analytical results · · · · · · · · · · · · · · · · · · · · · · · · · 3.2.3 Numerical results · · · · · · · · · · · · · · · · · · · · · · · · · 3.3 Stability and convergence of Crank-Nicholson method for fractional advection dispersion equation· · · · · · · · · · · · · · 3.3.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · 3.3.2 Problem formulation · · · · · · · · · · · · · · · · · · · · · · 3.3.3 Numerical formulation of the Crank-Nicholson method · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 3.3.4 Stability of the Crank-Nicholson method · · · · · · · · 3.3.5 Convergence · · · · · · · · · · · · · · · · · · · · · · · · · · · · 3.3.6 Radial flow problem · · · · · · · · · · · · · · · · · · · · · · · 3.3.7 Conclusions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Application of the Keller-box Method to Boundary Layer Problems· · · · · · · · · · · · · 4.1 Flow of a power-law fluid over a stretching sheet · · · 4.1.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · 4.1.2 Formulation of the problem · · · · · · · · · · · · · 4.1.3 Numerical solution method · · · · · · · · · · · · · · 4.1.4 Results and discussion · · · · · · · · · · · · · · · · · 4.1.5 Concluding remarks · · · · · · · · · · · · · · · · · · · 4.2 Hydromagnetic flow of a power-law fluid over a stretching sheet · · · · · · · · · · · · · · · · · · · · · · · · · · · 4.2.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · 4.2.2 Flow analysis · · · · · · · · · · · · · · · · · · · · · · · · 4.2.3 Numerical solution method · · · · · · · · · · · · · · 4.2.4 Results and discussion · · · · · · · · · · · · · · · · · 4.3 MHD Power-law fluid flow and heat transfer over a non-isothermal stretching sheet · · · · · · · · · · · · · · · · 4.3.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · 4.3.2 Governing equations and similarity analysis · · 4.3.3 Heat transfer · · · · · · · · · · · · · · · · · · · · · · · · 4.3.4 Numerical procedure · · · · · · · · · · · · · · · · · · 4.3.5 Results and discussion · · · · · · · · · · · · · · · · ·
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Contents
4.4 MHD flow and heat transfer of a Maxwell fluid over a non-isothermal stretching sheet · · · · · · · · · · · · · · · · · · 4.4.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · 4.4.2 Mathematical formulation· · · · · · · · · · · · · · · · · 4.4.3 Heat transfer analysis· · · · · · · · · · · · · · · · · · · · 4.4.4 Numerical procedure · · · · · · · · · · · · · · · · · · · · 4.4.5 Results and discussion · · · · · · · · · · · · · · · · · · · 4.4.6 Conclusions · · · · · · · · · · · · · · · · · · · · · · · · · · · 4.5 MHD boundary layer flow of a micropolar fluid past a wedge with constant wall heat flux · · · · · · · · · · · · · · · 4.5.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · 4.5.2 Flow analysis · · · · · · · · · · · · · · · · · · · · · · · · · · 4.5.3 Flat plate problem · · · · · · · · · · · · · · · · · · · · · · 4.5.4 Results and discussion · · · · · · · · · · · · · · · · · · · 4.5.5 Conclusions · · · · · · · · · · · · · · · · · · · · · · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
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Chapter 5
Application of the Keller-box Method to Fluid Flow and Heat Transfer Problems · · · · · · · 5.1 Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.1.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.1.2 Mathematical formulation· · · · · · · · · · · · · · · · · · · · · 5.1.3 Solution of the problem · · · · · · · · · · · · · · · · · · · · · · 5.1.4 Results and discussion · · · · · · · · · · · · · · · · · · · · · · · 5.1.5 Conclusions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.2 Convection flow and heat transfer of a Maxwell fluid over a non-isothermal surface · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.2.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.2.2 Mathematical formulation· · · · · · · · · · · · · · · · · · · · · 5.2.3 Skin friction· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.2.4 Nusselt number · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.2.5 Results and discussion · · · · · · · · · · · · · · · · · · · · · · · 5.2.6 Conclusion· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.3 The effects of variable fluid properties on the hydromagnetic flow and heat transfer over a nonlinearly stretching sheet · · · 5.3.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.3.2 Mathematical formulation· · · · · · · · · · · · · · · · · · · · · 5.3.3 Numerical procedure · · · · · · · · · · · · · · · · · · · · · · · ·
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Contents
5.3.4 Results and discussion · · · · · · · · · · · · · · · · · · · · · · · · 5.3.5 Conclusions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.4 Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid over a vertical stretching sheet· · · · · · · · · · · · 5.4.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.4.2 Mathematical formulation· · · · · · · · · · · · · · · · · · · · · · 5.4.3 Numerical procedure · · · · · · · · · · · · · · · · · · · · · · · · · 5.4.4 Results and discussion · · · · · · · · · · · · · · · · · · · · · · · · 5.5 The effects of linear/nonlinear convection on the non-Darcian flow and heat transfer along a permeable vertical surface · · · · 5.5.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.5.2 Mathematical formulation· · · · · · · · · · · · · · · · · · · · · · 5.5.3 Numerical procedure · · · · · · · · · · · · · · · · · · · · · · · · · 5.5.4 Results and discussion · · · · · · · · · · · · · · · · · · · · · · · · 5.6 Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid over a stretching surface · · · · · · · · · · · · · · · · · · 5.6.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.6.2 Mathematical formulation· · · · · · · · · · · · · · · · · · · · · · 5.6.3 Numerical procedure · · · · · · · · · · · · · · · · · · · · · · · · · 5.6.4 Results and discussion · · · · · · · · · · · · · · · · · · · · · · · · 5.6.5 Conclusions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Application of the Keller-box Method to More Advanced Problems · · · · · · · · · · · · · · · 6.1 Heat transfer phenomena in a moving nanofluid over a horizontal surface · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.1.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · 6.1.2 Mathematical formulation· · · · · · · · · · · · · · · · · 6.1.3 Similarity equations · · · · · · · · · · · · · · · · · · · · · 6.1.4 Numerical procedure · · · · · · · · · · · · · · · · · · · · 6.1.5 Results and discussion · · · · · · · · · · · · · · · · · · · 6.1.6 Conclusion· · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.2 Hydromagnetic fluid flow and heat transfer at a stretching sheet with fluid-particle suspension and
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Chapter 6
variable fluid properties · · · · · · · 6.2.1 Introduction · · · · · · · · · · 6.2.2 Mathematical formulation· 6.2.3 Solution for special cases ·
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Contents
6.2.4 Analytical solution by perturbation · · · · · · · · · · · · · · · 6.2.5 Numerical procedure · · · · · · · · · · · · · · · · · · · · · · · · · 6.2.6 Results and discussion · · · · · · · · · · · · · · · · · · · · · · · · 6.2.7 Conclusions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.3 Radiation effects on mixed convection over a wedge embedded in a porous medium filled with a nanofluid· · · · · · · 6.3.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.3.2 Problem formulation · · · · · · · · · · · · · · · · · · · · · · · · · 6.3.3 Numerical method and validation · · · · · · · · · · · · · · · · 6.3.4 Results and discussion · · · · · · · · · · · · · · · · · · · · · · · · 6.3.5 Conclusion· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.4 MHD mixed convection flow over a permeable non-isothermal wedge · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.4.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.4.2 Mathematical formulation· · · · · · · · · · · · · · · · · · · · · · 6.4.3 Numerical procedure · · · · · · · · · · · · · · · · · · · · · · · · · 6.4.4 Results and discussion · · · · · · · · · · · · · · · · · · · · · · · · 6.4.5 Concluding remarks · · · · · · · · · · · · · · · · · · · · · · · · · · 6.5 Mixed convection boundary layer flow about a solid sphere with Newtonian heating · · · · · · · · · · · · · · · · · · · · · · · 6.5.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.5.2 Mathematical formulation· · · · · · · · · · · · · · · · · · · · · · 6.5.3 Solution procedure · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.5.4 Results and discussion · · · · · · · · · · · · · · · · · · · · · · · · 6.5.5 Conclusions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.6 Flow and heat transfer of a viscoelastic fluid over a flat plate with a magnetic field and a pressure gradient · · · · · · · · 6.6.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.6.2 Governing equations · · · · · · · · · · · · · · · · · · · · · · · · · · 6.6.3 Results and discussion · · · · · · · · · · · · · · · · · · · · · · · · 6.6.4 Conclusions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
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Subject Index · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 391 Author Index· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 395
Chapter 0 Introduction
Nature is in essence nonlinear. Many fundamental laws in science and engineering are modeled by nonlinear differential equations. The origin of nonlinear differential equations is very old, but it has been undergone remarkable new developments in the field of nonlinear equations for last few decades. One of the main impulses, among others for developing nonlinear differential equations has been the study of boundary layer equations. At high Reynolds number, the effect of viscosity is confined to a layer near the wall where the velocity changes are very large. Prandtl was the pioneer in developing a theory by employing what are now called the boundary layer assumptions. Mathematical analyses based on these assumptions of many physical problems in fluid mechanics agree well with the experimental observations. These equations are derived from the Navier-Stokes equations which describe the behavior of the fluid using boundary layer approximation. Using similarity transformations, these equations can be converted into nonlinear coupled ordinary differential equations (ODEs) or partial differential equations (PDEs). Their solution structure demands sophisticated analytical or numerical schemes. The analysis of flow and heat transfer over an infinite range occurs in many branches of science and technology. The fluid velocity satisfies higher order nonlinear differential equations depending on the stress-strain relation. In some instances one is able to obtain exact analytical solutions. When exact or analytical solutions are obtained, one is often faced with difficulty generalizing such results to other nonlinear differential equations. In many situations one is compelled to develop a good numerical scheme, fast as well as accurate, in order to obtain approximate solution to these coupled equations. Obtaining such numerical schemes to solve these coupled ODEs/PDEs for all prevailing physical parameters is the key point of this book. Due to the difficulties of the problems, we frequently seek to obtain numerical solutions to a nonlinear problem, valid over some restricted region in the domain of the original problem. One such technique, which has shown a great promise over the past few years, is the Keller-box method. By use of the box method, numerous nonlinear differential equations have been studied
2
Chapter 0
Introduction
in great detail. Like many other finite difference methods, the box method is very useful as it allows us to obtain numerical solutions to systems of nonlinear differential equations. The finite difference method is unique among other numerical techniques as it allows us to effectively control the rate of convergence via an initial approximation and then proceeds as follows: • • • • •
reducing them to a system of first order equations; writing the difference equations using central differences; linearization of the algebraic equations by Newton’s method; writing them in matrix form; and finally solving the system by block tridiagonal elimination technique.
However, such great freedom comes with the dilemma of deciding just how to proceed. There have been a number of nonlinear differential equations to which the box method has been applied. However the selection of initial approximation varies greatly for different values of the non-dimensional parameters. That said, there are some underlying themes that become apparent when one examines the literature on the topic. Building on such themes, we hope to add some structure and formality to the application of nonlinear flow phenomena. In particular, we discuss several features of the method and the choices one can make in the initial approximation, far field conditions, and the convergence criteria. We hope that the book helps in achieving this long range goal. We present a number of ways in which one may select the initial conditions, far field boundary conditions, and the convergence criteria while solving a nonlinear differential equation by the finite difference method. We also focus our attention on the properties of solutions resulting from such a choice of the initial approximation, far field conditions, and the convergence criteria. These choices play a large role in the computational efficiency. We primarily discuss nonlinear ordinary differential equations associated with finite differences. However, such discussion is usually general enough to use for solving nonlinear partial differential equations as well as ordinary differential equations. We discuss many cases in general while still maintaining applicability of the results to actually computing solutions via the finite difference method. As frequent users of the method, we understand the importance of implementing the presented method. We note that a good companion to this book will be that of Cebeci and Bradshaw [1] which gives physical and computational aspects of convective heat transfer, and some guidelines to solve the boundary value problems. The first half of the book presents the finite difference method and how to implement the box method [2]–[4]. The outline of the book will be as follows: In Chapters 1–3, which comprise Part I of the book, we sketch the Kellerbox method. This first set of chapters serves as a summary to the method which can be directly employed by researchers in engineering, applied physics, and other applied sciences. We keep the discussion general enough so as to
References
3
provide a framework for researchers. In order to give the reader the best preparation for using the method, we realize that often the best way to convey information is through worked out examples. In Part II of the book, Chapters 4–6, we shift to examples by considering problems in fluid mechanics and heat transfer governed by nonlinear differential equations [5]–[10]. Such examples will benefit the reader in applying the methods to actual problems of physical relevance. We group such problem into three categories: general fluid flow and heat transfer problems (Chapter 4), coupled nonlinear problems (Chapter 5), and more advanced problems (Chapter 6).
References [1] [2] [3]
[4]
[5] [6] [7] [8] [9] [10]
T. Cebeci, P. Bradshaw, Physical and computational aspects of convective heat transfer, Springer-Verlag, New York, 1984. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980. T. Cebeci, K.C. Chang, P. Bradshaw, Solution of a hyperbolic system of turbulencemodel equations by the “box” scheme, Comput. Meth. Appl. Mech. Eng. 22 (1980) 213. H.B. Keller, A new difference scheme for parabolic problems. In: Numerical solutions of partial differential equations, II (B. Hubbard, Ed.), 327–350, Academic Press, New York, 1971. L. Fox, Numerical Solution of Two-point Boundary Value Problems in Ordinary Differential Equations, Clarendon Press, Oxford, 1957. H.I. Andersson, B.S. Dandapat, Flows of a power law fluid over a stretching sheet, Stability Appl. Anal. Continuous Media 1 (1992) 339. W.H. Press, S.A. Teukolsky, W.T. Vellering, B.P. Flannery, Numerical Recipes in Fortran, Cambridge University Press, Cambridge, 1993. S.D. Conte, C. de Boor, Elementary Numerical Analysis, McGraw-Hill, New York, 1972. T.Y. Na, Computational Methods in Engineering Boundary Value Problems, Academic Press, New York, 1979. S. Nakamura, Iterative finite difference schemes for similar and non-similar boundary layer equations, Advances in Engineering Software 21 (1994) 123.
Chapter 1 Basics of the Finite Difference Approximations
In this chapter we introduce the essentials of finite difference approximations for solving linear/nonlinear differential equations. In Section 1.2, we study the time-dependent differential equations beginning with the initial value problem and then present some theoretical issues pertaining to the equations. Oftentimes when dealing with nonlinear differential equations, the questions of the existence and uniqueness of a solution are of importance, and inveterate. Section 1.3, deals with the discretization of the differential equations and the solution processes for the coupled boundary value problems (BVPs). Further, it also explains the differences between the single-step and multistep methods of solving BVPs. In Section 1.4, we extend the ideas of the prior section to obtain the numerical solutions for partial differential equations (PDEs). Finally, in Section 1.5 we present the numerical solutions to the PDEs, viz., elliptic, parabolic and hyperbolic differential equations.
1.1 Finite difference approximations Our goal is to find approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to this function) which satisfies a given relationship between its derivatives on some given region of space and/or time, along with some boundary conditions at the edges of the domain. In general this is a difficult problem and only rarely can an analytic formula be found for the solution. A finite difference method proceeds by replacing the derivatives in the differential equations by finite difference approximations. This gives a large algebraic system of equations to be solved in place of the differential equation, something that is easily solvable on a computer. Before tackling this problem, we first consider the more basic question of how we can approximate the derivatives of a known function by finite difference formulas based only on values of the function itself at discrete points.
6
Chapter 1
Basics of the Finite Difference Approximations
Besides providing a basis for the later development of finite difference methods for solving differential equations, this allows us to investigate several key concepts such as the order of accuracy of an approximation in the simplest possible setting. Let u(x) represent a function of one variable that, unless otherwise stated, will always be assumed to be smooth, meaning that we can differentiate the function several times and each derivative is a well-defined bounded function over an interval containing a particular point of interest x ¯. Suppose we want to approximate u� (¯ x) by a finite difference approximation based only on values of u at a finite number of points near x ¯. One obvious choice would be to use x) ≡ D+ u(¯
u(¯ x + h) − u(¯ x) h
(1.1)
for some small value of h. This is motivated by the standard definition of the derivative as the limiting value of this expression as h → 0. Note that D+ u(¯ x) is the slope of the line interpolating u at the points x ¯ and x ¯ + h. The expression in (1.1) is a one-sided approximation to u� since u is evaluated only at values of x x ¯. Another one-sided approximation would be D− u(¯ x) ≡
u(¯ x − h) − u(¯ x) . h
(1.2)
x), Each of these formulas gives a first order accurate approximation to u� (¯ meaning that the size of the error is roughly proportional to h itself. Another possibility is to use the centered approximation x) ≡ D0 u(¯
u(¯ x + h) − u(¯ x − h) 1 x) + D− u(¯ x)). = (D+ u(¯ 2h 2
(1.3)
This is the slope of the line interpolating u at x ¯ − h and x¯ + h and is simply the average of the two one-sided approximations defined above. From Figure 1.1 it should be clear that we would expect D0 u(¯ x) to give a better approximation than either of the one-sided approximations. In fact this gives a second order accurate approximation, the error is proportional to h2 and
Fig. 1.1 Various approximations to u (¯ x), interpreted as slopes of secant lines
1.1
Finite difference approximations
7
hence is much smaller than the error in a first order approximation when h is small. Other approximations are also possible, for example x) ≡ D3 u(¯
1 (2u(¯ x + h) + 3u(¯ x) − 6u(¯ x − h) + u(¯ x − 2h)). 6h
(1.4)
It may not be clear where this came from or why it should approximate u� at all, but in fact it turns out to be a third order accurate approximation, the error is proportional to h3 when h is small. Our first goal is to develop systematic ways to derive such formulas and to analyze their accuracy and relative worth. First we will look at a typical example of how the errors in these formulas compare. Example 1.1. Let u(x) = sin x, x¯ = 1, so we are trying to approximate u� (1) = cos 1 = 0.5403023. Table 1.1 shows the error Du(¯ x) − u� (¯ x) for various values of h for each of the formulas above. Table 1.1
Errors in various finite difference approximations to u� (¯ x)
h
D+
D0
D3
1.0000e–01
–4.2939e–02
4.1138e–02
D−
–9.0005e–04
6.8207e–05
5.0000e–02
–2.1257e–02
2.0807e–02
–2.2510e–04
8.6491e–06
1.0000e–02
–4.2163e–03
4.1983e–03
–9.0050e–06
6.9941e–08
5.0000e–03
–2.1059e–03
2.1014e–03
–2.2513e–06
8.7540e–09
1.0000e–03
–4.2083e–04
4.2065e–04
–9.0050e–08
6.9979e–11
We see that D+ (u) and D− (u) behave similarly though one exhibits an error that is roughly the negative of the other. This is reasonable from Figure 1.1 and explains why D0 (u) the average of the two, has an error that is much smaller than either. We see that x) − u� (¯ x) ≈ −0.42h, D+ u(¯ � D0 u(¯ x) − u (¯ x) ≈ −0.09h2,
x) − u� (¯ x) ≈ 0.007h3, D3 u(¯
confirming that these methods are first order, second order, and third order, respectively. Figure 1.2 shows these errors plotted against h on a log-log scale. This is a good way to plot errors when we expect them to behave like some power of h, since if the error E(h) behaves like E(h) ≈ Chp then log |E(h)| ≈ log |C| + p log h. So on a log-log scale the error behaves linearly with a slope that is equal to p, the order of accuracy. Truncation errors The standard approach to analyzing the error in a finite difference approximation is to expand each of the function values of u in a Taylor series about
8
Chapter 1
Basics of the Finite Difference Approximations
Fig. 1.2 The errors in Table 1.1 are plotted against h on a log-log scale
the point x ¯, e.g., u(¯ x + h) = u(¯ x) + hu� (¯ x) +
h2 �� h3 u (¯ x) + u��� (¯ x) + O(h4 ), 2! 3!
(1.5a)
h2 �� h3 x) − u��� (¯ x) + O(h4 ). (1.5b) u (¯ 2! 3! These expansions are valid provided that u is sufficiently smooth. Using (1.5a), we can compute that x) + u(¯ x − h) = u(¯ x) − hu� (¯
D+ u(¯ x) ≡
u(¯ x + h) − u(¯ x) h h2 = u� (¯ x) + u�� (¯ x) + u��� (¯ x) + O(h3 ). h 2! 3!
x), u��� (¯ x), etc. are fixed constants Recall that x ¯ is a fixed point so that u�� (¯ independent of h. They depend on u of course, but the function is also fixed as we vary h. For h sufficiently small, the error will be dominated by the first term 0.5hu�� (¯ x) and all the other terms will be negligible compared to this term; we expect the error to behave roughly like a constant times h, where the constant has the value 0.5u�� (¯ x). Note that in Example 1.1, where u(x) = sin x, we have 0.5u�� (1) = −0.4207355 which agrees with the value seen in Table 1.1. Similarly, from (1.5b) we can compute that the error in x) as D− u(¯ D− u(¯ x) − u� (¯ x) = −
h �� h2 u (¯ x) + u��� (¯ x) + O(h3 ) 2! 3!
which also agrees with our expectations. Combining (1.5a) and (1.5b) we get u(¯ x + h) − u(¯ x − h) = 2hu� (¯ x) + so that x) − u� (¯ x) = D0 u(¯
h3 ��� u (¯ x) + O(h5 ), 3
h2 ��� u (¯ x) + O(h4 ). 6
(1.6)
1.1
Finite difference approximations
9
This confirms the second order accuracy of this approximation and again agrees with what is seen in Table 1.1, since in the context of Example 1.1 we have 1 1 u (¯ x) = − cos 1 = −0.09005038. 6 6 Note that all of the odd order terms drop out of the Taylor series expansion x). This is typical with centered approximations and usually (1.6) for D0 u(¯ leads to a higher order accuracy. In order to analyze D3 (u) we need to also expand u(¯ x − 2h) as u(¯ x − 2h) = u(¯ x) − 2hu (¯ x) +
(2h)2 (2h)3 u (¯ u (¯ x) − x) + O(h4 ). 2! 3!
(1.7)
Combining this with (1.5a) and (1.5b) we get D3 u(¯ x) = u (¯ x) +
1 3 h u (¯ x) + O(h4 ). 12
(1.8)
Deriving finite difference approximations x) based Suppose we want to derive a finite difference approximation to u (¯ on some given set of points: then we can use Taylor series to derive an appropriate formula, using the method of undetermined coefficients. x), based Example 1.2. Suppose we want a one-sided approximation to u (¯ on u(¯ x), u(¯ x − h) and u(¯ x − 2h), of the form D2 u(¯ x) = au(¯ x) + bu(¯ x − h) + cu(¯ x − 2h).
(1.9)
Then we can determine the coefficients a, b, and c to give the best possible accuracy by expanding in Taylor series and collecting terms. Using (1.5b) and (1.7) in (1.9) we get x) = (a + b + c)u(¯ x) − (b + 2c)hu (¯ x) D2 u(¯ 1 1 2 + (b + 4c)h u (¯ x) − (b + 8c)h3 u (¯ x) + · · · . 2 6 x) to high order then we need If this is going to agree with u (¯ a + b + c = 0,
b + 2c = −1/h,
b + 4c = 0.
(1.10)
We might like to require that higher order coefficients be zero as well, but since there are only three unknowns a, b and c we cannot in general hope to satisfy more than three such conditions. Solving the linear system (1.10) gives a = 3/(2h), b = −2/h, c = 1/(2h) so that the formula is D2 u(¯ x) =
1 (3u(¯ x) − 4u(¯ x − h) + u(¯ x − 2h)). 2h
(1.11)
The error in this approximation is clearly 1 1 2 h u (¯ x) − u (¯ x) = − (b + 8c)h3 u (¯ x) + · · · = x) + O(h3 ). D2 u(¯ 6 12
10
Chapter 1
Basics of the Finite Difference Approximations
Polynomial interpolation There are other ways to derive the same finite difference approximations. One way is to approximate the function u(x) by some polynomial p(x) and x) as an approximation to u� (¯ x). If we determine the polynomial then use p� (¯ by interpolating u at an appropriate set of points, then we obtain the same approximation formulas as in the finite difference methods above. Example 1.3. To derive the approximation formula of Example 1.2 in this way, let p(x) be the quadratic polynomial that interpolates u at x ¯, x ¯ − h and x ¯ − 2h and then compute p� (¯ x). The result is exactly (1.11). Second order derivatives Approximations to the second derivative u�� (¯ x) can be obtained in an analogous manner. The standard second order centered approximation is given by x) = D2 u(¯
1 1 (u(¯ x − h) − 2u(¯ x) + u(¯ x + h)) = u�� (¯ x) + h2 u�� (¯ x) + O(h4 ). 2 h 2
Again, since this is a symmetric centered approximation, all of the odd order terms drop out. This approximation can also be obtained by the method of undetermined coefficients, or alternatively by computing the second derivative of the quadratic polynomial interpolating u(x) at x ¯ − h, x ¯ and x ¯ + h. Another way to derive approximations to higher order derivatives is by repeatedly applying first order differences. Just as the second derivative is the derivative of u� , we can view D2 u(¯ x) as being a difference of first differences. In fact, x) = D+ D− u(¯ x). D2 u(¯ Hence 1 (D− u(¯ x + h) − D− u(¯ x)) h x + h) − u(¯ x) u(¯ x) − u(¯ x − h) 1 u(¯ − = D2 u(¯ x). = h h h
D+ D− u(¯ x) =
x) = D+D− u(¯ x) or we can also view it as a centered Alternatively, D2 u(¯ difference of centered differences, if we use a step size h/2 in each centered approximation to the first derivative. If we define 0 u(x) = 1 u x + h − u x − h D h 2 2 then we get that
1 0 (D 0 u(¯ D x)) = h
u(¯ x + h) − u(¯ x) u(¯ x) − u(¯ x − h) − h h
= D2 u(¯ x).
1.2
The initial value problem for ODEs
11
Higher order derivatives Finite difference approximations to higher order derivatives can also be obtained using any of the approaches outlined above. Repeatedly differencing approximations to lower order derivatives is a particularly simple way. Example 1.4. As an example, here are two different approximations to x). The first one is uncentered and with first order accuracy: u��� (¯ 1 (u(¯ x + 2h) − 3u(¯ x + h) + 3u(¯ x) − u(¯ x − h)) h h = u��� (¯ x) + u��� (¯ x) + O(h2 ). 2
D+ D 2 u(¯ x) =
The next approximation is centered and with second order accuracy: 1 (u(¯ x + 2h) − 2u(¯ x + h) + 2u(¯ x − h) − u(¯ x − 2h)) 2h3 2 h = u��� (¯ x) + u��� (¯ x) + O(h4 ). 4
x) = D0 D+ D− u(¯
Finite difference approximations derived above are the basic building blocks of finite difference methods for solving differential equations.
1.2 The initial value problem for ODEs In this section, we begin a study of time-dependent differential equations, beginning with the initial value problem (IVP) for a time-dependent ODE. Standard introductory texts are Lambert [1] and Gear [2]. Henrici [3] gives a more complete description of some theoretical issues, although stiff equations are not discussed. Hairer et al. [4]–[5] give a more recent and complete survey of the field. The initial value problem is of the form
with some initial data
u� (t) = f (u(t), t), for t > t0
(1.12)
u(t0 ) = η.
(1.13)
We will often assume t0 = 0 for simplicity. In general (1.12) may represent a system of ODEs, i.e., u may be a vector with s components u1 , u2 , . . . , us and then f (u, t) represents a vector with components f1 (u, t), f2 (u, t), . . . , fs (u, t) each of which can be a nonlinear function of all the components of u. The problem is linear if f (u, t) = A(t)u + B(t)
(1.14)
where A(t) ∈ �s×s and B(t) ∈ �s . We will consider only the first order equation (1.12) but in fact this is more general than it appears since we can reduce higher order equations to a system of first order equations.
12
Chapter 1
Basics of the Finite Difference Approximations
Example 1.5. Consider the initial value problem u��� (t) = u� (t)u(t) − 2t(u�� (t))2 for t > 0. This third order equation requires three initial conditions, typically specified as u(0) = η1 , (1.15)
u� (0) = η2 , u (0) = η3 . ��
We can rewrite this as a system of the form (1.12) and (1.13) by introducing the variables u1 (t) = u(t), u2 (t) = u� (t), u3 (t) = u�� (t). Then the above differential equation takes the form u�3 (t) = u1 (t)u2 (t) − 2tu23 (t). The initial condition is simply (1.14) where the three components of η come from (1.15). More generally, any single equation of order m can be reduced to m first order equations by defining uj (t) = u(j−1) (t), and an mth order system of s equations can be reduced to a system of sm first order equations. It is also sometimes useful to note that any explicit dependence of f on t can be eliminated by introducing a new variable that is simply equal to t. In the above example we could define u4 (t) = t so that u�4 (t) = 1 and u4 (t0 ) = t0 . The system then takes the form u� (t) = f (u(t)) with
⎡
u2
⎤
(1.16) ⎡
η1
⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ u3 ⎢ ⎥ and u(t0 ) = ⎢ η2 ⎥. f (u) = ⎢ ⎢η ⎥ 2⎥ ⎣ u1 u2 − 2u4 u3 ⎦ ⎣ 3⎦ 1 t0
Equation (1.16) is said to be autonomous since it does not depend explicitly on time t. It is often convenient to assume f is of this form since it simplifies notation. We will always assume that f is Lipschitz continuous in u as described in the next section, which implies that the initial value problem has a unique solution over some time interval. Lipschitz continuity The standard theory for the existence of a solution to the initial value problem u� (t) = f (u, t),
u(0) = η
(1.17)
1.2
The initial value problem for ODEs
13
is discussed in many text books, e.g., Coddington and Levinson [6]. To guarantee that there is a unique solution, it is necessary to require a certain amount of smoothness in the function f (u, t) of (1.17). We say that the function f (u, t) is Lipschitz continuous in u over some range of t and u if there exist some constant L > 0 such that |f (u, t) − f (u∗ , t)| L|u − u∗ |
(1.18)
for all u and u∗ in this range. This is slightly stronger than mere continuity, which only requires that |f (u, t) − f (u∗ , t)| → 0 as u → u∗ . Lipschitz continuity requires that |f (u, t) − f (u∗ , t)| = O(|u − u∗ |) as u → u∗ . The function is uniformly Lipschitz continuous over some time period if there is a single constant L that works for all u and u∗ and all t in this period. If f (u, t) is ∂f is bounded, then differentiable with respect to u and this derivative fu = ∂u we can take L = max |fu (u, t)|, since f (u, t) = f (u∗ , t) + fu (v, t)(u − u∗ ) for some value v between u and u∗ . Example 1.6. For the linear problem u� (t) = λu(t) + g(t), f � (u) = λ and we can take L = |λ|. This problem of course has a unique solution for any initial data η given by t λ(t−t0 ) u(t) = e η+ eλ(t−τ ) g(τ )dτ. (1.19) t0
In particular, if λ = 0 then L = 0. In this case f (t, u) = g(t) is independent of u. The solution is then obtained by simply integrating the function g(t), t g(τ )dτ . (1.20) u(t) = η + t0
Existence and uniqueness results The basic existence and uniqueness theorem states that if f is uniformly Lipschitz continuous over some time period 0 t T then there is a unique solution to the initial value problem (1.17) from any initial value η. If f is Lipschitz but not uniformly so, then there will be a unique solution through any value η over some finite time interval, but this solution may not exist past some time, as the next example shows. Example 1.7. Consider the initial value problem u� (t) = u2 (t) with initial conditions u(0) = η > 0. The function f (u) = u2 is Lipschitz continuous over any finite interval [η − a, η + a] with L = 2(η + a). From this it can be shown that the initial value problem has a unique solution over some time interval 0 t T with T > 0. However, since f is not uniformly Lipschitz for all u (i.e., there is not a single value of L that works for all u), we cannot prove that a solution exists for all time, and in fact it does not. The solution to the 1 1 initial value problem is u(t) = −1 and so u(t) → ∞ as t → . If the η −t η
14
Chapter 1
Basics of the Finite Difference Approximations
function f is not Lipschitz continuous at some point then the initial value problem may fail to have a unique solution over any time interval. √ Example 1.8. Consider the initial value problem u� (t) = t with initial √ conditions u(0) = 0. The function f (u) = u is not Lipschitz continuous √ near u = 0 since f � (u) = 1/2 u → ∞ as u → 0. We cannot find a constant L so that the bound (1.18) holds for all u and u� near 0. As a result, this initial value problem does not have a unique solution. In fact it has two distinct solutions u(t) = 0 and u(t) = (1/4)t2 . Systems of equations For systems of s > 1 ordinary differential equations, u(t) ∈ �s and f (u, t) is a function mapping �s ×� → �s . We say the function f is Lipschitz continuous in some norm � · � if there is a constant L such that �f (u, t) − f (u∗ , t)� L�u − u∗ �
(1.21)
for all u in a neighborhood of u∗ . By the equivalence of finite-dimensional norms, if f is Lipschitz continuous in one norm then it is Lipschitz continuous in any other norm, though the Lipschitz constant may depend on the norm chosen. The theorems on existence and uniqueness carry over to systems of equation. Significance of the Lipschitz constant The Lipschitz constant measures how much f (u, t) changes if we perturb u (at some fixed time t). Since f (u, t) = u� (t), the slope of the line tangent to the solution curve through the value u, this indicates how the slope of the solution curve will vary if we perturb u. The significance of this is best seen through some examples. Example 1.9. Consider the trivial equation u� (t) = g(t) which has Lipschitz constant L = 0. Several solution curves are sketched in Figure 1.3. Note
Fig. 1.3 Solution curves for Example 1.9 where L = 0
1.2
The initial value problem for ODEs
15
that all of these curves are “parallel”; they are simply shifted depending on the initial data. Tangent lines to the curves at any particular time are all parallel since f (u, t) = g(t) is independent of u. Example 1.10. Consider u (t) = λu(t) with λ constant and L = |λ|. Then u(t) = u(0) exp(λt). Two situations are shown in Figure 1.4 for negative and positive values of λ. Here the slope of the solution curve does vary depending on u. The variation in the slope with u (at fixed t) gives an indication of
Fig. 1.4 Solution curves for Example 1.10 with (a) λ = −3, (b) λ = 3
16
Chapter 1
Basics of the Finite Difference Approximations
how rapidly the solution curves are converging towards one another (in the case λ < 0) or diverging away from one another (in the case λ > 0). If the magnitude of λ is increased, the convergence or divergence would clearly be more rapid. The size of the Lipschitz constant is significant if we intend to solve the problem numerically since our numerical approximation will almost certainly produce a value U n at time tn that is not exactly equal to the true value u(tn ). Hence we are on a different solution curve than the true solution. The best we can hope for in the future is that we stay close to the solution curve that we are now on. The size of the Lipschitz constant gives an indication of whether solution curves that start close together can be expected to stay close together or to diverge rapidly.
1.3 Some basic numerical methods We begin by listing a few standard approaches to discretizing Eq. (1.12). Note that the IVP differs from the BVP considered before in that we are given all the data at the initial time t0 = 0 and from this we should be able to march forward in time, computing approximations at successive times t1 , t2 , . . . , tn . We will use k to denote the time step, so tn = nk for n 0. It is convenient to use the symbol k that is different from the spatial grid size h since we will soon study PDEs which involve both spatial and temporal discretizations. Often the symbols Δt and Δx are used. Suppose we are given initial data U0 = η
(1.22)
and want to compute approximations U 1 , U 2 , . . . satisfying U n ≈ u(tn ). We will use superscripts to denote the time step index, again anticipating the notation of PDEs where we will use subscripts for spatial indices. The simplest method is Euler’s method (also called Forward Euler), based on replacing u (tn ) by U n+1 − U n D+ U n = . k This implies U n+1 − U n (1.23) = f (U n ), n = 0, 1, 2, 3, . . . . k Rather than viewing this as a system of simultaneous equations as we did for the boundary value problem, it is possible to solve this explicitly for U n+1 in terms of U n : U n+1 = U n + kf (U n ). (1.24) From the initial data U 0 we can compute U 1 then U 2 and so on. This is called a time-marching method. The Backward Euler method is similar, but
1.3
Some basic numerical methods
17
is based on replacing u� (tn+1 ) by D− U n+1 . That is U n+1 − U n = f (U n+1 ) or U n+1 = U n + kf (U n+1 ). k
(1.25)
Again we can march forward in time since computing U n+1 only requires that we know the previous value U n . In the Backward Euler method, however, (1.25) is an equation that must be solved for U n+1 and in general f (u) is a nonlinear function. We can view this as looking for a zero of the function g(u) = u − kf (u) − U n which can be approximated using some iterative method such as Newton’s method. Because the Backward Euler method gives an equation that must be solved for U n+1 , it is called an implicit method, whereas the Forward Euler method (1.24) is an explicit method. Another implicit method is the Trapezoidal method, obtained by averaging the two Euler methods: U n+1 − U n 1 = (f (U n ) + f (U n+1 )). k 2
(1.26)
As one might expect, this symmetric approximation is second order accurate whereas the Euler methods are only first order accurate. The above methods are all one-step methods, meaning that U n+1 is determined from U n alone, and previous values of U are not needed. One way to get higher order accuracy is to use a multi-step method that involves other previous values. For example, using the approximation 1 u(t + k) − u(t − k) = u� (t) + k 2 u��� (t) + O(k 3 ) 2k 6 yields the Midpoint method (also called the Leapfrog method): U n+1 − U n−1 = f (U n ), 2k
(1.27)
U n+1 = U n−1 + 2kf (U n),
(1.28)
or which is a second order accurate explicit two-step method. Truncation errors The truncation error for these methods is defined in the same way as usual. We write the difference equation in a form that directly models the derivatives (e.g., in the form (1.27) rather than (1.28)) and then insert the true solution to the ODE into the difference equation. We then use a Taylor series expansion and cancel out common terms.
18
Chapter 1
Basics of the Finite Difference Approximations
Example 1.11. The local truncation error of the Midpoint method (1.27) is defined by u(tn+1 ) − u(tn−1 ) − f (u(tn )) 2k 1 1 = u� (tn ) + u��� (tn ) + O(k4 ) − u� (tn ) = u��� (tn ) + O(k4 ). (1.29) 6 6
τn =
Note that since u(t) is the true solution of the ODE, u� (tn ) = f (u(tn )). The O(k 3 ) term drops out by symmetry. The truncation error is O(k 2 ) and so we say the method is second order accurate, although it is not yet clear that the global error will have this behavior. As always, we need some form of stability to guarantee that the global error will exhibit the same rate of convergence as the local truncation error. This will be discussed below. One-step errors In much of the literature concerning numerical methods for ordinary differential equations, a slightly different definition of the local truncation error is used that is based on the form (1.28) rather than (1.27). Denoting this value by Ln we have Ln = u(tn+1 ) − u(tn−1 ) − 2kf (u(tn )) =
1 3 ��� k u (tn ) + O(k5 ). 3
Since Ln = 2kτ n this local error is O(k3 ) rather than O(k 2 ), but of course the global error remains the same and will have O(k 2 ). Using this alternative definition, many standard results in ODE theory say that a pth order accurate method should have a local truncation error (LTE) that is O(k p+1 ). With the notation we are using, a pth order accurate method has an LTE that is O(k p ). The notation used here is consistent with the standard practice for PDEs and leads to a more coherent theory, but one should be aware of this possible source of confusion. We prefer to call Ln the one-step error, since this can be viewed as the error that would be introduced in one time step if the past values U n , U n−1 , . . . were all taken to be the exact values from u(t). For example, in the Midpoint method (1.28) suppose that U n = u(tn ) and U n−1 = u(tn−1 ) and we now use these values to compute U n+1 , an approximation to U (tn+1 ). That is, U n+1 = u(tn−1 ) + 2kf (u(tn )) = u(tn−1 ) + 2ku� (tn ). Then the error is u(tn+1 ) − U n+1 = u(tn+1 ) − u(tn−1 ) − 2ku� (tn ) = Ln . From (1.29), we see that in one step the error introduced is O(k 3 ). This is consistent with second order accuracy in the global error if we think of trying
1.3
Some basic numerical methods
19
to compute an approximation to the true solution u(T ) at some fixed time T > 0. In order to compute from time t = 0 up to time T , we need to take T /k time steps of length k. A rough estimate of the error at time T might be obtained by assuming that a new error of size Ln is introduced in the nth time step, and is then simply carried along in later time steps without affecting the size of future local errors and without growing or diminishing itself. Then we would expect the resulting global error at time T to be simply the sum of all these local errors. Since each local error is O(k 3 ) and we are adding up T /k of them, we end up with a global error that is O(k 2 ). This viewpoint is in fact exactly right for the simplest ODE u (t) = f (t) in which f is independent of u and the solution is simply the integral of f . But it is a bit too simplistic for more interesting equations since the error at each time feeds back into the computation at the next step in the case where f depends on u. Nonetheless, it is essentially right in terms of the expected order of accuracy, provided the method is stable. In fact it is useful to think of stability as exactly what is needed to make this naive analysis essentially correct by insuring that the old errors from previous time steps do not grow too rapidly in future time steps. Taylor series methods The Forward Euler method (1.24) can be derived using a Taylor series expansion of u(tn+1 ) about u(tn ). That is, 1 u(tn+1 ) = u(tn ) + u (tn ) + k2 u (tn ) + · · · . 2
(1.30)
If we drop all terms of order k 2 and higher and use the differential equation to replace u (tn ) by f (u(tn ), tn ), we obtain u(tn+1 ) ≈ u(tn ) + kf (u(tn ), tn ). This suggests the method (1.24). The one-step error is O(k 2 ), since we dropped terms of this order. A Taylor series method of higher accuracy can be derived by keeping more terms in the Taylor series. If we keep the first p + 1 terms of the Taylor series expansion 1 1 u(tn+1 ) ≈ u(tn ) + ku (tn ) + k 2 u (tn ) + · · · + k p u(p) (tn ) 2 p! then we obtain a pth -order accurate method. The problem is that we are only given u (t) = f (u(t), t) and we must compute the higher derivatives by repeated differentiation of this function. For example, we can compute u (t) = fu (u(t), t)u (t) + ft (u(t), t) = fu (u(t), t)f (u(t), t) + ft (u(t), t).
(1.31)
This can result in very messy expressions that must be worked out for each equation, and as a result this approach is not often used. However, it is
20
Chapter 1
Basics of the Finite Difference Approximations
such an obvious approach that it is worth mentioning, and in some cases it may be useful. An example should suffice to illustrate the technique and its limitations. Example 1.12. Suppose we want to solve the equation u� (t) = t2 sin(u(t)).
(1.32)
Then we can compute u�� (t) = 2t sin(u(t)) + t2 cos(u(t))u� (t) = 2t sin(u(t)) + t4 cos(u(t)) sin(u(t)). A second order method is given by 1 U n+1 = U n + Kt2 sin(U n ) + k 2 (2tn sin(U n ) + t4n cos(U n ) sin(U n )). 2 Clearly higher order derivatives can be computed and used, but this is cumbersome even for this simple example. For systems of equations the method becomes still more complicated. Runge-Kutta methods Most methods we use in practice do not require that the user explicitly calculate higher order derivatives. Instead a higher order finite difference approximation is designed that typically models these terms automatically. A multi-step method, of the sort we will study can achieve high accuracy by using high order polynomial interpolation through several previous values of the solution and/or its derivative. To achieve the same effect with a one-step method it is typically necessary to use a multi-step method where intermediate values of the solution and its derivative are generated and used within a single time step. Example 1.13. A two-step explicit Runge-Kutta method is given by 1 (1.33) U ∗ = U n + kf (U n ), U n+1 = U n + kf (U ∗ ). 2 In the first step, an intermediate value is generated which approximates U (tn+1/2 ) via Euler’s method. In the second step, the function f is evaluated at this midpoint to estimate the slope over the full time step. Since this now looks like a centered approximation to the derivative, we might hope for second order accuracy, as we’ll now verify by computing the local truncation error. Combining the two steps above, we can rewrite 1 U n+1 = U n + kf U n + kf (U n ) . 2 Viewed this way this is clearly a one-step explicit method. The truncation error is 1 1 (1.34) τ n = (u(tn+1 ) − u(tn )) − f (u(tn ) + kf (u(tn))). k 2
1.3
Some basic numerical methods
21
f u(tn ) + = f u(tn ) +
1 kf (u(tn )) 2 1 � ku (tn ) 2 1 1 = f (u(tn )) + ku� (tn )f � (u(tn )) + k2 (u� (tn ))2 f �� (u(tn )) + · · · . 2 8 Since f (u(tn )) = u� (tn ), on differentiating we get f � (u)u� = u�� , and it can be written as 1 1 f u(tn ) + kf (u(tn )) = u� (tn ) + ku�� (tn ) + O(k2 ). 2 2 Using this in (1.34) gives 1 τ = k n
1 2 �� 1 �� � 3 � 2 ku (tn ) + k u (tn ) + O(k ) − u (tn ) + ku (tn ) + O(k ) 2 2
= O(k 2 ),
and the method is second order accurate. Remark. An easier way to determine the order of accuracy is to apply the method to the special test equation u� = λu which has solution u(tn+1 ) = eλk u(tn ) and determine the error for this problem. Here we obtain U
n+1
1 n n = U + kλ U + kλU 2 1 = U n + (kλ)U n + (kλ)2 U n = ekλ U n + O(k 3 ). 2 n
The one-step error is O(k 3 ) and hence the local truncation error is O(k2 ). Of course we have only checked that the local truncation error is O(k 2 ) on one particular function u(tn ) = eλt not on all smooth solutions. But it can be shown that this is more generally a reliable indication of the order. Example 1.14. The Runge-Kutta method (1.33) can also be applied to nonautonomous equations of the form u� (t) = f (u(t), t), 1 U ∗ = U n + kf (U n , tn ), 2 n+1 n = U + kf (U ∗ , tn + k/2). U
(1.35)
This is again second order accurate, as can be verified by expanding as above, but is slightly more complicated since Taylor series in two variables must be used.
22
Chapter 1
Basics of the Finite Difference Approximations
Example 1.15. One simple higher order Runge-Kutta method is the fourth order four-step method given by 1 1 F0 = f (U n , tn ), F1 = f U n + kF0 , tn + k , 2 2 1 1 F2 = f U n + kF1 , tn + k , F3 = f (U n + kF2 , tn+1 ), (1.36) 2 2 k U n+1 = U n + (F0 + 2F1 + 2F2 + F3 ). 6 This method was particularly popular in the pre-computer era when computations were done by hand because the coefficients are so simple. Today there is no need to keep the coefficients simple and other Runge-Kutta methods have advantages. A general explicit r-step Runge-Kutta method has the form F0 = f (U n , tn ), F1 = f (U n + kb10 F0 , tn + kb10 ), F2 = f (U n + k(b20 F0 + b21 F1 ), tn + k(b20 + b21 )), ············ Fn = f
Un + k
r−1 s=0
brs Fs ,tn + k
r−1 s=0
brs
(1.37) ,
U n+1 = U n + k(c0 F0 + c1 F1 + · · · + cr Fr ). Consistency requires
r
cs = 1 and there are typically many ways that the
s=0
coefficients bjs and cs can be chosen to achieve a given accuracy. Implicit Runge-Kutta methods can also be defined in which each intermediate Fj depends on all the intermediate values F0 , F1 , F2 , . . . , Fr rather than only on F0 , F1 , F2 , . . . , Fj−1 . These are typically expensive to implement, however, and not so often used in practice. One subclass of implicit methods that are simpler to implement are the diagonally implicit Runge-Kutta methods (DIRK methods) in which Fj depends on F0 , F1 , F2 , . . . , Fj but not on Fj+1 , Fj+2 , Fj+3 , . . . , Fr . For a system of m equations, DIRK methods require solving a sequence of r implicit systems, each of size m, rather than a coupled set of mr equations as would be required in a fully implicit Runge-Kutta method.
1.3
Some basic numerical methods
Example 1.16. A second order DIRK method is given by k k , F0 = f (U n , tn ), F1 = f U n + (F0 + F1 ), tn + 4 2 k F2 = f U n + (F0 + F1 + F2 ), tn + k , 3 k U n+1 = U n + (F0 + F1 + F2 ). 3 This method is known as the TR-BDF2 method.
23
(1.38)
One-step vs. multi-step methods Taylor series and Runge-Kutta methods are one-step methods; the approximation U n+1 depends on U n but not on previous values U n−1 , U n−2 , U n−3 , . . .. In the next section we will consider a class of multi-step methods where previous values are also used (one example is the Midpoint method (1.28)). One-step methods have several advantages over multi-step methods: • The methods are self-starting: From the initial data U 0 , the desired method can be applied immediately. Multi-step methods require that some other method be used initially and this will be discussed in the next section. • The time step k can be changed at any point, based on an error estimate. The time step can also be changed with a multi-step method but more care is required since the previous values are assumed to be equally spaced in the standard form of these methods. • If the solution u(t) is not smooth at some isolated point t∗ (for example, because f (u, t) is discontinuous at t∗ ), then with a one-step method it is often possible to get full accuracy simply by ensuring that t∗ is a grid point. With a multi-step method that uses data from both sides of t∗ in updating the solution nearby, a loss of accuracy may occur. On the other hand, one-step methods have some disadvantages. The disadvantage of Taylor series methods is that they require differentiating the given equation and are cumbersome, and often expensive to implement. RungeKutta methods only use evaluations of the function f , but a higher order multi-step method requires evaluating f several times in each time step. For simple equations this may not be a problem, but if function values are expensive to compute then high order Runge-Kutta methods may be quite expensive as well. An alternative is to use a multi-step method in which values of f already computed in previous time steps are reused to obtain higher order accuracy. Typically only one new f evaluation is required in each time step. The popular class of linear multi-step methods is discussed in the next section. Linear multi-step methods All of the methods introduced in above section are members of a class of methods called Linear Multi-step Methods (LMMs). In general, an r-step
24
Chapter 1
Basics of the Finite Difference Approximations
LMM has the form r j=0
αj U n+j = k
r
βj f (U n+j , tn+j ).
(1.39)
j=0
The value U n+r is computed from this equation in terms of the previous values U n+r−1 , U n+r−2 , . . . , U n and f values at these points (which can be stored and reused if f is expensive to evaluate). If βr = 0 then the method (1.39) is explicit, otherwise it is implicit. Note that we can multiply both sides by any constant and have essentially the same method, though the coefficients αj and βj would change. The normalization αr = 1 is often assumed to fix this scale factor. There are special classes of methods of this form that are particularly useful and have distinctive names. These will be written out for the autonomous case where f (t, u) = f (u) to simplify the formulas, but each can be used more generally by replacing f (U n+j ) = f (U n+j , tn+j ) in any of the formulas. Example 1.17. The Adams methods have the form U n+r = U n+r−1 + k
r
βj f (U n+j ).
(1.40)
j=0
These methods all have αr = 1, αr−1 = 1, and αj = 0 for j < r − 1. The βj coefficients are chosen to maximize the order of accuracy. If we require βr = 0 so the method is explicit, then the r coefficients β0 , β1 , β2 , β3 , . . . , βr−1 can be chosen so that the method has order r. This gives the r-step Adams-Bashforth method. The first few steps are given below: 1st -Step: U n+1 = U n + kf (U n ), k 2nd -Step: U n+2 = U n+1 + (−f (U n ) + 3f (U n+1)), 12 k rd n+3 n+2 3 -Step: U =U + (5f (U n ) − 16f (U n+1 ) + 23f (U n+2 )), 12 k th n+4 n+3 4 -Step: U =U + (−9f (U n)+37f (U n+1 )−59f (U n+2 )+55f (U n+3 )). 24 These can be derived by various approaches. If we allow βr to be nonzero, then we have one more free parameter and so we can eliminate an additional term in the local truncation error. This gives an implicit method of order r + 1 called the r-step Adams-Moulton method. The first few steps are given below: k 1st -Step: U n+1 = U n + (f (U n ) + f (U n+1 )), 2 k nd n+2 n+1 =U + (−f (U n ) + 8f (U n+1) + 5f (U n+2 )), 2 -Step: U 12 k rd n+3 n+2 3 -Step: U =U + (f (U n ) − 5f (U n+1) + 19f (U n+2 ) + 9f (U n+3)), 24
1.3
Some basic numerical methods
25
k (−19f (U n ) + 106f (U n+1) − 264f (U n+2)+ 270 646f (U n+3) + 251f (U n+4)).
4th -Step: U n+4 = U n+3 +
Example 1.18. The explicit Nystrom methods have the form U
n+r
=U
n+r−2
+k
r−1 �
βj f (U n+j )
j=0
with the βj chosen to give order r. The Midpoint method (1.27) is a two-step explicit Nystrom method. A two-step implicit Nystrom method is Simpson’s rule, 2k U n+2 = U n + (f (U n ) + 4f (U n+1 ) + f (U n+2 )). 6 This reduces to Simpson’s rule for quadrature if applied to the ODE u� (t) = f (t). Local truncation error For LMMs it is easy to derive a general formula for the local truncation error. We have ⎞ ⎛ r−1 r−1 � 1 ⎝� τ (tn+r ) = αj u(tn+j ) − k βj u� (tn+j )⎠ . k j=0 j=0 We used f (u(tn+j )) = u� (tn+j ), since u(t) is the exact solution of the ODE. Assuming u is smooth and expanding in Taylor series gives 1 u(tn+j ) = u(tn ) + jku� (tn ) + (jk)2 u�� (tn ) + · · · , 2 1 � � �� u (tn+j ) = u (tn ) + jku (tn ) + (jk)2 u��� (tn ) + · · · , 2 and so
⎛ ⎞ ⎞ ⎛ r r � � 1 αj ⎠ u(tn ) + ⎝ (jαj − βj )⎠ u� (tn ) τ (n + r) = ⎝ k j=0 j=0 ⎞ ⎛ � r � � 1 2 +k ⎝ j αj − jβj ⎠ u�� (tn ) 2 j=0 ⎛ ⎞ � r � � 1 1 + · · · + kq−1 ⎝ j q αj − j q−1 βj ⎠ u(q) (tn ). q! (q − 1)! j=0
The method is consistent if τ → 0 as k → 0 which requires that at least the first two terms in this expansion vanish: that is, r � j=0
αj = 0 and
r � j=0
jαj =
r � j=1
βj .
(1.41)
26
Chapter 1
Basics of the Finite Difference Approximations
If the first p + 1 terms vanish then the method will be pth order accurate. Note that these conditions depend only on the coefficients αj and βj of the method and not on the particular differential equation being solved.
1.4 Some basic PDEs In this section we will briefly develop some of the basic PDEs that will be used to illustrate the development of numerical methods. In solving a partial differential equation, we are looking for a function of more than one variable that satisfies some relations between different partial derivatives. Classification of differential equations First we review the classification of differential equations into elliptic, parabolic, and hyperbolic equations. Not all PDE’s fall into one of these classes, by any means, but many important equations that arise in practice do. These classes of equations model different sorts of phenomena, display different behavior, and require different numerical techniques for their solution. Standard texts on partial differential equations such as Kevorkian [7] give further discussion. Second order equations The classification of a linear second order differential equation of the form auxx + buxy + cuyy + dux + euy + f u = g in two independent variables depends on the sign of the discriminant: ⎧ ⎪ ⎪ ⎨ < 0 ⇒ elliptic, 2 b − 4ac = 0 ⇒ parabolic, ⎪ ⎪ ⎩ > 0 ⇒ hyperbolic.
The names arise by analogy with conic sections. The canonical examples are the Poisson problem uxx + uyy = g for an elliptic case, the heat equation ut = kuxx (with k > 0) for the parabolic case, and the wave equation utt = c2 uxx for the hyperbolic case. In the parabolic and hyperbolic cases, t is used instead of y since these are typically time-dependent problems. These can all be extended to more space dimensions. These equations describe different types of phenomena and require different techniques (both analytical and numerical), for their solution and so it is convenient to have names for classes of equations exhibiting the same general features. There are other equations that have some of the same features, and the classification scheme can be extended beyond the second order linear form given above. Some hint of this is given in the next few sections.
1.4
Some basic PDEs
27
Elliptic equations The classic example of an elliptic equation is the Poisson problem ∇2 u = f
(1.42)
where ∇2 is the Laplacian operator and f is a given function of x ¯ = (x, y) in some spatial domain Ω. We seek a function u(¯ x) in Ω satisfying (1.42) together with some boundary conditions all along the boundary of Ω. Elliptic equations typically model steady-state or equilibrium phenomena, so there is no temporal dependence. Elliptic equations may also arise in solving timedependent problems if we are modeling some phenomena that are always in local equilibrium and equilibrate on time scales that are much faster than the time scale being modeled. For example, in “incompressible” flow, the fast acoustic waves are not modeled and instead the pressure is computed by solving a Poisson problem at each time step. This models the global effect of these waves. Elliptic equations give boundary value problems (BVPs) where the solution at all points must be simultaneously determined based on the boundary conditions all around the domain. This typically leads to a very large sparse system of linear equations to be solved for the values of u at each grid point. If an elliptic equation must be solved in every time step of a time-dependent calculation, as in the examples above, then it is crucial that these systems be solved as efficiently as possible. More generally, a linear elliptic equation has the form Lu = f (1.43) where L is some elliptic operator. This notion will not be discussed further here, but the idea is that mathematical conditions are required on the differential operator L which ensure that the boundary value problem has a unique solution. Parabolic equations If L is an elliptic operator then the time-dependent equation ut = Lu − f
(1.44)
is called parabolic. If L = ∇2 is the Laplacian, then (1.44) is known as the heat equation or diffusion equation and models the diffusion of heat in a material. Now u(¯ x, t) varies with time and we require initial data u(¯ x, 0) for every x ¯ ∈ Ω as well as boundary conditions around the boundary at each time t > 0. If the boundary conditions are independent of time, then we might expect the heat distribution to reach a steady state in which u is independent of t. We could then solve for the steady state directly by setting ut = 0 in (1.44), which results in the elliptic equation (1.43). Marching to steady state by solving the time-dependent equation (1.44) numerically would be one approach to solving the elliptic equation (1.43), but this is typically not the fastest method to obtain the steady state solution.
28
Chapter 1
Basics of the Finite Difference Approximations
Hyperbolic equations Rather than discretizing second order hyperbolic equations such as the wave equation utt = c2 uxx , we will consider a related form of hyperbolic equations known as first order hyperbolic systems. The linear problem in one space dimension has the form (1.45) ut + Aux = 0 where u(x, t) ∈ �m and A is an m × n matrix. The problem is called hyperbolic if A has real eigen values and is diagonalizable, i.e., has a complete set of linearly independent eigenvectors. These conditions allow us to view the solution in terms of propagating waves, and indeed hyperbolic systems typically arise from physical processes that give wave motion or advective transport. The simplest example of a hyperbolic equation is the constantcoefficient advection equation ut + aux = 0
(1.46)
where u is the advection velocity. The solution is simply u(x, t) = u(x − at, 0) so any u profile simply advects with the flow at velocity a. As a simple example of a linear hyperbolic system, the equations of linearized acoustics arising from elasticity or gas dynamics can be written as a first order system of two equations in one space dimension as p p 0 k0 + =0 (1.47) u 1/ρ0 0 u t
x
in terms of pressure and velocity perturbations, where ρ0 is the background density and k0 is the “bulk modulus” of the material. Note that if we differentiate the first equation with respect to t, the second with respect to x, and then eliminate uxt = utx we obtain the second order wave equation for the pressure: ptt = c2 pxx where c = k0 /ρ is the speed of sound in the material. Derivation of PDEs from conservation principles
Many physically relevant partial differential equations can be derived based on the principle of conservation. We can view u(x, t) as a concentration or density function for some substance or chemical that is in dilute suspension in a liquid. The material presented here is meant to be a brief review, and more complete discussions are available in many sources: see, for example, Kevorkian [7] and Whitham [8]. A reasonable model to consider in one space dimension is the concentration or density of a contaminant in a stream or pipe, where the variable x represents distance along the pipe. The concentration is assumed to be constant across any cross-section, so that its value varies only with x. The density function u(x, t) is defined in such a way that integrating the function u(x, t) between any two points x1 and x2 gives the total mass of the substance in
1.4
Some basic PDEs
29
this section of the pipe at time t: total mass between x1 and x2 at time x2 t= u(x, t)dx. x1
The density functions is measured in units such as grams/meter. Note that this u really represents the integral over the cross section of the pipe of a density function that is properly measured in grams/meter3. The basic form of differential equation that models many physical processes can be derived in the following way: consider a section x1 < x < x2 and the manner in which x2 u(x, t)dx changes with time. This integral represents the total mass of x1
the substance in this section, so if we are studying a substance that is neither created nor destroyed within this section, then the total mass within this section can change only due to the flux or flow of particles through the endpoints of the section at x1 and x2 . This flux is given by some function f which, in the simplest case, depends only on the value of u at the corresponding point. Advection If the substance is simply carried along (advected) in a flow at some constant velocity a, then the flux function is f (u) = au.
(1.48)
The local density u(x, t) (in grams/meter, say) multiplied by the velocity (in meters/sec, say) gives the flux of material past the point x (in grams/sec). Since the total mass in [x1 , x2 ] changes only due to the flux at the endpoints, we have d x2 u(x, t)dx = f (u(x1 , t)) − f (u(x2 , t)). dt x1
The minus sign on the last term comes from the fact that f is, by definition, the flux to the right. If we assume that u and f are smooth functions, then this equation can be rewritten as x2 ∂ d x2 f (u(x, t))dx, u(x, t)dx = dt x1 ∂x x1 or, with some further modification, as x2 ∂ ∂ u(x, t) + f (u(x, t)) dx = 0. ∂t ∂x x1
Since this integral must be zero for all values of x1 and x2 , it follows that the integrand must be identically zero. This gives, finally, the differential equation ∂ ∂ u(x, t) + f (u(x, t)) = 0. (1.49) ∂t ∂x
30
Chapter 1
Basics of the Finite Difference Approximations
This form of equation is called a conservation law. (For further discussion see Whitham [8], Lax [9], and Le Veque [10].) For the case considered in the above section, f (u) = au with a constant and this equation becomes ut + aux = 0.
(1.50)
This is called the advection equation. This equation requires initial conditions and possibly also boundary conditions in order to determine a unique solution. The simplest case is the Cauchy problem on −∞ < x < ∞ (with no boundary), also called the pure initial value problem: then we only need to specify initial data u(x, 0) = η(x). (1.51) Physically, we would expect the initial profile of η to simply be carried along with the flow at speed a, so we should find u(x, t) = η(x − at).
(1.52)
It is easy to verify that this function satisfies the advection equation (1.50) and is the solution of the PDE. The curves x = x0 + at through each point x0 at time 0 are called the characteristics of Eq. (1.50). If we set U (t) = u(x0 + at, t), then by using (1.50) we get U (t) = au(x0 + at, t) + ut (x0 + at, t) = 0, and along these curves, the PDE reduces to a simple ODE U = 0. Hence the solution must be constant along each such curve, as is also seen from the solution (1.52). Diffusion Now suppose that the fluid in the pipe is not flowing, and has zero velocity. Then according to the above equation ut = 0 and the initial profile η(x) does not change with time. However, if η is not constant in space then in fact it will tend to change slowly due to molecular diffusion. The velocity “a” should really be thought of as a mean velocity, the average velocity of the roughly 1023 molecules in a given drop of water. But individual molecules are bouncing around in different directions and so molecules of the substance we are tracking will tend to get spread around in the water, much as a drop of ink spreads. There will be a net motion from regions where the density is large to regions where it is smaller, and in fact it can be shown that the flux (in 1D) is proportional to −ux . The flux at a point x now depends on the value of ux at this point, rather than on the value of u, so we write f (ux ) = −kux
(1.53)
where k is the diffusion coefficient. The relation (1.53) is known as Fick’s law. Using this flux in (1.49) we get ut = kuxx
(1.54)
1.4
Some basic PDEs
31
which is known as the diffusion equation. It is also called the heat equation since heat diffuses in much the same way. In this case we can think of the onedimensional equation as a model for the conduction of heat in a rod. The heat conduction coefficient k depends on the material and how well it conducts heat. The variable u is then the temperature and the relation (1.53) is known as Fourier’s law of heat conduction. In some problems the diffusion coefficient may vary with x, for example in a rod made of a composite of different materials. Then f = −k(x)ux and the equation becomes ut = (k(x)ux )x . Returning to the example of fluid flow, more generally there would be both advection and diffusion occurring simultaneously. Then the flux is f (u, ux) = au − kux giving the advection-diffusion equation ut + aux = kuxx .
(1.55)
The diffusion and advection-diffusion equations are examples of the general class of PDEs called parabolic. Source terms In some situations
x2
u(x, t)dx changes due to effects other than flux through
x1
the endpoints of the section, if there is some source or sink of the substance within the section (negative values correspond to a sink rather than a source), then the equation becomes x2 x2 d x2 ∂ f (u(x, t))dx + u(x, t)dx = − ψ(x, t)dx. dt x1 x1 ∂t x1 This leads to the PDE ut (x, t) + f (u(x, t))x = ψ(x, t).
(1.56)
For example, if we have heat conduction in a rod together with an external source of heat energy distributed along the rod with density, then we have ut = kuxx + ψ. In some cases the strength of the source may depend on the value of u. For example, if the rod is immersed in a liquid that is held at constant temperature u0 , then the flow of heat into the rod at the point (x, t) is proportional to u0 − u(x, t) and the equation becomes ut (x, t) = kuxx (x, t) + α(u0 − u(x, t)). Reaction-diffusion equations One common form of source term arises from chemical kinetics. If the components of u ∈ �m represent concentrations of m different species reacting with one another, then the kinetics equations have the form ut = ψ(u). This assumes that the different species are well-mixed at all times and so the concentrations vary only with time. If there are spatial variations in concentrations, then these equations may be combined with diffusion of each species.
32
Chapter 1
Basics of the Finite Difference Approximations
This would lead to a system of reaction-diffusion equations of the form ut = kuxx + ψ(u).
(1.57)
The diffusion coefficient could be different for each species, in which case k would be a diagonal matrix instead of a scalar. Advection terms might also be present if the reactions are taking place in a flowing fluid.
1.5 Numerical solution to partial differential equations Elliptic partial differential equations The elliptic partial differential equation which we consider is the Poisson equation ∂ 2u ∂ 2u (x, y) + 2 (x, y) = f (x, y), 2 ∂x ∂y (x, y) ∈ R and u(x, y) = g(x, y) for (x, y) ∈ S ∇2 u(x, y) ≡
(1.58)
where R = {(x, y)|a < x < b, c < y < d}
and S denotes the boundary of R. Here we assume that both f and g are continuous on their domains and a unique solution is ensured. The method used is an adaptation of the finite difference method for boundary value problem. The first step is to choose integers n and m and definite step sizes h and k by h = b − a/n and k = d − c/m. Partitioning the interval [a, b] into n-equal parts of width h and the interval [c, d] into m equal parts of width k provides a grid on the rectangle R by drawing vertical and horizontal lines through the points with coordinates (xi , yj ) where xi = a + ih for each i = 0, 1, 2, . . . , n and yj = c + jk for each j = 0, 1, 2, . . . , m. The lines x = xi and y = yj are grid lines and their intersections are the mesh points of the grid. For each mesh point in the interior of the grid (xi , yj ), we use the Taylor series in the variable x about xi to generate the central difference formula ∂ 2u u(xi+1 , yj ) − 2u(xi , yj ) + u(xi−1 , yj ) h2 ∂ 4 u (xi , yj ) = − (ξi , yj ) (1.59) 2 ∂x h2 12 ∂x4 where ξi ∈ (xi−1 , xi+1 ). We also use the Taylor series in the variable y about yj to generate the central difference formula ∂2u u(xi , yj+1 ) − 2u(xi , yj ) + u(xi , yj−1 ) k 2 ∂ 4 u (x , y ) = − (xi , ηj ) (1.60) i j ∂y 2 k2 12 ∂y 4
1.5
Numerical solution to partial differential equations
33
where ηj ∈ (yi−1 , yi+1 ). Use of these formulas in Eq. (1.58) allows us to express the Poisson equation at the points (xi , yj ) as u(xi+1 , yj ) − 2u(xi , yj ) + u(xi−1 , yj ) h2 u(xi , yj+1 ) − 2u(xi , yj ) + u(xi , yj−1 ) + k2 k2 ∂ 4 u h2 ∂ 4 u = f (xi , yj ) + (x , η ) + (ξi , yj ). i j 12 ∂y 4 12 ∂x4 For each i = 1, 2, . . . , n − 1 and j = 1, 2, . . . , m − 1, the boundary conditions can be written as u(x0 , yj ) = g(x0 , yj ) for each j = 0, 1, 2, . . . , m, u(xn , yj ) = g(xn , yj ) for each j = 0, 1, 2, . . . , m, u(xi , y0 ) = g(xi , y0 ) for each i = 0, 1, 2, . . . , n − 1,
u(xi , ym ) = g(xi , ym ) for each i = 0, 1, 2, . . . , n − 1.
In the difference equation form, these results in the central difference method with local truncation error are of order (k2 + h2 ) and 2 2 h h + 1 wij − (wi+1,j + wi−1,j ) − (wi,j+1 + wi,j−1 ) 2 k k = −h2 f (xi , yj ).
(1.61)
For each i = 1, 2, . . . , n − 1 and j = 1, 2, . . . , m − 1 we set
w0,j = g(x0 , yj ), for each j = 0, 1, 2, . . . , m, wn,j = g(xn , yj ), for each j = 0, 1, 2, . . . , m, wi,0 = g(xi , y0 ), for each i = 0, 1, 2, . . . , n − 1,
wi,m = g(xi , ym ), for each i = 0, 1, 2, . . . , n − 1
(1.62)
where wij approximates u(xi , yj ). The typical equation (1.61) involves approximations to u(x, y) at the points (xi−1 , yj ), (xi , yj ), (xi+1 , yj ), (xi , yj−1 ) and (xi , yj+1 ). Reducing the portion of the grid where these points are located shows that each equation involves approximations in a region about (xi , yj ). If we use the information from the boundary conditions (1.62) whenever appropriate in the system given by (1.61), that is at all points (xi , yj ) adjacent to a boundary mesh point, we have an (n − 1)(m − 1) by (n − 1)(m − 1) linear system with the unknowns being the approximations wij to u(xi , yj ) at the interior mesh points. The linear system involving these unknowns is expressed for matrix calculations more efficiently if relabeling of the interior points is introduced. A recommended labeling of these points is to let pl = (xi , yj ) and wl = wi,j where l = i + (m − 1 − j)(n − 1) for each i = 1, 2, . . . , n − 1 and j = 1, 2, . . . , m − 1. This labels the mesh points consecutively from left to right and top to bottom.
34
Chapter 1
Basics of the Finite Difference Approximations
Parabolic differential equations The parabolic differential equation we study is the heat or diffusion equation ∂u ∂ 2u (x, t) = α2 2 (x, t), 0 < x < l, t > 0, subject to the conditions (1.63) ∂t ∂x u(0, t) = u(l, t) = 0, t > 0, and u(x, 0) = f (x), 0 x l. The approach we use to approximate the solution to this problem involves finite differences and is similar to the method discussed in previous section. First select an integer m > 0 and a time step k > 0 and let h = l/m. The grid points for this situations are (xi , tj ) where xi = ih for i = 0, 1, . . . , m and tj = jk for j = 0, 1, . . .. We obtain the difference method by using the Taylor series in t to form the difference quotient u(xi , tj + k) − u(xi , yj ) k ∂ 2 u ∂u (xi , μj ) for some μj ∈ (tj , tj+1 ) (xi , tj ) = − ∂t k 2 ∂t2 (1.64) and the Taylor series in x to form the difference quotient ∂2u u(xi + h, yj ) − 2u(xi , yj ) + u(xi − h, yj ) h2 ∂ 4 u (x , y ) = − (ξi , tj ) i j ∂x2 h2 12 ∂x4 (1.65) where ξ ∈ (xi−1 , xi+1 ). The partial differential equation (1.63) implies that at the interior grid point (xi , tj ) for each i = 1, 2, . . . , m − 1 and j = 1, 2, . . . we have ∂u ∂2u (xi , tj ) − α2 2 (xi , tj ) = 0; ∂t ∂x so the difference method using the difference quotients (1.64) and (1.65) is wi+1,j − 2wi,j + wi−1,j wi,j+1 − wi,j =0 − α2 k h2
(1.66)
where wi,j approximates to u(xi , tj ). The local truncation error for this difference equation is τi,j =
2 4 k ∂ 2u 2h ∂ u (x , μ ) − α (ξi , tj ). i j 2 ∂t2 12 ∂x4
Solving Eq. (1.66) for wi,j+1 gives 2α2 k k wi,j+1 = 1 − 2 wi,j + α2 2 (wi+1,j + wi−1,j ) h h
(1.67)
(1.68)
for each i = 1, 2, . . . , m − 1 and j = 1, 2, . . .. Since the initial condition u(x, 0) = f (x) for each 0 x 1 implies that wi,0 = f (xi ) for each i = 0, 1, 2, . . . , m, these values can be used in Eq. (1.68) to find the value of wi,1 for each i = 1, 2, . . . , m−1. The additional condition u(0, t) = 0 and u(l, t) = 0 imply that w0,1 = wm,1 = 0 so all the entries wi,1 can be determined. If the
1.5
Numerical solution to partial differential equations
35
procedure is reapplied once all the approximations wi,1 are known, then the values wi,2 , wi,3 , wi,4 , . . . , wi,m−1 can be obtained in a similar manner. The explicit nature of the difference method implies that the (m − 1) by (n − 1) matrix associated with this system can be written in the tridiagonal form ⎡ ⎤ 1 − 2λ λ 0 ··· 0 0 ⎢ ⎥ ⎢ λ 1 − 2λ λ ··· 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ λ 1 − 2λ · · · 0 0 ⎥ ⎢ 0 ⎥ A=⎢ .. .. .. .. ⎥ ⎢ .. ⎢ . . . . . ⎥ ⎢ ⎥ ⎢ 0 0 0 · · · 1 − 2λ λ ⎥ ⎣ ⎦ 0 0 0 ··· λ 1 − 2λ
� k2 . If we let w(0) = (f (x1 ), f (x2 ), f (x3 ), . . . , f (xm−1 ))t h2 and w(j) = (w1,j , w2,j , w3,j , . . . , wm−1,j )t for each j = 1, 2, . . ., this is known as the forward difference method. If the solution to the partial differential equation has four continuous partial derivatives in x and two in t, then Eq. (1.67) implies that the method is of order (k 2 + h2 ).
where λ = α2
�
Hyperbolic differential equation In this section we consider the numerical solution of the hyperbolic differential equation. The wave equation is given by the differential equation 2 ∂ 2u 2∂ u (x, t) − α (x, t) = 0, ∂t2 ∂x2
0 < x < l, t > 0,
subject to the conditions u(0, t) = u(l, t) = 0, t > 0, u(x, 0) = f (x), 0 x l ∂u (x, 0) = g(x), 0 x l, where a is a constant. To set up the finite and ∂t difference method, select an integer m > 0 and time size k > 0. With h = l/m the mesh points (xi , tj ) are xi = ih for each i = 0, 1, 2, . . . , m and tj = jk for each j = 0, 1, 2, . . . at each mesh point (xi , tj ) the wave equation becomes ∂ 2u ∂ 2u (xi , tj ) − α2 2 (xi , tj ) = 0. 2 ∂t ∂x
(1.69)
The difference method obtained by using the centered-difference quotient for the second partial derivatives is given by ∂2u u(xi , tj+1 ) − 2u(xi , tj ) + u(xi , tj−1 ) k 2 ∂ 4 u (x , t ) = − (xi , μj ) i j ∂t2 k2 12 ∂t4 where μj ∈ (tj−1 , tj+1 ) and ∂2u u(xi+1 , tj ) − 2u(xi , tj ) + u(xi−1 , tj ) h2 ∂ 4 u (x , t ) = − (ξi , tj ) i j ∂x2 h2 12 ∂x4
36
Chapter 1
Basics of the Finite Difference Approximations
where ξi ∈ (xi−1 , xi+1 ). Substituting these into Eq. (1.69) gives u(xi , tj+1 ) − 2u(xi , tj ) + u(xi , tj−1 ) k2 u(xi+1 , tj ) − 2u(xi , tj ) + u(xi−1 , tj ) −α2 h2 � � 4 4 1 ∂ u 2 2 2∂ u k = (xi , μj ) − α h (ξi , tj ) . 12 ∂t4 ∂x4 Neglecting the error term τi,j =
1 12
� � ∂4u ∂ 4u k 2 4 (xi , μj ) − α2 h2 4 (ξi , tj ) ∂t ∂x
leads to the differential equation wi+1,j − 2wi,j + wi−1,j wi,j+1 − 2wi,j + wi,j−1 − α2 = 0. k2 h2 If λ =
αk , then we can write the difference equation as h wi,j+1 − 2wi,j + wi,j−1 − λ2 (wi+1,j − 2wi,j + wi−1,j ) = 0
and solve for wi,j+1 the most advanced time step approximation, to obtain wi,j+1 = 2(1 − λ2 )wi,j + λ2 (wi+1,j + wi−1,j ) − wi,j−1 .
(1.70)
The equation holds for each i = 1, 2, . . . , m−1 and j = 1, 2, . . .. The boundary conditions give w0,j = wm,j = 0 for each i = 1, 2, . . .
(1.71)
and the initial condition implies that wi,0 = f (xi ) for each i = 1, 2, . . . , m − 1. Writing this set of equations in matrix form gives ⎡
w1,j+1
⎤
⎢ ⎥ ⎢ w2,j+1 ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎣ ⎦ wm−1,j+1
(1.72)
1.5
Numerical solution to partial differential equations
⎡
⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡
2(1 − λ2 ) λ2
λ2
0
2(1 − λ2 )
λ2
λ2 .. .
2(1 − λ2 ) .. .
0
0
0
0
0
0
0 .. .
w1,j
⎤
⎡
w1,j−1
⎤
37
0
0
···
0
0
···
0 .. .
0 .. .
···
2(1 − λ2 )
λ2
···
···
λ
2
⎤
2(1 − λ2 )
⎢ ⎥ ⎢ ⎥ ⎢ w2,j ⎥ ⎢ w2,j−1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥. .. .. ⎢ ⎢ ⎥ ⎥ . . ⎣ ⎦ ⎣ ⎦ wm−1,j wm−1,j−1
⎥ ⎥ ⎥ ⎥ ⎥ ⎥· ⎥ ⎥ ⎥ ⎥ ⎦
(1.73)
Equations (1.70) and (1.71) imply that (j + 1) time step requires values from the j th and (j−1)th time steps. This procedure gives a minor starting problem since values for j = 0 are given by Eq. (1.72), but values for j = 1 which are needed in Eq. (1.70) to compute wi,2 must be obtained from the initial velocity condition ∂u (x, 0) = g(x), 0 x l. ∂t One approach is to replace ∂u/∂t by a forward difference approximation u(xi , t1 ) − u(xi , 0) k ∂ 2 u ∂u (xi , μi ), (xi , 0) = − ∂t k 2 ∂t2
0 < μ < t1 .
(1.74)
Solving for u(xi , t1 ) gives ∂u k2 ∂ 2 u (xi , μi ) (xi , 0) + ∂t 2 ∂t2 k2 ∂ 2u = u(xi , 0) + kg(xi ) + (xi , μi ). 2 ∂t2
u(xi , t1 ) = u(xi , 0) + k
As a consequence wi,1 = wi,0 + kg(xi ) for each i = 1, 2, . . . , m − 1.
(1.75)
From Eq. (1.74), however, we can see that this result gives an approximation that has local truncation error of only O(k). A better approximation to u(x, 0) can be obtained rather easily: particularity when the second derivative of f at xi can be determined. Using a second Taylor polynomial in t for u at (xi , 0), we write u(xi , t1 ) − u(xi , 0) ∂u k ∂ 2u k3 ∂ 3u = (xi , 0) + (x , 0) + (xi , μi ) i k ∂t 2 ∂t2 6 ∂t3
(1.76)
38
Chapter 1
Basics of the Finite Difference Approximations
for some 0 < μi < t1 . Suppose the wave equation also holds on the initial line; that is, ∂2u ∂ 2f ∂ 2u (xi , 0) = α2 2 (xi , 0) = α2 2 (xi ) = α2 f �� (xi ). 2 ∂t ∂x ∂x Substituting into Eq. (1.76) and solving for u(xi , t1 ) gives α2 k 2 �� k3 ∂ 3 u (xi , μi ) f (xi ) + 2 6 ∂t3 α2 k 2 �� f (xi ). = wi,0 + kg(xi ) + 2
u(xi , t1 ) = u(xi , 0) + kg(xi ) + and wi,1
(1.77)
This is an approximation with local truncation error O(k 2 ) for each i = 1, 2, . . . , m − 1. If f �� (xi ) is not readily available, we can use the difference equation to write f �� (xi ) =
f (xi+1 ) − 2f (xi ) + f (xi−1 ) h2 (4) − f (ξi ) h2 12
for some ξi ∈ (xi−1 , xi+1 ), provided that f ∈ C 4 [0, 1]. This implies that the approximation becomes k ∂u kα2 u(xi , t1 ) − u(xi , 0) = g(xi ) + (xi , 0) + 2 (f (xi+1 ) − 2f (xi ) + f (xi−1 )) k 2 ∂t 2h +O(k 2 + h2 ) or, letting λ =
kα , h
u(xi , t1 ) = u(xi , 0) + kg(xi ) + = (1 − λ2 )f (xi ) +
λ2 (f (xi+1 ) − 2f (xi ) + f (xi−1 )) + O(k 3 + h2 k 2 ) 2
λ2 (f (xi+1 ) + f (xi−1 )) + kg(xi ) + O(k3 + h2 k 2 ). 2
λ2 (f (xi+1 )+f (xi−1 ))+ 2 kg(xi ). This can be used to find wi,1 for each i = 1, 2, . . . , m−1. It is assumed that there is an upper bound for the value of t to be used in the stopping technique. Thus the difference equation is wi,1 = (1 − λ2 )f (xi )+
References [1]
J.D. Lambert, Computational Methods in Ordinary Differential Equations, Wiley, London/New York/Sydney/Toronto, 1973.
References [2] [3] [4] [5] [6] [7] [8] [9]
[10]
39
C.W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall Inc., Englewood Cliffs, N.J., 1971. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York, 1962. E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equations I. Non stiff Problems, Springer-Verlag, Berlin-Heidelberg, 1987. E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-algebraic Problems, Springer-Verlag, New York, 1993. E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGrawHill, New York, 1955. J. Kevorkian, Partial Differential Equations, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1990. G. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974. P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conference Series in Applied Mathematics, #11, 1972. R.J. Le Veque, Numerical Methods for Conservation Laws, Birkhauser-Verlag, Basel, 1990.
Chapter 2 Principles of the Implicit Keller-box Method
In this chapter, we introduce the basic principles of finite difference methods for solving a system of linear homogeneous first order differential equations. In Section 2.2, we solve the second and higher order linear/nonlinear differential equations using different types of boundary conditions; we also obtain convergence and stability results. In Section 2.3 we briefly explain the finite difference method, shooting method, and the Keller-box method.
2.1 Principles of implicit finite difference methods System of first order equation General Remarks. One of the fundamental concepts of analysis is that of a system of n simultaneous first order differential equations. If y1 (x), y2 (x), . . . , yn (x) are unknown functions of a single independent variable x, then the most general system of interest to us is one in which their derivatives y1 , y2 , . . . , yn are explicitly given as functions of x (Jain, et al. [1]) and y1 , y 2 , . . . , y n ; y1 = f1 (x, y1 , y2 , . . . , yn ), y2 = f2 (x, y1 , y2 , . . . , yn ),
(2.1)
············
yn = fn (x, y1 , y2 , . . . , yn ). Systems of differential equations arise quite naturally in many scientific problems. We have studied a system of two second order linear equations to describe the motion of coupled harmonic oscillators; in the example below we
42
Chapter 2
Principles of the Implicit Keller-box Method
shall see how they occur in connection with dynamical system having several degrees of freedom; and we will use them to analyze a simple biological community composed of different species of animals interacting with one another. An important mathematical reason for studying systems is that the single nth order equation y (n) = f (x, y, y � , . . . , y (n−1) )
(2.2)
can always be regarded as a special case of (2.1). To see this, we put y1 = y, y2 = y � , . . . , yn = y (n−1)
(2.3)
and observe that (2.2) is equivalent to the system y1� = y2 , y2� = y3 , ············
(2.4)
yn� = f (x, y1 , y2 , . . . , yn ) which is clearly a special case of (2.1). The statement that (2.2) and (2.4) are equivalent is understood to mean the following: If y(x) is a solution of Eq. (2.2), then the functions y1 (x), y2 (x), . . . , yn (x) defined by (2.3) satisfy (2.4); and conversely, if y1 (x), y2 (x), . . . , yn (x) satisfy (2.4), then y(x) = y1 (x) is a solution of (2.2). This reduction of an nth order equation to a system of first order equations has several advantages. We illustrate by considering the relation between the basic existence and uniqueness theorems for the system (2.1) and for Eq. (2.2). If a fixed point x = x0 is chosen and the values of the unknown functions y1 (x0 ) = a1 , y2 (x0 ) = a2 , . . . , yn (x0 ) = an
(2.5)
are assigned arbitrarily in such a way that the functions f1 , f2 , . . . , fn are defined, then (2.1) gives the values of the derivatives y1� (x0 ), y2� (x0 ), . . . , yn� (x0 ). The similarity between this situation and that discussed earlier suggests the following analog of Picard’s theorem. Lemma 2.1. Let the function f1 , f2 , . . . , fn and the partial derivatives ∂f1 ∂f1 ∂fn ∂fn ,..., ,..., ,..., ∂y1 ∂yn ∂y1 ∂yn be continuous in a region R of (x, y1 , y2 , . . . , yn ) space. If (x0 , a1 , a2 , . . . , an ) is an interior point of R, then the system (2.1) has a unique solution y1 (x), y2 (x), . . . , yn (x) that satisfies the initial conditions (2.5). Lemma 2.2. Let the function f and the partial derivatives ∂f ∂f ∂f , , . . . , (n−1) ∂y ∂y � ∂y
2.1
Principles of implicit finite difference methods
43
be continuous in a region R of (x, y, y � , . . . , y (n−1) ) space. If (x0 , a1 , a2 , . . . , an ) is an interior point of R, then Eq. (2.2) has a unique solution y(x) that satisfies the initial conditions y(x0 ) = a1 , y � (x0 ) = a2 , . . . , y (n−1) (x0 ) = an . As a further illustration of the value of reducing higher order equations to a system of first order equations, we consider the famous n-body problem of classical mechanics. Let n particles with masses mi be located at the points xi , yi , zi and assume that they attract one another according to Newton’s law of gravitation. If rij is the distance between mi and mj , and if θ is the angle from the positive x-axis to the segment joining them, Gmi mj then the x component of the force exerted on mi by mj is cos θ = 2 rij Gmi mj (xj − xi ) , where G is the gravitational constant. Since the sum of 3 rij these components for all j �= i equals mi (d2 xi /dt2 ), we have n second or mj (xj − xi ) d 2 xi d2 yi der differential equations = G , and similarly = 3 dt2 rij dt2 j�=i
mj (yj − yi ) mj (zj − zi ) d2 zi G and = G . If we put vxi = dxi /dt, 3 3 2 rij dt rij j�=i
j�=i
vyi = dyi /dt and vzi = dzi /dt, and apply the above reduction, then we obtain a system of 6n equations of the form (2.1) in the unknown functions x1 , vxi , . . . , xn , vxn , y1 , vyi , . . . , yn , vyn , z1 , vzi , . . . , zn , vzn . If we now make use 3 of the fact that rij = ((xi − xj )2 + (yi − yj )2 + (zi − zj )2 )3/2 , then Lemma 2.1 yields the following conclusion: if the initial positions and initial velocities of the particles, i.e., the values of the unknown functions at a certain instant t = t0 are given and if the particles do not collide in the sense that the rij do not vanish, then their subsequent positions and the velocities are uniquely determined. This conclusion underlies the once popular philosophy of mechanistic determinism, according to which the universe is nothing more than a gigantic machine whose future is inexorably fixed by its states at any given moment. Problem 2.1. Replace each of the following differential equations by an equivalent systems of first order equations: a) y �� − x2 y � − xy = 0, b) y ��� = y �� − x2 (y � )2 .
Problem 2.2. If a particle of mass m moves in the xy plane, its equations d2 y d2 x of motions are m 2 = f (t, x, y) and m 2 = g(t, x, y) where f and g dt dt represents the x and y components, respectively of the force acting on the particle. Replace this system of two second order equations by an equivalent system of four first order equations of the form (2.1).
44
Chapter 2
Principles of the Implicit Keller-box Method
Linear systems For the sake of convenience and clarity, we restrict our attention through the rest of this section to a system of only two first order equations in two unknown functions of the form ⎧ dx ⎪ ⎪ = F (t, x, y), ⎨ dt (2.6) ⎪ ⎪ ⎩ dy = G(t, x, y). dt
The brace notation is used to emphasize the fact that the equations are linked together, and the choice of the letter t for the independent variables x and y for the dependent variables is customary in this case for reasons that will appear later. In this and the next section we specialize even further, to linear systems, of the form ⎧ ⎪ ⎪ dx = a1 (t)x + b1 (t)y + f1 (t), ⎨ dt (2.7) ⎪ dy ⎪ ⎩ = a2 (t)x + b2 (t)y + f2 (t). dt
We shall assume in the present discussion, and in the lemmas presented below, that the functions ai (t), bi (t) and fi (t), i = 1, 2 are continuous on a certain closed interval [a, b] of the t-axis. If f1 (t) and f2 (t) are identically zero, then the system (2.7) is called homogeneous; otherwise it is said to be non-homogeneous. A solution of (2.7) on [a, b] is of course a pair of functions x(t) and y(t) that satisfy both equations � of (2.7) through this interval. We x = x(t) shall write such a solution in the form . Thus, it is easy to verify y = y(t) that the homogeneous linear system (with constants coefficients)
has both
�
⎧ dx ⎪ ⎪ = 4x − y, ⎨ dt ⎪ ⎪ ⎩ dy = 2x + y dt x = e3t , y = e3t
and
�
x = e2t , y = 2e2t
(2.8)
(2.9)
as solutions on any closed interval. We now give a brief sketch of the general theory of the linear system (2.7). We begin by stating the following fundamental existence and uniqueness lemmas.
2.1
Principles of implicit finite difference methods
45
Lemma 2.3. If t0 is any point in the interval [a, b], and if x0 and y0 are any numbers wherever, then (2.7) has one and only one solution �
x = x(t), y = y(t)
valid throughout [a, b], such that x(t0 ) = x0 and y(t0 ) = y0 . Our next step is to study the structure of the solutions of the homogeneous system obtained from (2.7) by removing the terms f1 (t) and f2 (t): ⎧ dx ⎪ ⎪ = a1 (t)x + b1 (t)y, ⎨ dt (2.10) ⎪ dy ⎪ ⎩ = a2 (t)x + b2 (t)y. dt It is obvious that (2.10) is satisfied by the so-called trivial solution in which x(t) and y(t) are both identically zero. Our main tool in constructing more useful solutions is the next lemma. Lemma 2.4. If the homogeneous system (2.10) has two solutions � on [a, b], then
x = x1 (t), y = y1 (t) �
and
�
x = x2 (t), y = y2 (t)
x = c1 x1 (t) + c2 x2 (t), y = c1 y1 (t) + c2 y2 (t)
(2.11)
(2.12)
is also a solution on [a, b] for any constants c1 and c2 . Proof: The proof is a routine verification, and is left to the reader. The solution (2.12) is obtained from the pair of solutions (2.11) by multiplying the first by c1 , the second by c2 , and adding; (2.12) is therefore called a linear combination of the solutions (2.11). With this terminology, we can restate Lemma 2.4 as follows: any linear combination of two solution of the homogeneous system (2.11) is also a solution. Accordingly, (2.8) has �
x = c1 e3t + c2 e2t , y = c1 e3t + 2c2 e2t
(2.13)
as a solution for every choice of the constants c1 and c2 . The next question we must settle is that of whether (2.12) contains all solutions of (2.10) on [a, b], that is, whether it is the general solution of (2.10) on [a, b]. By Lemma 2.1, (2.12) will be the general solution if the constants c1 and c2 can be chosen so as to satisfy arbitrary conditions x(t0 ) = x0 and
46
Chapter 2
Principles of the Implicit Keller-box Method
y(t0 ) = y0 at an arbitrary point t0 in [a, b], or equivalently, if the system of linear algebraic equations c1 x1 (t0 ) + c2 x2 (t0 ) = x0 , c1 y1 (t0 ) + c2 y2 (t0 ) = y0 in the unknowns c1 and c2 can be solved for each t0 in [a, b] and every pair of numbers x0 and y0 . By the elementary theory of determinants, this is possible wherever the determinant of the coefficients, x (t) x (t) 2 1 W (t) = y1 (t) y2 (t)
does not vanish on the interval [a, b]. This determinant is called the Wronskian of the two solutions (2.11). The above remarks prove the next lemma.
Lemma 2.5. If the two solutions (2.11) of the homogeneous system (2.10) have a Wronskian W (t) that does not vanish on [a, b], then (2.12) is the general solution of (2.10) on this interval. It follows from this theorem that (2.13) is the general solution of (2.8) on any closed interval, for the Wronskian of the two solutions (2.9) is e3t e2t W (t) = 3t = e5t , e 2e2t which never vanishes. It is useful to know, as this example suggests, that the vanishing or non-vanishing of the Wronskian W (t) of two solutions does not depend on the choice of t. To state it formally, we have
Lemma 2.6. If W (t) is the Wronskian of the two solutions (2.11) of the homogeneous system (2.10), then W (t) is either identically zero or nowhere zero on [a, b]. Proof: A simple calculation shows that W (t) satisfies the first order differential equation dW = (a1 (t) + b2 (t))W (2.14) dt from which it follows that W (t) = ce
R
(a1 (t)+b2 (t))dt
(2.15)
for some constant c. The conclusion of the lemma is now evident from the fact that the exponential factor in (2.15) never vanishes on [a, b]. Lemma 2.5 provides an adequate means of verifying that (2.12) is the general solution of (2.10); show that the Wronskian W (t) of the two solutions (2.11) does not vanish. We now develop an equivalent test that is often more direct and
2.1
Principles of implicit finite difference methods
47
convenient. The two solutions (2.11) are called linearly dependent on [a, b] if one is a constant multiple of the other in the sense that x1 (t) = kx2 (t), y1 (t) = ky2 (t)
or
x2 (t) = kx1 (t), y2 (t) = ky1 (t)
for some constant k and all t in [a, b], and linearly independent if neither is a constant multiple of the other. It is clear that linear dependence is equivalent to the condition that there exist two constants c1 and c2 , at least one of which is not zero, such that c1 x1 (t) + c2 x2 (t) = 0, (2.16) c1 y1 (t) + c2 y2 (t) = 0 for all t in [a, b]. We now have the next lemma. Lemma 2.7. If the two solutions (2.11) of the homogeneous system (2.10) are linearly independent on [a, b], then (2.12) is the general solution of (2.10) on this interval. Proof: In view of the Lemmas 2.5 and 2.6, it suffices to show that the solutions (2.11) are linearly independent if and only if their Wronskian W (t) is identically zero. We begin by assuming that they are linearly dependent. So that, say, x1 (t) = kx2 (t), (2.17) y1 (t) = ky2 (t), then x (t) x (t) kx (t) x (t) 2 2 1 2 W (t) = = = kx2 (t)y2 (t) − kx2 (t)y2 (t) = 0 y1 (t) y2 (t) ky2 (t) y2 (t)
for all t in [a, b] . The same argument works equally well if the constant k is on the other side of Eqs. (2.17). We now assume that W (t) is identically zero, and show that the solutions (2.11) are linearly dependent in the same of Eqs. (2.16). Let t0 be a fixed point in [a, b]. Since W (t0 ) = 0, the system of linear algebraic equations c1 x1 (t0 ) + c2 x2 (t0 ) = 0, c1 y1 (t0 ) + c2 y2 (t0 ) = 0 has a solution c1 , c2 in which these numbers are not both zero. Thus, the solution of (2.10) given by
x = c1 x1 (t) + c2 x2 (t), y = c1 y1 (t) + c2 y2 (t)
(2.18)
48
Chapter 2
Principles of the Implicit Keller-box Method
equals the trivial solution at t0 . It now follows from the uniqueness part of Lemma 2.3 that (2.18) must equal the trivial solution throughout the interval [a, b], so (2.16) holds and the proof is complete. The value of this test is that in specific problems it is usually a simple matter of inspection to decide whether two solutions of (2.10) are linearly independent or not. We now return to the non-homogeneous system (2.7) and conclude our discussion with Lemma 2.8. If the two solutions (2.11) of the homogeneous system (2.10) are linearly independent on [a, b], and if � x = xp (t), y = yp (t)
is any particular solution of (2.7) on this interval, then � x = c1 x1 (t) + c2 x2 (t) + xp (t), y = c1 y1 (t) + c2 y2 (t) + yp (t)
(2.19)
is the general solution of (2.7) on [a, b]. � x = x(t) Proof: It suffices to show that if is an arbitrary solution of (2.7), y = y(t) � x = x(t) − xp (t) is a solution of (2.10), and this we leave to the reader. then y = y(t) − yp (t) The above treatment of the linear system (2.7) shows how its general solution (2.19) can be built up of simpler pieces. But how do we find these pieces? Unfortunately as in the case of second order linear equations there does not exists any general method that always works. Homogeneous linear systems with constant coefficients We are now in a position to give a complete explicit solution of the simple system ⎧ dx ⎪ ⎪ = a1 (t)x + b1 (t)y, ⎨ dt (2.20) ⎪ ⎪ ⎩ dy = a2 (t)x + b2 (t)y dt where a1 , b1 , a2 and b2 are given constants. Some of the problems at the end of the previous section illustrate a procedure that can often be applied to this case: differentiate one equation, eliminate one of the dependent variables, and solve the resulting second order linear equation. The method we now describe is based instead on constructing a pair of linearly independent solutions directly from the given system. If we recall that the exponential function has the property that its derivative is constant multiples of the function itself,
2.1
Principles of implicit finite difference methods
then it is natural to seek solutions of (2.20) having the form x = Aemt , y = Bemt .
49
(2.21)
If we substitute (2.21) in to (2.20) we get Amemt = a1 Aemt + b1 Bemt , Bmemt = a2 Aemt + b2 Bemt , and dividing by emt yields the linear algebraic system (a1 − m)A + b1 B = 0,
a2 A + (b2 − m)B = 0
(2.22)
in the unknowns A and B. It is clear that (2.23) has the trivial solution A = B = 0, which makes (2.21) the trivial solution of (2.20). Since we are looking for nontrivial solutions of (2.20), this is no help at all. However, we know that (2.22) has non-trivial solutions whenever the determinant of the coefficients vanishes, i.e., whenever a − m b1 1 = 0. a2 b2 − m When this determinant is expanded, we get the quadratic equation m2 − (a1 + b2 )m + (a1 b2 − a2 b1 ) = 0
(2.23)
for the unknown m. By analogy with our previous work, we call this the auxiliary equation of the system (2.20). Let m1 and m2 be the roots of (2.23). If we replace m in (2.22) by m1 , then we know that the resulting equations have a nontrivial solution A1 , B1 , so x = A1 em1 t , (2.24) y = B1 e m 1 t is a nontrivial solution of the system (2.20). By proceeding similarly with m2 , we find another nontrivial solution x = A2 em2 t , (2.25) y = B2 e m 2 t . In order to make sure that we obtain two linearly independent solution and hence the general solution it is necessary to examine in detail each of the three possibilities of m1 and m2 .
50
Chapter 2
Principles of the Implicit Keller-box Method
Distinct real roots. When m1 and m2 are distinct real numbers, then (2.24) and (2.25) are easily seen to be linearly independent (why?) and x = c1 A1 em1 t + c2 A2 em2 t , y = c1 B1 em1 t + c2 B2 em2 t
(2.26)
is the general solution of (2.20). Example 2.3. The case of the system ⎧ dx ⎪ ⎪ = x + y, ⎨ dt ⎪ ⎪ ⎩ dy = 4x − 2y. dt
(2.27)
(1 − m)A + B = 0,
(2.28)
Solution: (2.22) is
4A + (−2 − m)B = 0.
The auxiliary equation here is
m2 + m − 6 = 0 or (m + 3)(m − 2) = 0 so m1 and m2 are −3 and −2 with m = −3. Then (2.28) becomes 4A + B = 0, 4A + B = 0. A simple nontrivial solution of this system is A = 1, B = −4, so we have � x = e−3t , (2.29) y = −4e−3t as a nontrivial solution of (2.27). With m = 2, (2.28) becomes −A + B = 0,
4A − 4B = 0.
A simple nontrivial solution is A = 1, B = 1. These yields � x = e2t , y = e2t
(2.30)
as another solution of (2.27); and since it is clear that (2.29) and (2.30) are linearly independent, � x = c1 e−3t + c2 e2t , (2.31) y = −4c1 e−3t + c2 e2t
2.1
Principles of implicit finite difference methods
51
is the general solution of (2.27). Distinct complex roots. If m1 and m2 are distinct complex numbers, then they can be written in the form a ± ib where a and b are real numbers and b �= 0. In this case we expect the A’s and B’s obtained from (2.22) to be complex numbers, and we have two linearly independent solutions x = A∗2 e(a−ib)t , x = A∗1 e(a+ib)t , and (2.32) y = B1∗ e(a+ib)t y = B2∗ e(a−ib)t . However, these are complex-valued solutions, and to extract a real-valued solution we proceed as follows. If we express the numbers A∗1 and B1∗ in the standard form A∗1 = A1 + iA2 and B1∗ = B1 + iB2 , and use Euler’s formula (2.26), then the first of the solutions (2.32) can be written as x = (A1 + iA2 )eat (cos bt + i sin bt), y = (B1 + iB2 )eat (cos bt + i sin bt)
or
x = eat ((A1 cos bt − A2 sin bt) + i(A1 sin bt + A2 cos bt)), y = eat ((B1 cos bt − B2 sin bt) + i(B1 sin bt + B2 cos bt)).
(2.33)
It is easy to see that if a pair of complex-valued functions is a solution of (2.20), in which the coefficients are real constants, then their two real parts and their two imaginary parts are real valued solutions. It follows from this that (2.33) yields the two real valued solutions x = eat (A1 cos bt − A2 sin bt), (2.34) y = eat (B1 cos bt − B2 sin bt)
and
x = eat (A1 sin bt + A2 cos bt), y = eat (B1 sin bt + B2 cos bt).
(2.35)
It can be shown that these solutions are linearly independent, so the general solution in this case is x = eat (c1 (A1 cos bt − A2 sin bt) + c2 (A1 sin bt + A2 cos bt)), (2.36) y = eat (c1 (B1 cos bt − B2 sin bt) + c2 (B1 sin bt + B2 cos bt)).
Since we have already found the general solution, it is not necessary to consider the second of the two solutions (2.32). Equal real roots. When m1 and m2 have the same value of m, then (2.24) and (2.25) are not linearly independent and we essentially have only one solution x = Aemt , (2.37) y = Bemt .
52
Chapter 2
Principles of the Implicit Keller-box Method
Our experience would lead us to expect a second linearly independent solution of the form � x = Atemt , y = Btemt .
Unfortunately the matter is not quite as simple as this, and we must actually look for a second solution of the form � x = (A1 + A2 t)emt , (2.38) y = (B1 + B2 t)emt so that the general solution is � x = c1 Aemt + c2 (A1 + A2 t)emt ,
y = c1 Bemt + c2 (B1 + B2 t)emt .
(2.39)
The constants A1 , A2 , B1 , and B2 are found by substituting (2.38) into the system (2.20). Instead of trying to carry this through in the general case, we illustrate the method by showing how it works in a simple example. Problem 2.4. Show that the condition a2 b1 > 0 is sufficient, but not necessary, for the system (2.20) to have two real-valued linearly independent solution of the form (2.21). Problem 2.5. Show that the Wronskian of the two solutions (2.34) and (2.35) is given by W (t) = (A1 B2 −A2 B1 )e2at , and prove that A1 B2 −A2 B1 �= 0. Problem 2.6. Show that in formula (2.39), the constants A2 and B2 satisfy the same linear algebraic system as the constants A and B, and that consequently we may put A2 = A and B2 = B without any loss of generality. Problem 2.7. Consider the non-homogeneous linear system ⎧ dx ⎪ ⎪ = a1 (t)x + b1 (t)y + f1 (t), ⎨ dt ⎪ ⎪ ⎩ dy = a2 (t)x + b2 (t)y + f2 (t) dt
and the corresponding homogeneous system ⎧ dx ⎪ ⎪ = a1 (t)x + b1 (t)y, ⎨ dt ⎪ ⎪ ⎩ dy = a2 (t)x + b2 (t)y. dt (a) If
�
x = x1 (t), y = y1 (t)
and
�
x = x2 (t), y = y2 (t)
(∗)
(∗∗)
2.2
Finite difference methods
53
are linearly independent solutions of (∗∗), so that � x = c1 x1 (t) + c2 x2 (t), y = c1 y1 (t) + c2 y2 (t)
is its general solution, show that � x = v1 (t)x1 (t) + v2 (t)x2 (t), y = v1 (t)y1 (t) + v2 (t)y2 (t)
will be a particular solution of (∗) if the functions v1 (t) and v2 (t) satisfy the system v1 x1 + v2 x2 = f1 , v1 y1 + v2 y2 = f2 . This technique for finding particular solutions of non-homogeneous linear systems is called the method of variation of parameters. (b) Apply the method outlined in (a) to find a particular solution of the non-homogeneous system ⎧ dx ⎪ ⎪ = x + y − 5t + 2, ⎨ dt ⎪ ⎪ ⎩ dy = 4x − 2y − 8t − 8 dt
whose corresponding homogeneous system is solved in previous example.
2.2 Finite difference methods The nodal points on an interval [a, b] may be defined by xj = a + jh, j = 0, 1, . . . , N + 1 where x0 = a, xN +1 = b and h = (b − a)/(N + 1). The k-step explicit and implicit methods both for first order and second order differential equations can been derived: the expressions for derivatives in terms of function values are discussed in the previous section. The finite difference solution of the boundary value problem is obtained by replacing the differential equation at each node by a difference equation. Incorporating the boundary conditions in the difference equations, the resulting algebraic system of equations is solved. This gives the approximate numerical solution of the boundary value problem. Linear second order differential equations We consider the linear second order differential equation of the form −u + p(x)u + q(x)u = r(x), a < x < b subject to the boundary conditions of first kind: u(a) = γ1 , u(b) = γ2 .
54
Chapter 2
Principles of the Implicit Keller-box Method
The function value u(x) at xj is denoted by Uj and its approximate value by uj . Using the Taylor series we can easily verify that u(xj+1 ) − u(xj−1 ) h2 ��� − u (ηj ) 2h 6
(2.40)
u(xj+1 ) − 2u(xj ) + u(xj−1 ) h2 (4) − u (ξj ) h2 12
(2.41)
u� (xj ) = where xj−1 < ηj < xj+1 , u�� (xj ) =
where xj−1 < ξj < xj+1 . We have assumed the continuity of u(4) in writing u(4) (ξj+ ) + u(4) (ξj− ) = 2u(4) (ξj ) where xj < ξj+ < xj+1 and xj−1 < ξj− < xj . Neglecting O(h2 ) terms in (2.40) and (2.41), the finite difference approximation of the differential equation at x = xj is given by � � � � uj+1 − 2uj + uj−1 uj+1 − uj−1 − + p(xj ) + q(xj )uj = r(xj ), h2 2h 1 j N. (2.42) The boundary conditions become u0 = γ1 ,
uN +1 = γ2 .
(2.43)
Upon multiplication by h2 /2, (2.42) may be written in the form Aj uj−1 + Bj uj + Cj uj+1 = where Aj = −
1 2
h2 r(xj ), 2
j = 1, 2, . . . , N
(2.44)
� � � � � � h h h2 1 1 + p(xj ) , Bj = 1 + q(xj ) , Cj = − 1 − p(xj ) . 2 2 2 2
The system (2.44) in matrix notation, after incorporating the boundary conditions, becomes Au = b (2.45) where
b=
h2 2
�
u = [u1 , u2 , . . . , uN ]T , 2A1 r1 2CN γ2 , r(x2 ), . . . , r(xN −1 ), r(xN ) − 2 h h2 ⎡ ⎤ B1 C1 0 ⎢ ⎥ ⎢ A2 B2 C2 ⎥ ⎢ ⎥ ⎢ ⎥ . . . .. .. .. A=⎢ ⎥. ⎢ ⎥ ⎢ ⎥ AN −1 BN −1 CN −1 ⎦ ⎣
r(x1 ) −
0
AN
�T
,
BN
The solution of the system of linear equations (2.45) will give the finite difference solution.
2.2
Finite difference methods
55
Local truncation error The local truncation error Tj of (2.44) is defined by Tj = Aj Uj−1 + Bj Uj + Cj Uj+1 − h2 /2r(xj ),
where Uj = u(xj ). (2.46)
Expanding each term on the right side of (2.46) in Taylor’s series about xj , we get Tj = −
h4 (4) + (u (ξj ) − 2p(xj )u(3) (ξj− )), 24
j = 1, 2, . . . , N
(2.47)
where ξj+ ∈ (xj−1 , xj+1 ), ξj− ∈ (xj−1 , xj+1 ). Derivative boundary conditions We now consider the boundary conditions as a0 u(a) − a1 u� (a) = γ1 ,
b0 u(b) + b1 u� (b) = γ2 .
The difference approximation of the differential equation at the internal nodes, j = 1, 2, . . . , N is given by (2.44) which has N +2 unknowns in N equations. We need to find two more equations corresponding to the boundary conditions. Ignoring O(h2 ) terms in (2.40), the finite difference approximations are u1 − u−1 = γ1 a 0 u0 − a 1 2h 2ha0 2h or u−1 = − u0 + u 1 + γ1 at x = xN +1 (2.48) a1 a1 uN +2 − uN = γ2 , or b0 uN +1 + b1 2h 2hb0 2h uN +2 = uN − uN +1 + γ2 at (2.49) b1 b1 where u−1 and uN +2 are the function values at x−1 and xN +2 . The nodes x−1 and xN +2 lie outside the interval [a, b] and are called fictitious nodes. The values u−1 and uN +2 may be eliminated by assuming that the difference equation (2.44) holds also for j = 0 and N + 1, i.e., at the boundary points x0 and xN +1 . Substituting the values u−1 and uN +2 from (2.48) and (2.49) into the Eqs. (2.44) for j = 0 and j = N + 1, we obtain 2ha0 h2 2h r(x0 ) − (i) B0 − A0 u0 + (A0 + C0 )u1 = γ 1 A0 , a1 2 a1 2hb0 CN +1 uN +1 (2.50) (ii) (AN +1 + CN +1 )uN + BN +1 − b1 h2 2h = r(xN +1 ) − γ2 CN +1 . 2 b1
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The equations (2.50 i), (2.44), j = 1, 2, . . . , N and (2.50 ii) form a tridiagonal system of equations. Solution of tridiagonal system The solution of the linear second order differential equation subject to the boundary conditions leads to the solution of the system of algebraic equations in N or N + 2 unknowns whose coefficients give rise to a tridiagonal system. The LU decomposition of a tridiagonal matrix is performed by Gaussian elimination. We consider the solution of the tridiagonal system of the form ⎡ ⎤⎡ ⎤ ⎡ ⎤ b1 c 1 x1 r1 0 ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ a2 b2 c2 ⎥ ⎢ x2 ⎥ ⎢ r2 ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ a3 b3 c3 ⎢ ⎥ ⎢ x3 ⎥ ⎢ r3 ⎥ ⎢ ⎥⎢ . ⎥ = ⎢ . ⎥. (2.51) .. .. .. ⎢ ⎥⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥⎢ . ⎥ ⎢ . ⎥ . . . ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ aN −1 bN −1 cN −1 ⎥ ⎣ ⎦ ⎣ xN −1 ⎦ ⎣ rN −1 ⎦ 0 aN bN xN rN
We have
α1 = c1 /b1 ,
(2.52)
γ1 = r1 /b1 ,
(2.53)
αj = cj /bj − aj αj−1 , j = 2, 3, . . . , N γj = (rj − aj rj−1 )/(bj − aj αj−1 ), j = 2, 3, . . . , N
(2.54) (2.55)
xN = γN , xj = rj − αj xj+1 , j = N − 1, N − 2, . . . , 2, 1.
(2.56) (2.57)
with cN = 0. The solution is given by
Equations (2.54) and (2.55) are the LU decomposition and forward substitution respectively. Equation (2.57) is the backward substitution. Example 2.8. Solve the boundary value problem u�� = u + x, u(0) = 0, u(1) = 0 with h = 1/4. Use the following methods: (i) the second order method, (ii) the Numerov method. We divide the interval [0, 1] into four subintervals. The nodal points are xj = jh, 0 j 4 and h = 1/4. (i) The Second order method gives the following system of equations: uj−1 − 2uj + uj+1 = uj + xj , h2 Multiplying by −h2 , we obtain
1 j 3.
−uj−1 + 2uj − uj+1 = −h2 (uj + xj ), 1 j 3, or − 16uj−1 + 33uj − 16uj+1 = −xj .
2.2
Finite difference methods
57
We have −16u0 + 33u1 − 16u2 = −1/4 for j = 1, −16u1 + 33u2 − 16u3 = −1/2 for j = 2,
−16u2 + 33u3 − 16u4 = −3/4 for j = 3.
Using the boundary conditions ⎡ 33 −16 ⎢ ⎢ −16 33 ⎣ 0 −16
which gives
u1 = −0.034885,
u0 = u4 = 0, we get the system of equations ⎤⎡ ⎤ ⎡ ⎤ 0 u1 0.25 ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −16 ⎥ ⎦ ⎣ u2 ⎦ = − ⎣ 0.50 ⎦ 33 u3 0.75
u2 = −0.056326,
u3 = −0.050037.
(ii) The Numerov method gives the following system of equations uj−1 − 2uj + uj+1 = or We have
1 ((uj−1 + xj−1 ) + 10(uj + xj ) + (uj+1 + xj+1 )), 192 1 j 3
191uj−1 − 394uj + 191uj+1 = xj−1 + 10xj + xj+1 . 191u0 − 394u1 + 191u2 = 3 for j = 1, 191u1 − 394u2 + 191u3 = 6 for j = 2, 191u2 − 394u3 + 191u4 = 9 for j = 3.
Using the boundary conditions u0 = u4 = 0, we get the system of equations, −394u1 + 191u2 = 3,
191u1 − 394u2 + 191u3 = 6, 191u2 − 394u3 = 9, which gives u(0.25) ≈ u1 = −1136118/32415956 = −0.0350481, u(0.50) ≈ u2 = −4656/82274 = −0.0565914, u(0.75) ≈ u3 = −1629762/32415956 = −0.0502765.
The exact solution is u(x) = (sinh x/ sinh 1) − x and the close-to-exact values are u(0.25) = −0.0350476,
u(0.50) = −0.0565906,
u(0.75) = −0.0502758.
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Example 2.9. Solve the boundary value problem u�� = ux, u(0) + u� (0) = 1, u(1) = 1 with h = 1/3. Using the second order method, we obtain uj−1 − 2uj + uj+1 = h2 fj . With h = 1/3, we have four nodal points xj = jh, 0 j 3. The second order method gives the following system of equations: uj−1 − 2uj + uj+1 = (1/9)uj xj , 0 j 2. We have u−1 − 2u0 + u1 = 0 for j = 0,
u0 − 2u1 + u2 = (1/27)u1 for j = 1, u1 − 2u2 + u3 = (2/27)u2 for j = 2. Since the method is of second order, we may replace u� (0) in the boundary condition by the relation u� (0) = (u1 − u−1)/2h which is also of second order. Thus the boundary conditions become u0 + (3/2)(u1 − u−1 ) = 1 and u3 = 1. Thus we get the equations −2u0 + 3u1 = 1,
u0 − (55/27)u1 + u2 = 0,
u1 − (55/27)u2 = −1.
Solving the system of equations we get u(0) ≈ u0 = −0.9879518,
u(1/3) ≈ u1 = −0.3253012,
u(2/3) ≈ u2 = 0.3253012.
Nonlinear second order differential equations We consider the nonlinear second order differential equation u�� = f (x, u),
a J, it follows that 0 < M −1 < J −1 . (2.78) From (2.77), we have �E� �M −1 ��T � �J −1 ��T �.
(2.79)
In order to simplify (2.79) further we determine J −1 = (ji,j ) explicitly. On multiplying the rows of J by the j th column of J −1 , we have the following equations: (i) 2j1,j − j2,j = 0,
(ii) − ji−1,j + 2ji,j − ji+1,j = 0,
2 i j − 1,
(iv) − ji−1,j + 2ji,j − ji+1,j = 0,
j + 1 i N − 1,
(iii) − ji−1,j + 2ji,j − ji+1,j = 1,
(v) − jN −1,j + 2jN,j = 0.
(2.80)
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The solution of (2.80 ii) using (2.80 i) is given by ji,j = c2 i,
2 i j−1
(2.81)
where c2 is independent of i but may depend on j. Similarly, the solution of (2.80 iv) using (2.80 v) is given by ji,j = c1 (1 − i/N + 1),
j + 1 i N − 1.
(2.82)
The constant c1 depends only on j. On equating the expression for ji,j obtained from (2.81) and (2.82) for i = j, we get c2 j = c1 (1 − j/N + 1).
(2.83)
c2 + c1 /N + 1 = 1.
(2.84)
Also, on substituting the values of ji,j (i = j − 1, j + 1) obtained from (2.81) and (2.82) in (2.80 iii), we have
Finally from (2.84) and (2.83), we get c1 = j,
c2 = N − j + 1/N + 1.
On substituting the values of c1 and c2 , we have ⎧ i(N − j + 1) ⎪ ⎪ , i j, ⎨ N +1 ji,j = ⎪ ⎪ ⎩ j(N − j + 1) , i j. N +1
(2.85)
(2.86)
From (2.86) we see that J −1 is symmetric. The row sum of J −1 is given as N �
ji,j =
j=1
Hence we obtain
(xi − a)(b − xi ) i(N − i + 1) = . 2 2h2
�J −1 � = max
1 i N
Equation (2.79) becomes
N � j=1
|ji,j |
(b − a)2 . 8h2
(b − a)2 �T �. 8h2
(2.87)
1 (b − a)2 h2 M4 = O(h2 ) 96
(2.88)
�E� Substituting for �T � , we obtain �E� where M4 = max |u(4) (ξj )|. ξj ∈[a,b]
From Eq. (2.88) it follows that �E� → 0 or uj → u(xj ) as h → 0. This establishes the convergence of the second order method.
2.2
Finite difference methods
65
Stability of finite difference schemes Now we examine the stability of finite difference formulations. We take a simple second order differential equation with a significant first derivative u�� + ku� = 0
(2.89)
where k is the constant such that |k| >> 1. Three different approximations for (2.89) in which the first derivative is replaced by central, backward or forward difference respectively are 1 k (uj+1 − uj−1 ) = 0, (uj+1 − 2uj + uj−1 ) + h2 2h 1 k (ii) 2 (uj+1 − 2uj + uj−1 ) + (uj − uj−1 ) = 0, h h k 1 (iii) 2 (uj+1 − 2uj + uj−1 ) + (uj+1 − uj ) = 0. h h (i)
(2.90)
The analytical solutions of (2.89) given by u(x) = A1 + B1 e−kx
(2.91)
where A1 and B1 are arbitrary constants to be determined with the help of boundary conditions. Each of the three representations (2.90 i)–(2.90 iii) has A1 as a solution, so we examine how close are the non constant components of their solutions to e−kx . The very least that we expect of the finite difference solutions is that they behave monotonically as e−kx for k > 0 and k < 0. Equation (2.90 i) has the solution y1 = A1 + B1
2 − kh 2 + kh
j
.
If the behavior of the exponential term is analyzed, it is seen that it only gives the correct monotonic behavior for k > 0 and k < 0 if the condition h < |2/k| is satisfied. This is the condition for stability of the difference equation (2.90 i). For k very large, the stability conditions will make a central difference scheme computationally infeasible. Next, we write the solution of (2.90 ii) as uj = A1 + B1 (1 − kh)j . The analysis of the exponential term gives that if k > 0 then h < 1/k for stability. On the other hand, if k < 0 then there is no condition on h and the difference scheme (2.90 ii) is unconditionally stable. If k becomes very large and positive, then a backward difference scheme becomes infeasible. Finally, we write the solution of (2.90 iii) in the form uj = A1 + B1 (1/1 + kh)j . Again, if k < 0 then h < −1/k is the condition of stability. If k > 0, there is no condition on h and proper behavior is guaranteed for all h. Thus, if k becomes very large and negative, then a forward difference scheme becomes infeasible. Hence, for stability it is
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necessary that the different difference quotient for the first order term must be used depending on the sign of h ⎧u −u j j−1 ⎪ , k > 0, ⎨ h u� (xj ) = ⎪ ⎩ uj+1 − uj , k < 0. h
The one-sided difference is unconditionally stable and it is always on the upstream or upwind side of xj . However, it suffers from the disadvantage that it is first order accurate. Example 2.12. Solve the difference equation Δ2 yj + 3Δyj − 4yj = j 2 with the initial conditions y0 = 0, y2 = 2. We substitute for the forward differences in the difference equation and yj+2 + yj+1 − 6yj = j 2 . The general solution of the difference equation is of the form yj = yc + yp where yc is the complimentary solution and yp is the particular solution. To determine the solution yc we assume the solution is in the form yj = Aξ j and substitute into the homogenous difference equation and get the characteristic equation as ξ 2 + ξ − 6 = 0. The roots of the characteristic equation are ξ = −3 and 2. Then yc becomes yc = c1 (−3)j + c2 (2)j . We assume the particular solution is in the form yp = aj 2 + bj + c and substitute into the difference equations and obtain (a(j + 2)2 + b(j + 2) + c) + a(j + 1)2 + b(j + 1) + c − 6(aj 2 + bj + c) = j 2 . Comparing the right hand side with the left hand side we get the following equations: −4a = 1,
6a − 4b = 0,
5a + 3b − 4c = 0,
1 or a = − , 4
3 19 b=− , c=− . 8 32
The general solution becomes 1 3 19 yj = c1 (−3)j + c2 (2)j − j 2 − j − . 4 8 32 The initial conditions give c1 + c2 −
19 75 = 0 for n = 0, 9c1 + 4c2 − = 2 for n = 2, 32 32 63 32 , c2 = . or c1 = 160 160
Substituting c1 and c2 , we obtain yj =
1 (63(−3)j + 32(2)j − 40j 2 − 60j − 95). 160
2.2
Finite difference methods
67
√ Example 2.13. The system y � = z, z � = −by − az where 0 < a < 2 b, b > 0 is to be integrated by Euler’s method with known values. What is the largest step length h for which all the solutions of the corresponding difference equations are bounded? (Royal Inst. Tech., Stockholm Sweden, BIT 7 (1967), 247) Applying the Euler’s method, we obtain uj+1 = yj + hzj , zj+1 = zj + h(−byj − azj )
= −bhyj + (1 − ah)zj
which may be written as
yj+1 zj+1
=
1
h
−bh 1 − ah
yj zj
where uj = [yj
zj ]
T
and A =
or uj+1 = Auj
1
h
−bh 1 − ah
.
The characteristic equation of the matrix A is given by ξ2 −(2−ah)ξ+1−ah+ bh2 = 0. Putting ξ = (1 + z)/(1 − z) in the characteristic equation, we get the reduced characteristic equation as (4 − 2ah + bh)z 2 +2(a−bh)z +bh2 = 0. The roots of the characteristic equation will lie within the unit circle or that of the reduced characteristic equation on the left half plane of the z-axis if and only if the following conditions are satisfied: 4 − 2ah + bh2 > 0,
a − bh > 0, bh2 > 0.
√ √ This may be written as (2 − bh)2 + 2h(2 b − a) > 0 and h a/b. Thus the largest step length is given by h a/b. Example 2.14. We want to find the numerical methods for the differential equation y � = f (t, y), y(t0 ) = y0 which produces exact results for (i) y(t) = a + be−t . (ii) y(t) = a + b cos t + c sin t. (i) Since the function y(t) depends upon two arbitrary parameters a and b, we need three relations in y(t) and y � (t) to eliminate a and b to get the required numerical method. We assume a relation between y(tj+1 ), y(tj ) and y � (tj ) to obtain the single-step method. We have y(tj+1 ) = a+be−tj+1 , y(tj ) =
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a + be−tj , y � (tj ) = −be−tj or y(tj+1 ) −1 −e−tj+1 y(tj+1 ) −1 −e−tj = 0. � y (tj+1 ) 0 −e−tj
Simplifying, we get the numerical method yj+1 = yj + (1 − e−h )yj� . (ii) We have the three arbitrary parameters a, b and c and we require four relations to eliminate the parameters. We have y(tj+1 ) = a + b cos tj+1 + c sin tj+1 , y � (tj+1 ) = −b sin tj+1 + c cos tj+1 , y(tj ) = a + b cos tj + c sin tj ,
y � (tj ) = −b sin tj + c cos tj .
We eliminate a and get y(tj+1 )−y(tj ) = b(cos tj+1 − cos tj )+c(sin tj+1 − sin tj ). Eliminating b and c we obtain y(tj+1 ) − y(tj ) −(cos tj+1 − cos tj ) −(sin tj+1 − sin tj ) = 0. sin tj+1 − cos tj+1 y � (tj+1 ) � y (tj ) sin tj − cos tj
1 − cos h � (yj+1 + yj� ). sin h Example 2.15. We want to find the implicit Runge-Kutta method of the form
Simplifying, we have yj+1 = yj +
yj+1 = yj + W1 K1 + W2 K2 ,
K1 = hf (yj ),
K2 = hf (yj + a(K1 + K2 ))
for the initial value problem y � = f (y), y(t0 ) = y0 , obtain the interval of absolute stability for y � = λy, λ < 0. We have K1 = hfj , K2 = hA1 + h2 A2 + h3 A3 + · · · where Ai ’s are independent of h. Expanding K2 in Taylor’s series we get h h K2 = hfj +ha(K1 +K2 )fjy + a2 (K1 +K2 )2 fjyy + a3 (K1 +K2 )3 fjyyy +· · · . 2 6 Substituting for K1 and K2 , we obtain hA1 + h2 A2 + h3 A3 + · · ·
= hfj + ha(hfj + hA1 + h2 A2 + h3 A3 + · · ·)fjy h h + a2 (hfj + hA1 + h2 A2 + · · ·)2 fjyy + a3 (hfj + hA1 + · · ·)3 fjyyy + · · · . 2 6 Equating the coefficients of various powers of h, we have 2 A1 = fj , A2 = 2afj fjy , A3 = aA2 fjy + 2a2 fjyy fj2 = 2a2 fj fjy + 2a2 fjyy fj2 .
2.2
Finite difference methods
69
Hence, h2 �� h3 ��� y + yj + · · · 2 j 6 h2 h3 2 = yj + hfj + fj fjy + (fj fjy + fjyy fj2 ) + · · · 2 6 2 + 2a2 fjyy fj2 ) + · · ·). = yj + W1 hfj + W2 (hfj + 2ah2 fj fjy + h3 (2a2 fj fjy
yj+1 = yj + hyj� +
Comparing the terms corresponding to various powers of h, we obtain W1 + W2 = 1,
2aW2 = 1/2,
2a2 W2 = 1/6
whose solution is given by a = 1/3, W1 = 1/4, W2 = 3/4. The implicit RungeKutta method becomes 1 K1 = hf (yj ), K2 = hf yj + (K1 + K2 ) , 3 1 yj+1 = yj + (K1 + 3K2 ). 4 The order of the method is 3. Now we apply the method to the test equation y � = λy, λ < 0. We have 1 1+ h 3 K2 = hyj , K1 = hyj , 1 1− h 3 1 2 1 2 1+ h+ h 1 3 1 + 3h 3 6 y yj = yj+1 = yj + hyj + h j 1 1 4 4 1− h 1− h 3 3 1 K2 = h yj + (hyj + K2 ) , 3
where h = hλ. This is the first order difference equation and characteristic equation is given by 2 1 2 1+ h+ h 3 6 . ξ= 1 1− h 3 For absolute stability (λ < 0), we require |ξ| 1 which gives h ∈ (−6, 0).
Example 2.16. In order to illustrate the significance of the fact that even the boundary conditions of the differential equation are to be accurately approximated when difference methods are used, we examine the differential equation y �� = y with boundary condition y � (0) = 0, y(1) = 1 which has the solution y(x) = cosh x/cosh 1. We put xj = jh, assume that 1/h is an integer, and use the difference approximation yj�� ≈ (yj+1 − 2yj + yj−1 )/h2 . Two different approximations for the boundary conditions are 1. Symmetric case: y−1 = y1 ; yN = 1, N = 1/h. 2. Non-symmetric case: y0 = y1 ; yN = 1.
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(a) We show that the error y(0) − y0 asymptotically approaches ah2 in the first case, and bh in the second, where a and b are constants to be determined. (b) We show that the truncation error in the first case is O(h2 ) in the closed interval [0, 1] (Stockholm Univ., Sweden, BIT 5 (1965), 294). 1. The difference equations become h2 2 yj + yj−1 = 0. yj+1 − 2yj + yj−1 = h yj or yj+1 − 2 1 + 2 j j The solution of the difference equation is given by yj = c1 ξ1h + c2 ξ2h where ξ1h and ξ2h are the roots of the characteristic equation h2 ξ + 1 = 0, ξ2 − 2 1 + 2
where ξ1h = 1 +
1/2 1/2 h2 h2 h2 h2 , ξ2h = 1 + , +h 1+ −h 1+ 2 4 2 4 and ξ1h ξ2h = 1.
Simplifying we get ξ1h = 1+h+
h2 h 3 h3 h3 + +O(h4 ) = eh − +O(h4 ) and ξ2h = e−h + +O(h4 ). 2 8 24 24
−1 −1 Satisfying the symmetric boundary condition c1 ξ1h + c2 ξ2h = c1 ξ1h + c1 ξ2h and noting that ξ1h ξ2h = 1, we get c1 = c2 . The second boundary condition N N gives yN = c1 ξ1h + c2 ξ2h = 1. Hence,
c1 =
1 h2 N N sinh 1 + O(h3 ). , where ξ + ξ = 2 cosh 1 − 1h 2h N N 12 ξ1h + ξ2h
Thus we obtain c1 =
1 1 h2 sinh 1 h2 sinh 1 + O(h3 ) and y0 = + O(h3 ). + + 2 2 cosh 1 48 cosh 1 cosh 1 24 cosh2 1
The error at x = 0 is given by y(0) − y0 = −
h2 sinh 1 1 sinh 1 = ah2 where a = − = −0.020565. 2 24 cosh 1 24 cosh2 1
2. Satisfying the non-symmetric approximation y0 = y1 , we get ξ2h − 1 h2 c2 = 1 − h + + O(h3 ) c2 . c1 + c2 = c1 ξ1h + c2 ξ2h or c1 = − ξ1h − 1 2
2.3
Boundary value problems in ordinary differential equations
71
We also have h2 h2 N N + c2 ξ2h = c1 e 1 − yN = c1 ξ1h + O(h3 ) + c2 e−1 1 + + O(h3 ) 12 12 2 h2 h h2 = c2 e 1 − h + 1− + e−1 1 + + O(h3 ) 2 12 12 = c2 (2 cosh 1 − eh + O(h2 )) = 1.
Hence, e 1 1+ h + O(h2 ) , 2 cosh 1 2 cosh 1 e 1 1−h+ h + O(h2 ) , c1 = 2 cosh 1 2 cosh 1 h sinh 1 1 2+ , y 0 = c1 + c2 = 2 cosh 1 cosh 1 1 sinh 1 , y(0) − y0 = − y(0) = h = 12ah. cosh 1 2 cosh2 1 c2 =
Thus b = 12a = −0.24678.
2.3 Boundary value problems in ordinary differential equations We were considering so far numerical methods for solving initial value problems. In such problems all the initial conditions are given at a single point. In this section we consider problems in which the conditions are specified at more than a point. A simple example of a second order boundary value problem (Simmons [2]) is y �� (x) = y(x),
y(0) = 0,
y(1) = 1.
(2.92)
An example of a fourth order boundary problem is (i) y (4) (x) + ky(x) = q, (ii) y(0) = y � (0) = 0, (iii) y(L) = y �� (L) = 0.
(2.93)
Here y may represent the deflection of a beam of length L which is subjected to a uniform load q. Condition (2.93 ii) states that the end x = 0 is built in while (2.93 iii) states that the end x = L is simply supported. We shall consider two methods for solving such problem; the method of finite differences and an adaptation of the methods, which we shall call “shooting” methods.
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Second order equation We assume that we have a linear differential equation of order greater than one with conditions specified at the end points of an interval [a, b]. We divide the interval [a, b] into N equal parts of width h. We set x0 = a, xN = b and define xn = x0 + nh, n = 1, 2, . . . , N − 1 as the interior mesh points. The corresponding values of y at these mesh points are denoted by yn = y(x0 + nh), n = 0, 1, 2, . . . , N − 1. We shall sometimes have to deal with points outside the interval [a, b]. These will be called exterior mesh points, those to the left of x0 being denoted by x−1 = x0 − h, x−2 = x0 − 2h, etc. and those to the right of xN being denoted by xN +1 = xN + h, xN +2 = xN + 2h etc. The corresponding values of y at the exterior mesh points are denoted in the obvious way as y−1, y−2 , . . . , yN +1 , yn+2 , etc. To solve a boundary value problem by the method of finite differences, every derivative appearing in the equation, as well as in the boundary conditions, is replaced by an appropriate difference approximation. Central differences are usually preferred because they lead to greater accuracy. Some typical central difference approximations are the following yn+1 − yn−1 , 2h yn+1 − 2yn + yn−1 y �� (xn ) ≈ , h2 yn+2 − 4yn+1 + 6yn − 4yn−1 + yn−2 . y (4) (xn ) ≈ h4 y � (xn ) ≈
(2.94)
In each case the finite difference representation is an O(h2 ) approximation to the respective derivative. To illustrate the procedure, we consider the linear second order differential equation y �� (x) + f (x)y � + g(x)y = q(x)
(2.95)
under the boundary conditions y(x0 ) = α,
(2.96)
y(xN ) = β.
(2.97)
The finite difference approximation to (2.95) is yn+1 − 2yn + yn−1 f (xn )(yn+1 − yn−1 ) + g(xn )yn = q(xn ), + 2 h 2h n = 1, 2, . . . , N − 1. Multiplying through by h2 , setting f (xn ) = fn , etc. and grouping terms, we have h h 1 − fn yn−1 + (−2 + h2 gn )yn + 1 + fn yn+1 = h2 qn , 2 2 n = 1, 2, . . . , N − 1. (2.98)
2.3
Boundary value problems in ordinary differential equations
73
Since y0 and yN are specified by the conditions (2.96) and (2.97), (2.98) is a linear system of N −1 equations in the N −1 unknowns yn (n = 1, . . . , N −1). Writing out (2.98) and replacing y0 by α and yN by β, the system takes the form � � � � h h 2 2 (−2 + h g1 )y1 + 1 + f1 y2 = h q1 − 1 − f1 α, 2 2 � � � � h h 1 − f2 y1 + (−2 + h2 g2 )y2 + 1 + f2 y3 = h2 q2 , 2 2 � � � � h h 2 1 − f3 y2 + (−2 + h g3 )y3 + 1 + f3 y4 = h2 q3 , 2 2 ············ (2.99) � � � � h h 1 − fN −2 yN −3 + (−2 + h2 gN −2 )yN −2 + 1 + fN −2 yN −1 2 2 = h2 qN −2 , � � � � h h 2 2 1 − fN −1 yN −2 + (−2 + h gN −1 )yN −1 = h qN −1 − 1 + fN −1 β. 2 2
The coefficients in (2.99) can, of course, be computed since f (x), g(x), and q(x) are known functions of x. This linear system can now be solved by any of the methods. In matrix form we have Ay = b [y = (y1 , y2 , . . . , yN −1 )], representing the vector of unknowns; b, representing the vector of known quantities on the right hand side of (2.99); and A, the matrix of coefficients. The matrix A in this case is tridiagonal and of order N − 1. It has the special form ⎡ ⎤ B1 C1 ⎢ ⎥ ⎢ A2 B2 C2 ⎥ ⎢ ⎥ ⎢ ⎥ A3 B3 C3 ⎢ ⎥ ⎥. A=⎢ ⎢ ⎥ .. .. .. ⎢ ⎥ . . . ⎢ ⎥ ⎢ AN −2 BN −2 CN −2 ⎥ ⎣ ⎦ AN −1 BN −1
The system Ay = b can be solved directly using Algorithm 2.1. We need only to replace n and N − 1, identify x and y, and apply the recursion formulas of the algorithm. Returning to the boundary conditions, let us see how the system (2.99) is affected if in place of (2.96) we prescribe the following condition at x = x0 : y � (x0 ) + γy(x0 ) = 0. (2.100) If we replace y � (x0 ) by a forward difference, we will have y(x0 + h) − y(x0 ) + γy(x0 ) = 0 h
74
or, on rearranging,
Chapter 2
Principles of the Implicit Keller-box Method
y1 + (−1 + γh)y0 = 0.
(2.100)�
If we now write out (2.98) for n = 1 and then replace y0 by y1 /(1 − γh), we will have 1 − (h/2)f1 h y1 + 1 + f1 y2 = h2 q1 . (−2 + h2 g1 ) + (2.101) 1 − γh 2 The first equation of (2.99) can now be replaced by (2.101). All other equations of (2.99) will remain unchanged, and the resulting system can again be solved, using Algorithm 2.1. We note, however, that (2.100)� is only an O(h) approximation to the boundary condition (2.100). The accuracy of the solution will also then be of order h. To obtain a solution which is everywhere of order h2 , we replace (2.100) by the approximation y(x0 + h) − y(x0 − h) + γy(x0 ) = 0 h or, on rearranging,
y1 − y−1 + 2γhy0 = 0.
(2.100)��
Since we have introduced an exterior point y−1 , we must now consider y0 as well as y1 , y2 , . . . , yN −1 as unknowns. Since we now have N unknowns, we must have N equations. We can obtain an additional equation by taking n = 0 in (2.98). If we then eliminate y−1 using (2.100)�� , we will have for the first two equations h 2hγ 1 − f0 + (−2 + h2 g0 )y0 + 2y1 = h2 q0 , n = 0, 2 h h 2 1 − f0 y0 + (−2 + h g1 )y1 + 1 + f1 y1 = h2 q1 , n = 1. 2 2
The remaining equations will be the same as those appearing in (2.99). The system is still tridiagonal but now of order N . It can again be solved explicitly with the aid of the numerical algorithm. The accuracy attainable with finite difference methods will clearly depend upon the fineness of the mesh and upon the order of the finite difference approximation. As the mesh is refined, the number of equations to be solved will increase. As a result, the amount of computer time required may become excessive, and good accuracy may be difficult to achieve. The use of higher order approximations will yield greater accuracy for the same mesh size but results in considerable complication, especially near the end points of the interval where the exterior values will not be known. In practice, it is advisable to solve the linear system for several different values of h. A comparison of the solutions at the same mesh points will then indicate the accuracy being obtained. In addition, the extrapolation process, can usually be applied to yield further improvement. As adapted to
2.3
Boundary value problems in ordinary differential equations
75
the solution of finite difference systems, extrapolation to the limit proceeds as follows. Let yn (h) (n = 1, . . . , N − 1) denote the approximate solution of the boundary value problem based on N subdivision of the interval [a, b]. Let ym (h/2) (m = 1, 2, . . . , 2N − 1) be the approximate solution of the same problem based on 2N subdivision of the interval [a, b]. At the N − 1 points x1 = a + h, x2 = a + 2h, . . . , xN −1 = a + (N − 1)h, we now have two approximations, yn (h), yn (h/2). Applying extrapolation to these, we obtain 4yn(h/2) − yn (h) n = 1, 2, . . . , N − 1. 3 This extrapolation will usually produce a significant improvement in the approximation. y(xn ) ≈ yn(1) =
Example 2.17. Solve by difference method the boundary value problem d2 y + y = 0, y(0) = 0; y(1) = 1. dx2 Solution: Taking f (x) = 0, g(x) = −1, q(x) = 0, and setting y0 = 0, yn = 1 in (2.99), we obtain the system (−2 + h2 )y1 + y2 = 0, yn−1 + (−2 − h2 )yn + yn+1 = 0, yN −2 + (−2 − h )yN −1 = −1. 2
n = 2, 3, . . . , N − 2,
This is a system of N − 1 equations in the N − 1 unknowns. This system was solved on the IBM 7090 with h = 0.1 and h = 0.05 using a subroutine based on the usual algorithm. The results are given in Table 2.1. The fourth column gives the extrapolated values at intervals of 0.1 obtained from the formula 4yh/2 (xn ) − yh (xn ) . 3 The values in the last column are obtained from the exact solution to the problem y(x) = sinh x/sinh 1. These results show that for h = 0.1 the solution is correct to three to four significant decimals and for h = 0.05 to four to five decimals, while the extrapolated solution is correct to about seven decimals. To obtain seven significant decimals of accuracy without extrapolation would require a subdivision of the interval [0, 1] into approximately 100 mesh points (h = 0.01). yn (1) =
Table 2.1
Computer results for Example 2.17
xn
yn (h = 0.05)
yn (h = 0.1)
yn (1)
y(xn )
0 0.05 0.10 0.15 0.20 0.25
0 0.04256502 0.08523646 0.12812098 0.17132582 0.21495896
0
0
0.08524469
0.08523372
0.17134184
0.17132048
0 0.04256363 0.08533369 0.12811689 0.17132045 0.21495239
76
Chapter 2
Principles of the Implicit Keller-box Method
Continued xn
yn (h = 0.05)
yn (h = 0.1)
yn (1)
y(xn )
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.0
0.25912950 0.30394787 0.34952610 0.39597815 0.44342014 0.49197068 0.54175115 0.59288599 0.64550304 0.69973386 0.75571401 0.81358345 0.87348684 0.93557395 1
0.25915240
0.25912187
0.34955449
0.34951663
0.44345213
0.44340946
0.54178427
0.54174010
0.64553425
0.64549263
0.75573958
0.75570550
0.87350228
0.87348166
1
1
0.25912183 0.30393920 0.34951659 0.39596794 0.44340942 0.49195965 0.54174004 0.59287506 0.64549258 0.69972418 0.75570543 0.81357635 0.87348163 0.93557107 1
Problem 2.18. Solve by difference method the boundary value problem d2 y + y = 0, dx2
y(0) = 0; y(1) = 1.
Take h = 1/4, and solve the resulting system, using a desk calculator. Solution: y1 = 0.2943, y2 = 0.5702, y3 = 0.8104. Compare this solution with the exact solution y = sin x/ sin 1. Problem 2.19. Solve the boundary value problem (2.92) with the condition y(0) = 0 replaced by the condition y (0) + y(0) = 0, using a mesh h = 0.1. Problem 2.20. Write an O(h2 ) finite difference system for approximating the solution of the boundary value problem y +xy +y = 2x, y(0) = 1; y(1) = 0. Let h = 0.1 and write the system in matrix form. Then solve this system, using computer program (Krishna Murthy and Sen [3]). Problem 2.21. Show that the Gauss-Seidal iterative method can also be used to solve the system of (2.92) and obtain this solution by iteration to four significant figures of accuracy. For this problem, is the direct method more efficient than the iterative method? Fourth order equation We now solve the boundary value problem (2.93) by difference methods. Using the difference approximations in (2.94) with x0 = 0, xN = L, and N h = L,
2.3
Boundary value problems in ordinary differential equations
77
we will have (i) yn−2 − 4yn−1 + (6 + kh4 )yn − 4yn+1 + yn+2 = h4 q, n = 1, 2, . . . , N − 1,
(ii) y0 = 0,
(iii) yN = 0,
(2.102)
y−1 = y1 ,
yN +1 = −yN −1 .
Writing out (2.102 i) for n = 1, 2, . . ., N − 1, and using the boundary conditions (2.102 ii) and (2.102 iii), we obtain (7 + kh4 )y1 − 4y2 + y3 = h4 q,
−4y1 + (6 + kh4 )y2 − 4y3 + y4 = h4 q,
yn−2 − 4yn−1 + (6 + kh4 )yn − 4yn−1 + yn−2 = h4 q, n = 3, 4, . . . , N − 3,
(2.103)
yN −4 − 4yN −3 + (6 + kh4 )yN −2 − 4yN −1 = h4 q,
yN −3 − 4yN −2 + (5 + kh4 )yN −1 = h4 q.
In this case the matrix is again band type, the nonzero elements appearing only along the principal five diagonals. An algorithm is again available for solving directly such systems. To describe this algorithm, we write (2.103) in matrix form Ay = b: ⎡
⎤
C1 D1 E1
⎢ ⎢ B2 C2 ⎢ ⎢ ⎢ A3 B3 ⎢ ⎢ A4 Ay = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
D2
E2
C3
D3
E3
B4 .. .
C4 .. .
D4 .. .
E4
AN −2 BN −2 CN −2
AN −1 BN −1
⎥⎡ ⎤ ⎡ ⎤ ⎥ y1 b1 ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎢ y2 ⎥ ⎢ b 2 ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎥⎢ . ⎥ = ⎢ . ⎥. ⎥⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎣ yN −2 ⎦ ⎣ bN −2 ⎦ ⎥ DN −2 ⎥ bN −1 ⎦ yN −1 CN −1 (2.104)
Algorithm 2.1. Direct solution of the five-diagonal system. The algorithm consists in applying the following steps: 1. The initial values ω1 = C1 ,
β1 = D1 ω1 , γ0 = 0,
β0 = 0,
βN −1 = 0,
γN −1 = γN −2 = 0.
γ 1 = D 1 ω1 ,
78
Chapter 2
Principles of the Implicit Keller-box Method
2. Compute recursively δn = Bn − An βn−2 ,
ωn = Cn − An γn−2 − δn βn−1 ,
βn = (Dn − δn γn−1 )/ωn ,
n = 2, 3, . . . , N − 1,
γn = En /ωn .
3. Compute h0 = 0,
h1 = b1 /ω1 , hn = (bn − An hn−2 − δn hn−1 )/ωn , n = 2, 3, . . . , N − 1.
4. Compute the values of y backward using yN −1 = hN −1 ,
yn = hn − βn yn+1 − γn yn+2 ,
n = N − 2, N − 3, . . . , 1.
Example 2.22. Solve the boundary value problem, Example 2.17, with the condition y(0) = 0 replaced by the condition y � (0) + y(0) = 0, using a mesh h = 0.1. get
Using finite difference method and taking k = 1, L = 1, q = 16 · 104 , we Table 2.2 n
yn (h = 0.05)
yn (h = 0.025)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0 24.812481 87.296463 176.498330 282.463930 396.237960 509.863350 616.380620 709.827130 785.236390 838.637520 867.054840 868.507460 842.009190 787.568410 706.188280 599.867140 471.598860 325.373700 166.178990 0
0 23.362666 84.603290 172.758530 277.864600 390.956600 504.067960 610.229860 703.470540 778.814480 832.282220 860.890000 862.649340 836.866950 782.644590 701.879050 596.262590 468.783160 323.425320 165.170940 0
yn (1) 22.879394 83.705566 171.511623 276.331490 389.196146 502.213616
2.3
Boundary value problems in ordinary differential equations
79
In the matrix form the system (2.103) then becomes ⎡
7 + h4
⎢ ⎢ −4 ⎢ ⎢ ⎢ 1 ⎢ ⎢ Ay = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎡ ⎤ 1 ⎢ ⎥ ⎢1⎥ ⎢ ⎥ 4 ⎢ .. ⎥ = h q⎢ . ⎥. ⎢ ⎥ ⎢ ⎥ ⎣1⎦ 1
−4
6 + h4 −4 1
⎤
1 −4
6 + h4 −4 .. .
1 −4
6+h .. . 1
1 4
−4 .. .
1 .. .
−4 6 + h4 1
−4
⎥⎡ ⎤ ⎥ y1 ⎥ ⎥⎢ ⎥ ⎥ ⎢ y2 ⎥ ⎥⎢ ⎥ ⎥ ⎢ .. ⎥ ⎥⎢ . ⎥ ⎥⎢ ⎥ ⎥⎢ ⎥ ⎥ ⎣ yN −2 ⎦ ⎥ −4 ⎥ ⎦ yN −1 5 + h4
(2.105)
On identifying the elements of this coefficient matrix with those of (2.104), we can now apply a suitable algorithm to obtain the solution. The second column gives the results for h = 0.05 (N = 20 subdivisions), and the third column the results for h = 0.025 at the points corresponding to n = 20 subdivisions. There are at most two significant figures of agreement between the two solutions, indicating that the results for h = 0.025 are probably correct to at most two or three significant figures. More accurate results can be obtained by increasing the number of subdivisions or by applying h2 extrapolation. The first few extrapolated values are given under the column headed yN (1). Problem 2.23. The deflection y of an elastic beam under a transverse load q(x) is governed by the equation EIy (4) = q(x). If the beam is simply supported at the end x = 0 and rigidly supported at the end x = L, the boundary conditions are y(0) = 0, y �� (0) = 0; y(L) = 0, y � (L) = 0. (a) Write a finite difference approximation to this boundary value problem. (b) Let E = I = 1, and let q(x) = 1. Solve the system for h = 0.1, using a suitable algorithm. Compare this with the exact solution y = (q/48)(2x4 − 3x3 + x).
Problem 2.24. Use extrapolation to the limit to complete the fourth column of the table in Example 2.22.
Problem 2.25. Write an O(h2 ) difference approximation to the boundary value problem y (4) + 25x2 y = 25x; y(0) = y � (0) = 0, y(1) = y �� (1) = 0. Solve the system of equations using h = 1/16, h = 1/32 and then improve the results by extrapolation.
80
Chapter 2
Principles of the Implicit Keller-box Method
Shooting method For linear boundary-problems, a number of methods can be used. The method of differences described above works reasonably well in such cases. Other methods attempt to obtain linearly independent solutions of the differential equation and to combine them in such a way as to satisfy the boundary conditions. (Conte, de Boor [4]) For nonlinear equations, the latter method cannot be used. Difference methods can be adapted to nonlinear problems, but they require guessing at a tentative solution and then improving this by an iterative process. In addition to complexity of the programming required, there is no guarantee of convergence of the iterations. The shooting method to be described in this section applies equally well to linear and nonlinear problems. Again, there is no guarantee of convergence, but the method is easy to apply, and when it does converge, it is usually more efficient than the other methods. Consider again the problem given in (2.92). We wish to apply the initial value methods, but to do so we must know both y(0) and y (0). Since y (0) is not prescribed, we consider it as an unknown parameter, say α, which must be determined so that the resulting solution yields the prescribed value y(1) to some desired accuracy. We therefore guess at the initial slope and set up an iterative procedure for converging to the correct slope. Let α0 , α1 be two guesses at the initial slope y (0), and let y(α0 ; 1), y(α1 ; 1) be the values of y at x = 1 obtained from integrating the differential equation. y(α; 1) is plotted as a function of α: Now, a better approximation to α can now be obtained by linear interpolation. The intersection of the line joining p0 to p1 with the line y(1) = 1 has its α coordinate given by α2 = α0 + (α1 − α0 )
y(1) − y(α0 ; 1) . y(α1 ; 1) − y(α0 ; 1)
(2.106)
We now integrate the differential equation using the initial values y(0) = 0, y (0) = α2 , to obtain y(α2 ; 1). Again, using linear interpolation based on α1 , α2 , we can obtain the next approximation α3 . The process is repeated until convergence has been obtained, i.e., until y(α1 ; 1) agrees with y(1) = 1 to the desired number of places. There is no guarantee that this iterative procedure will converge. The rapidity of convergence will clearly depend upon how good the initial guesses are. Estimates are sometimes available from physical considerations, and sometimes from simple graphical representations of the solution. (See Figures 2.1 and 2.2.) For a general second order boundary value problem y = f (x, y, y ),
y(0) = y0 ,
y(b) = yb ,
(2.107)
the procedure is summarized in the algorithm. Algorithm 2.2. The shooting method for second order boundary value problems:
2.3
Boundary value problems in ordinary differential equations
81
1. Let αk be an approximation to the unknown initial slope y � (0) = α2 (choose the first two α0 , α1 using physical intuition). 2. Solve the initial-value problem. 3. y �� = f (x, y, y � ), y(0) = y0 , y � (0) = αk from x = 0 to x = b, using any of the methods. Call the solution y(αk , b) at x = b. 4. Obtain the next approximation from the linear interpolation αk+1 = αk−1 + (αk − αk−1 )
y(b) − y(α0 ; b) , y(α1 ; b) − y(α0 ; b)
k = 1, 2, . . . .
5. Repeat Step 2 and 3 until |y(αk , b) − yb | < ε for a prescribed ε.
The iteration used in the above is an application of the secant method. For systems of equations of higher order, this procedure becomes considerably more complicated, and convergence more difficult to obtain. The general situation for a nonlinear system may be represented as follows. We consider a system of four equations in four unknowns: x� = f (x, y, z, w, t),
y � = g(x, y, z, w, t),
z = h(x, y, z, w, t),
w� = l(x, y, z, w, t)
�
Fig. 2.1 Approximate solution curves at x = 1
Fig. 2.2 Approximations to α
(2.108)
82
Chapter 2
Principles of the Implicit Keller-box Method
where now t represents the independent variable. We are given two conditions at t = 0, say x(0) = x0 , y(0) = y0 and two conditions at t = T , say z(T ) = zT and w(T ) = wT . Let z(0) = α, w(0) = β be the correct initial values of z(0), w(0), and let α0 , β0 be guesses for these initial values. Now integrate the system (2.108), and denote the values of z and w obtained at t = T by z(α0 , β0 ; T ) and w(α0 , β0 ; T ). Since z and w at t = T are clearly functions of α and β, we may expand z(α0 , β0 ; T ) and w(α0 , β0 ; T ) into a Taylor series for two variables through linear terms z(α, β; T ) = z(α0 , β0 ; T ) + (α − α0 ) +(β − β0 )
∂z (α0 , β0 ; T ) ∂α
∂z (α0 , β0 ; T ), ∂β
w(α, β; T ) = w(α0 , β0 ; T ) + (α − α0 ) +(β − β0 )
∂w (α0 , β0 ; T ) ∂α
(2.109)
∂w (α0 , β0 ; T ). ∂β
We may set z(α0 , β0 ; T ) and w(α0 , β0 ; T ) to their desired values zr and wr , but before we can solve (2.109) for the corrections α − α0 and β − β0 we must obtain the partial derivatives in (2.109). We do not know the solutions z and w and therefore cannot find these derivatives analytically. However, we can find approximate numerical values for them. To do so, we solve (2.108) once with the initial conditions x0 , y0 , α0 , β0 , once with the conditions x0 , y0 , α0 + Δα0 , β0 , and then with conditions x0 , y0 , α0 , β0 + Δβ0 , where Δα0 and Δβ0 are small increments. Omitting the variables x0 , y0 which remain fixed, we then form the difference quotients: ∂z z(α0 , β0 + Δβ0 ; T ) − z(α0 , β0 ; T ) (α0 , β0 ; T ), ≈ Δβ0 ∂β w(α0 , β0 + Δβ0 ; T ) − w(α0 , β0 ; T ) ∂w (α0 , β0 ; T ), ≈ Δβ0 ∂β ∂z z(α0 + Δα0 , β0 ; T ) − z(α0 , β0 ; T ) ≈ (α0 , β0 ; T ), Δα0 ∂α w(α0 + Δα0 , β0 ; T ) − w(α0 , β0 ; T ) ∂w (α0 , β0 ; T ). ≈ Δα0 ∂α After replacing z(α, β; T ) by zr and w(α, β; T ) by wr , we can solve (2.109) for the corrections δα0 = α − α0 and δβ0 = β − β0 , to obtain new estimates α1 = α0 + δα0 and β1 = β0 + δβ0 for the parameters α and β. The entire process is now repeated, starting with x0 , y0 , α1 , β1 as the initial conditions. Each iteration thus consists in solving the system (2.108) three times. In general, if there are n unknown initial parameters then each of the iterations will require n + 1 solutions of the original system. The method used here is equivalent to a modified Newton’s method for finding the roots of equations
2.3
Boundary value problems in ordinary differential equations
83
in several variables. Boundary value problems constitute one of the most difficult classes of problems to solve on a computer. Convergence is by no means assured, good initial guesses must be available, and considerable trial and error as well as large amounts of machine time, are usually required. Example 2.26. Solve the problem (2.92) using the shooting method. Start with the initial approximations α0 = 0.3 and α1 = 0.4 to y � (0) and h = 0.1. The solution given below was obtained using the AMRK (Adam Moulton Runge Kutta) differential equation solver described, combined with linear interpolation based on (2.106) (Table 2.3). The iteration is stopped by the condition |αk+1 − αk | < 1.10−6 . Table 2.3 Iteration no.
αk
y(αk ; 1)
1 2 3 4
0.30000000 0.40000000 0.85091712 0.85091712
0.35256077 0.47008103 0.99999999 0.99999999
The correct value of y � at x = 0 is sinh−1 1 ≈ 0.85091813. Convergence for this problem is very rapid. Moreover, the indicated accuracy is exceptionally good, considering the coarse step size used. To obtain comparable accuracy using the finite difference method of the earlier section it would requires a step size h = 0.01. Nevertheless, the finite difference method might still be computationally more difficult. Example 2.27. Solve the nonlinear boundary value problem y �� + 1 + y �2 = 0, y(0) = 1, y(1) = 2 by the shooting method. Solution: Let α0 = 0.5, α1 = 1.0 be two approximations to the unknown slope y � (0). Using again the AMRK package and linear interpolation with a step size h = 1/0.4, the following results were obtained: Table 2.4 α1
y(α1 ; 1)
0.5000000 0.9999999 1.7071071 1.9554118 1.9982968 1.9999940 2.0000035
0.9999999 1.4142133 1.8477582 1.9775786 1.9991463 1.9999952 2.0000000
The correct slope at x = 0 is y � (0) = 2. After seven iterations, the initial slope is seen to be correct to six significant figures, while the value of y at x = 1 is correct to at least seven significant figures. After the first three iterations, convergence could have been stopped up by using quadratic interpolation.
84
Chapter 2
Principles of the Implicit Keller-box Method
The required number of iterations will clearly depend upon the choice of the initial approximations α0 and α1 . These approximations can sometimes be obtained from graphical or physical considerations. Keller-box method An alternative implicit method due to Keller is now described and is referred to as the Box method. This method has several very desirable features that make it appropriate for the solution of all parabolic partial differential equations. The main features of this method are 1. Only slightly more arithmetic to solve than the Crank-Nicolson method. 2. Second order accuracy with arbitrary (non-uniform) x and y spacing. 3. Allows very rapid variations. 4. Allows easy programming of the solution of large numbers of coupled equations. The solution of an equation by this method can be obtained by the following four steps: 1. Reduce the equation or equations to a first order system. 2. Write difference equations using central differences. 3. Linearize the resulting algebraic equations (if they are nonlinear), and write them in matrix-vector form. 4. Solve the linear system by the block-tridiagonal-elimination method. To solve equations by this method, we first express it in terms of a system of two first order equations by letting (i) T � = p
and (ii) p� =
Pr ∂T u . ν ∂x
(2.110)
Here the primes denote differentiation with respect to y. The finite difference form of the ordinary differential equation (2.110 i) is written for the midpoint (xn , yj−1/2 ) of the segment p1 p2 shown in Figure 2.3, and the finite difference form of the partial differential equation (2.110 ii) is written for the midpoint (xn−1/2 , yj−1/2 ) of the rectangle p1 p2 p3 p4 .
Fig. 2.3 Finite difference grid for the Box method
2.3
Boundary value problems in ordinary differential equations
85
This gives n pnj + pnj−1 Tjn − Tj−1 = = pnj−1/2 , hj 2 � � (2.111) n−1 n − pn−1 pn−1 Pr n−1/2 Tj−1/2 − Tj−1/2 1 pnj − pnj−1 j j−1 = uj−1/2 (ii) + . 2 hj hj ν kn
(i)
Note that both h and k can be non-uniform. Here xn−1/2 = 1/2(xn + xn−1/2 ) and yj−1/2 = 1/2(yj + yj−1 ). Rearranging both expressions we can write them in the form hj n (p − pnj−1 ) = 0, 2 j n−1 n (S1 )j pnj + (S2 )j pnj−1 + (S3 )j (Tjn − Tj−1 ) = Rj−1/2 . n − Tjn − Tj−1
Here
(S1 )j = 1,
(S2 )j = −1,
(S3 )j = −λj /2,
n−1 n−1 n−1 Rj−1/2 = −λj + Tj−1/2 + pn−1 , j−1 − pj 2 Pr n−1/2 hj . λj = u ν j−1/2 kn
(2.112)
(2.113)
As before, the superscript on uj−1/2 is not necessary but is included for generality. Equations (2.112) are imposed for j = 1, 2, . . . , J − 1. Aj = 0 and for J, we have (2.114) T0 = Tw , TJ = Te respectively. Since Eq. (2.112) are linear, as are the corresponding boundary conditions given by Eq. (2.114), the system may be written at once in matrix vector form as shown below, without the linearization needed in the case of the finite difference equations for the velocity field. ⎡ ⎤ 1 0 0 0 ⎢ ⎥ −h1 ⎢ −1 −h1 ⎥ ⎡ �T � ⎤ ⎡ �(r1 )0 � ⎤ 1 0 ⎢ ⎥ p0 2 2 ⎢ ⎥ ⎢ �(r2 )0 � ⎥ ⎢ (S ) (S ) (S ) (S ) ⎢ ⎥ � ⎥ 0 0 ⎥ ⎢ Tj � ⎥ 2 j 3 j 1 j ⎢ 3 j ⎥ ⎢ ⎢ (r1 )j ⎥ . = ⎢ ⎥ ⎢ ⎥ p j (r ) ⎢ −hj+1 ⎥ ⎣ −hj+1 ⎢ 0 ⎦ ⎣� 2 j �⎥ ⎦ 1 0 −1 ⎢ ⎥ � � (r1 )J 2 2 ⎥ ⎢ TJ ⎢ pJ (r2 )J (S3 )J (S2 )J (S3 )J (S1 )J ⎥ ⎣ ⎦ 0 0 1 0 (2.115) (r1 )0 = Tw , (r2 )j = 0,
n−1 (r1 )j = Rj−1/2 ,
1 j J − 1,
1 j J,
(r2 )J = Te .
(2.116)
86
Chapter 2
Principles of the Implicit Keller-box Method
The system of equations given by Eq. (2.115) can be written as Aδ = r,
(2.117)
where ⎡
⎤
A0 C0
⎢ ⎥ ⎢ B1 A1 C1 ⎥ ⎢ ⎥ ⎢ ⎥ . . . .. .. .. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A=⎢ ⎥, Bj Aj Cj ⎢ ⎥ ⎢ ⎥ .. .. .. ⎢ ⎥ . . . ⎢ ⎥ ⎢ ⎥ BJ−1 AJ−1 CJ−1 ⎦ ⎣ BJ AJ δj =
�
Tj pj
�
,
rj =
�
⎡
δ0
⎤
⎢ ⎥ ⎢ δ1 ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎢ ⎥ δ = ⎢ ⎥, ⎢ δj ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎣ ⎦ δJ
(r1 )j (r2 )j
�
⎡
r0
⎤
⎢ ⎥ ⎢ r1 ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎢ ⎥ r = ⎢ ⎥, ⎢ rj ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎣ ⎦ rJ
(2.118) (2.119)
and Aj , Bj , Cj are 2 × 2 matrices defined as follows ⎡ ⎡ ⎤ ⎤ 1 0 (S3 )j (S1 )j A0 ≡ ⎣ −h1 ⎦ , Aj ≡ ⎣ −hj+1 ⎦ , 1 j J − 1, −1 −1 2 2 � � � � (S3 )J (S1 )J (S3 )J (S2 )J AJ ≡ , BJ ≡ , 1 j J, (2.120) −1 0 0 0 ⎡ ⎤ 0 0 Cj ≡ ⎣ −hj+1 ⎦ , 1 j J − 1. 1 2
Note that, as in the Crank-Nicolson method, the implicit nature of the method has again generated a tridiagonal matrix, but the entries are 2 × 2 blocks rather than scalars. The solution of Eq. (2.117) by the block-elimination method consists of two sweeps. In the forward sweep we compute Γj , Δj , and wj from the recursion formulas given by Δ0 = A0 ,
Γj Δj−1 = Bj , w0 = r0 ,
Δj = Aj − Γj Cj−1 ,
wj = rj − Γj wj−1 ,
1 j J, (2.121)
1 j J.
Here Γj has the same structure as Bj , that is, � � (γ11 )j (γ12 )j Γj ≡ , 0 0
(2.122)
References
87
and although the second row of Δj has the same structure as the second row of Aj , ⎤ ⎡ (α11 )j (α12 )j Δj ≡ ⎣ −hj+1 ⎦ . −1 2 For generality we write it as � � (α11 )j (α12 )j Δj ≡ . (α21 )j (α22 )j In the backward sweep, δj is computed from the following recursion formulas: ΔJ δJ = wJ ,
Δj δj = wj − Cj δj+1 ,
j = J − 1, J − 2, . . . , 0.
References [1]
[2] [3] [4]
M.K. Jain, S.R.K. Iyengar, R.K. Jain, Numerical Methods for Scientific and Engineering Computation, 3rd edition, New Age International publishers, New Delhi, 1995. G.F. Simmons, Differential Equations with Applications and Historical notes, 2nd edition, Tata McGraw-Hill, New York, 2003. E.V. Krishna Murthy, S.K. Sen, Numerical Algorithms, Computations in Science and Engineering, 2nd edition, East-west Press limited, New Delhi, 1986. S.D. Conte, C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, 3rd ed., McGraw-Hill, New York, 1980.
Chapter 3 Stability and Convergence of the Implicit Keller-box Method
In this chapter, we study in detail the stability and convergence of an implicit Keller-box method. Classical solutions of initial-boundary value problems are obtained in Section 3.1. Also, a complete convergence analysis is presented. It is shown through an example that the new method is considerably better than the explicit scheme. The proof of stability is based on a theorem on difference functional inequalities. Nonlinear estimates of the Perron type for given functions (with respect to functional variables) are used. Finite difference schemes for some two point boundary value problems are analyzed in Section 3.2. It is found that for schemes defined on non-uniform grids, the order of the local truncation error does not greatly effect the rate of convergence of the numerical approximation. Numerical results presented indicate that this is also the case for higher dimensional problems. Stability and convergence of numerical algorithms used to solve one dimensional fractional advection-dispersion equation are presented in Section 3.3. Furthermore, the Crank-Nicholson algorithm is applied to a one-dimensional fractional advection-dispersion equation with variable coefficients on a finite domain. Application of these results is illustrated by modeling a radial flow problem. Use of the space-fractional derivative allows the model equation to capture the early arrival of the tracer observed at a field site.
90
Chapter 3
Stability and Convergence of the Implicit Keller-box Method
3.1 Convergence of implicit difference methods for parabolic functional differential equations 3.1.1
Introduction
For any two metric spaces X and Y we denote by C(X, Y ) the class of all continuous functions defined on X and taking values in Y . Let M [n] denote the set of all n × n real matrices. We will use vectorial inequalities, understanding that the same inequalities hold between their corresponding components. Let E = [0, a] × (−b, b),
D = [−d0 , 0] × [−d, d],
where a > 0, b = (b1 , . . . , bn), bi > 0 for 1 i n, d0 ∈ R+ , d = (d1 , . . . , dn ) ∈ Rn+ and R+ = [0, +∞). We put c = (c1 , . . . , cn ) = b + d and ∂0 E = [0, a] × ([−c, c]\(−b, b)),
E0 = [−d0 , 0] × [−c, c],
Ω = E ∪ E0 ∪ ∂0 E.
For a function z : Ω → R and for a point (t, x) ∈ [0, a] × [−b, b] we define a function z(t,x) : D → R as follows: z(t,x) (τ, y) = z(t + τ, x + y) for (τ, y) ∈ D. The function z(t,x) is the restriction of z to the set [t − d0 , t]×[x − d, x + d], and this restriction is shifted to the set D. Elements of space C(D, R) will be denoted by w, w and so on. The maximum norm in the space C(D, R) will be denoted by �.�D . Write Σ = E × C(D, R) × Rn and suppose that f : Σ → M [n], f = [fij ]i,j=1,...,n , g : Σ→ R, φ : E0 ∪ ∂0 E → R, are given functions. We consider the functional differential equation ∂t z(t, x) =
n
fij (t, x, z(t,x), ∂x z(t, x))∂xi xj z(t, x) + g(t, x, z(t,x) , ∂x z(t, x))
i,j=1
with the initial boundary condition z(t, x) = φ(t, x) for (t, x) ∈ E0 ∪ ∂0 E,
(3.1) (3.2)
where ∂x z = (∂x1 z, . . . , ∂xn z). We are interested in establishing a method of numerical approximation of classical solutions of problem (3.1) subject to (3.2) by means of solutions of associated difference functional equations and in estimating of the difference between exact and approximate solutions. In recent years a number of papers concerned with numerical methods for parabolic differential or functional differential equations were published. Difference approximations of nonlinear equations with initial boundary conditions of Dirichlet type were considered
3.1
Convergence of implicit difference methods for parabolic functional ...
91
in [1], [2], [3], [4]. The convergence results for a general class of difference methods related to parabolic problems and solutions defined on unbounded domains can be found in [5], [6]. Various monotone iterative methods and finite difference schemes for computing of numerical solutions of reaction diffusion equations with time delay were presented in [7], [8]. Difference methods for nonlinear parabolic problems have the following property: It is easy to construct an explicit Euler’s type difference scheme, which satisfies consistency conditions on all classical solutions of the original problems. The main task in these considerations is to find a finite difference scheme, which is stable. The methods of difference inequalities or simple theorems on recurrent inequalities are used in the investigations of the stability. The convergence results are also based on a general theorem on the error estimate of numerical solutions for functional difference equations of the Volterra type with an unknown function of several variables. Implicit difference methods for nonlinear parabolic differential equations were considered in [9]–[11]. Classical solutions of initial boundary value problems of the Dirichlet type for nonlinear equations without mixed derivatives are approximated in [9], [10] by solutions of difference schemes, which are implicit with respect to time variables. The paper [11] deals with initialboundary value problems of the Neumann type for nonlinear equations with mixed derivatives. Semi-linear parabolic equations with initial boundary conditions of the Dirichlet type were considered in [12]. It is shown that there are implicit difference schemes which are convergent. Classical solutions of quasi-linear parabolic differential functional equations and implicit difference methods are investigated in [13]. The proofs of the stability of implicit difference methods considered in [2], [9], [11], [12] are based on a comparison technique. Implicit difference functional inequalities generated by nonlinear parabolic differential functional equations were investigated in [14]. The results presented in [13], [14], [9]–[11], [12] are not applicable to problems (3.1), (3.2). In this book we present a new class of numerical methods for parabolic functional differential equations with initial-boundary conditions of the Dirichlet type. The numerical methods are difference schemes, which are implicit with respect to the time variable. We give sufficient conditions for the convergence of the methods and we show by examples that the new methods are considerably better than the classical schemes. The proof of the convergence is based on a comparison technique with nonlinear estimates of the Perron type with respect to the functional variable.
3.1.2
Discretization of mixed problems
We will denote by F (X, Y ) the class of all functions defined on X and taking values in Y , where X and Y are arbitrary sets. We formulate now a difference problem corresponding to (3.1), (3.2). We will denote by N and Z the set of
92
Chapter 3
Stability and Convergence of the Implicit Keller-box Method
natural numbers and the set of integers, respectively. For x, y ∈ Rn , U ∈ M [n] where x = (x1 , . . . , xn ), We write �x� =
n i=1
y = (y1 , . . . , yn ),
|xi |, �U � =
n
i,j=1
U = [uij ]ij=1,...,n .
|uij | and x ∗ y = (x1 y1 , . . . , xn yn ). We
define a mesh on Ω in the following way. Let (h0 , h ) where h = (h1 , . . . , hn ) stands for steps of the mesh. For h = (h0 , h ) and (r, m) ∈ Z1+n where m = (m1 , . . . , mn ) we define nodal points as follows: t(r) = rh0 ,
x(m) = m ∗ h ,
(m1 )
x(m) = (x1
, . . . , xn(mn ) ).
Let us denote by H the set of all h = (h0 , h ) such that there exist K0 ∈ Z and K = (K1 , . . . , Kn ) ∈ Zn satisfying the conditions K0 h0 = d0 and K ∗ h = d. We define N = (N1 , . . . , Nn) ∈ Nn as follows. Let 1 i n. If di = 0, then Ki = 0 and we assume that there is Ni ∈ N such that Ni hi = bi = ci . If di > 0, then there is Ni ∈ N such that (Ni − 1)hi < bi Ni hi . There is N0 ∈ N such that N0 h0 a < (N0 + 1)h0 . For h ∈ H we put R1+n = h {(t(r) , x(m) ) : (r, m) ∈ Z1+n } and , Dh = D ∩ R1+n h
Eh = E ∩ R1+n , h
, ∂0 Eh = ∂0 E ∩ R1+n h
E0.h = E0 ∩ R1+n , h
Ωh = Ω ∩ R1+n . h
Let �.�Dh denote the maximum norm in the space F (Dh , R). Put Ωr,h = Ω ∩ ([−d0 , t(r) ] × Rn ), where 0 r N0 . For z : Ωh → R we write z (r,m) = z(t(r) , x(m) ). Let
Eh = {(t(r) , x(m) ) ∈ Eh : (t(r+1) , x(m) ) ∈ Eh }.
For a function z : Ωh → R and for a point (t(r) , x(m) ) ∈ Eh we define the function z[r,m] : Dh → R by z[r,m] (τ, y) = z(t(r) + τ, x(m) + y),
(τ, y) ∈ Dh .
The function z[r,m] is the restriction of z to the set ([t(r) − d0 , t(r) ] × [x(m) − d, x(m) + d]) ∩ Ωh , and this restriction is shifted to that set Dh . Write J = {(i, j) : 1 i, j n, i �= j}.
3.1
Convergence of implicit difference methods for parabolic functional ...
93
Let ei = (0, . . . , 0, 1, 0, . . . , 0) ∈ Rn be a vector with 1 on the i-th position. We define δi+ z (r,m) = (1/hi )(z (r,m+ei ) − z (r,m) ),
δi− z (r,m) = (1/hi )(z (r,m) − z (r,m−ei ) ),
where 1 i n. Let us denote by δ0 , δ = (δ1 , . . . , δn ), δ(2) = [δi,j ]i,j=1,...,n difference operators corresponding to partial deviations ∂t , ∂x = (∂x1 , . . . , ∂xn ),
δxx = [∂xi xj ]i,j=1,...,n
respectively. Write Σh = Eh� × F (Dh , R) × Rn and suppose that fh : Σh → M [n], gh : Σh → R,
fh = [fh,ij ]i,j=1,...,n , φh : E0.h ∪ ∂0 Eh → R
are given functions. Let z be an unknown function of the variables (t(r) , x(m) ) ∈ Ωh . Set
P (r,m) [z] = (t(r) , x(m) , z[r,m] , δz (r,m) )
where δz = (δ1 z, . . . , δn z). We consider the difference functional equation δ0 z (r,m) =
n
fh,ij (P (r,m) [z])δij z (r+1,m) + gh (P (r,m) [z])
(3.3)
i,j=1
with the initial-boundary condition (r,m)
z (r,m) = φh
on E0.h ∪ ∂0 Eh ,
(3.4)
where δ0 z
(r,m)
1 = h0
z
(r+1,m)
n 1 (r,m+ei ) (r,m−ei ) − (z +z ) , 2n i=1
(3.5)
1 + (r,m) + δi− z (r,m) ), 1 i n. (3.6) (δ z 2 i Difference operators corresponding to partial derivatives of the second order are defined in the following way. Put δi z (r,m) =
δii z (r+1,m) = δi+ δi− z (r+1,m) for 1 i n.
(3.7)
Suppose that (tr , x(m) ) ∈ Eh� and that the function z is known on the set Ωr.h . Let (r,m)
J−
[z] = {(i, j) ∈ J : fh.ij (P (r,m) [z]) 0},
(r,m)
J+
(r,m)
[z] = J\J−
[z]. (3.8)
94
Chapter 3
Stability and Convergence of the Implicit Keller-box Method
Write δij z (r+1,m) =
1 + − (r+1,m) (r,m) (δ δ z + δi− δj+ z (r+1,m) ) for (i, j) ∈ J− [z], (3.9) 2 i j
1 + + (r+1,m) (r,m) + δi− δj− z (r+1,m) ) for (i, j) ∈ J+ [z]. (δ δ z 2 i j (3.10) The difference functional problem (3.3) subject to (3.4) is considered as an implicit numerical formulation of problem (3.1), (3.2). The corresponding explicit difference scheme has the form δij z (r+1,m) =
δ0 z (r,m) =
n
fh,ij (P (r,m) [z])δij z (r+1,m) + gh (P (r,m) [z]).
(3.11)
i,j=1
It is clear that there exists exactly one solution u ¯h : Ωh → R of problem (3.11). We prove that under natural assumptions on given functions and on the mesh there exists exactly one solution uh : Ωh → R of the implicit difference scheme (3.3), (3.4). Solutions of (3.3), (3.4) are considered as approximate solutions of (3.1), (3.2). We give sufficient conditions for the convergence of sequences of approximate solutions to a classical solution of (3.1), (3.2).
3.1.3
Solvability of implicit difference functional problems
We prove a maximum principle for implicit parabolic difference functional inequalities. The difference functional equation z (r+1,m) = h0
n
fh,ij (P (r,m) [z])δij z (r+1,m)
(3.12)
i,j=1
is a principal part of (3.3). The maximum principle asserts that a solution of difference functional inequalities corresponding to (3.12) cannot have a positive maximum (or negative minimum) on the set Eh . Assumption H[fh ]. Suppose that the matrix fh : Σh → M [n] is symmetric and the estimate n 1 fh.ii (P ) − hi
j=1,j�=i
1 |fh.ij (P )| 0, hi
i = 1, . . . , n
(3.13)
is satisfied for P = (t, x, w, q) ∈ Σh and for h ∈ H. Remark 3.1. Suppose that fh.ii (P )
n
j=1,j�=i
|fh.ij (P )|, P ∈ Σh , i = 1, . . . , n,
3.1
Convergence of implicit difference methods for parabolic functional ...
95
and h1 = h2 = · · · = hn . Then condition (3.13) is satisfied .Write B
(r,m)
n n 1 (r,m) [z] = −2 fh,ii (P [z]) + h2i i=1
(r,m)
Ci
[z] =
j=1,j�=i
1 fh,ii (P (r,m) [z]) − h2i
n
j=1,j�=i
1 |fh,ij (P (r,m) [z])|, (3.14) hi hj
1 |fh,ij (P (r,m) [z])|, hi hj
1 i n. (3.15)
Theorem 3.1. Suppose that Assumption H[fh ] is satisfied and 0 r N0 − 1 is fixed. If zh : Ωr+1.h → R satisfies the implicit difference inequality (r+1,m)
zh
h0
n
(r+1,m)
fh.ij (P (r,m) [zh ])δij zh
(3.16)
i,j=1
for (t(r+1) , x(m) ) ∈ Eh and μ = (μ1 , . . . , μk ) ∈ Z n is such that (r+1,μ)
zh (r+1,μ)
and zh
(r+1,m)
= max{zh
: (t(r+1) , x(m) ) ∈ Ωr+1.h }
> 0, then (t(r+1) , x(m) ) ∈ ∂0 Eh .
Proof: The assumption (3.16) is equivalent to z (r+1,m) (r+1,m)
h0 z h +
h0 2
h0 − 2
B (r,m) [zh ] + h0
(r,m)
(i,j)∈J+
(r,m)
(i,j)∈J+
n
(r,m)
Ci
(r+1,m+ei )
[zh ](zh
(r+1,m−ei )
+ zh
)
i=1
[zh ]
[zh ]
1 (r+1,m+ei +ej ) (r+1,m−ei −ej ) fh.ij (P (r,m) [zh ])(zh + zh ) hi hj 1 (r+1,m+ei −ej ) (r+1,m−ei +ej ) fh.ij (P (r,m) [zh ])(zh + zh ). hi hj (3.17)
Let us suppose that (t(r+1) , x(μ) ) ∈ Eh . We conclude from (3.17) that n (r+1,μ) (r+1,μ) (r,μ) (r,μ) B zh h0 zh [zh ] + 2 Ci [zh ] +
[zh ]
[zh ]
(r,m)
(i,j)∈J+
−
i=1
(r,m)
(i,j)∈J−
1 fh.ij (P (r,μ) [zh ]) hi hj
1 (r,μ) fh.ij (P [zh ]) = 0. hi hj (r+1,m)
This contradicts our assumption that zh
> 0. This proves the theorem.
96
Chapter 3
Stability and Convergence of the Implicit Keller-box Method
Remark 3.2. Suppose that Assumption H[fh ] is satisfied and h ∈ H. Then Theorem 3.1 asserts that solutions of the implicit difference inequality (3.16) cannot have a positive maximum on Eh . It is clear that solutions of inverse implicit difference inequalities cannot have a negative minimum on Eh . Lemma 3.1. Suppose that fh : Σh → M [n] and gh : Σh → R, φh : E0.h ∪ ∂0 Eh → R and Assumption H[fh ] is satisfied. Then there is exactly one solution uh : Ωh → R of problem (3.3), (3.4). Proof: Suppose that 0 r N0 − 1 is fixed and that uh is a solution of problem (3.3), (3.4) on Ωr.h . Consider the difference problem δ0 z (r,m) =
n
fh,i,j (P (r,m) [uh ])δij z (r+1,m) + gh (P (r,m) [uh ]),
(3.18)
i,j=1
for (t(r+1) , x(m) ) ∈ ∂0 Bh . z (r+1,m) = φ(r+1,m) r
(3.19)
It follows from Theorem 3.1 that the problem consisting of the difference equation (3.12) with P (r,m) [uh ] instead of P (r,m) [z] and boundary condition (r+1,m) =0 z (r+1,m) = 0 for (t(r+1) , x(m) ) ∈ ∂0 Bh has exactly one solution zˆh (r+1,m) for −N < m < N. Then there is exactly one solution uh , −N < m < N of (3.18), (3.19) and uh is defined on Ωr+1.h . Since uh is given by (3.4) on Ω0.h , then we obtain the lemma by induction with respect to r, 0 r N0 .
3.1.4
Approximate solutions of difference functional problems
We will denote by Fh the Niemycki operator corresponding to (3.3), i.e., Fh [z](r,m) =
n
fh,ij (P (r,m) [z])δij z (r+1,m) + gh (P (r,m) [z]).
i,j=1
Let us suppose that uh : Ωh → R is the solution of problem (3.3), (3.4) and uh : Ωh → R satisfies the following conditions: (r,m)
|δ0 νh
− Fh [νh ](r,m) | γ(h) on Eh� ,
|(νh − φh )(r,m) | α0 (h) on E0.h ∪ ∂0 Eh ,
(3.20) (3.21)
where γ, α0 : H → R+ and
lim γ(h) = 0,
h→0
lim α0 (h) = 0.
h→0
(3.22)
The function uh satisfying the above conditions is considered as an approximate solution of problem (3.3), (3.4). We prove a theorem on an estimate
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Convergence of implicit difference methods for parabolic functional ...
97
of the difference between the exact and approximate solutions of (3.3), (3.4). Put Ih = {t(r) : 0 r N0 )}, Ih� = Ih \{t(N0 ) }. For a function η : Ih → R we write η (r) = η(t(r) ). Assumption H[σh ]. The function σh : Ih� × R+ → R+ is such that 1) σh is non decreasing with respect to the second variable and σh (t, 0) = 0 for t ∈ Ih� . 2) For c 1 the difference problem η (r+1) = η (r) + h0 cσh (t(r) , η (r) ), η
(0)
=0
0 r N0 − 1,
(3.23) (3.24)
is stable in the following sense: If γ, α0 : H → R+ are such functions that lim γ(h) = 0,
h→0
lim α0 (h) = 0,
h→0
and ηh : Ih → R+ is a solution of the difference problem η(r+1) = η(r) + h0 cσh (t(r) , η (r) ) + h0 γ(h), η(0) = α0 (h),
0 r N0 − 1,
(3.25)
(3.26) (3.27)
then there is α : H → R+ such that (r)
ηh α0 (h) for t(r) ∈ Ih , ˆ (h = 0). lim α
h→0
Assumption H[fh gh ]. The functions fh : Σh → M [n] and qh : Σh → R of variables (t, x, w, p) are such that 1) Assumption H[fh ] is satisfied. 2) fh (t, x, w, ·) ∈ C(Rn , M [n]), gh (t, x, w, ·) ∈ C(Rn , R) where (t, x, w, ·) ∈ � Eh × F (Dh , R); the derivatives ∂pk fh = [∂pk fh.ij ]i,j=1,...,n ,
1 k n,
∂p gh = (∂p1 gh , . . . , ∂pn gh )
(3.28) (3.29)
exist on Σh and ∂pk fh (t, x, w, ·) ∈ C(Rn , M [n]),
1 i n,
3) Assumption H[σh ] is satisfied and
∂p gh (t, x, w, ·) ∈ C(Rn , Rn ).
�fh (t, x, w, q) − fh (t, x, w, ¯ q)� σh (t, �w − w� ¯ Dh ), on Σh . Let
¯ q)� σh (t, �w − w� ¯ Dh ) �gh (t, x, w, q) − gh (t, x, w, Ak = sup{�∂pk fh (t, x, w, q)� : (t, x, w, q) ∈ Σh },
Bk = sup{�∂pk gh (t, x, w, q)� : (t, x, w, q) ∈ Σh },
where 1 k n.
98
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Stability and Convergence of the Implicit Keller-box Method
Theorem 3.2. Suppose that H[fh , gh ] is satisfied and 1) uh : Ωh → R is a solution of (3.3), (3.4). 2) The function vh : Ωh → R satisfies (3.20)–(3.21), and there is c˜ ∈ R+ such that (r,m) |δij νh | c˜, i, j = 1, . . . , n, h ∈ H. 3) The following estimates are satisfied
1 h0 (˜ cAk + Bk ) 0, − n hk Then there is α : H → R+ such that
1 k n.
|(uh − νh )(r,m) | α(h) on Eh ,
(3.30)
lim α(h) = 0.
(3.31)
h→0
(r,m)
= Fh [νh ](r,m) + Proof: Let Γh : Eh� → R be defined by the relation δ0 νh (r,m) � Γh on Eh . (r,m) | γ(h) on Eh� . Write zh = νh − uh . It follows from (3.20) that |Γh Then we have n (r,m) (r+1,m) δ 0 zh = fh.ij (P (r,m) [uh ])δij zh i,j=1 n
+
i,j=1
(r+1,m)
(fh.ij (P (r,m) [vh ]) − fh.ij (P (r,m) [uh ]))δij vh (r,m)
+gh (P (r,m) [vh ]) − gh (P (r,m) [uh ]) + Γh
Let (r,m) (P, P˜ ) = Ξk
n
(r+1,m)
∂pk fh.ij (P )δij vh
+ ∂pk gh (P˜ ),
i,j=1
.
1 k n, P, P˜ ∈ Σh .
The above relations and (3.5)–(3.10) imply (r+1,m)
zh
(1 − B (r,m) [uh ]) = h0
+h0
(r,m)
(i,j)∈J−
−h0
(r,m)
(i,j)∈J+
[uh ]
[uh ]
n
(r,m)
Ci
(r+1,m+ei )
[uh ](zh
i=1
1 (r+1,m+ei +ej ) (r+1,m−ei −ej ) fh.ij (P (r,m) )[uh ])(zh + zh ) 2hi hj
k=1
(r,m+ek )
zh
)
1 (r+1,m+ei −ej ) (r+1,m−ei +ej ) fh.ij (P (r,m) )[uh ])(zh + zh ) 2hi hj
1 h0 (r,m) ˜ + Ξk Q, Q 2 n hk k=1 n h0 (r,m) 1 (r,m−ek ) 1 ˜ . − zh Ξk Q, Q + 2 n hk (r,m) 1
+h0 Γh
n
(r+1,m−ei )
− zh
(3.32)
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Convergence of implicit difference methods for parabolic functional ...
99
We conclude from (3.14), (3.15) that (r,m)
1 − B (r,m) [uh ] 0, Ci
[uh ] 0 for 1 i n
(3.33)
and n
(r,m)
Ci
[uh ] +
i=1
(i,j)∈J
1 |fh.ij (P (r,m) [uh ])| + B (r,m) [uh ] = 0. (3.34) hi hj
Write (r)
(˜ r,m)
εh = max{|zh
| : r˜ r, (t(˜r) , x(m) ) ∈ Eh },
0 r N0 .
It follows from (3.32), (3.34) that the function εh satisfies the recurrent inequality (r+1)
εh
(r)
(r)
εh + ch0 σ(t(r) , εh ) + h0 γ(h),
0 r N0 ,
(r)
where c = (˜ c + 1) and εh α0 (h). Let us denote by ηh : Ih → R+ the solution of (3.26), (3.27) with c = c˜+1. (r) (r) Then εh ηh for 0 r N0 . The stability of problem (3.23), (3.24) gives the existence of such a function α ˜ : H → R+ , that lim α ˜ (h) = 0 and h→0
(r,m)
|zh
| α ˜ (h). Thus (3.31) is satisfied with α = α ˜.
3.1.5
Convergence of implicit difference methods
Now we give an example of the operator fh corresponding to (3.1), (3.2), and we prove that the implicit difference method is convergent. Equation (3.1) contains the functional variable z(t,x) which is an element of space C(D, R). Then we need an interpolating operator Th : F (Dh , R) → F (D, R). We give an example of such operator as follows. Put � = {λ = (λ1 , . . . , λn ) : λi ∈ {0, 1} for 0 i n}. Let ω ∈ F (Dh , R) and (t, x) ∈ D. There exists (r, m) ∈ Z1+n such that t(r) t t(r+1) , x(m) x x(m+1) and (t(r) , x(m) ), (t(r+1) , x(m+1) ) ∈ Dh where m + 1 = (m1 + 1, . . . , mn + 1). We define λ 1−λ t − t(r) (r+1,m+λ) x − x(m) x − x(m) ω Th ω(t, x) = 1− h0 h� h� λ∈� 1−λ λ t − t(r) (r,m+λ) x − x(m) x − x(m) + 1− ω 1− h0 h� h� λ∈�
where
x − x(m) h�
λ
=
λi n xi − x(mi )
i=1
hi
,
100
Chapter 3
Stability and Convergence of the Implicit Keller-box Method
1−λ 1−λi n x − x(m) xi − x(mi ) = 1− 1− h� hi i=1
and we take 00 = 1 in the above formulas. We approximate classical solutions of (3.1), (3.2) with solutions of the difference equation δ0 z (r,m) =
n
fij (t(r) , x(m) , Th z[r,m] , δz (r,m) )δij z (r+1,m)
i,j=1
+g(t(r) , x(m) , Th z[r,m] , δz (r,m) )
(3.35)
with initial boundary condition (3.4). The operators δ0 and δ = (δ1 , . . . , δn ), δii , 1 i n are defined by (3.5)–(3.7) respectively. Difference operators δij for (i, j) ∈ J are given by (3.8)–(3.10) with fh (t, x, w, p) = f (t, x, Th w, p).
Assumption H[σ]. Suppose that σ : [0, a] × R+ → R+ is a continuous function such that 1) σ is non-decreasing with respect to both variables. 2) σ(t, 0) = 0 for t ∈ [0, a] and the maximal solution of the Cauchy problem (3.36) ζ � (t) = cσ(t, ζ(t)), ζ(0) = 0 is ζ(t) = 0 for t ∈ [0, a], c > 1.
Assumption H[f, q]. For the functions f and q of variables (t, x, w, q), the following conditions hold: 1) For P = (t, x, w, q) ∈ E × C(D, R) × Rn and for h ∈ H, we have n 1 fii (P ) − hi
j=1,j�=i
1 |fii (P )| 0, hj
i = 1, . . . , n.
(3.37)
2) f = (t, x, w, ·) ∈ C(Rn , M [n]), g(t, x, w, ·) ∈ C(Rn , R), the derivatives ∂pk f = [∂pk fij ]i,j=1,...,n ,
1 k n,
∂p g = (∂p1 g, . . . , ∂pn g)
(3.38)
exist on Σ, and for each (t, x, w, ·) ∈ E × C(D, R) we have
∂pk f (t, x, w, ·) ∈ C(Rn , M [n]), and ∂p g(t, x, w, ·) ∈ C(Rn , Rn )
for i, j = 1, . . . , n, and 1 k n. 3) Assumption H[σ] is satisfied and the estimates
�f (t, x, w, q) − f (t, x, w, q)� σ(t, �w − w�D ), �g(t, x, w, q) − g(t, x, w, q)� σ(t, �w − w�D )
hold on Σ. Remark 3.3. If fii (P )
n
j=1,j�=j
|fij (P )|,
P ∈ Σ, 1 i n
(3.39)
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Convergence of implicit difference methods for parabolic functional ...
101
and h1 = h2 = · · · = hn , then the condition (3.37) is satisfied. Note that conn fij (P )ξi ξj 0, ξ = (ξ1 , . . . , ξn ) ∈ Rn , which dition (3.39) implies that i,j=1
means that problem (3.1), (3.2) is parabolic in the sense of Walter [15].
Lemma 3.2. Suppose that the function w : D → Rk , w = (w1 , . . . , wk ) is of class C 2 and C˜ ∈ R+ is a constant such that |∂tt w(t, x)|,
|∂xi xj w(t, x)|,
˜ |∂xi t w(t, x)| C,
(t, x) ∈ D,
1 i, j n.
2 ˜ Then �Th [wh ] − w�D C�h� where wh is the restriction of w to the set Dh . The proof of the lemma is given in [16], Chapter 5. Write
Ak = sup{�∂pk f (t, x, w, q)� : (t, x, w, q) ∈ Σ}, Bk = sup{�∂pg(t, x, w, q)� : (t, x, w, q) ∈ Σ}, 1 k n. Now we can prove a theorem on the convergence of method (3.5), (3.4). Theorem 3.3. Suppose that assumption H[f, g] is satisfied and the following hold: 1) The function v : Ω → R is the solution of (3.1), (3.2) and v is of class C 2 on Ω, and h0 1 − (˜ cAk + Bk ) 0, k = 1, . . . , n (3.40) n hk where h ∈ H and c˜ ∈ R+ is defined by the relations |∂xi xj v(t, x)| c˜,
i, j = 1, . . . , n, (t, x) ∈ E.
2) There is c0 > 0 such that hi h−1 j c0 , i, j = 1, . . . , n. 3) The function uh : Ωh → R is a solution of (3.3), (3.4) and there is α0 : H → R+ such that (r,m)
|v(r,m) − uh
| α0 (h) on E0.h ∪ ∂0 Eh and lim α(h) = 0. h→0
(3.41)
Then there exists a function α : H → R+ such that we have (r,m)
|uh
(r,m)
− vh
| α(h) on Eh� and lim α(h) = 0 h→0
(3.42)
where vh is the restriction of v to the set Ωh and h ∈ H.
Proof: We first show that the problem
η (r+1) = η (r) + h0 cσ(t(r) , η (r) ), η (0) = 0
0 r N0 − 1,
(3.43) (3.44)
is stable for c 1. Let the functions γ, α0 : H → R+ be such that the condition (3.25) is satisfied. Consider the difference problem η (r+1) = η (r) + h0 cσ(t(r) , η (r) ) + h0 γ(h),
0 r N0 − 1,
η (0) = α0 (h)
102
Chapter 3
Stability and Convergence of the Implicit Keller-box Method
and its solution ηh : Ih → R+ . Denote by ωh the maximal solution of the Cauchy problem ζ � (t) = cσ(t, ζ(t)) + γ(h), ζ(0) = α0 (h). There is ε0 > 0 such that for �h� < ε0 the function ωh is defined on [0, a] and lim ωh (t) = 0 uniformly on [0, a]. It is easily seen that ηhi ωhi ωh (a) h→0
for 0 i n0 . Then problem (3.43), (3.44) is stable. Put fh (t, x, w, p) = f (t, x, Th w, p),
gh (t, x, w, p) = g(t, x, Th w, p)
where (t, x, w, p) ∈ Σh . Then �fh (t, x, w, p) − fh (t, x, w, ¯ p)� σ(t, �Th (w − w)� ¯ D) σ(t, �(w − w)� ¯ Dh ) on Σh and
|gh (t, x, w, p) − gh (t, x, w, p)| σ(t, �w − w�Dh ) on Σh .
Write
F˜h [vh ](r,m) =
n
(r,m)
fij (t(r) , x(m) , Th (vh )[r,m] , δvh
(r+1,m)
)δij vh
i,j=1 (r,m)
+g(t(r) , x(m) , Th (vh )[r,m] , δvh
).
It follows from Lemma 3.2 and from Condition 1 of Theorem 3.3 that there is γ : H → R+ such that (r,m)
|δ0 vh
− F˜h [vh ](r,m) | γ(h),
(t(r) , x(x) ) ∈ Eh�
and lim γ(h) = 0. The above relations and (3.41) imply that vh is an aph→0
proximate solution of (3.5), (3.4). Then all the assumptions of Theorem 3.3 are satisfied and we obtain (3.42) with α(h) = ωh (a). Remark 3.4. In Theorem 3.3 we need estimates of the derivatives ∂xx v of the solution of problem (3.1), (3.2). One may obtain them by the method of differential inequalities ([17], Chapter XIII and [15], Chapter IV). Remark 3.5. Suppose that Assumption H[f, g] is satisfied with σ(t, p) = Lp, (t, p) ∈ [0, a] × R+ where L ∈ R+ . Then we have assumed that f and g satisfy the Lipschitz condition with respect to the functional variable. We obtain the following error estimates: ecLa − 1 on Eh if L > 0, and cL (i,m) (i,m) �uh − vh � α(h) ˜ + a˜ γ (h) on Eh if L = 0. (i,m)
�uh
(i,m)
− vh
cLa � α(h)e ˜ + γ˜ (h)
The above inequalities follows from (3.42) with α(h) = ωh (a) where wh : [0, a] → R+ is a solution of the problem ζ � (t) = cLζ(t) + γ˜(h), ζ(0) = α0 (h).
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Convergence of implicit difference methods for parabolic functional ...
103
Remark 3.6. There are the following consequences of our approach: In classical theorems concerning explicit difference methods for (3.1), (3.2) it is assumed that (see [1]) n 1 fij (t, x, z(t,x) , ∂x z(t, x)) h2 i=1 i 1 |fii (t, x, z(t,x) , ∂x z(t, x))| 0. hi hj
1 − 2h0 +h0
(3.45)
(i,j)∈J
It is important in our considerations that we have omitted the above assumption for Eq. (3.1). The above condition is replaced by (3.40).
3.1.6
Numerical examples
Consider E = [0, 0.5] × [−1, 1] × [−1, 1], E0 = {0} × [−1, 1] × [−1, 1], ∂0 E = [0, 0.5] × [([−1, 1] × [−1, 1])]/((−1, 1) × (−1, 1)).
Consider the differential equation with deviated variables x y ∂xy z(t, x, y) ∂t z(t, x, y) = ∂xx z(t, x, y) + ∂yy z(t, x, y) + sin xyz t, , 2 2 x+y x−y , + f (t, x, y) (3.46) +z t, 2 2 and the initial-boundary condition z(t, x, y) = 0 on ∂0 E ∪ E0
(3.47)
where f (t, x, y) = 16(x2 − 1)(y 2 − 1) − 32t(x2 + y 2 − 2) −t((x + y)2 − 4)((x + y)2 − 4) − sin(txy(x2 − 4)(y 2 − 4)). The solution of (3.46), (3.47) is known, it is v(t, x, y) = 16t(x2 − 1)(y 2 − 1). We found the approximate solutions of (3.46), (3.47), using both implicit and explicit numerical method and taking the following steps of the mesh: h0 = 0.005, h1 = 0.005, h2 = 0.005. Note that the function f and the steps of the mesh do not satisfy condition (3.6), which is necessary for the explicit method to be convergent. In our numerical example, the average errors of the explicit method exceeded 1020 , while the average errors εh for fixed t(r) of implicit method are given in Table 3.1.
104
Chapter 3
Table 3.1
Stability and Convergence of the Implicit Keller-box Method
errors (εh ) when h0 = 0.005, h1 = 0.005, h2 = 0.005
t = 0.001
t = 0.050
t = 0.100
t = 0.150
t = 0.200
0.00004
0.00166
0.00295
0.00403
0.00495
Let us consider the integral-differential equation ∂t z(t, x, y) = ∂xx z(t, x, y) + ∂yy z(t, x, y) + xy∂xy z(t, x, y) π2 π2 x y z(t, τ, s)dsdτ + 1+ z(t, x, y) − xy 2 64 −x −y π π + cos xt cos y (3.48) 2 2 and the initial-boundary condition (3.49)
z(t, x, y) = 0 on ∂0 E ∪ E0
where E, E0 , ∂E0 are defined as earlier. The solution of (3.48), (3.49) is known, it is π π v(t, x, y) = (et − 1) cos x cos y. 2 2 Likewise, in the previous numerical example, we chose mesh step which does not satisfy the condition (3.6). In accordance with our expectations, the explicit method is not convergent, and the average errors are so big, that it is impossible for the personal computer to count them, while the implicit method is convergent and gives the following average errors (Table 3.2): Table 3.2
errors (εh ) when h0 = 0.005, h1 = 0.005, h2 = 0.005
t = 0.001
t = 0.050
t = 0.100
t = 0.150
t = 0.200
0.000002
0.000305
0.000924
0.001808
0.003081
The above examples show that there are implicit difference schemes which are convergent and the corresponding classical methods are not convergent. This is due to the fact that we need the relation (3.6) for steps of the mesh in the classical case. We do not need this condition in our implicit method. Implicit difference methods in earlier sections have the potential for applications in the numerical solving of differential integral equations or equations with deviated variables.
3.2
Rate of convergence of finite difference scheme on uniform/non-uniform ...
105
3.2 Rate of convergence of finite difference scheme on uniform/non-uniform grids
3.2.1
Introduction
It is well known that finite difference schemes which use uniform spacings are unsatisfactory for problems where the solution has rapid local variation such as boundary layers. The basic problem is that if enough points are used to resolve the layers, the computational effort becomes unacceptably large. On the other hand, if no enough points are used to resolve the layers, the numerical solution may be a poor approximation even in the interior. It is therefore necessary to consider finite difference schemes based on irregular grid points. For problems where only first order derivatives need to be discretized, simple finite difference schemes such as the Keller-box scheme (Keller [18]) appear to work well, and although a general theory is not available, it has been shown by Weiss [19] that for some classes of problems, it is necessary only to resolve the layer regions adequately. However it has been found that finite difference schemes with grid spacings that change discontinuously do not yield satisfactory results for second and higher order equations. In fact, Crowder and Dalton [20] claim that even if a large number of points are used in the boundary layers, such schemes can lead to numerical solutions that are worse overall than the numerical solutions obtained using the corresponding finite difference scheme on a regular grid with the same number of points. The explanation given for this phenomenon (see for example Bolttner [21], de Rives [22], Jones and Thompson [23]) is that the local truncation error may be much larger for a scheme defined on an even grid than for the corresponding scheme defined on an even grid. For example (see also Section 2), an elementary Taylor series argument can be used to show that a three point approximation to the second derivative of a function evaluated at a grid point can be second order accurate if the grid is uniform but is only first order accurate for an arbitrary grid. Thus, for second order problems, many finite difference schemes used in practice will have a local truncation error that is an order less accurate than the corresponding scheme on a uniform grid. While it is usually possible to devises finite difference scheme on non-uniform grids that are second order accurate, such schemes may have other undesirable properties such as being non-conservative. This fact has motivated the use of smooth coordinate transformations to generate nonuniform grids. For such grids, the local truncation error of a finite difference scheme is of the same order as the local truncation error of the corresponding scheme on a regular grid. Examples of such coordinate transformations may be found in Bolttner [21], de Rivas [22], Jones and Thompson [23], and Orszag and Israeli [24].
106
Chapter 3
Stability and Convergence of the Implicit Keller-box Method
Although the technique of using coordinate transformations to generate non-uniform grids works well in practice, the justification for its use is misleading in the sense that an assessment of the accuracy of a numerical solution based solely on the order of the local truncation error of the scheme can be incorrect. To support this claim, we examine a class of three point finite difference schemes for the numerical solution of a second order system of ordinary differential equations. It is found that the asymptotic rate of convergence of the approximate solutions obtained by the finite difference schemes with an arbitrary nonuniform grid is the same as the asymptotic rate of convergence of the approximate solutions obtained by finite difference schemes with a nonuniform grid generated by a smooth coordinate transformation. Of course the order of the local truncation error will not be the same. However it is pointed out that it is possible for a scheme to have zero order accuracy for the local truncation error and still be convergent. These analytical results are verified numerically and further calculations are presented that indicate that the results extend to higher dimensions.
3.2.2
Analytical results
Consider the second order system of ordinary differential equation (Ly)(x) := y �� (x) + p(x)y � (x) + Q(x)y(x) = f (x),
0 0. Two equal and opposite forces are applied along the x-axis so that the wall is stretched keeping the origin fixed, as shown schematically in Figure 4.1. The steady two-dimensional viscous boundary layer equations are ∂u ∂v + = 0, ∂x ∂y ∂u ∂u 1 ∂τxx ∂τxy u +v = + , ∂x ∂y ρ ∂x ∂y
(4.1) (4.2)
where u and v are the velocity components along the x and y directions,
4.1
Flow of a power-law fluid over a stretching sheet
123
Fig. 4.1 A sketch of the physical problem
respectively. ρ is the density and τij is the stress tensor defined by τij = 2K(2Dkl Dkl )(n−1)/2 Dij . Here Dij ≡
1 2
∂uj ∂ui + ∂xj ∂xi
(4.3)
denotes the strain-rate tesnsor, K is called the consistency coefficient, and n is the power-law index. The n = 1 case represents a Newtonian fluid with constant dynamic coefficient of viscosity K, while n < 1 and n > 1 correspond to the cases of pseudoplastic and dilatants fluids, respectively. A discussion of literature on boundary layer flow of power-law fluids can be found in [8]. When the tangential stress in Eq. (4.2) is at least of the same order of magnitude as the normal stress, ∂τxx /∂x is, by usual boundary layer arguments, negligible in comparison with ∂τxy /∂y. For the problem in hand, we have ∂u/∂y 0. This gives the shear stress as τxy = −K(−∂u/∂y)n.
(4.4)
Using the boundary layer approximations, the momentum equation (4.2) reduces to the form n ∂u K ∂ ∂u ∂u +v =− − u . (4.5) ∂x ∂y ρ ∂y ∂y It should be pointed out here that due to the entrainment of the ambient fluid, the boundary layer flow over a stretching sheet is different from the Blasius boundary layer flow over a semi-infinite rigid flat plate. It is also important to note that the flow is caused solely by the stretching of the wall and there is no free stream velocity outside the boundary layer. The corresponding boundary conditions are therefore u = Cx and v = 0 at y = 0, u → 0 as y → ∞
(4.6)
124
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
where the constant C is positive. It can be easily shown that the above system consisting of (4.1) and (4.5) subject to the boundary conditions (4.6) admits similarity solutions in terms of the similarity function f and the similarity variable η defined by
and where
ψ = (C 1−2n /(K/ρ))−1/(n+1)x2n/(n+1) f (η)
(4.7)
η = y(C 2−n /(K/ρ))1/(n+1) x(1−n)/(n+1)
(4.8)
u = ∂ψ/∂y and v = −∂ψ/∂x
(4.9)
n(−f �� )(n−1) f ��� − (f � )2 + (2n/(n + 1))f f �� = 0.
(4.10)
define the physical stream function in this problem. Using the transformation (4.7)–(4.8) in (4.5), we obtain
The boundary conditions (4.6) become f � (0) = 1 and f (0) = 0,
f � (∞) = 0.
(4.11)
One can readily obtain the equation for a Newtonian fluid as a special case by putting n = 1 in Eqs. (4.10) and (4.11). In this case (4.10) reduces to f ��� − (f � )2 + f f �� = 0
(4.12)
with the boundary conditions (4.11). Crane [1] obtained the exact analytical solution of (4.12) satisfying (4.11) as f = 1 − exp(−η).
4.1.3
(4.13)
Numerical solution method
For n �= 1, we have not been able to find an exact analytical solution of (4.10) satisfying (4.11). Hence, we solved Eq. (4.10) subject to the boundary conditions (4.11) numerically by the efficient shooting method developed by Keller [9] and described, for instance, by Cebeci and Bradshaw [10]. This numerical scheme is based on a standard fourth order Runge-Kutta integration technique for first order ordinary differential equations. A Newton iteration procedure is employed to provide quadratic convergence of the iteration required to satisfy the outer boundary condition (4.11), and the solution is accepted when (4.11) is satisfied to within a tolerance error of 10−5 . Depending on the boundary layer thickness δ, two different grid spacings Δη = 0.05 and 0.10 were used. For the special case n = 1, the numerical results differ from the exact analytical solution (4.13) due to Crane [1] by not more than 0.0001%.
4.1
Flow of a power-law fluid over a stretching sheet
4.1.4
125
Results and discussion
Converged numerical solutions were obtained for several values of the powerlaw index in the range 0.4 n 2.0. The dimensionless wall shear stress τxy (0) can be related to the dimensionless wall friction coefficient cf according to −1/(n+1) −2τxy (0) (Cx)2−n xn �� n = 2(−f (0)) cf = ρ(Cx)2 K/ρ where (−f �� (0))n is given in Table 4.1. Here (Cx)2−n xn /(K/ρ) is a local Reynolds number based on the velocity Cx of the stretching sheet. It can be observed that the magnitudes of f �� (0) as well as (−f �� (0))n are monotonically decreasing with increasing values of n. It is also observed that the dimensionless boundary layer thickness ηδ defined as the value of η for which the normalized velocity f � (η) becomes equal to 0.01, is a decreasing function of the power-law index n. Table 4.1 Variation of dimensionless wall velocity gradient, skin-friction and boundary layer thickness with power-law index n n
0.4
0.6
0.8
1.0
1.2
1.5
2.0
−f (0)
1.273 1.101 22.57
1.096 1.057 13.05
1.029 1.023 7.52
1.000 1.000 4.60
0.987 0.985 3.31
0.981 0.971 2.45
0.979 0.959 1.91
(−f (0))n ηδ
According to Eq. (4.8), the boundary layer thickness δ(x) varies as δ = ηδ ((K/ρ)/C 2−n )1/(n+1) x(n−1)/(n+1) . Thus, the boundary layer thickness is constant for a Newtonian fluid, as already pointed out by Crane [1]. For n �= 1, on the other hand, the boundary layer thickens in the downsteam direction for dilatants fluids (n > 1), while δ decreases with x for pseudoplastics (n < 1). Figure 4.2 shows the dimen-
Fig. 4.2 Similarity velocity profiles for n 1
126
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
sionless velocity profile f � (η), for several values of n > 1 which correspond to the case of dilatants (shear thickening) fluids. It is clearly observed from Figure 4.2 that the velocity component u = cxf � parallel to the stretching sheet decreases with increasing n. Figure 4.3 shows the corresponding profiles for a pseudoplastic (shear thinning) fluid for several n < 1. Here, u increases with increasing pseudoplasticity. A point to notice, which cannot be observed from the displayed f � (η) profiles, is that the profiles intersect each other close to the η-axis. This behaviour is consistent with the predicted variation of the wall velocity gradient f �� (0) in Table 4.1.
Fig. 4.3 Similarity velocity profiles for n 1
4.1.5
Concluding remarks
Although the model of a power-law fluid due to Ostwald-de-Waele used in the present analysis can explain the non-Newtonian behaviour over some range of shear rate, it should be noted that one defect of the model is that it leads to an infinite value of the “apparent viscosity” K|∂u/∂y|(n−1) for pseudoplastic fluids (n < 1) in the limit of zero shear rate. This is in keeping with our numerical calculations, which failed to converge for n < 0.4.
4.2 Hydromagnetic flow of a power-law fluid over a stretching sheet 4.2.1
Introduction
The boundary layer flow of a viscous fluid due to the motion of a plane sheet in its own plane was first investigated by Sakiadis [11]. Erickson et al. [12]
4.2
Hydromagnetic flow of a power-law fluid over a stretching sheet
127
extended this problem to study the temperature distribution in the boundary layer when the sheet is maintained at a constant temperature with suction or blowing. These investigations have applications in the polymer industry when a polymer sheet is extruded continuously from die, with a tacit assumption that the sheet is inextensible. However, in real situations one has to encounter the boundary layer flow over a stretching sheet. For example, in a meltingspinning process, the extrudate is stretched into a filament or sheet while it is drawn from the die. Finally, this sheet solidifies while it passes through the controlled cooling system. Crane [1] and McCormack and Crane [2] studied the boundary layer flow of a Newtonian fluid caused by stretching of an elastic flat sheet which moves in its own plane with a velocity varying linearly with distance from a fixed point due to the application of a uniform stress. Following two different approaches, the uniqueness of the exact analytical solution provided in [1], [2] was proved simultaneously by McLeod and Rajagopal [13] and Troy et al. [14]. Rajagopal et al. [4] studied the boundary layer flow over a stretching sheet for a special class of non-Newtonian fluids, known as second order fluids, and obtained similarity solutions of the boundary layer equation numerically. Dandapat and Gupta [5] examined the same problem with heat transfer and found an exact analytical solution of the non linear equation governing the self-similar flow which is consistent with the numerical results in [4]. The non-similar boundary layer flow of a second order fluid over a stretching sheet in the presence of a uniform free stream was studied by Rajagopal et al. [15]. Unlike the case in which the free stream is absent, it was demonstrated that the boundary layer may separate from the sheet under certain conditions. Recently, Andersson and Dandapat [16] extended the Newtonian boundary layer flow problem considered by Crane [1] to an important class of non-Newtonian fluids obeying the power-law model. Pavlov [17] studied the MHD boundary layer flow of an electrically conducting fluid due to the stretching of a plane elastic surface in the presence of a uniform transverse magnetic field and obtained an exact similarity solution of this problem. Chakrabarti and Gupta [18] extended their investigation to study the temperature distribution in this MHD boundary layer over a stretching sheet in the presence of uniform suction. Motion of non-Newtonian fluids in the presence of a magnetic field was studied earlier by several authors (Sarpkaya [19], Djukic [20], [21], etc.) in different contexts, but the MHD boundary layer flow of an electrically conducting non-Newtonian fluid over a stretching sheet in the presence of a magnetic field has not been dealt with so far. In this section, we present the results of Andersson et al. [40] for the study of boundary layer flow of an electrically conducting incompressible fluid obeying the Ostwald-de-Waele power-law model in the presence of a transverse magnetic field due to the stretching of a plane sheet. The corresponding MHD flow of a viscoelastic fluid past a stretching surface is considered in an accompanying paper [22].
128
4.2.2
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
Flow analysis
Consider the flow of an electrically conducting fluid obeying the power-law model in the presence of a transverse magnetic field past a flat sheet coinciding with the plane y = 0, the flow being confined to y > 0. Two equal and opposite forces are applied along the x-axis, so that the wall is stretched keeping the origin fixed. The basic equations for the boundary layer flow are, in the usual notation, ∂u ∂v + = 0, ∂x ∂y u
(4.14)
∂u 1 ∂ σB02 ∂u +v = (τxy ) − u ∂x ∂y ρ ∂y ρ
(4.15)
where u and v are the velocity exponents along the x and y directions, respectively. ρ, σ, B0 and τxy are density, electrical conductivity, magnetic field, and shear stress, respectively. The stress tensor is defined as τij = 2K(2Dkl Dkl )(n−1)/2 Dij where
1 Dij = 2
∂ui ∂uj + ∂xj ∂xi
(4.16)
denotes the stretching tensor, K is called the consistency coefficient, and n is the power law index. The two-parameter rheological equation (4.16) is known as the Ostwald-de-Waele model or more commonly, the power-law model. With n = 1, Equation (4.16) represents a Newtonian fluid with dynamic coefficient of viscosity K. Therefore, the deviation of n from unity indicates the degree of deviation from Newtonian behavior (see Andersson and Irgens [8]). With n �= 1, the constitutive equation (4.16) represents shear thinning (n < 1) and shear thickening (n > 1) fluids. However, unlike the second order viscoelastic fluid studied in [4], [5], [15], the inelastic power-law model (4.16) does not exhibit normal-stress differences. In the present problem, we have ∂u/∂y 0; this gives the shear stress as τxy = −K(−∂u/∂y)n.
(4.17)
Now the momentum equation (4.15) reduces to the form u
∂u K ∂ ∂u +v =− ∂x ∂y ρ ∂y
n ∂u σB02 − u. − ∂y ρ
(4.18)
It should be noted here that we have neglected the induced magnetic field under the justification for flow at small magnetic Reynolds number (Shercliff [23]). Moreover, this flow is caused solely by the stretching of the wall and,
4.2
Hydromagnetic flow of a power-law fluid over a stretching sheet
129
so, there is no free-stream velocity outside the boundary layer. The boundary conditions are u = Cx,
v = 0,
u → 0,
at y = 0,
as y → ∞
(4.19) (4.20)
where C is a positive constant. Following [16], it can be easily shown that the present system consisting of Eqs. (4.14) and (4.18) subject to the boundary conditions (4.19) and (4.20) admits similarity solutions in terms of the similarity function f and the similarity variable η defined as −1
2n
ψ = (C 1−2n /(K/ρ)) n+1 x n+1 f (η), 1
1−n
ξ = (C 2−n /(K/ρ)) n+1 x n+1 y,
(4.21) (4.22)
where u = ∂ψ/∂y
and v = −∂ψ/∂x
(4.23)
define the physical stream function in this problem. After using the transformation defined by Eqs. (4.21) and (4.22) in Eq. (4.18), we obtain n(−f �� )(n−1) f ��� − (−f � )2 + (2n/n + 1)f f �� − M f � = 0
(4.24)
where M (= σB02 /ρC) is the magnetic parameter. The boundary conditions (4.19) and (4.20) become f � (0) = 1 and f (0) = 0,
(4.25a)
f � (∞) = 0.
(4.25b)
The equation for Newtonian fluid can be obtained as a special case if one puts n = 1 in Eqs. (4.24) and (4.25). In this case we have f ��� − (−f � )2 + f f �� − M f � = 0
(4.26)
with the boundary conditions (4.25). It is interesting to note that the above special case (cf. Eqs. (4.26) and (4.25)) has a fairly simple exact analytic solution of the form f � (η) = e−βη , β > 0 (4.27) satisfying f � (0) = 1 and f � (∞) = 0 in Eqs. (4.25). Integration of Eq. (4.27) gives, on using f (0) = 0 of Eqs. (4.25), f (η) = β −1 (1 − e−βη )
(4.28)
where β = (M + 1)1/2 (see Pavlov [17]). It should be noted here that by setting M = 0 in either Eq. (4.24) or (4.26) one can arrive at Eq. (4.23) or Eqs. (4.25) of Andersson and Dandapat [16].
130
4.2.3
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
Numerical solution method
We are unable to find an exact analytical solution of the nonlinear equation (4.24) with the boundary condition (4.25) for the case n �= 1. So, the transformed momentum equation (4.24) subject to the boundary condition (4.25) was solved numerically by an efficient shooting method due to Keller [9] for some different values of the parameters n and M . First, however, Equation (4.24) was written as a system of three first order equations solved by the Runge-Kutta integration technique. Then a Newton iteration procedure was employed to assure quadratic convergence. The same numerical method has recently been applied to the non-magnetic case in [16].
4.2.4
Results and discussion
The above non-Newtonian MHD flow problem was solved numerically for five different values of the magnetic parameter (M 2.0) and for seven values of the power-law index in the range 0.4 n 2.0. Some of the computed velocity profiles f � (η) are presented in Figures 4.4–4.6 for shear thinning, Newtonian and shear thickening fluids, respectively. First of all, it should be noted that the exact analytical solution (4.27) is accurately reproduced by numerical computations for n = 1 in Figure 4.5. A common feature of all the displayed velocity profiles, irrespective of n, is that f � decreases with increasing M . For a given fluid the effect of the magnetic field is therefore, to reduce the velocity component u = cxf � parallel to the stretching surface. An important boundary layer characteristic is the skin friction coefficient Cf , which is a dimensionless form of the shear stress τxy at the sheet y = 0. According to its definition, Cf =
−2τxy (0) = 2(−f �� (0))n ρ(Cx)2
(Cx)2−n xn K/ρ
−1/(n+1)
(4.29)
where (Cx)2−n xn /(K/ρ) is a local Reynolds number based on the sheet velocity Cx . Cf is directly related to the dimensionless velocity gradient f �� at the wall. Since f �� (0) is negative, the computed variation of −f �� (0) with M and n has been summarized in Table 4.2 and displayed graphically in Figure 4.7. In accordance with the near-wall behavior of the velocity distributions f � (η) in Figures 4.4–4.6, it is observed that −f �� (0) increases monotonically with M for a given fluid. Figure 4.7, moreover, demonstrates that the magnitude of the wall gradient decreases gradually with increasing n for any fixed M -value. The observation is consistent with recent findings [16] for M = 0. The presence of a magnetic field, however, makes the difference in f �� (0) between shear thickening (n > 1) and shear thinning (n < 1) fluids significantly more pronounced. The variation of the dimensionless boundary layer thickness ηδ
4.2
Hydromagnetic flow of a power-law fluid over a stretching sheet
131
Fig. 4.4 Similarity velocity profile for a shear thinning fluid (n = 0.4) for different values of M
Fig. 4.5 Similarity velocity profile for a Newtonian fluid (n = 1.0) for different values of M . The symbol denotes the exact analytical solution (4.27) due to Pavlov
with M and n is presented in Figure 4.8 and the accompanying Table 4.3. Here, ηδ is defined as the value of the similarity variable η at which the dimensionless velocity f (η) equals 0.01. An alternative measure of the thickness of the various layer adjacent to the sheet is the dimensionless displacement thickness, which can be defined as
132
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
Fig. 4.6 Similarity velocity profile for a shear thickening fluid (n = 2.0) for different values of M
Fig. 4.7 Dimensionless velocity gradient at the wall
Table 4.2 Numerical results for −f (0) n 0.4 0.6 0.8 1.0 1.2 1.5 2.0
0.0
0.5
M 1.0
1.5
2.0
1.273 1.096 1.029 1.000 0.987 0.981 0.980
1.811 1.463 1.309 1.225 1.175 1.131 1.039
2.284 1.777 1.444 1.414 1.333 1.257 1.187
2.719 2.060 1.754 1.581 1.472 1.367 1.269
3.127 2.321 1.945 1.732 1.596 1.466 1.342
4.2
Hydromagnetic flow of a power-law fluid over a stretching sheet
δ1 ≡
∞ 0
f � (η)dη = lim f − f (0)
133
(4.30)
n→∞
in the present situation. According to the boundary condition (4.25a), f (0) is equal to zero. The data for δ1 in Table 4.4, which are plotted in Figure 4.9, show that the two global boundary layer characteristics ηδ and δ1 vary with n and M in about the same way. It is readily observed that the thickness of the boundary layer decreases with the magnetic parameter for all the fluids considered. Table 4.3 n 0.4 0.6 0.8 1.0 1.2 1.5 2.0
Numerical results for ηδ 0.0
0.5
M 1.0
1.5
2.0
22.57 13.64 7.45 4.61 3.32 2.45 1.87
11.32 7.54 5.14 3.75 2.94 2.27 1.78
8.04 5.53 4.10 3.22 2.63 2.12 1.69
6.32 4.52 3.50 2.85 2.40 1.96 1.63
5.31 3.94 3.09 2.57 2.24 1.83 1.57
Table 4.4 Numerical results for δ1 n 0.4 0.6 0.8 1.0 1.2 1.5 2.0
0.0
0.5
M 1.0
1.5
2.0
2.183 1.600 1.220 0.999 0.878 0.775 0.688
1.163 1.024 0.903 0.815 0.754 0.693 0.635
0.857 0.799 0.747 0.704 0.669 0.632 0.594
0.692 0.673 0.650 0.628 0.609 0.585 0.562
0.590 0.590 0.582 0.572 0.563 0.548 0.537
This tendency is also seen from Figures 4.4 to 4.6. For the most shear thickening fluid (n = 2.0) the boundary layer thickness, being measured in terms of δ1 or ηδ , is reduced about 20% when M = 2.0 as compared to the nonmagnetic case. For the highly shear thinning fluid (n = 0.4) a more dramatic reduction in δ1 or ηδ , of about 75% takes place. In the non-magnetic case (M = 0) Andersson and Dandapat [16] observed that ηδ is a monotonically decreasing function of the power law index, ηδ being reduced by a factor greater than 10 as n varies from 0.4 to 0.2. The data in Figures 4.8 and 4.9 demonstrate that imposition of a magnetic field reduces significantly the sensitivity of δ1 or ηδ with respect to the rheological parameter n, and the displacement thickness becomes nearly independent of n for the case M = 2.0. The so-called form factor ηδ /δ1 is a useful
134
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
parameter in classical, i.e, Newtonian, boundary layer theory. It is therefore, interesting to observe from the data in Tables 4.3 and 4.4 that ηδ /δ1 remain fairly independent of the magnetic field for a given fluid, although this form factor varies from less than three for the most shear thickening fluid to 10 for the shear thinning substances.
Fig. 4.8 Dimensionless boundary layer thickness
Fig. 4.9 Dimensionless displacement thickness
It can be concluded that the magnetic field tends to make the boundary layer thinner, thereby increasing the wall friction, and that this effect is more pronounced for shear thinning than for shear thickening fluids. The imposition of a magnetic field also makes the difference between the various fluids more distinct in the near-wall region, albeit less influential on the global thickness characteristics.
4.3
MHD Power-law fluid flow and heat transfer over a non-isothermal ...
135
4.3 MHD Power-law fluid flow and heat transfer over a non-isothermal stretching sheet
4.3.1
Introduction
The study of two-dimensional boundary layer flow over a stretching sheet has gained much interest in recent times because of its numerous industrial applications viz. in the polymer processing of a chemical engineering plant and in metallurgy for metal processing. Crane [1] was first to formulate this problem to study a steady two-dimensional boundary layer flow caused by stretching of a sheet that moves in its plane with a velocity which varies linearly with the distance from a fixed point on the sheet. Many investigators have extended the work of Crane [1] to study heat and mass transfer under different physical situations (e.g., Gupta and Gupta [24], Chen and Char [25], Datta et al. [26], McLeod and Rajagopal [27], Chiam [28], [29]) by including quadratic and higher order stretching velocity. All these works are restricted to Newtonian fluid flows which have received much attention in the last three decades due to their occurrence in nature and their increasing importance in industry. Different types of non-Newtonian fluids are viscoelastic fluid, couple stress fluid, micropolar fluid and power-law fluid. Rajagopal et al. [4] and Siddappa and Abel [30] studied the flow of a viscoelastic fluid flow over a stretching sheet. Troy et al. [14], Chang [31], Lawrence and Rao [32], McLeod and Rajagopal [27] have discussed the problem of uniqueness/nonuniqueness of the flow of a non-Newtonian viscoelastic fluid over a stretching sheet. Abel and Veena [33] studied the heat transfer of a viscoelastic fluid over a stretching sheet. Bujurke et al. [34] have investigated heat transfer phenomena in a second order fluid flow over a stretching sheet with internal heat generation and viscous dissipation. Prasad et al. [35] analyzed the problem of a viscoelastic fluid flow and heat transfer in a porous medium over a non-isothermal stretching sheet with variable thermal conductivity. Prasad et al. [36] have investigated the diffusion of a chemically reactive species of a non-Newtonian fluid immersed in a porous medium over a stretching sheet. All the above-mentioned research does not however consider the situation where hydromagnetic effects arise. The study of hydrodynamic flow and heat transfer over a stretching sheet may find its applications in polymer technology related to the stretching of plastic sheets. Also, many metallurgical processes involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid, and while drawing, these strips are sometimes stretched. The rate of cooling can be controlled by drawing such strips in an electrically conducting fluid subjected to a magnetic field in order to get the final products of desired characteristics as the final product greatly
136
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
depends on the rate of cooling. In view of this, the study of MHD flow of Newtonian/non-Newtonian flow over a stretching sheet was carried out by many researchers (Sarpkaya [19], Pavlov [17], Chakrabarti and Gupta [18], Char [37], Andersson [22], Datti et al. [38]). These works however do not consider the study of a non-Newtonian power-law fluid model. The simplest and most common type of such a model is the Ostwald-de Waele model, i.e. power-law fluid for which the rheological equation of the state between the stress components τij and strain components eij is defined by (Vujannovic et al. [39]) n−1 3 3 2 elm elm eij , τij = −pδij + K m=1 l=1
where p is the pressure, δij is Kronecker delta, K and n are respectively, the consistency coefficient and power-law index of the fluid. Such fluids are known as power-law fluid. For n > 1, fluid is said to be dilatant or shear thickening; for n < 1, the fluid is called shear thinning or pseudoplastic fluid; and for n = 1, the fluid is simply the Newtonian fluid. Several fluids studied in the literature suggest the range 0 < n 2 for the value of power-law index n. Keeping this in view, in the present paper, we study the effect of variable thermal conductivity on the heat transfer of a non-Newtonian power-law fluid, subjected to a magnetic field, over a non-isothermal stretching sheet with internal heat generation/absorption. This is in contrast to the work of Andersson et al. [40], Jadhav and Waghmode [41], where constant thermal conductivity was considered. In Savvas et al. [42] it has been observed that for liquid metals, the thermal conductivity varies linearly with temperature in the range 0◦ F to 400◦ F. Hence, we have assumed that the thermal conductivity is a linear function of the temperature. Further, we consider two cases of non-isothermal boundary conditions, namely, (1) surface with Prescribed surface temperature (PST case) and (2) surface with Prescribed wall heat flux (PHF case). Because of the rheological equation of state Eq. (4.16), the momentum and energy equations are highly nonlinear and coupled partial differential equations (PDEs). These PDEs are then converted to coupled nonlinear ordinary differential equations (ODEs) by using the similarity variables along with the appropriate boundary conditions. In this section, we propose to solve these ordinary differential equations numerically by the Keller-box method [43], [44]. One of the important findings is that the horizontal boundary layer thickness decreases with the increase of power-law index. The thickness is much larger for shear thinning (pseudoplastic) fluids (n < 1) than that of Newtonian (n = 1) and shear thickening (dilatants) fluids (n > 1). In this section we present the results of [55] for the MHD power-law fluid flow and heat transfer over a non-isothermal stretching sheet.
4.3
MHD Power-law fluid flow and heat transfer over a non-isothermal ...
4.3.2
137
Governing equations and similarity analysis
We consider two-dimensional steady, laminar flow of an incompressible and electrically conducting fluid obeying a power-law model in the presence of a uniform transverse magnetic field over a non-isothermal stretching sheet. The flow is generated due to the stretching of the sheet by applying two equal and opposite forces along the x-axis while keeping the origin fixed and considering the flow to be confined to the region y > 0. In order to obtain the temperature difference between the surface and the ambient fluid, we consider the temperature dependent heat source/sink in the flow. In this situation, the basic governing equations of continuity and momentum (Andersson [22], Chiam [45]) take the following form: ∂u ∂v + = 0, ∂x ∂y ∂u ∂u ∂ u +v = −ν ∂x ∂y ∂y
∂u − ∂y
n
(4.31)
−
σμ2m H02 u ρ
(4.32)
where u and v are the flow velocity components in the stream-wise (x) and cross-stream (y) directions, respectively. ν is the kinematic viscosity of the fluid, σ is the electrical conductivity, μm is the magnetic permeability, H0 is the applied transverse magnetic field and ρ is the fluid density. The first term in the right hand side of Eq. (4.32) i.e. the shear rate (∂u/∂y) has been assumed to be negative throughout the entire boundary layer since the stream-wise velocity component u decreases monotonically with the distance y from the moving surface. A rigorous derivation and subsequent analysis of the boundary layer equations for power-law fluids were recently provided by Denier and Dabrowski [46]. They focused on boundary layer flow driven by free stream U (x) ≈ xm which is of the Falkner-Skan type. Such boundary layer flows are driven by a stream wise pressure gradient −dp/dx = ρ(du/dx) set up by the external free stream outside the viscous boundary layer. In the present context no driving pressure gradient is present. Instead the flow is driven solely by a flat surface which moves with a prescribed velocity U (x) = bx, where x denotes the distance from the slit from which the surface emerges and b > 0. Thus, the relevant boundary conditions applicable to the flow are u(x, 0) = U (x), (4.33a) v(x, 0) = 0,
(4.33b)
u(x, y) → 0 as y → ∞.
(4.33c)
Here, Equation (4.33c) claims that the stream-wise velocity vanishes outside the boundary layer, the requirement equation (4.33b) signifies the importance of impermeability of the stretching surface whereas (4.33a) assures no slip at
138
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
the surface. The following transformation is introduced in accordance with Andersson and Dandapat [16]: η=
1 y (Rex ) n+1 , x
−1
ψ(x, y) = U x(Rex ) n+1 f (η),
(4.34)
where η is the similarity variable and ψ(x, y) is the stream function. The velocity components u and v are given by u=
∂ψ , ∂y
v=−
∂ψ . ∂x
(4.35)
The local Reynolds number is defined by Rex =
U 2−n xn . ν
(4.36)
The mass conservation equation (4.31) is automatically satisfied by Eq. (4.35). By assuming the similarity function f (η) to depend on the similarity variable η, the momentum equation (4.32) is transformed into ordinary differential equation 2n n| − f �� |n−1 f ��� − f �2 + f f �� − Mn f � = 0 (4.37) n+1 where Mn = σμ2m H02 /ρb is the magnetic parameter. Equation (4.37) is solved numerically subject to the following boundary conditions obtained from Eq. (4.33) using Eq. (4.34) as f (η) = 0 at η = 0,
(4.38a)
f � (η) = 1 at η = 0,
(4.38b)
f (η) → 0 as η → ∞.
(4.38c)
f ��� − f �2 + f f �� − Mn f � = 0
(4.39)
�
It should be noted that the velocity U = U (x) which is used to define the dimensionless stream function ψ in Eq. (4.34) and the local Reynolds number in Eq. (4.36) describes the velocity of the moving surface that drives the flow. This choice coincides with the conventional boundary layer analysis, in which the free stream velocity is taken as the velocity scale. The transformations defined in Eq. (4.34) and Eq. (4.35) can be used for arbitrary variation of U (x), so the transformation results in a true similarity problem only if U varies as bx. Therefore, such surface velocity variations are required for the ordinary differential equation (4.37) to be valid. Non-similar stretching sheet problems which require the solution of partial differential equations rather than ordinary differential equations were considered by several researchers for Newtonian fluids. Three boundary conditions (4.38) are sufficient for solving the third order equation which results for transformed momentum equations for power-law fluids. The equation for a Newtonian fluid can be obtained as a special case if one puts n = 1 in Eq. (4.37). In this case we have
4.3
MHD Power-law fluid flow and heat transfer over a non-isothermal ...
139
with boundary conditions (4.38). It is interesting to note that (4.39) has an exact analytical solution of the form f � = e−αη ,
α > 0,
(4.40)
satisfying the boundary conditions (4.38). Integration of Eq. (4.40) and using (4.38a) gives f=
1 (1 − e−αη ) where α = (1 + Mn ). α
(4.41)
The skin friction coefficient Cf at the sheet is given by
−1
Cf = −(2τxy /ρ(bx)2 )y=0 = 2(−f �� (0))n (Rex ) n+1
(4.42)
where τxy is the shear stress and Rex is the local Reynolds number. We now discuss the heat transport in the above flow due to a stretching sheet.
4.3.3
Heat transfer
The energy equation for a fluid with variable thermal conductivity in the presence of internal heat generation/absorption for the two-dimensional flow is given by (Chiam [45]) ∂2T ∂T ∂κ(T ) ∂T + ρcp v − = κ(T ) 2 + Qs (T − T∞ ), ρcp u (4.43) ∂x ∂y ∂y ∂y where cp is the specific heat at constant pressure, T is the temperature of the fluid, T∞ is the constant temperature of the fluid far away from the sheet, and κ(T ) is the temperature-dependent thermal conductivity. We consider the temperature-thermal conductivity relationship of the following form (Chiam [45]): ε (T − T∞ ) , κ(T ) = κ∞ 1 + (4.44) ΔT where ΔT = Tw − T∞ , Tw is the sheet temperature, ε is a small parameter, and κ∞ is the conductivity of the fluid far away from the sheet. The term containing Qs in Eq. (4.43) represents the temperature-dependent volumetric rate of heat source when Qs > 0 and heat sink when Qs < 0. These deal with the situation of exothermic and endothermic chemical reactions respectively. Substituting Eq. (4.44) into Eq. (4.43), we get ∂T k∞ ε ∂T ∂T ρcp u + ρcp v − ∂x ΔT ∂y ∂y ∂2T ε (T − T∞ ) + Qs (T − T∞ ). = κ∞ 1 + ΔT ∂y 2
(4.45)
140
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Application of the Keller-box Method to Boundary Layer Problems
The thermal boundary conditions depend on the type of heating process under consideration. Here, we consider two different heating processes, namely, (i) prescribed surface temperature and (ii) prescribed wall heat flux, varying with the distance. The boundary conditions assumed for solving Eq. (4.45) are ⎫ � x �� � T = Tw = T ∞ + A (PST case) ⎪ ⎪ ⎬ l at y = 0 (4.46) � � x ∂T ⎪ = qw = D (PHF case) ⎪ −k ⎭ ∂y l T → T∞ as y → ∞ (4.47)
where A is a constant. It is obvious now that ⎧ � � x ⎪ ⎪ ⎨A l , ΔT = Tw − T∞ = −1 D �x� ⎪ ⎪ (Rex ) n+1 x, ⎩ κ∞ l
PST case, (4.48) PHF case.
We now use a scaled η-dependent temperature of the form (4.49)
θ(η) = T − T∞ /ΔT.
The advantage of using Eq. (4.46) is that the temperature-dependent thermal conductivity turns out to be x-independent. Equation (4.45) reduces to a nonlinear differential equation using Eq. (4.34): (1 + εθ)θ�� + εθ�2 + Pr(2n/(n + 1)(f θ� − (f � − β)θ)) = 0 (4.50) � �1/(n+1) is the generalized Prandtl numwhere Pr = γ 2 b3(n−1) x2(n−1) /αn+1 ber for a power-law fluid and β = Qs /ρcp b is the heat source/sink parameter. Equation (4.46), on using Eqs. (4.47) and (4.48), can be written as � θ(0) = 1 (PST case) , θ(∞) = 0. (4.51) θ� (0) = −1 (PHF case) The local Nusselt number is given by
N ux =
h(x) K∞
(4.52)
qw (x) ΔT
(4.53)
where the heat transfer coefficient h(x) is of the form h(x) = and the local heat flux at the sheet is 1
qw = −K∞ (∂T /∂y)y=0 = −K∞ Ax(Rex ) n+1 θ� (0).
(4.54)
Substituting Eqs. (4.46), (4.53) and (4.54) into Eq. (4.52), we get 1
N ux = −Rexn+1 θ� (0).
(4.55)
4.3
MHD Power-law fluid flow and heat transfer over a non-isothermal ...
4.3.4
141
Numerical procedure
Since the equation for f does not involve θ(η), we use the Keller-box method (Cebeci and Bradshaw [43], Keller [44] and Press et al. [47]) to compute f numerically. The resulting system of algebraic equations has been solved by a tridiagonal block solver [43]. The choice of numerical value of η∞ , which obviously depends on the physical parameters n and Mn , is very crucial in this numerical procedure. The initial guess was made from the known exact solution for n = 1, and several trial and error runs were made to obtain accurate values of f, fη , etc. up to a significant number of decimal places that satisfy the boundary condition at η∞ . Similarly, to obtain θ(η) numerically, we use the values of f obtained from Keller-box method and employ a shooting technique (Conte and de Boor [48]). Here also, the initial guess was made with the help of the known exact solution for ε = 0, Pr = 1 and n = 1. The numerically computed values are presented in Figures 4.10– 4.15 and Tables 4.5–4.8.
Fig. 4.10 Horizontal velocity profiles fη (η) for different values of n with M n = 0.0
142
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Application of the Keller-box Method to Boundary Layer Problems
Fig. 4.11 Horizontal velocity profiles fη (η) for different values of M n and n
Fig. 4.12a Variation of temperature profiles θ(η) for different values of n
4.3
MHD Power-law fluid flow and heat transfer over a non-isothermal ...
143
Fig. 4.12b Variation of temperature profiles θ(η) for different values of n
Fig. 4.13a Variation of θ(η) for various values of M n with Pr = 1.0, β = 0.1, n = 0.8
144
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
Fig. 4.13b Variation of θ(η) for various values of M n and ε for n = 0.8, Pr = 1.0, β = 0.1
Fig. 4.14a Variation of θ(η) for different values of β and n
4.3
MHD Power-law fluid flow and heat transfer over a non-isothermal ...
Fig. 4.14b Temperature profiles θ(η) for different values of β and n
Fig. 4.15a Variation of θ(η) for different values of Prandtl number
145
146
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
Fig. 4.15b Variation of θ(η) for different values of Prandtl number
Table 4.5 Comparison of skin friction −fηη (0) with Andersson et al. [40] Andersson et al. [40] Present Study
n = 0.4
n = 0.6
n = 0.8
n = 1.0
n = 1.2
n = 1.5
n = 2.0
1.273
1.096
1.029
1.00
0.987
0.981
0.980
1.27968 1.09838 1.02897 1.00000 0.98738 0.98058 0.98035
Table 4.6 Values of skin friction −fηη (0) for different values of Mn and n n
Mn = 0.0
Mn = 0.5
Mn = 1.0
Mn = 1.5
Mn = 2.0
0.4
1.292
1.8151
2.28536
2.71942
3.12702
0.6
1.107
1.4649
1.77762
2.06012
2.32088
0.8
1.034
1.3086
1.54429
1.75406
1.94588
1.0
1.0
1.2249
1.41440
1.58100
1.73200
1.2
0.989
1.1752
1.33306
1.47150
1.59599
1.4
0.982
1.1441
1.27858
1.39653
1.50229
1.6
0.980
1.1207
1.23901
1.34266
1.43425
1.8
0.979
1.1047
1.20995
1.30106
1.38230
2.0
0.978
1.0926
1.18711
1.26904
1.34174
4.3
MHD Power-law fluid flow and heat transfer over a non-isothermal ...
147
148
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
4.3
MHD Power-law fluid flow and heat transfer over a non-isothermal ...
4.3.5
149
Results and discussion
The numerical computation is carried out for different values of the magnetic parameter M n, power-law index n, Prandtl number Pr, variable thermal conductivity parameter ε and heat source/sink parameter β using the numerical procedure discussed in the previous section. In order to get a clear insight of the physical problem, the horizontal velocity profiles fη (η) and temperature profiles θ(η) for both PST and PHF cases have been discussed by assigning the numerical values to the non-dimensional parameters encountered in the problem. The numerical results are shown graphically in Figures 4.10–4.15. To assess the accuracy of the numerical method used, the computed value of the skin friction coefficient and rate of heat transfer were compared with those obtained by Andersson et al. [22] for different values of physical parameters. It is observed that our results are in good agreement with the results obtained by the previous investigators as seen from the tabulated results in Table 4.5. Figures 4.10 and 4.11 illustrate the effect of the power-law index n and magnetic parameter M n on the horizontal velocity profiles fη (η). It is noticed from these figures that the horizontal velocity profiles decrease with increasing values of power-law index n and magnetic parameter Mn in the boundary layer, but this effect is not very prominent near the wall. The effect of increasing the value of the power-law index parameter n is to reduce the horizontal velocity and thereby reduce boundary layer thickness, i.e. the thickness is much larger for shear thinning (pseudoplastic) fluids (0 < n < 1) than that of Newtonian (n = 1) and shear thickening (dilatant) fluids (1 < n < 2), as clearly seen from Figure 4.10. The effect of the magnetic parameter M n on the horizontal velocity profile is depicted in Figure 4.11. It is observed that the horizontal velocity profile decreases with increase in the magnetic field parameters due to the fact that, the introduction of a transverse magnetic field (normal to the flow direction) has a tendency to create a drag, known as Lorentz force, which tends to resist the flow. This behavior is even true in the case of shear thickening and shear thinning fluids. The effect of the power-law index, n, on temperature profiles θ(η) in the boundary layer for both PST and PHF cases is shown in Figures 4.12a and 4.12b, respectively. It is observed that the temperature distribution θ(η) is unity at the wall in PST case and is less than the unity at the wall in PHF case for n 1. However, the temperature distribution θ(η) for both PST and PHF cases decreases asymptotically to zero in the boundary layer. The effect of increasing the values of the power-law index n leads to thinning of the thermal boundary thickness. This behavior is more noticeable in shear thinning and shear thickening fluids. The effect of the magnetic parameter M n on the temperature profile θ(η) in the boundary layer in presence/absence of variable thermal conductivity parameter ε for both PST and PHF cases is depicted in Figures 4.13a and 4.13b, respectively. It is observed that the effect of the magnetic field param-
150
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
eter M n is to increase the temperature profile θ(η) and θ(η) tends to zero as the space variable η increases in the boundary layer. As explained above, the introduction of a transverse magnetic field to an electrically conducting fluid gives rise to a resistive type of force known as Lorentz force. This force makes the fluid experience a resistance by increasing the friction between its layers, due to which there is increase in the temperature profile θ(η). This behavior is even true in the presence of non-zero values of the variable thermal conductivity parameter. The effect of the variable thermal conductivity parameter is to increase the temperature profile which in turn increases the thermal boundary layer thickness for both PST and PHF cases. Figures 4.14a and 4.14b exhibit the temperature distribution θ(η) with η for different values of heat source/sink parameter β in PST and PHF cases, respectively. From these graphs we observe that the temperature distribution is lower throughout the boundary layer for negative values of β (heat sink) and higher for positive values of β (heat source) as compared with the temperature distribution in the absence of heat source/sink parameter, i.e., β = 0. Physically β > 0 implies Tw > T∞ , i.e., the supply of heat to the flow region from the wall. Similarity, β < 0 implies Tw < T∞ , i.e., the transfer of heat is from flow to the wall. The effect of increasing the value of the heat source/sink parameter β is to increase the temperature profile θ(η) for both PST and PHF cases. However, the minimum temperature distribution is observed in PHF case compared to PST case. The variations of temperature profile θ(η) with η for the various values of modified Prandtl number Pr are displayed in Figures 4.15a and 4.15b for PST and PHF cases, respectively. Both the graphs demonstrate that the increase of Prandtl number Pr results in decrease of temperature distribution which tends to zero as the space variable η increases from the wall; hence thermal boundary layer thickness decreases as Prandtl number Pr increases for both PST and PHF cases. The values of−fηη (0), which signifies the local skin friction coefficient, Cf , are recorded in Table 4.6 for different values of the physical parameters n and M n. From Table 4.6, we observe that −fηη (0) increases monotonically with increase in the magnetic field parameter M n for various values of n. It is interesting to note that the magnitude of the wall surface gradient decreases gradually with increasing power-law index for a fixed value of magnetic parameter M n. The effect of the power-law index on −fηη (0) is more significant in a shear thinning fluid (n < 1) than in a shear thickening fluid (n > 1). The heat transfer phenomenon is usually analyzed from the numerical values of the two physical parameters, i.e., (1) the wall temperature gradient −θη (0) in PST case which in turn helps in the computation of the local Nusselt number N ux and (2) the wall temperature θ(0) in PHF case. Numerical results for the wall temperature gradient in PST case and wall temperature in PHF case are recorded in Tables 4.7 and 4.8, respectively for different non-dimensional physical parameters n, M n, Pr, ε, β. It is observed that the effect of the power-law index n is to increase the wall temperature gradient in PST case
4.4
MHD flow and heat transfer of a Maxwell fluid over a non-isothermal ...
151
and is to decrease wall temperature in PHF case whereas the reverse trend is seen with the magnetic parameter M n. This result has a significant role in industrial applications to reduce expenditure on power supply in stretching the sheet just by increasing the magnetic parameter M n. Further, it is analyzed from Table 4.7 that the effect of the Prandtl number Pr is to decrease the wall temperature gradient in PST case and the wall temperature in PHF case. In addition, the effect of increasing values of heat source/sink parameter β is to decrease the wall temperature gradient in PST case, whereas its effect is to increase the wall temperature in PHF case. All the results obtained here are consistent with the physical situations. In the present study, we have investigated MHD non-Newtonian flow over a semi-infinite non-isothermal stretching sheet with internal heat generation or absorption using the Keller-box method. Temperature profiles are obtained for two types of heating processes namely, PST and PHF for various values of physical parameters. As expected, the power-law index and magnetic parameters decrease the velocity profile and reduce the boundary layer thickness as they increase. Also, an increase in the value of the power-law index, n, leads to thinning of the thermal boundary layer thickness whereas the effect of increasing values of the magnetic parameter is to increase the temperature and the thermal boundary layer thickness, for both PST and PHF cases. It is noteworthy that the effect of increasing the value of the heat source/sink parameter leads to increase in the temperature profile for both PST and PHF cases. Finally, it is concluded that the thermal boundary layer thickness decreases with increase in Prandtl number for both PST and PHF cases.
4.4 MHD flow and heat transfer of a Maxwell fluid over a non-isothermal stretching sheet
4.4.1
Introduction
During the past three decades, the flow of an incompressible viscous fluid over a stretching sheet has acquired special attention because of its many industrial applications (see for details, Agassant et al. [49] and Bird et al. [50]). In particular, flow of this kind occurs in a cooling bath, the boundary layer along a material handling conveyers, the aerodynamic extrusion of plastic sheets, the boundary layer along liquid film and condensation processes, the cooling or drying of papers and textiles, and glass fiber production. This type of flow investigation was initiated by Sakiadis [11] and extended by Crane [1] to fluid flow over a linearly stretched sheet. Later works on the stretching
152
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
sheet problems with Newtonian fluid models by taking into account different physical situations are extensively analyzed by several authors (see [25], [51]– [59]). However, many industrial fluids are non-Newtonian such as molten plastics, polymers, suspension, foods, slurries, paints, glues, printing inks, blood. That is, they may exhibit dynamic deviation from Newtonian behavior depending upon the flow configuration and/or the rate of deformation. These fluids often obey nonlinear constitutive equations and the complexity of their constitutive equation is the main culprit for the lack of exact analytical solution. For example, viscoelastic fluid models used in these works are simple models; second order fluid model and Walters’ model (see [4], [30], [60], [62], [63]) are known to be good for weakly elastic fluids subject to slowly varying flows. These two models are known to violate certain rules of thermodynamics, and virtually all of them are based on the boundary layer theory which is still incomplete for non-Newtonian fluids. Therefore significance of the results reported in the above works is limited as far as the polymer industry is concerned. Obviously for the theoretical results to be of any industrial importance, more general viscoelastic fluid models such as an upper convected Maxwell (UCM) model or Oldroyd B model should be invoked in the analysis. Indeed, these two fluid models have been recently used to study the flow of viscoelastic fluids above stretching and non-stretching sheets with or without heat transfer [61], [64]–[68]. In all the above mentioned studies, the thermophysical properties of the ambient fluids were assumed to be constant. However it is well known that these properties may change with temperature, especially the thermal conductivity. Available literature [29], [44], [69], [70] on variable thermal conductivity shows that this type of work has not been carried out for nonNewtonian UCM fluid in the presence of a transverse magnetic field. This type of flow finds applications in polymer industry (where one deals with stretching of plastic sheets) and metallurgy where hydromagnetic techniques are being used. To be more specific, it may be pointed out that many metallurgical processes involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid and that in the process of drawing, these strips are sometimes stretched. Mention may be made of drawing, annealing, and thinning of copper wires. In all these cases, the properties of the final product depend to a great extent on the rate of cooling by drawing such strips in an electrically conducting fluid subject to a uniform magnetic field. Another important application of hydromagnetic flow to metallurgy lies in the purification of molten metals from non-metallic inclusion. Therefore, in the present section we present the results of K.Vajravelu, K.V. Prasad, A. Sujatha, (in press) Journal of Applied Fluid Mechanics, for the effects of variable thermal conductivity, and thermal radiation on the heat transfer of a non-Newtonian UCM fluid over a nonisothermal stretching sheet in the presence of internal heat generation/absorption and viscous dissipation, subject to a transverse magnetic field. Savvas et al. [42] suggested that for liquid metals, the thermal conductivity varies
4.4
MHD flow and heat transfer of a Maxwell fluid over a non-isothermal ...
153
linearly with temperature in the range 0◦ F to 400◦ F. Hence, we assume that the thermal conductivity is a linear function of the temperature. Because of the rheological equation of state, the momentum and energy equations are highly nonlinear partial differential equations (PDEs). These PDEs are converted into nonlinear ordinary differential equations (ODEs) by using a similarity transformation. Because of the complexity and the nonlinearity in the proposed problem, it is solved numerically by the Keller-box method. Numerical computation is carried out for temperature and horizontal velocity fields, the Nusselt number and the skin friction for two general cases of nonisothermal boundary conditions. The effects of different physical parameters on the flow phenomenon and heat transfer process are presented through graphs, and the results are discussed.
4.4.2
Mathematical formulation
We consider a steady, laminar, two-dimensional flow of an incompressible, electrically conducting, non-Newtonian, upper convected Maxwell fluid (in the presence of a transverse magnetic field) over a non-isothermal stretching sheet. The flow is generated due to the stretching of an elastic sheet caused by the simultaneous application of two equal and opposite forces along the x-axis, keeping the origin fixed and considering the flow to be confined to the region y > 0. The thermophysical properties of the sheet and the ambient fluid are assumed to be constant. The flow is subject to a uniform magnetic field of strength B0 applied normal to the surface. It is assumed that the sheet is being stretched with a linear velocity u = bx, where b is the linear stretching rate and x is the distance from the slit. It is also assumed that the magnetic Reynolds number is very small, further since there is no electric field, the electric field due to polarization of charges is negligible. It is assumed that boundary layer approximation is applicable in our case (Gupta and Wineman [67]). Therefore, the first step would be to derive the boundary layer equations for our fluid of interest in this particular geometry, and this can be done starting from Cauchy equations of motion in which a source term due to the magnetic field should also be included (Bird et al. [50]). For a two-dimensional flow, the equations of continuity and the momentum (with no pressure gradient present) can be written as ∂u ∂v + = 0, ∂x ∂y
(4.56)
∂u ∂τxx ∂τxy ∂u ρ u = +v + − σB02 u, ∂x ∂y ∂x ∂y
(4.57)
∂v ∂τxx ∂τyy ∂v +v = + , ρ u ∂x ∂y ∂x ∂y
(4.58)
154
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
where u and v are the velocity components along the x and y axes respectively, ρ is the fluid density, σ is the electrical conductivity, and B0 is the uniform magnetic field. As mentioned above, the fluid of interest in the present work obeys an upper convected Maxwell model. For a Maxwell fluid the extra tensor τij can be related to the deformation rate tensor dij by an equation of the form Δ (4.59) τij + λ τij = 2δdij , Δt where δ is the coefficient of viscosity and λ is the relaxation time of the period. The time derivative Δ/Δt appearing in the above equation is the so called upper convected time derivative devised to satisfy the requirements of the continuum mechanics (i.e., material objectivity and frame difference). This time derivative when applied to stress tensor reads as follows [50]: Δ D τij = τij − Ljk τik − Lik τkj , Δt Dt
(4.60)
where Lij is the velocity gradient tensor. For an incompressible fluid obeying the upper convected Maxwell model, the x-momentum equation can be simplified using the boundary layer theory as (Sadeghy et al. [62]) ∂u ∂ 2 u σB02 u ∂u ∂2u ∂ 2u ∂ 2v +v + λ u2 2 + v 2 2 + 2uv =γ 2 − , (4.61) u ∂x ∂y ∂x ∂y ∂x∂y ∂y ρ where γ is the kinematic viscosity of the fluid. The appropriate boundary conditions for the problem are u(x, 0) = bx,
v(x, 0) = 0,
u(x, y) → 0 as y → ∞.
(4.62)
To solve the above boundary layer problem, the following similarity transformation are introduced: u = bxf � (η), v = − bγf (η), η = b/γy. (4.63)
Here, f (η) is the dimensionless stream function and η is the similarity variable. The velocity components u and v in Eq. (4.63) automatically satisfy the continuity equation (4.56). In terms of f , the momentum equation (4.61) can be written as f ��� − M nf � − f �2 + f f �� + β(2f f � f �� − f 2 f ��� ) = 0,
(4.64)
where M n = σB02 /ρb is the magnetic parameter and β = λb is the Maxwell parameter. In view of the transformations, the boundary conditions (4.62) can be written as f (η) = 0,
f � (η) = 1 at η = 0,
lim f � (η) → 0.
η→∞
(4.65)
4.4
MHD flow and heat transfer of a Maxwell fluid over a non-isothermal ...
155
It is worth mentioning here that for the Sakiadis flow of a second grade fluid, we would get a fourth order differential equation with only three boundary conditions. For the second grade fluid flow case, augmenting the needed boundary conditions to match the order of the differential equation has turned out to be a big issue (for details see Rajagopal and Gupta [68], Grag and Rajagopal [69]). Fortunately, here in spite of the fact that the Maxwell model is much more involved than the second-grade model, Sakiadis flow provides a much simpler fluid mechanical problem to be solved. The exact solution of Eq. (4.64) with the boundary conditions (4.65) for β = 0 is obtained as f (η) =
√ 1 − e−αη , α > 0 where α = 1 + M n. α
(4.66)
The shear stress at the sheet is τ0 = −μ(∂u/∂y)y=0,
(4.67)
and its dimensionless form is τ = τ0 /b2 x2 ρ.
4.4.3
(4.68)
Heat transfer analysis
The energy equation with variable thermal conductivity in the presence of internal heat generation/absorption, viscous dissipation and thermal radiation for two-dimensional boundary layer UCM fluid flow ([29], [72], [71], [70]) is given by ∂T ∂T 1 ∂ ∂T Qs (T − T∞ ) u +v = k(T ) + ∂x ∂y ρcp ∂y ∂y ρcp +
μ ρcp
∂u ∂y
2
−
1 ∂qr , ρcp ∂y
(4.69)
where T is the temperature, cp is the specific heat at constant pressure, k is the thermal conductivity. In this section thermal conductivity is assumed to vary as a linear function of temperature [28] as ε (T − T∞ ) . (4.70) k(T ) = k∞ 1 + ΔT
In Eq. (4.70), ΔT = Tw − T∞ , Tw is the sheet temperature, ε is a small parameter and κ∞ is the conductivity of the fluid far away from the sheet. The second term containing Qs in the right hand side (RHS) of Eq. (4.69) represents the temperature dependent volumetric rate of heat source when Qs > 0 and heat sink when Qs < 0. These heat sources and sinks deal with
156
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
the situations of exothermic and endothermic chemical reactions respectively. Viscous dissipation or frictional heating term occurs in the heat transfer analysis as � �2 μ ∂u ρcp ∂y with the assumption that UCM fluid is more viscous in nature than elastic: due to this assumption, we neglect elastic deformation in comparison with the viscous dissipation. The last term qr in the RHS of Eq. (4.69) is the radiative heat flux and is given by qr =
−4σ ∗ ∂T 4 , 3K ∗ ∂y
(4.71)
where σ ∗ and K ∗ are respectively the Stephan-Boltzmann constant and the mean absorption coefficient. We assume that the temperature field within the fluid is of the form T 4 and may be expanded in Taylor series about T∞ . 3 4 Neglecting the higher order terms, we obtain T 4 ≈ 4T∞ − 3T∞ and using this 4 expression for T in Eq. (4.71) we get qr =
3 ∂T −16σ∗ T∞ . 3K ∗ ∂y
(4.72)
Substituting quantities in (4.70), (4.71) and (4.72) into Eq. (4.69) we get � � ∂T k∞ ε ∂T ∂T + ρcp v − ρcp u ∂x ΔT ∂y ∂y � � 2 3 1 16σ∗ T∞ ε ∂ T = κ∞ 1 + (T − T∞ ) + ΔT ρcp 3K ∗ ∂y 2 � �2 ∂u μ +Qs (T − T∞) + . (4.73) ρcp ∂y
From Eq. (4.73) it is observed that the effect of the variable thermal conductivity parameter ε and the thermal radiation parameter is to enhance the thermal diffusivity. The appropriate non-isothermal boundary conditions are � ⎫ � x �2 � ⎪ T = Tw = T∞ + A (PST case) ⎪ ⎬ l at y = 0, (4.74) � � ⎪ x 2 ∂T ⎪ = qw = D (PHF case) ⎭ −k ∂y l T → T∞
as y → ∞,
where A and D are constants. It is obvious now that, ⎧ � �2 x ⎪ PST case, ⎪ ⎨A l , � ΔT = Tw − T∞ = ⎪ D � x �2 γ ⎪ ⎩ , PHF case. κ∞ l b
(4.75)
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157
We now use a scaled η-dependent temperature of the form (4.76)
θ(η) = T − T∞ /ΔT.
The advantage of Eq. (4.74) is that the temperature-dependent thermal conductivity turns out to be x-independent. Using Eq. (4.63) we reduce Eq. (4.73) to ((1 + εθ + N r)θ� )� + Prf θ� − Pr(2f � − α)θ + EcPrf ��2 = 0.
(4.77)
The parameters Pr, α, Ec and N r are the Prandtl number, heat source/sink parameter, the Eckert number, and the thermal radiation parameter respectively, and are defined by Pr =
μCp , k∞
α=
Qs , ρcp b
Ec =
b2 l 2 , Acp
Nr =
3 16σ ∗ T∞ . 3K ∗ k∞
Using quantities in (4.76), (4.75) one can reduce Eq. (4.74) to θ(0) = 1 (PST case) , θ(∞) = 0. θ � (0) = −1 (PHF case)
(4.78)
The local Nusselt number is given by N ux =
h(x) , K∞
where the heat transfer coefficient h(x) is of the form γ 12 K∞ ∂T qw (x) = −K∞ θ� (0), =− h(x) = ΔT ΔT ∂y y=0 b
(4.79)
(4.80)
Substituting quantities in (4.74), (4.76) and (4.80) into Eq. (4.79), we get 1
N ux = −(γ/b) 2 θ � (0).
(4.81)
Exact solution for a special case: perturbation analysis for the Eq. (4.77) in the absence of Maxwell parameter (β = 0 and ε �= 0)
Here, we present analytical solutions for certain special cases. Such solutions are useful and serve as a baseline for comparison with the solutions obtained via numerical schemes. We follow a perturbation expansion approach to solve Eq. (4.77) for PST case [Chiam [29]]. Suppose θ(η) = θ0 (η) + εθ1 (η) + ε2 θ2 (η) + · · · .
(4.82)
Substituting this into Eq. (4.73) and equating like powers of ε ignoring quadratic and higher order terms in ε, we obtain ((1 + N r)θ0� )� + Prf θ0� − Pr(2f � − α)θ0 + EcPrf ��2 = 0
(4.83)
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with the boundary conditions θ0 (0) = 1, and
θ0 (∞) = 0
(4.84)
((1 + N r)θ1� )� + Prf θ1� − Pr(2f � − α)θ1 = −θ0 θ0�� − θ0�2
with the boundary conditions
θ1 (0) = 0,
θ1 (∞) = 0.
The equation can be solved explicitly for θ0 in terms of Kummer’s function: θ0 (η) =
(1 − C1 Pr2 /α41 ) exp(−α1 (a0 + b0 )η/2)M (a1 , b1 , z) M (a1 , b1 , −Pr/α21 )
where a1 = (a0 + b0 − 2)/2, b1 = 1 + b0 , a0 = Pr/α21 (1 + N r), b0 = a20 − (4Prα/α21 ), C1 = Ecα21 /Pr(4 − 2a0 + Prα/α21 ).
A similar analysis may be carried out for Eq. (4.75) for PHF case.
4.4.4
Numerical procedure
The transformed nonlinear coupled ordinary differential equations (4.64) and (4.73) with the boundary conditions (4.65) and (4.78) are solved numerically by the Keller-box method [43]–[44]. For the sake of brevity further details on the solution process are not presented here. It is also important to note that the computational time for each set of input parameter values should be short. Because the physical domain in this problem is unbounded, whereas the computational domain has to be finite, we applied the far field boundary conditions for the similarity variable η at a finite value denoted by ηmax . We ran our bulk computations with the value ηmax = 10, which is sufficient to achieve the far field boundary conditions asymptotically for all values of the parameters considered. For numerical calculations, a uniform step size of Δη = 0.01 was found to be satisfactory and the solutions were obtained with an error tolerance of 10−6 in all the cases. The accuracy of the numerical scheme has been validated by comparing the skin friction and the wall temperature gradient results to those reported in the previous studies: these results agree very well and are shown in Tables 4.9 and 4.10. In order to get a clear insight into the physical problem, the numerical results for the horizontal velocity field f � (η) and temperature field θ(η), skin friction f �� (0) and wall-temperature gradient in PST case and wall temperature in PHF case, are presented in Figures 4.16–4.23. Furthermore, the Salient features are discussed in Section 4.4.5.
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Table 4.9 Comparison of skin friction f (0) for different values of the magnetic parameter when β = 0.0 Analytical solution Numerical solution
M n = 0.0
M n = 0.5
M n = 1.0
M n = 1.5
M n = 2.0
–1.0 –1.000
–1.224 –1.2241
–1.414 –1.4140
–1.581 –1.5812
–1.732 –1.7320
Table 4.10 Comparison of wall temperature gradient θ (0) for different values of Prandtl number when β = 0.0, M n = 0.0, α = 0.0, Ec = 0.0, N r = 0.0 and ε = 0.0 Pr
0.01
0.72
1.0
3.0
5.0
10.0
100.0
Present results —— —— —— —— —— —— —— Grubka and Bobba [51] –0.0099 –0.4631 –0.5820 –1.1652 —— –2.3080 –7.7657 Ali [52] —— –0.4617 –0.5801 –1.1599 —— –2.2960 —— Chen [53] 0.0091 –0.46315 –0.58199 –1.16523 —— –2.30796 ——
Fig. 4.16 Horizontal velocity profile f Vs η for different values of β and M n
4.4.5
Results and discussion
Figure 4.16 illustrates the effects of the Maxwell parameter β and the magnetic parameter M n on the horizontal velocity profiles f (η). It is noticed from the figure that the horizontal velocity profiles f (η) decrease with increasing values of β and M n in the boundary layer, but this effect is not very prominent near the wall. The effect of increasing the value of β is to reduce the horizontal velocity f (η) and thereby reduce boundary layer thickness.
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Fig. 4.17a Temperature profile θ Vs η for different values of β and M n when Pr = 1.0, ε = α = Ec = N r = 0.0
Fig. 4.17b Temperature profile θ Vs η for different values of β and M n when Pr = 1.0, ε = α = Ec = N r = 0.0
That is, the thickness is much larger for non-zero values of β, as clearly seen from Figure 4.16. Further, from Figure 4.16, it can be seen that the horizontal velocity f (η) decreases with an increase in the magnetic parameter M n. This is due to the fact that the introduction of a transverse magnetic field (normal to the flow direction) has a tendency to create a drag force, known
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161
Fig. 4.18 Temperature profile θ Vs η for different values of ε with β = 0.4, M n = 0.5, Ec = N r = α = 0.0, Pr = 1.0
Fig. 4.19 Temperature profile θ Vs η for different values of α with β = 0.4, M n = 0.5, Ec = N r = 0.0, ε = 0.1, Pr = 1.0
as the Lorentz force, which tends to resist the flow. This behavior is even true in the case of increased values of the Maxwell parameter. The effects of the Maxwell parameter β, the magnetic parameter M n, the thermal conductivity parameter ε, the heat source/sink parameter α, the
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Fig. 4.20 Temperature profile θ Vs η for different values of Ec with β = 0.4, M n = 0.5, α = N r = 0.0, ε = 0.1, Pr = 1.0
Fig. 4.21 Temperature profile θ Vs η for different values of N r with β = 0.4, M n = 0.5, α = Ec = 0.0, ε = 0.1, Pr = 1.0
thermal radiation parameter N r, and the Eckert number Ec on temperature profile θ(η) for non-isothermal boundary conditions (both PST and PHF cases) are shown graphically in Figures 4.17–4.22. The general trend is that the temperature distribution θ(η) is unity at the wall in PST case and is not
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163
Fig. 4.22 Temperature profile θ Vs η for different values of Pr with β = 0.4, M n = 0.5, α = Ec = 0.0, ε = 0.1
Fig. 4.23a Values of skin friction f (0) Vs β for different values of M n
unity at the wall in PHF case. However, the temperature distribution θ(η) for both PST and PHF cases decreases asymptotically to zero as the distance increases from the boundary. The effects of the Maxwell parameter β and the magnetic parameter M n on temperature profiles θ(η) in the boundary layer for both PST and PHF cases are shown respectively, in Figures 4.17a and 4.17b. The effect of increasing the values of β leads to enhanced thermal boundary thickness. This is because of the fact that the thickening of the
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Fig. 4.23b Values of skin friction θ (0) Vs β (PST case), θ(0) Vs β (PHF case) for different values of M n with Ec = N r = α = 0.0, Pr = 1.0
thermal boundary layer occurs due to an increase in the elasticity stress parameter. The behavior is even true for non-zero values of the magnetic parameter M n. As explained above, the introduction of a transverse magnetic field to an electrically conducting fluid gives rise to a resistive type of force known as Lorentz force. This force makes the fluid experience a resistance by increasing the friction between its layers, due to which there is increase in the temperature profile θ(η). The effect of the thermal conductivity parameter ε on the temperature profile θ(η) in the boundary layer for non-isothermal boundary conditions is shown graphically in Figure 4.18. The profiles demonstrate quite clearly that an increase in the value of ε results in an increase in the temperature profile θ(η) and hence the thermal boundary layer thickness increases as ε increases. This is due to the fact that the assumption of temperature-dependent thermal conductivity causes a reduction in the magnitude of the transverse velocity by a quantity ∂k(T )/∂y as can be seen from the heat transfer equation (4.73). This phenomenon holds for PHF case; however, thickness of the thermal boundary layer is smaller in comparison with PST case. In Figure 4.19 the temperature distribution θ(η) for different values of the heat source/sink parameter α is shown. The dimensionless temperature attains unity at the wall for prescribed surface temperature and reduces to zero in the free stream for different values of heat source/sink parameter. However the temperature distribution for prescribed wall heat flux is different (less than unity) for different values of ε at the surface and reduces to zero in the free stream. From this figure we examine that the temperature profile is lower throughout the boundary layer for negative values of α (heat sink) and higher for positive values of α (heat source). Physically α > 0 implies Tw > T∞ , i.e., there is
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165
a supply of heat to the flow region from the wall. Similarity α < 0 implies Tw < T∞ and there is a transfer of heat from the fluid to the wall. The effect of increasing value of heat source/sink parameter α is to increase the temperature θ(η) in both PST and PHF cases. The graphs for the temperature distribution θ(η) for different values of the Eckert number Ec, the thermal radiation parameter N r, and the Prandtl number Pr, for non-isothermal boundary conditions are plotted graphically in Figures 4.20–4.22. We noticed from the curves that the effect of increasing values of Ec is to enhance the temperature distribution θ(η). This is in conformity with the fact that the energy is stored in the fluid region, as a consequence of dissipation due to viscosity and elastic deformation, as shown in Figure 4.20. The effect of increasing values of N r is to increase the temperature profile θ(η) and hence increase the thermal boundary layer thickness as depicted in Figure 4.21. This result qualitatively agrees well with the fact that the effect of thermal radiation is to enhance the rate of transport to the fluid, thereby increasing the temperature of the fluid. Further it is observed from the Figures 4.20 and 4.21 that, the region of the thermal boundary layer is more pronounced for PHF case in comparison with PST case for non-zero values of the Eckert number and the thermal radiation parameter. The profiles in Figure 4.22 exhibit the role of the Prandtl number on the temperature profile θ(η). Increasing values of Pr results in decrease of the temperature distribution and hence the thermal boundary layer thickness decreases as Pr increases. This phenomenon is true in both PST and PHF cases. However, from the Figure 4.22, it is noticeable that the thickness of the thermal boundary layer is larger in PST case as compared to PHF case. Figures 4.23a and 4.23b display the variation of skin friction f �� (0), wall temperature gradient θ� (0) (PST case), and wall temperature θ(0) (PHF case) versus the Maxwell parameter for zero and non-zero values of the magnetic parameter. It can be noted that the skin friction decreases with an increase in the Maxwell parameter as well as with the magnetic parameter; whereas quite the opposite holds for non-isothermal temperature boundary conditions.
4.4.6
Conclusions
In this section, a theoretical analysis has been carried out to study the effects of thermal radiation on the MHD boundary layer flow and heat transfer of a UCM fluid over a non-isothermal stretching sheet in the presence of internal heat generation/absorption. The thermal conductivity is assumed to vary linearly with temperature. Here, the flow is generated, due to the stretching of an elastic sheet caused by the simultaneous application of two equal and opposite forces along the x-axis, keeping the origin fixed. The sheet is then stretched with a speed varying linearly with distance from the slit. The governing equations are transformed using appropriate similarity variables and then solved numerically using a second order finite difference scheme. A
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systematic study on the effects of variable thermal conductivity and other physical parameters controlling the flow and heat transfer characteristics is carried out. Some of the important findings of the study are presented herewith. • The effect of increasing Maxwell parameter and the magnetic parameter is to decrease the velocity throughout the boundary layer. However, quite the opposite is true with the thermal boundary layer. • The effect of increasing the Prandtl number is to decrease the thermal boundary layer thickness and the wall temperature gradient. • The effects of increasing the variable thermal conductivity parameter, the thermal radiation parameter and the heat source/sink parameter are to enhance the temperature in the flow region. • The results obtained for the flow and heat transfer characteristics reveal many interesting behaviors that warrant further study of UCM fluid flow and heat transfer. This may help in reducing the skin friction and increasing the heat transfer. • The results for PHF case are qualitatively similar to that of PST case, but quantitatively reduced in magnitude in the presence of variable thermal conductivity. • The effect of viscous dissipation and the thermal radiation is to enhance the temperature profile in PHF case as compared to PST case, while the reverse is true in the presence of internal heat generation/absorption and increase of the Prandtl number.
4.5 MHD boundary layer flow of a micropolar fluid past a wedge with constant wall heat flux 4.5.1
Introduction
Boundary layer theory has been successfully applied to non-Newtonian fluid models and has received substantial attention during the last few decades. One of the important non-Newtonian fluids is the micropolar fluid in which the theory was first introduced by Eringen [73], [74]. This theory takes into account the microscopic effects arising from the local structure and micromotions of the fluid elements. The theory is expected to provide a mathematical model for non-Newtonian fluid behavior. This can be used to analyze the behavior of exotic lubricants, polymeric fluids, liquid crystals, animal blood, colloidal fluids, ferro-liquid, real fluids with suspensions, etc., for which the classical Navier-Stokes theory is inadequate. Extensive reviews of the theory
4.5
MHD boundary layer flow of a micropolar fluid past a wedge with ...
167
and applications micropolar fluids can be found in the review articles by Ariman et al. [75], [76] and the recent books by Lukaszewicz [77] and Eringen [78]. The micropolar fluid theory requires that one must solve an additional transport equation representing the principle of conservation of local angular momentum as well as the usual transport equations for the conservation of mass and momentum. Recently, Kim [79] and Kim and Kim [80] have considered the steady boundary layer flow of a micropolar fluid past a fixed wedge with constant surface temperature and constant surface heat flux, respectively. The similarity variables found by Falkner and Skan [81] were employed to reduce the governing partial differential equations to ordinary differential equations. Unfortunately, the angular momentum equation was not correctly derived in the papers by Kim [79] and Kim and Kim [80] so that the results of these papers are inaccurate. Therefore, the objective of this paper is to improve and extend the work of Kim and Kim [80] by considering the effect of variable magnetic field on the fluid flow and heat transfer characteristics for a fixed wedge with constant surface heat flux. The effect of a transverse magnetic field on a permeable wedge placed symmetrically with respect to the flow direction in a non-Newtonian fluid has been considered by Hady and Hassanien [82]. Watanabe and Pop [83] presented the numerical results for MHD free convection flow over a wedge in the presence of a magnetic field, while Kafoussias and Nanousis [84] investigated the MHD laminar boundary-layer flow over a permeable wedge. Both of these papers considered a wedge immersed in a Newtonian fluid. Later, Yih [85] extended the work of Watanabe and Pop [86] by considering the MHD forced convection flow adjacent to a non-isothermal wedge. The former considered MHD boundary layer flow over a flat plate in the presence of a transverse magnetic field. The effects of variable magnetic field on the fluid flow and heat transfer characteristics were considered by Cobble [87], [88], Helmy [89], [90], Chiam [91], Anjali Devi and Thiyagarajan [92] and very recently by Zhang and Wang [93], [94], Amkadni et al. [95], and Hoernel [96]. On the other hand, the existence of the similarity solutions for the case of a variable magnetic field has been established by an experiment reported by Papailiou and Lykoudis [97] when they reexamined the theoretical work done by Lykoudis [98]. They found that similarity solutions exist when the intensity of the magnetic field changes with x−1/4 , where x is the coordinate measured in the direction of the flow. In this section we present the results of [109] for the MHD boundary-layer flow of a micropolar fluid past a wedge with constant wall heat flux.
4.5.2
Flow analysis
Consider the steady laminar boundary layer flow past a wedge in an electrically conducting micropolar fluid in the presence of a magnetic field B(x) applied normal to the walls of the wedge, as shown in Figure 4.24. The in-
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duced magnetic field is assumed to be small. This implies a small magnetic Reynolds number Rem = μ0 σU (x) � 1 where μ0 is the magnetic permeability, σ is the electrical conductivity, and U (x) is the free stream velocity. It is also assumed that the viscous dissipation, the induced magnetic field, and the Hall effects are neglected. Under these assumptions, the boundary layer equations are
Fig. 4.24 Physical model and coordinate system
∂u ∂v + = 0, ∂x ∂y u
(4.85)
∂u ∂u dU μ + k ∂ 2 u k ∂N σB 2 (x) + +v =U + + (U − u), 2 ∂x ∂y dx ρ ∂y ρ ∂y ρ ∂ ∂N ∂u ∂N ∂N = γ − k 2N + , +v ρj u ∂x ∂y ∂y ∂y ∂y
(4.87)
∂j ∂j +v = 0, ∂x ∂y
(4.88)
∂T ∂T ∂ 2T +v =α 2, ∂x ∂y ∂y
(4.89)
u u
(4.86)
where x-and y-axes are measured along the surface of the wedge and normal to it, respectively, with u and v being the respective velocity components. Further, N, j, γ, α, μ, k and ρ are the components of the microrotation vector normal to the xy-plane, microinertia, spin-gradient viscosity, thermal diffusivity, dynamic viscosity, vortex viscosity, and fluid density, respectively. We shall assume that the boundary conditions of these equations are of the following form: u = 0,
v = 0,
qw = −k
∂T at y = 0, ∂y
u → U (x),
j = 0,
N → 0,
N =−
1 ∂u , 2 ∂y (4.90)
T → T∞ as y → ∞.
Following Falkner and Skan [81], we assume that the free stream velocity is of the form U (x) = axm , where m = 2/(2 − β) and β is the Hartree pressure gradient parameter which corresponds to β = Ω/π for a total angle
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169
Ω of the wedge, and a is a positive constant. We notice that 0 m 1 with m = 0 for the boundary layer flow over a stationary flat plate (Blasius problem) and m = 1 for the flow near the stagnation point on an infinite wall. In order to obtain similarity solutions of the problem described by Eqs. (4.85)–(4.90) we assume that the variable magnetic field B(x) is of the form B(x) = B0 xm−1/2 , where B0 is the uniform magnetic field. This form of B(x) has also been used by Cobble [87], [88], Helmy [89], Chiam [91], Anjali Devi and Thiyagarajan [92], and very recently by Zhang and Wang [93], [94] and Hoernel [96] in their MHD flow problems. Following Ahmadi [99] and Kline [100], we assume that the spin-gradient viscosity γ is defined by γ(x, y) = (μ + k/2)j(x, y) = μ(1 + K/2)j(x, y)
(4.91)
where K = k/μ denotes the dimensionless viscosity ratio and is called the material parameter. This assumption is invoked to allow the field of equations to predict the correct behaviour in the limiting case when the microstructure effects become negligible and the total spin N reduces to the angular velocity (see Ahmadi [99] or Y¨ ucel [101]). Relation (4.91) has also been used by Gorla [102] and Ishak et al. [103] to study different problems of convective flow of micropolar fluids. It is stated by Ahmadi [99] that for a non-constant microinertia, it is possible using Eq. (4.91) to find similar and self-similar solutions for a large number of problems of micropolar fluids. It is also worth mentioning that the case K = 0 describes the classical Navier-Stokes equations for a viscous and incompressible fluid. Following Kim and Kim [80] and Falkner and Skan [81], we introduce now the following similarity variables:
1 1 2γxU 2 (m + 1)U 2 ψ(x, y) = f (η), N (x, y) = h(η), m+1 2γx 2γx j(x, y) = i(η), (m + 1)U 1 (m + 1)U k(T − T∞ ) (m + 1)U 2 y, θ(η) = η= 2γx 2γx qw
(4.92)
where γ is the kinematic viscosity and ψ is the stream function defined in the usual way as u = ∂ψ/∂y, and v = −∂ψ/∂x so as to identically satisfy Eq. (4.85). Substituting (4.92) into Eqs. (4.86)–(4.89) we get the following ordinary differential equations: 2m (1 − f �2 ) + Kh� + M (1 − f � ) = 0, m+1 3m − 1 � K 1+ (ih� )� + i f h� − f h − K(2h + f �� ) = 0, 2 m+1
(1 + K)f ��� + f f �� +
1 (1 − m)f � i − m + f i� = 0, 2
(4.93) (4.94) (4.95)
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pr−1 θ �� + f θ� +
m−1 � f θ=0 m+1
(4.96)
subject to the boundary conditions f (0) = 0,
f � (0) = 0,
h(0) = −0.5f (0), ��
f � (∞) → 1,
i(0) = 0,
θ (0) = −1, �
h(∞) → 0,
θ(∞) → 0
(4.97) (4.98)
where primes denote differentiation with respect to η. M = 2σB02 / aρ(m + 1) is the magnetic parameter and Pr = γ/α is the Prandtl number. If we integrate Eq. (4.95) subjected to (4.97), we get i = Af 2(1−m)/(1+m) where A is a non-dimensional constant of integration. We notice that Eqs. (4.93)–(4.96) were also derived by Kim and Kim [80]. However, Equation (4.94) was wrongly derived in Kim and Kim [80] because (4.97) in their paper contained the extra term (m/(m + 1))ηf � h� . Thus the results reported in that paper are not accurate. The physical quantities of interest are the skin friction coefficient and the local Nusselt number which are, respectively, defined as τw xqw , N ux = (4.99) Cf = ρU 2 /2 k(Tw − T∞ )
where the skin friction τw and the heat transfer from the plate qw are defined as ∂u ∂T + kN τw = (μ + k) , qw = − . (4.100) ∂y ∂y y=0 y=0 Using variables (4.92), we get
K 1 = (m + 1)/2 1 + Cf Re1/2 f �� (0), x 2 2 N ux /Re1/2 x
1 = (m + 1)/2 θ(0)
(4.101)
= U x/γ is the local Reynolds number. where Re1/2 x
4.5.3
Flat plate problem
In this case, m = 0 and thus, Equations (4.93) and (4.94) reduce to (1 + K)f ��� + f f �� + Kh� + M (1 − f � ) = 0, 3m − 1 � K � � � 1+ (ih ) + i f h − f h − K(2h + f �� ) = 0 2 m+1
(4.102) (4.103)
subject to the boundary conditions (4.97). Also, (4.98) becomes i = Af 2 .
(4.104)
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171
The solution of Eqs. (4.102)–(4.104) subject to the boundary conditions (4.97) in the absence of the transverse magnetic field (M = 0) can be found in Ahmadi [99]. If K �= 0, but A = 0, i = 0, from Eq. (4.103), we find 1 h = − f �� , 2
(4.105)
that is, gyration is identical to the angular velocity. Then (4.102) becomes (1 + K)f ��� + f f �� + M (1 − f � ) = 0.
(4.106)
Following Rees and Bassom [104], we take f˜��� + f˜f˜�� + M (1 − f˜� ) = 0
(4.107)
with the boundary conditions f˜(0) = 0,
f˜� (0) = 0,
f˜� (∞) → 1
(4.108)
where now primes denote differentiation with respect to η . The problem (4.107), (4.108) describe the MHD boundary layer flow of a Newtonian fluid over a flat plate in the presence of an applied magnetic field which was first studied by Rossow [105]. The skin friction coefficient given by (4.101) now becomes −1/2 1 K 1/2 (4.109) f˜�� (0). Cf Rex = (m + 1)/2 1 + 2 2 Wedge problem
The problem when M = 0 (absence of magnetic field) has been considered by Kim and Kim [80], but their angular momentum equation was not adequately derived. Thus, we could not compare our results with the results reported by Kim and Kim [80]. Furthermore, if A = 0, i.e., i = 0, it can be easily shown that on using (4.107), Equations (4.5.9) and (4.93) reduce to f˜��� + f˜f˜�� + β(1 − f˜�2 ) + M (1 − f˜� ) = 0 where β = 2m/(m + 1) subject to the boundary conditions (4.109). This equation has also been derived by Soundalgekar et al. [106].
4.5.4
Results and discussion
The nonlinear ordinary differential equations (4.93), (4.94), (4.96), and (4.98) subject to the boundary conditions (4.97) have been solved numerically by means of the Keller-box method, described in the book by Cebeci and Bradshaw [10], for several values of Pr, K, m and M while the non-dimensional constant A is fixed to be unity. The value of A = 1 was also used by Ahmadi [99], Kim [79], and Kim and Kim [80]. The respective system of ordinary differential equations has been integrated forwards in η until a predetermined
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Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
large value of η is reached, say η∞ , where we assume the infinity boundary condition may be enforced. The step size of η, Δη, and the edge of the boundary layer, η∞ are adjusted for different ranges of parameters to maintain the accuracy. In order to verify the accuracy of the present method used on the simulation model, the results are compared with those cases reported by Yih [85], Cebeci and Bradshaw [43], Lin and Lin [107], and Chamkha et al. [108], as shown in Tables 4.11 and 4.12, and the comparisons are found to be in a very good agreement. Therefore, the developed code can be used with great confidence to study the problem considered in this section. 1/2
Table 4.11 Values of Cf Rex
for various values of K, M and m
K
M
m
Yih [85]
Cebeci and Bradshaw [43]
Chamkha et al. [108]
Present results
0
0
0 1/3 1
0.332057 0.757448 1.232588
0.33206 0.75745 1.23259
0.332206 0.757586 1.232710
0.3321 0.7575 1.2326
1
1
0 1/3 1
0.9599 1.3983 0.8670
1/2
Table 4.12 Values of N ux /Rex K
for various values of K, M and Pr when m = 0
M
Pr
Lin and Lin [107]
Present results
0
0
0.0001 0.001 0.01 0.1 1 10 100 1000 10000
0.00873 0.02676 0.07756 0.20066 0.45897 0.99789 2.15197 4.63674 9.98965
0.0087 0.0268 0.0776 0.2007 0.4590 0.9980 2.1520 4.6367 9.9897
1
1
1
0.5251
Figures 4.25–4.27 display the dimensionless velocity profiles f � (η) for various values of K, m and M , respectively, while the other parameters are fixed. It is observed that the velocity f � (η) across the boundary layer increases with m and M but decreases with the material parameter K. These results show that increasing the wedge angle Ω as well as the magnetic parameter M increases the velocity, while increasing the material parameter K decreases it. Further, the boundary layer thickness decreases with an increase in m or M which in turn increases the velocity gradient at the surface (η = 0), and hence produces an increase in the skin friction coefficient f �� (0). The opposite trend is observed for the effect of K, i.e. increasing K decreases the skin friction coefficient f �� (0). Thus, micropolar fluids show drag reduction
4.5
MHD boundary layer flow of a micropolar fluid past a wedge with ...
173
compared to Newtonian fluids. These figures show also that the boundary conditions (4.97) are satisfied, which supports the validity of the numerical results obtained.
Fig. 4.25 Velocity profiles for different values of m with K = 1 and M = 1
Fig. 4.26 Velocity profiles for different values of M with K = 1
The variations of the non-dimensional temperature θ(η) with η for different values of M, K and Pr are displayed in Figures 4.28–4.30, respectively. The temperature at the surface θ(0) is positive and decreases with an increase in M or Pr, but increases with K. Thus, the local Nusselt number 1/θ(0), which represents the heat transfer rate at the surface increases with M or Pr,
174
Chapter 4
Application of the Keller-box Method to Boundary Layer Problems
Fig. 4.27 Velocity profiles for different values of K with M = 1 and m = 0
Fig. 4.28 Temperature profiles for different values of M when Pr = 1, m = 1/3 and K = 1
but it decreases with K. The results presented in Figures 4.28–4.30 indicate that for m = 1/3 and Pr = 1, increasing the skin friction coefficient increases the heat transfer rate at the surface.
4.5
MHD boundary layer flow of a micropolar fluid past a wedge with ...
175
Fig. 4.29 Temperature profiles for different values of K when Pr = 1, m = 1/3 and M = 0
Fig. 4.30 Temperature profiles for different values of Pr when m = 1/3
A sample of microrotation profiles for various values of M when m = 1/3 (Ω = 90◦ ) and K = 1 is displayed in Figure 4.31. It is observed that the
176
Chapter 4
Fig. 4.31 K=1
Application of the Keller-box Method to Boundary Layer Problems
Microrotation profiles for different values of M when m = 1/3 and
absolute value of the dimensionless micro rotation or angular velocity |h(η)| continuously decreases with η and becomes zero far away from the surface, which satisfies the boundary conditions (4.97). As expected, the microrotation effects are more dominant near the wall. Also, |h(η)| increases as M increases in the vicinity of the wedge, but the reverse happens as one moves away from it.
4.5.5
Conclusions
We have studied the theory of steady two-dimensional laminar fluid flow past a fixed wedge immersed in an electrically conducting micropolar fluids. The governing partial differential equations were transformed using suitable transformations to a more convenient form for numerical computation. The numerical results for the velocity, microrotation and temperature profiles were illustrated in some graphs, while the values of the skin friction coefficient and the local Nusselt number were presented in tables for various values of parameters. It was shown that the numerical values of the skin friction coefficient and the local Nusselt number compared well with previously published results for some particular cases of the present problem. The numerical results showed that micropolar fluids display drag reduction compared to the classical Newtonian fluid and consequently reduced the heat transfer rate at the surface. The opposite trends were observed for the effects of the transverse magnetic field on the fluid flow and heat transfer characteristics. Moreover the skin friction coefficient increases as the wedge angle (i.e. m) increases, and increasing Pr increases the heat transfer rate at the surface.
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Chapter 5 Application of the Keller-box Method to Fluid Flow and Heat Transfer Problems
In this chapter, we discuss the application of the Keller-box method to solve coupled nonlinear boundary value problems. In Sections 5.1–5.5, we study the boundary layer convective flow of a Newtonian/non-Newtonian fluid flow over a semi-infinite, stretching permeable sheet with variable fluid properties. The prescribed surface temperature is assumed to vary with the distance from the origin. In Section 5.6, we study the effects of viscous dissipation and the temperature-dependent thermal conductivity on an unsteady flow and heat transfer in a thin liquid film of a non-Newtonian fluid over a horizontal porous stretching surface.
5.1 Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet
5.1.1
Introduction
The study of fluid dynamics due to a stretching surface is important in extrusion processes. The production of sheeting material arises in a number of industrial manufacturing processes and includes both metal and polymer sheets. Examples are numerous, and they include the cooling of an infinite metallic plate in a cooling bath, the boundary layer along material handling conveyers, the aerodynamic extrusion of plastic sheets, the boundary layer along a liquid film, in condensation processes, paper production, glass blowing, metal spinning, drawing plastic films and polymer extrusion, to name
184
Chapter 5
Application of the Keller-box Method to Fluid Flow and ...
just a few. The quality of the final product depends on the rate of heat transfer at the stretching surface. Sakiadis [1] was the first to consider the boundary layer flow on a moving continuous solid surface. Crane [2] extended this concept to a stretching sheet with linearly varying surface speed and presented an exact analytical solution for the steady two-dimensional stretching of a surface in a quiescent fluid. Since then many authors have considered various aspects of this problem and obtained similarity solutions. The boundary layer flow due to a stretching vertical surface in a quiescent viscous and incompressible fluid when the buoyancy forces are taken into account have been considered in the papers by Daskalakis [3], Ali and Al-Yousef [4], Chen [5], [6], Lin and Chen [7], Ali [8], and very recently by Partha et al. [9]. Recently, a significant number of investigations have been carried out on the effects of electrically conducting fluids such as liquid metals, water mixed with a little acid, and others in the presence of a magnetic field on the flow and heat transfer of a viscous and incompressible fluid past a moving surface or a stretching sheet in a quiescent fluid. Chamkha [10] and Abo-Eldahab [11] considered the problems related to hydromagnetic three-dimensional flow on a stretching surface, while Ishak et al. [12] studied the effect of a uniform transverse magnetic field on the stagnation-point flow towards a stretching vertical sheet. Very recently, Anjali Devi and Thiyagarajan [13] investigated the effect of a transverse magnetic field on the flow and heat transfer characteristics over a stretching surface by assuming that the magnetic strength is nonlinear, and they obtained similarity solutions. Motivated by the abovementioned investigations and applications, we present in this section the results of A. Ishak R. Nazar I. Pop, Heat Mass Transfer 44 , 2008, 921–927 for the study of the hydromagnetic flow and heat transfer on a vertical surface of variable temperature that is stretched with a power-law velocity. This study may be regarded as the extension of [13] to a vertical plate.
5.1.2
Mathematical formulation
Consider the mixed convection boundary layer flow due to a stretching vertical heated sheet in a quiescent viscous, incompressible, and electrically conducting fluid in the presence of a transverse variable magnetic field B(x), as shown in Figure 5.1. Two equal and opposite forces are impulsively applied along the x-axis so that the sheet is stretched, keeping the origin fixed in the fluid of ambient temperature T∞ . The stationary coordinate system has its origin located at the centre of the sheet with the x-axis extending along the sheet, while the y-axis is measured normal to the surface of the sheet and is positive in the direction from the sheet to the fluid. The continuous stretching surface is assumed to have velocity and temperature of the form U (x) = axm and Tw = T∞ + bxn , respectively, where a and b are constants. It is also assumed that the magnetic Reynolds number Rex is very small, i.e. Rex = μ0 σaL 1, where μ0 is the magnetic permeability, r is the electrical
5.1
Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet
185
conductivity and L is a characteristic length. Under this assumption, it is possible to neglect the induced magnetic field in comparison to the applied magnetic field. The viscous dissipation and Ohmic heating terms are not included in the energy equation since they are, generally small. Under these assumptions along with the Boussinesq and boundary layer approximations, the basic equations are (see [5] and [13]) ∂u ∂v + = 0, ∂x ∂y ∂u ∂u ∂ 2 u σB 2 (x) u +v =ν 2 − u ± gβ(T − T∞ ), ∂x ∂y ∂y ρ ∂T ∂2T ∂T +v =α 2. u ∂x ∂y ∂y
(5.1) (5.2) (5.3)
Fig. 5.1 Physical model and coordinate system (heated sheet)
Thus the relevant boundary conditions to the problem are u(x, 0) = U (x)(= bx), u(x, y) → 0,
v(x, 0) = 0,
T (x, 0) = Tw (x),
T (x, y) → T∞ as y → ∞
(5.4)
where u and v are the velocity components along the x and y-axes, respectively, g is the acceleration due to gravity, a is the thermal diffusivity of the fluid, m is the kinematic viscosity, b is the coefficient of thermal expansion and q is the fluid density. The last term on the right-hand side of Eq. (5.2) represents the influence of the thermal buoyancy force on the flow field, with
186
Chapter 5
Application of the Keller-box Method to Fluid Flow and ...
“+” and “−” signs pertaining to the buoyancy assisting and the buoyancy opposing flow regions respectively. Figure 5.1 illustrates such a flow field for a stretching vertical heated sheet with the upper half of the flow field being assisted and the lower half of the flow field being opposed by the buoyancy force. For the assisting flow, the x-axis points upwards in the direction of the stretching hot surface such that the stretching induced flow and the thermal buoyant flow assist each other. For the opposing flow, the x-axis points vertically downwards in the direction of the stretching hot surface but in this case the stretching induced flow and the thermal buoyant flow oppose each other. The reverse trend will occur if the sheet is cooled below the ambient temperature. To obtain the similarity solutions of Eqs. (5.1)–(5.4), we assume that the variable magnetic field B(x) is of the form B(x) = B0 (x)(m−1)/2 . This form of B(x) has also been considered by Anjali Devi and Thiyagarajan [13], Chiam [14], and Helmy [15] in their MHD flow problems past moving or fixed flat plates. The continuity equation can be satisfied by introducing a stream function w, such that u = ∂ψ/∂y, v = −∂ψ/∂x. The momentum and energy equations can be transformed to the corresponding ordinary differential equations by the following transformation [5]: η(x, y) = U (x)/(vx)1/2 y,
ψ(x, y) = vxU (x)1/2 f (η),
T = T∞ + (Tw − T∞ )θ(η)
(5.5)
where the functions f (η) and θ(η) are given by the ordinary differential equations m+1 f f �� − mf �2 − M f � + λθ = 0, f ��� + (5.6) 2 m+1 1 �� (5.7) f θ� − nf � θ = 0. θ + Pr 2
Here, primes denote differentiation with respect to η, λ = ±Grx /Re2x , [with “±” sign has the same meaning as in Eq. (5.2)] is the buoyancy or mixed convection parameter, M 2 = σB02 /(ρa) is the magnetic parameter, Pr = v/α is the Prandtl number, Grx = gβ(Tw −T∞ )x3 /v 2 is the local Grashof number and Rex = U x/v is the local Reynolds number. It can be shown that λ is independent of x if n = 2m − 1. Thus, the similarity solutions are obtained under this limitation when λ = 0. We notice that when n = 2m − 1, λ is a constant, with λ > 0 and λ < 0 corresponding to the assisting flow and opposing flow, respectively (see [16]). Under the limitation n = 2m − 1, Equation (5.7) becomes 1 �� m+1 θ + f θ� − (2m − 1)f � θ = 0. (5.8) Pr 2 The transformed boundary conditions are f (0) = 0, f (∞) → 0, �
f � (0) = 1,
θ(0) = 1,
θ(∞) → 0.
(5.9)
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Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet
187
We notice that in the absence of the buoyancy parameter λ, Equations (5.6) and (5.7) reduce to those of Anjali Devi and Thiyagarajan [13] (when the same definition of similarity variable is used), while when the magnetic parameter M is absent, Equations (5.6) and (5.8) reduce to those of Chen [5]. Further, when both buoyancy and magnetic parameters are absent, the analytical solution of Eq. (5.6) for m = 1 is given by Crane [2] and the analytical solution of Eq. (5.8) can be found in Grubka and Bobba [17]. The physical quantities of interest are the skin friction coefficient and the local Nusselt number which are defined as Cf =
τw , ρU 2 /2
N ux =
xqw k(Tw − T∞ )
(5.10)
respectively, where the skin friction τw and heat transfer from the sheet qw are given by ∂u ∂T τw = μ qw = −k (5.11) ∂y y=0 ∂y y=0 with μ and k being the dynamic viscosity and thermal conductivity, respectively. Using the non-dimensional variables (Eq. (5.5)), we get = 2f �� (0), Cf Re1/2 x
N ux /Re1/2 = −θ� (0). x
(5.12)
On the other hand, integrating equation (5.8) subject to boundary conditions (5.9) gives ∞ 5m − 1 � −θ (0) = Pr f � θdη (5.13) 2 0
and it shows that m = 1/5 (n = −3/5) represents a stretching surface subject to an adiabatic situation, i.e. θ� (0) = 0.
5.1.3
Solution of the problem
Equations (5.6) and (5.8) subject to the boundary conditions (5.9) are integrated numerically using the Keller-box method, which is described in [18]. The solution is obtained in the following four steps: • Reduce Eqs. (5.6) and (5.8) to a first order system. • Write the difference equations using central differences. • Linearize the resulting algebraic equations by Newton’s method and write them in matrix-vector form. • Solve the linear system by the block tridiagonal elimination technique.
The step size Δη, and the position of the edge of the boundary layer η∞ had to be adjusted for different values of parameters to maintain accuracy. To conserve space, the details of the solution procedure are not presented here.
188
5.1.4
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Results and discussion
The transformed system of Eqs. (5.6) and (5.8) subject to Eq. (5.9) has been solved numerically using the Keller-box method. This method is unconditionally stable and has second order accuracy. In the presence of buoyancy force (λ = 0), similarity solutions to the present problem may arise if the temperature exponent parameter n and the velocity exponent parameter m are related by n = 2m − 1. Numerical computation has been carried out under this limitation, when λ �= 0. Moreover, for simplicity we considered Prandtl number as unity throughout the section, except for comparisons with the previously investigated cases. To verify the validity and accuracy of the present analysis, results for the skin friction coefficient and the local Nusselt number are compared with those of previous investigations for several values of parameters. The comparison, as shown in Table 5.1 shows a very good agreement. It is noticed that the limitation n = 2m − 1 does not apply in Table 5.1 since Eqs. (5.6) and (5.8) are uncoupled when the buoyancy force is absent (λ = 0), i.e. the solutions to the flow field are not affected by the thermal field. The values of the skin friction coefficient and the local Nusselt number in terms of f �� (0) and −θ � (0), respectively, are presented in Table 5.2, for the case of linearly stretching sheet (m = 1) and the surface temperature proportional to the streamwise coordinate (n = 1). The values of −θ� (0) as shown in Table 5.2 are always positive, which follows from the integral relationship given by Eq. (5.13), i.e. −θ� (0)Pr
∞
0
f � θdη when m = 1. Conversely, the
values of f �� (0) are always negative, for all values of parameters considered. Physically, a positive sign for f �� (0) implies that the fluid exerts a drag force on the sheet, and a negative sign implies the opposite. Figures 5.2 and 5.3 present some samples of velocity and temperature profiles, respectively, for different values of M when Pr = 1, λ = 1, m = 1 and n = 1. The velocity curves in these figures show that the rate of transport is considerably reduced with the increase of M . It clearly indicates that the transverse magnetic field opposes the transport phenomena. This is due to the fact that the variation of M leads to the variation of the Lorentz force due to magnetic field and the Lorentz force produces more resistance to the transport phenomena. It can also be seen from Figure 5.2 that the momentum boundary layer thickness decreases as M increases, and hence induces an increase in the absolute value of the velocity gradient at the surface. Conversely, the absolute value of the temperature gradient at the surface decreases with an increase in M , as shown in Figure 5.3. Thus, the heat transfer rate at the surface decreases with increasing M . We notice that for the same values of parameters, the thickness of the thermal boundary layer is slightly larger than those of the momentum boundary layer. The effects of m and n on the velocity and temperature distributions are displayed in Figures 5.4 and 5.5, respectively. A different characteristic is observed when n = −1, i.e. when the temperature is inversely proportional to the distance from the origin. A detailed discussion
5.1
Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet
189
for this case when the effect of magnetic field is lacking can be found in [19]. Figures 5.2, 5.3, 5.4 and 5.5 also show that the boundary conditions (5.9) are satisfied. The variations of the skin friction coefficient f �� (0) and the local Nusselt number −θ � (0) as a function of k for various values of M are shown in Figures 5.6 and 5.7, respectively. Figure 5.6 shows that for small values of k, the values of f �� (0) are negative, which means that the sheet exerts a drag force on the fluid (and vice versa). It is observed from Figure 5.7 that for a particular value of M the local Nusselt number is increased as the buoyancy parameter is increased from a negative value to a positive value. This is due to the fact that a positive buoyancy force produces a favourable pressure gradient that enhances the fluid motion in the boundary layer, which in turn increases the surface friction and the heat transfer at the surface, while a negative k produces an adverse pressure gradient that slows down the fluid flow and thus reduces the heat transfer rate. Furthermore, the effects of M on the local Nusselt number can be examined from Figure 5.7 that decreasing M enhances the local Nusselt number. This observation is consistent with the surface temperature gradient displayed in Figure 5.3. The numerical results for the effects of m (and n) on the surface friction and heat transfer rate at the surface are presented in Figures 5.8 and 5.9, respectively. As for the variation with M , the value of f �� (0) for a particular value of m is negative for small buoyancy force, and becomes positive if it is large enough. For λ = 1, the absolute value of f �� (0) is larger for a larger value of m which is consistent with the velocity gradient at the surface as shown in Figure 5.4. Furthermore, a decrease in the skin friction coefficient is observed from Figure 5.8 when the velocity exponent parameter m is increased. This behavior can be explained by using the definition of the skin friction coefficient given by Eq. (5.10) with U (x) = axm , so that a larger value of m yields a smaller surface friction. As can be seen from Figure 5.9, for a particular value of m, the heat transfer rate at the surface increases as the buoyancy parameter k increases. The same behavior is observed for a particular value of k, i.e. the local Nusselt number is increased with increasing m, in other words, enlarging the surface temperature variation enhances the heat transfer rate Tw = T∞ + bx2m−1 . Further, the heat transfer rate is negative when m = 0, i.e. when the sheet is stretched with uniform velocity. This numerical result is in agreement with the integral relationship given by Eq. (5.13) which im∞ 1 plies −θ� (0) = − Pr f � θdη. Negative values of −θ� (0) indicate that heat 2 0 is transferred from the fluid to the stretching surface in spite of the excess of surface temperature over that of the free-stream fluid. This phenomenon can be explained as a fluid particle heated to nearly the wall temperature moving downstream to a location where the wall temperature is lower [6]. It is interesting to note that −θ� (0) = 0 for all values of λ (see Figure 5.9) when m = 0.2. This result, which is in agreement with Eq. (5.13) indicates that there is no local heat transfer at the surface of the stretching sheet for all x > 0 even when the fluid and the sheet are at different temperatures. The
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paradox is resolved by recalling that the similarity solution requires a singular behavior of the temperature at x = 0, cf. Tw = T∞ + bx−0.6 . Nevertheless, although dissipation has been neglected, the temperature of the fluid keeps changing during the flow process. Thus, all the necessary heat to change the fluid temperature must be transferred at the singular point x = 0 [20], [21]. Table 5.1 Local Nusselt number −θ (0) for λ = 0, M = 0, m = 1 and various values of Pr and n n
Chen [5]
Grubka and Bobba [17]
Present results
Pr = 1.0 Pr = 3.0 Pr = 10.0 Pr = 1.0 Pr = 3.0 Pr = 10.0 Pr = 1.0 Pr = 3.0 Pr = 10.0 −2 −1.0003 −3.0046 −10.0047 −1.000 −3.000 −10.000 −1.000 −3.000 −10.000 −1 0 1 2
—
—
—
0.0
0.0
0.0
0.0
0.0
0.0
0.5819
1.16523
2.30796
0.5830
1.1652
2.3080
0.5830
1.1652
2.3080
—
—
—
1.00
1.9237
3.7207
1.00
1.9237
3.7207
1.3333
2.5097
4.7969
1.333
2.5097
4.7969
1.3334 2.509972 4.79686
Table 5.2 Values of f (0) and −θ (0) for various values of Pr = 1, λ = 1, n = 1, m = 1 M
f (0)
0 0.1 0.2 0.5 1 2 5
−0.5607 −0.5658 −0.5810 −0.6830 −1.000 −1.18968 −4.9155
−θ (0)
1.0873 1.0863 1.0833 1.0630 1.000 0.8311 0.4702
Fig. 5.2 Influence of the magnetic parameter M on the dimensionless velocity profiles f (η)
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Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet
191
Fig. 5.3 Influence of the magnetic parameter M on the dimensionless temperature profiles θ(η)
Fig. 5.4 Influence of m (and n) on the dimensionless velocity profiles f (η)
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Fig. 5.5 Influence of m (and n) on the dimensionless temperature profiles θ(η)
Fig. 5.6 Influence of the magnetic parameter M on the dimensionless skin friction coefficient f (0)
5.1
Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet
193
Fig. 5.7 Influence of the magnetic parameter M on the dimensionless Nusselt number −θ (0)
Fig. 5.8 Influence of m (and n) on the dimensionless skin friction f (0)
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Fig. 5.9 Influence of m (and n) on the dimensionless Nusselt number −θ (0)
5.1.5
Conclusions
The problem of mixed convection adjacent to a vertical, continuously stretching sheet in the presence of a variable magnetic field has been investigated. The effects of various governing parameters such as M, m, n and λ on the flow and heat transfer characteristics were examined for a fixed value of Pr, namely Pr = 1. Similarity solutions were obtained in the presence of buoyancy force if n = 2m − 1. It can be concluded that both the skin friction coefficient and the local Nusselt number decrease as the magnetic parameter M increases for any given λ and m. Conversely, both of them increase as the buoyancy parameter k increases for fixed values of M and m. Also, for fixed values of M and λ, increasing m is to decrease the skin friction coefficient, while it increases the local Nusselt number. For m = 0.2 (i.e. n = −0.6), although the surface temperature of the sheet is different from the free stream temperature, there is no local heat transfer at the surface except at the singular point of the origin (fixed point).
5.2 Convection flow and heat transfer of a Maxwell fluid over a non-isothermal surface 5.2.1
Introduction
The problem of steady free convection boundary layer flow and heat transfer from a continuously stretching surface finds application in several manufac-
5.2
Convection flow and heat transfer of a Maxwell fluid over a ...
195
turing industries. For example, materials manufactured by extrusion processes and heat treated materials travelling between a feed roll and a wind up roll or on a conveyer belt possess the characteristics of a continuously moving surface. Some of the other examples of these processes include glass blowing, continuous casting, cooling of metallic sheets and electronic chips, crystal blowing, and melts spinning. The classical problem of a steady flow on a continuous moving surface extruded from a slit was first initiated by Sakiadis [1]. He developed a numerical solution using a similarity transformation. Erickson et al. [22] extended the work of Sakiadis by including the suction or injection at the stretching surface and investigating its effect on heat and mass transfer. Tsou et al. [23] showed experimentally that such a flow is physically realizable and explored its basic characteristics. Crane [2] extended this concept to a stretching sheet with linearly varying surface speed and presented an exact analytical solution. Since then the heat and mass transfer aspects have been studied by several authors (such as Gupta and Gupta [24], Chen and Char [25], Grubka and Bobba [17], Vleggaar [26], Jeng et al. [27], Datta et al. [28], and Soundalgekar and Ramana Murthy [29]) under different physical situations. In all these studies the fluid was assumed to be Newtonian. However, many industrial fluids are non-Newtonian or rheological in their flow characteristics, such as molten plastics, polymers, suspensions, foods, slurries, paints, glues, printing inks, blood. That is, they might exhibit dynamic deviation from Newtonian behavior depending upon the flow configuration and/or the rate of deformation. These fluids often obey nonlinear constitutive equations, and the complexity of these constitutive equations is the main culprit for the lack of exact analytical solutions. For example, viscoelastic fluid models considered in these works are simple models, such as second order fluid model and Walters’ model (Rajagopal et al. [30], Siddappa and Abel [31], Dandapat and Gupta [32], Andersson [33], Char [34], Cortell [35], Vajravelu and Rollins [36]), which are known to be good for weakly elastic fluids subjected to slowly varying flows. These two models are known to violate certain rules of thermodynamics. Therefore significance of the results reported in the above works is limited as far as the polymer industry is concerned. Obviously for the theoretical results to become of any industrial importance, more general viscoelastic fluid models such as upper convected Maxwell model or Oldroyd B model should be invoked in the analysis. Indeed these two fluid models are being used recently to study the viscoelastic fluid flow above stretching and non-stretching sheets with or without heat transfer (Bhatnagar et al. [37], Renardy [38], Sadeghy et al. [39], Hayat et al. [40], Aliakbar et al. [41]). All the above studies deal with Newtonian/non-Newtonian flows and heat transfer in the absence of a buoyancy forces. In many practical situations the material moves in a quiescent fluid due to the fluid flow induced by the motion of the solid material and by the thermal buoyancy. Therefore the resulting flow and the thermal field are determined by these two mechanisms, i.e., surface motion and thermal buoyancy. It is well known that the buoyancy force
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stemming from the heating or cooling of the continuous stretching sheet alter the flow and the thermal field and thereby the heat transfer characteristics of the manufacturing processes. However, the buoyancy force effects were not considered in the afore-mentioned studies. Effects of thermal buoyancy force on Newtonian flow and heat transfer over a stretching sheet were reported by several investigators (Acrivos [42], Vajravelu [43], Chen and Strobel [44], Moutsoglou and Chen [45], Chen [5], Afifty et al. [46], Chen [47], Ali and Al-Yousef [4], Vajravelu and Hadjinicolaou [48]). Combined free and forced convection heat transfer at a stretching sheet with variable temperature and linear velocity was investigated by Vajravelu [43]. Similar analyses were performed numerically by Chen and Strobel [44], and Moutsoglou and Chen [45] for Newtonian fluids under different physical situations. Recently an analysis has been carried out by Chen [47] for laminar mixed convection in boundary layer adjacent to a vertical continuously stretching sheet. In all these studies, the thermophysical properties of the ambient fluids were assumed to be constant. However, it is well known that these properties may change with temperature, especially the thermal conductivity. Available literature on variable thermal conductivity (Chiam [49], Datti et al. [50], Prasad et al. [51], Abel et al. [52]) shows that this type of work has not been investigated for UCM fluid flow and heat transfer over a stretching sheet. Motivated by these practical applications, in this section we explores the effects of thermal buoyancy and variable thermal conductivity on a nonNewtonian UCM fluid flow past a vertical continuously stretching sheet (for details see K. Vajravelu, K.V. Prasad, A Sujatha, Central European journal of Physics 9 , 2011, 807–815). In contrast to the work of Sadeghy et al. [39], the present work considers the effects of thermal buoyancy and variable thermal conductivity on the boundary layer UCM fluid flow and heat transfer. In addition to this, we consider a variable temperature distribution at the surface. The governing coupled, nonlinear partial differential equations of the flow and heat transfer problem are transformed into dimensionless equations by using a similarity transformation. These dimensionless nonlinear coupled ordinary differential equations are solved numerically by the Keller-box method for different values of the physical parameters.
5.2.2
Mathematical formulation
The physical problem considered for investigation here is that of a steady mixed convection boundary layer flow of an upper convected Maxwell fluid at a heated vertical stretching sheet, as shown in Figure 5.10. The flow is generated as a consequence of linear stretching of the boundary and the temperature difference between the fluid and the boundary. The positive x-coordinate is measured along the direction of the motion, with the slot at the origin, and the positive y-coordinate is measured normal to the surface and is positive from the sheet to the fluid. The continuous stretch-
5.2
Convection flow and heat transfer of a Maxwell fluid over a ...
197
Fig. 5.10 Schematic of the heat transfer process from a stretching surface
ing sheet is assumed to have a linear velocity and variable wall temperature of the forms U (x) = bx and Tw (x) = T∞ + A(x/l) respectively, where T∞ is the ambient temperature. Here b > 0 is the linear stretching constant, x is the distance from the slit, A is a constant whose value depends upon the properties of the fluid and l is the characteristic length. The thermophysical properties of the sheet and the ambient fluid are assumed to be constant. Under these assumptions (with constant physical properties along with Boussinesq approximation), the governing equations for the convective flow and heat transfer of the upper convected Maxwell fluid (see for details Sadeghy et al. [39]) are ∂u ∂v + = 0, ∂x ∂y 2 2 ∂u ∂2u ∂u 2∂ u 2∂ v u +v + 2uv +v +λ u ∂x ∂y ∂x2 ∂y 2 ∂x∂y ∂2u = ν 2 ± gβ(T − T∞ ), ∂y ∂T ∂T ∂ ∂T +v = k(T ) . ρcp u ∂x ∂y ∂y ∂y
(5.14)
(5.15)
(5.16)
In the above equations, u and v are the velocity components along the x and y-axes respectively, λ is the relaxation time, ν is the kinematic viscosity,
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ρ is the density, g is the acceleration due to gravity, β is the coefficient of thermal expansion. The last term in right hand side of Eq. (5.15) represents the buoyancy force on the flow field. The “+” and “−” signs refer to the buoyancy assisting and buoyancy opposing flow regions, respectively. Figure 5.10 provides the necessary information of such a flow field for a heated vertical stretching sheet with the upper half of the flow field being assisted and the lower half of the flow field being opposed by the buoyancy force. For the assisting flow, the x-axis points upwards in the direction of the hot stretching surface. For the opposing flow, the x-axis points vertically downward in the direction of the stretching surface. Here the stretching induced flow and the thermal buoyant flow oppose each other. The opposite trend occurs if the sheet is cooled below the ambient temperature. T is the temperature of the fluid, Cp is the specific heat at constant pressure, k(T ) is the variable thermal conductivity. In this section thermal conductivity is assumed to vary linearly with temperature (Chiam [49]) as ε k = k∞ 1 + (T − T∞ ) , (5.17) ΔT where ΔT = Tw − T∞ , Tw is the surface temperature, ε is a small parameter, and k∞ is the thermal conductivity of the fluid far away from the sheet. Thus the relevant boundary conditions to the problem are u(x, 0) = U (x)(= bx),
(5.18)
v(x, 0) = 0, T (x, 0) = Tw (x) = T∞ + A(x/l),
(5.19) (5.20)
u(x, y) → 0,
T (x, y) → T∞ as y → ∞.
(5.21)
Here, Equation (5.21) implies that the stream wise velocity and the temperature vanish outside the boundary layer. Equation (5.20) is the variable surface temperature at the wall. The requirement of Eq. (5.19) signifies the importance of impermeability of the stretching surface, and Eq. (5.18) assures no slip at the surface. To facilitate the analysis, the governing partial differential equations are reduced to non-dimensional form with a suitable similarity transformation and an order of magnitude analysis. We define the dimensionless variables as √ η = y b/ν, ψ(x, y) = x bνf (η), T − T∞ = (Tw − T∞ )θ(η), (5.22) where η is the similarity variable, ψ(x, y) is the stream function, f is the dimensionless similarity function, and θ is the dimensionless temperature. The velocity components u and v are given by u=
∂ψ , ∂y
v=−
∂ψ . ∂x
(5.23)
The mass conservation equation (5.14) is automatically satisfied by Eq. (5.23).
5.2
Convection flow and heat transfer of a Maxwell fluid over a ...
199
The momentum equation (5.15) and the energy equation (5.16) are transformed into the coupled nonlinear ordinary differential equation as f ��� − f �2 + f f �� + β(2f f � f �� − f 2 f ��� ) + Grθ = 0, (1 + εθ)θ�� + εθ�2 + Pr(f θ� − f �� θ) = 0
(5.24) (5.25)
where β = λb is the Maxwell parameter, Gr = gβA/(lb2 ) is the free convection parameter or the Grashof number, and Pr = μCp /k∞ is the Prandtl number. It is worth mentioning that Gr > 0 and Gr < 0 correspond to the assisting and opposing the flows respectively, while Gr = 0 (i.e., Tw = T∞ ) represents the case when the buoyancy force is absent (pure forced convective flow). On other hand, if Gr is of a significantly greater order of magnitude than one, the buoyancy forces will predominant and the flow will essentially be free convective. Hence, combined convective flow exists when Gr = 0(1). Equations (5.24) and (5.25) are solved numerically subject to the following transformed boundary conditions (obtained from Eqs. (5.18)–(5.21) using (5.22)): f (η) = 0 at η = 0,
(5.26)
f � (η) = 1 at η = 0, θ(η) = 1 at η = 0,
(5.27) (5.28)
f � (η) → 0,
θ(η) → 0 as η → ∞.
(5.29)
We notice that in the absence of the buoyancy parameter, Equations (5.24) and (5.25) reduce to those of Sadeghy et al. [39], while in the absence of buoyancy parameter and no heat transfer, the equations reduce to those of Pahlavan and Sadeghy [53] in the presence of a magnetic field. Further, when the Maxwell parameter and free convection parameter are absent, the analytical solution of Eqs. (5.24) and (5.25) with the corresponding boundary conditions (5.26)–(5.29) represents the Newtonian case. This agrees well with the results of Crane [2] and Grubka and Bobba [17]. The local Reynolds number describes the velocity of the moving surface that drives the flow. This choice contracts with the conventional boundary layer analysis, in which the free stream velocity is taken as the velocity scale. The physical quantities of interest include the skin friction and the Nusselt number.
5.2.3
Skin friction
The shear stress at the wall is τ0 = −μ(∂u/∂y)y=0 .
(5.30)
The non-dimensional form of shear stress is τ = τ0 /b2 x2 ρ.
(5.31)
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Application of the Keller-box Method to Fluid Flow and ...
Nusselt number
The Nusselt number is given by N ux = h(x)/K∞
(5.32)
where the heat transfer coefficient h(x) is of the form h(x) = qw (x)/ΔT ,
(5.33)
and the rate of heat transfer at the wall is 12 b ∂T x � = −K∞ A qw = −K∞ θ (0). ∂y y=0 ν l
(5.34)
Substituting (5.22) and (5.33) in to Eq. (5.32), we get 1/2 b θ � (0). N ux = − ν
5.2.5
(5.35)
Results and discussion
The system of Eqs. (5.24)–(5.25) subject to the boundary conditions (5.26)– (5.29) is solved numerically by the Keller-box method (for details see Cebeci and Bradshaw [18], Keller [54], Prasad et al. [55], [56]). For numerical calculations, uniform step size of Δη = 0.01 is found to be satisfactory and the solutions are obtained with an error tolerance of 10−6 . The accuracy of the numerical scheme was validated by comparing the skin friction and the rate of heat transfer results to those reported in the previous studies for the special cases; these results agree very well. The results presented here are applicable to highly conducting fluid (such as liquid metal) flow. It can be seen that the solutions are affected by the major five non-dimensional parameters, namely, the Maxwell parameter β, free convection parameter Gr, Prandtl number Pr, variable thermal conductivity parameter ε, and the wall temperature r. In order to get a clear insight into the physical problem, the numerical results for the horizontal velocity f � and the temperature θ are presented through Tables 5.3 and 5.4 and Figures 5.11–5.15, and are discussed. Table 5.3 Skin friction and wall temperature gradient for different values of β when Gr = 0.0, Pr = 1.0, ε = 0.0 β = 0.0
β = 0.2
Present Study −1.0001743 −1.051975 fηη (0) Sadeghy −1.0000 −1.0549 et al. [39]
β = 0.4
β = 0.6
β = 0.8
−1.1019475 −1.1501625 −1.1967279 −1.10084
−1.0015016
−1.19872
θη (0) −1.0001743 −0.98009229 −0.96078789 −0.94231808 −0.92469829
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Convection flow and heat transfer of a Maxwell fluid over a ...
201
Table 5.4 Skin friction and wall temperature gradient for different values of physical parameters Pr ε
β
Gr = −0.1 fηη (0) θη (0)
Gr = 0.0 fηη (0) θη (0)
0.0 −1.0513058 −0.98607433 −1.0001742 0.2 0.0 −1.0544401 −0.86070991 −1.0001742 −0.87531209 0.4 −1.0574566 −0.76738989 −0.78273326 1.0 0.0 −1.1045023 −0.96416777 −0.98009229 0.2 0.2 −1.1079024 −0.83985567 −1.0519750 −0.85657138 0.4 −1.1112329 −0.74710703 −0.76493949
Gr = 0.5 fηη (0) θη (0) −0.76957822 −0.75865120 −0.74876678 −0.81865060 −0.80750287 −0.79740208
−1.0508404 −0.9254475 −0.8328852 −1.0997956 −0.9098025 −0.8182708
1.0 −1.1045023 −0.9641677 −0.98009229 −0.81865066 2.0 −1.0887885 −1.4963832 −1.5042328 −0.87578404 3.0 0.0 0.2 −1.0823967 −1.8999150 −1.0519750 −1.9057292 −0.90377259 4.0 −1.0786309 −2.2387125 −2.2435417 −0.92113376 5.0 −1.0760539 −2.5365765 −2.5407991 −0.93329191
−1.0337958 −1.5387468 −1.9327776 −2.2665184 −2.5611334
1.0 −1.0513057 −0.98607445 −1.0001742 −0.76957822 2.0 −1.0366924 −1.5155694 −1.5230896 −0.82484460 3.0 0.0 0.0 −1.0305252 −1.9179201 −1.0001742 −1.9236088 −0.85212934 4.0 −1.0268393 −2.2560031 −2.2607696 −0.86919987 5.0 −1.0242969 −2.5533717 −2.5575602 −0.88122320
−1.0508404 −1.5565587 −1.9051928 −2.2835028 −2.5777552
Fig. 5.11a Velocity profile fη (η) Vs η for different values of β
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Fig. 5.11b Velocity profile fη (η) Vs η for different values of β and Gr
Fig. 5.12 Temperature profiles θ(η) Vs η for different values of Maxwell parameter β
5.2
Convection flow and heat transfer of a Maxwell fluid over a ...
Fig. 5.13 Temperature profiles θ(η) Vs η for different values of β and Gr
Fig. 5.14a Temperature profiles θ(η) Vs η for different values of ε and β
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Fig. 5.14b Temperature profiles θ(η) Vs η for different values of ε and β
Fig. 5.15a Temperature profiles θ(η) Vs η for different values of Pr and β
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205
Fig. 5.15b Temperature profiles θ(η) Vs η for different values of Pr and β
Figures 5.11a and 5.11b illustrate the changes in the horizontal velocity profiles f � for different values of the Maxwell parameter and the free convection parameter. It is noticed from the figures that the horizontal velocity profiles decrease with increasing the values of the Maxwell parameter and the free convection parameter. That is, the effect of increasing the Maxwell parameter and the free convection parameter is to reduce the horizontal velocity and thereby reduce the boundary layer thickness and hence induce an increase in the absolute value of the surface velocity gradient. This phenomenon is even true in the absence of the free convection parameter. Physically Gr > 0 means heating of the fluid or cooling of the surface, Gr < 0 means cooling of the fluid or heating of the surface, Gr = 0 corresponds to the absence of free convection currents. From Figure 5.11b we also notice that an increase in Gr leads to an increase of f � . Increase of Gr means an increase in the temperature difference Tw − T∞ . This leads to an enhancement in the horizontal velocity and this enhances the convection currents and thus increases the boundary layer thickness. In Figures 5.12–5.15, temperature profiles θ are plotted for different values of the physical parameters. Figure 5.12 presents the effects of the Maxwell parameter on the temperature profile θ. The general trend of these temperature profiles is that, the temperature distribution is unity at the wall as the physical parameters change and tends to zero as the distance increases from
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the boundary. The effect of increasing values of the Maxwell parameter is to increase the temperature profile. This is due to the fact that the thickening of the thermal boundary layer occurs due to an increase in the elasticity stress parameter; however the temperature distribution asymptotically tends to zero as the distance increases from the boundary. Figure 5.13 depicts the temperature profiles for different values of the free convection parameter. Increasing the values of Gr results in a decrease in the thermal boundary layer thickness and hence produces an increase in the surface heat transfer rate. This observation is even true in the presence/absence of the Maxwell parameter. The effect of the variable thermal conductivity parameter on the temperature profiles for different values of convection parameter may be seen in the Figures 5.14a and 5.14b. These figures demonstrate the fact that an increase in the value of the thermal conductivity parameter and the Maxwell parameter results in an increase in the temperature. This is due to the fact that the assumption of a linear form of the temperature dependent thermal conductivity implies a reduction in the magnitude of the transverse velocity by a quantity ∂k(T )/∂y as can be seen from the heat transfer equation. Figures 5.15a and 5.15b, respectively, depict the temperature profiles for different values of Prandtl number in the presence of the Maxwell parameter for opposing forced convection. The figures demonstrate the fact that an increase in the Prandtl number Pr results in a decrease in the temperature distribution and tends to zero as the space variable increases from the wall. That is, the thermal boundary layer thickness decreases as Prandtl number Pr increases for all values of convection parameter. The impact of all the physical parameters on the skin friction −f (0) and the wall temperature gradient −θ (0) may be analyzed from the Table 5.4. Analysis of the tabular data shows that the effect of the Maxwell parameter and the free convection parameter (from negative to positive) is to decrease the skin friction and enhance the rate of heat transfer. This phenomenon is even true in the presence of a variable thermal conductivity parameter. The effect of the Prandtl number is to decrease the wall temperature gradient in the presence of the Maxwell parameter.
5.2.6
Conclusion
In this section, the boundary layer mixed convection flow and heat transfer of a UCM fluid over a non-isothermal stretching sheet is analyzed. The continuous surface is assumed to move with a linear velocity and maintained at a variable surface temperature. The governing equations are transformed into a system of coupled nonlinear ordinary differential equations, and the system is solved numerically by the Keller-box method. A parametric study is performed to explore the effects of various governing parameters (including the Maxwell parameter, the free convection parameter, the variable thermal conductivity parameter, and the Prandtl number) on the fluid flow and heat
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transfer characteristics. Numerical results are presented in tables for the skin friction coefficient and the Nusselt number to reveal the tendency of the solutions. Representative velocity and temperature profiles are illustrated and discussed for different controlling parameters. The numerical results presented reveal the salient features of the upper convected Maxwell (non-Newtonian) fluid flow and heat transfer characteristics.
5.3 The effects of variable fluid properties on the hydromagnetic flow and heat transfer over a nonlinearly stretching sheet
5.3.1
Introduction
The problem of flow and heat transfer in the boundary layer adjacent to a continuous moving surface has attracted many researchers because of its numerous applications in engineering/manufacturing processes, namely continuous casting, glass fiber production, metal extrusion, hot rolling of paper and textiles, and wire drawing. The physical situation was recognized as a backward boundary layer problem by Sakiadis [1], [57]. He was the first, among others, to investigate the flow behavior for this class of boundary layer problems. In his pioneering papers, solutions were obtained to the boundary layer flows on continuous moving surfaces which are substantially different from those of boundary layer flows on stationary surfaces. The thermal behavior of the problem was studied by Erickson et al. [22] using finite difference and integral methods and experimentally verified by Tsou et al. [23]. But in recent years, we find several applications in the polymer industry (where one deals with stretching of plastic sheets) and metallurgy where hydromagnetic techniques are being used. To be more specific, it may be pointed out that many metallurgical processes involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid and that in the process of drawing, these strips are sometimes stretched. Mention may be made of drawing, annealing, and thinning of copper wires. In all these cases, the properties of the final product depend to a great extent on the rate of cooling by drawing such strips in an electrically conducting fluid subject to a magnetic field and the characteristics desired in the final product. In view of these applications, Pavlov [58] investigated the flow of an electrically conducting fluid caused solely by the stretching of an elastic sheet in the presence of a uniform magnetic field. Chakrabarti and Gupta [59] considered the flow and heat transfer of an electrically conducting fluid past a porous stretching sheet and presented the analytical solution for the flow and the numerical
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solution for the heat transfer problem. Andersson [60] extended the work of Chakrabarti and Gupta [59] to MHD flow of a non-Newtonian viscoelastic fluid over an impermeable stretching sheet and found that the magnetic parameter has the same effect as the viscoelastic parameter. Andersson [60] obtained an analytical solution of the magneto-hydromagnetic flow using a similarity transformation for the velocity and temperature fields. Chiam [49] investigated the boundary layer flow of Newtonian fluid: The flow is caused by sheet stretching according to a power law velocity in the presence of a transverse magnetic field. Chamkha [10] and Abo-Eldahab [11] considered the problems related to hydromagnetic three-dimensional flow on a stretching surface. Further, Ishak et al. [12] studied the effect of a uniform transverse magnetic field on the stagnation point flow toward a vertical stretching sheet. Ali [61] extended the work of Chiam [49] to heat transfer characteristics by assuming nonlinear magnetic field strength and obtained similarity solutions for different thermal boundary conditions. In all the above mentioned papers, the thermophysical properties of the ambient fluid were assumed to be constant. However, it is well known that (Herwig and Wickern [62], Lai and Kulacki [63], Takhar et al. [64], Pop et al. [65], Hassanien [66], Abel et al. [67], Seddeek [68], Ali [16], Andersson and Aarseth [69], Prasad et al. [55]) these physical properties may change with temperature, especially for fluid viscosity and thermal conductivity. For lubricating fluids, heat generated by internal friction and the corresponding rise in the temperature affects the physical properties of the fluid, and the properties of the fluid are no longer assumed to be constant. The increase in temperature leads to an increase in the transport phenomena by reducing the physical properties across the thermal boundary layer, and so the heat transfer at the wall is also affected. Therefore to predict the flow and heat transfer rates, it is necessary to take into account the variable fluid properties. In view of this, the problem studied here extends the work of Vajravelu [70] by considering the temperature dependent variable fluid properties. Thus in the present paper, we study the effects of variable viscosity and variable thermal conductivity on the hydromagnetic flow and heat transfer over a nonlinear stretching sheet (for details see K.V. Prasad, K. Vajravelu, P.S. Datti, Int. J. Thermal Sciences 49 , 2010, 603–610). The coupled nonlinear partial differential equations governing the problem are reduced to a system of coupled nonlinear ordinary differential equations by applying a suitable similarity transformation. These nonlinear coupled differential equations are solved numerically by the Keller-box method for different values of the parameters.
5.3.2
Mathematical formulation
Consider a steady, two-dimensional boundary layer flow of an incompressible electrically conducting fluid, in the presence of a transverse magnetic field
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209
B(x) with variable fluid properties, past an impermeable stretching sheet coinciding with the plane y = 0. The origin is located at the slit, through which the sheet is drawn through the fluid medium. The x-axis is taken in the direction of the main flow along the sheet, and the y-axis is normal to it. Two equal and opposite forces are applied along the x-axis so that the wall is stretched, keeping the origin fixed. The continuous stretching surface is assumed to have a power law velocity u = uw = bxm , where b is a constant and m is an exponent. Here we assume that the induced magnetic field produced by the motion of an electrically conducting fluid is negligible. This assumption is valid for a small magnetic Reynolds number. Further, since there is no external electric field, the electric field due to polarization of charges is negligible. The viscous dissipation and the ohmic heating terms are not included in the energy equation since they are generally small. Under the foregoing assumptions and invoking the usual boundary layer approximation, the governing equations of mass, momentum and energy for the problem under consideration, in the presence of variable fluid properties (i.e. fluid viscosity and thermal conductivity), can be written as ∂u ∂v + = 0, ∂x ∂y ∂u ∂u 1 ∂ ∂u σB 2 (x) u +v = μ − u, ∂x ∂y ρ∞ ∂y ∂y ρ∞ ∂T ∂T ∂ ∂T ρ∞ cp u +v = k(T ) , ∂x ∂y ∂y ∂y
(5.37)
1 1 = (1 + δ(T − T∞ )). μ μ∞
(5.39)
(5.36)
(5.38)
where u and v are the velocity components in the stream wise x and crossstream y directions, respectively. Here ρ∞ is the constant fluid density and μ is the coefficient of viscosity, and μ is considered to vary as an inverse function of temperature (Lai and Kulacki [63]) as
This can be rewritten as
1 = a(T − Tr ) μ
where
1 δ and Tr = T∞ − . (5.40) μ∞ δ Here both a and Tr are constants, and their values depend on the reference state and the small parameter δ reflecting a thermal property of the fluid. In general, a > 0 corresponds to liquids and a < 0 to gases when the temperature at the sheet Tw is larger than that of the temperature at far away from the sheet T∞ . The correlations between the viscosity and the temperature for air and water are given below: for air, a=
1 = −123.2(T − 742.6), based on T∞ = 293 K (20◦ C), μ
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and for water, 1 = −29.83(T − 258.6), based on T∞ = 288 K (15◦ C). μ The reference temperatures selected here for the correlations are practically meaningful. The viscosity of a liquid usually decreases with increase in temperature while it increases for gases, when Tw − T∞ is positive. In the above equations, σ is the electrical conductivity, and B 2 (x) is the strength of the magnetic field. The special form of the magnetic field B(x) = B0 x(m−1)/2 is chosen to obtain a similarity solution. This form of B 2 (x) has also been considered by Chiam [14] in MHD flow past a moving flat plate. Here, cp is the specific heat at constant pressure and k(T ) is the temperature-dependent thermal conductivity. We consider the temperature-dependent thermal conductivity relationship in the form (Chiam [49]) ε k(T ) = k∞ 1 + (T − T∞ ) (5.41) ΔT
where ΔT = Tw − T∞ , ε = (kw − k∞ )/k∞ is assumed to be small in magnitude, and kw and k∞ are respectively the thermal conductivities of the fluid at the sheet and far away from the sheet. Substituting Eqs. (5.39), (5.40), and (5.41) in Eqs. (5.37) and (5.38), we obtain ∂u 1 ∂ μ∞ ∂u σB 2 (x) ∂u +v = − u u, (5.42) ∂x ∂y ρ∞ ∂y 1 + δ(T − T∞ ) ∂y ρ∞ ∂ 2 T ∂T k∞ ε ∂T ∂T ε + ρcp v − = k∞ 1 + (T − T∞ ) . (5.43) ρcp u ∂x ΔT ∂y ∂y ΔT ∂y 2 The appropriate boundary conditions on the velocity and the temperature fields are u = uw = bxm , v = 0, T = Tw at y = 0, (5.44) u → 0, T → T∞ as y → ∞,
where b is a stretching rate [(1/sec) for m = 1]. It should be noted that the positive or negative m indicates respectively that the surface is accelerated or decelerated from the extruded slit. Now we transform the system of Eqs. (5.36)–(5.38) into a dimensionless form. To this end, let the dimensionless similarity variable be y m + 1 uw (x) η= Rex , where Rex = x, (5.45) x 2 γ∞ and the dimensionless stream function f (η) and dimensionless temperature θ(η) are 1
f (η) = ψ(x, y)/(uw x(Rex )− 2 ), θ(η) = (T − T∞ )/(Tw − T∞ ),
(5.46) (5.47)
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211
where the dimensionless stream function ψ(x, y) identically satisfies the continuity equation (5.36) where u=
∂ψ ∂ψ and v = − . ∂y ∂x
(5.48)
By using (5.45)–(5.47), the momentum equation (5.42) and the energy equation (5.43) can be written as 1 f �� θ� (5.49) f ��� + βf �2 − f f �� = − M nf � , θ θr − θ 1− θr (1 + εθ)θ�� = −εθ�2 − Prf θ� , (5.50) and they are subject to boundary conditions f � = 1, f = 0, θ = 1 at η = 0, f � = 0, θ = 0 as η → ∞.
(5.51) (5.52)
Here, the � denotes the differentiation with respect to η. The parameters β, θr , M n, and Pr are respectively, the stretching parameter, fluid viscosity parameter, magnetic parameter, and the Prandtl number, which are defined as follows: 2m 2 Tr − T ∞ 1 , θr = , =− m+1 T w − T∞ δ(Tw − T∞) 1 + m 2σB02 μ∞ cp . Mn = , Pr = ρ∞ b(m + 1) k∞
β=
(5.53)
The value of θr is determined by the viscosity of the fluid under consideration and the operating temperature difference. If θr is large, in other words, if T∞ − Tw is small, the effects of variable viscosity on the flow can be neglected. On other hand, for smaller values of θr , either the fluid viscosity changes markedly with temperature or the operating temperature difference is high. In either case, the effect of the variable fluid viscosity is expected to be very important. Also let us keep in mind that the liquid viscosity varies differently with temperature compared to the gas viscosity. Therefore it is important to note that θr is negative for liquids and positive for gases. It should be noted that the velocity u = uw (x) used to define the dimensionless stream function f in Eq. (5.46) and the local Reynolds number in Eq. (5.45) is the velocity of the moving surface that drives the flow. This choice contrasts with conventional boundary layer analysis in which the free stream velocity is taken as the velocity scale. Although the transformation defined in (5.45) and (5.46) can be used for arbitrary variations of uw (x), the transformation results in a true similarity problem only if uw varies as bxm . Here m is an arbitrary constant, not necessarily an integer. Such surface velocity variations are therefore required for the ODE (5.49) to be valid. Non-similar
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stretching sheet problems, which require the solution of partial differential equations rather than ODEs, were considered by Jeng et al. [27] for Newtonian fluids. It is worth mentioning here that θr → ∞ as δ → 0. In this situation for the constant magnetic field case, Equations (5.49) and (5.50) reduce to those of Chakrabarti and Gupta [59], and for m = 0 those of Vajravelu [70]. In the presence of a variable magnetic field and when there is no heat transfer, Equation (5.49) reduces to that of Chiam [14]. Further, when the variable thermal conductivity parameter and the magnetic parameter are absent, Equations (5.49) and (5.50) are similar to the ones studied by Crane [2] and Grubka and Bobba [17]. The physical quantities of interest here are the skin friction coefficient cf and the Nusselt number N u; and they are defined by 2τw (x) qw x , N ux = , (5.54) ρu2w k∞ (Tw − T∞ ) ∂u ∂T and qw (x) = −k∞ . where τw (x) = −μw ∂y y=0 ∂y y=0 Using Eqs. (5.39), (5.40), (5.45)–(5.47) and (5.53), the skin friction and the Nusselt number can be written as √ 2(m + 1)θr �� cf Rex = − f (0, θr ), θr − 1 Nu m+1 � √ =− θ (0, θr ). 2 Rex cf x =
5.3.3
Numerical procedure
By applying similarity transformation to the governing equations and the boundary conditions, the governing equations are reduced to a system of coupled, nonlinear differential equations with appropriate boundary conditions. Finally the system of similarity equations with the boundary conditions is solved numerically by the Keller-box method (Cebeci and Bradshaw [18], Prasad et al. [55], Datti and Prasad [71]). This method is unconditionally stable and has a second order accuracy with arbitrary spacing. First, we write the transformed differential equations and the boundary conditions in terms of a first order system, which is then converted to a set of finite difference equations using central differences. Then the nonlinear algebraic equations are linearized by Newton’s method and the resulting linear system of equations is then solved by block tridiagonal elimination technique. For the sake of brevity, the details of the numerical solution procedure are not presented here. It is worth mentioning that a uniform grid of Δη = 0.01 is satisfactory in obtaining sufficient accuracy with an error tolerance less than 10−6 . To validate the present results, a comparison is made with the known results of Crane [2] and Soundalgekar and Murthy [29].
5.3 The effects of variable fluid properties on the hydromagnetic flow and heat ...
5.3.4
213
Results and discussion
In order to have an insight into the effects of the parameters on the MHD flow and heat transfer characteristics, we present the numerical results graphically in Figures 5.16–5.23 and in Table 5.5 for several sets of values of the temperature dependent fluid property parameters. We consider only the case of a liquid for which θr < 0. We can have a glimpse of the physical layout of the boundary layer structure which develops near the slit by observing the horizontal profiles in Figures 5.16–5.18.
Fig. 5.16 Horizontal velocity profiles for different values of magnetic parameter
Figure 5.16 illustrates the effect of the magnetic parameter M n on the horizontal velocity f in the presence/absence of stretching parameter. From Figure 5.16 we see that f is considerably reduced with an increase in the magnetic parameter. It clearly indicates that the transverse magnetic field opposes the transport phenomena. This is due to the fact that, the transverse magnetic field has a tendency to create a drag force, known as the Lorentz force, and hence an increase in the absolute value of the velocity gradient at the surface. That is, the thickness of the boundary layer is reduced for higher values of the magnetic parameter M n. This behavior is clearly noticeable when the surface is accelerated (β > 0) from the extruded slit. The effect of the stretching parameter β on the horizontal velocity f in the presence/absence of the magnetic parameter M n is depicted in Figure 5.17. It is observed that an increase in the stretching parameter β is to re-
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Fig. 5.17 Horizontal velocity profiles for different values of stretching parameter and magnetic parameter
duce the momentum boundary layer thickness, which tends to zero as the variable η increases from the boundary. Physically, β < 0 implies the surface decelerating case, β = 0 implies the continuous movement of a flat surface, and β > 0 implies the surface accelerating case. A decrease in f discloses the fact that the effect of β is to decrease the velocity and hence reduces the momentum boundary layer thickness. Figures 5.18a–5.18c respectively, show the effects of a decelerating surface (β < 0), continuously moving surface (β = 0), and an accelerating surface (β > 0) from the slit, on the horizontal velocity f for various values of the fluid viscosity parameter θr with Pr = 1.0. From these figures it can be seen that f decreases asymptotically to zero as the variable η increases. However, in the decelerating surface case (β < 0), the velocity profile increases from its value one and then decays to zero. The effect of increasing values of the fluid viscosity parameter θr is to decrease the momentum boundary layer thickness. Also, as θr → 0, the boundary layer thickness decreases and the velocity distribution is asymptotically tends to zero (see Figure 5.18d). This is due to the fact that, for a given fluid (air or water), when δ is fixed, smaller θr implies higher temperature difference between the wall and the ambient fluid. The results presented in this section demonstrate clearly that θr , the indicator of the variation of fluid viscosity with temperature, has a substantial effect on the horizontal velocity f and hence on the skin friction.
5.3 The effects of variable fluid properties on the hydromagnetic flow and heat ...
215
Fig. 5.18a Horizontal velocity profiles for different values of fluid vicosity parameter
Fig. 5.18b Horizontal velocity profiles for different values of fluid vicosity parameter
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Fig. 5.18c Horizontal velocity profiles for different values of fluid vicosity parameter
Fig. 5.18d Horizontal velocity profiles for different values of fluid vicosity parameter
5.3 The effects of variable fluid properties on the hydromagnetic flow and heat ...
217
In Figures 5.19–5.22 the numerical results for the temperature θ for several sets of values of the governing parameters are presented. Figure 5.19 illustrates the effect of the stretching parameter β and the magnetic parameter M n on θ. The effect of increasing values of the stretching parameter β is to increase the temperature θ. This is true even in the presence of the magnetic field. The effect of increasing values of the magnetic parameter M n is to increase the temperature θ. Of course, the effect of M n on the thermal transport, if any, is only an indirect effect through the changes in f and f � . Figures 5.20a and 5.20b exhibit the temperature distribution θ for several sets of values of the fluid viscosity parameter θr , the stretching parameter β, and the Prandtl number Pr. From the graphical representation, we observe that the effect of increasing values of the fluid viscosity parameter θr is to enhance the temperature. This is due to the fact that an increase in the fluid viscosity parameter θr results in an increase in the thermal boundary layer thickness. This is even true for the higher values of the Prandtl number Pr (see Figure 5.20b). The variations of θ for different values of the Prandtl number Pr and the stretching parameter β are displayed in Figure 5.21. The effect of increasing Pr is to decrease θ. That is, an increase in Pr means decrease in the thermal conductivity k∞ and hence, there would be a decrease of thermal boundary layer thickness. The effect of the variable thermal conductivity parameter ε on θ can be seen in Figure 5.22. From this figure we observe that θ increases with increasing ε.
Fig. 5.19 Temperature profiles for different values of stretching parameter and magnetic parameter
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Fig. 5.20a Temperature profiles for different values of fluid viscosity parameter
Fig. 5.20b Temperature profiles for different values of fluid viscosity parameter
5.3 The effects of variable fluid properties on the hydromagnetic flow and heat ...
219
Fig. 5.21 Temperature profiles for different values of Prandtl number (Pr)
Fig. 5.22 Temperature profiles for different values of variable thermal conductivity parameter (ε)
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Figure 5.23 displays the variation of skin friction −f (0) against the fluid viscosity parameter for several sets of values of the stretching parameter and the magnetic parameter. It can be noted that the skin friction decreases with an increase in the viscosity parameter or the stretching parameter. This observation is true even in the presence of the magnetic field. The impact of all the physical parameters on the skin friction −f (0) and the wall temperature gradient −θ (0) may be analyzed from Table 5.5. From Table 5.5 it can be seen that the effect of the magnetic parameter, stretching parameter, and the fluid viscosity parameter is to decrease the skin friction and to enhance the wall temperature gradient. This phenomenon is true even in the presence of the variable thermal conductivity. The effect of the Prandtl number is to decrease the wall temperature gradient even in the presence of the variable viscosity and the variable thermal conductivity.
Fig. 5.23 Influence of variable viscosity parameter, stretching parameter and magnetic parameter on the dimensionless skin friction coefficient
5.3 The effects of variable fluid properties on the hydromagnetic flow and heat ...
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5.4 Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid ...
5.3.5
223
Conclusions
(a) The effect of the magnetic field and the variable viscosity is to decrease the velocity and the skin friction. However, they have opposite effects on the dimensionless temperature and the rate of heat transfer. (b) The velocity and the skin friction are reduced by the stretching parameter; while its effect is to increase the dimensionless temperature and the rate of heat transfer. This is true even in the presence of the temperature dependent fluid properties. (c) The effect of the Prandtl number is to decrease the thermal boundary layer thickness and the wall temperature gradient in the presence of the other physical parameters of the model.
5.4 Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid over a vertical stretching sheet
5.4.1
Introduction
The stretching sheet concept in Crane’s [2] boundary layer flow of a Newtonian fluid over a stretching sheet has been investigated/extended by many authors [17], [24], [25], [27], [28], [29], [72], [73] under different physical situations, due to its important applications in the polymer industry. These studies restrict their analyses to Newtonian fluids. Flow due to a stretching sheet also occurs in thermal and moisture treatment of materials, particularly in processes involving continuous pulling of a sheet through a reaction zone, as in metallurgy, textile and paper industries, in the manufacture of polymeric sheets, and sheet glass and crystalline materials. It is well known that a number of industrial fluids such as molten plastics, polymeric liquids, food stuffs, or slurries exhibit non-Newtonian character. Therefore flow and heat transfer in non-Newtonian fluid is of practical importance. Many different types of non-Newtonian fluids exist but the simplest and most common type is the power law fluid for which the rheological equation of state between the stress components τij and the strain components eij defined by Vujannovic et al [74] is n−1 2 elm elm τij = −P δij + K eij , where P is the pressure, δij is the Kronecker delta, and K and n are consistency and flow behaviour indices of the fluid. Such fluids are known as power
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law fluids and n is referred to as the power law index. When n > 1 the fluid is described as dilatant, n < 1 pseudoplastic, and when n = 1 it is simply the Newtonian fluid. Many papers in this field with or without heat transfer, have been published by a number of authors. To mention a few, Schowalter [75] studied the application of boundary layer theory to power law pseudoplastic fluids. Similarly solutions for non-Newtonian power law fluids were obtained by Kapur and Srivastava [76] and Lee and Ames [77]. The flow of a power law fluid over a continuous moving flat plate with constant surface velocity and temperature distribution has been considered by Fox et al. [78], employing the similarity and integral methods. Andersson and Dandapat [79] extended the work of Crane [2] to non-Newtonian power law fluids over a linear stretching sheet and further extended by Hassanien et al. [80] to include heat transfer analysis. Sahu et al. [81] considered the momentum and heat transfer in a power law fluid from a continuous moving plate. The above studies dealt with Newtonian flows and heat transfer in the absence of a magnetic field. But in recent years several industrial applications such as in polymer technology contain the application of a magnetic field in the power law fluid flows. In view of this, some researchers [82]–[87] have presented works on MHD flow and heat transfer in an electrically conducting power law fluid over a stretching sheet. In many practical situations, the material moves in a quiescent fluid with the fluid flow induced by the motion of the solid material and by thermal buoyancy. Therefore the resulting flow and the thermal fields are determined by these two mechanisms. It is well known that the buoyancy force stemming from the heating or cooling of the continuous stretching sheet alter the flow and the thermal fields and thereby heat transfer characteristics of the manufacturing processes. However, the buoyancy force effects were not considered in the afore-mentioned studies. Effects of thermal buoyancy force on Newtonian flow and heat transfer over a stretching sheet have been reported by several investigators [42]–[48]. Combined free and forced convection heat transfer at a stretching sheet maintained at a variable temperature and moving with a linear velocity was investigated by Vajravelu [43]. Similar analyses were performed numerically by Chen and Strobel [44] and Moutsoglou and Chen [45] for Newtonian fluids under different physical situations. Recently an analysis has been carried out by Chen [47] for laminar mixed convection in boundary layers adjacent to a vertical continuously stretching sheet, and the generated the numerical solutions by a finite difference method. Motivated by these practical applications, the present study explores the effects of thermal buoyancy on non-Newtonian power law fluid flow past a vertical continuous stretching sheet. The surface velocity and temperature distribution at the surface are assumed to vary linearly. In contrast to the work of Hassanien et al. [80] the present work considers the effects of buoyancy and Lorentz force on boundary layer power law fluid flow and heat transfer (K.V. Prasad, P.S. Datti, K. Vajravelu, Int. J. Heat and Mass Transfer. 53 , 2010, 879–888). The governing coupled, nonlinear partial differential equations governing the flow and heat transfer are transformed into a dimensionless form by using
5.4 Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid ...
225
the similarity transformation. These dimensionless forms of nonlinear coupled differential equations are solved numerically by Keller-box method for different values of the physical parameters.
5.4.2
Mathematical formulation
The physical situation considered for the investigation is that of a steady state, mixed convection boundary layer flow of an electrically conducting fluid obeying a power law model in the presence of a transverse magnetic field B0 due to a stretching vertical heated sheet, as shown in Figure 5.24. The flow is generated as a consequence of linear stretching of the boundary sheet, caused by simultaneous application of two equal and opposite forces along the x-axis, while keeping the origin fixed in a fluid of ambient temperature T∞ . The positive x-coordinate is measured along the direction of the motion with the slot as the origin, and the positive y-coordinate is measured normal to the surface of the sheet and is of positive direction from the sheet to the fluid. The continuous stretching sheet is assumed to have a linear velocity and temperature of the form U (x) = bx and Tw (x) = T∞ + A(x/l) respectively, where b is the linear stretching rate and is constant, x denotes the distance from the slit, A is a constant whose value depends upon the properties of the fluid, and l is the characteristic length. It is also assumed that the magnetic Reynolds number Rem is very small, i.e., Rem = μ0 σbl � 1, where μ0 is the magnetic permeability and σ is the electric conductivity. Under these notations, it is possible to neglect the induced magnetic field in comparison to the applied magnetic field. Under these assumptions for an incompressible viscous fluid environment with constant physical properties along with Boussinesq approximation, the basic equations governing this convective flow and heat transfer (Andersson et al. [83], Hassanien et al. [80]) are ∂u ∂v + = 0, ∂x ∂y n ∂u ∂ ∂u ∂u σB02 +v = −γ − u ± gβ(T − T∞ ), u − ∂x ∂y ∂y ∂y ρ ∂T ∂T ∂ 2T u +v =α 2. ∂x ∂y ∂y
(5.55) (5.56) (5.57)
In the above equations, u and v are the flow velocity components along the x and y-axes respectively, γ is the kinematic viscosity of the fluid, n is the power law index, ρ is the fluid density, g is the acceleration due to gravity, and β is the coefficient of thermal expansion. The first term in the right hand side of Eq. (5.55), the shear rate ∂u/∂y has been assumed to be negative throughout the entire boundary layer since the streamwise velocity component u decreases monotonically with the distance y from the moving surface (for a continuous stretching surface). A rigorous derivation and subsequent analysis
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Fig. 5.24 Schematic of mixed convection heat transfer from a stretching surface
of the boundary layer equations for power law fluids were recently provided by Denier and Dabrowski [88]. They focused on boundary layer flow driven by a free stream U (x) ≈ xm i.e., Falkner-Skan type. Such boundary layer flows are driven by a streamwise pressure gradient −dp/dx = ρdu/dx set up by the external free stream outside the viscous boundary layer. In the present context, no driving pressure gradient is present. Instead the flow is driven solely by a flat surface which moves with a prescribed velocity U (x) = bx. The last term in the right hand side of Eq. (5.56) represents the influence of thermal buoyancy force on the flow field, with “+” and “−” sign refers to the buoyancy assisting and buoyancy opposing flow region respectively. Figure 5.24 provides the necessary information of such a flow field for a stretching vertical heated sheet with the upper half of the flow field being assisted and the lower half of the flow field being opposed by the buoyancy force. For the assisting flow, the x-axis points upwards in the direction of the stretching hot surface such that the stretching induced flow and the thermal buoyant flow assist each other. For the opposing flow, the x-axis points vertically downwards in the direction of the stretching hot surface, but in this case the stretching induced flow and the thermal buoyant flow oppose each other. The reverse trend occurs if the sheet is cooled below the ambient temperature. T is the temperature of the fluid and α is the thermal diffusivity of the fluid. Thus the relevant boundary conditions applicable to the flow are u(x, 0) = U (x)(= bx),
(5.58a)
v(x, 0) = 0,
(5.58b)
5.4 Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid ...
T (x, 0) = Tw (x)(= T∞ + A(x/l)),
(5.58c)
T (x, y) → T∞ as y → ∞.
u(x, y) → 0,
227
(5.58d)
Here, (5.58d) claims that the streamwise velocity and the temperature vanishes outside the boundary layer, (5.58c) is the variable prescribed surface temperature at the wall, the requirement (5.58b) signifies the importance of impermeability of the stretching surface, whereas (5.58a) assures no slip at the surface. As in Andersson and Dandapat [79] and Hassanien et al. [80] the following transformation is introduced: η=
1 y (Rex ) n+1 , x
−1
ψ(x, y) = U x(Rex ) n+1 f (η),
θ(η) =
T − T∞ , (5.59) Tw − T∞
where η is the similarity variable, ψ(x, y) is the stream function f and θ are the dimensionless similarity function and temperature, respectively. The velocity components u and v are given by u=
∂ψ , ∂y
v=−
∂ψ . ∂x
(5.60)
The local Reynolds number is defined by Rex = U 2−n xn /γ.
(5.61)
The mass conservation equation (5.55) is automatically satisfied by Eq. (5.60). By assuming the similarity function f (η) to depend on the similarity variable η, the momentum equation (5.56) and the heat equation (5.57) transform into the coupled nonlinear ordinary differential equations 2n − f �2 + f f �� − M nf � + λθ = 0, n+1 2n θ �� + N pr f θ� − f � θ = 0, n+1
n(−f �� )n−1 f
���
(5.62) (5.63)
and M n = σB02 /ρb is the magnetic parameter, λ = ±Grx /Rex is the buoy2 bx2 (Rex ) n+1 is the modified ancy or mixed convection parameter, N pr = α Prandtl number for power law fluids, Grx = gβ(Tw − T∞ )xb−n /γ is the local Grashof number. It is worth mentioning that λ > 0 and λ < 0 correspond to assisting and opposing the flows respectively, while λ = 0 i.e., Tw = T∞ represent the case when the buoyancy force is absent (pure forced convective flow). On other hand, if λ is of a significantly greater order of magnitude than one, the buoyancy forces will predominate and the flow will essentially be free convective. Hence, combined convective flow exists when λ = O(1). A consideration of Eq. (5.62) shows that λ is a function of x. Equations (5.62) and (5.63) are solved numerically subject to the following transformed boundary conditions obtained from Eq. (5.58) using Eq. (5.59) as f = 0 at η = 0,
(5.64a)
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f � = 1 at η = 0,
(5.64b)
θ = 1 at η = 0,
(5.64c)
f � → 0,
θ → 0 as η → ∞.
(5.64d)
We notice that in the absence of buoyancy parameter and magnetic parameter, Equations (5.62) and (5.63) reduce to those of Hassanien et al [80], while in the absence of buoyancy parameter and heat transfer equation, they reduce to those of Andersson and Dandapat [79]. In the presence of the magnetic parameter equation they reduce to those of Andersson et al. [83] and Cortell [35]. Further, when the buoyancy parameter and the magnetic parameter are absent, the analytical solution of Eqs. (5.62) and (5.63) with the corresponding boundary conditions (5.64) can be obtained for Newtonian fluid (n = 1) and is found in Crane [2] and Grubka and Bobba [17]. It should be noted that the velocity U = U (x) is used to define the dimensionless stream function f in Eq. (5.62), and the local Reynolds number in Eq. (5.61) describes the velocity of the moving surface that drives the flow. This choice contracts with the conventional boundary layer analysis, in which the free stream velocity is taken as the velocity scale. Although the transformation defined in Eqs. (5.59) and (5.61) can be used for arbitrary variation of U (x), the transformation results in a true similarity problem only if U varies as bx. Such surface velocity variations are therefore required for the ordinary differential equation (5.62) to be valid. Non-similar stretching sheet problems which require the solution of partial differential equations rather than ordinary differential equations were considered by several researchers for Newtonian fluids. Three boundary conditions, Equations (5.62), are sufficient for solving the third order equation which results for transformed momentum equations for power-law fluids. The physical quantities of our interest include the velocity components u and v, the temperature T , the local surface heat flux qw (x) = −k(∂T /∂y)y=0 or the local Nusselt number N ux = h(x)/K where h(x) = qw (x)/(Tw − T∞ ), and the local skin friction coefficient Cfx = τw (x)/(ρu2w /2) with τw (x) = μ(∂u/∂y)y=0 denoting the local wall shear stress. In terms of the transformation variables, these quantities can be written as
−1
v = −U Rex n+1
u = U f �, 2n 1−n f +η f� , n+1 1+n
1 1 Cf Rexn+1 = (−f �� (0))n , 2
−
1
N ux Rex n+1 = −θ� (0).
(5.65) (5.66) (5.67) (5.68)
5.4 Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid ...
5.4.3
229
Numerical procedure
The transformed coupled, nonlinear ordinary differential equations (5.62) and (5.63) subject to the boundary conditions (5.64) were solved numerically by the Keller-box method (Cebeci and Bradshaw [18] and Keller [54]). This method is second order accurate and allows non-uniform grid size. The numerical algorithm to solve the coupled boundary value problem is as explained below. First, the coupled boundary value problem of (5.62) and (5.63) in f and in θ are reduced to a first order system of five simultaneous ordinary differential equations: dfi = Fi (η, f1 , f2 , . . . , fn ), dη
1 i n (= 5),
where f1 = f,
f2 = f � ,
f3 = f �� ,
f4 = θ,
f5 = θ � ,
F1 = f2 ,
F2 = f3 , F4 = f5 , 2n f1 f3 + M nf22 + λf4 , n(−f3 )n−1 F3 = f22 − n+1 2n f 1 f5 − f 2 f4 . F5 = −N pr n+1
Here prime denotes differentiation with respect to η. Next, after choosing η∞ , the “numerical infinity” (see below), a grid for the closed interval [0, η∞ ] is chosen and the above system of first order equations are transformed into a system of finite difference equations (FDEs) by replacing the differential terms by forward difference approximation and the non-differential terms by the average of two adjacent grid points. The numerical method gives approximate values of f, f � , f �� , θ and θ� at all the grid points. By adding the boundary conditions (5.64) to the system of FDEs, we obtain a nonlinear system of algebraic equations in which the number of equations and unknowns are the same. Subsequently, the linearization of these FDEs was done by Newton’s method. The resulting systems of linear equations were solved by block tridiagonal solver. The step size Δη and the position of the edge of the boundary layer η∞ had to be adjusted for different values of the parameters to maintain accuracy. To conserve space and since the remaining details of the solution procedure are standard, further explanations are not presented here. It is worth mentioning that a uniform grid of Δη = 0.01 was found to be satisfactory for a convergence criterion of 10−6 in nearly all the cases.
5.4.4
Results and discussion
The numerical computation has been carried out for various values of power law index n, Magnetic parameter M n, buoyancy parameter λ and modified Prandtl number N pr using the numerical algorithm discussed in the
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previous section which is unconditionally stable. In order to illustrate the results graphically, the numerical values are plotted in Figures 5.25–5.32. These figures depict the horizontal velocity profiles (Figures 5.25–5.27), temperature profiles (Figures 5.28–5.30), the variation of local skin friction at the wall (Figure 5.31), and the local Nusselt number (Figure 5.32) at the wall for different values of the physical parameters governing the flow and heat transfer.
Fig. 5.25 Influence of Grashof number (λ) on the dimensionless velocity profiles for n = 1.0, with Pr = 1.0
Figures 5.25–5.27 are the graphical representation of the horizontal velocity profiles f for different values of n (i.e, Newtonian (n = 1.0, Figure 5.25), shear thinning fluids (n = 0.4, Figure 5.26), and shear thickening fluids (n = 2.0, Figure 5.27). Figure 5.25 is the graphical representation of f for different values of M n and λ for Newtonian (linear) fluids. Increase of M n leads to a decrease of f in the absence of λ. The velocity profiles in this figure show that the rate of transport is considerably reduced with increase in M n. It clearly indicates that the transverse magnetic field opposes the transport phenomena. This is due to the fact that the variation of M n leads to the variation of Lorentz force, due to magnetic field, and the Lorentz force produces more resistance to the transport phenomena. These results are fully consistent with the results of Andersson et al. [83]. This phenomenon
5.4 Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid ...
231
Fig. 5.26 Influence of Grashof number (λ) on the dimensionless velocity profiles for n = 0.4, with Pr = 1.0
Fig. 5.27 Influence of Grashof number (λ) on the dimensionless velocity profiles for n = 2.0, with Pr = 1.0
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Fig. 5.28a Influence of modified Prandtl number (N pr) on the dimensionless temperature profiles for n = 1.0 with λ = 0.0
Fig. 5.28b Influence of modified Prandtl number (N pr) on the dimensionless temperature profiles for n = 1.0, with M n = 0.0
5.4 Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid ...
233
Fig. 5.29a Influence of modified Prandtl number (N pr) on the dimensionless temperature profiles for n = 0.4, with λ = 0.0
Fig. 5.29b Influence of modified Prandtl number (N pr) on the dimensionless temperature profiles for n = 0.4, with M n = 0.0
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Fig. 5.30a Influence of modified Prandtl number (N pr) on the dimensionless temperature profiles for n = 2.0, with λ = 0.0
Fig. 5.30b Influence of modified Prandtl number (N pr) on the dimensionless temperature profiles for n = 2.0, with M n = 0.0
5.4 Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid ...
235
Fig. 5.31 Influence of modified Prandtl number and Grashof number on the dimensionless skin friction coefficient for M n = 0.0
Fig. 5.32 Influence of modified Prandtl number and Grashof number on the dimensionless Nusselt number for M n = 0.0
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is even true in the presence of λ. Physically λ > 0 means heating of the fluid or cooling of the surface, λ < 0 means cooling of the fluid or heating of the surface, λ = 0 corresponds to the absence of free convection currents. From Figure 5.27 we also noticed that the increase of λ leads to the increase of f � . Increase of λ means increase of temperature gradient Tw − T∞ which leads to the enhancement of horizontal velocity profile due to the enhanced convection and thus increases the boundary layer thickness. This behavior can also be noticed in shear thinning (pseudoplastic) fluids for n < 1 and shear thickening (dilatant) fluids for n > 1, shown in Figure 5.26 and Figure 5.27 respectively. Comparison of Figures 5.26–5.27 reveals that the boundary layer thickness is larger for shear thinning i.e., pseudoplastic (n < 1) fluids as compared to Newtonian (n = 1) and shear thickening i.e., dilatant fluids (n > 1). In Figures 5.28–5.30, temperature profiles θ(η) are plotted for different values of physical parameters. Figure 5.28a is plotted for the effect of M n on temperature profile. Increase of M n leads to increase of the temperature profile. This is due to the fact that Lorentz force increases the friction between its layers and is responsible for an increase in temperature profile θ(η). Increasing the values of N pr leads to decrease of the thickness of the thermal boundary layer. This is consistent with the physical situation that the thermal boundary layer thickness decreases as N pr increases. In Figure 5.28b temperature profiles are plotted for different values of λ. Increasing the values of Gr results in a decrease in the thermal boundary layer thickness associated with an increase in the magnitude of the wall temperature gradient and hence produces an increase in the surface heat transfer rate. This phenomenon is even true for all the fluids considered here. This notation is also seen from Figure 5.29 and Figure 5.30 for shear thinning (n < 1) and shear thickening (n > 1) fluids, respectively. Comparison of these figures on n is that the thermal boundary layer thickness is larger for shear thinning fluids (n < 1) as compared to Newtonian (n = 1) and shear thickening fluids (n > 1). Numerical results for the local skin friction coefficient in terms of f �� (0) and the local Nusselt number in terms of θ� (0) as a function of λ and n for a wide range of modified Prandtl number N pr are shown in Figure 5.31 and Figure 5.32 respectively. Figure 5.31 shows that for different values of λ, the values of f �� (0) are negative which means that the surface exerts a drag force on the fluid. As the parameters λ and n increase, the local skin friction at the sheet increases. It is observed from Table 5.6 that for a particular value of M n the local Nusselt number is increased as λ increases from a negative to a positive value. This is due to the fact that a positive λ produces an increase in the local skin friction coefficient, while negative values of λ give rise to a decrease in the local skin friction coefficient. Increase in N pr increases the skin friction and decreases the local Nusselt number as shown in Figures 5.31 and 5.32.
5.4 Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid ...
237
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Table 5.7 depicts the numerical values of local skin friction and the local Nusselt number for different values of λ and M n. Increase of λ and n leads to increase of skin friction, and decrease the local Nusselt number in the presence/absence of M n. The presence of M n however makes the difference in f (0) between shear thinning and shear thickening fluids significantly more pronounced. Table 5.7 Values of skin friction and wall temperature gradient for different values of Prandtl number with Gr = 0.0 Mn Pr
n = 0.4 n = 1.0 n = 2.0 Skin friction Heat transfer Skin friction Heat transfer Skin friction Heat transfer
1 −0.907636762 2 −1.36556995 3 −1.72155309 0.0 −1.27727556 4 −2.02345419 5 −2.29036236 10 −3.34255242 1 2 3 1.0 −2.283916171 4 5 10
−1.0000
−1.00029039 −1.11737406 −1.52302241 −1.62238181 −1.92355382 −2.04277802 −0.980356753 −2.26072097 −2.40330625 −2.55751467 −2.72152424 −3.7206955 −3.96524167
−0.74202013 −0.895445228 −1.1801548 −1.41426599 −1.52660227 −1.81591523 −1.41421461 −1.82221198 −2.15412116 −2.0852809 −2.45171952 −3.12665081 −3.61707139
−1.1875
−1.10862422 −1.58217645 −1.99022532 −2.34678769 −2.664485 −3.91059351
5.5 The effects of linear/nonlinear convection on the non-Darcian flow and heat transfer along a permeable vertical surface
5.5.1
Introduction
The study of transport processes through porous media has gained interest in the recent past due to its important roles in diverse applications, such as in geothermal operations, cooling of nuclear reactors, petroleum industries, moisture transport in thermal insulation, design of solid-matrix heat exchange, chemical catalytic reactors, underground nuclear waste storage sites, grain storage installations, and many others. Comprehensive literature surveys concerning the subject of porous media can be found in the most recent books by Nield and Bejan [89], Vafai [90], Bejan and Kraus [91]. Cheng and Minkowycz [92] have used the Darcy law
5.5
The effects of linear/nonlinear convection on the non-Darcian flow and ...
239
in their study on free convection about a vertical impermeable flat plate in porous media by the similarity method. Cheng [93] extended the work of [92] by studying the effect of lateral mass flux with prescribed temperature and velocity on vertical free convection boundary layers in a saturated porous medium. The necessary and sufficient conditions under which similarity solutions exist for free convection boundary layers adjacent to flat plates in porous media were reported by Johnson and Cheng [94] using a power law form. Later, Merkin [95]–[96] reported the dual solutions occurring in the problem of the mixed convection flow over a vertical surface in a porous medium with constant surface temperature for the case of opposing flow. The existence of dual solutions in the porous medium for mixed convection boundary layer flow problems has been investigated by Aly et al. [97]. Furthermore, exact analytical solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium were reported by Magyari and Keller [98]. All the above-mentioned investigators have used the Darcy law, which is a linear empirical relationship between the Darcian velocity and the pressure drop across the porous medium and is limited to slow flows. However, for high velocity flow situations, the Darcy law is inapplicable because it does not account for the effects of resulting inertia of the porous medium. In this situation, the relationship between the velocity and the pressure drop is quadratic. The high flow situation is established when the Reynolds number based on the pore size is greater than unity. The Darcy-Forchheimer model is often employed in porous media heat transfer studies and incorporates a second order quadratic drag term which provides a fairly accurate estimation of porous drag effects at higher Reynolds numbers. Vafai and Tien [99] gave an excellent discussion of the importance of inertia effects for flows in porous media. Later, Muralidhar and Kulacki [100] studied thermal convection in a nonDarcian horizontal porous annulus system. Das and Morsi [101] studied the natural convection flow and heat transfer in a dome-shaped heat-generating porous enclosure. Chen et al. [102] presented a numerical investigation for steady free convection inside a cavity filled with a porous medium. B´eg et al. [103] studied computationally the viscous dissipation effects on thermal boundary layers in Darcy-Brinkman porous media. More recently Takhar et al. [104] studied the hydromagnetic convection from an inclined surface in a non-Darcian regime. The physical situation considered by Cheng [93] and Cheng and Minkowycz [92] is one of the possible cases. Another physical phenomenon is the case in which the temperature difference between the plate and the ambient fluid may be appreciably large. The nonlinear density temperature variation in the buoyancy force term (for details see Barrow and Rao [105], Vajravelu and Sastri [106], and Vajravelu et al. [107]) may exert a strong influence on the flow and heat transfer characteristics. This physical concept has a number of geothermal and engineering applications. For example, the residual warm water discharged from a geothermal power plant is usually
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disposed of by wells through subsurface reinjection. This can be idealized as vertical plane sources in a porous medium. If the temperature difference between the injected fluid and the receiving ground water is appreciably large, then the injected fluid experiences a positive or negative buoyancy force which results in a convective movement of ground water near the well. Also, convection currents will set in, in the ground water along the vertical fissures or cracks during the natural recharge of aquifer whenever the temperature of the ground water discharged from the fissures and cracks differs from that of the receiving water in the aquifer. Motivated by these applications, in the present section the authors (K. V. Prasad, K. Vajravelu, Robert A. Van Gorder, Acta Mechanica 220 , 2011, 139–154) investigate numerically the effects of non-Darcy convective flow and heat transfer along a vertical permeable surface in a saturated porous medium with nonlinear variations of density with temperature. The boundary layer and Boussinesq approximations are used to simplify the momentum and heat transfer equations. However, the flow and heat transfer characteristics depend on the inertia parameter, linear/nonlinear density temperature parameter, Rayleigh number, Eckert number, Prandtl number, and lateral mass flux parameter. Hence, the momentum and energy equations are coupled and nonlinear. These equations are in turn solved numerically by the Keller-box method. Thus for the solution of highly nonlinear coupled boundary value problem, computer simulation is a powerful technique to predict the flow behavior.
5.5.2
Mathematical formulation
Consider the problem of nonlinear convection from a vertical flat plate embedded in a fluid saturated porous medium of ambient temperature T∞ in the presence of viscous dissipation. We assume that the plate is maintained at a constant temperature Tw where Tw > T∞ is for a heated plate and Tw < T∞ corresponds to a cooled plate. The temperature between the plate and the medium is assumed to be sufficiently large; hence the convection region is thick. The vertical plate is immersed in a Newtonian, saturated, isotropic, homogeneous, and high porosity non-Darcian porous medium which is stable and thermally stratified. The temperature of the plate is given by a constant quantity Tw and the discharge or withdrawal rate is given by v = −axn (here n = −1/2), where a > 0 for the discharge of the fluid and a < 0 for withdrawal of the fluid. The origin is located at the leading edge of the vertical plate and the y-coordinate is directed normal to the plate. It is assumed that the fluid and the porous medium have constant physical properties except for the density variation in the buoyancy term (which varies linear/nonlinear with temperature) in the momentum equation. The fluid flow is moderate and the permeability of the medium is low, so the Forchheimer flow model is employed to simulate the resistance of the porous
5.5
The effects of linear/nonlinear convection on the non-Darcian flow and ...
241
medium fibres on the buoyant flow, following Bhargava [108], [109], Takhar [110]. With the usual Boussinesq and the boundary layer approximations, the governing equations for the convective flow from the wall to the fluidsaturated porous medium can be written (Cheng [93] and Vajravelu et al. [107]) in Cartesian co-ordinates as ∂u ∂v + = 0, ∂x ∂y 1 ∂P μ C 1 ρg, u + u2 = − − ρ∞ K K ρ∞ ∂x ρ∞ μ 1 ∂P C v+ v 2 = − , ρ∞ K K ρ∞ ∂y 2 ∂T ∂2T μ ∂T ∂u +v =α 2 + , u ∂x ∂y ∂y ρ∞ Cp ∂y
(5.69) (5.70) (5.71) (5.72)
where u and v are the flow velocities in x- and y-directions respectively, ρ∞ is the constant fluid density, μ is the coefficient of viscosity, K is the permeability of the porous medium, C is the inertial coefficient (i.e. Forchheimer parameter which is related to the geometry of the porous medium), P is the pressure, g is the acceleration due to gravity, T is the temperature, α = k/ρ∞ Cp is the effective thermal diffusivity, k is the thermal conductivity, and Cp is the specific heat at constant pressure. The fluid density ρ is assumed to vary with temperature in nonlinear form and is defined as follows (see, Bird et al. [111], Streeter [112]): 2 ∂ρ ∂ ρ (T − T∞ ) + (T − T∞ )2 + · · · . (5.73) ρ(T ) = ρ(T∞ ) + ∂T ∞ ∂T 2 ∞
By taking into account the terms up to (T − T∞ )2 , we write the above relation as ρ = ρ∞ (1 − β0 (T − T∞ ) − β1 (T − T∞ )2 ), (5.74)
where β0 and β1 are the coefficients of thermal expansion of the first and second orders respectively. The above relation accommodates the linear density temperature variation and the quadratic density temperature variation (used by Barrow and Rao [105] and Brown [113]), as well as the nonlinear density temperature variation. This relation can be used to analyze flow of water between 0◦ C to 20◦ C (see for details [106]). The appropriate boundary conditions for the flow regime are defined as v = ax−1/2 , u → 0,
T = Tw at y = 0,
T → T∞ as y → ∞.
(5.75a) (5.75b)
Let us make use of the following non-dimensional quantities: α u= U, d
v=
α V, d
c=
d C, α
x = Xd,
y = Y d,
T − T∞ = T ∗ ΔT.
(5.76)
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Here ΔT = (Tw − T∞ ) and invoking the usual Boussinesq and boundary layer approximations, the governing equations in terms of stream function ((U, V ) = (∂ψ/∂y, −∂ψ/∂x)) can be written as ∗ ∗ ∂2ψ ∂T ∂ψ ∂ 2 ψ ∗ ∂T + δT , (5.77) + k1 = Ra ∂Y 2 ∂Y ∂Y 2 ∂Y ∂Y 2 ∂ψ ∂T ∗ ∂ψ ∂T ∗ ∂ 2T ∗ Ec ∂ 2 ψ − = + . (5.78) ∂Y ∂X ∂X ∂Y ∂Y 2 Pr ∂Y 2 The stream function ψ automatically satisfies the continuity equation (5.69). In the above equation, k1 , Ra, δ, Ec and Pr denote respectively, the inertial parameter, the Rayleigh number, the nonlinear density variation with temperature (NDT) parameter, the Eckert number, and the Prandtl number, and they are defined as k1 = 2 Ec =
C , γ
Ra =
ρ∞ gβ0 (Tw − T∞ )Kd , αμ
γ2 , cp (Tw − T∞ )d2
δ=
2β1 (Tw − T∞ ) , β0
(5.79)
Pr = γ/α.
With properly chosen similarity variables, the above partial differential equations (5.77) and (5.78) can be transformed into a set of coupled nonlinear ordinary differential equations. The suitable similarity transformation for the problem is √ (5.80) ψ = x1/2 f (η), η = y/ x, T ∗ = θ(η). Now, the governing equations (5.77) and (5.78) in terms of the new variables are f �� + k1 f � f �� − Ra(1 + δθ)θ� = 0, Ec �� 2 1 � (f ) + f θ = 0, θ �� + Pr 2
(5.81) (5.82)
and the boundary conditions for the problem in the non dimensional form are f (η) = fw , θ(η) = 1 at η = 0, (5.83a) fη (η) → 0,
θ(η) → 0 as η → ∞.
(5.83b)
Here fw = −2a is positive for the withdrawal of the fluid and negative for the discharge of the fluid. Therefore fw = 0 corresponds to an impermeable surface. Furthermore the plate is permeable with suction or injection according to fw > 0 or fw < 0 respectively. The physical quantity of interest in this problem is the local Nusselt number, which is defined as 1 x ∂T Qx =− k = x 2 (−θ� (0)). N ux = k(Tw − T∞ ) k(Tw − T∞ ) ∂y y=0
5.5
The effects of linear/nonlinear convection on the non-Darcian flow and ...
5.5.3
243
Numerical procedure
The transformed ordinary differential equations (5.81) and (5.82) are coupled and nonlinear, their analytical solution may not be possible, and therefore, we employ a numerical technique to solve the equations. Here, the solutions for f and θ are obtained by the Keller-box method [18], [54]. The numerical solutions are obtained in four steps as follows: • reduce Eqs. (5.81) and (5.82) to a system of first order equations; • write the difference equations using central differences; • linearize the algebraic equations by Newton’s method, and write them in matrix-vector form; and • solve the linear system using the conditions (5.83a) and (5.83b) by a block tridiagonal elimination technique. To maintain the accuracy, in calculations, a step size of η = 0.01 were found to be satisfactory in obtaining sufficient accuracy within a tolerance less than 10−6 in nearly all cases. The step size Δη, and the position of the edge of the boundary layer η∞ are adjusted for several sets of values of the parameters. For brevity, the details of the solution procedure are not presented here. Numerical results are obtained for slip velocity and the temperature gradient at the wall. The pertinent parameters are the inertia parameter, the nonlinear density temperature variation parameter (or the NDT parameter), the linear density temperature variation parameter (or the LDT parameter), Eckert number (Ec), Prandtl number (Pr), and the lateral mass flux parameter (fw ). Effects of these parameters on the velocity and the temperature fields are shown graphically in Figures 5.33–5.37. Velocity and temperature profiles are presented respectively in Figures 5.33–5.34 and 5.35–5.37. The numerical results for the slip velocity and the wall-temperature gradient are respectively, presented in Tables 5.8 and 5.9. Table 5.8 Values of the slip velocity f (0) for different values of the physical parameters with Ec = 0.1, Pr = 1.0 Ec 0.1
Ec 0.0
fw 0.0
k1 0.1
0.0
0.0
1.0
0.0 0.2 0.4 k1 0.1
fw −1.0
Ra 1.0 5.0 10.0 15.0 1.0 5.0 10.0 15.0 5.0
Ra 1.0 5.0 10.0 15.0
δ = −0.2 0.862781 3.784047 6.733196 9.235379 0.900000 4.49999 8.99999 13.49999 5.000000 3.660254 3.090170 δ = −0.5 0.723810 3.228763 5.811389 8.027752
δ = 0.0 0.954451 4.142136 7.320508 10.00000 1.00000 5.000000 10.00000 15.00000 4.49999 3.366596 2.861897 δ = 0.0 0.954451 4.142136 7.320508 10.0000
δ = 0.2 1.045362 4.491381 7.888541 10.736441 1.099999 5.50000 11.00005 16.50000 5.50000 3.944270 3.309474 δ = 0.5 1.180341 5.000 8.708285 11.794494
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5.5
The effects of linear/nonlinear convection on the non-Darcian flow and ...
245
Fig. 5.33a Velocity profiles for different values of the inertia parameter k1
Fig. 5.33b δ = −0.2
Velocity profiles for different values of the Rayleigh number with
246
Chapter 5
Application of the Keller-box Method to Fluid Flow and ...
Fig. 5.33c Velocity profiles for different values of the Rayleigh number with δ = 0.0
Fig. 5.33d δ = 0.2
Velocity profiles for different values of the Rayleigh number with
5.5
The effects of linear/nonlinear convection on the non-Darcian flow and ...
247
Fig. 5.34a Slip velocity profiles for different values of the inertia parameter k1
Fig. 5.34b Slip velocity profiles for different values of the Rayleigh number with δ = −0.2
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Fig. 5.34c Slip velocity profiles for different values of the Rayleigh number with δ = 0.0
Fig. 5.34d Slip velocity profiles for different values of the Rayleigh number with δ = 0.2
5.5
The effects of linear/nonlinear convection on the non-Darcian flow and ...
249
Fig. 5.35a Temperature profiles for different values of lateral mass flux parameter fw
Fig. 5.35b Temperature profiles for different values of inertia parameter k1
250
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Application of the Keller-box Method to Fluid Flow and ...
Fig. 5.35c Temperature profiles for different values of NDT parameter δ = −0.5
Fig. 5.35d Temperature profiles for different values of NDT parameter δ = 0.5
5.5
The effects of linear/nonlinear convection on the non-Darcian flow and ...
251
Fig. 5.36a Temperature profiles for different values of Eckert number with Ra = 5.0
Fig. 5.36b Temperature profiles for different values of Eckert number with Ra = 10.0
252
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Application of the Keller-box Method to Fluid Flow and ...
Fig. 5.37a Temperature profiles for different values of Prandtl number with Ra = 5.0
Fig. 5.37b Temperature profiles for different values of Prandtl number with Ra = 10.0
5.5
The effects of linear/nonlinear convection on the non-Darcian flow and ...
253
The boundary layer approximations used here are valid for (i) ∂/∂y � −1/2 ) and ∂/∂y and (ii) v � u. From Eq. (5.80) it follows that y/x = O(x −1/2 v/u = O(x ). Thus the conditions (i) and (ii) are satisfied. However, near the leading edge (x = 0), the boundary layer approximations are not expected to be valid. The expression for thermal and momentum boundary layer thickness can be obtained if the edges of the boundary layers are defined as the points where θ or u/uw have a value 0.01. It is of some interest to note that although u → 0 outside the momentum boundary layer, the value of the vertical velocity in general not zero there. This can be seen from Eq. (5.80). That is, 1 v∞ = − α∞ 2
gρ∞ β0 K(Tw − T∞ ) α∞ μ∞
1/2
f (∞).
It is worth mentioning here that the value f (∞) is positive and finite as shown in Figures 5.33a–5.33d.
5.5.4
Results and discussion
In this section, we have presented the graphical results: the numerical values of the slip velocity at the wall and temperature gradient at the wall for different values of the pertinent parameters. Velocity and slip velocity profiles are shown graphically in Figures 5.33–5.34 for different values of the pertinent parameters when Pr = 1.0, Ec = 0.1. The general behavior from these profiles is that the velocity profile f increases initially at the plate and attains a maximum as the distance η increases from the boundary, whereas the slip velocity f � is not uniform at the plate but shows an exponential decay as η increases. The effect of increasing values of the lateral mass flux parameter fw is to increase the velocity (Figure 5.33) and to decrease the slip velocity (Figure 5.34). This is as expected, because, positive values of fw (suction) reduce the boundary layer thickness when compared to zero (impermeable) and negative (blowing) values. This phenomenon is true even in the presence of increasing values of the Rayleigh number (Ra). The effect of increasing values of Ra is to increase the velocity as well as the slip velocity. Figures 5.33b–5.33d and Figures 5.34b–5.34d exhibit respectively, the behavior of NDT parameter δ on the velocity and the slip velocity profiles. From the figures we noticed that the effect of δ is to increase the velocity as well as the slip velocity profiles. It is also seen from the figures that the slip velocity is lower throughout the boundary layer for negative values of δ and higher for positive values of δ as compared with the velocity distribution with LDT (δ = 0) parameter. Physically δ > 0 implies Tw > T∞ , there will be a supply of heat to the flow region from the wall. Similarly δ < 0 implies Tw < T∞ , and there will be a transfer of heat from the flow to the wall. Again this increase in velocity with positive values of δ is more prominent in the presence/absence
254
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of the inertia parameter. Figure 5.34a shows the behavior of the slip velocity profiles for increasing values of inertia parameter k1 . The effect of k1 is to decrease the slip velocity profile f . This is due to the fact that the porous medium opposes the flow and thus leads to enhanced deceleration of the flow. This behavior is even true for all the values of lateral mass flux parameter. The numerical results for the temperature profile θ are depicted in Figures 5.35–5.36 for different values of the governing parameters. It is observed from these figures that temperature distribution is unity at the wall with the change of physical parameters and tends asymptotically to zero in the free stream region. Figure 5.35a exhibit the effect of the lateral mass flux parameter fw and NDT parameter δ on the temperature profiles. The effect of increasing values of fw is to decrease the thermal boundary layer thickness. This observation is even true for all the values of δ. The negative values of δ increase the temperature profile as compared to zero and positive values of δ. These results are consistent with the physical situation (as explained above). The effects of the inertia parameter k1 on the temperature profile for different values of lateral mass flux parameter fw are shown graphically in Figure 5.35b. From the figures we deduce that the effect of increasing values of k1 is to increase the the temperature profile. This is expected because increase of inertia parameter introduces some more additional stress which is responsible for thickening of the thermal boundary layer and hence thermal boundary layer increases as the inertia parameter increases. Figures 5.35c– 5.35d exihibit respectively, the behavior of temperature for δ = −0.5 and δ = 0.5 for different values of the Rayleigh number Ra. The effect of increasing values of Ra is to decrease the thermal boundary layer thickness when the Eckert number is zero. The Rayleigh number describes the relative intensity of the buoyancy forces. Thus, when Ra increases, the relative intensity of the buoyancy force increases, consequently the thickness of the thermal boundary layer is reduced. That is, buoyancy induced upward flow adjacent to the vertical plate increases as Ra increases and is hence capable of transporting more heat energy from the wall, resulting in reduction in thermal boundary layer thickness. In other words, heat energy does not have enough time to build up the temperature normal to the vertical wall. It is further noted that as Ra increases, the temperature distribution normal to the wall decreases indicating a reduction in the thermal boundary layer. Figures 5.36a–5.36b show changes in the temperature distribution θ(η) for different values of Eckert number Ec with Ra = 5.0 and Ra = 10.0 respectively. From these profiles we observe that the temperature profile is lower throughout the boundary layer when Ec is zero. The effect of increasing values of Ec is to increase the temperature profile. This is noticeable for higher values of Rayleigh number. This is in conformity with the fact that energy is stored in the fluid region due to frictional heating as a consequence of dissipation due to viscosity, and hence temperature increases as Ec increases. Samples of temperature profiles are presented in Figures 5.37a–5.37b for different values of Prandtl number with Ra = 5.0 and Ra = 10.0 respectively.
5.6
Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid ...
255
We notice that the effect of increasing values of Pr is to decrease the temperature profile in the flow field. Physically, increase of Pr means decrease of thermal conductivity: that results in decrease of thermal boundary layer thickness. Hence, the temperature profile is higher for smaller values of Pr and smaller for larger values of Pr. From the same graph we also notice that the effect of Rayleigh number is to increase the temperature profile for all values of the Prandtl number considered here. The effects of all the physical parameters involved, on the slip velocity and on the wall temperature gradient respectively are recorded in Tables 5.8 and 5.9. The effect of increasing values of Ra and δ is to increase the slip velocity and decrease the wall temperature gradient. This is even true for all the values of the lateral mass flux parameter and the inertia parameter. The effect of increasing values of the inertia parameter is to decrease the slip velocity and increase the wall temperature gradient. The effects of the Eckert and Prandtl numbers respectively are to increase and decrease the wall temperature gradient at the wall.
5.6 Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid over a stretching surface
5.6.1
Introduction
During the past two decades, due to its applications to several areas in science and engineering, considerable attention has been devoted to the study of flow and heat transfer within a thin liquid film on an unsteady stretching sheet. These areas include extrusion processes, wire and fiber coating, polymer processing, food stuff processing, design of various heat exchangers, and chemical processing equipment. In particular, in melt-spinning processes, the extrudate from the die is generally drawn and simultaneously stretched into a filament or sheet, which is then solidified through rapid quenching or gradual cooling by direct contact with water or chilled metal rolls. In fact, stretching imparts a unidirectional orientation to the extrudate and, as a consequence, the quality of the final product depends considerably on the flow and heat transfer mechanism. Therefore, the analysis of momentum and thermal transport within a thin liquid film on a continuously stretching surface is important for gaining some fundamental understanding of such processes. Motivated by the process of polymer extrusion, in which the extrudate emerges from a narrow slit, Crane [2] examined the Newtonian fluid flow induced by the stretching of an elastic flat sheet. Subsequently, several extensions related to
256
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Application of the Keller-box Method to Fluid Flow and ...
Crane’s [2] flow problem were made for different physical situations (see [31], [32], [115], [126]). In these studies [2], [31], [32], [115], [126], the boundary layer equation is considered and the boundary conditions are prescribed at the sheet and on the fluid at infinity. Imposition of a similarity transformation reduced the system to a set of ordinary differential equations (ODEs), which was then solved analytically or numerically. All the above mentioned studies deal with flow and/or heat transfer from a stretching sheet in a fluid medium extending to infinity. However, in real physical situations involving coating processes, one needs to consider the fluid adhering to the stretching sheet as a finite liquid film. Wang [116] was the first to consider such a flow problem with a finite liquid film of a Newtonian fluid over an unsteady stretching sheet. Later, Usha and Sridharan [117] considered a similar problem of axi-symmetric flow in a liquid film. Dandapat et al. [118] investigated the effects of variable fluid properties and thermocapillarity on the flow and heat transfer in a liquid film on a horizontal stretching sheet. Further, Liu and Andersson [119] explored the work of [116] to study the thermal characteristics of liquid film on an unsteady stretching surface. Abel et al. [120] studied the heat transfer problem for a thin liquid film in the presence of an external magnetic field with viscous dissipation. Nadeem and Awais [121] analyzed the effect of a thin film flow over an unsteady shrinking sheet with variable viscosity. Recently, Aziz et al. [122] addressed the influence of internal heat generation/absorption on the flow and heat transfer in a thin film on an unsteady stretching sheet. It should be noted that the flow and heat transfer characteristics are affected not only by the velocity and the thermal boundary conditions but also by the physical properties of the liquid-film. Furthermore, the study of non-Newtonian fluid flow on an unsteady stretching surface is important. Although the fluid employed in material processing or protective castings are generally non-Newtonian (example, most of the paints), there has been a very little work done on the flow and heat transfer of a non-Newtonian liquid film over a stretching surface. Among the most popular rheological models of non-Newtonian fluids is the power-law or Ostwald-de Waele model. This model deals with a simple nonlinear equation of state for inelastic fluids; this includes linear Newton-fluids as a special case. The power-law model provides an adequate representation of many non-Newtonian fluids for a range of shear rates. For instance, Andersson et al. [123] carried out a numerical study for the hydro-dynamical problem of a power-law fluid flow with in a liquid film over a stretching sheet. Here, the thermophysical properties of the ambient fluid are assumed to be constant. However, it is well known that these properties may change with temperature, especially the thermal conductivity. Available literature [55], [124], [125] on variable thermal conductivity shows that this type of work has not been carried out for non-Newtonian fluid obeying the Ostwald-de Waele power-law model. The purpose of the present section is to explore the effects of thermophysical property, namely, the variable thermal conductivity and viscous dissipa-
5.6
Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid ...
257
tion on the heat transfer of an incompressible power-law liquid thin film on an unsteady porous stretching surface (see for details K. Vajravelu, K. V. Prasad and Chiu-On Ng, Communications in Nonlinear Sci. Numerical Simulation 17 , 2012, 4163–4173). In non-Newtonian liquid thin film flow, the effects of variable thermal conductivity, power law index, and viscous dissipation play a significant role in the heat transfer process. Here, the momentum and energy equations are highly nonlinear. Hence, a similarity transformation is used to transform the nonlinear partial differential equations into nonlinear ordinary differential equations. Due to its complexity and nonlinearity, the proposed problem is solved numerically by the Keller-box method. The obtained numerical results are used to analyze the flow and heat transfer characteristics of the power-law liquid film that would find applications in manufacturing industries.
5.6.2
Mathematical formulation
Consider an unsteady, two-dimensional, viscous, laminar flow and heat transfer of an incompressible non-Newtonian thin fluid film obeying a power-law model. The flow is due to the stretching of a porous elastic sheet parallel to the x-axis at y = 0. Two equal and opposite forces are applied along the x-axis, keeping the origin fixed. A schematic representation of the physical model is presented in Figure 5.38. The continuous stretching sheet is assumed to have a prescribed velocity Us (x, t) and temperature Ts (x, t). Further, a thin liquid film of uniform thickness h(t) rests on the horizontal sheet. With the above assumptions, the equations of conservation of mass, momentum, and energy can be written as ∂u ∂v + = 0, (5.84) ∂x ∂y ∂u ∂u ∂u 1 ∂τxy +u +v = , (5.85) ∂t ∂x ∂y ρ ∂y n+1 ∂T ∂T ∂ ∂T ∂u ∂T +u +v = κ(T ) +K , (5.86) ρcp ∂t ∂x ∂y ∂y ∂y ∂y where u and v are the velocity components along the x and y directions respectively, ρ is the density, and τxy is the shear stress. Here, we assume the shear stress as τxy = −K(−∂u/∂y)n , (5.87)
where K is the consistency coefficient and n is the flow behavior index, namely, the power-law index. The fluid is Newtonian for n = 1 with K = μ (the absolute viscosity). As n deviates from unity, the fluid becomes nonNewtonian: for example, n < 1 and n > 1 correspond to shear thinning (pseudoplastic) and shear thickening (dilatants) fluids, respectively. Further, Cp is the specific heat at constant pressure, T is the temperature, and κ(T ) is
258
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Fig. 5.38 Schematic of a liquid film on an unsteady stretching sheet
the temperature-dependent variable thermal conductivity. For liquid metals, the thermal conductivity varies linearly with temperature in the range 0◦ F to 400◦ F (see for details Savvas et al. [124]). In the present study, the thermal conductivity is assumed to vary linearly with temperature (Chiam [125]) as ε κ(T ) = κ0 1 + (T − T0 ) . (5.88) ΔT Here, ΔT = Ts − T0 , Ts is the temperature of the stretching sheet, ε is a small parameter known as the variable thermal conductivity parameter, and κ0 is the thermal conductivity of the fluid. The last term in Eq. (5.86) is due to the viscous dissipation. Substituting (5.87)–(5.88) into Eqs. (5.85)–(5.86), we obtain n ∂u ∂u ∂u K ∂ ∂u +u +v =− − , (5.89) ∂t ∂x ∂y ρ ∂y ∂y
∂T ∂T ∂T +u +v ∂t ∂x ∂y n+1 T − T0 ∂T ∂ ∂u . = κ0 1 + ε +K ∂y ΔT ∂y ∂y ρcp
(5.90)
In the derivation of the above governing equations, the conventional boundary layer approximation has been invoked. This is justified by the assumption that the film thickness h is much smaller than the characteristic length L (in the direction along the sheet). The mass conservation equation (5.84) then implies that the ratio v/u between the two velocity components is of order h/L2 . Also, streamwise diffusion of momentum and thermal energy is of order h/L2 , smaller than the corresponding diffusion perpendicular to the sheet.
5.6
Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid ...
259
For this reason the streamwise diffusion terms are neglected in Eqs. (5.89) and (5.90). Assuming that the interface of the planar liquid film is smooth and free of surface waves and the viscous shear stress and the heat flux vanish at the adiabatic free surface, the boundary conditions become u = Us , v = vs , T = Ts at y = 0, ∂u ∂T dh = = 0, v = at y → h(t), ∂y ∂y dt
(5.91) (5.92)
where Us and Ts are the surface velocity and temperature of the stretching sheet respectively, and vs is the injection parameter. Here h(t) is the free surface elevation of the liquid film. That is, the film thickness. In this section, the flow is caused by the linear stretching of the elastic sheet at y = 0 with a velocity of the form Us = bx/1 − αt,
(5.93)
where b and α are both positive constants with dimension reciprocal of time t. Here b is the initial stretching rate, whereas b/(1 − αt) is the effective stretching. In the context of polymer extrusion, the material properties, in particular the elasticity of the extruded sheet, may vary with time even though the sheet is being pulled by a constant force. The dimensionless ratio S ∼ = α/b, is the only parameter in Wang’s [116] analysis; in the limiting case as S → 0, Wang’s case reduces to the steady-state problem of Crane [2]. With unsteady stretching (α �= 0), however, α−1 becomes the representative time scale of the resulting unsteady boundary layer problem. The adopted formulation of the sheet velocity Us (x, t) in Eq. (5.93) is valid for t < α−1 only, unless α = 0. Further, it should be noted that the end effects and gravity are negligible, and the surface tension is sufficiently large such that the film surface remains smooth and stable throughout the motion. The surface temperature Ts of the sheet varies with the distance x from the slot and time t: Ts = T0 − Tref
b2−n x2 2(K/ρ)
(1 − αt)n−5/2 ,
(5.94)
where T0 is the fixed temperature at the slit, and Tref is the reference temperature, which can be taken as Tref = T0 in the present study. The constant of proportionality α is assumed to be positive with dimension time−1 . Equation (5.94) represents a situation in which the sheet temperature decreases from T0 and is proportion to x2 , and the amount of temperature reduction along the sheet increases with time. It should be noted that the expressions given by Eqs. (5.93) and (5.94) are valid for time t < (1/α). The assumptions about Us and Ts in Eqs. (5.93) and (5.94) respectively allow us to develop a similarity transformation which converts the partial differential equations (PDEs) into a set of ODEs. We introduce the following dimensionless variable
260
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Application of the Keller-box Method to Fluid Flow and ...
f (ξ) and θ(ξ) as well as the similarity variable ξ thus: ψ=
b1−2n (K/ρ)
1 n+1
T = T0 − Tref
ξ=
b2−n x2 (K/ρ)
2n
x n+1 (1 − αt)
b2−n x2 2(K/ρ)
1−2n n+1
β −1 f (ξ),
(5.95)
5
(5.96)
(1 − αt)n− 2 θ(ξ),
x(1−n)/(1+n) (1 − αt)(n−2)/(n+1) β −1 y.
(5.97)
In Eq. (5.95) the stream function ψ(x, y, t) is defined by u = ∂ψ/∂y and v = −∂ψ/∂x, such that the continuity equation (5.84) is satisfied automatically, and β is a constant denoting the dimensionless film thickness. In terms of these new variables, the momentum and the energy equations together with the boundary conditions become 2n �� n−1 ��� n+1 �� �2 n(−f ) ff − f f +β n+1 2 − n �� ξf −Sβ n+1 f � + = 0, (5.98) n+1 2n � f f 1 ((1 + εθ)θ� )� + β 2 n + 1 Pr 2θ θ� 5 2−n 2 −Sβ −n θ− ξθ� + Ecβ 1−n (f �� )n+1 = 0, (5.99) 2 n+1 and f (0) = fw ,
f � (0) = 1,
f �� (1) = 0,
θ(0) = 1,
f (1) =
θ� (1) = 0.
2−n 2n
S,
(5.100) (5.101)
Here a prime denotes the differentiation with respect to ξ, S = α/b is a dimenis the generalized sionless measure of the unsteadiness, Pr = (Us /α)Re−2/n+1 x Prandtl number, fw = −(vs /Us )(2n/n + 1)Re1/n+1 is the suction/injection x parameter (namely, fw > 0 corresponds to suction whereas fw < 0 corresponds to injection), and Ec = Us2 /Cp (Ts − T0 ) is the Eckert number. The parameter β is an unknown constant which must be determined as a part of the boundary value problem. Although the dimensionless film thickness is a constant for fixed values of S and n, the actual film thickness depends on time t and the streamwise location x. From Eq. (5.97) we find that the film thickness h(x, t) can be expressed as (K/ρ) x(n−1)/(n+1) (1 − αt)(2−n)/(1+n) . (5.102) h(x, t) = β b2−n
5.6
Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid ...
261
In the Newtonian case (n = 1), h becomes a function of time only; whereas for non-Newtonian films, the thickness decreases with x for pseudoplastics (n < 1), while the film thickness decreases in the streamwise direction for dilatant fluids (n > 1). It is worth mentioning here that the momentum boundary layer problem defined by the ODE (5.98) subject to the relevant boundary conditions (5.100) is decoupled from the thermal boundary layer problem, while the temperature field is on the other hand coupled with the velocity field. For practical purposes, the physical quantities of interest include the velocity components u and v, the local skin friction coefficient Cf x , and the local Nusselt number N ux . These quantities can be written as u = Us f � , 2n 1−n � −1/(n+1) v = −Us Rex f+ ξf , n+1 1+n (−f �� (0))n , Cf x = 2Re−1/(n+1) x N ux =
1 (1 − αt)−1/2 Re(n+2)/(n+1) θ� (0), x 2
where Rex = ρUs2−n xn /K is the local Reynolds number.
5.6.3
Numerical procedure
The system of Eqs. (5.98) and (5.99) are highly nonlinear ordinary differential equations of third-order and second-order, respectively. Exact analytical solutions are not possible for the complete set of Eqs. (5.98) and (5.99). Hence, we use the Keller-box method [18]–[54]. First, we write the differential equations and the boundary conditions in terms of first order system, which is then converted to a set of finite difference equations using central differences. Then the nonlinear algebraic equations are linearized by Newton’s method and the resulting linear system of equations is solved by block tridiagonal elimination technique. For the sake of brevity, the details of the solution process are not presented here. For numerical calculations, a uniform step size of Δξ = 0.005 is found to be satisfactory and the solutions are obtained with an error tolerance of 10−6 in all the cases. To demonstrate the accuracy of the present method, results for the dimensionless film thickness and the skin friction are compared with the available results in the literature for a special case: that is, for a Newtonian fluid (n = 1), and it was found from Table 5.10 that the present results agree very well with those of Aziz et al. [122] and Wang [116].
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Table 5.10 Variation of the dimensionless film thickness and the skin friction with respect to the unsteady parameter for a Newtonian fluid (n = 1) when Pr = 1.0 S 0.8 1.0 1.2 1.4 1.6 1.8
5.6.4
β
Present work f (0)
2.149956 1.540905 1.124422 0.816898 0.570868 0.348569
−2.677546 −1.967298 −1.435752 −1.003991 −0.631578 −0.296197
Aziz et al. [122] β f (0) 2.151994 1.543616 1.127780 0.821032 0.576173 0.356389
−2.680943 −1.972384 −1.442625 −1.012784 −0.642397 −0.309137
β
Wang [116] f (0)
2.15199 1.54362 1.127780 0.821032 0.567173 0.356389
−2.68094 −1.97238 −1.442631 −1.012784 −0.642397 −0.309137
Results and discussion
In order to analyze the effects of the pertinent parameters, namely, the powerlaw index n, the dimensionless film thickness β, the unsteady parameter S, the injection parameter fw , the variable thermal conductivity parameter ε, the modified Prandtl number Pr, and the Eckert number Ec on the flow and heat transfer characteristics, the numerical solutions are obtained. Also, in order to get a clear insight into the physical problem, the velocity and the temperature fields are presented graphically in Figures 5.39–5.40. Values of the skin friction, the dimensionless film thickness, and the wall-temperature gradient for different values of the physical parameters are recorded in Tables 5.11 and 5.12. For this hydrodynamic problem, there exists a critical value of S, above which no solution could be obtained: Wang [116] noticed the critical value of S = 2 for Newtonian fluid. It may be noted here that (for positive values of S), S → 0 stands for the case of an infinitely thick fluid layer (i.e., β → ∞), whereas the limiting case of S → 2 represents a liquid film of infinitesimal thickness (i.e., β → 0). In the case of non-Newtonian fluids, the present calculations show that the critical value of S = 1.35 stands for shear thinning fluids and the critical value of S = 3.03 for that of shear thickening fluids when β → 0. However, it is difficult to perform these calculations for the limiting case of β → ∞. The transverse velocity profiles f and the horizontal velocity profiles f for blowing and suction cases are shown graphically in Figures 5.39–5.40 with different values of S, fw and n. The general trend is that f decreases monotonically, whereas f increases monotonically as the distance increases from the stretching sheet. The effect of increasing values of S is to increase f and f and thereby reduce the horizontal boundary layer thickness. This phenomenon is true even for shear thinning (n = 0.8), Newtonian (n = 1) and shear thickening (n = 1.2) fluids. We further notice from these figures that a moderate deviation from Newtonian rheology (n = 1) has a significant influence on the horizontal velocity component f across the fluid film. For a given value of S, the pseudoplastic (shear thinning fluids) film is thinner
5.6
Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid ...
263
Fig. 5.39a Transverse velocity profiles for different values of n and S with fw = −0.1
Fig. 5.39b Transverse velocity profiles for different values of n and S with fw = 0.1
264
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Fig. 5.40a Horizontal velocity profiles for different values of n and S with fw = −0.1
Fig. 5.40b Horizontal velocity profiles for different values of n and S with fw = 0.1
5.6
Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid ...
265
266
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5.6
Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid ...
267
and exhibits a greater surface velocity than a Newtonian film, while quite reverse behavior is true for shear thickening (dilatant) fluids. In shear thinning fluids, viscosity is reduced with increasing shear rates; whereas for dilatants, viscosity increases with shear rate and becomes more viscous and will thicken with an increasing rate of shear. It is therefore not surprising to observe that the pseudoplastics are more likely to flow nearly as an inviscid layer on top of the stretching sheet than as in the case of shear thickening or dilatants fluids. These results are in good agreement with the physical situations. Comparison of Figure 5.40a with Figure 5.40b reveals that suction (fw > 0) reduces the horizontal velocity boundary layer thickness whereas blowing (fw < 0) has quite the opposite effect on the velocity boundary layer. The effects of the power law index parameter on the temperature profiles for fw < 0 and fw > 0 are shown graphically in Figures 5.41a–5.41b. It is observed that the temperature distribution is unity at the wall. With changes in the physical parameters, it decreases as the distance increases from the sheet. Further, the effect of increasing S with different values of n (namely, shear thinning (n = 0.8), Newtonian (n = 1), and shear thickening (n = 1.2) fluids) is to reduce the temperature, and hence the thermal boundary layer thickness. Comparison of Figure 5.41a with Figure 5.41b reveals that the effect of the injection parameter is to reduce the thermal boundary layer thickness. Figures 5.42a and 5.42b exhibit the temperature distribution θ(ξ) for different values of Ec in blowing and suction cases respectively. From these
Fig. 5.41a Temperature profiles for different values of n and S with Pr = 1.0, Ec = 0.0, ε = 0.0, fw = −0.1
268
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Fig. 5.41b Temperature profiles for different values of n and S with Pr = 1.0, Ec = 0.0, ε = 0.0, fw = 0.1
figures we see that the effect of increasing Ec is to increase the temperature distribution θ(ξ). This is in conformity with the fact that energy is stored in the fluid region as a consequence of dissipation due to viscosity and elastic deformation. The effects of ε on the temperature profile in the boundary layer for both fw < 0 and fw > 0 are depicted in Figures 5.43a and 5.43b, respectively. From these figures, we observe that the temperature distribution is lower throughout the boundary layer for zero values of ε as compared with nonzero values of ε. This is due to the fact that the presence of temperaturedependent thermal conductivity results in reducing the magnitude of the transverse velocity by a quantity ∂K(T )/∂y, and this can be seen from the energy equation. This behavior holds for all types of fluids considered, namely, pseudoplastic, Newtonian, and dilatant fluids. The variations of temperature profile θ(ξ) for various values of the modified Prandtl number Pr are shown in Figures 5.44a and 5.44b for both fw < 0 and fw > 0 respectively. Both figures demonstrate that an increase in Pr results in a monotonic decrease in the temperature distribution, and it tends to zero as the distance increases from the sheet. That is, the thermal boundary layer thickness decreases for higher values of the Prandtl number. This holds good for all values of n and fw .
5.6
Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid ...
269
Fig. 5.42a Temperature profiles for different values of n and Ec with Pr = 1.0, S = 0.5, ε = 0.0, fw = −0.1
Fig. 5.42b Temperature profiles for different values of n and Ec with Pr = 1.0, S = 0.5, ε = 0.0, fw = 0.1
270
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Fig. 5.43a Temperature profiles for different values of n and ε with Pr = 1.0, Ec = 0.0, S = 0.8, fw = −0.1
Fig. 5.43b Temperature profiles for different values of n and ε with Pr = 1.0, Ec = 0.0, S = 0.8, fw = 0.1
5.6
Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid ...
271
Fig. 5.44a Temperature profiles for different values of n and Pr with ε = 0.1, Ec = 0.0, S = 0.8, fw = −0.1
Fig. 5.44b Temperature profiles for different values of n and Pr with ε = 0.1, Ec = 0.0, S = 0.8, fw = 0.1
272
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The values of f (0), θ (0), S and fw are recorded in Table 5.11. It is interesting to note that β, |f (0)| and |θ (0)| decrease gradually with increasing S. This is true for all values of n. Further, the effect of increasing n and fw is to enhance β, |f (0)| and |θ (0)|. From Table 5.12, we see that the effect of Ec and ε is to decrease the magnitude of the wall temperature gradient, whereas the effect of Pr is to enhance it. This is true for all values of n and fw .
5.6.5
Conclusions
The purpose of the present work is to obtain numerical solutions to the problem of flow and heat transfer in a power-law liquid film on an unsteady porous stretching sheet in the presence of viscous dissipation and temperature-dependent thermal conductivity. Results for the velocity and the temperature distributions across the liquid film, the free surface velocity f (1), the temperature θ(1), the wall-shear stress, and the wall-temperature gradient are presented for different values of the governing parameters. The results obtained might be useful for the material processing industries. We summarize some of the interesting results below: 1) In comparison with the Newtonian fluid, the free surface temperature is enhanced for shear thinning fluid, while it is decreased for shear thickening fluid. Also free surface temperature approaches zero for higher values of the Prandtl number. 2) The effect of suction is to reduce the thermal boundary layer thickness as compared to blowing. This holds for all values of the power-law index, the variable thermal conductivity, and the Eckert number. 3) The effect of viscous dissipation is found to increase the dimensionless free surface temperature θ(1). These observations are true for all values of the power-law index.
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Application of the Keller-box Method to Fluid Flow and ...
S. Nadeem, M. Awais, Thin film flow of an unsteady shrinking sheet, through medium with variable viscosity, Phys. Lett. A. 372 (2008) 4695. R.C. Aziz, I. Hasim, A.K. Almari, Thin film flow and heat transfer on an unsteady stretching sheet with internal heating, Meccanica 46 (2011) 349. H.I. Andersson, J.B. Aaresh, N. Braud, B.S. Dandapat, Flow of a power law fluid on an unsteady stretching surface, J. Non-Newtonian Fluid Mech. 62 (1996) 1. T.A. Savvas, N.C. Markatos, C.D. Papaspyrides, On the flow of non-Newtonian polymer solutions, Appl. Math. Modelling 18 (1994) 14. T.C. Chiam, Heat transfer in a fluid with variable thermal conductivity over a linearly stretching sheet, Acta Mech. 129 (1998) 63. M.E. Ali, On the thermal boundary layer on a power law stretched surface with suction or injection, Int. J. Heat and Fluid Flow 16 (1995) 280.
Chapter 6 Application of the Keller-box Method to More Advanced Problems
In this chapter, we discuss the application of the Keller-box method to solve more advanced coupled nonlinear boundary value problems of one or more independent variables. In Sections 6.1 and 6.3, the effects of variable fluid properties on the boundary layer flow and heat transfer of a fluid over a flat sheet are studied; whereas in Section 6.2, we study dusty fluid flow over a stretching sheet. The hydromagnetic mixed convection boundary layer flow of an electrically conducting fluid over a non-isothermal wedge and sphere under different physical situations are analyzed in Sections 6.4 and 6.5. In Section 6.6, we study the flow and heat transfer of an electrically conducting viscoelastic fluid past a semi-infinite plate.
6.1 Heat transfer phenomena in a moving nanofluid over a horizontal surface
6.1.1
Introduction
Enhancement of heat transfer in nanofluids has generated much interest over the years due to its wide range of applications to nano technological industries, and to public endeavors in biological and physical sciences, electronics cooling, transportation, the environment, and national security. The word nanofluid refers to a liquid containing a suspension of submicronic solid particles. The commonly used fluids, such as oil, water, and ethylene glycol mixtures are poor in transmitting heat since the thermal conductivity of these fluids is low. Therefore numerous methods were proposed to improve the thermal conductivity of these fluids by suspending nano/micro sized par-
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ticles of copper, silver, titanium oxide, and aluminum oxide in liquids. The effective thermal conductivity of nanofluids is expected to enhance the rate of heat transfer compared with the convectional liquids. This phenomenon suggests the possibility for the use of nanofluids in advanced nuclear systems (Choi [1], [2], Masuda et al. [3], and Buongiorno and Hu [4]). Choi [2] was the first among the others to initiate a study on “nanoscale particle added fluids” called nanofluids and observed that the addition of a small amount (less than 1% by volume) of nanoparticles to conventional liquids increased the thermal conductivity of the fluid up to approximately two times. The characteristic feature of nanofluid is the thermal conductivity enhancement, a phenomenon observed by Masuda et al. [3]. Numerous methods were proposed to improve the thermal conductivity of these fluids by suspending nano/micro size particle materials in liquids (Kaka¸c and Pramuanjaroenkij [5]). Buongiorno [6] and several others noticed that convective heat transfer enhancement could be due to the dispersion of the suspended nanoparticles and argued that this effect is too small to explain the observed enhancement. They also concluded that turbulence is not affected by the presence of nanoparticles, so this cannot explain the observed enhancement. In another paper Buongiorno [6] has pointed out that nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and the relative velocity (he called it the slip velocity). Numerous models and methods were proposed by many authors to study the convective flows of nanofluids: we mention here the papers by Kuznetsov and Nield [7], Aminossadati and Ghasemi [8], Khanafer et al. [9], Ghasemi and Aminossadati [10], Khan and Pop [11], Bachok et al. [12], and Vajravelu et al. [13]. We also mention the valuable recently published book by Das et al. [14]. In all the above mentioned papers, the thermophysical properties of the ambient fluid were assumed to be constant. However, from the studies of Herwig and Wickern [15], Lai and Kulacki [16], Takhar et al. [17], Pop et al. [18], Hassanien [19], Abel et al. [20], Seddeek [21], Ali [22], and Prasad et al. [23], we notice that these physical properties may change with temperature, especially the fluid viscosity and the thermal conductivity. For lubricating fluids, heat generated by internal friction and the corresponding rise in the temperature affects the physical properties of the fluid, and the properties of the fluid are no longer assumed to be constant. The increase in temperature leads to an increase in the transport phenomena. Therefore to predict the flow and heat transfer rates, it is necessary to take into account the variable fluid properties. In view of this, we extend the work of Kuznetsov and Nield [7] by considering the effects of temperature dependent variable fluid properties. That is, in this section, we envisage the effects of variable viscosity and variable thermal conductivity on the flow and heat transfer characteristics of a nanofluid over a moving horizontal surface in the presence of viscous dissipation (for details see K.Vajravelu, K.V. Prasad, Meccanica 28 , 2012, 391–400). The study may have a variety of applications to nano-industry because materials with sizes of nanometers possess unique physical and chem-
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Heat transfer phenomena in a moving nanofluid over a horizontal surface
281
ical properties. Also, it is worth mentioning that the flow and heat transfer of a viscous fluid over a moving surface has many important applications in polymer industry. Highly nonlinear, coupled partial differential equations governing the momentum, energy, and mass transfer of the model problem are reduced to a coupled system of nonlinear ordinary differential equations by using a suitable similarity transformation. These coupled nonlinear ordinary differential equations are solved numerically by the Keller-box method for different values of the parameters. The effects of various parameters on the velocity, temperature, and nanoparticle volume fraction profiles are presented. Also, the skin friction and the rate of heat transfer are presented in graphical and tabular forms. It is our belief that the results presented here will provide useful information for applications and complement the previous studies.
6.1.2
Mathematical formulation
Consider a steady, two-dimensional, viscous and incompressible boundary layer flow of a nanofluid past a continuous moving sheet in a uniform free stream as shown in Figure 6.1. The thermophysical nanofluid properties are assumed to be isotropic and constant, except for the fluid viscosity and the fluid thermal conductivity which are assumed to vary with temperature in the following forms: 1 1 (1 + γ(T − T∞ )), = μ μ∞ T − T∞ k(T ) = k∞ 1 + ε , ΔT
(6.1) (6.2)
Fig. 6.1 Physical model and coordinate system
where μ∞ and k∞ are the ambient fluid viscosity and thermal conductivity
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respectively. ε is a small parameter known as the variable thermal conductivity parameter, T is the temperature of the fluid, and ΔT = (Tw − T∞ ). Equation (6.1) can be written as 1 = a(T − Tr ) μ
(6.3)
where
1 γ (6.4) and Tr = T∞ − . μ∞ γ Both a and Tr are constants and their values depend on the reference state and the thermal property of the fluid, i.e. γ (a constant); in general, a > 0 for liquids and a < 0 for gases. Consider the uniform velocity of the free stream flow U and temperature T∞ through a nanofluid bounded by a semi-infinite flat plate parallel to the flow. Also θr is a constant which is defined by a=
θr =
1 T r − T∞ =− . ΔT γΔT
(6.5)
It is worth mentioning here that for γ → 0, θr → ∞. It is also important to note that θr is negative for liquids and positive for gases. This is due to the fact that viscosity of a liquid usually decreases with increasing temperature while it increases for gases. The reference temperatures selected here for the correlations are very useful for most applications [24]. It is assumed that the velocity of the uniform free stream is U and that of the flat plate is Uw = λU , where λ is the plate velocity parameter. The flow takes place at y 0, where the positive y-coordinate is measured normal to the plate in the upward direction, towards the fluid. It is also assumed that at the moving surface, the temperature T and the nanoparticle volume fraction C take constant values Tw and Cw , respectively, while the values of T and C in the ambient fluid are denoted by T∞ and C∞ respectively. By making use of the usual boundary layer approximations, the governing equations for the nanofluid flow over a semi-infinite flat plate with variable thermophysical properties, in usual notation, can be written as ∂u ∂v + = 0, ∂x ∂y ∂u 1 ∂ ∂u ∂u +v = μ , u ∂x ∂y ρ∞ ∂y ∂y ∂T ∂T +v ∂x ∂y 2 ∂ ∂C ∂T ∂T DT ∂T = α + τ DB + ∂y ∂y ∂y ∂y T∞ ∂y 2 ∂u μ , + ρ∞ Cp ∂y
(6.6) (6.7)
u
(6.8)
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Heat transfer phenomena in a moving nanofluid over a horizontal surface
u
∂C DT ∂ 2 T ∂ 2C ∂C , +v = DB 2 + ∂x ∂y ∂y T∞ ∂y 2
283
(6.9)
where u and v are the velocities components along the x and y axes respectively, ρ∞ is the fluid density, Cp is the specific heat at constant pressure, α = k(T )/(ρ∞ Cp )f is the variable thermal diffusivity of the fluid, and τ = (ρ∞ Cp )p /(ρ∞ Cp )f is the ratio of the heat capacity of the nanoparticle material to the effective heat capacity of the fluid. Further, the coefficients that appear in Eqs. (6.8) and (6.9) are the Brownian diffusion coefficient DB and the thermophoretic diffusion coefficient DT . The last term in Eq. (6.8) represents the viscous dissipation due to frictional heating. The appropriate boundary condition on the velocity and temperature fields and the nanoparticle volume fraction are u = Uw = λU, u → U,
v = 0,
T → T∞ ,
T = Tw ,
C = Cw at y = 0,
C → C∞ as y → ∞,
(6.10)
where U is the uniform velocity of the free stream. It is worth mentioning here that the moving plate velocity parameter λ > 0 corresponds to downstream movement of the plate.
6.1.3
Similarity equations
From the numerical solutions of the forced convection nanofluid flow and heat transfer, it is observed that the nanoparticle volume fraction, thermal, and momentum boundary layers exist along a horizontal impermeable surface whenever the wall temperature and the wall nanoparticle temperature differs from that of the surrounding fluid temperature. Using the boundary layer approximations and the variable fluid properties the governing equations (6.7)–(6.9) in terms of stream function ψ can be written as ∂ψ ∂ 2 ψ ∂ψ ∂ 2 ψ − ∂y ∂x∂y ∂x ∂y 2 1 ∂ 2 ψ ∂ψ ∂ 1 = ν∞ + , 1 + γ(T − T∞ ) ∂y 2 ∂y ∂y 1 + γ(T − T∞ )
(6.11)
∂ψ k∞ ∂ T − T∞ ∂T ∂ψ ∂T − + 1+ε ∂y ∂x ∂x ρ∞ cp ∂y ΔT ∂y 2 2 ∂C ∂T T − T∞ ∂ T ∂T k∞ DT + τ DB 1+ε = + ρ ∞ cp ΔT ∂y 2 ∂y ∂y T∞ ∂y 2 2 ∂ ψ μ , (6.12) + ρ∞ Cp ∂y 2
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DT ∂ 2 T ∂ψ ∂C ∂ 2C ∂ψ ∂C . − = DB 2 + ∂y ∂x ∂x ∂y ∂y T∞ ∂y 2
(6.13)
The stream function ψ given by
(u, v) =
∂ψ ∂ψ ,− ∂y ∂x
automatically satisfies the continuity equation (6.6). With a properly chosen similarity variable, the above equations can be transformed into ordinary differential equations: the suitable similarity transformations for the problem are 12 U 1 η= y, ψ = (2U ν∞ x) 2 f (η), 2ν∞x (6.14) C − C∞ T − T∞ , φ(η) = . θ(η) = T w − T∞ Cw − C∞ In terms of these new variables, the velocity components can be written as u = U f (η), �
v=−
U ν∞ 2x
12
(f (η) − ηf � (η)),
(6.15)
and the governing equations (6.11)–(6.13) in terms of the new variables are −1 � θ f �� 1 − + f f �� = 0, (6.16) θr −1 θ � � � � � ((1 + εθ)θ ) + Prθ (f + N bφ + N tθ ) + EcPr 1 − f ��2 = 0, (6.17) θr N t �� θ = 0, φ�� + Scf φ� + (6.18) Nb
where a prime denotes the differentiation with respect to η, and the six parameters are defined by 1 τ DB (Cw − C∞ ) v∞ , Nb = , , Pr = γ(Tw − T∞ ) α∞ ν∞ v∞ τ DT (Tw − T∞ ) U2 , Sc = Nt = , Ec = . T∞ ν∞ Cp (Tw − T∞ ) DB θr =
(6.19)
Here θr , Pr, N b, N t, Ec, and Sc denote the fluid viscosity parameter, the Prandtl number, the Brownian motion parameter, the thermophoresis parameter, the Eckert number, and the Schmidt number, respectively. In view of the above transformations, the boundary conditions (6.10) can be written as f (η) = 0, f � (η) = λ, θ(η) = 1, φ(η) = 1 at η = 0, (6.20a)
6.1
Heat transfer phenomena in a moving nanofluid over a horizontal surface
f � (η) → 1,
θ(η) → 0,
φ(η) → 0 as η → ∞.
285
(6.20b)
We notice that in the absence of thermophysical properties, namely, the fluid viscosity and variable thermal conductivity parameter, and viscous dissipation effects, Equations (6.16)–(6.18) reduce to those of Bachok et al. [12]. Further, when the Brownian motion parameter and thermophoresis parameter are zero, the problem reduces to the classical problem of Weidman et al. [25] for an impermeable moving surface in a Newtonian fluid. From the engineering point of view, the important characteristics of the flow are the skin friction coefficient Cf , the local Nusselt number N ux , and the local Sherwood number Shx which are defined as Cf =
τw , ρ∞ U 2
N ux =
xqw , kΔT
Shx =
xqm , DB (Cw − C∞ )
where τw , qw , qm are the shear stress, heat flux and mass flux at the surface. Using the variables in (6.14), we obtain 1 1 = f �� (0), (2Rex ) 2 Cf 1 − θr 1
(Rex /2)− 2 N ux = −θ� (0), 1
(Rex /2)− 2 Shx = −φ� (0), 1
where Rex = U x/ν∞ is the local Reynolds number. Here, (Rex /2)− 2 N ux 1 and (Rex /2)− 2 Shx are referred to as the modified Nusselt number and the modified Sherwood number. They are denoted by N ur and Shr, and are represented by −θ� (0), and −φ� (0), respectively.
6.1.4
Numerical procedure
Equations (6.16), (6.17) and (6.18) are coupled nonlinear ordinary differential equations of third order in f , second order in θ, and second order in φ. Exact analytical solutions are hard to get for the complete set of physical parameters and therefore we use the Keller-box method [26]–[29]. First, we write the transformed differential equations and the boundary conditions in terms of the first order system, which is then converted to a set of finite difference equations using central differences. Then the nonlinear algebraic equations are linearized by Newton’s method and the resulting linear system of equations is then solved by block tridiagonal elimination technique. For the sake of brevity, the details of the numerical solution procedure are not presented here. It is also important to note that the computational time for each set of input parameters should be short. Because the physical domain in this problem is unbounded, whereas the computational domain has to be finite, we apply the far field boundary conditions for the similarity variable η at finite value denoted by ηmax . We ran our bulk computations with the value
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ηmax = 10, which is sufficient to achieve the far field boundary conditions asymptotically for all values of the parameters considered. For numerical calculations, a uniform step size of Δη = 0.01 is found to be satisfactory, and the solutions are obtained with an error tolerance of 10−6 in all the cases.
6.1.5
Results and discussion
Employing the above numerical method, the governing equations of the problem are solved for several sets of values of the pertinent parameters, namely, the plate velocity parameter λ, the variable viscosity parameter θr , the variable thermal conductivity parameter ε, the Brownian motion parameter N b, the thermophoresis parameter N t, the Prandtl number Pr, the Eckert number Ec, and the Schmidt number Sc. The numerical results thus obtained are presented for horizontal velocity, temperature, and nanoparticle volume fraction in Figures 6.2–6.4. Also the numerical results for the skin friction, modified Nusselt number, and the modified Sherwood number are presented in Tables 6.1 and 6.2. Since it is not possible to present the results here for all possible permutations and combinations of all the physical parameters, we focus our attention on the effects of the new parameters (related to the thermophysical transport properties) on the nanofluid flow and heat transfer. Horizontal velocity, temperature, and nanoparticle volume fraction profiles are, respectively, depicted in Figures 6.2a–6.2b, Figures 6.3a–6.3f and Figures 6.4a–6.4d. Horizontal velocity profiles f � (η) are shown graphically in Figures 6.2a– 6.2b for different values of λ and θr when all the other parameters are fixed. The general trend is that the velocity profiles show an exponential increase for λ < 1, i.e., velocity of the plate is less than the uniform velocity of the free stream and shows an exponential decay for λ > 1. That is, the velocity of the plate is greater than the uniform velocity of the free stream and both attain to unity as the distance increases from the boundary. Further velocity profiles f � (η) are lower for negative values of λ as compared to positive values of λ. Physically λ < 0 means the plate and the free stream velocity move in opposite directions, λ > 0 means the plate and the free stream velocity move in the same direction. In addition to this, the values of f �� (0) are positive for λ < 1, while they are negative for λ > 1. Physically, positive sign for f �� (0) implies the fluid exerts a drag force on the plate. The value f �� (0) = 0.469604 for a fixed plate as given in Table 6.1 is in good agreement with 0.4694 reported Apelblat [30], while Davies [31] reported it as 0.47. The zero skin friction when λ = 1 does not mean separation of the boundary layer from the solid surface, but it refers to the case when both the fluid and the plate move at the same velocity and thus no friction occurs between them. It is evident from these curves that all the profiles satisfy the far field boundary conditi-
6.1
Heat transfer phenomena in a moving nanofluid over a horizontal surface
287
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6.1
Heat transfer phenomena in a moving nanofluid over a horizontal surface
289
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Fig. 6.2a Horizontal velocity profiles for different values of λ with N t = N b = 0.5, Pr = Sc = 1.0, Ec = 0.1, ε = 0.1
Fig. 6.2b Horizontal velocity profiles for different values of λ and θr with N t = N b = 0.5, Pr = Sc = 1.0, Ec = 0.1, ε = 0.1
6.1
Heat transfer phenomena in a moving nanofluid over a horizontal surface
291
Fig. 6.3a Temperature profiles for different values of λ and θr with N t = N b = 0.5, Pr = Sc = 1.0, Ec = 0.1, ε = 0.1
Fig. 6.3b Temperature profiles for different values of N b and λ with N t = 0.5, Pr = Sc = 1.0, θr = −5.0, Ec = 0.1, ε = 0.1
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Fig. 6.3c Temperature profiles for different values of N t and λ with N b = 0.5, Pr = Sc = 1.0, θr = −5.0, Ec = 0.1, ε = 0.1
Fig. 6.3d Temperature profiles for different values of Pr and λ with N t = N b = 0.5, Sc = 1.0, θr = −5.0, Ec = 0.1, ε = 0.1
6.1
Heat transfer phenomena in a moving nanofluid over a horizontal surface
293
Fig. 6.3e Temperature profiles for different values of Ec and λ with N t = N b = 0.5, Sc = 1.0, θr = −5.0, Pr = 1.0, ε = 0.1
Fig. 6.3f Temperature profiles for different values of ε and λ with N t = N b = 0.5, Sc = 1.0, θr = −5.0, Pr = 1.0, Ec = 0.1
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Fig. 6.4a Concentration profiles for different values of λ and θr with N t = N b = 0.5, Sc = 1.0, Pr = 2.0, ε = Ec = 0.1
Fig. 6.4b Concentration profiles for different values of N b and λ with N t = 0.5, Sc = 1.0, Pr = 2.0, θr = −5.0, ε = Ec = 0.1
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Heat transfer phenomena in a moving nanofluid over a horizontal surface
295
Fig. 6.4c Concentration profiles for different values of N t and λ with N b = 0.5, Sc = 1.0, Pr = 1.0, θr = −5.0, ε = Ec = 0.1
Fig. 6.4d Concentration profiles for different values of Sc and λ with N t = N b = 0.5, Pr = 1.0, θr = −5.0, ε = Ec = 0.1
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ons asymptotically and thus support the numerical results obtained. The effect of the fluid viscosity parameter θr on the velocity profiles f � (η) is shown graphically in Figure 6.2b for different values of λ. The effect of increasing values of θr is to decrease the velocity profile for λ > 1 and hence reduce the momentum boundary layer thickness. Also, as θr → 0, the boundary layer thickness decreases and the velocity distribution asymptotically tends to unity. This is due to the fact that, for a given fluid (water), when δ is fixed, smaller θr implies higher temperature difference between the wall and the ambient fluid. The results presented here demonstrate clearly that θr , the indicator of the variation of fluid viscosity with temperature, has a substantial effect on the horizontal velocity f � (η) and hence on the skin friction. The reverse trend is true whenever the plate velocity parameter is less than the velocity of the free stream (λ < 1). Temperature profiles θ(η) with η are shown graphically in Figures 6.3a– 6.3f for different values of the pertinent parameters. The general trend is that the temperature distribution is unity at the wall and tends asymptotically to zero in the free stream region. Figure 6.3a shows the effect of the plate velocity parameter λ in the presence/absence of the fluid viscosity parameter θr on the temperature profile θ(η). The effect of increasing values of λ is to decrease the temperature distribution in the flow region. This is in conformity with the fact that increase of λ leads to decrease of the thermal boundary layer thickness. This is true even in the presence of θr . The effect of increasing the value of θr is to enhance or decelerate θr respectively, whenever the plate velocity is greater or less than the uniform velocity of the free stream flow. This is because of the fact that an increase in θr results in an increase in the thermal boundary layer thickness for λ > 1. The physical parameters of greatest interest here are the N b (the Brownian motion parameter) and N t (the thermophoresis parameter), both of which depend on the type of the nanoparticle present through the parameters DB and DT respectively, while N b depends on the nanoparticle concentration, namely Cw − C∞ . An increase in either parameter namely, the Brownian motion parameter or the thermophoresis parameter, leads to a decrease in the magnitude of the wall temperature gradient and hence an increase in the thermal boundary layer thickness; this is shown graphically in Figures 6.2b–6.2c respectively. The variation of temperature profile θ(η) for different values of Pr and the velocity plate parameter λ is displayed in Figure 6.3d. This figure demonstrates that the increase in the Prandtl number Pr results in a decrease in the thermal conductivity k∞ leading to a decrease in the temperature distribution θ(η). Hence the thermal boundary layer thickness decreases as the Prandtl number increases. For Pr > 1, the temperature nearer the plate increases for negative values of the plate velocity parameter and decreases as the distance increases from the boundary. Figure 6.3e shows changes in the temperature distribution θ(η) for different values of the Eckert number Ec, and velocity plate parameter λ. The effect of increasing values of Ec is to increase the temperature profile. This is in conformity with the fact that energy is stored
6.1
Heat transfer phenomena in a moving nanofluid over a horizontal surface
297
in the fluid region due to frictional heating as a consequence of dissipation due to viscosity, and hence temperature increases as Ec increases. The effects of the variable thermal conductivity parameter ε on the temperature profile θ(η) for different values of λ are shown graphically in Figure 6.3f. It is observed that the temperature distribution is lower throughout the boundary layer in the absence of ε and becomes higher when we increase the values of the variable thermal conductivity parameter ε. In either case the temperature distribution tends to zero as the space variable η increases from the boundary. This is due to the fact that the temperature dependent thermal conductivity reduces the magnitude of the transverse velocity by a quantity ∂k(T )/∂y which can be seen from (6.12). The dimensionless nanoparticle volume fraction profiles φ(η) are depicted in Figures 6.4a–6.4d for different values of the governing parameters. Figure 6.4a shows the dependence of nanoparticle volume fraction on θr and λ. The parameters, namely, the fluid viscosity and plate velocity parameters, have similar effects on the nanoparticle volume fraction as those seen in temperature profiles (see Figure 6.3a). The effect of the Brownian motion and thermophoresis parameter on the nanoparticle volume fraction profiles are depicted graphically in Figures 6.4a–6.4c. The effects of the Brownian motion parameter N b and the thermophoresis parameter N t, respectively, are to reduce and enhance the thickness of the nanoparticle volume fraction boundary layer. In the presence of the Brownian diffusion coefficient, the effect of Schmidt number Sc on the nanoparticle volume fraction profiles φ(η) is to decrease it. The values of the skin friction coefficient, the Nusselt number, and the Sherwood number for various values of the pertinent parameters are recorded in Tables 6.1 and 6.2. The skin friction coefficient is found to increase in the upstream movement of the plate. The effect of fluid viscosity parameter is to increase the skin friction coefficient for λ < 1 and decrease it for λ > 1. The effect of the variable thermal conductivity parameter, the Eckert number, the Brownian motion parameter, and the thermophoresis parameter is to enhance the temperature gradient at the wall. The effect of the Schmidt number and the Brownian motion parameter is to reduce the Sherwood number, whereas the effect of thermophoresis parameter is to enhance it.
6.1.6
Conclusion
In this section we have studied the effects of the temperature-dependent thermophysical properties on the boundary layer flow and heat transfer of a nanofluid past a moving semi-infinite horizontal flat plate in a uniform free stream. The effects of Brownian motion, thermophoresis and viscous dissipation due to frictional heating are considered simultaneously. The governing partial differential equations are transformed into ordinary differential equations by using an appropriate similarity transformation and the resulting boundary value problem is solved numerically by the Keller-box method. The
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numerical result for the skin friction is in good agreement with Apelblat [30] and Davies [31] for a fixed plate in the absence of thermophysical properties. Numerical results for the skin friction coefficient, the local Nusselt number, the local Sherwood number, the velocity and temperature fields, and the nanoparticle volume fraction profiles are presented in graphs for several sets of values of the governing parameters. The following conclusions are drawn from the computed numerical values: • The effect of increasing variable viscosity parameter is to decrease the velocity profile for λ > 1 and the reverse trend is true for λ < 1. • The effect of increasing values of the plate velocity parameter and the Prandtl number is to decrease the temperature distribution and hence the thermal boundary layer thickness. • The effect of the Brownian motion parameter and the thermophoresis parameter is to decrease the magnitude of the wall-temperature gradient.
6.2 Hydromagnetic fluid flow and heat transfer at a stretching sheet with fluid-particle suspension and variable fluid properties
6.2.1
Introduction
The study of two-dimensional boundary layer flow and heat transfer induced by continuous stretching surfaces attracted interest due to its various applications to engineering and industrial disciplines. These applications include extrusion process, wire and fiber coating, polymer processing, food-stuff processing, design of heat exchangers, and chemical processing equipment. The concept of continuous stretching will bring in a unidirectional orientation to extrude; consequently the quality of the final product depends considerably on the flow and heat transfer mechanism. To that end, the analysis of momentum and thermal transports within the fluid on a continuously stretching surface is important for gaining some fundamental understanding of such processes. Keeping these practical applications in view, Crane [32] initiated the study of steady two-dimensional boundary layer flow due to the stretching of an elastic sheet. Subsequently, several extensions related to Crane’s [32] flow problem were made for different physical situations (see Gupta and Gupta [33], Grubka and Bobba [35], Siddappa and Abel [36], Vleggaar [34], Chen [37], Dutta et al. [38], Ali [39], Cortell [40], and Liu [41]). In these studies the boundary layer approximation is considered and the boundary conditions are prescribed at the sheet and on the fluid at infinity.
6.2
Hydromagnetic fluid flow and heat transfer at a stretching sheet with ...
299
All the above investigators restrict their analyses to the flow induced by a linear stretching sheet in the absence of fluid-particle suspension. The analysis of two-phase flows in which solid spherical particles are distributed in a fluid are of interest in a wide range of technical problems such as flow through packed beds, sedimentation, environmental pollution, centrifugal separation of particles and blood rheology. The study of the boundary layer of fluidparticle suspension flow is important in determining the particle accumulation and impingement of the particle on the surface. In view of these applications, Chakrabarti [42] analyzed the boundary layer in a dusty gas. Datta and Mishra [43] investigated boundary layer flow of a dusty fluid over a semiinfinite flat plate. Further, research in this field has been carried out by Agranat [44], Kumar and Sharma [45], Vajravelu and Nayfeh [46], Asmolov and Manuilovich [47], Palani et al. [48], and Gireesha et al. [49]. Kumar and Sharma [45] studied the fluid-particle suspension flow over a stretching sheet by using the least square finite element method. Recently, Gireesha et al. [49] analyzed the flow and heat transfer of a dusty fluid over a non-isothermal stretching sheet in the presence of a non-uniform heat source/sink. In all the papers above, the thermophysical properties of the ambient fluid particle suspension were assumed to be constant. However, it is well known that (Herwig and Wickern [15], Lai and Kulacki [16], Takhar et al. [17], Pop et al. [18], Hassanien [19], Abel et al. [20], Seddeek [21], Ali [22], Prasad et al. [51]) these physical properties may change with temperature, especially the viscosity and the thermal conductivity. For lubricating fluids, heat generated by internal friction and the corresponding rise in the temperature affects the physical properties of the fluid, and the properties of the fluid are no longer assumed to be constant. The increase in temperature leads to increase in the transport phenomena by altering the physical properties across the thermal boundary layer, and so the heat transfer at the wall is also affected. Therefore to predict the flow and heat transfer rates, it is necessary to take into account the variable fluid properties. Motivated by these analyses, we extend the work of Vajravelu and Nayfeh [46] by considering the temperature-dependent variable fluid properties. Thus in the present section, we study the effects of variable viscosity and variable thermal conductivity on the hydromagnetic, fluid-particle suspension flow and heat transfer over a stretching sheet (for details see K. Vajravelu, K.V. Prasad and P.S. Datti, Transactions of ASME J. Fluid Engineering 135 , 2013, 011101, 9 pages). The coupled nonlinear partial differential equations governing the problem are reduced to a system of coupled nonlinear ordinary differential equations by applying a suitable similarity transformation. These nonlinear coupled differential equations are solved numerically by the Kellerbox method for different values of the pertinent parameters. The numerical results are presented through tables and graphs. Further, the salient features of the flow and heat transfer characteristics are discussed.
300
6.2.2
Chapter 6
Application of the Keller-box Method to More Advanced Problems
Mathematical formulation
Consider the steady flow of a viscous, incompressible and electrically conducting dusty fluid over a horizontal stretching sheet with a stretching linear velocity Uw (x) = bx, and prescribed surface temperature Tw (x) = A1 (x/l), where b (> 0) and A1 are constants. The thermophysical fluid properties are assumed to be isotropic and constant, except for the fluid viscosity and the fluid thermal conductivity which are assumed to vary as a function of temperature in the following forms: 1 1 = (1 + γ(T − T∞ )), μ μ∞ T − T∞ , K(T ) = K∞ 1 + ε ΔT
(6.21) (6.22)
where μ∞ and K∞ are the ambient fluid viscosity and thermal conductivity respectively. ε is a small parameter known as the variable thermal conductivity parameter, T is the temperature of the fluid and ΔT = Tw −T∞ . Equation (6.21) can be written as 1 (6.23) = a(T − Tr ), μ where a=
γ 1 and Tr = T∞ − . μ∞ γ
(6.24)
T r − T∞ 1 =− . ΔT γΔT
(6.25)
Both a and Tr are constants and their values depend on the reference state and the thermal property of the fluid, i.e., γ (a constant). In general, a > 0 for liquids and a < 0 for gases, when Tw > T∞ . The correlations between the viscosity and the temperature for air and water are given as follows: For air, 1/μ = −123.2 (T − 742.6), based on T∞ = 293 K (20◦C); and for water, 1/μ = −29.83 (T − 258.6), based on T∞ = 288 K (15◦C). Also, let θr be the constant which is defined by θr =
It is worth mentioning here that for γ → 0 i.e., μ = μ∞ (constant), θr → ∞. It is also important to note that θr is negative for liquids and positive for gases. This is due to the fact that viscosity of a liquid usually decreases with increasing temperature while it increases for gases. The reference temperatures selected here for the correlations are very useful for most applications (see for details Refs. [51]–[52]). The flow region is exposed under a uniform transverse magnetic field B = (0, B0 , 0), and the imposition of such a magnetic field, stabilizes the boundary layer flow. It is assumed that the flow is generated by stretching of an elastic sheet from a slit by imposing two equal and opposite forces in such a way that sheet is intact. It is also assumed that the magnetic Reynolds number is very small and the electric
6.2
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301
field due to polarization of charges is negligible. Under these conditions, the basic boundary layer equations for continuity, conservation of mass (with no pressure gradient), and energy can be written as ∂u ∂v + = 0, ∂x ∂y ρp ∂u ∂u ∂ ∂u ρ∞ u +v = μ − σB02 u − (u − up ), ∂x ∂y ∂y ∂y τ 1 ∂up ∂up + vp = (u − up ), up ∂x ∂y τ 1 ∂vp ∂vp + vp = (v − vp ), up ∂x ∂y τ ∂ ∂ (ρp up ) + (ρp vp ) = 0, ∂x ∂y ∂T ∂ ∂T ρp cs ∂T (Tp − T ), +v = α(T ) + u ∂x ∂y ∂y ∂y γT ρ∞ cp up
∂Tp ∂Tp 1 + vp = − (Tp − T ), ∂x ∂y γT
(6.26) (6.27) (6.28) (6.29) (6.30) (6.31) (6.32)
where (u, v) and (up , vp ) are the velocities components of the fluid and particle phases along the x and y axes respectively. Furthermore μ and ρ∞ are the coefficients of viscosity of the fluid and the density of the fluid. Here τ = 1/k is the relaxation time of particles, k = 6πμ∞ D is the Stokes constant, and D is the average radius of the dust particles. Further, σ is the electrical conductivity, B0 is the uniform magnetic field, and ρp is the mass of the dust particles per unit volume of the fluid. T and Tp are respectively the temperatures of the fluid and the dust particles. Further, cp and cs are respectively the specific heat capacity of the fluid and specific heat capacity of the dust particles, γT is the temperature relaxation time (= 3Prγp cs /2cp ), γp is the velocity relaxation time (= 1/k), and Pr is the usual Prandtl number. The last term in Eq. (6.27) represents the force due to the relative motion between the fluid and the dust particles. In such a case the force between dust and fluid is proportional to the relative velocity. α(T ) = K(T )/ρ∞ cp is the thermal diffusivity of the fluid; it varies as a linear function of temperature. In deriving these equations the Stokesian drag force is considered for the interaction between the fluid and the particle phases. The appropriate boundary condition on velocity and temperature are u = Uw (x) = bx, u → 0,
T → T∞ ,
up → 0,
v = 0, vp → v,
T = Tw = A1 (x/l) at y = 0, ρp → kρ∞ ,
Tp → T∞ as y → ∞.
(6.33)
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Application of the Keller-box Method to More Advanced Problems
Here b is a constant known as stretching rate, A1 is a constant, and l is the characteristic length. Now, let the dimensionless similarity variable be (6.34) η = b/ν∞ y
and the dimensionless similarity functions are u = bxf � (η), ρr = H(η),
√ v = − bν∞ f (η),
up = bxF (η),
vp =
√
T − T∞ = (Tw − T∞ )θ(η),
Tp − T∞ = (Tw − T∞ )θp (η),
bν∞ G(η),
(6.35)
(Tw − T∞ ) = A(x/l).
Substituting the expressions for variable fluid viscosity and the variable fluid thermal conductivity from Eqs. (6.21) and (6.22) into Eqs. (6.26) to (6.33) and making use of similarity equations from (6.34)–(6.35), we obtain � f �� + f f �� − f �2 − M nf � + βH(F − f � ) = 0, (1 − θ/θr ) GF � + F 2 + β(F − f � ) = 0,
GG� + β(f + G) = 0,
GH � + HG� + F H = 0,
(6.36)
2 ((1 + εθ)θ � )� + Pr(f θ� − f � θ) + βH(θp − θ) = 0, 3 � 2F θp + Gθp + L0 β(θp − θ) = 0, and
f � = 1,
f = 0,
f → 0,
F → 0,
�
θ → 0,
θ = 1,
y = 0,
G → −f,
θp → 0 as y → ∞
H → k,
(6.37)
where a prime denotes differentiation with respect to η. Here ρr = ρp /ρ∞ is the relative density, M n = σB02 /ρ∞ b is the magnetic parameter, β = 1/bτ is the fluid particle interaction parameter, θr = 1/γ(ΔT ) is the fluid viscosity parameter, which is negative for liquids and positive for gases, Pr = ν∞ /α∞ is the Prandtl number, ε is the variable thermal conductivity parameter and L0 = τ /γT is the temperature relaxation parameter. The value of θr is determined by the viscosity of the fluid under consideration and the operating temperature difference. If θr is large, in other words, if T∞ − Tw is small, the effects of variable viscosity on the flow can be neglected. On the other hand, for smaller values of θr , either the fluid viscosity changes markedly with temperature or the operating temperature difference is high. In either case, the effect of the variable fluid viscosity is expected to be very important. Also let us keep in mind that the liquid viscosity varies differently with temperature compared to the gas viscosity. Therefore it is important to note that θr is negative for liquids and positive for gases.
6.2
Hydromagnetic fluid flow and heat transfer at a stretching sheet with ...
6.2.3
303
Solution for special cases
In the limiting case of θr → ∞ and ε = 0, the system of Eqs. (6.36) reduces to those of Gireesha et al. [49], those of Vajravelu and Nayfeh [46] (when no heat transfer is considered), and when β = 0, to those of Chakrabarti and Gupta [53]. In the presence of variable fluid properties, when there is no fluid interaction and no magnetic field, the system of Eqs. (6.36) reduces to those of Pop et al. [18]. Further, when the variable thermophysical properties, fluid particle interaction and the magnetic field are absent, the equations are similar to the ones studied by Crane [32], and Grubka and Bobba [35]. In the absence of variable fluid properties, the hydromagnetic boundary layer flow and heat transfer problem degenerates. In this case, the approximate analytical solutions for the velocity field and temperature fields are obtained via perturbation analysis. These solutions are useful and serve as a baseline for comparison with the solutions obtained via numerical schemes.
6.2.4
Analytical solution by perturbation
For small β that is for low particle interaction, let us perturb the flow and heat transfer fields as f = f0 + βf1 + o(β 2 ), F = F0 + βF1 + o(β 2 ), G = G0 + βG1 + o(β 2 ), H = H0 + βH1 + o(β 2 ),
(6.38)
θ = θ0 + βθ1 + o(β 2 ), θ = θp0 + βθp1 + o(β 2 ), where the perturbations are small compared with the mean or the zerothorder quantities. With the help of (6.38), Equations (6.36) and the boundary conditions (6.37) become f0 + f0 f0 − (f0 )2 − M nf0 = 0, G0 F0 + F02 = 0, G0 G 0 = 0, G0 H0 + H0 G 0 + F0 H0 = 0, θ0
+
Pr(f0 θ0
2F0 θp0 + f0 f0
−
G0 θp0
f0 θ)
(6.39)
= 0,
= 0,
= 1,
f0 = 0,
→ 0,
F0 → 0,
θ0 = 1,
y = 0,
G0 → −f0 ,
H0 → k,
θ0 → 0,
θp0 → 0 as y → ∞,
304
Chapter 6
Application of the Keller-box Method to More Advanced Problems
to the zeroth order, and f1��� + f1 f0�� − 2f0� f1� − M nf1� + H0 (F0 − f0� ) = 0,
G0 F1� + G1 F0� + 2F0 F1 + F0 − f0� = 0, G0 G�1 + G�0 G1 + f0 + G0 = 0,
G0 H1� + H0� G1 + H0 G�1 + G�0 H1 + F0 H1 + H0 F1 = 0, 2 θ1�� + Pr(f0 θ1� + f0� θ1 ) = H0 (θp0 − θ0 ) − Pr(f1 θ0� − f1� θ0 ), 3 � � + 2F0 θp1 = −2F1 θp0 − G1 θp0 + L0 (θ0 − θp0 ), G0 θp1 f1� = 0, f1�
→ 0,
H1 → k,
f1 = 0,
θ1 = 0,
F1 → 0,
θ1 → 0,
y = 0, G1 → −f1 ,
(6.40)
θp1 → 0
as y → ∞,
to the first order. The exact solutions for the zeroth order velocity components f0 , G0 , F0 , particle density H0 and temperature θ0 satisfying the boundary conditions are f0 = A2 + B2 exp(−αη), F0 = 0, G0 = −A2 , H0 = k, Pr Pr −α1 η Pr M − 1, 1 + 2 , − 2 e (6.41) Pr α2 α α , θp0 (η) = 0 θ0 (η) = exp η Pr Pr Pr α M − 1, 1 + 2 , − 2 α2 α α
where
A2 = 1/α,
B2 = −1/α,
α=
√ 1 + M n.
Similarly the exact solutions for the first order velocity components, firstorder particle density and temperature are f1 = (−kB2 /(A2 α − B2 α + 2M n))e−αη + αηe−αη − 1, F1 = −(B2 /A2 )e−αη ,
G1 = −(B2 /A2 )e−αη − (−kB2 /(A2 α − B2 α + 2M n)), ∞ 2 θp0 (z)dz. H1 = 0, θp1 = − 3PrG0 η
(6.42)
The solution θ1 may be obtained by solving the inhomogeneous equation it satisfies, using the standard variation of constant method. The results of the present work, in the absence of β are compared with the available results in the literature, and are shown in Tables 6.3 and 6.4. The results in Tables 6.3 and 6.4 reveal very good agreement between the numerical results and the available results in the literature. The physical quantities of interest here in the study are, the skin friction coefficient cf and the Nusselt number. They are defined by cf x =
2τw (x) , ρu2w
N ux =
qw x , k∞ (Tw − T∞ )
(6.43)
6.2
Hydromagnetic fluid flow and heat transfer at a stretching sheet with ...
305
where τw (x) = −μw
∂u ∂y
y=0
and qw (x) = −k∞
∂T ∂y
. y=0
Table 6.3 Comparison of skin friction for different values of the magnetic parameter with the exact solution and the numerical solution given by Andersson et al. [28] when βr = ε = 0.0 and θr → ∞ Mn
Present results
Andersson et al. [28]
Exact Solution
0.0 0.5 1.0 1.5 2.0
−1.0001 −1.2249 −1.414 −1.581 −1.73205
−1.0000 −1.2247 −1.414 −1.581 −1.73350
−1.0000 −1.2247 −1.414 −1.582 −1.73205
Table 6.4 Comparison of wall-temperature gradient θ (0) for different values of the Prandtl number for constant surface temperature when θr → ∞, ε = 0.0, β = 0 and M n = 0.0 Pr
Present
Grubka and Bobba
Chen
Ali
0.01 0.72 1.0 3.0 5.0 10.0 100.
—— —— —— —— —— −2.308029 −7.769667
−0.0099 −0.4631 −0.5820 −1.1652 —— −2.3080 ——
−0.0091 −0.46315 −0.58199 −1.16523 —— −2.30796 ——
−0.4617 −0.5801 −1.1599 —— −2.2960 ——
6.2.5
Numerical procedure
The system of Eqs. (6.36) is coupled and highly nonlinear. Exact analytical solutions are not possible for the complete set of equations and therefore we use Keller-box method [51]. The coupled boundary value problem (6.36), (6.37) of third order in f (η), first order in F (η), G(η), H(η), and θp (η) and second order in θ(η), respectively is reduced to a system of nine simultaneous ordinary differential equations of first order with nine unknowns, by assuming f = f1 , f � = f2 , f �� = f3 , θ = θ1 , θ� = θ2 . To solve this system of equations we require nine initial conditions while we have only two initial conditions f (0), f � (0) on f (η) and one initial condition θ(0) on θ(η). The other six initial conditions f �� (0), F (0), G(0), H(0), θ� (0) and θp (0) are not prescribed. However, the values of f � (η), F (η), G(η), H(η), θ(η) and θp (η) are known as η → ∞. Now, we employ the Keller-box scheme where these six boundary conditions are utilized to produce six unknown initial conditions
306
Chapter 6
Application of the Keller-box Method to More Advanced Problems
at η = 0. To select η∞ , we begin with some initial guess values and solve the boundary value problem with some particular set of parameters to obtain f �� (0), F (0), G(0), H(0), θ� (0) and θp (0). Thus, we start with the initial approximations as f �� (0) = δ1 , F (0) = δ2 , G(0) = δ3 , H(0) = δ4 and θp (0) = δ6 . Let δi∗ (i = 1, 2, 3, 4, 5, 6) be the correct values of f �� (0), F (0), G(0), H(0), θ� (0) and θp (0). We integrate the resulting system of nine ordinary differential equations using the fourth order Runge-Kutta method and obtain the values of f �� (0), F (0), G(0), H(0), θ� (0) and θp (0). The solution process is repeated with another larger value of η∞ until two successive values of f �� (0), F (0), G(0), H(0), θ� (0) and θp (0) differ only after some desired digit signifying the limit of the boundary along η. The last value of η∞ is chosen as an appropriate value for that particular set of parameters. Finally, the problem can be solved numerically using the Keller-box method (for details see Prasad et al. [51]). It is also important to note that the computational time for each set of input parameters should be short. Because the physical domain in this problem is unbounded, whereas the computational domain has to be finite, we apply the far field boundary conditions for the similarity variable η at a finite value denoted by ηmax . We ran the our bulk of our computations with the value ηmax = 7, which is sufficient to achieve the far field boundary conditions asymptotically for all values of the parameters considered. For numerical calculations, a uniform step size of Δη = 0.01 is found to be satisfactory and the solutions are obtained with an error tolerance of 10−6 in all the cases. To assess the accuracy of the present method, comparison of the skin friction f �� (0) and the wall-temperature gradient θ� (0) between the present results and previously published results is made, for several special cases in which the fluid-particle interaction parameter and thermophysical fluid properties are neglected (see Table 6.3). It was found from Tables 6.3 and 6.4 that the present results agree very well with those of analytical solutions given by Andersson et al. [28], Grubka and Bobba [35], Chen [37] and Ali [39].
6.2.6
Results and discussion
In this section, we analyze the effects of the pertinent parameters, namely, the fluid-particle interaction parameter β, the magnetic parameter M n, the variable fluid viscosity parameter θr , the variable thermal conductivity parameter ε, and the Prandtl number Pr on the flow and heat transfer of fluid-particle suspension over a horizontal stretching sheet. Approximate analytical solutions are obtained via the perturbation method for the special case when θr → ∞ and ε = 0. The temperature relaxation parameter L0 is chosen as unity throughout the computations. The numerical solution for the general case is obtained by the Keller-box method (Cebeci and Bradshaw [26]). Also, in order to get a clear insight into the physical problem, we present the numerical results graphically in Figures 6.5–6.10. These figures depict respe-
6.2
Hydromagnetic fluid flow and heat transfer at a stretching sheet with ...
307
Fig. 6.5a Temperature profiles for different values of β and M n with θr−1 = 0.0, ε = 0.0, Pr = 1.0
Fig. 6.5b Horizontal velocity profiles for different values of β and M n with θr−1 = 0.0, ε = 0.0, Pr = 1.0
308
Chapter 6
Application of the Keller-box Method to More Advanced Problems
Fig. 6.5c Particle velocity components F for different values of β and M n with θr−1 = 0.0, ε = 0.0, Pr = 1.0
Fig. 6.5d Particle velocity components G for different values of β and M n with θr−1 = 0.0, ε = 0.0, Pr = 1.0
6.2
Hydromagnetic fluid flow and heat transfer at a stretching sheet with ...
309
Fig. 6.6a Fluid transverse velocity profiles for different values of β and θr with M n = 0.5, ε = 0.1, Pr = 1.0
Fig. 6.6b Fluid horizontal velocity profiles for different values of β and θr with M n = 0.5, ε = 0.1, Pr = 1.0
310
Chapter 6
Application of the Keller-box Method to More Advanced Problems
Fig. 6.7a Fluid temperature profiles for different values of β and M n with θr−1 = 0.0, ε = 0.0, Pr = 1.0
Fig. 6.7b Dust phase temperature profiles for different values of β and M n with θr−1 = 0.0, ε = 0.0, Pr = 1.0
6.2
Hydromagnetic fluid flow and heat transfer at a stretching sheet with ...
311
Fig. 6.8a Fluid temperature profiles for different values of β and θr with M n = 0.5, ε = 0.1, Pr = 1.0
Fig. 6.8b Dust phase temperature profiles for different values of β and θr with M n = 0.5, ε = 0.1, Pr = 1.0
312
Chapter 6
Application of the Keller-box Method to More Advanced Problems
Fig. 6.9a Fluid temperature profiles for different values of β and ε with M n = 0.5, θr = −0.5, Pr = 1.0
Fig. 6.9b Dust phase temperature profiles for different values of β and ε with M n = 0.5, θr = −0.5, Pr = 1.0
6.2
Hydromagnetic fluid flow and heat transfer at a stretching sheet with ...
313
Fig. 6.10a Fluid temperature profiles for different values of β and Pr with M n = 0.5, θr = −0.5, ε = 0.1
Fig. 6.10b Dust phase temperature profiles for different values of β and Pr with M n = 0.5, θr = −0.5, ε = 0.1
314
Chapter 6
Application of the Keller-box Method to More Advanced Problems
ctively the velocity profiles (f, f ); the particle-suspension velocity profiles (F, G); and the temperature of the fluid and the dust phase profiles (θ, θp ). The computed numerical results are recorded in Table 6.5 to show the behavior of the skin friction, the particle velocity and the density components, the temperature gradient, and the dust-phase temperature at the sheet for different values of the governing parameters. Table 6.5 Values of the skin friction f (0), the particle velocity components F (0), G(0), the particle density component H(0), the temperature gradient θ (0) and the temperature of the dust particle θp (0) at the wall for different values of the physical parameters Pr
ε
θr
Mn
0.0
0.5 1.0 0.0
0.0 1.0
2.0
β
f (0)
F (0)
G(0)
H(0)
θ (0)
θp (0)
0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0
−1.00015 −1.05395 −1.07784 −1.09489 −1.22498 −1.26851 −1.28854 −1.30292 −1.41432 −1.45223 −1.46979 −1.48246 −1.73215 −1.76326 −1.77779 −1.78831
0.00000 0.52159 0.70818 0.78917 0.00000 0.52146 0.70817 0.78909 0.00000 0.52145 0.70817 0.78901 0.0000 0.52145 0.70816 0.78889
−0.97382 −0.32762 −0.07737 −0.00661 −0.81064 −0.27277 −0.06468 −0.00577 −0.70535 −0.23825 −0.05667 −0.00524 −0.57708 −0.19616 −0.04686 −0.00459
0.20000 0.26572 0.52056 1.67711 0.20000 0.26609 0.52052 1.59296 0.20000 0.26615 0.52003 1.52775 0.20000 0.26614 0.51900 1.43136
−1.000140 −1.21223 −1.290571 −1.331141 −0.94656 −1.162988 −1.239744 −1.28042 −0.90194 −1.121731 −1.196771 −1.237092 −0.832988 −1.05494 −1.12634 −1.165365
0.00000 0.39785 0.60459 0.71187 0.00000 0.43138 0.61700 0.71317 0.00000 0.45608 0.62525 0.71404 0.00000 0.49087 0.63584 0.71518
0.0 1.0 −10.0 0.5 2.0 3.0 0.0 1.0 1.0 0.1 −5.0 0.5 2.0 3.0 0.0 1.0 −1.5 0.5 2.0 3.0
−1.21744 −1.34587 −1.36781 −1.38334 −1.36441 −1.41778 −1.44157 −1.45818 −1.63335 −1.72756 −1.76318 −1.78529
0.00000 0.52082 0.70797 0.78904 0.00000 0.52031 0.70782 0.78900 0.00000 0.51875 0.70734 0.78883
−0.78636 0.20000 −0.26190 0.26700 −0.06166 0.52328 −0.00554 1.57683 −0.76382 0.20000 −0.25211 0.26777 −0.05901 0.52549 −0.00535 1.56064 −0.87817 0.20000 −0.21609 0.27031 −0.04960 −0.53236 −0.00468 1.48823
−0.869809 −1.073854 −1.144771 −1.182618 −0.858874 −1.06338 −1.133600 −1.171181 −0.815650 −1.019397 −1.086136 −1.12225
0.00000 0.44564 0.62294 0.71392 0.00000 0.45230 0.62550 0.71421 0.00000 0.47868 0.63481 0.71528
6.2
Hydromagnetic fluid flow and heat transfer at a stretching sheet with ...
315
Continued Pr
1.0
ε
θr
Mn
β
f (0)
F (0)
0.0 −1.36578
0.0000
0.0 −1.36317
0.0000
0.0 −1.36101
0.0000
G(0)
H(0)
θ (0)
θp (0)
−0.76469 0.20000 −0.923053 0.00000
0.0 −5.0
0.5 1.0 −1.41926 0.52026 −0.25230 0.26781 −1.140651 0.44507 2.0 −1.44310 0.70780 −0.05903 0.52570 −1.215927 0.62262
0.2 −5.0
0.5 1.0 −1.41644 0.52036 −0.25194 0.26772 −0.997784 0.45899 2.0 −1.44017 0.70784 −0.059 0.52530 −1.063699 0.62805
0.4 −5.0
0.5 1.0 −1.41407 0.52044 −0.25167 0.26763 −0.891964 0.47102 2.0 −1.43770 0.70787 −0.05899 0.52494 −0.950924 0.63236
−0.76301 0.20000 −0.804383
0.0000
−0.76157 0.20000 −0.716511 0.00000
0.0 −1.35983 0.00000 −0.76055 0.20000
−0.69308
0.00000
0.72 0.1 −5.0
0.5 1.0 −1.41330 0.52047 −0.25164 0.26757 −0.88966 0.55769 2.0 −1.43667 0.70788 −0.05901 0.52473 −0.942570 0.70601
2.0
0.1 −5.0
0.5 1.0 −1.42826 0.51994 −0.25337 0.26817 −1.51995 0.24584 2.0 −1.45243 0.70768 −0.05907 0.52711 −1.618615 0.42808
5.0
0.1 −5.0
0.5 1.0 −1.44461 0.51943 −0.25580 0.26857 −2.426292 0.08421 2.0 −1.46821 0.70745 −0.05931 0.52912 −2.521481 0.19810
10.0 0.1 −5.0
0.0 −1.37637
0.0000
−0.77201 0.20000 −1.341275 0.00000
0.0 −1.39453 0.00000 −0.78259 0.20000 −2.311434 0.00000
0.0 −1.40843
0.0000
−0.78921 0.20000 −3.402150 0.00000
0.5 1.0 −1.45811 0.51912 −0.25817 0.26864 −3.478130 0.03380 2.0 −1.48129 0.70724 −0.05966 0.53028 −3.552070 0.09539
The transverse velocity f , the horizontal velocity f , and the particle transverse velocity and horizontal velocity (F (η), G(η)) profiles are shown graphically in Figures 6.5a–6.5d for different values of the magnetic parameter M n and the fluid-particle interaction parameter β. The general trend is that f , F and G decrease monotonically, whereas f increases as the distance increases from stretching sheet. It is observed from these figures that the horizontal velocity and transverse velocity profiles decrease with an increase in the magnetic parameter. This observation holds true even with particle velocity component F (η), but quite the opposite is true with G(η). Physically it means that the induction of transverse magnetic field (normal to the flow direction) has a tendency to induce a drag, known as the Lorentz force, which tends to resist the flow. It is noticed that the effect of increasing values of the fluid-particle interaction parameter β is to reduce the fluid velocity in the boundary layer and increase the dust phase transverse velocity as well as the horizontal velocity F (η). Figures 6.6a and 6.6b exhibit the velocity profiles for several sets of values of the fluid viscosity parameter θr and the fluid-particle interaction parameter β. From the graphical representation, we infer that the effect of increasing values of the fluid viscosity parameter θr is to decrease the momentum boundary layer thickness. Also, as θr approaches zero, the boundary layer
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thickness decreases and the horizontal velocity distribution tends to zero (see Figure 6.6b) asymptotically. This is due to the fact that for a given fluid (air or water), when δ is fixed, a smaller θr implies a higher temperature difference between the wall and the ambient fluid. The results presented here demonstrate clearly that θr , the indicator of the variation of fluid viscosity with temperature, has a substantial effect on the horizontal velocity components f , as well as the transverse velocity f and hence on the skin friction. This phenomenon is true with zero and non-zero values of the fluid-particle interaction parameter β. In Figures 6.7–6.10, the numerical results for the fluid temperature and the dust-phase temperature (θ(η), θp (η)) are presented for several sets of values of the governing parameters. The general trend is that the fluidtemperature distribution is unity at the wall whereas the dust-phase temperature is not. However, with changes in the governing parameters, both asymptotically tend to zero as the distance increases from the boundary. Figure 6.7 illustrates the effect of the magnetic parameter and the fluid-interaction parameter on θ(η). The effect of increasing values of the magnetic parameter M n is to increase the fluid temperature θ(η) and also the dust-phase temperature θp (η). As explained above, the induction of a transverse magnetic field to an electrically conducting fluid gives rise to the Lorentz force. This force makes the fluid experience a resistance by increasing the friction between its layers. Hence, there is an increase in the temperature profile θ(η) as well as the dust-phase profile. The effect of the fluid interaction parameter is to decrease the temperature profile which in turn reduces the thermal boundary layer thickness whereas it enhances the dust phase temperature at the wall and hence increases the thickness of the dust phase temperature. Figures 6.8a and 6.8b exhibit the fluid-temperature distribution and dustphase temperature distribution for several sets of values of the variable viscosity parameter θr and the fluid-particle interaction parameter. From the graphical representation, we observe that the effect of increasing values of the variable viscosity parameter θr is to enhance both the fluid-temperature as well as the dust-phase temperature. This is due to the fact that an increase in the variable viscosity parameter θr results in an increase in the thermal boundary layer thickness. This is very much noticeable for zero values of fluid-particle interaction parameter as compared to the larger values. The graphs for the fluid-temperature profile θ(η) and dust-phase temperature θp (η) for different values of the variable thermal conductivity parameter ε are respectively shown in Figures 6.9a and 6.9b. These figures demonstrate that an increase in the value of thermal conductivity parameter ε results in increasing the temperature θ(η). This is due to the fact that the assumption of temperature-dependent thermal conductivity (linear form) implies a reduction in the magnitude of the transverse velocity by a quantity ∂K(T )/∂y as can be seen from heat transfer equation. Figures 6.10a and 6.10b are drawn to display the fluid-temperature profile θ(η) and dust-phase temperature θp (η) for different values of the Prandtl
6.2
Hydromagnetic fluid flow and heat transfer at a stretching sheet with ...
317
number in the absence of the fluid-interaction parameter. We observe that the effect of increasing values of the Prandtl number Pr is to decrease both θ(η) as well as θp (η). Physically it means that an increase in the Prandtl number means a decrease in the thermal conductivity K∞ . Hence, there is a decrease in the thermal boundary layer thickness. This behavior can be seen even in the presence of the fluid interaction parameter. Finally, the effects of all the physical parameters on the surface-velocity gradient, the particle-velocity components, particle-density component, the temperature gradient, and the dust-phase temperature, at the sheet are depicted in Table 6.5. It is of interest to note that the effect of increasing values of the variable viscosity parameter, the magnetic parameter, and the fluidparticle interaction parameter is to increase the magnitude of the skin friction coefficient. The effects of the variable thermal conductivity parameter, the variable viscosity parameter and the magnetic parameter are to decrease the magnitude of the temperature gradient at the sheet whereas the reverse trend is observed with an increase in the Prandtl number as well as the fluid interaction parameter. From Table 6.5, it is further noticed that the effect of the fluid interaction parameter is to increase the dust-phase temperature as well as the particle transverse velocity component F (0). This observation is true for zero and non-zero values of the magnetic field parameter.
6.2.7
Conclusions
In this section, the effects of temperature dependent thermophysical properties on the MHD boundary layer flow and heat transfer of a fluid-particle suspension over a stretching sheet are investigated. The governing partial differential equations are converted into ordinary differential equations by similarity transformations. The transformed equations are solved numerically by the Keller-box method. The effects of the physical parameters on the fluid velocity, and the temperature, and the dust phase are shown graphically and discussed. Some of the important findings are listed below: • In the presence of temperature-dependent thermophysical properties, the effect of increasing values of the fluid interaction parameter and the magnetic parameter is to decrease the velocity throughout the boundary layer. However, quite the opposite is true with dust phase velocity profiles. • The effect of increasing values of the fluid viscosity parameter is to decrease the velocity boundary layer thickness. However, it enhances the thermal boundary layer thickness. The effect of variable thermal conductivity parameter is to enhance the fluid temperature as well as the particle phase temperature in the flow region. • The thermal boundary layers of the fluid and the dust phase are highly influenced by the Prandtl number. The effect of Pr is to decrease the thermal boundary layer thickness.
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• Of all the parameters, the variable thermophysical fluid property parameters have strong effects on the drag, heat transfer characteristics, the horizontal velocity, and the temperature.
6.3 Radiation effects on mixed convection over a wedge embedded in a porous medium filled with a nanofluid 6.3.1
Introduction
Fluid flow and heat transfer in porous media received considerable interest during the last several decades. This is primarily because of the numerous applications of flow through porous media, such as storage of radioactive nuclear waste, transpiration cooling, separation processes in chemical industries, filtration, transport processes in aquifers, groundwater pollution, geothermal extraction, fiber insulation, etc. Theories and experiments of thermal convection in porous media and state-of-the-art reviews, with special emphasis on practical applications, are presented in the recent books by Nield and Bejan [68], Pop and Ingham [70], Ingham and Pop [66], Bejan et al. [58] and Vafai [73]. Nanofluid is envisioned as a fluid, in which nanometer-sized particles are suspended (see [1]). Convectional heat transfer fluids, including oil, water, and ethylene glycol mixture are poor heat transfer fluids, since the thermal conductivity of these fluids plays an important role on the heat transfer coefficient between the heat transfer medium and the heat transfer surface. Therefore numerous methods are proposed to improve the thermal conductivity of these fluids by suspending nano/micro sized particle materials in liquids. Recently, several numerical studies on the modeling of natural convection heat transfer in nanofluids have been published. Duangthongsuk and Wongwises [62] studied the influence of thermophysical properties of nanofluids on the convective heat transfer and summarized various models used in the literature for predicting the thermophysical properties of nanofluids. The problem of the thermal instability in a porous medium layer saturated by a nanofluid was investigated by Nield and Kuznetsov [69]. Abu-Nada and Oztop [54] studied the effects of inclination angle on natural convection in enclosures filled with Cu-water nanofluid. Nield and Kuznetsov [69] suggested natural convective boundary layer flow of a nanofluid past a vertical plate. Chamkha et al. [59] studied the mixed convection MHD flow of a nanofluid past a stretching permeable surface in the presence of Brownian motion and thermopherosis effects. Also, Chamkha et al. [59] analyzed the natural convection
6.3 Radiation effects on mixed convection over a wedge embedded in a porous ...
319
flow past a sphere embedded in a porous medium saturated by a nanofluid. Gorla et al. [63] studied the steady boundary layer flow of a nanofluid on a stretching circular cylinder in a stagnant free stream. Furthermore, Gorla et al. [64] analyzed the mixed convection past a vertical wedge embedded in a porous medium saturated by a nanofluid. However, the thermal radiation effect on mixed convection heat transfer in porous media has many important applications such as space technology, and processes involving high temperatures such as geothermal engineering, the sensible heat storage bed, the nuclear reactor cooling system, and underground nuclear wastes disposal. Yih [75] studied radiation effect on mixed convection over an isothermal wedge/cone in porous media. Bakier [57] presented an analysis of the thermal radiation effect on stationary mixed convection from vertical surfaces in saturated porous media. Kumari and Nath [67] studied the radiation effect on the non-Darcy mixed convection flow over a non-isothermal horizontal surface in a porous medium. Chamkha and Ben-Nakhi [60] studied the mixed convection-radiation interaction along a permeable surface immersed in a porous medium. The problem of hydromagnetic heat transfer by mixed convection from melting of a vertical plate in a liquid saturated porous medium, taking into account the effects of thermal radiation, was investigated by Bakier et al. [56]. Also, the study of convection heat transfer from a cone/wedge is of special interest and has a wide range of practical applications. Mainly, these types of heat transfer problems deal with the design of space-craft, nuclear reactors, solar power collectors, power transformers, steam generators and others. Many investigations (Pop et al. [71], Takhar et al. [72], Alam et al. [55], Vajravelu and Nayfeh [74]) have developed similarity and non-similarity solutions for natural convection flows over a vertical cone/wedge in steady state. Motivated by these studies, the problem of steady, laminar, mixed convection boundary layer flow over an isothermal vertical wedge embedded in a porous medium saturated with a nanofluid, in the presence of thermal radiation, is studied in this section (for details see A.J. Chamkha, S. Abbasbandy, A.M. Rashad, K. Vajravelu, Transport in Porous Media 91 , 2012, 261–279). The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis with Rosseland diffusion approximation. The resulting governing equations are non-dimensionalized and transformed into a non-similar form and then solved by the Keller-box method. A comparison is made with the available results in the literature, and the salient features of the new results are analyzed and discussed.
6.3.2
Problem formulation
Consider the problem of the radiation effect on mixed-convection boundary layer flow of optically dense viscous incompressible nanofluid over an isothermal wedge embedded in a saturated porous medium. The model used for the
320
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nanofluid incorporates the effects of Brownian motion and thermophoresis. The uniform wall temperature of the wedge Tw and uniform nanoparticle volume fraction Cw are higher than the ambient temperature T∞ and ambient nanoparticle volume fraction C∞ , respectively. The flow over the wedge is assumed to be two-dimensional, laminar, steady, and incompressible (see Figure 6.11 for the flow model and the physical coordinate system). The porous medium is assumed to be uniform, isotropic and is in local thermal equilibrium with the fluid. All fluid properties are assumed to be constant. Under the Boussinesq and the Rosseland diffusion approximations, the governing equations based on the Darcy law proposed by Hsieh et al. [65] and Yih [76] can be written as ∂u ∂v + = 0, ∂x ∂y ∂u (1 − C∞ )ρf ∞ βgK ∂T (ρp − ρf ∞ )gK ∂C = − , ∂y μ ∂y μ ∂y
u
(6.45)
2 ∂C ∂T ∂T DT DB + ∂y ∂y T∞ ∂y ∂ ∂T 16σ T3 , + 3(ar + σs )(cp ρ)f ∂y ∂y
∂T ∂ 2T ∂T u +v = αe 2 + τ ∂x ∂y ∂y
(6.44)
∂C ∂2C ∂C +v = DB 2 + ∂x ∂y ∂y
DT T∞
∂2T , ∂y 2
(6.46)
(6.47)
Fig. 6.11 Flow model and coordinate system
where x and y denote the vertical and horizontal directions, respectively. u, v, T and C are the x and y components of velocity, temperature and nanoparticle volume fraction, respectively. K, β, DB and DT are the permeability of the porous medium, volumetric expansion coefficient of fluid, the Brownian diffusion coefficient and thermophoretic diffusion coefficient, respectively. μ, ρf and ρp are the fluid viscosity, fluid density and the nanoparticle mass
6.3 Radiation effects on mixed convection over a wedge embedded in a porous ...
321
density, respectively. g, σ, σs , and ar are the acceleration due to gravity, the Stefan-Boltzmann constant, scattering coefficient, and the Rosseland mean extinction coefficient, respectively. αe = k/(ρc)f and τ = (ρc)p /(ρc)f are the thermal diffusivity of the porous medium and the ratio of heat capacities, respectively. k, (ρc)f and (ρc)p are thermal conductivity, heat capacity of the fluid, and the effective heat capacity of the nanoparticle material, respectively. The last term on the right side of the energy equation (6.46) is the thermal radiation heat flux and is approximated using the Roseland diffusion equation. The appropriate boundary conditions suggested by the physics of the problem are (6.48a) y = 0 : v(x, 0) = 0, T = Tw , C = Cw , y→∞:
u = U∞ ,
T = T∞ ,
C = C∞ ,
(6.48b)
where Tw and Cw are the wall temperature and wall nanoparticle volume fraction, respectively. U∞ , T∞ and C∞ are the free stream velocity, temperature and nanoparticle volume fraction, respectively. It is convenient to transform the governing equations into a non-similar dimensionless form which can be studied as an initial-value problem. This can be done by introducing the stream function: u = ∂ψ/∂y, v = −∂ψ/∂x and using y 1/2 η = (P ex )χ−1 , x
χ= 1 +
Rax P ex
1/2 −1 ,
1/2
ψ = αe (P ex )χ−1 f (χ, η),
T − T∞ C − C∞ ϕ , , φ= , λ= T w − T∞ Cw − C∞ π−ϕ U∞ = Bxλ , P ex = U∞ x/αe , Rax = (1 − C∞ )ρf ∞ gβT K(Tw − T∞ )x/μαe ,
θ=
(6.49)
where P ex and Rax are the local Peclet and modified Rayleigh numbers respectively. The parameters a, ϕ and λ are the free stream velocity constant, half-wedge angle and the free stream velocity exponent respectively. Using the expressions in (6.49), we can write Eqs. (6.44)–(6.48) as f �� = (1 − χ)2 (θ� − N rφ� ),
(6.50)
1 4Rd � (θ ((H − 1)θ + 1)3 )� θ�� + (1 + λχ)f θ� + N bφ� θ� + N tθ�2 + 2 3 λ � ∂θ � ∂f = χ(1 − χ) f −θ , (6.51) 2 ∂χ ∂χ Le N t �� Le ∂φ ∂f (1 + λχ)f φ� + θ = λχ(1 − χ) f � − φ� , (6.52) φ�� + 2 Nb 2 ∂χ ∂χ (1+λχ)f (χ, 0)−λχ(1−χ)
∂f (χ, 0) = 0, ∂χ
θ(χ, 0) = 1,
φ(χ, 0) = 1, (6.53a)
322
where
Chapter 6
Application of the Keller-box Method to More Advanced Problems
f � (χ, ∞) = χ2 ,
θ(χ, ∞) = 0,
φ(χ, ∞) = 0,
(6.53b)
ε(ρc)p DB (Cw − C∞ ) αe (ρp − ρf ∞ )(Cw − C∞ ) , Nb = , Nr = , DB (1 − C∞ )ρf ∞ β(Tw − T∞ ) (ρc)f αe ε(ρc)p DT (Tw − T∞ ) 3 Nt = , Rd = 4σT∞ /(k(ar + σs )), H = Tw /T∞ (ρc)f αe T∞ (6.54) are the Lewis number, buoyancy ratio, Brownian motion parameter, thermophoresis parameter, conduction-radiation parameter, and the surface temperature excess ratio, respectively. It should be noted that χ = 0 (P ex = 0) corresponds to pure free convection while χ = 1 (Rax = 0) corresponds to pure forced convection. The entire regime of mixed convection corresponds to the values of χ between 0 and 1. Of special significance for this problem are the local Nusselt and Sherwood numbers. These physical quantities can be defined as N ux 4RdH 3 � , (6.55) = −θ (χ, 0) 1 + 1/2 1/2 3 Rax + P ex Shx = −φ� (χ, 0). (6.56) 1/2 1/2 Rax + P ex Le =
6.3.3
Numerical method and validation
The governing equations (6.50), (6.51) and (6.52) with the boundary conditions (6.53) are nonlinear partial differential equations. The system of Eqs. (6.50)–(6.52) is solved numerically by the Keller box method as described by Cebeci and Bradshaw [26]. The computations were carried out with Δχ = 0.01 and Δη = 0.01 (uniform grids). The value of η∞ = 50 is found to be sufficiently large to obtain the accuracy of |θ� (0)| < 10−5. In order to validate the numerical results, comparisons with the previously published results of Hsieh et al. [65] and Yih [76] for the case of Newtonian fluid are made when Rd = N r = N b = N t = 0. These comparisons are presented in Table 6.6. It is easy to see from the table that an excellent agreement exists between the results. Table 6.6 Comparison of values of −θ (ξ, 0) for various values of λ and χ in the absence of nanoparticles volume fraction, Brownian motion and thermophoresis effects (N r = N b = N t = 0) χ
Hsieh et al. [65] λ=0
λ=0
Yih [76] λ = 1/3
λ=1
λ=0
1.0 0.9 0.8
0.5642 0.5098 0.4603
0.5642 0.5097 0.4602
0.6515 0.5878 0.5278
0.7979 0.7181 0.6385
0.5642 0.5098 0.4602
Present results λ = 1/3 λ=1 0.6516 0.5879 0.5280
0.7979 0.7181 0.6385
6.3 Radiation effects on mixed convection over a wedge embedded in a porous ...
323
Continued χ 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
6.3.4
Hsieh et al. [65]
Yih [76]
Present results
λ=0
λ=0
λ = 1/3
λ=1
λ=0
λ = 1/3
λ=1
0.4174 0.3832 0.3603 0.3506 0.3550 0.3732 0.4035 0.4438
0.4173 0.3832 0.3603 0.3505 0.3550 0.3732 0.4035 0.4437
0.4731 0.4261 0.3900 0.3686 0.3643 0.3769 0.4044 0.4437
0.5599 0.4854 0.4227 0.3823 0.3697 0.3786 0.4049 0.4437
0.4173 0.3832 0.3603 0.3506 0.3550 0.3732 0.4035 0.4437
0.4732 0.4261 0.3901 0.3687 0.3643 0.3769 0.4043 0.4437
0.5599 0.4854 0.4227 0.3823 0.3697 0.3786 0.4049 0.4437
Results and discussion
In this section, representative numerical results are displayed with the help of graphical illustrations. Computations were carried out for various values of physical parameters such as the wedge angle parameter λ, buoyancy ratio N r, the Brownian motion parameter N b, thermophoresis parameter N t, surface temperature parameter H, radiation-conduction parameter Rd, the Lewis number Le and the mixed convection parameter χ. Figures 6.12a–6.12c show representatively the velocity f , the temperature θ, and the nanoparticles volume fraction φ profiles for different values of the wedge angle parameter λ. It is clear that the fluid velocity, the temperature and the volume fraction decrease while the negative values of their wall slopes increase as λ increases. This has the enhancing effect on both heat and mass transfer.
Fig. 6.12a Effect of λ on the velocity profiles
In Figures 6.13a–6.13b, we present, respectively, the effects of the wedge
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Fig. 6.12b Effect of λ on the temperature profiles
Fig. 6.12c Effect of λ on the volume fraction profiles
angle parameter λ on the local Nusselt number −(1 + (4RdH 3 /3))θ (χ, 0) and on the local Sherwood number −φ (χ, 0) in the entire range of the mixed convection parameter 0 χ 1. From these figures, we see that an increase in the wedge angle parameter λ causes enhancements in both the heat and mass transfer and, as a result, in the local Nusselt and Sherwood numbers. This phenomenon is true for the entire range of 0 < χ < 1. However, it is noticed that the effect of λ on the local Nusselt and the Sherwood numbers is almost negligible. Figures 6.14a–6.14c show the effect of the surface temperature parameter H on the velocity f , temperature θ, and the nanoparticles volume fraction φ. It is observed that the temperature field increases with an increase in the surface temperature parameter H. This is due to the fact that as the value of H increases, radiation absorption in the boundary layer increases, causing the temperature to increase. In addition, as H increases, both the fluid velocity and the nanoparticles volume fraction increase.
6.3 Radiation effects on mixed convection over a wedge embedded in a porous ...
Fig. 6.13a Effect of λ on the local Nusselt number
Fig. 6.13b Effect of λ on the local Sherwood number
Fig. 6.14a Effect of H on the velocity profiles
325
326
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Fig. 6.14b Effect of H on the temperature profiles
Fig. 6.14c Effect of H on the volume fraction profiles
Figures 6.15a and 6.15b illustrate the effect of the surface temperature parameter H on the values of the local Nusselt and the Sherwood numbers in the entire range of the mixed convection parameter 0 χ 1 respectively. It can be observed from these figures that the values of both the local Nusselt and the Sherwood numbers increase as the value of H increases. This is true in the entire range 0 < χ < 1. The effect of H is almost negligible on the local Sherwood number but more pronounced with the local Nusselt number. Figures 6.16a–6.16c show the effects of the buoyancy ratio parameter N r on the velocity f , the temperature θ, and the nanoparticles volume fraction φ, respectively. It is found that an increase in N r decreases the fluid velocity in the immediate vicinity of the wedge surface. This behavior in the velocity is accompanied by a slight increase in the fluid temperature and in the nanoparticles volume fraction as N r increases.
6.3 Radiation effects on mixed convection over a wedge embedded in a porous ...
Fig. 6.15a Effect of H on the local Nusselt number
Fig. 6.15b Effect of H on the local Sherwood number
Fig. 6.16a Effect of N r on the velocity profiles
327
328
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Fig. 6.16b Effect of N r on the temperature profiles
Fig. 6.16c Effect of N r on the volume fraction profiles
Moreover, Figures 6.17a and 6.17b illustrate the changes in the local Nusselt number and the local Sherwood number for the entire range of the mixed convection parameter 0 χ 1 for various values of N r. It is observed that an increase in the buoyancy ratio enhances both the local Nusselt and the Sherwood numbers. However, for χ = 1 (forced convection limit), the flow is uncoupled from the thermal and volume fraction buoyancy effects, and hence there is no change in the local Nusselt and the Sherwood numbers for all values of N r. From the definition of χ it is seen that an increase in the value of the parameter Rax /P ex causes the mixed convection parameter χ to decrease. Thus, small values of Rax /P ex correspond to values of χ close to unity, which indicate almost pure forced convection regime. On the other hand, high values of Rax /P ex correspond to values of χ close to zero, indicating almost pure free convection regime. Furthermore, moderate values of Rax /P ex represent values of χ between 0 and 1 which correspond to the mixed convection regime. For the forced convection limit (χ = 1), it is clear
6.3 Radiation effects on mixed convection over a wedge embedded in a porous ...
329
from Eq. (6.50) that the velocity in the boundary layer is uniform. However, for smaller values of χ (higher values of Rax /P ex) at a fixed value of N r, the buoyancy effects increase. As this occurs, the fluid velocity close to the wall 0.5 due to the buoyancy effect, and becomes maximum for increases for χ < 0.5 (free convection limit). This decrease and increase in the fluid velocity f as χ decreases from unity to zero is accompanied by a respective increase and a decrease in the fluid temperature and concentration. As a result, the local Nusselt and the Sherwood numbers will be affected.
Fig. 6.17a Effect of N r on the local Nusselt number
Fig. 6.17b Effect of N r on the local Sherwood number
Figures 6.18a–6.18c present the effects of an increase in the Brownian motion parameter N b on the velocity, the temperature, and the nanoparticles volume fraction profiles, respectively. It can be seen that as the Brownian motion parameter N b increases, both the velocity and the temperature increase, especially in the region close to the wedge surface. However, we observe a slight increase in the nanoparticles volume fraction.
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Fig. 6.18a Effect of N b on the velocity profiles
Fig. 6.18b Effect of N b on the temperature profiles
Fig. 6.18c Effect of N b on the volume fraction profiles
6.3 Radiation effects on mixed convection over a wedge embedded in a porous ...
331
In Figures 6.19a and 6.19b, we present, respectively, the values of the local Nusselt number −(1 + (4RdH 3 /3))θ (χ, 0) and local Sherwood number −φ (χ, 0) for different values of the Brownian motion parameter N b in the entire range of the mixed convection parameter 0 χ 1. As observed before, an increase in the Brownian motion parameter N b increases the fluid temperature and the nanoparticles volume fraction. As a result, we see an enhancement in the local Nusselt number or local Sherwood number.
Fig. 6.19a Effect of N b on the local Nusselt number
Fig. 6.19b Effect of N b on the local Sherwood number
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Figures 6.20a–6.20c display the typical velocity, the temperature, and the nanoparticles volume fraction profiles for various values of the thermophoresis parameter N t, respectively. An increase in the thermophoresis parameter N t has the tendency to increase slightly the fluid velocity, the temperature and the nanoparticles volume fraction. Figures 6.21a–6.21b depict the influence of the thermophoresis parameter N t on N u and Sh, respectively. An increase in the thermophoresis parameter N t results in an increase in the temperature and the volume fraction. This causes the value of −(1 + (4RdH 3 /3))θ (χ, 0) to increase and −φ (χ, 0) to decrease.
Fig. 6.20a Effect of N t on the velocity profiles
Fig. 6.20b Effect of N t on the temperature profiles
6.3 Radiation effects on mixed convection over a wedge embedded in a porous ...
Fig. 6.20c Effect of N t on the volume fraction profiles
Fig. 6.21a Effect of N t on the local Nusselt number
Fig. 6.21b Effect of N t on the local Sherwood number
333
334
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Figures 6.22a–6.22c show the velocity, the temperature, and the nanoparticles volume fraction profiles for different values of the Lewis number Le, respectively. It is clearly observed that the fluid velocity and the temperature increase while the nanoparticles volume fraction and its boundary layer thickness decrease considerably as the Lewis number Le increases. This results in the enhancement of heat and mass transfer.
Fig. 6.22a Effect of Le on the velocity profiles
Fig. 6.22b Effect of Le on the temperature profiles
Figures 6.23a and 6.23b illustrate the effects of the Lewis number Le on the local Nusselt number and the local Sherwood number for the entire range of the mixed convection parameter 0 χ 1. An increase in the Lewis number Le causes the nanoparticles volume fraction to increase. As a consequence, a reduction in the local Nusselt number and an enhancement in the local Sherwood number are observed.
6.3 Radiation effects on mixed convection over a wedge embedded in a porous ...
335
Fig. 6.22c Effect of Le on the volume fraction profiles
Fig. 6.23a Effect of Le on the local Nusselt number
Figure 6.24a and 6.24b show, respectively, the effect of the radiationconduction parameter Rd on the local Nusselt number for the Newtonian fluid and the nanofluids, in the entire range of the mixed convection parameter 0 χ 1. It is found that the local Nusselt number and the Sherwood number increase with increasing Rd, for χ = 0 (pure-convection heat transfer). Since the local Nusselt number is proportional to the wall temperature gradient −(1 + (4RdH 3 /3))θ (χ, 0), the local Nusselt number is found to be more sensitive to H and Rd than −φ (χ, 0), as revealed in Eqs. (6.55) and (6.56). Moreover, for χ = 1 (forced convection limit), the flow is uncoupled from the thermal and volume fraction buoyancy effects, and hence, the local Sherwood number does not depend on Rd.
336
Chapter 6
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Fig. 6.23b Effect of Le on the local Sherwood number
Fig. 6.24a Effect of Rd on the local Nusselt number
Fig. 6.24b Effect of Rd on the local Sherwood number
6.4
MHD mixed convection flow over a permeable non-isothermal wedge
6.3.5
337
Conclusion
The non-similar solution of steady mixed convection flow of a nanofluid adjacent to an isothermal wedge embedded in a saturated porous medium in the presence of thermal radiation with Rosseland diffusion approximation is investigated. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The entire regime of mixed convection is included, as the combined convection parameter varies from 0 (pure free convection) to 1 (pure forced convection). The transformed nonlinear system of equations is solved by the Keller-box method. A comparison between the present (for some special cases) and the previously published results are found to be in very good agreement. The numerical results are presented for the local Nusselt and the Sherwood numbers with several sets of values of the buoyancy ratio, the Brownian motion parameter, the thermophoresis parameter, the wedge angle parameter, the radiation-conduction parameter, the surface temperature parameter, and the Lewis number. It was found that the local Nusselt number increases when any of the buoyancy ratio, the Brownian motion, the thermophoresis, the radiation-conduction, the surface temperature parameters, and the Lewis number increases. In addition, the local Sherwood number was increased as the buoyancy ratio, Brownian motion parameter, Lewis number, wedge angle parameter, radiation-conduction parameter, or the surface temperature parameter increase. But quite the opposite is seen as the thermophoresis parameter increases. Furthermore, both the local Nusselt and the Sherwood numbers decrease initially, reaching to a minimum for the intermediate value of the mixed convection parameter, and then increase gradually. Moreover, it is observed that the effects of the Lewis number, and the thermophoresis parameters are stronger on the local Sherwood number than that on the local Nusselt number. However, the effects of the radiation-conduction parameter and the surface temperature parameter are significantly stronger on the local Nusselt number than that on the local Sherwood number.
6.4 MHD mixed convection flow over a permeable non-isothermal wedge
6.4.1
Introduction
In recent years a great deal of interest has been generated in the study of magneto-hydrodynamic (MHD) incompressible, steady viscous flow over a non-isothermal wedge. This is due to its extensive practical applications in
338
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technological processes such as MHD power generator designs, design for cooling of nuclear reactors, construction of heat exchangers, installation of nuclear accelerators, and blood flow measurement techniques. Watanabe [96], [97] studied theoretically the characteristics of MHD boundary layer flow past a flat plate with/without pressure gradient. Watanabe and Pop (1994) extended the work of Watanabe [96] to the thermal field. Yih [99], [100] investigated the effects of viscous dissipation and the work done by stress on the heat transfer characteristics. Chandrasekar and Baskaran [80] studied the effects of transverse magnetic field, viscous dissipation, stress work, shear stress, and surface heat transfer over a non-isothermal wedge. An approximate numerical solution for thermal stratification on MHD steady laminar boundary layer flow over a wedge with suction or injection was investigated by Anjali Devi and Kandasamy [79]. Recently, Loganathan and Arasu [88] analyzed the effects of thermophoresis particle deposition on the non-Darcy mixed convective heat and mass transfer past a porous wedge in the presence of suction/blowing. The physical situation discussed by Watanabe [96] is one of the possible cases. Another physical phenomenon is the case in which the difference between the surface temperature and the free stream temperature namely, Tw − T∞ , is appreciably large. The findings of such a physical phenomenon will have a definite bearing on technological industries. The problem of mixed convection flow over a heated vertical plate is of considerable interest. There are many examples of relevant studies in the articles reported by Ali and Al-Youself [78], Chen [82], Kumari et al. [87] and Patil et al. [90]. Mixed convection heat transfer at a stretching sheet with variable temperature and linear velocity was investigated by Vajravelu [95]. Similar analyses were performed numerically by Chen and Strobel [81], and Moutsoglou and Chen [89] for fluids under different physical situations. All the above researchers restricted their analyses to hydromagnetic flow and heat transfer over a vertical plate. The role of thermal radiation on the flow and heat transfer processes is of major importance in the design of many advanced energy conversion systems. In view of this, Raptis [93] studied the thermal radiation and free convection flow through a porous medium. Chamkha et al. [83] generalized the work of Yih [99] by considering the effects of suction and thermal radiation. In these studies, the thermophysical properties of the ambient fluids are assumed to be constant. However, it is well known that these properties may change with temperature, especially the thermal conductivity. Available literature on variable thermal conductivity and thermal radiation (Chiam [84], Raptis [93], Prasad et al. [91], Chamkha et al. [83], Datti et al. [86], Abel et al. [94], and Aydın and Kaya [77]) shows that not much work has been carried out on mixed convection flow over a non-isothermal permeable wedge with variable thermal conductivity. In view of this, we analyze in this section MHD mixed convection flow over a permeable non-isothermal wedge in the presence of variable thermal conductivity (for details see K. Vajravelu, K.V. Prasad, P.S. Datti, Jour-
6.4
MHD mixed convection flow over a permeable non-isothermal wedge
339
nal of King Soud University, in press). Further, we include the effects of internal heat generation/absorption, viscous dissipation, and work done by stress. While deriving the basic equations, a temperature dependent heat source/sink term is added and the Roseland approximation for the thermal radiation term is assumed. The governing coupled, nonlinear partial differential equations for the flow and heat transfer are solved numerically by the Keller-box method.
6.4.2
Mathematical formulation
Consider a steady, two-dimensional, viscous, incompressible, mixed convection boundary layer flow over a permeable non-isothermal wedge in the presence of thermal radiation and heat generation/absorption. Let the x-axis be taken horizontally along the wedge and y-axis normal to it. A uniform magnetic field of strength B0 is applied parallel to the y-axis. Fluid suction or injection is imposed at the surface of the wedge and the surface of the wedge is maintained at a variable temperature proportional to the power of the distance along the surface from the origin as shown in Figure 6.25. The induced magnetic field is assumed to be uniform and is in the direction normal to the surface. It is also assumed that the magnetic Reynolds number is small and the electric field due to polarization of charges is negligible. All the thermophysical properties of the fluid are assumed to be constant except the density variation in the body force term and the thermal conductivity. Effects due to viscous dissipation, Joule heating, and the work due to stress are included. Under these assumptions and the usual Boussinesq approximation, the governing boundary layer equations for the conservation of mass, momentum and energy can be written as ∂u ∂v + = 0, ∂x ∂y ∂u ∂v ∂2u σB02 dU∞ u +v = ν 2 + U∞ + (U∞ − u) ∂x ∂y ∂y dx ρ ±gβ0 (T − T∞ ) sin Ω/2, ∂T Q0 (T − T∞ ) k(T ) + ∂y ρcp 2 σB02 1 ∂qr ν ∂u + (u − U∞ )2 . + + ρcp ∂y cp ∂y ρcp
(6.57)
(6.58)
∂T 1 ∂ ∂T +v = u ∂x ∂y ρcp ∂y
(6.59)
The physical boundary conditions for the problem are given by u = 0,
v = ±v0 ,
u → U∞ = Cxm ,
T = Tw (x) = T∞ + Ax2m−1 at y = 0,
T → T∞ as y → ∞,
(6.60)
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Fig. 6.25 Flow analysis along the wall of the wedge
where x and y are coordinates measured along and normal to the surface respectively. u and v are the velocity components in the x and y directions, ν is the kinematic viscosity, v0 is the suction or injection velocity, U∞ = Cxm is the free stream velocity, and m = β1 /(2 − β1 ) is the Hartree pressure gradient parameter which corresponds to β1 = Ω/π for a total angle Ω of the wedge. C is a positive number, σ is the electrical conductivity, B0 is the externally imposed magnetic field, and ρ is the density. The last term in right hand side of Eq. (6.58) represents the influence of thermal buoyancy force on the flow field, with “+” and “–” signs referring to the buoyancy assisting and buoyancy opposing flow region respectively. T is the temperature, cp is the specific heat at constant pressure, A is a positive number, Q0 is the temperaturedependent volumetric rate of heat source when Q0 > 0 and heat sink when Q0 < 0. They deal with the situation of exothermic and endothermic chemical reactions respectively. T∞ is the free stream temperature. k(T ) is the thermal conductivity. The thermal conductivity is assumed to vary as a linear function of temperature (Chiam [84]) in the form k(T ) =
ε k∞ 1+ (T − T∞ ) . ρcp ΔT
(6.61)
Here ε is a small parameter known as the variable thermal conductivity parameter and ΔT = Tw − T∞ is the surface temperature. In addition, the radiative heat flux qr is employed in accordance with the Rosseland approximation qr =
−4σ ∗ ∂T 4 , 3K ∗ ∂y
(6.62)
where σ ∗ and K ∗ are respectively the Stephan-Boltzmann constant and the mean absorption coefficient. We assume that the temperature field within the fluid is of the form T 4 and may be expanded in Taylor series about T∞ . 3 4 Neglecting the higher order terms, we obtain T 4 =4T ˜ ∞ T − 3T∞ , and using
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341
this expression for T 4 in Eq. (6.62) we get qr =
3 −16σ∗ T∞ ∂T . ∗ 3K ∂y
(6.63)
With the help of (6.62) and (6.26), Equation (6.59) can be written as u
∂T k(T ) ∂ 2 T Q0 1 ∂T ∂k(T ) ∂T +v = + + (T − T∞ ) 2 ∂x ∂y ρcp ∂y ρcp ∂y ∂y ρcp 2 3 ν ∂u σB02 ∂2T 16σ ∗ T∞ + + (u − U∞ )2 . (6.64) − 3Kρcp ∂y 2 cp ∂y ρcp
Defining the stream function in the usual way such that u = ∂ψ/∂y and v = −∂ψ/∂x and introducing the non-similar dimensionless variables (Chamkha et al. [83]) 1/2 σB02 x y U∞ x ξ= , η= , ρU∞ x ν (6.65) T − T∞ ψ , θ(ξ, η) = f (ξ, η) = (Tw − T∞ ) (U∞ xν)1/2
into Eqs. (6.57), (6.58) and (6.64), we get 1+m f f �� + m(1 − (f � )2 ) + ξ(1 − f � ) f ��� + 2 ∂f � ∂f − f �� ± λ sin Ω/2θ, (6.66) = (1 − m)ξ f � ∂ξ ∂ξ 1+m f θ� − (2m − 1)Prf � θ + Prβξθ (1 + εθ + N r)θ�� + εθ�2 + Pr 2 ∂θ ∂f − θ� . (6.67) +EcPr((f �� )2 + ξ(f � − 1)2 ) = Pr(1 − m)ξ f � ∂ξ ∂ξ
The parameters Pr, N r, β, Ec and λ are the Prandtl number, the thermal radiation parameter, the heat source/sink parameter, the Eckert number, and the mixed convection parameter respectively, and are defined by 3 2 16σ ∗ T∞ Q0 U∞ ν , Nr = , β= Ec = , 2, α∞ 3Kk∞ σCp B0 Cp (Tw − T∞ ) (6.68) gβ(Tw − T∞ )x3 U∞ x Grx , Gr = , Re = λ= . x x ν2 ν Re2x
Pr =
The dimensionless forms of the boundary conditions are √ 2 f0 ξ, θ(ξ, η) = 1 at η = 0, f � (ξ, η) = 0, f (ξ, η) = 1+m f � (ξ, η) → 1, θ(ξ, η) → 0 at η → ∞,
(6.69)
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where f0 = (ρ/(νσB02 ))1/2 v0 is the dimensionless suction or injection parameter; such that f0 > 0 indicates suction and f0 < 0 indicates injection or blowing. Equations (6.66) and (6.67) are self-similar for ξ = 0 and hence we generate for ξ = 0 the starting profiles (for the velocity and temperature fields) and use them for the numerical computations. In addition, it is also observed that for m = 1, the terms containing derivatives with respect to ξ vanish in Eqs. (6.66) and (6.67). The important physical parameters for the flow and heat transfer characteristics are the shearing stress and the heat flux at the surface of the wedge, and they are defined as Cf = and
N ux =
6.4.3
1 μ∂u/∂y|y=0 = 2Rex2 f �� (ξ, 0) 1 ρU∞ ν 2x
3 16σT∞ ∂T k∞ + 3K ∗ ∂y y=0 k∞ (Tw − T∞ )/x
3 16σT∞ θ� (ξ, 0). = −Rex 1 + 3K ∗ 1 2
Numerical procedure
Equations (6.66) and (6.67) are highly nonlinear, coupled partial differential equations. Exact analytical solutions are not possible for the complete set of equations subject to the boundary conditions (6.69). Hence we use an efficient implicit finite difference for the solution process. The implicit finite difference scheme discussed by Cebeci and Bradshaw [26] is chosen for this purpose because it has been proven to be more than adequate to give accurate results for coupled boundary layer equations. The coupled boundary value problem of third order in f and second order in θ are reduced to a system of five simultaneous differential equations of first order with respect to η by assuming f = f1 , f � = f2 , f �� = f3 , θ = θ1 , θ� = θ2 . Initially all first order derivatives with respect to ξ are replaced by two-point backward difference formulae of the form j+ 12
(f )i ∂f = ∂ξ
j+ 1
− (f )i−12 ; Δξ
j+ 12
(f )i
=
1 j+1 (f + fij ) and fij ≈ f (iΔξ, jΔη), 2 i
denoting an approximate value of f at the grid point (iΔξ, jΔη). To solve this system of equations, we require five initial conditions whilst we have only two initial conditions f (ξ, 0), f � (ξ, 0) on f and one initial condition θ(ξ, 0) on θ. The other two initial conditions f �� (ξ, 0) and θ� (ξ, 0) which are not prescribed; however, the values of f � (ξ, η) and θ(ξ, η) are known for η at infinity. Hence, we employ the numerical Keller-box scheme where these two boundary conditions are utilized to produce two unknown initial conditions at η = 0. To select η∞ , we begin with some initial guess value and solve
6.4
MHD mixed convection flow over a permeable non-isothermal wedge
343
the boundary value problem for a set of parameters to obtain f �� (ξ, 0) and θ� (ξ, 0). Thus we start with the initial approximation as f3 (ξ, 0) = α0 and θ2 (ξ, 0) = β0 and then let α and β be the correct values of f3 (ξ, 0) and θ2 (ξ, 0) respectively. We integrate the resulting system of five differential equations using the fourth order Runge-Kutta method and obtain the values of f3 (ξ, 0) and θ2 (ξ, 0) respectively. Finally the problem is solved numerically using the Keller-box method (Prasad et al. [91], [92]). The solution process is repeated with another larger value of η∞ until two successive values of f �� (ξ, 0) and θ� (ξ, 0) agree up to the desired decimal level signifying the limit of the boundary along η. The last value of η∞ is chosen as the appropriate value for that set of parameters. The numerical solutions are obtained in four steps as follows: • reduce equations (6.66) and (6.67) to a system of first order equations; • write the difference equations using central differences; • linearize the algebraic equations by Newton’s method, and write them in matrix-vector form; and • solve the linear system by the block tridiagonal elimination technique.
For each value of ξ, we get a set of algebraic equations. With each of the nonlinear terms evaluated at the previous iteration, the algebraic equations are solved with iteration by the well-known Thomas algorithm. This process is repeated for the next ξ value and the problem is solved line by line until the desired ξ value is reached. For the sake of brevity further details on the solution process are not presented here. It is also important to note that the computational time for each set of input parameters should be as short as possible. Since the physical domain in this problem is unbounded, whereas the computational domain has to be finite, we apply the far field boundary conditions for the pseudosimilarity variable η at the finite value denoted by ηmax . We ran our bulk of computations with ηmax = 7, which is sufficient to achieve asymptotically the far field boundary conditions for all values of the parameters considered. For numerical calculations, a uniform step size of Δη = 0.01 and Δξ = 0.01 are found to be satisfactory and the solutions are obtained with an error tolerance of 10−6 in all the cases. To assess the accuracy of the present method, comparison of the skin friction and the wall-temperature gradient between the present results and the previously published results are presented for several special cases in which the buoyancy parameter and the variable thermal conductivity parameter are neglected (see Tables 6.7 and 6.8). Table 6.7 fw = 0.0 m −0.05 0.0 0.3333 1.0
Comparison of the values of f (0, 0) for various values of m with
Present results 0.214342 0.333228 0.757158 1.232578
Yih [99] 0.213484 0.332057 0.757448 1.232588
Cebeci and Bradshaw [26] 0.21351 0.33206 0.75745 1.23259
Chamkha et al. [83] 0.213802 0.332206 0.757586 1.232710
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Table 6.8 Comparison of the values of θ � (0, 0) for various values of Pr with Ec = m = N r = ε = fw = Ω = 0.0 ξ
Pr=0.733 Pr=1.0 Watanabe Watanabe Present Chamkha Present Chamkha Yih [99] and Pop Yih [99] and Pop results et al. [83] results et al. [83] [98] [98]
0.0 0.305517 0.297526
0.29755
0.297600 0.336228 0.332057
0.33206
0.332173
0.5 0.363473 0.357022
0.35699
0.357040 0.405721 0.402864
0.40280
0.403103
1.0 0.388155 0.382588
0.38336
0.383191 0.435844 0.433607
0.43446
0.433901
1.5 0.403345 0.398264
0.39959
0.399980 0.454605 0.452634
0.45413
0.452808
2.0 0.413890 0.409168
0.41091
0.409450 0.467772 0.465987
0.46798
0.466111
6.4.4
Results and discussion
In order to analyze the physical model, numerical computations are carried out by the method described above for several sets of values of the pressure gradient parameter m, suction or injection parameter fw , the magnetic parameter ξ, the mixed convection parameter λ, the Prandtl number Pr, the thermal radiation parameter N r, heat source/sink parameter β, variable thermal conductivity parameter ε, and the Eckert number Ec. Since it is not possible to present the results here for all possible permutations and combinations of all the physical parameters, we focus our attention on the effect of new parameters on the flow and heat transfer fields. The numerical results are presented in Figures 6.26–6.37. These figures depict the changes in the wall-normal velocity and the fluid temperature. Changes in the skin friction and the wall-temperature gradient for several sets of the pertinent parameters are recorded in Tables 6.9 and 6.10. Table 6.9 Values of skin friction f �� (0, 0) for different values of the pertinent parameters when Ω = 30◦ Nr
ε
β
Pr Ec
fw
m
λ
ξ = 0.0
ξ = 0.5
ξ = 1.0
ξ = 1.5
0.5 0.1 −1.0 1.0 0.1
0.5
−0.5 0.5244012 1.1871145 1.5484974 1.8652896 0.333 0.0 0.7571389 1.3590680 1.6787713 1.9794817 0.5 0.9682134 1.5288637 1.8098274 2.0938566
0.2 0.1 −0.5 1.0 0.0
0.5
0.333 1.0
0.1
0.7993567 1.3930539 1.7054529 2.0030243 1.2584687 1.6472000 1.8966861 2.1075742
−0.5 0.0 0.333 0.5
0.1
0.8008918 0.7819469 0.9331067 1.0274590 0.8008918 1.0579530 1.2655948 1.4461085 0.8008918 1.3931195 1.7049015 2.0023298
0.5 0.1 −1.0 1.0 0.1
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345
Fig. 6.26a Wall normal velocity profiles for different values of m and ζ with λ = N r = fw = ε = 0.0, Pr = 1.0, β = 0.0, Ω = 30◦
Fig. 6.26b Wall normal velocity profiles for different values of λ and ζ with m = 0.333, N r = 0.5, fw = 0.5, ε = 0.1, Pr = 1.0, β = 1.0, Ω = 30◦ , Ec = 0.1
346
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Fig. 6.27a Wall normal velocity profiles for different values of fw and ζ with m = 0.333, N r = 0.5, λ = 0.1, ε = 0.1, Pr = 1.0, β = −1.0, Ω = 30◦ , Ec = 0.1
Fig. 6.27b Wall normal velocity profiles for different values of m and ζ with fw = 0.5, N r = 0.5, λ = 0.1, ε = 0.1, Pr = 1.0, β = −0.1, Ω = 30◦ , Ec = 0.1
6.4
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347
Fig. 6.28 Temperature profiles for different values of m and ζ with fw = 0.0, N r = 0.0, λ = 0.0, ε = 0.0, Pr = 1.0, β = 0.01, Ω = 30◦ , Ec = 0.0
Fig. 6.29 Temperature profiles for different values of fw and ζ with m = 0.33, N r = 0.0, λ = 0.1, ε = 0.1, Pr = 1.0, β = −1.0, Ω = 30◦ , Ec = 0.0
348
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Fig. 6.30 Temperature profiles for different values of λ and ζ with m = 0.33, N r = 0.5, fw = 0.5, ε = 0.1, Pr = 1.0, β = −1.0, Ω = 30◦ , Ec = 0.0
Fig. 6.31 Temperature profiles for different values of ε and ζ with m = 0.33, N r = 0.5, fw = 0.5, λ = 0.1, Pr = 1.0, β = −1.0, Ω = 30◦ , Ec = 0.0
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349
Fig. 6.32 Temperature profiles for different values of β and ζ with m = 0.33, N r = 0.5, fw = 0.5, λ = 0.1, Pr = 1.0, ε = 0.1, Ω = 30◦ , Ec = 0.0
Fig. 6.33 Temperature profiles for different values of Ec and ζ with m = 0.33, N r = 0.5, fw = 0.5, λ = 0.1, Pr = 1.0, ε = 0.1, Ω = 30◦ , β = 0.1
350
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Fig. 6.34 Temperature profiles for different values of N r and ζ with m = 0.33, Ec = 0.01, fw = 0.5, λ = 0.1, Pr = 1.0, ε = 0.1, Ω = 30◦ , β = 0.1
Fig. 6.35 Temperature profiles for different values of Pr and ζ with m = 0.33, Ec = 0.01, fw = 0.5, λ = 0.1, N r = 0.5, ε = 0.1, Ω = 30◦ , β = 0.1
6.4
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351
Fig. 6.36 Skin friction f (0, 0) Vs ζ for different values of λ and fw
Fig. 6.37 Wall temperature gradient θ (0, 0) Vs ζ for different values of N r, Pr and ε
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Table 6.10 Values of the wall temperature gradient θ� (0, 0) for different values of the pertinent parameters when Ω = 30◦ Nr ε
β
Pr Ec
fw
m
λ
ξ = 0.0
ξ = 0.5
ξ = 1.0
ξ = 1.5
−0.5 −0.2769064 −0.48430869 −0.64858717 −0.78895611 1.0 0.1 −0.5 1.0 0.1 0.5 0.333 0.0 −0.2910083 −0.47778577 −0.64210945 −0.78296512 0.5 −0.3005078 −0.47159243 −0.63560712 −0.77671635 0.0 −0.1725166 −0.21111232 −0.41617125 −0.60140407 0.333 1.0 0.1 −0.5 1.0 0.1 0.5 0.1 −0.2932055 −0.47653311 −0.64081323 −0.78173470 1.0 −0.5565393 −0.78539824 −0.85341299 −0.98960036 −0.5 −0.3236902 −0.20794937 −0.29920900 −0.35990244 0.0 0.5 0.1 −0.5 1.0 0.1 0.333 0.1 −0.3236902 −0.35920355 −0.48724070 −0.59055054 0.5 −0.3236902 −0.56201142 −0.76259756 −0.93418515 0.0 −0.3117204 −0.61486977 −0.82890773 −1.0129173 0.1 0.5 0.1 0.2 1.0 0.5 0.333 0.1 −0.3236902 −0.56201142 −0.73000848 −0.93418515 0.2 −0.3357611 −0.50909376 −0.69624943 −0.85542172 1.0 −0.3236960 −0.56200647 −0.76259184 −0.93418515 2.0 0.5 0.1 0.2 0.01 0.5 0.333 0.1 −0.4214001 −0.89351082 −1.2352148 −1.5255585 3.0 −0.4880211 −1.1992261 −1.6753783 −2.0771673 −0.5 −0.3236951 −0.56201142 −0.76259756 −0.93418515 0.0 0.5 0.1 1.0 0.1 0.5 0.333 0.1 −0.3236951 −0.42021284 −0.51066387 −0.58872575 0.5 −0.3236951 −0.25678539 −0.18132886 −0.089466 0.0 −0.3369952 −0.59147191 −0.80304921 −0.98427737 0.2 0.5 −1.0 1.0 0.1 0.5 0.333 0.1 −0.3119196 −0.53593725 −0.72678328 −0.88984710 0.4 −0.2917960 −0.49181601 −0.66615480 −0.81481564 0.0 −0.3711618 −0.71473402 −0.98029852 −1.2065980 0.5 −0.3236903 −0.56201142 −0.76259756 −0.93418515 1.0 0.1 −1.0 1.0 0.1 0.5 0.333 0.1 −0.2933673 −0.47676486 −0.64098734 −0.78186834 2.0 −0.2580753 −0.38481852 −0.50761569 −0.61371660
Figures 6.26a and 6.26b respectively illustrate the effects of the pressure gradient parameter m and the magnetic parameter ξ on the wall-normal velocity profiles f (η) for zero and non-zero values of the other pertinent parameters. It is noticed from Figure 6.26a that the velocity profiles decrease with increasing values of the pressure gradient parameter m and magnetic parameter ξ in the boundary layer. The effect of increasing values of the pressure gradient parameter m is to reduce the normal velocity and thereby reduce the boundary layer thickness, i.e., the thickness is much larger for negative values of pressure gradient parameter m than for zero or positive values of m as clearly seen in Figure 6.26b. This observation holds for all values of the magnetic parameter ξ. It is observed that the normal velocity profile decreases with an increase in the magnetic parameter. This is due to the fact that the introduction of a transverse magnetic field, normal to the flow direction, has a tendency to create a drag force, the Lorentz force, which tends to resist the flow. This behavior is seen even in the presence of other parameters as shown in Figure 6.26b. The effect of the magnetic parameter
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on the wall-normal velocity profile f � (η) for both the mixed convection parameter λ and suction/injection parameter are shown in Figures 6.27a–6.27b, respectively. The effect of increasing values of the mixed convection parameter is to reduce the velocity profile. Physically λ > 0 means heating of the fluid or cooling of the wedge surface, λ < 0 means cooling of the fluid or heating of the wedge surface, and λ = 0 means the absence of free convection currents. From Figure 6.28, we noticed that an increase in λ leads to a decrease in the wall-normal velocity f � (η). Also, an increase in the value λ leads to an increase in the temperature difference Tw − T∞ . This reduces the velocity profile due to the enhanced convection and thus decreases the velocity boundary layer thickness. The effect of the suction/injection parameter on the normal velocity profile for different values of ξ is shown in Figure 6.28. It can be seen that the suction (fw > 0) reduces the velocity boundary layer thickness whereas the blowing (fw < 0) has the opposite effect on the velocity boundary layer. These results are consistent with the physical situation. In Figures 6.28–6.35, the numerical results for the temperature θ(η) for several sets of values of the governing parameters are presented. Figure 6.28 illustrates the effect of the pressure gradient parameter m and the magnetic parameter ξ on θ(η). The effect of increasing values of the pressure gradient parameter β is to decrease the temperature θ(η). This is true even for different values of magnetic parameter. The effect of increasing values of the magnetic parameter ξ is to reduce the temperature θ(η). Of course, as explained above, the transverse magnetic field gives rise to a resistive force known as the Lorentz force. This force makes the fluid experience a resistance by increasing the friction between its layers and thus decreases its temperature. Figure 6.29 depicts the effects of the suction parameter on the temperature distribution. The thermal boundary layer becomes thicker for suction and thinner for blowing. Figure 6.30 depicts the temperature profiles for different values of λ. Increasing the values of λ results in a decrease in the thermal boundary layer thickness and an increase in the magnitude of the wall-temperature gradient, and hence produces an increase in the surface heat transfer rate. The effect of the variable thermal conductivity parameter ε on the temperature distribution θ(η) is shown in Figure 6.31. The effect of the variable thermal conductivity parameter ε is to enhance the temperature, and this behavior holds for all values of the magnetic parameter. This is due to the fact that the presence of temperature-dependent thermal conductivity results in a reduction in the magnitude of the transverse velocity by a quantity ∂k(T )/∂y, and this can be seen from energy equation. In Figure 6.32 the temperature distribution for different values of the heat source parameter are drawn. The direction of heat flow depends both on the temperature difference Tw − T∞ and the temperature gradient θ� (0). We observe that the temperature distribution is lower throughout the boundary layer for negative values of β (heat sink) and higher for positive values of β (heat source) as compared with the temperature distribution in absence of heat source/sink parameter. Physically β > 0 implies Tw > T∞ , i.e., the sup-
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ply of heat to the flow region from the surface. Physically these correspond, respectively, to recombination and dissociation within the boundary layer. Similarly, β < 0 implies Tw < T∞, i.e., the transfer of heat is from flow to the surface. This corresponds to combustion and an endothermic chemical reaction. The effect of increasing the value of the heat source/sink parameter β is to increase the temperature profile for zero and non-zero values of the magnetic parameter. The variations in the temperature profiles for various values of the Eckert number Ec are displayed in Figure 6.33. The effect of increasing values of Ec is to increase the temperature profile. This is in conformity with the fact that energy is stored in the fluid region due to frictional heating as a consequence of dissipation due to viscosity, and hence temperature increases as Ec increases. Figure 6.34 shows the effect of thermal radiation on temperature profiles in the boundary layer. It is observed that an increase in the thermal radiation parameter produces a significant increase in the thickness of the thermal boundary layer of the fluid, and as a consequence the temperature profiles increase. The temperature gradient at the surface increases as the thermal radiation parameter increase which can be observed in Table 6.10. Figure 6.35 exhibits the temperature distribution θ(η) for different values of the Prandtl number. The figure demonstrates that an increase in the Prandtl number Pr is to decrease the temperature distribution. That is, the thermal boundary layer thickness decreases as Pr increases. Numerical results for the skin friction coefficient f �� (0, 0), and the Nusselt number θ � (0, 0) as a function of fw and λ for a wide range of magnetic parameter ξ are shown in Figures 6.36 and 6.37 respectively. Figure 6.36 shows that for different values of λ the values of f �� (0, 0) are positive and increase as the parameters increase, namely, the mixed convection parameter, the suction/injection parameter and the magnetic parameter. From Figure 6.37, it is noticed that the effect of the variable thermal conductivity parameter and thermal radiation parameter is to increase the wall-temperature gradient, whereas the reverse is true with the Prandtl number and the Eckert number. The impact of all the physical parameters on the local Nusselt number θ� (0, 0) may be analyzed from the Table 6.10. It is of interest to note that the local Nusselt number monotonically decreases as the pressure gradient parameter increases. Further, the effect of the heat source/sink parameter or the variable thermal conductivity parameter is to enhance the wall-temperature gradient.
6.4.5
Concluding remarks
Based on the numerical results, some of the interesting results are as follows: • The effect of suction is to reduce the thermal boundary layer thickness. This holds for all values of the magnetic parameter, the variable thermal conductivity parameter, and the Eckert number.
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• The effect of increasing values of the mixed convection parameter is to increase the momentum boundary layer thickness as well as the thermal boundary layer thickness. • The effect of the Prandtl number is to decrease the thermal boundary layer thickness and the wall-temperature gradient. • The effects of the variable thermal conductivity parameter, the thermal radiation parameter, the Eckert number, and the heat source/sink parameter are to enhance the temperature field. • Of all the parameters, the mixed convection parameter has the strongest effect on the drag, heat transfer rate, the wall-normal velocity, and the temperature field of the MHD flow over a permeable non-isothermal wedge.
6.5 Mixed convection boundary layer flow about a solid sphere with Newtonian heating 6.5.1
Introduction
The analysis of heat transfer through a laminar boundary layer in the free, forced, and mixed convection flow over a body of arbitrary shape and arbitrarily specified surface temperature or surface heat flux, constitutes a very important problem in the field of heat transfer and has received extensive attention. The prediction of heat transfer under such conditions encompasses a wide range of technological applications, such as the cooling problems in turbine blades or electronic systems, the calculation of heat transfer from bodies moving through the atmosphere, manufacturing processes, process industries, etc. (see Yaho [101]). To the best of our knowledge, the only such studies which have been reported are the pioneering experimental work of Yuge [102] and the analytical work of Hieber and Gebhart [103]. These studies, both experimental and analytical, were conducted under the action of very small Reynolds and Grashof numbers. Chen and Mucoglu [104] have later studied mixed convection over a sphere with uniform surface temperature and uniform surface heat flux for very large Reynolds Re and Grashof numbers Gr, using the boundary layer approximations. The solution depends on the nondimensional mixed convection parameter λ = Gr/Re2 . The Prandtl number considered is 0.7. Later, the mixed convection boundary layer flow about a solid sphere has been considered by many investigators in various ways. Wong et al. [106] solved the full Navier-Stokes and energy equation of an isothermal sphere in combined convection by a finite element method. Minkowycz et al. [107] considered the mixed convection about a non-isothermal cylinder and sphere in a porous medium and mixed convection boundary layer flow about
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a solid sphere. Later, Kumari and Nath [108] studied the unsteady mixed convection with double diffusion over a horizontal cylinder and sphere within a porous medium. In 2002, Antar and El-Shaarawi [109] studied the mixed convection around a liquid sphere in an air stream, in which they considered both aiding and opposing natural convection and the effect of the controlling parameters on engineering quantities such as the shear stress and the Nusselt number. Nazar et al. [110], [111], [112] studied the mixed convection boundary layer flow about a solid sphere with constant surface temperature and constant heat flux in viscous and micropolar fluids respectively. They carried out a very similar study to that of Chen and Mucoglu [104] for two values of the Prandtl number Pr = 0.7 and 7. Quite recently, Yacob and Nazar [113] considered the mixed convection boundary layer on a solid sphere with constant surface heat flux, followed by Kotouˇc et al. [114] who studied the loss of axisymmetry in the mixed convection (assisting flow) past a heated sphere. We mention also the relatively recent papers on this problem by Jenny and Duˇsek [115], Jenny et al. [116], Mograbi and Bar-Ziv [117], [118] and Mebarek et al. [119]. A detailed list of references on convective heat transfer problems can also be found in the recent book by Pop and Ingham [120]. In general, there are three common heating processes representing the constant wall temperature (CWT), constant heat flux (CHF), and conjugate conditions, where the heat transfer through a bounding surface of finite thickness and finite heat capacity is specified. The interface temperature is not known a priori but depends on the intrinsic properties of the system, namely, the thermal conductivities of the fluid and solid. In Newtonian heating (NH), the rate of heat transfer from the bounding surface with a finite heat capacity is proportional to the local surface temperature, and it is usually termed the conjugate convective flow. The Newtonian heating conditions have been used only recently by Merkin [121], Lesnic et al. [122], [123], [124] and Pop et al. [125], to study the free convection boundary layer over vertical and horizontal surfaces as well as over a small inclined flat plate from the horizontal surface embedded in a porous medium. The asymptotic solution near the leading edge and the full numerical solution along the whole plate domain have been obtained numerically, whilst the asymptotic solution far downstream along the plate has been obtained analytically. Chaudhary and Jain [126], [127] studied the unsteady free convection boundary layer flow past an impulsively started, vertical infinite flat plate with Newtonian heating. Recently, Salleh and Nazar [128], Salleh et al. [129], [130] employed the Keller-box method to obtain numerical solutions for the free convection boundary layer flow over a horizontal circular cylinder and sphere with Newtonian heating and forced convection boundary layer flow at a forward stagnation point with Newtonian heating respectively. Therefore, the aim of the present section is to study the mixed convection boundary layer flow about a solid sphere with Newtonian heating (see for details M.Z. Salleh, R. Nazar, I.Pop, Arch. Mech. 62 , 2010, 283–303). The governing boundary layer equations are first transformed into a system of
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non-dimensional equations via the non-dimensional variables, and then into non-similar equations before they are solved numerically by the Keller-box method, as described in the books by Na [29] and Cebeci and Bradshaw [26]. To the best of our knowledge, this problem has not been considered before for the case of Newtonian heating so that the reported results are new.
6.5.2
Mathematical formulation
Here we consider the problem of steady mixed convection boundary layer flow about a solid sphere for the case of Newtonian heating, where the heat transfer rate from the bounding surface with a finite heat capacity is proportional to the local surface temperature. As it was first modelled by Merkin [121], ∂T = −hs Tw (6.70) ∂ y¯ y¯=0 where Tw is the unknown local surface temperature and hs is a coefficient of proportionality for the surface heat flux. This configuration can arise in many important engineering devices. (See [131], [132] for example.) Alternatively, this set-up can model the heat transfer when there is a weak exothermic catalytic reaction taking place on the surface, generating heat at a rate proportional to the surface temperature. This is a reasonable assumption when the difference between the surface temperatures arising from the reaction and the ambient temperature are small, the situation envisaged in this section. Other situations can occur in heat exchanger systems where the conduction in solid tube wall is greatly influenced by the convection in the fluid flowing over it; in conjugate heat transfer around fins where the conduction within the fin and the convection in the fluid surrounding it must be simultaneously analyzed in order to obtain vital design information; and in a convective flow set-up when the bounding surfaces absorbs heat by solar radiation (Lesnic et al. [124]). The convective forced flow is assumed to be moving upwards, while the gravity vector g acts downwards in the opposite direction as shown in Figure 6.38, where the coordinates x¯ and y¯ are chosen such that x ¯ measures the distance along the surface of the sphere from the lower stagnation point and y¯ measures the distance normal to the surface of the sphere. Under the Boussinesq and boundary layer approximations, the basic equations are ∂(¯ ru ¯) ∂(¯ r v¯) + = 0, ∂x ¯ ∂ y¯ x ¯ d¯ ue ¯ ∂u ¯ ∂2u ∂u ¯ + v¯ =u ¯e + v 2 + gβ(T − T∞ ) sin , u ¯ ∂x ¯ ∂ y¯ d¯ x ∂ y¯ a ¯ ∂T ∂T ∂2T u ¯ + v¯ =α 2 ∂x ¯ ∂ y¯ ∂ y¯
(6.71) (6.72) (6.73)
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Fig. 6.38 Physical model and coordinate system
subject to the boundary conditions u ¯ = v¯ = 0, u ¯ → 0,
∂T /∂ y¯ = −hs T at y¯ = 0,
T → T∞ as y¯ → ∞
(6.74)
where u ¯ and v¯ are the velocity components along the x ¯ and y¯ directions, respectively, T is the local temperature, T∞ is the temperature of the ambient fluid, g is the gravity acceleration, α = γ/Pr is the thermal diffusivity, β is the thermal expansion coefficient, ν = μ/ρ is the kinematic viscosity, μ is the dynamic viscosity, ρ is the density and Pr is the Prandtl number. Let r¯(¯ x) be the radial distance from the symmetrical axis to the surface of the sphere x) be the local free stream velocity, which are given by and u ¯e (¯ r¯(¯ x) = a sin(¯ x/a) =
3 x/a). U∞ sin(¯ 2
(6.75)
However, for the sake of comparison, we shall also consider the classical cases of constant wall temperature (CWT), T = Tw and constant surface heat flux (CHF), ∂T /∂y = −qw /k at y = 0, where Tw is the constant wall temperature, qw is the constant heat flux from the wall and k is the thermal conductivity. We introduce now the following non-dimensional variables: x = x¯/a,
y = Re1/2 y¯/a,
¯e (¯ x)/U∞ , ue (x) = u
r(x) = r¯(¯ x)/a,
v = Re
1/2
U∞ ,
u=u ¯/U∞ ,
θ = T − T∞ /T∞
(6.76)
where Re = U∞ a/v is the Reynolds number and we use θ = T − T∞ /T∞ (for CWT) and θ = (k/(aqw ))Re1/2 (T − T∞ ) (for CHF). Substituting variables (6.76) into Eqs. (6.71)–(6.73), they become ∂(ru) ∂(rv) + = 0, ∂x ∂y due ∂u ∂u ∂2u u +v = ue + v 2 + λθ sin x, ∂x ∂y dx ∂y ∂θ 1 ∂2θ ∂θ +v = , u ∂x ∂y Pr ∂y 2
(6.77) (6.78) (6.79)
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and the boundary conditions (6.74) become (see [133]) u = v = 0, ∂θ/∂y = −γ(1 + θ)(NH) at y = 0, 3 ue (x) → sin x, θ → 0 as y → ∞, 2
(6.80)
where γ = ahs Re−1/2 represents the conjugate parameter for Newtonian heating. We have noticed that (6.80) gives θ = 0 when γ = 0, corresponding to having hs = 0 and hence no heating from the sphere exists. On the other hand, λ is the mixed convection parameter which is given by λ = Gr/Re2 (NH, CWT) or λ = Gr/Re5/2
(6.81)
and Gr is the Grashof number which is given by Gr = gβT∞ a3 /ν 2 (NH) or Gr = gβ(Tw − T∞ )a3 /ν 2 (CWT) or
Gr = gβ(aqw /k)a3 /ν 2 (CHF).
(6.82)
It is worth mentioning that in both cases of CWT and CHF, λ > 0 is for the aiding or assisting flow (heated sphere) and λ < 0 is for the opposing flow (cooled sphere), while for the present case of NH, the value of λ considered is only for λ > 0. For very small |λ|, forced convection effects dominate, while for large |λ| it is the natural or free convection which is important. To solve Eqs. (6.77)–(6.79), subject to the boundary conditions (6.80), we assume the following variables: ψ = xr(x)f (x, y),
θ = θ(x, y)
(6.83)
1 ∂ψ . r ∂x
(6.84)
where ψ is the stream function defined as u=
1 ∂ψ , r ∂y
v=−
So that Eqs. (6.78) and (6.79) then become ∂ 2 f ∂f 2 9 sin x cos x ∂ 3f x sin x cos x f 2 − θ+ + 1+ +λ 3 ∂y sin x ∂y ∂y x 4 x ∂f ∂ 2 f ∂f ∂ 2 f =x , (6.85) − ∂y ∂x∂y ∂x ∂y 2 ∂θ ∂f ∂θ x ∂f ∂θ 1 ∂2θ + 1 + cos x f = − (6.86) Pr ∂y 2 sin x ∂y ∂y ∂x ∂x ∂y subject to the boundary conditions
∂θ ∂f = 0, = −γ(1 + θ)(NH) at y = 0, ∂y ∂y ∂f 3 sin x → , θ → 0 as y → ∞ ∂y 2 x
f=
(6.87)
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along with θ(0) = 1 (CWT) and θ� (0) = −1 (CHF). It can be seen that at the lower stagnation point of the sphere, x ≈ 0, Equations (6.86) and (6.87) reduce to the following ordinary differential equations: f ��� + 2f f �� − f �2 + λθ + 9/4 = 0, θ�� + 2Prf θ� = 0,
(6.88) (6.89)
and the boundary conditions become f (0) = f � (0) = 0, f � → 3/2,
θ� (0) = −γ(1 + θ(0))(NH),
θ → 0 as y → ∞,
(6.90)
where primes denote differentiation with respect to y. The quantities of physical interest in this problem are the skin friction coefficient Cf , and the wall temperature θw (x), which are given by Re1/2 x Cf = x
6.5.3
∂2f (x, 0), ∂y 2
θw (x) = −1 −
∂θ (x, 0). ∂y
Solution procedure
Equations (6.85) and (6.86) subject to boundary conditions (6.87) are solved numerically using the Keller-box method as described in the books by Na [29] and Cebeci and Bradshaw [26]. The solution is obtained in the following four steps: • reduce Eqs. (6.85) and (6.86) to a first order system, • write the difference equations using central differences, • linearize the resulting algebraic equations by Newton’s method and write them in the matrix vector form, • solve the linear system by the block tridiagonal elimination technique (see Salleh et al. [133], [134] and Ishak et al. [135] for the details of this method).
6.5.4
Results and discussion
Equations (6.85) and (6.86) subject to the boundary conditions (6.87) were solved numerically using the Keller-box scheme for the cases of CWT, CHF, and NH with several parameters considered, namely, the mixed convection parameter λ, the Prandtl number Pr, the conjugate parameter γ, and the coordinate running along the surface of the sphere, x. The numerical solutions start at the lower stagnation point of the sphere, x ≈ 0, with initial profiles as given by Eqs. (6.88) and (6.89), and proceed round the sphere up to 120◦ (see Nazar et al. [110], [111]). Values of Pr considered are Pr = 0.7, 1.0 and 7.0.
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It is worth mentioning that small values of Pr(� 1) physically correspond to liquid metals, which have high thermal conductivity but low viscosity, while large values of Pr(� 1) correspond to high-viscosity oils. It is also worthwhile to point out that, the Prandtl numbers considered in this study, namely Pr = 0.7, 1.0 and 7.0, correspond to air, electrolyte solution, and water respectively. At the lower stagnation point of the sphere, x ≈ 0, due to the decoupled boundary layer equations (6.88) and (6.89), when the mixed convection parameter λ = 0 (forced convection), there is a unique value of the reduced skin friction coefficient, f (0) = 2.4104 for all Prandtl numbers Pr, which is in good agreement with the value f (0) = 2.4104 found by Nazar et al. [110], [111] by using the Keller-box method as well as the series solutions. The values of f (0), −θ (0) and θ(0) for the cases of CWT and CHF, are shown in Tables 6.11 and 6.12, respectively. Some numerical results obtained by an implicit finite-difference scheme as reported by Nazar et al. [110], [111] for the cases of CWT and CHF, are also included in these tables for comparison purposes. It is found that the agreement between the previously published results with the present ones is very good. We can conclude that this numerical method works efficiently for the present problem and we are also confident that the results presented here are accurate. For the case of NH, the values of f (0) and θ(0) obtained numerically by solving Eqs. (6.88) and (6.89) subject to the boundary conditions (6.90) for various values of λ when γ = 1 and Pr = 0.7, 1.0 and 7.0 are presented in Tables 6.13 and 6.14 respectively. It can be seen from these tables that the values of f (0) and θ(0) are higher for Pr = 0.7 than those for Pr = 1.0 and 7.0. Further, Tables 6.15 to 6.20 show the numerical values of Cf and θw (x) at the different positions of x for different values of λ when γ = 1 and Pr = 0.7, 1.0 and 7.0, respectively. It can be seen from these tables that the values of Cf and θw (x) are higher for Pr = 0.7 than Pr = 7.0, when the parameter x and λ are fixed. It is also seen from these tables that θw (x) decreases as the mixed convection parameter λ increases. Also, for a given value of λ, the skin friction coefficient Cf and the wall temperature θw (x) are seen to increase with increasing the distance x from the stagnation point. Further, we can see from these tables that, increasing of λ delays the separation and that separation can be completely suppressed in the range 0 x 120 for sufficiently large values of λ (> 0). The actual value of λ = λk , which first gives no separation, is difficult to determine exactly as it has to be found by successive integrations of the equations. However, the numerical solutions indicate that the value of λ which first gives no separation, lies between 0.02 and 0.04 for Pr = 0.7. It lies between 0.06 and 0.1 for Pr = 1.0, while for Pr = 7.0 the value of λ lies between 0.02 and 0.03. Figure 6.39 illustrates the variation of the wall temperature θw (x) with Prandtl number Pr when λ = 1 and γ = 1. To get a physically acceptable solution, Pr must be greater than the critical value, say Prc = Prc (γ) i.e. Pr > Pr(γ). It can be seen from this figure that θw (x) becomes large (unbounded) as Pr approaches the critical value Prc = 0.0169 when λ = 1 and γ = 1.
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Fig. 6.39 Variation of wall temperature with Prandtl number Pr when λ = 1 and γ=1
Figure 6.40 shows the variation of the wall temperature θw (x) with the conjugate parameter γ when Pr = 1 and λ = 1. Also, to get a physically acceptable solution, γ must be less than a certain critical value, say γc = γc (Pr) i.e. γ < γc (Pr). It can be seen from this figure that θw (x) becomes large (unbounded) as γ approaches the critical value γc = 3.522 when Pr = 1 and λ = 1. It should be pointed out that from the boundary conditions (6.74), we must have (∂T /∂ y¯)y¯=0 < 0, as the applied heating condition is given in terms of the physical fluid temperature T , not of a temperature difference. Therefore, we can only have physically acceptable solutions of the Eqs. (6.85) and (6.86) subject to the boundary conditions (6.87), which have (∂θ/∂ y¯)y¯=0 < 0. But we will further refer to Eqs. (6.88) and (6.89) with the boundary conditions (6.90). From these equations, in order to have θ (0) < 0 it means that we can have solutions only when γ < γc (γc is the critical value of γ), where the solutions become unbounded, for the existence of mixed convection solution with the Newtonian heating given by (6.80) or (6.90). This is shown in Figures 6.39 and 6.40. The velocity and temperature profiles near the lower stagnation point x ≈ 0 are given in Figures 6.41– 6.43 for some values of λ when Pr = 0.7 and γ = 1. We found that for fixed values of Pr and γ, the velocity profiles increase, while the temperature profiles decrease when the mixed convection parameter λ increases. From Figure 6.41 it is noticed that there are overshoots of the velocity profiles when λ 1 where these overshoots take place higher for λ = 10 than for λ = 1. The velocity and temperature profiles near the lower stagnation point of the sphere, x ≈ 0, for some values of Pr when λ = 1.0 and γ = 1 are plotted in Figures 6.44 and 6.45. It can be seen from these figures that, as Pr increases, both the velocity and temperature profiles decrease. At large Pr, the thermal boundary layer is thinner than at a smaller Pr. This is because for small values of Pr (� 1), the fluid is highly conductive. Physically, if Pr increases, the thermal diffusivity decreases and these phenomena lead to
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decreasing of the energy ability that reduces the thermal boundary layer. From Figure 6.44 it is also noticed that there are overshoots of the velocity profiles when Pr 1 where these overshoots take place higher for Pr = 0.7 than for Pr = 1.0. Further, Figures 6.46 and 6.47 illustrate the variation of θw (x) at different positions x for different values of λ, when γ = 1.0 and Pr = 0.7 and 1.0, respectively. It can be seen from these figures that the values of θw (x) are higher for Pr = 0.7 than for Pr = 1.0, when the parameters x and λ are fixed. It is also found that θw (x) decreases as the mixed convection parameter increases and θw (x) is seen to increase with increasing of the distance x from the stagnation point of the sphere, x ≈ 0.
Fig. 6.40 Variation of the wall temperature with conjugate parameter γ when λ=1
Fig. 6.41 Velocity profile near the lower stagnation point of sphere, x ≈ 0, for various values of λ when Pr = 0.7 and γ = 1
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Fig. 6.42 Temperature profile near the lower stagnation point of sphere, x ≈ 0, for various values of λ(< 1) when Pr = 0.7 and γ = 1
Fig. 6.43 Temperature profile near the lower stagnation point of sphere, x ≈ 0, for various values of λ( 1) when Pr = 0.7 and γ = 1
Fig. 6.44 Velocity profile near the lower stagnation point of sphere, x ≈ 0, for various values of Pr when λ = 1.0 and γ = 1
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Fig. 6.45 Temperature profile near the lower stagnation point of sphere, x ≈ 0, for various values of Pr when λ = 1.0 and γ = 1
Fig. 6.46 Variation of wall temperature θw (x) with x for various values of λ when Pr = 1.0 and γ = 1
Fig. 6.47 Variation of wall temperature θw (x) with x for various values of λ when Pr = 1.0 and γ = 1
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Conclusions
In this section, we have numerically studied the problem of mixed convection boundary layer flow of a solid sphere with Newtonian heating (NH). It is shown how the mixed convection parameter λ, the Prandtl number Pr and the conjugate parameter γ affect the skin friction coefficient, the wall temperature and the velocity and temperature profiles. We can conclude that (for the case of NH): • an increase of the value of Pr leads to a decrease of both the velocity and temperature profiles; • near the lower stagnation point of the sphere, when λ increases, the velocity profiles increase but the temperature profiles decrease; • there are overshoots of the velocity profiles near the lower stagnation point of the sphere from the free stream velocity; • an increase of the value of Pr and λ leads to a decrease of the wall temperature θw (x); • to get a physically acceptable solution, Pr must be greater than Prc (critical value of Pr) depending on γ, and also γ must be less than γc (critical value of γ) depending on Pr. Table 6.11 Values of f (0) and −θ (0) for various values of λ when Pr = 0.7 (CWT) λ −4.6 −4.5 −4.0 −3.0 −2.0 −1.0 −0.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0
f (0) Nazar et al. [110] 0.0770 0.1566 0.5028 1.0700 1.5581 2.0016 2.2115 2.4151 2.8064 3.1804 3.5401 3.8880 4.2257 4.5546 4.8756 5.1896 5.4974 5.7995 8.5876
Present 0.0699 0.1544 0.4996 1.0664 1.5542 1.9973 2.2070 2.4104 2.8012 3.1745 3.5336 3.8807 4.2177 4.5457 4.8659 5.1791 5.4859 5.7870 8.5647
−θ (0) Nazar et al. [110] 0.6011 0.6117 0.6534 0.7108 0.7529 0.7870 0.8021 0.8162 0.8463 0.8648 0.8857 0.9050 0.9230 0.9397 0.9555 0.9704 0.9846 0.9981 1.1077
Present 0.5990 0.6115 0.6528 0.7099 0.7519 0.7860 0.8010 0.8150 0.8406 0.8636 0.8845 0.9038 0.9217 0.9385 0.9542 0.9691 0.9833 0.9967 1.1061
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Table 6.12 Values of f (0) and −θ(0) for various values of λ when Pr = 0.7 (CHF) λ −2.8 −1.5 −1.0 −0.5 0.0 0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0
f (0) Nazar et al. [111] 0.0791 1.5620 1.8785 2.1592 2.4151 2.6526 2.8756 3.2881 —— —— 4.3515 —— —— —— —— 5.8046 8.1431
Present
θ(0) Nazar et al. [111]
Present
0.0669 1.5560 1.8731 2.1541 2.4104 2.6478 2.8707 3.2830 3.6613 4.0136 4.3451 4.6602 4.9606 5.249 5.5272 5.7954 8.1273
1.6504 1.3277 1.2856 1.2525 1.2252 1.2020 1.1818 1.1479 —— —— 1.0759 —— —— —— —— 1.0017 0.9160
1.6567 1.3302 1.2878 1.2545 1.2270 1.2038 1.1834 1.1494 1.1214 1.0977 1.0773 1.0591 1.0430 1.0284 1.0151 1.0029 0.9171
Table 6.13 Values of f (0) and θ(0) for various values of λ when Pr = 0.7, 1.0 and γ = 1 (NH) Pr = 0.7
Pr = 1.0
λ
f (0)
θ(0)
f (0)
θ(0)
0.01 0.02 0.03 0.04 0.05 0.1 0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0
5.8834 5.9061 5.9269 5.9491 5.9694 6.0716 6.7525 7.4152 8.4538 9.2945 10.0216 10.6723 11.2669 11.8173 12.3355 12.8236 13.2876 17.0966
1032.4011 520.1127 349.0227 263.6847 212.2873 109.6156 26.6476 15.7022 9.7883 7.6114 6.4336 5.6782 5.1456 4.7431 4.4287 4.1733 3.9609 2.8658
3.3748 3.4225 3.4683 3.5101 3.5485 3.7186 4.3491 5.2161 6.1773 6.9258 7.5625 8.1273 8.6407 9.1151 9.5584 9.9763 10.3730 13.6177
274.6189 144.4513 100.9444 78.7904 65.4268 37.9501 28.9103 8.7364 6.0929 5.0014 4.3714 3.9495 3.6419 3.4047 3.2147 3.0580 2.9258 2.2136
368
Chapter 6
Application of the Keller-box Method to More Advanced Problems
Table 6.14 Values of f �� (0) and θ(0) for various values of λ when Pr = 7.0 and γ = 1 (NH) Pr = 7.0
λ
f �� (0)
θ(0)
0.05 0.1 0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0
2.4235 2.4365 2.5378 2.6579 2.8808 3.0854 3.2760 3.4552 3.6249 3.7867 3.9415 4.0904 4.2340 5.4682
1.1174 1.1141 1.0933 1.0709 1.0326 1.0009 0.9739 0.9503 0.9296 0.9110 0.8943 0.8790 0.8651 0.7679
Table 6.15 Values of Cf at the different positions x for various values of λ when Pr = 0.7 and γ = 1 λ xs 0◦ 10◦ 20◦ 30◦ 40◦ 50◦ 60◦ 70◦ 80◦ 90◦ 100◦ 110◦ 120◦
0.01
0.02
0.0000 2.4247 5.1300 7.7607 10.4076 12.8475 15.2483 17.3454 19.2011
0.0000 2.4143 5.1086 7.7283 10.3645 12.7946 15.0594 17.2749 19.1229 20.8530 22.0944
Cf 0.04
1
7
0.0000 2.4192 5.1168 7.7395 10.3784 12.8110 15.0779 17.2954 19.1451 20.8766 22.1192 23.1678 23.5698
0.0000 2.5549 5.3448 8.0487 10.7600 13.2593 15.5847 17.8562 19.7677 21.5858 22.9133 24.0142 24.4255
0.0000 3.0281 6.2020 9.2646 12.3109 15.1916 18.0592 20.8610 23.1955 25.2947 26.7892 27.9605 28.3528
Table 6.16 Values of θw (x) at the different positions x for various values of λ when Pr = 0.7 and γ = 1 λ xs
0.01
0.02
0◦ 10◦ 20◦
1032.4011 4593.8650 4932.8780
520.1127 2279.8779 2448.5298
θw (x) 0.04 263.6847 1143.2143 1227.1952
1
7
15.7022 49.4303 52.3580
4.7431 9.1384 9.4377
6.5
Mixed convection boundary layer flow about a solid sphere with ...
369
Continued λ
θw (x)
xs
0.01
0.02
0.04
1
7
30◦
5135.8530
2549.5688
1277.5526
54.1713
9.6484
40◦
5309.2326
2635.8811
1320.6215
55.7855
9.8618
50◦
5484.3107
2723.0335
1364.5369
57.4712
10.0882
60◦
5675.1214
2815.1750
1410.2106
59.2949
10.2630
70◦
5885.0581
2922.5078
1463.8869
61.4519
10.4711
80◦
6134.9920
3046.9182
1526.1240
63.9607
10.7660
90◦
3193.0798
1599.2579
66.8564
11.1646
100◦
3367.0696
1686.3283
70.2466
11.6790
110◦
1791.3590
74.3284
12.3286
120◦
1920.3443
79.3798
13.1477
Table 6.17 Values of Cf at the different positions x for various values of λ when Pr = 1.0 and γ = 1 λ xs 0◦ 10◦ 20◦ 30◦ 40◦ 50◦ 60◦ 70◦ 80◦ 90◦ 100◦ 110◦ 120◦
0.01
0.03
0.0000 1.4834 3.2056 4.8554 6.5608 8.0932 9.5454 10.9992
0.0000 1.4934 3.2216 4.8765 6.5861 8.1223 9.5777 11.0342 12.1980 13.3567
Cf 0.06 0.0000 1.5062 3.2422 4.9038 6.6190 8.1603 9.6200 11.0803 12.3642 13.4085 14.1970 14.9453
0.1
2
0.0000 1.5214 3.2667 4.9363 6.6584 8.2059 9.6709 11.1359 12.3070 13.4714 14.2624 15.0122 15.2655
0.0000 1.7821 3.7124 5.5491 7.4212 9.1097 10.7082 12.3647 13.7480 15.0774 15.9685 16.7488 16.9774
Table 6.18 Values of θw (x) at the different positions x for various values of λ when Pr = 1.0 and γ = 1 λ xs
0.01
0.03
0◦ 10◦ 20◦ 30◦ 40◦ 50◦ 60◦
2742.3120 2556.4810 2793.4136 2931.8003 3045.9184 3158.1121 3274.3028
100.8235 859.8126 937.5609 983.1001 1020.7964 1057.9828 1096.5915
θw (x) 0.06 56.4181 434.8734 472.9940 495.4030 514.0432 532.5109 551.7461
0.1
2
37.9501 264.4599 286.8403 300.0423 311.0866 322.0844 333.5819
6.0929 16.6862 17.4648 17.9706 18.4440 18.9597 19.5281
370
Chapter 6
Application of the Keller-box Method to More Advanced Problems
Continued λ
θw (x)
xs
0.01
0.03
0.06
0.1
2
70◦
3407.9159
1141.0687
573.9547
346.8912
20.1585
80◦
1192.2899
600.3492
362.2686
20.8382
90◦
1252.1177
629.5182
380.2685
21.6373
100◦
664.9468
401.5768
22.6267
110◦
707.3675
120◦
427.1015
23.8709
458.3454
25.4522
Table 6.19 Values of Cf at the different positions x for various values of λ when Pr = 7.0 and γ = 1 λ xs 0◦ 10◦ 20◦ 30◦ 40◦ 50◦ 60◦ 70◦ 80◦ 90◦ 100◦ 110◦ 120◦
0.01
0.02
0.0000
0.0000 0.1119 0.3326 0.3945 0.6322 0.6543 0.8914 1.0050
Cf 0.03
0.1
3
0.0000 0.1139 0.3361 0.3992 0.6337 0.6572 0.7710 1.0096 0.9784 1.2065 1.1202 1.3184 1.1603
0.0000 0.1352 0.3614 0.4319 0.6718 0.7091 0.8397 1.0922 1.0762 1.3108 1.2352 1.4360 1.2842
0.0000 0.4705 1.0377 1.4934 2.0614 2.4761 2.9584 3.5252 3.9618 4.3058 4.4679 4.7778 4.7417
Table 6.20 Values of θw (x) at the different positions x for various values of λ when Pr = 7.0 and γ = 1 λ xs 0◦ 10◦ 20◦ 30◦ 40◦ 50◦ 60◦ 70◦ 80◦ 90◦ 100◦ 110◦ 120◦
0.01
0.02
1.1118
1.1219 43.1846 46.2051 48.8039 50.2278 51.6119 53.1400 55.0546
θw (x) 0.03
0.1
3
1.1193 30.1700 32.2326 33.7935 34.3635 35.2579 36.3028 37.5999 39.2765 41.2613 43.8152 46.9797 51.1374
1.1139 10.5022 10.2346 10.1297 10.1304 10.2852 10.4855 10.7531 11.1334 11.6064 12.2376 13.0352 14.0961
2.1872 8.8129 9.2373 9.5529 9.7932 10.0785 10.3337 10.6022 10.9429 11.3547 11.8991 12.5949 13.4929
6.6
Flow and heat transfer of a viscoelastic fluid over a flat plate with ...
371
6.6 Flow and heat transfer of a viscoelastic fluid over a flat plate with a magnetic field and a pressure gradient
6.6.1
Introduction
In recent years the non-Newtonian fluids in the presence as well as in the absence of a magnetic field find increasing applications in industry and technology. A few examples are the flow of nuclear fuel slurries, flow of liquid metals and alloys such as the flow of gallium at ordinary temperature (30◦ C), flow of plasma, flow of a mercury and amalgams, flow of blood, a Bingham fluid with some thixotropic behavior, coating paper, and plastic extrusion and lubrication with heavy oils and greases [136]. Another important field of application is electromagnetic propulsion (see Sarpakaya [136]). During the past two decades there have been several studies on such fluids and the reviews are presented in Skelland [137], Bird et al. [138] and Tanner [139]. The non-Newtonian fluids characterized by a power-law model have some limitations as they do not exhibit any elastic properties such as normal stress differences in shear flow. The viscoelastic fluid is one of the many models that has been proposed to describe the non-Newtonian behaviour of such fluids. This model was first proposed by Rivlin and Ericksen [140]. A detailed discussion on the history of this model, and some of the controversy that surrounds it, are presented by Dunn and Fosdick [141]. These issues are not discussed here. In the present study, it is assumed that this model models the fluid exactly. For viscoelastic fluid, the equations of motion are found to be one order higher than the Navier-Stoke or boundary layer equations and the boundary conditions are not sufficient to determine the solution completely. The mathematical problem reduces to a singular perturbation problem. In order to overcome this difficulty, Beard and Walters [142] have used a regular perturbation parameter which occurs as a coefficient of the highest derivative. This reduces the order of the equation, but it treats the singular perturbation problem as a regular perturbation problem. In recent years, this difficulty has been overcome by augmenting the boundary conditions on the basis of physically reasonable assumptions. For unbounded domain, it is assumed that the solution is bounded or has been discussed in detail by Rajagopal [143], [144] and Rajagopal and Kaloni [145]. They showed that it is unnecessary to make the assumption a priori, rather the extra boundary condition is a natural consequence of the differential equations and the boundary conditions. Much work has been done on the generalization of well known viscous flow solutions of Newtonian fluids to take into account the effect of a magnetic field when the fluid is electrically conducting. An important class of such solutions
372
Chapter 6
Application of the Keller-box Method to More Advanced Problems
deals with the flow in the boundary layers both viscous and magnetic. The steady incompressible and two-dimensional flow of an electrically conducting fluid past a semi-infinite plate with or without a magnetodynamic pressure gradient has been studied by Greenspan and Carrier [146], Glauert [147], Davies [148], Gribben [149], Tan and Wang [150], Na [151], and Nath [152]. The flow and heat transfer problem of Newtonian fluids over a wedge and the flow problem of a viscoelastic fluid over a wedge have been studied by Watanabe [153] and Garg and Rajagopal [154] respectively. The problem investigated here in this section is the flow and heat transfer of a viscous, electrically conducting viscoelastic fluid past a fixed, semiinfinite, unmagnetized but conducting plate with a magnetic field and a magnetodynamic pressure gradient (see for details M. Kumari, H.S. Takhar, G. Nath, Ind. J. Pure and Applied math. 28 , 1997, 109–121). The boundary layer equations governing the flow are solved numerically. Particular cases of the present results have been compared with those of Glauert [147], Tan and Wang [150], Na [151], Nath [152], Watanabe [153], and Garg and Rajagopal [154].
6.6.2
Governing equations
Let us consider the steady laminar incompressible viscous electrically conducting fluid flow with constant properties over a fixed, semi-finite, unmagnetized but conducting flat plate with a magnetic field and a magnetodynamic pressure gradient. The fluid flow U0 and the applied magnetic field H0 outside the boundary layer are both assumed to be parallel to the plate. The effects of the induced magnetic field, viscous dissipation and Joule heating have been included in the analyses, but the Hall effect is neglected. The Renolds number Re(= U x/v) and the magnetic Reynolds number Rm(= U x/α) are assumed to be large enough for momentum and magnetic boundary layers to have developed. It is assumed that there is no applied voltage, which implies the absence of an electric field (E = 0). The electrical currents flowing in the fluid give rise to an induced magnetic field which would exist if the fluid was an electrical insulator. Here it is assumed that the normal component of the induced magnetic field H2 vanishes at the wall and the parallel component H1 approaches its given value H0 at the edge at the boundary layer [149]. The magnetodynamic pressure gradient and the temperature at the wall are taken to be proportional to a power of distance along the plate measured from the leading edge. The temperature in the free stream is constant. Under the above assumptions, the boundary layer equations governing the flow can be expressed (see [149], [151], [153]) as ux + uy = 0, (H1 )x + (H2 )y = 0,
(6.91) (6.92)
6.6
Flow and heat transfer of a viscoelastic fluid over a flat plate with ...
uux + vuy = −ρ−1 px + vuyy + (μ0 /ρ)(H1 (H1 )x + H2 (H2 )y ) +(K1 /ρ)((uuyy )x + uy vyy + vy uyyy ),
373
(6.93)
u(H1 )x + v(H1 )y − H1 ux − H2 uy = α(H1 )yy , uTx + uTy = νPr−1 + (ν/cp )u2y + (ρcp σ)−1 ((H1 )y )2 ,
(6.94) (6.95)
−ρ−1 Px = U0 (U0 )x − (μ0 /ρ)H0 (H0 )x .
(6.96)
The boundary conditions are given by u(x, 0) = v(x, 0) = H1y (x, 0) = 0, u(x, ∞) → u0 (x),
uy (x, ∞) → 0,
H2 (x, 0) = 0,
T (x, 0) = Tw (x),
H1 (x, ∞) → H0 (x),
T (x, ∞) → T∞ . (6.97) It may be remarked that Eq. (6.93) is one order higher than the classical boundary layer equation and the usual boundary conditions are less. Hence we have augmented the boundary (see for details [143] – [145], [154]) conditions by taking uy (x, ∞) → 0. Here x and y are the distance along and perpendicular to the plate respectively; u and v are the components of velocity along x and y directions, respectively; u0 and H0 are the velocity and the applied magnetic field in the x-direction at the edge of the boundary layer respectively; H1 and H2 are the components of the induced magnetic field in the x and y directions respectively; p is the pressure; Pr the prandtl number, T is the temperature, K1 the viscoelastic parameter, ρ is the density, ν the kinematic viscosity, μ0 the magnetic permeability, α the magnetic diffusivity, cp the specific heat at a constant pressure, σ the electrical conductivity, U and H are the characteristic velocity and magnetic field respectively and the subscripts x and y denote partial derivatives with respect to x and y respectively. In order to reduce the system of partial differential equations to a system of ordinary differential equations, we apply the following transformation (see [149]) η = 2−1 (U (n + 3)/v)1/2 yx(n−1)/4 , u = ψy , H1 = φy ,
v = −ψx ,
H2 = −φx ,
x ¯ = x/L,
U0 = U x(n+1)/2 ,
H0 = Hx(n+1)/2 ,
ψ(x, y) = 2(U v/(n + 3))1/2 x(n+3)/4 f (η), φ(x, y) = 2(Hv/(n + 3))1/2 x(n+3)/4 g(η), u = U x(n+1)/2 f � (η), v = −(U/2)(v/U (n + 3))1/2 x(n−1)/4 ((n + 3)f + (n − 1)ηf � ),
H1 = Hx(n+1)/2 g � (η),
H2 = −(H/2)(v/U (n + 3))1/2 x(n−1)/4 ((n + 3)g + (n − 1)ηg � ), β = 2(n + 1)/(n + 3),
S = μ0 H 2 /ρU 2 , K = K1 U L(n−1)/2 /μ,
T − T∞ = (Tw − T∞ )θ(η),
ε = ν/α,
Tw − T∞ = (Tw0 − T∞ )x(n+1) ,
Ec = U 2 /(cp (Tw0 − T∞ )),
α = (μ0 σ)−1
(6.98)
374
Chapter 6
Application of the Keller-box Method to More Advanced Problems
to Eqs. (6.91) and (6.95). We find that Eqs. (6.91) and (6.92) are identically satisfied, and Eqs. (6.93) to (6.95) reduce to f ��� + f f �� + β(1 − f �2 ) − S(gg �� + β(1 − g �2 ))
+K x ¯(n−1)/2 (2 − β)−1 ((2β − 1)(2f � f ��� − f ��2 ) − f f (4) ) = 0, −1 ��
Pr
g ��� + ε(f g �� − f �� g) = 0,
θ + f θ − 2βf θ + Ec((f ) + (S/ε)(g ) ) = 0. �
�
�� 2
�� 2
(6.99) (6.100) (6.101)
The boundary conditions (6.97) can be written as f = f � = g = g �� = 0, f → 1, �
f → 0, ��
θ = 1 at η = 0,
g → 1, �
θ → 0 as η → ∞.
(6.102)
Here η is the similarity variable ; x¯ is the dimensionless stream wise distance; ψ and φ are the dimensional velocity and the magnitude stream functions respectively; f and g are respectively the dimensionless velocity and magnetic stream functions respectively, f � is the dimensionless velocity in the x direction; g � is the dimensionless induced magnetic field in the x direction; θ is the dimensionless temperature; β is the dimensionless pressure gradient parameter; S is the dimensionless magnetic parameter; K is the dimensionless viscoelastic parameter; Ec is the Eckert number (viscous dissipation parameter); ε is the magnetic Prandtl number; n is the index in the power-law variation of velocity or applied magnetic field with the distance x measured from the leading edge; and prime denotes the derivative with respect to η. Equation (6.99) is not self-similar due to the presence of the dimensionless stream wise distance x ¯. However, it is locally self similar and can be solved locally at a given x ¯. When n = 1 (which corresponds to two dimensional stagnation point flow), Equation (6.99) becomes self- similar. Also it is self-similar for Newtonian fluids (k = 0). Here we have taken the fluid as finitely conducting and the plate as non-conducting. Therefore, there is surface current sheet or equivalently, the x component of the induced magnetic field (H1 ) is continuous across the interface. This condition is expressed as ∂H1 /∂y = 0 when y = 0 (i.e., in dimensionless form g �� = 0 when η = 0). For S = 0 (without magnetic field) and n = 4m + 1, Equation (6.99) is the same as that of Garg and Rajagopal [154] who studied the flow of viscoelastic fluid over a wedge for β = K = 0 (Newtonian fluid without pressure gradient). Equations (6.99)–(6.101) are essentially same as those of Glauert [147], Tan and Wang [150], and Na [151] except for some scaling factors. Further for K = 0, Equation (6.99) and (6.100) reduce to those of Davies [148] and Gribben [149] and Nath [152] who studied the two dimensional MHD flow over a flat plate with a pressure gradient. Also for K = S = Ec = 0, n = 2m − 1, Equations (6.99) and (6.100) for an isothermal wall (that is f � G = 0 in Eq. (6.101)) are the same as those of Watanabe when the buoyancy parameter ξ = 0 in his equations.
6.6
Flow and heat transfer of a viscoelastic fluid over a flat plate with ...
375
For ε → 0 and ε → ∞, Equations (6.99)–(6.101) under conditions (6.102) constitute a singular perturbation problem and can be solved by matched asymptotic expansion (see for details [149], [155]). Hence we have taken in the range 0.01 ε 100 for our numerical computation. It has been shown by Greenspan and Carrier [146], Glauert [147], and Davies[148] that if a uniform magnetic field is applied to the flow past a semi-infinite flat plate without pressure gradient (β = 0), the boundary layer thickens with increase in S until S = 1 when the enter the flow is plugged. The induced current produces a counter magnetic field which ultimately annuls the entire fluid flow and the magnetic field, i.e., f (η) = g(η) = 0 when S = 1. Glauert [147] and Gribben [149] found that the boundary layer solutions break down as S → 1. Hence we have taken the range of S as 0 S < 1. The skin friction coefficient cf and heat transfer coefficient N u can be expressed as cf = 2μ(uy )y=0 /ρU02 = (n + 3)1/2 (Rex )−1/2 f �� (0), N u = x(Ty )y=0 /(Tw − T∞ ) = 2−1 (n + 3)1/2 (Rex )1/2 θ� (0),
(6.103)
where μ is the coefficient of viscosity and Rex (= U0 x/v) is the local Reynolds number.
6.6.3
Results and discussion
Equations (6.99)–(6.101) under conditions (6.102) have been solved numerically using the Keller-box method [156]. In order to check the accuracy of our analyses, we have compared the surface the shear stress of f �� (0) for S = 0 with that of Garg and Rajagopal [154]. Also, we have compared our results (f �� (0), g � (0)) for K = β = 0 with those of Glauert [147] and Na [151]. The heat transfer parameter (−θ� (0) for β = K = Ec = 0) is compared with that of Tan and Wang [150]. The shear stress at the wall f �� (0) and the induced magnetic field in the x-direction, g � (0), for K = 0 have been compared with the results of Nath [152]. Further, the surface shear stress f �� (0) and the heat transfer at the wall (−θ � (0) for β = K = Ec = 0) have been compared with Watanabe [153]. In all these cases the results are found to be in very good agreement. The comparison is presented in Tables 6.21–6.23 and Figures 6.48 and 6.49. The computations have been carried for several values of the parameters β, S, K, ε, Ec and Pr. However the results are presented only for some represented values of these parameters. The results are given in Figures 6.50–6.53 and Tables 6.24–6.25.
376
Chapter 6
Application of the Keller-box Method to More Advanced Problems
Table 6.21 Comparison of surface shear stress (−f �� (0)) and induced magnetic field in the x direction at the wall g � (0), k = β = 0.0, ε = 10 f �� (0) S
Present calculation
0.0060 0.0181 0.0602 0.1792 0.2960 0.3535 0.4103 0.4659
1.3239 1.3149 1.2829 1.1891 1.0910 1.0404 0.9883 0.9343
g � (0)
Glauert [147] Na [151] 1.3238 1.3151 1.2839 1.1918 1.0946 1.0438 0.9915 0.9379
1.3239 1.3149 1.2829 1.1892 1.0911 1.0402 0.9880 0.9341
Present calculation
Glauert [147] Na [151]
0.4903 0.4882 0.4811 0.4593 0.4358 0.4231 0.4099 0.3963
0.4915 0.4892 0.4808 0.4557 0.4284 0.4139 0.3986 0.3828
0.4903 0.4882 0.4810 0.4592 0.4356 0.4229 0.4097 0.3960
Table 6.22 Comparison of surface shear stress (−f �� (0)) and induced magnetic field in the x direction at the wall g � (0), k = 0.0, ε = 0.1 S
β
0.05 0.05 0.05 0.05 0.125 0.125 0.125 0.125 0.25 0.25 0.25 0.25
0 0.125 0.25 −0.10 0 0.125 0.25 −0.10 0 0.125 0.25 −0.10
Present calculation f �� (0) g � (0)
Nath [152] f �� (0) g � (0)
0.4575 0.5998 0.7174 0.3079 0.4383 0.5788 0.6942 0.2898 0.4033 0.5403 0.6525 0.2567
0.4575 0.5999 0.7173 0.3078 0.4383 0.5787 0.6943 0.2897 0.4033 0.5403 0.6523 0.2565
0.6231 0.6566 0.6801 0.5803 0.6115 0.6462 0.6709 0.5667 0.5901 0.6265 0.6520 0.5413
0.6230 0.6565 0.6802 0.5803 0.6115 0.6461 0.6708 0.5668 0.5901 0.6266 0.6519 0.5414
Table 6.23 Comparison of surface shear stress (−f �� (0)) and induced magnetic field in the x direction at the wall g � (0), S = k = 0.0, Ec = 0.0, Pr = 0.73 m∗ 0 0.0141 0.0435 0.0909 0.1429 0.2000 0.3333
Present calculation f �� (0) −θ � (0)
0.46960 0.50463 0.56896 0.65499 0.73202 0.80211 0.92767
0.42014 0.42579 0.43546 0.44732 0.45696 0.46505 0.47817
Watanabe [153] f �� (0) −θ � (0)
0.46960 0.50461 0.56898 0.65498 0.73200 0.80213 0.92765
0.42015 0.42578 0.43548 0.44730 0.45693 0.46503 0.47814
6.6
Flow and heat transfer of a viscoelastic fluid over a flat plate with ...
377
Fig. 6.48 Comparison of the heat transfer parameter (−θ (0)) for k = β = Ec = 0, Pr = 0.7
Fig. 6.49 Comparison of the heat transfer parameter (−f (0)) for S = 0
The variations of the surface shear stress f �� (0), the x-component of the induced magnetic field at the wall g � (0), and the local heat transfer parameter at the wall −θ� (0) with the magnetic parameter S for several values of the product of the viscoelastic parameter K and streamwise distance x ¯(n−1)/2 are presented in Figure 6.50. The surface shear stress, the x-component of the induced magnetic field at the wall, and the local heat transfer parameter at the wall, f �� (0), g � (0), −θ� (0), in general decrease as S or K x ¯(n−1)/2 increases. A similar trend has been observed by Glauert [147], Tan and Wang [150] and Na [151] for the flat plate case when the fluid is Newtonian. This trend is due to thickening of the momentum, magnetic, and thermal boundary layer thickness. The variation of the surface shear stress f �� (0), the x-component of the induced magnetic field at the surface g � (0), and the heat transfer parameter at the surface −θ� (0) with pressure gradient parameter β for several values of the K x ¯(n−1)/2 is shown in the Figure 6.51. It is found that for β < 0.4, f �� (0), g � (0), −θ� (0), increase as K x ¯(n−1)/2 increases, but for β > 0.4, they decrease. Also, they increase with β for K x ¯(n−1)/2 < 1, but they decrease for K x ¯(n−1)/2 1.
378
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Application of the Keller-box Method to More Advanced Problems
Fig. 6.50a Variation of surface shear stress (−f (0)) with S for ε = 0.1, Ec = 0.0, Pr = 0.7, f (0) or β = 0.5
Fig. 6.50b Variation of surface shear stress (−f (0)) with S for ε = 0.1, Ec = 0.0, Pr = 0.7, f (0) or β = 1.0
Fig. 6.51a Variation of shear stress (−f (0)) with β for S = 0.5, ε = 0.1, Ec = 0.0, Pr = 0.7
6.6
Flow and heat transfer of a viscoelastic fluid over a flat plate with ...
379
Fig. 6.51b Variation of the induced magnetic field in the x direction at the wall with β for S = 0.5, ε = 0.1, Ec = 0.0, Pr = 0.7
Fig. 6.51c Variation of heat transfer parameter with β for S = 0.5, ε = 0.1, Ec = 0.0, Pr = 0.7
The effect of the magnetic Prandtl number ε on the surface shear stress f (0), the x-component of the induced magnetic field g (0) and the heat transfer parameter −θ (0) is given in Table 6.24. The wall shear stress f (0) decreases as ε increases to 2.0. Beyond this value, f (0) slightly increases. The reason for such a behavior can be explained as follows. Large values of ε correspond to large electrical conductivity σ so that the magnetic lines of forces are more or less frozen into the fluid for large ε. As ε increases the electrical conductivity σ is increased which results in increase in the ycomponent of the induced magnetic field H2 and hence in ponder-motive force which resists the motion parallel to the body. This tends to reduce the surface shear stress f (0). Also by increasing ε the magnetic diffusivity α becomes less resulting in a smaller x-component of the induced magnetic field H1 near the wall. Since the y-component of the induced magnetic field H2 is proportional to the x-component of the induced magnetic field H1 the former also reduces as ε increases. This causes less resistance to fluid motion
380
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Application of the Keller-box Method to More Advanced Problems
and increases wall shear stress f (0). For large ε the second effect dominates the first effect and for small ε, it is the other way around. A similar trend has been observed by Glauert [147] for the Newtonian fluid flow over the flat plate (β = 0). The x-component of the induced magnetic field at the surface g (0) decreases as ε increases. The heat transfer parameter −θ (0) changes little with ε. The effect of ε is found to be more pronounced on the x-component of the induced magnetic field g (0) as compared to that on the shear stress and the heat transfer −θ (0). Table 6.24 Surface shear stress in the induced magnetic field in the x-direction at the wall and heat transfer parameter for K = 1, S = 0.5, β = 1.0, Ec = 0.0, Pr = 0.7 ε
f (0)
g (0)
0.01 0.02 0.1 1 2 3 5 10 20 50 100
0.7255 0.7031 0.6238 0.5516 0.5377 0.5373 0.5334 0.5301 0.5282 0.5273 0.5271
0.9415 0.8895 0.6839 0.4169 0.3501 0.3209 0.2817 0.2339 0.1926 0.1473 0.1195
−θ (0) 0.7297 0.7219 0.6937 0.6687 0.6633 0.6655 0.6650 0.6650 0.6654 0.6659 0.6663
The effects of the Prandtl number Pr and the Eckert number Ec on the heat transfer parameter at the wall −θ (0) are presented in Table 6.25 and Figure 6.52 respectively. The heat transfer parameter −θ (0) is found to decrease as the Prandtl number Pr decreases, since a low Prandtl number fluid has a relatively high thermal conductivity which promotes conduction and thereby reduces the variations. This results in increase in the thermal boundary layer and reduction in the convective heat transfer at the wall. The effect of high Pr is just the reverse. The Eckert number Ec (viscous dissipation parameter) is a measure of the heat produced by friction. The positive and negative values of Ec correspond to cooling and heating, respectively. The heat transfer parameter −θ (0) is found to decrease for positive values of Ec and increase for negative values. Table 6.25 Heat transfer parameter K = 1, S = 0.5, β = 1.0, Ec = 0.0 Pr
ε = 0.01
ε = 0.5
ε = 1.0
0.7 3 7 10 20 50
0.7297 1.2262 1.6475 1.8636 2.3647 3.2328
0.6731 1.1296 1.5172 1.7160 2.1772 2.9806
0.6687 1.1212 1.5054 1.7026 2.1598 2.9518
6.6
Flow and heat transfer of a viscoelastic fluid over a flat plate with ...
381
Fig. 6.52 Variation of heat transfer parameter with Ec for S = 0.5, k¯ x(n−1)/2 = 1.0, β = 1.0, ε = 0.1, Pr = 0.7
The velocity, induced magnetic field in the x-direction, and temperature profiles (f , g , θ) are shown in Figure 6.53. The thermal boundary layer thickness is found to be less than the momentum and magnetic boundary layer thickness. The velocity f and the induced magnetic field in the x-direction, g are more affected by the parameters S and ε than the temperature θ.
Fig. 6.53a Velocity profiles for k¯ x(n−1)/2 = 1.0, Ec = 0.0, Pr = 0.7, β = 1.0
Fig. 6.53b Induced magnetic field in the x direction for k¯ x(n−1)/2 = 1.0, Ec = 0.0, Pr = 0.7, β = 1.0
382
Chapter 6
Fig. 6.53c 0.5, β = 1.0
6.6.4
Application of the Keller-box Method to More Advanced Problems
Temperature profiles for k¯ x(n−1)/2 = 1.0, Ec = 0.0, Pr = 0.7, β =
Conclusions
The surface shear stress, the heat transfer parameter, and the x-component of the induced magnetic field decrease as the magnetic parameter of the viscoelastic parameter increases. The heat transfer at the wall increases with the Prandtl number, but decreases as the Eckert number increases. The Prandtl number strongly affects the induced magnetic field in the x-direction, whereas the Prandtl number and Eckert number strongly affect the heat transfer.
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Chapter 6
Application of the Keller-box Method to More Advanced Problems
D. Lesnic, D.B. Ingham, I. Pop, Free convection from a horizontal surface in a porous medium with Newtonian heating, J. Porous Media 3 (2010) 227. D. Lesnic, D.B. Ingham, I. Pop, C. Storr, Free convection boundary-layer flow above a nearly horizontal surface in a porous medium with Newtonian heating, Heat Mass Transfer 40 (2004) 665. I. Pop, D. Lesnic, D.B. Ingham, Asymptotic solutions for the free convection boundary-layer flow along a vertical surface in a porous medium with Newtonian heating, Hybrid Methods Eng. 2 (2000) 31. R.C. Chaudhary, P. Jain, Unsteady free convection boundary-layer flow past an impulsively started vertical surface with Newtonian heating, Romanian J. Phys. 9 (2006) 911. R.C. Chaudhary, P. Jain, An exact solution to the unsteady free convection boundary-layer flow past an impulsively started vertical surface with Newtonian heating, Journal of Engineering Physics and Thermophysics 80 (2007) 954. M.Z. Salleh, R. Nazar, Free convection boundary layer flow over a horizontal circular cylinder with Newtonian heating, Sains Malaysiana 39 (2010) 671. M.Z. Salleh, R. Nazar, I. Pop, Forced convection boundary layer flow at a forward stagnation point with Newtonian heating, Chem. Eng. Comm. 196 (2009) 987. M.Z. Salleh, R. Nazar, I. Pop, Modeling of free convection boundary layer flow on a sphere with Newtonian heating, Acta Appl. Math. 112 (2010) 263. I. Pop, J.K. Sunada, P. Cheng, W.J. Minkowycz, Conjugate free convection from long vertical plate fins embedded in a porous medium, Int. J. Heat Mass Transfer 28 (1985) 1629. Pozzi, M. Lupo, The coupling of conduction with laminar natural convection along a flat plate, Int. J. Heat Mass Transfer 31 (1988) 1807. M.Z. Salleh, R. Nazar, N.M. Arifin, I. Pop, J.H. Merkin, Forced convection heat transfer over a horizontal circular cylinder with Newtonian heating, J. Eng. Math. 69 (2011) 101. M.Z. Salleh, S. Ahmad, R. Nazar, Numerical solution of the forced convection boundary layer flow at a forward stagnation point, European J. Scientific Research 19 (2008) 644. A. Ishak, R. Nazar, I. Pop, Post stagnation-point boundary layer flow and mixed convection heat transfer over a vertical, linearly stretching sheet, Archives of Mech. 60 (2008) 303. T. Sarpakaya, Flow of non-Newtonian fluids in a magnetic field, AIChE. J. 7 (1961) 324. A.H.D. Skelland, Non-Newtonian Flow and Heat Transfer, John Wiley, New York, 1967. R.B. Bird, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, Fluid Mechanics, 2nd Ed. John Wiley, New York, 1987. R.I. Tanner, Engineering Rheology, Oxford University Press, New York, 1988. R.S. Rivlin, J.L. Ericksen, Stress-deformations relations for isotropic material, J. Rational Mech. Anal. 4 (1955) 323. J.E. Dunn, R.L. Fosdick, Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Rational Mech. Anal. 56 (1974) 191. D.W. Beard, K. Walters. Elastico-viscous boundary-layer flows. I. Two-dimensional flow near a stagnation point, Proc. Camb. Phil. Soc. 60 (1964) 667. K.R. Rajagopal, On the creeping flow of the second order fluid. J. Non-Newtonian Fluid Mech. 15 (1984) 239. K.R. Rajagopal, Flow of viseoelastie fluids between rotating disks, Theoret. Comput. Fluid Dynamics 3 (1974) 185. K.R. Rajagopal, P.N. Kaloni, Continum Mechanics and Its Applications, Hemisphere Press, Washinton DC, 1989.
References [146] [147] [148]
[149] [150] [151] [152] [153] [154] [155] [156]
389
H.P. Greeenspan, G.F. Carrier, The magnetohydrodynamic flow past a flat plate, J. Fluid Mech. 6 (1959) 77. M.B. Glauert, A study of the magnetohydrodynamic boundary layer on a flat, J. Fluid Mech. 10 (1961) 276. T.V. Davies, The magneto-hydrodynamic boundary layer in the two-dimensional steady flow past a semi-infinite flat plate, I. Uniform conditions at infinity, Proc. Roy. Soc. 273A (1963) 496. R.J. Gribben, The magnetohydrodynamic boundary layer in the presence of pressure gradient, Proc Roy. Soc. 287A (1965) 123. C.W. Tan, C.T. Wang, Heat transfer in aligned-field magnetohydrodynamic flow past a flat plate, Int. J. Heat Mass Transfer 11 (1968) 319. T.Y. Na, An initial value method for the solution of a class of nonlinear equations in fluid mechanics, J. Aeronaut. Quart. 21 (1970) 91. G. Nath, Hydrazine complexes of some first row transition metal (ii) picrates and picramates, Def. Sci., J. 26 (1976) 153. T. Watanabe, Thermal boundary layers over a wedge with uniform suction or injection in forced flow, Acta Mech. 83 (1990) 119. V.K. Garg, K.R. Rajagopal, Flow of a non-Newtonian fluid past a wedge, Acta Mech. 88 (1991) 113. D.R. Smith, Singular Perturbation Theory, Cambridge University Press, Cambridge, 1985. T. Cebeci, A.M.O. Smith, Analysis of Turbulant Boundary Layers, Academics Press, New York, 1974.
Subject Index
Colloidal fluid, 166 Conservation law, 30 Continuous function, 90, 100 Acceleration due to gravity, 105, 198, Convection flow, 166, 194, 206, 239, 225, 241, 321 319, 337, 355 Adams method, 24 Coupled boundary value problem Adams-Bashforth method, 24 (BVP), 5, 229, 240, 305, 342 Adams-Moulton method, 24 Crank-Nicholson algorithm, 89, 113 Ambient temperature, 184, 186, 197, 225, 240, 320, 357 D Animal blood, 166 Arbitrary grid, 105 Darcy-Forchheimer model, 239
A
Difference approximation, 5, 10, 60, 90, 112 Diffusion, 27, 30, 113, 135, 259 Backward Euler method, 16, 17 Diffusion coefficient, 30, 283, 297, 320 Block-elimination method, 86 Dimension less temperature profile, Block-tridiagonal-elimination 198, 210, 223, 374 method, 84 Dirichlet, 90, 111 Blood rheology, 299 Discretization, 5, 16, 59, 91, 107 Boussinesq approximation, 185, 197, Dispersion model, 113 225, 240, 242, 339 Drag force, 160, 188, 213, 233, 286, Brownian diffusion coefficient, 283, 301, 352 297, 320 Dynamic viscosity, 168, 187, 355 Brownian motion, 113, 284, 319, 322, 329, 331, 337 E Buoyancy force, 184, 186, 189, 240, 340 Eckert number, 157, 162, 165, 240, Buoyancy parameter, 187, 189, 194, 245, 254, 260, 272, 284, 296, 228, 343, 374 341, 344, 354, 382 Electric conductivity, 225 C Elliptic operator, 27 Elliptic PDE, 32 Cauchy problem, 30, 100, 102
B
392
Subject Index
Energy equation, 136, 140, 153, 155, 185, 206, 211, 257, 267, 321, 354, 355 Euler method, 16, 17, 112
F Falkner-Skan type, 137, 226 Ferro liquid, 166 Finite difference approximation, 5, 7, 9, 10, 74, 79, 112 Finite difference method, 6, 7, 10, 32, 35, 41, 53 Flat plate problem, 170 Fluid density, 137, 154, 168, 185, 241, 283, 320 Forchheimer flow, 240 Fourier’s law, 30
G Gauss-Seidal iterative method, 76 Global error, 18, 19
H Hall effect, 168, 372 Hartree pressure, 168, 340 Hartree pressure gradient parameter, 168, 340 Heat flux, 121, 140, 156, 164, 166, 285, 321, 340, 342, 355, 358 Heat transfer, 1, 3, 122, 135, 152, 156, 183, 225, 312, 355, 372 Hydromagnetic flow, 126, 152, 184, 208, 223, 338 Hyperbolic equation, 26, 28
342, 344, 353
K Keller-box method, 1, 41, 84, 89, 121, 151, 153, 158, 183, 279, 356, 357, 360 Kinematic viscosity, 137, 154, 170 Kronecker delta, 136, 185, 223
L Laminar fluid flow, 176 Laplacian operator, 27 Lateral mass flux parameter, 240, 243, 253 Lewis number, 322, 323, 334, 337 Linear multi-step Method (LMM), 23, 25 Liouville definition, 113 Lipschitz constant, 14, 16 Lipschitz continuity, 12 Liquid crystal, 166 Liquid film, 183, 255, 256, 259, 262, 272 Lorentz force, 150, 164, 224, 230, 236, 253, 316, 352, 353
M
Magnetic parameter, 129, 130, 149, 150, 170, 187, 194, 208, 213, 227, 316, 317 Maxwell fluid, 151, 194, 196, 197 MHD flow, 127, 130, 151, 210, 213, 224, 355, 374 Microinertia, 168, 169 Micropolar fluid, 121, 135, 166, 167, 169, 176 I Microrotation profile, 176 Midpoint method, 17, 23, 25 Implicit method, 17, 24, 53, 84, 103 Mixed convection parameter, 186, Incompressible flow, 27 227, 323, 331 Initial value problem (IVP), 5, 11, 12, 30, 71, 81, 321 Molten plastic, 152, 195, 223 Injection parameter, 259, 260, 267,
Subject Index
393
N Nano fluid, 279, 280, 283, 286, 297, 318, 319 Nanoparticles, 282, 322, 324, 329, 331, 332, 334 Navier-Stokes equation, 1, 169 Neumann type, 91 Newton-Raphson method, 60, 62 Newtonian fluid, 121, 127, 173, 176, 208, 223, 224, 228, 256, 371 Niemycki operator, 96 Non-Darcian flow, 238 Non-Newtonian fluid, 121, 127, 166, 183, 207, 223, 371 Numerov method, 56 Nusselt number, 140, 150, 153, 157, 170, 187, 200, 207, 212, 326 Nystrom method, 25
O
338, 344, 371 Pseudo plastic, 122, 136 Pseudosimilarity variable, 343
R Radial flow, 89, 112, 113, 117, 118 Radiation conduction parameter, 323, 337 Rayleigh number, 240, 242, 253, 321 Reaction-diffusion equations, 31, 32 Real fluid, 166 Reynolds number, 1, 125, 128, 130, 138, 153, 168, 170, 199, 239, 375 Riemann and Liouville form, 113 Rosseland diffusion approximation, 320, 337 Runge-Kutta method, 20, 23, 68, 306, 343
S
Schmidt number, 284, 286, 297 One-step error, 18, 19, 21 Ostwald-de Waele liquid, 121, 127, Sedimentation, 299 Shear stress, 285, 338, 356, 375, 377, 255 379, 380, 382 Sherwood number, 285, 298, 322, P 328, 334, 337 Paints, 152, 195, 256 Shooting method, 41, 71, 80, 83, 124, Parabolic differential equation, 34, 91 130 Parabolic equation, 27, 91 Simpson’s rule, 25 Perron type, 89, 91 Sink, 137, 149, 150, 151, 157, 299, Picard’s theorem, 42 344, 355 Poisson problem, 26, 27 Skin friction, 121, 125, 139, 149, 153, Polymer, 121, 135, 152, 195, 223, 224, 165, 176, 189, 199 255, 281, 298 Slip velocity, 243, 254, 255, 256 Polymer extrusion, 183, 255, 259 Slurries, 152, 223, 371 Polymeric fluid, 166 Solid matrix heat exchange, 238 Polynomial interpolation, 10, 20 Specific heat, 139, 155, 198, 210, 241, Porous medium, 135, 159, 240, 318, 257, 283, 301, 340, 373 319, 355 Stability, 18, 68, 90, 99, 107, 115 Power-law fluid, 121, 126, 135, 223 Stokesian drag force, 301 Prandtl number, 122, 140, 151, 186 Stress tensor, 123, 128, 154 Pressure gradient, 137, 153, 189, 301, Stretching sheet, 121, 126, 135, 151,
394
Subject Index
Truncation error, 7, 17, 25, 55 207, 223, 255, 298 Suction parameter, 353 Surface temperature parameter, 323, U 326, 337 Suspension, 28, 152, 166, 195, 279, UCM fluid, 121, 152, 155, 206 Unsteady flow, 183, 255 298, 299, 306, 314, 317
T Taylor series, 7, 9, 19, 32, 54, 107, 156, 340 Taylor polynomial, 37 Thermal conductivity, 135, 140, 152, 187, 198, 338, 361 Thermal diffusivity, 156, 168, 187, 226, 241, 283, 301, 321, 358, 362 Thermal buoyancy, 185, 195, 224, 226, 340 Thermophoresis parameter, 285, 286, 297, 298, 322, 332, 337 Thermophoretic diffusion coefficient, 283, 320 Thin film, 255, 261 Transverse magnetic field, 128, 137, 300, 315, 316, 338, 352, 353 Trapezoidal method, 17, 25, 70, 107
V Velocity gradient, 125, 130, 154, 188, 189, 205, 213, 317 Velocity profile, 149, 230, 314, 366 Viscoelastic fluid, 122, 208, 371, 374 Viscosity parameter, 211, 217, 298, 306, 316 Volterra type, 91
W Wall friction, 125, 134 Walters’ model, 152, 195 Wave equation, 26, 28, 35, 38 Wedge, 166, 171, 318, 337 Wronskian, 46, 47, 52
172, 352, 372, 220,
Author Index
Amkadni, M., 167 Andersson, H. I., 122, 127, 128, 129, 130, 133, 136, 137, 146, 149, Aaresh, J. B., 256 195, 208, 224, 225, 227, 230, Abbas, Z., 152, 195 256, 285, 305 Abdullaah, A., 224, 225, 227 Anjali Devi, S. P., 167, 169, 184, 185, Abel, M. S., 122, 135, 136, 152, 195, 186, 187, 338 209, 255, 256, 280, 298, 299, Antar, M. E., 356 303, 333, 343 Apelblat, A., 286, 298 Abo-Eldahab, E. M., 184, 196, 208, Arifin, N. M., 360 224 Ariman, T., 167 Abu-Nada, E., 318 Armaly, B. F., 320, 322 Acrivos, A., 196, 224 Armstrong, R. C., 151, 153, 154, 371 Afifty, A. A., 196, 224 Asmolov, E. S., 299 Agassant, G. F., 151 Astle, M. J., 282 Agranat, V. M., 299 Avens, P., 151 Ahmad, S., 360 Awais, M., 256 Ahmadi, G., 169, 171 Aydin, O., 338 Alam, M. M., 319 Aziz, R, C., 256, 261, 262 Ali, M., 184, 196, 338 Azzouzi, A., 167 Ali, M. E., 184, 186, 208, 256, 280, 298, 299, 306 B Aliakbar, V., 195 Alim, M. A., 319 Bachok, N., 280, 285 Alizadeh-Pahlavan, A., 195 Bagewadi, C. S., 299, 303 Almari, A. K., 256, 261, 262 Bakier, A. Y., 319 Al-Mudhaf, H., 318 Barrow, H., 239, 241 Aly, A. M., 318 Bar-Ziv, 356 Aly, E. H., 239 Baskaran, S., 338 Al-Yousef, F., 184, 196, 338 Bassom, A. P., 171 Ames, W. F., 224 Beard, D. W., 371 Amin, N., 356, 360, 361, 366, 367 Bech, K. H., 127, 136, 146, 224, 225, Aminossadati, S. M., 280 230, 285, 305
A
396
Author Index
Beg, O. A., 239, 241 Beg, T. A., 241 Bejan, A., 238, 318 Ben-Nakhi, A., 319 Benson, D. 112 Beyer, W. H., 280 Bhargava, R., 241 Bhatnagar, R. K., 152, 195 Biradar, S. N., 135 Bird, R. B., 151, 153, 154, 241, 371 Blottner, F. G., 105 Bobba, K. M., 187, 190, 195, 199, 212, 223, 298, 303, 306 Bouchet, G., 356 Boyce, W., 116 Bradshaw, P., 124, 136, 141, 158, 171, 172, 200, 212, 229, 243, 261, 285, 306, 322, 342, 357 Braud, N., 256 Brewster, M. Q., 155 Brown, A., 241 Bujurke, N. M., 135 Buongiorno, J., 280
C Canon, J. R., 239, 241 Carreau, P. J., 151 Cebeci, T., 36, 41, 58, 121, 124, 171, 172, 200, 229, 243, 261, 285, 306, 322, 342, 357 Chakrabarti, A., 127, 136, 207, 208, 212, 298, 303 Chamkha, A. J., 172, 184, 208, 239, 318, 319, 338, 340, 344 Chandrashekar, M., 338 Chang, C. H., 355 Chang, T. C. A., 195, 223 Chang, W. D., 135 Changhoon, L., 280 Chapman, J., 117 Char, M. I., 135, 136, 152, 195, 223 Chaturani, P., 224 Chaudhary, R. C., 356 Chen, C. H., 152, 184, 185, 186, 187,
190, 196, 224, 298, 306, 338 Chen, C. K., 135, 152, 184, 195, 223, 355 Chen, T. S., 196, 224, 320, 322, 338, 340, 355, 356 Chen, X. B., 239 Cheng, P., 238, 239, 241, 357 Chiam, T. C., 135, 137, 139, 152, 155, 157, 167, 169, 186, 196, 198, 207, 208, 210, 212, 256, 258, 338 Choi, S., 280, 318 Choi, S. U. S., 280 Chowdhury, M. M. K., 319 Cobble, M. H., 167, 169 Cobble, W. H., 167, 169 Coddington, E. A., 13 Conte, S. D., 80, 141 Cortell, R., 136, 152, 195, 256, 298 Crane, L. J., 121, 122, 124, 125, 127, 135, 151, 184, 187, 195, 199, 212, 223, 224, 255, 256, 259, 298, 303 Crowder, H. J., 105 Czernous, W., 91
D Dabrowski, P. P., 137, 226 Dalton, C., 105 Dandapat, B. S., 122, 127, 128, 129, 130, 136, 138, 146, 195, 224, 225, 227, 230, 256, 285, 305 Das, S., 239 Das, S. K., 280 Daskalakis, J. E., 184 Datta, B. K., 135, 195, 223, 298 Datta, N., 298 Datti, P. S., 136, 196, 200, 208, 212, 256, 280, 299, 300, 338 Davies, T. V., 286, 298, 372, 374, 375 De-Boor, C., 80, 141 Denier, J. P., 137 De-Rivas, K. E., 105 Dewitt, K., 195, 223
Author Index
397
Diethelm, K., 112 Dincer, I., 318 Diprima, R., 116 Djukiv, D. J., 127, 136 Draper, M., 239, 241 Duangthongsuk, W., 318 Dunn, J. E., 371 Dusek, J., 356
E
Green Span, H. P., 372, 375 Gribben, R. J., 372, 373, 374, 375 Grosan, T., 319 Grubka, L. G., 187, 190, 195, 199, 212, 223, 298, 303, 306 Grulke, E. A., 280 Gupta, A. S., 122, 127, 128, 135, 136, 152, 153, 155, 195, 207, 208, 212, 223, 256, 298, 303 Gupta, G., 152, 195 Gupta, P. S., 135, 152, 195, 223, 298
Ebata, A., 280 Elliott, L., 239 H El-Shaarawi, M. A. I., 356 Erickson, L. E., 126, 195, 207, 224, Hadjinicolaou, A., 196, 224 Hady, F. M., 167 371 Haga, M., 117 Eringen, A. C., 166, 167 Hammock, D., 239, 241 Ermentrout, G. B., 127 Hammouch, Z., 167 Hasim, I., 256, 261, 262 F Hassager, O., 151, 153, 154, 371 Falkner, V. M., 167, 168, 169 Hassanien, I. A., 167, 208, 224, 225, Fan, L. T., 126, 195, 107, 224 227, 280, 299 Flannery, B. P., 141 Hayat, T., 152, 195 Fosdick, R. L., 371 Helmy, K. A., 167, 169, 186 Fox, V. G., 126, 195, 107, 224 Henrici, P., 11 Freed, A., 112 Herwig, H., 280, 299 Hieber, C. A., 355 Hiremath, P. S., 135 G Hishinuma, N., 280 Ganeshan, P., 299 Hoernel, J. D., 167, 169 Gear, C. W., 11 Hsieh, J. C., 320, 322 Gebhart, B., 355 Hu, W., 280 Gerald, C., 116 Gerber, H., 116 I Ghasemi, B., 280 Gireesha, B. J., 299, 303 Ingham, D. B., 189, 318, 356, 357 Glauert, M, B., 372, 374, 375, 376, Irgens, F., 122, 128 377, 380 Ishaka, A., 167, 169, 184, 187, 190, Goldstein, R. J., 195, 207 280, 285 Gorenflo, R., 112, 113 Israeli, M., 105 Gorla, R. S. R., 169, 208, 224, 225, Issa, C., 172 227, 280, 299, 303, 319 Iyenger, S. R. K., 41 Govindarajalu, T., 152 Grag, V. K., 152, 155, 372, 373, 375
398
Author Index
J
L
Lai, F. C., 209, 299 Lambert, J. D., 11 Lawrence, P. S., 135 Lax, P. D., 29 Lee, J., 280 Lee, S. C., 355 Lee, S. Y., 224 Lesnic, D., 356, 357 Leszczynski, H., 91, 167 Leto, J., 239, 241 Le-Veque, R. J., 29 K Levinson, N., 13 Kafoussias, N. G., 167 Liao, S. J., 224 Kakac, S., 280 Lightfoot, E. N., 241 Kaloni, P. N., 371 Lightstone, M., 280 Kamont, Z., 91, 101, 103 Lin, C .R., 184 Kandaswamy, R., 338 Lin, H. T., 172 Kapur, J. N., 224 Lin, L. K., 172 Kaya, A., 338 Liu, I. C., 152, 256, 298 Kaya, W. M., 300 Lockwood, F. E., 280 Keller, B., 239 Lorente, S., 318 Keller, H. B., 105, 124, 130, 136, 141, Low, H. T., 239 152, 158, 200, 229, 243, 261, Lupo, M., 357 285 Lykoudis, P. S., 167 Kevorkian, J., 26, 28 Lyles, B., 117 Khan, S. K., 35, 196, 208, 280, 299, 338, 343 M Khan, W. A., 280 Magyari, E., 239 Khanafer, K., 280 Mahantesh, M. N., 152 Khapate, B. S., 122 Mahesha, N., 152, 156 Kim, T. A., 167, 169, 170, 171 Mainardi, F., 112, 113 Kim, Y. J., 167, 169, 170, 171 Malec, M., 91 Kline, K. A., 169 Mansour, M. A., 319 Kosinski, R., 117 Manuilovich, S. V., 299 Kraus, A. D., 238 Markatos, N. C., 136, 152, 256, 258 Krishnamurthy, E. V., 76 Martinson, L. K., 224 Kropielnicka, K., 91 Masuda, H., 280 Kulacki, F. A., 208, 209, 280, 299 Mathur, M. N., 224 Kumar, S. K., 299 Mc-Cormack, P. D., 121, 122, 127 Kumari, N., 318, 319, 338, 355 Mc-Leod, B., 135 Kuznetsov, A. V., 280, 318 Mc-Leod, J. B., 127 Jadhav, J. P., 136 Jain, M. K., 41 Jain, P., 356 Jain, R. K., 41 Jeng, D. R., 195, 212 Jenny, M., 356 Johnson, C. H., 239 Jones, I. P., 105 Joshi, A., 136, 196, 338
Author Index
399
Palani, G., 299 Pao, C. V., 91 Papailiou, D. D., 167 Papas-Pyrides, C. D., 136, 152, 256, 258 Partha, M. K., 184 Patil, P. M., 338 Pavlov, K. B., 127, 129, 207, 224 Perdikis, C., 155 Pohll, G., 117 Pop, I., 167, 169, 184, 190, 208, 280, 285, 299, 303, 318, 319, 344, 356, 357, 360, 361, 366, 367 Pozzi, C., 357 Praddep, T., 280 N Pramuanjaroenbij, A., 280 Na, T. Y., 122, 127, 128, 152, 195, Prasad, K. V., 135, 136, 196, 200, 285, 357, 372, 374, 375, 376, 208, 212, 257, 280, 299, 300, 377 338, 343 Nadeem, S., 256 Prasad, V., 239 Najafi, A. H., 152, 154, 195, 197, 199 Press, W. H., 141 Nandeppanavar, M. M., 152 Nanousis, N. D., 167 Q Nath, G., 239, 318, 319, 338, 355, 372, 374, 375, 376 Quadri, A., 172, 338, 340, 344 Nathan, S., 239, 241 Nayfeh, J., 298, 303, 319 R Nazar, R., 167, 169, 184, 190, 208, Rajagopal, K. R., 122, 127, 128, 135, 356, 360, 361, 366, 367 152, 155, 195, 371, 372, 373 Nield, D. A., 238, 280, 318 Rajasekhar, G. P., 184 Niswonger, R., 117 Ramana-Murthy, T. V., 195, 212, 223 Nitu, S., 208, 280, 299 Ramesh, G. K., 299, 303 Norsett, S. P., 11 Rao, B. N., 135 Rao, T. L. S., 239, 241 O Raptis, A., 152, 155, 338 Rashad, A. M., 319 Oldham, K ., 112, 113 Rashidi, M., 208, 280, 299, 303 Orszag, S. A., 105 Rawat, S., 241 Overman, E. A., 127 Rees, D. A. S., 171 Oztop, H. F., 318 Reimus, P., 117 Renardy, M., 152, 195 P Rivlin, R. S., 371 Pachner, J., 116 Rollins, D., 195 Pal, D., 200, 208, 212, 256, 280 Ross, B., 112 Meerschaert, M., 112, 113, 114, 117, 118 Merkin, J. H., 239, 356, 357, 360 Mihevc, T., 117 Minkowycz, W. J., 238, 239, 355, 357 Mishra, S. K., 299 Mohammed, E. S., 196, 224 Morsi, Y. S., 239 Moutsoglou, A., 196, 224, 338 Mucoglu, A., 355, 356 Mujtoba, M., 172, 338, 340, 344 Muralidhar, K., 239 Murthy, P. V. S. N., 184
400
Author Index
Stauss, A. M., 136, 223 Stewar, W. E., 241 Storr, C., 356, 357 Streeter, V. L., 241 Strobel, F. F., 196, 224, 338 S Sunada, J. K., 357 Sadeghy, K., 152, 154, 195, 197, 199 Sylvester, N. D., 167 Saffaripour, M., 152, 154, 195, 197, Szarski, J., 102 199 Sahu, A. K., 224 T Sajid, M., 152, 195 Sakiadis, B. C., 126, 151, 184, 195, Tadjeran, C., 112, 113, 114, 117, 118 207 Takhar, H. S., 171, 208, 239, 241, 280, Salleh, M. Z., 356, 360 299, 319, 338, 343 Samokhen, V. N., 224 Tan, C. W., 372, 374, 375, 377 Sanders, P., 117 Tanner, R., 371 Sanjayanand, E., 152 Tawade, J., 256 Santra, B., 256 Teramae, K., 280 Saponkov, Y., 224 Teukolsky, S. A., 141 Sarpakaya, T., 127, 136, 224, 371 Thangaraj, C. J., 152 Sastri, K. S., 239, 241 Thompson, C. P., 105 Savvas, T. A., 136, 152, 256, 258 Thyagarajan, M., 167, 169, 184, 185, Saxena, S. B., 224 186, 187 Scheffler, H. G., 112, 118 Tien, C. L., 239 Schneider, W., 190 Troy, W. C., 127, 135 Schowalter, W. R., 224 Tsou, F. K., 195, 207 Seddeek, M. A., 208, 280, 299 Turk, M. A., 167 Semmoum, R. 239, 241 Sen, S. K., 76 U Sergent, J., 151 Usha, R., 256 Sharma, L. V. K. V., 299 Shercliff, J. A., 128 Siddappa, B., 122, 135, 152, 195, 255, V 256, 298 Vafai, K., 238, 239, 280, 318 Simmons, G. F., 71 Vajravelu, K., 152, 195, 196, 200, 212, Singh, M., 171 223, 224, 239, 241, 256, 280, Skan, S. W., 167, 168, 169 298, 299, 300, 303, 319, 338, Skelland, A. H. D., 371 343 Smith, D. R., 375 Soundalgekar, V. M., 171, 195, 212, Van Gorder, R. A., 240, 280, 343 Veena, P. H., 135 223, 239 Vellering, W. T., 141 Spanier, J., 112, 113 Vleggaar, J., 195, 298 Sparrow, E. M., 195, 207 Voigt, W., 91 Sridharan, R., 256 Vujannovic, V., 136, 223 Srivastava, R. C., 224 Rossow, V., 171 Roy, P., 135, 195, 223 Roy, S., 338
Author Index
401
W Waghmode, B. B., 136 Walter, W., 101, 102 Walters, K., 122, 371 Wang, C. T., 372, 374, 375, 377 Wang, C. Y., 256, 259, 261, 262 Wang, J., 167, 169 Wanner, G., 11 Watanabe, T., 167, 338, 372, 375 Weast, R. C., 282 Weiss, R., 105 Wheat Craft, S., 112 Wheatley, P., 116 Whitham, G. B., 28, 30 Wickern, G., 280, 299 Wineman, A. S., 153 Winoto, S. H., 239
Wong, K. L., 355 Wongwises, S., 318
Y Yacob, N. A. M., 356 Yaho, L. S., 355 Yih, K. A., 167, 172, 319, 320, 322, 338 Yu, P., 239 Yu, W., 280 Yu, W. Y., 280 Yucel, A., 169 Yuge, T., 355
Z Zhang, Z., 167, 169 Zhang, Z. C., 280