Thermophysics of rheological fluids: educational manual for master course students 9786010429628

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AL-FARABI KAZAKH NATIONAL UNIVERSITY

A. S. Askarova S. A. Bolegenova S. A. Bolegenova

THERMOPHYSICS OF RHEOLOGICAL FLUIDS Educational manual for master course students

Almaty «Qazaq university» 2017

UDC 532 (075) LBC 22.253 я 73 А 85 Recommended for publication by the decision of the Academic Council of the Faculty of Physics and Technology, Editorial and Publishing Council of Al-Farabi Kazakh National University (Protocol №2 dated 03.11.2017) Reviewers: Doctor of technical sciences, Professor B.N. Absadykov Doctor of technical sciences, Professor V.E. Messerle

Translation from Russian: Sh.B. Gumarova, L.E. Strautman

А 85

Askarova A.S. Thermophysics of rheological fluids: educational manual for master course students / A.S. Askarova, S.A. Bolegenova, S.A. Bolegenova; translation from Russian: Sh.B. Gumarova, L.E. Strautman. – Almaty: Qazaq university, 2017. – 136 p. ISBN 978-601-04-2962-8 The textbook is based on the lectures of Professor V.P. Kashkarov who has read them for several years on the special course «Hydrodynamics of non-Newtonian Fluids» for students specializing in the department of physical hydrodynamics. These lectures were revised by the authors in accordance with the Master's course curriculum for reading the special discipline «Thermophysics of Rheological Fluids» in the specialization «Thermophysics and Theoretical Heat Engineering«. The textbook is intended primarily for graduate students of the department of thermal physics, but it can be useful for students of engineering specialties «Thermophysics», «Hydroaerodynamics», «Techniques and Physics of Low Temperatures», «Material Science» and others. Published in authorial release.

UDC 532 (075) LBC 22.253 я 73 ISBN 978-601-04-2962-8

© Askarova A.S., Bolegenova S.A., Bolegenova S.A., 2017 © Al-Farabi KazNU, 2017

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CONTENT Introduction ............................................................................................. 5 1. Classification of non-Newtonian fluids............................................... 7 1.1. Non-Newtonian fluids with rheological characteristics independent of time ................................................................................... 10 1.2. Non-Newtonian fluids with time-dependent rheological characteristics.......................................................................... 17 1.3. Viscoelastic materials ......................................................................... 22 2. Experimental determination of characteristics of rheological fluids ...................................................................................... 28 2.1. Methods of studying of stationary rheological fluids .......................... 28 2.2. Coaxial-cylindrical viscosimeters ....................................................... 28 2.2.1 Newtonian fluid ................................................................................ 31 2.2.2. Pseudoplastic and dilatant fluids ...................................................... 32 2.2.3 Shvedov-Bingham plastics................................................................ 34 2.3. Cone-plate viscosimeter ...................................................................... 38 2.4. Stationary rheological fluids in viscosimeters with a capillary tube........................................................................................... 39 2.5. Methods for experimental study of characteristics of rheological non-stationary fluids ........................................................... 42 2.6. Experimental study of viscoelastic materials ...................................... 43 2.7. Fluid sliding near the surface of a solid body ..................................... 47 3. Flow of non-Newtonian fluids in pipes ............................................... 50 3.1. Liquid flow rate in the round tube....................................................... 51 3.2. The flow of a Newtonian fluid ............................................................ 53 3.3. Flow of Shvedov-Bingham plastics .................................................... 54 3.4. The flow of «power» liquids ............................................................... 57 3.5. Axial flow in the annular channel (basic ratio) ................................... 59 3.6. Flow of Shvedov-Bingham plastics in the annular channel ................ 61 3.7. Flow of «power» liquids in the annular channel ................................. 67 3.8. Heat exchange in the laminar flow in the pipe .................................... 70 4. Boundary layer in non-Newtonian liquids ......................................... 77 4.1. Equations of the boundary layer of «power» liquids........................... 79 4.2. Boundary conditions ........................................................................... 84 4.3. Automodel problems of the boundary layer ........................................ 87 5. Exact solutions to the problems of the theory of a tationary boundary layer ................................................................. 89

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5.1. The boundary layer with a power-law velocity distribution at the outer boundary ................................................................................. 89 1

5.2. Wedge flow for U = Ax 3 ............................................................... 95 5.3. Flow of a homogeneous fluid flow around a flat permeable plate .......................................................................................... 98 5.4. The temperature boundary layer ......................................................... 107 5.5. Flat (radial) submerged stream ........................................................... 117 5.6. Liquid jet with a free surface ............................................................. 127 References ................................................................................................ 135

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INTRODUCTION The rapid development of chemistry in the field of oil refining, production of polymers, plastics and new building materials attracted special attention to theoretical and applied aspects of rheology (i.e., the science of deformations and fluidity), hydrodynamics and heat transfer in rheological systems. Hydrodynamics and heat-and-mass transfer of non-Newtonian fluids is a further development of classical hydrodynamics and theory of heat-and-mass transfer of viscous liquids. In recent decades, the interest of researchers and engineers to this problem has sharply increased, and it is intensively developed in the CIS and foreign countries. Demands of power-generating industry (high-temperature heat carriers based on polymers and suspensions, pastes and suspensions of nuclear fuel, highly concentrated filled rocket fuels and fuel mixtures, etc.), mass production and processing of synthetic and natural materials (for example, construction), oil production, petrochemical, pharmaceutical, food, paper, paint and varnish production stimulate the research and engineering developments in rheodynamics and heat-and-mass transfer of rheologically complex environments. In terms of power-generation, a scientifically-based choice and optimal calculation of efficient operational (for example, heat exchangers), transport and processing systems for rheologically complex fluids (mainly, high-viscosity fluids) is an important problem of national economy. For example, when designing pipeline transportation of typical non-Newtonian fuels such as black oil, the power engineer has to find optimal characteristics providing minimum annual economic costs for transportation of a given amount of black oil at a certain distance. The cost of transportation (including pipes, pumps, heating) in some cases can reach 50% of the cost of basic energy equipment. At a constant mass flow rate of the pumped liquid, the pressure loss due to friction, i.e., energy costs, progressively increase with a decrease in the diameter of the pipe and a decrease in temperature. 5

Therefore, the cost of pumping increases. On the other hand, initial costs on creation of the pipeline system are lower for pipes of smaller diameter. In this case it is much more difficult to find optimal values of Dopt and Topt as it is necessary to take into account nonNewtonian properties correctly, i.e. to know real rheological «head flow» curves for different temperature conditions of laminar or turbulent flow with temperature-dependent characteristics. Only on the basis of reliable quantitative estimates of the rheological factor is it possible to choose the most economical combination of costs for pipes, pumps, heaters, thermal insulation, etc. It is possible to intensify and increase productivity and cost effectiveness of commercial production and processing of rheological fluids only on the basis of new scientific and technical developments. Therefore, it is clear how important it is to study the problems of hydrodynamics, heat and mass transfer in rheological complex media. Below we will consider only laminar regimes, as turbulent flows are not typical of structured viscous-plastic, elastic-viscous and highly viscous non-Newtonian media.

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1

CLASSIFICATION OF NON-NEWTONIAN FLUIDS Classical hydromechanics of the XVIII-XІX centuries relied on a model of a nonviscous fluid, which could move with a shear without energy loss. The mathematical expression of this idea was Euler's equations of motion. Then, in the middle of the XIX century, the theory of a viscous liquid, based on the Newton model, establishing a linear connection between tangential stresses and the corresponding rates of irreversible shear deformation, appeared. In the tensor form, the model of a linearly viscous incompressible fluid is expressed by the generalized Newton law:

τ ij = − pσ ij + 2 µlij ,

(1.1)

where τіј are components of the stress tensor, lіј are components of the tensor of strain rate, p is the pressure, μ is the viscosity coefficient, σij is the Kronecker symbol, equal to unity for the same indices i = j and zero for different i≠j. For the conditions of a one-dimensional shear flow of an incompressible fluid, Newton's law is formulated in the form:

τ =µ Here ν =

du d =ν ( ρu ) . dy dy

(1.2)

µ is kinematic viscosity, ρ is the liquid density, u is ρ

the speed perpendicular to the y axis. Formula (1.2) is convenient as it allows us to consider the Newton postulate as a particular expression of a more general linear transport law including Fick's diffusion law. The value ρu can be identified with the volume concentration of the impulse (momentum 7

density). Hence, Newton's law, in accordance with the transfer laws mentioned above, establishes the proportionality of the momentum flux τ (tangential stress) to the gradient concentration ρu. The proportionality coefficient ν has the same dimension (L·T-1), as the other phenomenological transport coefficients – thermal diffusivity α and diffusion D: dI , (1.3) q = −α dy m = − D

dC . dy

(1.4)

Here I is the enthalpy of the unit of volume, С is the concentration, q and m  are densities of heat and mass fluxes, respectively. We can rewrite formula (1.2) in a different form. As dγ γ , d dl d dlx dU = = ( ) = ( x) = dt dt dy dy dy dt

where lx is the path length in the direction of speed u, γ = dl x is a dy relative shear, γ is a shear rate for the layered motion (Figure 1.1), then (1.5) τ = µ γ .

Figure 1.1. A scheme of the layered flow between parallel plates Figure

In this formula, μ is the tangential force per unit area applied to the fluid layers spaced from each other at a distance equal to one unit length with a unit difference in velocities between them. 8

Figure 1.2. The flow curve for a Newtonian fluid

The coefficient of Newtonian viscosity μ depends on the nature of liquid and its temperature, but does not depend on the shear rate. The relationship between the frictional stress and the shear rate, the so-called «flow curve», is a straight line (Figure 1.2) with a slope tangent equal to μ, and this single constant completely characterizes the liquid. Newtonian behavior is typical of liquids in which viscous dissipation of energy is due to the collision of small molecules. All gases, liquids and solutions with a small molar mass fall into this category. However, more complex liquids, for example, solutions and melts of polymers, suspensions, emulsions, pastes, etc., differ considerably from Newtonian liquids. Non-Newtonian fluids are liquids whose «flow curve» is not a straight line, i.e., viscosity of a non-Newtonian fluid is not constant at a given temperature, but also depends on the other factors: the rate of shear deformation, the structural features of the equipment in which the liquid is located, and the history of the flow. Real fluids with a nonlinear flow curve can be divided into three large groups: 1. Systems for which the shear rate in each point is a function of only the shear stress in this point. 2. More complex systems in which the relationship between voltage and shear rate depends on the time of action of the voltage or on the prehistory of the fluid. 3. Systems having properties of a solid and a liquid, and partially showing an elastic recovery of the shape after stress removal (socalled viscoelastic liquids). Let us consider the properties of these three types of liquids in more detail. 9

1.1. Non-Newtonian fluids with rheological characteristics independent of time Systems of the first type, whose properties do not depend on time (such fluids are sometimes called rheologically stationary or rheostable), can be described by a rheological equation:

γ = f (t ) ,

(1.1.1)

from which it follows that the shear rate in each point of the fluid is a function of only the shear stress in the same point. Such substances are called non-Newtonian viscous fluids (or nonlinear viscous fluids).

Figure 1.3. Flow curves for different types of rheologically stationary non-Newtonian fluids: 1 – Shvedov-Bingham plastic, 2 – pseudoplastic, 3 – Newtonian fluid, 4 – dilatant fluid

They can be divided into three groups, depending on the form of the function f (τ) in equation (1.1.1): A) Shvedov-Bingham plastic fluids (plastics); B) pseudoplastic fluids (pseudoplastics); C) dilatant fluids. The flow curves typical of these three groups of liquids are given in Fig. 1.3; for comparison, linear dependencies for Newtonian fluids are given. a) Shvedov-Bingham plastics The flow curve for these materials is a straight line intersecting the shear stress axis at a distance τ0 from its origin. Consequently, in 10

such media the flow begins only after exceeding a certain threshold, called the flow limit τ0. The quantity τ0 characterizes the plastic properties of the medium. The rheological equation for Shvedov-Bingham plastics (also called viscoplastic materials) can be written as:

τ − τ 0 = µ ρ γ (τ>τ0),

(1.1.2)

where μp is the plastic viscosity or shear stiffness coefficient, numerically equal to the tangent of the slope angle of the flow curve to the γ axis. This value characterizes fluid mobility. We note that the dependence similar to (1.1.2) was proposed by F.N. Shvedov in 1889 long before the appearance of the work of E. Bingham (1916). Therefore, plastic fluids satisfying the equation (1.1.2) are called Shvedov-Bingham fluids. The concept of Shvedov-Bingham idealized plastic body is very convenient in practice, since many real fluids are very close to this type. The examples include sludge, (small-sized crushing product in the enrichment of ore or coal), drilling muds, oil paints, toothpaste, sewage, concentrated lubricants, pulps (finely ground ores, liquefied by water or liquid solvents for metal recovery or enrichment), rocket fuels, blood. Peat masses and aqueous suspensions of nuclear fuel (oxides of uranium and thorium) also refer to this group. Below the flow limit, Shvedov-Bingham plastic medium can behave either as a solid body or as an ideally elastic body. The explanation of the behavior of viscoplastic materials is based on the assumption that the liquid at rest has a spatial rigid structure, which is strong enough to resist any shear stress exceeding τ0, the stress, structure is completely destroyed and the system behaves like an ordinary Newtonian fluid at shear stresses τ-τ0. When the shear stress becomes less than τ0, the spatial rigid structure is again restored. By the analogy with the Newtonian fluid, we will introduce the concept of viscosity as the ratio of the shear stress to the shear rate:

µa =

τ . γ

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(1.1.3)

The quantity μа is called an apparent or effective viscosity of a non-Newtonian fluid. In the case of viscoplastic fluid

µa = µ p +

τ0 . γ

(1.1.4)

Hence it follows that the apparent and plastic viscosities do not coincide with each other, although they have the same dimensions. They can differ considerably at small and moderate shear rates (especially for large values of τ0). For γ → 0 we get an unrealistic result: µ a → ∞ . This property can be attributed only to an absolutely rigid body. If the shear rate is γ → ∞ , then μа and μр (μа=μр) coincide. Consequently, the actual fluidity of viscoplastic media is variable and is not a characteristic of the substance, but strongly depends on the rate of its deformation, and the dependence of μа on γ is a decreasing dependence. It means that the formula (1.1.4) is

valid in a limited interval of γ variation. The second rheological characteristic of viscoplastic fluids is the flow limit τ0. If the dependence of τ on γ is studied in detail, then in the region of small γ values it turns out to be nonlinear (Figure 1.4.). Therefore, static τс and dynamic τ0 limiting shear stresses have different values. The first of them characterizes the strength of the internal structure of the material and is numerically equal to the shear stress at which the liquid starts flowing from the equilibrium position. Thus, the rheological parameter τc has the meaning of a physical or, more accurately, physicochemical structural characteristic.

Figure 1.4. Dynamic and static limiting shear stresses

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While the technique of geometry and equipment were imperfect, the area of very slow currents, i.e., small and very small shear rates was not available for investigation, and the obtained flow curves of viscoplastic media were presented as rectilinear. Later, it was found out that in the range of low shear rates the dependence τ( γ ) turned out to be nonlinear. Such a behavior of the real flow curve contradicted the trend predicted by the Shvedov-Bingham linear model used in calculations. For this reason, it was suggested to ignore the curvature of the initial section and to extrapolate the rectilinear part of τ( γ ) dependence to the zero shear rate. Then, on the stress axis, a certain segment τ0 is cut off – the dynamic yield strength. Consequently, the value of τ0, unlike τc, is a conditional, purely calculated characteristic that cannot be directly measured. b) Pseudoplastics Pseudoplastic liquids do not show any yield stress, and their flow curve shows that the ratio of shear stress to shear rate, i.e. the apparent viscosity μa, gradually decreases with increasing shear rate (curve 2 in Fig. 1.3). The flow curve has a linear section at low shear rates, i.e., the substance behaves like a Newtonian fluid. In this range of shear rates, the liquid is characterized by the apparent viscosity at zero shear μ0. Another linear section on the flow curve is observed at very high shear rates. Corresponding limiting slope of the curve determines viscosity at infinite shear μ It should be noted that for both limiting sections, the values of μ0 and μ∞ are real viscosity characteristics of non-Newtonian fluids. The relationship between the shear stress and its rate in logarithmic coordinates for pseudoplastic materials is often linear. Then, to describe the fluids of this type, one can establish an empirical functional dependence in the form of a power law. Such a dependence, first proposed by Ostwald and then improved by Rayner, can be written in the form

τ = k γ 13

n −1

γ ,

(1.1.5)

where k and n are constants (n 1) it decreases, and for pseudoplastic liquids (n τ2). When the torque acting on the surface of the outer cylinder is small, the shear stress is below the yield strength of the ShvedovBingham plastic τ0. Since the condition τ1> τ2 is preserved, τ2 is smaller than τ0 (τ2> 1 . Under this condition, the expression for B is simplified: n −1 n −1  ∂u  B = Re n + 1    ∂y 

and equations (4.1.3)-(4.1.5) take the form:

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2 n −1   − ∂u  ∂Ρ ∂  ∂u  ∂u ∂u ∂u + + 2 Re n + 1   =− +v +u ∂x  ∂x ∂x  ∂y  ∂y ∂x ∂t    2  n − 1 −  ∂u ∂ ∂  ∂u  v 1 +  + Re n , +   ∂x  ∂y  ∂y   ∂y     2 − n + 1  ∂v + u ∂v + v ∂v  = − ∂Ρ + Re  ∂t ∂x ∂y  ∂y   2 2   −   ∂u  n − 1 ∂u v ∂ ∂   +  + Re n + 1 + Re n + 1   ∂x  ∂y  ∂x   ∂y      2 n −1   − ∂v  n + 1 ∂  ∂u  + 2 Re , ∂x  ∂y  ∂y    −

∂u ∂v =0 + ∂x ∂y 2   ∂T ∂T ∂T 1  − n + 1 ∂ 2T ∂ 2T  .  + =  Re +u +v 2 ∂y 2  ∂y Pr  ∂t ∂x ∂ x    

For Re >> 1 these equations are transformed into: dΡ ∂  ∂u  ∂u ∂u ∂u =− +   +v +u dx ∂y  ∂y  ∂y ∂x ∂t ∂u ∂v ∂Ρ = 0, + = 0, ∂x ∂y ∂y ∂T ∂T ∂T 1 ∂ 2T . = +u +v ∂t ∂x ∂y Pr ∂y 2 83

n

(4.1.15)

Let us consider the properties of these equations. As can be seen from (4.1.10), the finite values of the dimensionless variable y for large Reynolds numbers correspond to small values of the dimensional coordinate y. The same tendency was detected for the transverse components of the velocity v and v . Hence, the resulting equations (4.1.15) describe the motion in a thin fluid layer near the zero current line. In this region v 0 and α are constants, and the dimensionless constant α characterizes the rate of change of the velocity U (x) and the intensity of the pressure gradient in the external potential flow beyond the boundary layer. The validity of the last assertion follows directly from equation (4.2.3) for a stationary flow: ρU

dU dP , =− dx dx

(5.1.2)

whence dP = − ραA2 x 2α − 1 . dx

89

(5.1.3)

It can be seen from (5.1.1) and (5.1.3) that for α=0 the velocity U and the pressure P are constant. This condition corresponds to the well-known Blasius problem of the flow of a homogeneous fluid around a semi-infinite plate. If α0, the dependence is the opposite. The linear increase in the velocity of the external potential flow (α=0) is realized in the case of a transverse flow around a frontal critical point on the blunt cylindrical body. Of special interest is the value α = 1 , which as will be shown 3

below corresponds to the flow around a rectangular wedge located symmetrically with respect to the incoming flow. In this case, the frictional stress on the streamlined surface remains constant, and the temperature field is automodel. To transform the equations of a stationary dynamic boundary layer (4.1.17) into an ordinary differential equation, we introduce, as usual, automodel variables: u =U

β dF = Axα F ′(ϕ ) , ϕ = Byx . dϕ

(5.1.4)

The transverse velocity component is found from the continuity equation: y ∂u A α − β −1 [(α − β )F + βϕF ′] . (5.1.5) V = − ∫ dy = − x B ∂ x 0 Substituting u and V from (5.1.4) and (5.1.5) into the first equation of the system (4.1.17), and taking into account (5.1.3) we obtain ∂u  for the stationary flow  = 0 : ∂ t  

αA2 x 2α −1F ′2 − (α − β )A2 x 2α −1FF ′′ = =n

k

ρ

An B n +1 xαn + β (n +1) F ′′ 90

n −1

F ′′′ + αA2 x 2α −1

.

(5.1.6)

It follows that

β=

2α − αn − 1 n +1

(5.1.7)

and as B is an arbitrary constant, we can take it equal to 1  2αn − α + 1 ρ 2 − n  n + 1 . A B=  nk  n +1 

(5.1.8)

Then the equation (5.1.6) takes the form:

F ′′

n − 1 ′′′ F + FF ′′ = λ  F ′2 − 1 ,  

where

λ=

α (n + 1) . 2αn − α + 1

(5.1.9)

(5.1.10)

The boundary conditions for equation (5.1.9) are: F (0 ) = N , F ′(0 ) = 0 , F ′(∞ ) = 1 .

(5.1.11)

In the case of an impermeable wall N = 0. For a wall boundary layer on a streamlined body (an external problem) ∂u > 0 and ∂y , therefore, equation (5.1.9) takes the form: ′ ′ F (0 ) > 0

(F ′′)n − 1 F ′′′ + FF ′′ = λ  F ′2 − 1 . 



(5.1.12)

Taking into account (5.1.7) and (5.1.8), we have for the dimensionless variable ϕ :

1  2αn − α + 1 ρ 2 − n 2α − αn − 1 n + 1 . ϕ = y A x   n(n + 1) k  91

(5.1.13)

Equation (5.1.12) contains a single parameter λ combining geometric and rheological characteristics of the problem. For a Newtonian fluid (n = 1), 2α λ= α +1 coincides with the Fokner-Scan parameter for the problem of the flow past a flat wedge with a cone angle πλ by a flow homogeneous far from the wedge. If in our case (n≠1) we follow this interpretation, it should be noted that for a fixed value of the exponent α the angle of the wedge is determined by the current index n. Only in one case, 1 1 for all values of the rheological namely, α = , we obtain λ = 2 3 parameter n. The angle of the wedge is equal to

π

2

.

Figure 5.1 shows the dependences of λ(n) for a number of fixed 1 values of α. It is seen that for all α > , the value of λ decreases 3 with increasing n. For n → ∞ the curves λ(n) approach the general asymptote λ=

1 1 from the above. In the case 0 < α < , the curves asymptoti3 2

cally approach the value λ = 1 from below. 2

Figure 5.1. The dependence of the parameter λ on the flow index n for different values of α

92

For α=0 (a longitudinally streamlined plate) we have λ=0 regardless of the flow index n. In the case of negative α (which can be interpreted, for example, for A