Lie Group Analysis of Equations Arising in non-newtonian Fluids

Transcription

1 Lie Group Analsis of Equations Arising in non-newtonian Fluids Hermane Mambili Mamboundou A thesis submitted to the Facult of Science, Universit of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of PhD Johannesburg, Feb 28

2 ABSTRACT It is known now that the Navier-Stokes equations cannot describe the behaviour of fluids having high molecular weights. Due to the variet of such fluids it is ver difficult to suggest a single constitutive equation which can describe the properties of all non-newtonian fluids. Therefore man models of non-newtonian fluids have been proposed. The flow of non-newtonian fluids offer special challenges to the engineers, modellers, mathematicians, numerical simulists, computer scientists and phsicists alike. In general the equations of non-newtonian fluids are of higher order and much more complicated than the Newtonian fluids. The adherence boundar conditions are insufficient and one requires additional conditions for a unique solution. Also the flow characteristics of non-newtonian fluids are quite different from those of the Newtonian fluids. Therefore, in practical applications, one cannot replace the behaviour of non-newtonian fluids with Newtonian fluids and it is necessar to examine the flow behaviour of non-newtonian fluids in order to obtain a thorough understanding and improve the utilization in various manufactures. Although the non-newtonian behaviour of man fluids has been recognized for a long time, the science of rheolog is, in man respects, still in its infanc, and new phenomena are constantl being discovered and new theories proposed. Analsis of fluid flow operations is tpicall performed b examining local conservation relations, conservation of mass, momentum and energ. This analsis gives rise to highl non-linear relationships given in terms of differential equations, which are solved using special non-linear techniques. i

3 Advancements in computational techniques are making easier the derivation of solutions to linear problems. However, it is still difficult to solve non-linear problems analticall. Engineers, chemists, phsicists, and mathematicians are activel developing non-linear analtical techniques, and one such method which is known for sstematicall searching for exact solutions of differential equations is the Lie smmetr approach for differential equations. Lie theor of differential equations originated in the 87s and was introduced b the Norwegian mathematician Marius Sophus Lie ( ). However it was the Russian scientist Ovsannikov b his work of 958 who awakened interest in modern group analsis. Toda, the Lie group approach to differential equations is widel applied in various fields of mathematics, mechanics, and theoretical phsics and man results published in these area demonstrates that Lie s theor is an efficient tool for solving intricate problems formulated in terms of differential equations. The conditional smmetr approach or what is called the non-classical smmetr approach is an extension of the Lie approach. It was proposed b Bluman and Cole 969. Man equations arising in applications have a paucit of Lie smmetries but have conditional smmetries. Thus this method is powerful in obtaining exact solutions of such equations. Numerical methods for the solutions of non-linear differential equations are important and nowadas there several software packages to obtain such solutions. Some of the common ones are included in Maple, Mathematica and Matlab. This thesis is divided into six chapters and an introduction and conclusion. The first chapter deals with basic concepts of fluids dnamics and an introduction to smmetr approaches to differential equations. In Chapter 2 we investigate the influence of a time-dependent magnetic field on the flow of an incompressible third grade fluid bounded b a rigid plate. Chapter 3 describes the modelling of a fourth grade flow caused b a rigid plate moving in its own plane. The resulting fifth order partial differential equation is reduced using smmetries and conditional smmetries. In Chapter 4 we present a Lie group analsis of the third oder PDE ii

4 obtained b investigating the unstead flow of third grade fluid using the modified Darc s law. Chapter 5 looks at the magnetohdrodnamic (MHD) flow of a Sisko fluid over a moving plate. The flow of a fourth grade fluid in a porous medium is analzed in Chapter 6. The flow is induced b a moving plate. Several graphs are included in the ensuing discussions. Chapters 2 to 6 have been published or submitted for publication. Details are given in the references at the end of the thesis. iii

5 DECLARATION I declare that the contents of this thesis are original except where due references have been made. It has not been submitted before for an degree to an other institution. H. Mambili Mamboundou iv

6 DEDICATION To m Famil and m Friends v

7 ACKNOWLEDGEMENTS Firstl I would like to thank m supervisor Prof F M Mahomed who encouraged and challenged me throughout the writing up of this thesis. He guided and supported me through the entire process and I cannot thank him enough for that. I am also grateful to Prof Tasawar Haat whose enriching visits to the School of Computational and Applied Mathematics have been important and invaluable for the completion of this thesis. I am indebted to him for providing all the models on which this work is based without which this thesis would not have been possible. I also take this opportunit to thank m famil for their moral support and especiall m elder brother whose constant phone calls have given me strength to go through this work. Finall, I am grateful to the Universit of Witwatersrand for financial assistance under the Postgraduate Merit Award and the School of Computational and Applied Mathematics for awarding me a research grant. vi

8 Contents Introduction Fluid Dnamics and Computational Approaches 4. Definitions Compressible and incompressible flows Unstead and stead flows Laminar and turbulent flows Newtonian and non-newtonian fluids Basic equations Equation of continuit Equation of motion Governing equations for unidirectional flow of a fourth grade fluid Differential Equations: Algebraic and Computational Approaches Classical smmetr method for partial differential equations Example on classical smmetr method Non-classical smmetr method Group invariant solution The Raleigh problem for a third grade electricall conducting fluid with variable magnetic field 2 2. Basic equations Mathematical formulation vii

9 2.3 Smmetr analsis Case : a 3 =, a Case 2: a 2 =, a Phsical invariant solutions Invariant solution corresponding to case Invariant solution corresponding to case Numerical solution of the PDE (2.4) Concluding remarks Effect of magnetic field on the flow of a fourth order fluid Mathematical formulation Solutions of the problem Numerical solutions Results and discussion Solutions in a third grade fluid filling the porous space Problem formulation Solutions of the problem Lie smmetr analsis Travelling wave solutions Group invariant solutions corresponding to X Group invariant solutions corresponding to X Numerical solution Results and discussion Reduction and solutions for MHD flow of a Sisko fluid in a porous medium 6 5. Governing equations Problem formulation Exact solutions Exact solutions with magnetic field and porosit viii

10 5.5 Results and discussion A note on some solutions for the flow of a fourth grade fluid in a porous space 9 6. Flow development Solutions of the problem Conditional smmetr solutions Numerical solution of the PDE (6.4) Summar and Conclusions Conclusions References 3 ix

11 Introduction The stud of non-newtonian fluids involves the modelling of flow with dense molecular structure such as polmer solutions, slurries, pastes, blood and paints. These materials exhibit both viscous properties like liquids and elastic properties like solids and the understanding of their complex behaviour is crucial in man industrial applications. Due to increasing importance of non-newtonian fluids in modern technolog and industries, the investigation of such fluids is desirable. The flows of non-newtonian fluids occur in a variet of applications, for example from oil and gas well drilling to well completion operations, from industrial processes involving waste fluids, snthetic fibers, foodstuffs, extrusion of molten plastic and as well as in some flows of polmer solutions. Some important studies dealing with the flows of non- Newtonian fluids are made b Abel-Malek et al. [], Ariel et al. [3], Chen et al. [6], Fetecau and Fetecau [], [], [2], Haat and Ali [8], Haat and Kara [9], Haat et al. [2], [22], [27], [34], Rajagopal and Gupta [46], Rajagopal and Na [48], [49] and Wafo-Soh [58]. Modelling visco-elastic flows is important for understanding and predicting the behaviour of processes and thus for designing optimal flow configurations and for selecting operating conditions. Because of the complex nature of these fluids there is not a single constitutive equation available in the literature which describes the flow properties of all non-newtonian fluids. For this reason various models have been suggested and among those models, power-law and differential tpe fluids have acquired a great deal of attention. Some relevant contributions dealing with this tpe of fluids are given in references [3], [4], [5], [9], [25], [35], [38], [39], [55] and [56].

12 Power-law fluids, also referred to as fluids of grade one are the simplest models of non- Newtonian fluids and it is well-known for accuratel moodelling the shear stress and shear rate of non-newtonian fluids, but it does not properl predict the normal stress differences that are observed in phenomena like die-swell and rod-climbing [53] which are manifestation of the stresses that develop orthogonal to the plane of shear which can be well modelled b extending the stud to the fluid of grade two. In turn this does not fit shear thining and shear thickening fluids. The third grade fluid model represents a further, although inconclusive, attempt towards a more comprehensive description of the behavior of viscoelastic fluids. Accordingl certain effects ma well be described b flow of fourth grade fluids [36]. On the other hand, the governing equations resulting from non-newtonian fluid models are non-linear high order equations whose analsis present a particular challenge to researchers. Hence progress was limited until recent times and closed-form solutions are available to more problems of particular interest than before. Also the stud of such flows in porous media are quite important in man engineering fields such as enhanced oil recover, paper and textile, but little work seems to be available in the literature. Few recent studies [28], [37], [38], [55], [56] mabe mentioned in this direction. Similarl the stud of hdrodnamic flows with application of magnetic field (MHD flows) is of particular interest in chemical engineering, electromagnetic propulsions and the stud of the flow of blood and et again the literature is scarce. Mention mabe made here to the recent stud of the topic b Haat et al.[2], [23], [24], [25] and [33]. Motivated b the above, in Chapter 2 we take a look at the Raleigh problem for the flow of third grade fluid, when the initial profile is arbitrar. This chapter covers the mathematical modelling and solutions of the applied magnetic field on the viscoelastic flow of order three bounded b a rigid plate. The findings of this chapter has been written up as a paper and accepted in the Journal of Nonlinear Mathematical Phsics [29]. In Chapter 3 the unstead unidirectional MHD flow of an incompressible fourth order fluid 2

13 bounded b a rigid plate is discussed. The flow is induced due to arbitrar velocit V (t) of an insulated plate. The fluid is electricall conducting in the presence of a uniform magnetic field. The stud of the motion of non-newtonian fluids b the application of a magnetic field has man applications, such as the flow of liquid metals and allos, flow of plasma, flow of mercur amalgams, flow of blood and lubrication with heav oils and greases. We have submitted this work to the Journal of Nonlinear Analsis Series B for publication [3]. In the remaining Chapters, 4, 5 and 6, we have concentrated on the stud of flows in a porous space. Recentl Tan and Masuoka [55] analzed the Stokes first problem for a second grade fluid in a porous medium. In another paper, Tan and Masuoka [56] studied the Stokes first problem for an Oldrod-B fluid in a porous medium. In these investigations, the authors have used the modified Darc s law. Chapter 4 s goal is to determine the analtical solution for an unstead flow of a third grade fluid over a moving plate. The relevant problem is formulated using modified Darc s law of third grade fluid. Two tpes of analtical solutions are presented and discussed, see [3]. Chapter 5 deals with the investigation of a MHD flow of a power law fluid filling the porous half-space. The formulation of the problem is given using modified Darc s law for a Sisko fluid. The exact analtical and numerical solutions are constructed. The phsical interpretation of the obtained results is made through graphs. This work was accepted in the Journal of Porous Media [4]. Chapter 6 concentrates on the exact analtical and numerical solutions for MHD flow of a non-newtonian fluid filling the porous half-space. The formulation of the problem is given using modified Darc s law for a Sisko fluid. The fluid is electricall conducting and a uniform magnetic field is applied normal to the flow b neglecting the induced magnetic field. The exact analtical solutions have been constructed using the similarit approach. We have reported this work in [32]. Finall Chapter 7 is concerned with the analsis of obtained results and the conclusion of the thesis. 3

14 Chapter Fluid Dnamics and Computational Approaches This chapter deals with some useful concepts in the field of fluid mechanics and the description of reduction methods of differential equations. We start b defining some notion of fluids. Then we present how the equations governing the motion of unidirectional flow of third and fourth grade fluids are derived and finall we review some techniques of solving nonlinear differential equations.. Definitions This section contains some basic definitions necessar for the subsequent chapters. Some of this are taken from [7]. Others are referred to in the text... Compressible and incompressible flows An incompressible flow is a flow in which the variation of the mass per unit volume (densit) within the flow is considered constant. In general, all liquids are treated as the incompressible fluids. On the contrar, flows which are characterised b a varing densit are said to be compressible. Gases are normall used as the compressible fluids. However, all fluids in realit are compressible since an change in temperature or pressure result in changes in 4

15 densit. In man situations, though, the changes in temperature and pressure are so small that the resulting changes in densit is negligible. The mathematical equation that describes the incompressibilit propert of the fluid is given b Dρ Dt =, (.) where D/Dt is the substantive derivative defined b D Dt = t + V., (.2) in which V represents the velocit of the flow...2 Unstead and stead flows A stead flow is one for which the velocit does not depend on time. When the velocit varies with respect to time then the flow is called unstead...3 Laminar and turbulent flows Laminar flow is one in which each fluid particule has a definite path. In such flow, the paths of fluid particules do not intersect each other. In turbulent flow the paths of fluid particules ma intersect each other (see figures below, taken from [8] and [9]). Figure.: Laminar flow 5

16 Figure.2: Turbulent flow..4 Newtonian and non-newtonian fluids Fluids such as water, air and hone are described as Newtonian fluids. These fluids are essentiall modelled b the Navier-Stokes equations, which generall describe a linear relation between the stress and the strain rate. On the other hand, there is a large number of fluids that do not fall in the categor of Newtonian fluids. Common examples like maonnaise, peanut butter, toothpaste, egg whites, liquid soaps, and multi grade engine oils are non-newtonian. A distinguishing feature of man non-newtonian fluids is that the exhibit both viscous and elastic properties and the relationship between the stress and the strain rate is non-linear. Contrar to Newtonian fluids, there is not a single model that can describe the behaviour of all the non-newtonian fluids. For this reason man models have been proposed and among those models, the fluid of differential tpe and rate tpe have acquired particular attention. In this thesis we discuss some models of differential tpe namel the third and fourth grade fluids. 6

17 .2 Basic equations In this section, we derive the equations that describe the motion of fluids in general. These equations are the basic laws of conservation of mass and momentum. The flow modeling of the subsequent chapters is based upon these two conservation laws..2. Equation of continuit Let Ω be a controlled volume and Ω its bounded surface, so that the fluid can move in and out of the boundar. The conservation of mass states that the rate of change of matter into the controlled volume is constant. Therefore we have d ρdω =, dt Ω (.3) in which ρ is the densit of the fluid and d/dt is the ordinar time derivative. Using the Renold s transport theorem, the left hand side of equation (.3) gives ( ) d ρ ρdω = +. (ρv) dω, (.4) dt Ω Ω t where V is the velocit at which the fluid is moving. Equations (.3) and (.4) ield Ω ( ρ t ) +. (ρv) dω =. (.5) For an arbitrar controlled volume Ω, we can drop the integral in equation (.5) to obtain the continuit equation ρ t +. (ρv) =. (.6) In the case of an incompressible fluid, the densit ρ does not var and the continuit equation (.6) is equivalent to the following equation.v =. (.7) 7

18 .2.2 Equation of motion The motion of a fluid is generall governed b the the equation of continuit derived above and the conservation of linear momentum. The conservation law of momentum states that the rate of change of linear momentum over a control volume Ω bounded b Ω must equal what is created b external forces acting on the control volume, minus what is lost b the fluid moving out of the boundar. Thus d ρvdω = ρfdω Π.n d Ω, (.8) dt Ω Ω Ω where V is the velocit of the fluid, ρ is the densit, f is the bod force per unit mass and Π is the linear momentum current densit given b Π = (ρv) V T, (.9) in which T is the Cauch stress tensor and denotes the tensor product. Using the divergence theorem, the last part of the right hand side of equation (.8) can be written as Π.n d Ω = (.Π) dω. (.) Ω Ω Substituting equation (.) into equation (.8) we have d ρvdω = ρfdω (.Π) dω. (.) dt Ω Ω Ω Since the control volume Ω is invariant in time, we can take the d/dt under the integral and equation (.) becomes ( ) (ρv) +.Π dω = t Ω For an arbitrar volume Ω we can drop the integral and we have Ω ρfdω. (.2) (ρv) +.Π = ρf. (.3) t The equation of motion in general is commonl represented b [ ] V ρ t + (V. )V =.T + ρf. (.4) In the above equation the force f is the bod force, for example, gravit and electromagnetic forces. In this thesis, flows of non-newtonian fluids with applied magnetic field are considered. 8

19 .3 Governing equations for unidirectional flow of a fourth grade fluid In this section we derive the equation which governs the unstead unidirectional flow of an incompressible fourth grade fluid. We make the derivation of such an equation b emploing the following equations of continuit and momentum: divv =, (.5) ( ) ρ t + (V. ) V = p + divs + r, (.6) where V is the velocit, ρ is the fluid densit, t is the time, p is the pressure, S is the extra stress tensor and r is the Darc s resistance in a fourth grade fluid. The constitutive equation in an incompressible fourth grade fluid is S = µa + α A 2 + α 2 A 2 + S + S 2, (.7) S ( ) = β A 3 + β 2 (A 2 A + A A 2 ) + β 3 tra 2, (.8) S 2 = γ A 4 + γ 2 (A 3 A + A A 3 ) + γ 3 A γ ( 4 A2 A 2 + A2 A ) 2 (.9) + γ 5 (tra 2 )A 2 + γ 6 (tra 2 )A 2 + (γ 7trA 3 + γ 8 tr (A 2 A ))A, (.2) in which µ is the dnamic viscosit and α i (i =, 2), β j (j = 3) and γ k (k = 8) are the material constants. It should be noted that if α i = β j = γ k =, we recover the Navier-Stokes model. For γ k =, we recover the third grade fluid and when γ k = β j = then we have a second grade fluid. The Rivlin-Ericksen tensors A i (i = 4) are defined through the following relations: A = ( V) + ( V) T, (.2) ( ) A n = t + V. A n + A n ( V) + ( V) T A n, n 2. (.22) For unidirectional flow, the velocit is given b V = (u(, t),, ), (.23) 9

20 where u is the velocit in x-direction. Note that the velocit defined in Eq. (.23) satisfies the incompressibilit condition automaticall. It is a well-known fact that Darc s law holds for viscous flows having low speed [55], [56]. This law provides a relationship between the pressure drop and the velocit. Moreover, this law is invalid for a porous medium having boundaries. In such a situation, Brinkman law is valid. On the basis of the Oldrod constitutive equation, the following expression of pressure drop has been suggested [56]: ( + λ ) p = µφ t k ( ) + λ r V, (.24) t in which Φ is the porosit, k is the permeabilit of the porous medium, λ is the relaxation time and λ r is the retardation time. The pressure gradient in the above expression is also interpreted as the measure of the resistance to the flow. Therefore, the x-component of the Darc s resistance through Eq. (.24) is given b r x = µφ k ( ) + λ r u. (.25) t Using Eq. (.24) we have through Eqs. (.2) and (.22) the following expressions A = A 2 = A 3 = u u u t (.26) 2 u ( t ) 2 u 2,, (.27) 3 u 2 t 6 u 2 u, (.28) t 3 u 2 t

21 A 3 = A 4 = 3 u t 2 6 u 2 u, (.29) t 4 u t u 2 u 3 u, (.3) t t 2 3 u t 2 4 u t 3 with u V =, (.3) ( V) T = u In view of Eqs. (.6) and (.23) one has ρ u t. (.32) = p x + S x + r x, (.33) = p + S + r, (.34) = p z + S z + r z (.35) where S x, S and S z can be computed through relations (.7)-(.2) and r x, r and r z are the components of the Darc s resistance in the x, and z-directions. Emploing Eqs. (.7)-(.2) we have

22 S x = µ u + α 2 u t + β 3 u t 2 + 2(β 2 + β 3 ) ( ) 3 u ( ) 4 2 u u +γ t 3 + (6γ 2 u 2 + 2γ 3 + 2γ 4 + 2γ 5 + 6γ 7 + 2γ 8 ) t, (.36) ( ) 2 ( ) u 3 u S = p + (2α + α 2 ) + 2(3β + β 2 ) t ( ) u 2 u +6γ + 2(γ + γ 2 ) u ( ) 3 u t t t 2 ) 2 ) 4 + 2(2γ 3 + 2γ 4 + 2γ 5 + γ 6 ), (.37) S z =. +γ 3 ( 2 t ( u (.38) Utilizing the same idea as in Eq. (.25) we have the following expression of the Darc s resistance in the x-direction ( r x = µ + α t + β 2 t + 2 (β β 3 ) + (6γ 2 + 2γ 3 + 2γ 4 + 2γ 5 + 6γ 7 + 2γ 8 ) u ( ) 2 u 3 + γ t 3 2 u t ) Φu k, (.39) and r = r z = (since u = w = ). Here u and w are the and z components of the velocit. Substituting Eqs. (.36) and (.39) into Eq. (.33) and then neglecting the pressure gradient we get ρ u ( ) t = u µ 2 + α 3 u 2 2 t + β 4 2 u u 2 t + 6(β 2 u β 3 ) [ 2 ( u 5 u +γ 2 t + (6γ γ 3 + 2γ 4 + 2γ 5 + 6γ 7 + 2γ 8 ) ) ] 2 2 u t [ ( ) µ + α t + β 2 2 u 3 t 22(β 2 + β 3 ) + γ t + (6γ γ 3 + 2γ 4 +2γ 5 + 6γ 7 + 2γ 8 ) u ] 2 u φu t k. (.4) 2

23 .4 Differential Equations: Algebraic and Computational Approaches In this section we present some features, and useful methods for solving partial differential equations. We start b presenting the Lie classical approach for solving differential equations followed b the nonclassical method and later on we utilise computational methods based on finite differences and other associated methods. Here, we restrict the analsis for our use of smmetr method with respect to one dependent variable. The reader is referred to the books [5], [4] and [42]..4. Classical smmetr method for partial differential equations We consider the p th order partial differential equation in n independent variables x = (x,..., x n ) and one dependent variable u, E(x, u, u (),..., u (p) ) =, (.4) where u (k), k p, represent the set of all k th order derivatives of u, with respect to the independent variables defined b: k u u (k) = { }, (.42) x i,..., x ik with i, i 2,..., i k n. (.43) To find the smmetries of equation (.4), we fist construct the group of invertible transformations depending on a real parameter a, that leaves equation (.4) invariant, namel x = f (x, u, a), x 2 = f 2 (x, u, a),..., x n = f n (x, u, a), ū = g(x, u, a). (.44) The transformations (.44) have the closure propert, are associative, admit inverses and identit transformation and are said to form a one-parameter group. 3

24 Since a is a small parameter the transformations (.44) can be written b means of Talor series expansions as: x = x + aξ (x, u) + O(a 2 ),..., x n = x n + aξ n (x, u) + O(a 2 ), ū = u + aη(x, u) + O(a 2 ). (.45) The transformations (.45) above are the infinitesimal transformations, and the finite transformations are found b solving the Lie equations ξ ( x, ū) = d x da,..., ξ n( x, ū) = d x n dū, η( x, ū) = da da, (.46) subject to the initial conditions x ( x, ū, a) a= = x,..., x n ( x, ū, a) a= = x n, ū( x, ū, a) a= = u, (.47) where x = ( x, x 2,..., x n ). The transformations (.44) can also be represented b the Lie smmetr generator χ = ξ (x, u) x ξ n (x, u) x n + η(x, u) u. (.48) The functions ξ i and η are the coefficient functions of the operator χ which is also referred to as the infinitesimal generator or operator, and equation (.45) can be represented using (.48) b the following x ( + aχ)x,..., x n ( + aχ)x n, ū ( + aχ)u. (.49) The operator (.48) is a smmetr generator of Equation (.4) if: χ [p] E E= =, (.5) where χ [p] denotes the p th prolongation of the operator χ and is given b χ [] = χ + n i= η x i, (.5) u xi 4

25 χ [2] = χ [] +. n i= n j= χ [p] = χ [] χ [p ] + where u xi = u x i, u xi...x ik = η x i = D xi (η) η x ix j 2 u xi x j, (.52) n... i = n i p= η x i...x ip p u xi...x ip, (.53) k u x i...x ik and the additional coefficient functions satisf n u xj D xi (ξ j ), (.54) j= η x ix j = D xj (η x i ). n u xi x k D xj (ξ k ), (.55) k= n η x i...x ip = Dxip (η x i...x ip ) u xi...x ip x j D xip (ξ j ), (.56) j= where D xi is the total derivative operator given b D xi = n + u xi x i u + j= u xi x j x j +... (.57) The determining equation (.5) gives a polnomial in the derivatives of the dependent variable u, and according to Lie s theor those derivatives are taken independent. After separation of equation (.5) with respect to the partial derivatives of u and their powers, one gets an overdetermined sstem of linear homogeneous partial differential equations for the coefficient functions ξ i s and η. Solving the overdetermined sstem leads to the following cases: There is no smmetr, which means that the Lie point smmetr ξ i and η are all zero. The point smmetr has r arbitrar constants, in which case we obtain a r-dimensional Lie Algebra. The point smmetr admits some finite number of arbitrar constants and arbitrar functions; in this case we obtain a infinite-dimensional Lie Algebra. 5

26 .4.2 Example on classical smmetr method In this section, we illustrate the use of the classical smmetr method on the well-known Korteweg-de Vries equation given b u t + u xxx + uu x =. (.58) We look for the operator of the form χ = ξ (t, x, u) t + ξ2 (t, x, u) + η(t, x, u) x u. (.59) Equation (.59) is a smmetr operator of equation (.58) if χ [3] (u t + u xxx + uu x ) ut = uxx uux =, (.6) where the third prolongation in this case is: χ [3] = η u + ηt + η x + η xxx. (.6) u t u x u xxx Thus the determining equation (.6) becomes ( η t + η xxx + u x η + uη x) uxxx= ut uux =, (.62) and using equations (.54) to (.56) we get η t = D t (η) u t D t (ξ ) u x D t (ξ 2 ) = η t + u t η u u t (ξt + u tξu ) u x(ξt 2 + u tξu 2 ), (.63) η x = D x (η) u t D x (ξ ) u x D x (ξ 2 ) = η x + u x η u u t (ξ x + u x ξ u) u x (ξ 2 x + u x ξ 2 u), (.64) η xx = D x (η x ) u tx D x (ξ ) u xx D x (ξ 2 ) = η xx + u x (2η xu ξxx) 2 u t ξxx + u 2 x(η uu 2ξxu) 2 2u t u x ξxu u3 x ξ2 uu u2 x u tξuu + (η u 2ξx 2 )u xx 2u tx ξx 3u x u xx ξu 2 u t u xx ξu 2ξuu x u tx, (.65) 6

27 η xxx = D x (η xx ) u xxt D x ( ξ ) u xxx D x ( ξ 2 ) = η xxx + u x ( 2ηxxu ξ 2 xxx) ut ξ xxx + u 2 x (η xuu ξ xxu ) 2u t u x ξ xxu u 3 x ξ2 xuu u2 x u tξ xuu + u xx ( ηxu 2ξ 2 xx) 2utx ξ xx 3u xu xx ξ 2 xu u t u xx ξ xu 2u x u tx ξ xu + u x ( ηxxu + u x (2η xuu ξ 2 xxu) u t ξ xxu +u 2 x (η uuu 2ξxuu 2 ) 2u tu x ξxuu u3 x ξ2 uuu u2 x u tξuuu ) +u xx (η uu 2ξxu) 2 2u tx ξxu 3u x u xx ξuu 2 u t u xx ξuu u x u tx ξuu ( +u tx ξ xx 2u x ξxu u2 x ξ uu u ( xxξu) + uxx 2ηxu ξxx 2 ) +2u x (η uu 2ξxu) 2 2u t ξxu 3u 2 xξuu 2 2u x u t ξuu 3u xx ξu 2 2u tx ξu ( ( +u txx 3ξ x 3u x ξu) + uxxx ηu 3ξx 2 4u xξu 2 u tξu). (.66) We substitute the coefficient functions into the determining equation (.62) and the resulting is a polnomial in the derivatives of u. Since the unknown functions ξ, ξ 2 and η are independent of the derivatives of u, we can then separate with respect to the derivatives and their powers. One gets after simplifications the following linear sstem of partial differential equations u txx u x : ξ u = u txx : ξ x = u 2 xx : ξ 2 u = (.67) (.68) (.69) u xx u x : η uu = (.7) u xx : 3η xu 3ξxx 2 = (.7) u t : 3ξx 2 + ξt = (.72) u x : η uξx 2 + 3η xxu ξxxu 2 + uξ t ξ2 t = (.73) : uη x + η xxx + η t = (.74) 7

28 We solve the sstem of equations (.67) to (.74) and obtain ξ = c 3 2 c 4t, (.75) ξ 2 = c 2 + c 3 t 2 c 4x, (.76) η = c 3 + c 4 u, (.77) where c to c 4 are arbitrar constants of integration. We obtain a 4-dimensional Lie smmetr algebra generated b the operators χ = t χ 2 = x χ 3 = t x + u χ 4 = 3t t x x + 2u u (.78) (.79) (.8) (.8) where each χ i is obtained b substituting (.75) to (.77) into (.59) and letting c i to be some non zero constant, while c j = with j i. For instance χ is obtained b taking c = and c 2 = c 3 = c 4 =..4.3 Non-classical smmetr method There exist equations whose exact solutions cannot be obtained b using the classical Lie smmetr method. The non-classical method was first introduced b Bluman and Cole [4]. As compared to the classical smmetries, non-classical smmetries are smmetries of the equation along side an auxiliar condition, also called the invariant surface condition. Considering the p th order equation (.4) augmented b a side condition as given below E(x, u, u (),..., u (p) ) = ξ (x, u)u x + ξ 2 (x, u)u x ξ n (x, u)u xn η(x, u) =, (.82) one onl requires the subset of the solutions of (.4) to be invariant. Moreover, this provides a more general approach than the classical method. The general method for finding nonclassical smmetries consists in computing the p th prolongation of (.4) χ ( [p] E(x, u, u (),..., u (p) ) ) =, (.83) 8

29 and then reduce the resulting prolongation using the invariant surface condition. In general, man difficulties arise and the algorithm proposed b Peter Clarkson and Elizabeth Mansfield [7], avoid some of the difficulties encountered. The algorithm of calculating non-classical smmetr is not as straightforward as for the classical method. The computations are not eas to perform b hand and the resulting determining equation is an overdetermined sstem of non-linear equations. There are usuall n cases to be considered ξ u x + ξ 2 u x2 + + ξ n u xn + u xn = η, (.84) ξ u x + ξ 2 u x2 + + u xn = η, (.85). (.86) u x = η. (.87).4.4 Group invariant solution In this section, we state the use of smmetr for finding special exact solutions to differential equations. Given a smmetr operator of the form χ = ξ (x, u) x ξ n (x, u) x n + η(x, u) u, (.88) the group invariant solution u = u(x) is obtained from X(u u(x)) u=u(x) = and is deduced b solving the corresponding characteristic sstem defined as dx ξ (x, u) = dx 2 ξ 2 (x, u) =... = dx n ξ n (x, u) = du η(x, u). (.89) 9

30 Chapter 2 The Raleigh problem for a third grade electricall conducting fluid with variable magnetic field In this chapter the influence of time-dependent magnetic field on the flow of an incompressible third grade fluid bounded b a rigid plate is investigated. The flow is induced due to motion of a plate in its own plane with an arbitrar velocit. The solution of the equations of mass and momentum is obtained analticall. We also present a numerical solution with particular choices of the magnetic field and boundar conditions. Emphasis has been given to stud the effects of various parameters on the flow characteristics. In the present analsis we extend Wafo Soh s [58] analsis for the flow of a third grade fluid, when the initial profile is g(). Section 2. contains the basic equations. Section 2.2 deals with the formulation of the problem. Analtical solutions are presented in sections 2.3 and 2.4. Numerical solutions are presented in section 2.5. Section 2.6 contains the concluding remarks. 2. Basic equations Emploing Eqs. (.7) and (.4), the fundamental equations governing the MHD flow of an incompressible electricall conducting fluid are 2

31 divv =, ρ DV Dt = divt + J B, J = σ (E + V B) (2.) where V is the fluid velocit, ρ is the densit of the fluid, σ is the fluid electrical conductivit, J is the current densit, B is the magnetic induction so that B = B + b (B and b are the applied and induced magnetic fields respectivel), and D/Dt is the material time derivative. Emploing Eqs. (.7)-(.22), the Cauch stress tensor for a third grade fluid satisfies the following constitutive equation T = pi + µa + α A 2 + α 2 A 2 + β A 3 + β 2 [A A 2 + A 2 A ] + β 3 (tra 2 )A, A = gradv + (gradv) T, (2.2) ( ) A n = t + V. A n + A n gradv + (gradv) T A n, where the isotropic stress p I is due to the constraint of incompressibilit, µ denotes the dnamic viscosit, α i (i =, 2) and β i (i = 3) are the material constants, T indicates the matrix transpose and A i (i = 3) are the first three Rivlin-Ericksen tensors. Moreover, the Clausius-Duhem inequalit and the result that the Helmholtz free energ is minimum in equilibrium provide the following restrictions [6]: µ, α, β = β 2 =, β 3, α + α 2 24µβ 3. (2.3) 2.2 Mathematical formulation Consider the unidirectional flow of a third grade fluid, obeing equations (2.2) and (2.3), maintained above a non-conducting plate b its motion in its own plane with arbitrar velocit V (t). The fluid is magnetohdrodnamic with small magnetic Renold s ( number so ) that the induced magnetic field is negligible. B taking the velocit field u(, t),,, the conservation of mass equation is identicall satisfied and in the absence of modified pressure gradient the momentum balance equation (2.) along with equations (2.2) and (2.3) ields ( ) 2 u t u ν 2 α 3 u u 2 t ǫ 2 u MH2 (t)u =, (2.4) 2

32 where ν = µ ρ, α = α ρ, ǫ = 6β 3 ρ, M = σ µ2 ρ,h = B µ. (2.5) In equations (2.4) and (2.5) ν is the dnamic viscosit and µ is the magnetic permeabilit of the fluid. B neglecting the modified pressure gradient the dependence of (2.4) on α 2 has been removed. The relevant boundar and initial conditions are u(, t) = V (t), t >, u(, t) as, t >, (2.6) u(, ) = g(), < <, where V (t) and g() are as et arbitrar functions. These functions are constrained in the next section when we seek exact solutions using the Lie point smmetries method. Also in section 2.5, for the numerical solution, we choose specific functions for V (t) and g(). Section 2.6 contains the concluding remarks. 2.3 Smmetr analsis We present a complete Lie point smmetr analsis of the nonlinear partial differential equation (2.4). We find two cases for which equation (2.4) admits a Lie point smmetr algebra. These algebras are used to reduce the initial and boundar value problem (2.4)-(2.6) to solvable form. An operator (see Chapter, section.4) χ = τ t + ξ + η u, (2.7) where τ, ξ and η are functions of t, and u, is a Lie point smmetr generator of the partial differential equation (2.4) if (see Chapter, section.4) χ [3] {u t νu αu t ǫu 2 u + MH 2 (t)u} 22 (2.4) =, (2.8)

33 where χ [3] = τ t + ξ + η u + ηt + η + η + η t (2.9) u t u u u t is the third prolongation of the operator (2.7) in which the additional coefficient functions satisf η t = D t (η) u t D t (τ) u D t (ξ), η = D (η) u t D (τ) u D (ξ), η = D (η ) u t D (τ) u D (ξ), η t = D t (η ) u t D t (τ) u D t (ξ) with total derivative operators given b D = + u u + u t + u t D t = t + u t u + u tt +. u t The determining equation (2.8), after separation with respect to the partial derivatives of u and their products and their powers, gives rise to an overdetermined sstem of linear homogeneous partial differential equations for the coefficient functions τ, ξ and η. Solution of this sstem gives ξ = a, where a is a constant, and that H 2 is constrained b the ordinar differential equation dh 2 dt + 2βa 3e 2βt a 2 + a 3 e 2βtH2 = where a 2, a 3 are further constants and β = ν/α. 2β 2 a 3 e 2βt M(a 2 + a 3 e 2βt ), (2.) The solution for H 2 gives two sets of Lie point smmetries depending on the values of the constants in the ordinar differential equation (2.). 23

34 2.3. Case : a 3 =, a 2 Equation (2.) ields H 2 = C, where C is a constant and the solutions for the coefficient functions are ξ = a, τ = a 2, η = a 4 e MH2t, where a 4 is a constant Case 2: a 2 =, a 3 When a 2 =, a 3, H 2 = β/m + C 2 e 2βt where C 2 is a constant and the Lie point smmetr coefficients are ξ = a, τ = a 3 e 2βt, η = a 3 βue 2βt + a 5 L(t), where a 5 is a constant with ( L(t) = exp M t ) H 2 (s)ds. Thus we obtain two sets of three-dimensional Lie algebras, generated in each case b Case : X =, X 2 = t, X 3 = L(t) u, (2.) Case 2: X =, X 2 = e 2βt t βue2βt u, X 3 = L (t) u. (2.2) 24

35 2.4 Phsical invariant solutions Given the generator (2.7) the invariant solutions corresponding to χ are obtained b solving the characteristic sstem (cf. Chapter, section.4) d ξ = dt τ = du η. We use onl the operators which give meaningful phsical solutions of the problem consisting of Eqs. (2.4) and (2.6). This means that onl X 2 in each case is considered Invariant solution corresponding to case The form of the invariant solution in this case corresponding to X 2 is the stead state u(, t) = F(). (2.3) The substitution of (2.3) into the partial differential equation (2.4) results in the following reduced second-order ordinar differential equation ( ( ) 2 ) ν + ǫ F () F () C MF() =. (2.4) The boundar conditions impl F() = u, F( ) =, (2.5) and V (t) = u, (2.6) where u is a constant. Note that no stead state solution exists when the magnetic field is zero as equation (2.4) reduces to F =. Appling the boundar condition (2.5) on this equation results in the trivial solution F =. The double integration of (2.4) leads to ± ǫdf [ ν ± [ν 2 + 2ǫ(C MF 2 + λ 2 )] /2] /2 = + λ, (2.7) 25

36 where λ and λ 2 are constants that need to be fixed b (2.5). We observe that we cannot impose the boundar condition u(, ) = g() for arbitrar g(). The expression of the vorticit is w z = df d ν ± ν = ± 2 + 2ǫ(C MF 2 + λ 2 ). (2.8) ǫ We focus our attention on the solution (2.7). For a real solution we require the positive sign in the integrand. From (2.5) the boundar condition F() = u implies ± u ǫ df [(ν 2 + 2ǫC MF 2 + 2ǫλ 2 ) /2 ν] /2 = λ. (2.9) Thus equation (2.7) becomes ± F df ǫ =. (2.2) u [(ν 2 + 2ǫC MF 2 + 2ǫλ 2 ) /2 /2 ν] We now investigate when the boundar condition F( ) = of (2.4) is satisfied for the solution (2.2). Clearl the integral of (2.2) must be divergent as F so as to ensure that. It is seen that if λ 2, then the integrand behaves like [ ] /2 ν2 + 2ǫλ 2 ν as F (2.2) which means that the integral in (2.2) is convergent as F. For λ 2 =, the integral in (2.2) is divergent since the left hand side of (2.2) is ( ǫ ) /2 F df ± ν u [( + 2ǫν 2 C MF 2 ) /2 ] /2 ( ) /2 ν F ( ) = ± C M u F + O(F) df as F (2.22) ( ) /2 ( ) ν F = ± ln + O(F 2 ) as F. (2.23) C M u This tends to infinit if the negative sign is taken. Therefore the solution (2.2) subject to (2.5) is ( ǫ ν ) /2 u F df =. (2.24) [( + 2ǫν 2 C MF 2 ) /2 /2 ] 26

37 The vorticit (2.8) upon using (2.24) is ( ν w z = ǫ ) /2 [ ( + 2ǫν 2 C MF 2) ] /2 /2 (2.25) We present the numerical solution of (2.4) for various values of parameters in Figures 2. to 2.4. We have plotted Figures 2. to 2.4 to observe the behaviour of the flow when varing different emerging parameters. Figure 2. just leads to the numerical solution of the problem consisting of Eqs. (2.4) and (2.5). This indicates that u decreases as increases. It can be noted from Figure 2.2 that the velocit increases when the kinematic viscosit ν increases. Figure 2.3 depicts that the velocit is an increasing function of the parameter ǫ. In Figure 2.4 it is observed that the velocit decreases as the Hartmann number M increases Invariant solution corresponding to case 2 The invariant solution for this case for X 2 is given b u(, t) = F()e βt. (2.26) The insertion of (2.26) into (2.4) gives the reduced equation ( 2 γ F ()) F () F() =, (2.27) where γ = ǫ/(c 2 M). Through Eqs. (2.6) and (2.26) and selecting V (t) = C 2 e βt, the corresponding boundar conditions are F() = C 2, F( ) =. (2.28) To integrate the boundar value problem (2.27) and (2.28), we let F () = K(F) (2.29) and substitute (2.29) into (2.27) to obtain F γk 3dK df =. (2.3) 27

38 u Figure 2.: Numerical solution of equations (2.4) and (2.5) when M =, C =, ǫ =., ν = ν =.5 ν = 2 ν = 5 u Figure 2.2: Numerical solution of equations (2.4) and (2.5) varing the kinematic viscosit ν when M =, C = and ǫ = 2 28

39 .9.8 ε =.5 ε = 8 ε = 25 u Figure 2.3: Numerical solution of equations (2.4) and (2.5) varing the parameter ǫ when M =, C = and ν =.9.8 M = M = 2 M = 5 u Figure 2.4: Numerical solution of equations (2.4) and (2.5) varing the Hartmann number M with ν = 2, C = and ǫ = 2 29

40 The integration of (2.3) gives K(F) = ( ) /4 2 γ F()2 + B, (2.3) where B is a constant. Equation (2.29) together with (2.3) ields Thus ( ) /4 df 2 d = γ F()2 + B. (2.32) ( ) /4 2 γ F 2 + B df = + B 2, (2.33) where B 2 is another constant. Now we impose the boundar (2.28) on the solution (2.33). The boundar condition F() = C 2 of (2.28) imposed on the solution (2.33) ields C2 df (B + 2γ F 2 ) /4 = B 2. (2.34) Therefore using the condition (2.34), solution (2.33) becomes F C 2 df (B + 2γ F 2 ) /4 =. (2.35) Now we invoke F( ) = of (2.28) on the solution (2.35). We first consider the case B. The integral (2.35) needs to be divergent as F as. For small F, the left hand side of (2.35) is convergent. Hence the boundar condition F( ) = is not satisfied. For the case B =, the solution (2.35) gives ( γ /4 [ ] 2F 2) /2 C /2 2 = (2.36) which means that the integral is convergent. Thus the solution (2.35) does not satisf the boundar condition F( ) =. Therefore for this case we do not obtain a time-dependant solution using smmetr. 3

41 2.5 Numerical solution of the PDE (2.4) In this section we present a numerical solution of the partial differential equation (2.4) subject to (2.6) at various times for different values of the magnetic field and for some emerging parameters. In Figures 2.5 to 2.7, we have plotted numericall the velocit profile for different values of time, when the magnetic field is first taken to be zero (Figure 2.5), then constant (Figure 2.6) and finall dependent on time (Figure 2.7). In the case of H =, the velocit decreases and then increases when the value of t is increased. The velocities in Figures 2.5 to 2.7 are similar in a qualitative sense. However the shape of velocit differs slightl depending on the form of the magnetic field considered. It can be said that the shape of the velocit is more parabolic in the case when the strength of the magnetic field is non zero. Finall the variation of ǫ on the velocit is seen in Figure 2.8. It shows that velocit first decreases slightl and then increases when ǫ is increased. 2.6 Concluding remarks In this chapter we have considered the Raleigh problem of an incompressible non-newtonian fluid of third grade with variable magnetic field. After having formulated the problem, we obtained a non-linear third-order partial differential equation. The Lie smmetr approach was emploed for the reduction of this equation. We obtained a first-order differential equation that the magnetic field H had to satisf in order for the existence of point smmetries. Two cases arose and one smmetr was invoked for each case. Invoking the relevant smmetr, we were able to reduce the partial differential equation to a second-order non-linear ordinar differential equation for each of the cases. In the first case, we presented the stead state for various values of the emerging parameters and in the second case, the analtical solution that was obtained did not satisf the boundar conditions. Thus we have shown that static solutions onl exist for H b using smmetr methods. Finall, we presented numerical 3

42 u t.5 t t Figure 2.5: Numerical solution of (2.4) when there is no magnetic field (H = ), with ǫ =., ν =.85, α =, M =, V (t) = e t and g() = e 2 u t.5 t t Figure 2.6: Numerical solution of (2.4) when the magnetic field is taken constant (H = ), with ǫ =., ν =.85, α =, M =, V (t) = e t and g() = e 2 32

43 t.5 t u.3 t Figure 2.7: Numerical solution of (2.4) for a time dependent magnetic field (H = + 2e 2t ), with ǫ =., ν =.85, α =, M =, V (t) = e t and g() = e Ε.5 Ε 8 u.4.2 Ε Figure 2.8: Numerical solution of (2.4) varing ǫ, with ν =.85, H = + 2e 2t, α =, M =, t =.2, V (t) = e t and g() = e 2 33

44 solutions of the PDE with a choice of variable magnetic field as well as suitable boundar conditions. These solutions were presented graphicall in Figures 2.5 to 2.8. We briefl comment on the characteristic diffusion distance. The velocit diffuses a characteristic distance (νt) /2 in time t. This is clearl illustrated in figures 2.5 to 2.8. This ma explain wh u decas significantl after t. 34

45 Chapter 3 Effect of magnetic field on the flow of a fourth order fluid This chapter deals with an unstead flow engendered in a semi-infinite expanse of an incompressible fluid b an infinite rigid plate moving with an arbitrar velocit in its own plane. The fluid is considered to be fourth order and electricall conducting. The magnetic field is applied in the transverse direction to the flow. The stud of the motion of non-newtonian fluids with magnetic field has importance in the flow of liquid metals and allos, flow of plasma, flow of mercur amalgams, flow of blood and lubrication with heav oils and greases. Both analtical and numerical solutions are developed. Lie smmetr analsis is performed for an analtical solution. Limiting cases of interest are deduced as the special cases of the presented analsis b choosing suitable parameters values. This chapter is organized in the following fashion. Section 3. deals with the position of the problem. In sections 3.2 and 3.3 the analtical and numerical solutions of the problem are given. Section 3.4 deals with the concluding remarks. 3. Mathematical formulation Let us consider an infinite plate located at = and incompressible fourth grade fluid which is in contact with the plate and occupies the region >. The fluid is electricall conducting 35

46 and plate is insulating. The magnetic field s strength is taken as constant. The plate is moving in its own plane with time-dependent velocit for t >. Initiall, both the fluid and the plate are at rest. The unstead motion of the conducting fluid in the Cartesian coordinate sstem is governed b the conservation laws of momentum and mass which are presented in Eq. (2.) i.e. ρ DV Dt = divt + J B, (3.) divv =, (3.2) where all the quantities appearing in the above equations have been alread defined in the previous chapter. Through Eqs. (.7)-(.2), the Cauch stress T for a fourth order fluid [36] is related to the fluid motion in the form T = pi + S, (3.3) S = S + S 2 + S 3 + S 4, (3.4) S = µa, (3.5) S 2 = α A 2 + α 2 A 2, (3.6) S 3 = β A 3 + β 2 (A A 2 + A 2 A ) + β 3 (tra 2 )A, (3.7) S 4 = γ A 4 + γ 2 (A 3 A + A A 3 ) + γ 3 A γ 4(A 2 A 2 + A2 A 2) +γ 5 (tra 2 )A 2 + γ 6 (tra 2 )A 2 + [γ 7 tra 3 + γ 8 tr (A 2 A )]A. (3.8) Here µ is the dnamic viscosit, α i (i =, 2), β i (i =, 2, 3) and γ i (i =, 2,, 8) are material constants. The kinematical tensors (A n ) are defined via Eqs. (.2) and (.22) in the following form A = (gradv) + (gradv) T, (3.9) ( ) A n = t + V. A n + A n (gradv) + (gradv) T A n, n >, (3.) in which grad is the gradient operator. In our analsis the fluid is electricall conducting and there is an imposed magnetic field in the direction normal to the plate. In the lowmagnetic-renolds-number consideration (see, e.g., Shercliff [52]), in which the induced 36