CHAPTER 4 ANALYTICAL SOLUTIONS OF COUPLE STRESS FLUID FLOWS THROUGH POROUS MEDIUM BETWEEN PARALLEL PLATES WITH SLIP BOUNDARY CONDITIONS Introduction: The objective of this chapter is to establish analytical solutions of three unidirectional flows of an incompressible couple stress fluid namely Poiseuille, Couette and generalized Couette flows through porous medium between infinite parallel plates with slip boundary conditions at the plates. In addition to the slip boundary conditions, Stokes boundary condition has been used at the boundary to obtain the solutions. The effect of emerging flow parameters on the fluid velocity is discussed for each flow situation. The results specify that the porosity parameter has a decreasing effect on the velocity field in all flow problems. It is also found that, the presence of couple stresses has no influence on the velocity field in Couette flow case while it has decreasing effect on the fluid velocity in other two flow problems. The study of unidirectional flows of non-newtonian fluids through porous medium has gained increasing interest as they have enormous engineering applications. Few such unidirectional flows that are Couette flow and generalized Couette flow or Couette-Poiseuille flow. Several researchers have studied these unidirectional flows for diverse non-newtonian fluids. Danish et.al [40] obtained the exact solutions of Poiseuille and Couette-Poiseuille flow of third grade fluid between parallel plates. 69
Awartani and Hamdan [4] made complete study on the Couette, Poiseuille and Couette- Poiseuille flows through porous medium. Ramana Murthy and Srinivas [41] studied the second analysis for Poiseuille flow of immiscible micropolar fluids in a channel. Malik et.al [43] analyzed the Poiseuille and Couette flows of Carreau fluid with pressure dependent viscosity in a variable porous medium. Equations Governing the flow: The equations governing the flow of an incompressible couple stress fluid are given by [, 9] q 0 (4.1) dq 1 f c p q q q dt k (4.) where q and are the velocity and the density of the fluid respectively, p is the fluid pressure at any point, f and c are the body force per unit mass and body couple per unit mass respectively and k is the permeability of the porous medium In the absence of body forces and couples, the field equations governing the flow of an incompressible couple stress fluid reduce to q 0 (4.3) dq p q q q (4.4) dt k 70
Mathematical Formulation and Solution: For unidirectional steady flow between parallel plates, the velocity field q ( u( y),0,0) and it automatically satisfies the incompressibility condition (4.3). The momentum equation (4.4) governing the flow reduces to 4 d u d u dp u 0 4 (4.5) dy dy dx k Fig.4.1: Flow Configuration It can be noted that, in the absence of couple stresses 0 and the above equation reduces to the equations of the classical Navier-Stokes model. As the governing differential equation (4.5) is higher order than the Navier- Stokes equation(s), extra boundary conditions are required in order to obtain a solution. The information relevant to the additional boundary conditions is described thoroughly by Stokes in []. One of the additional boundary conditions introduced by Stokes is the vanishing of couple stresses on the solid boundary. We use this additional boundary condition to obtain the solutions of the problems under consideration. 71
Plane Poiseuille flow: Consider the steady laminar flow of an incompressible couple stress fluid through homogeneous porous medium between two infinite horizontal parallel plates distance h apart. In this case, both the plates are assumed to be at rest and the flow is due to the constant pressure gradient G in the positive x direction. With these, the governing equation (4.5) is to be solved subject to the following slip boundary conditions: 3 du d u u( h) 0, 3 dy dy yh 3 du d u uh ( ) 0, 3 dy dy yh (4.6) (Stokes boundary conditions): du 0 dy at y = h and y =h (4.7) where is slip constant. Introducing the following non-dimensional parameters y h h h y*, u* u, p* p, a, = and h h k (4.8) h the boundary value problem (4.5) (4.7), after dropping * s and using dp G reduce dx to 4 d u d u a u G (4.9) 4 dy dy subject to the slip boundary conditions, 3 3 du d u du d u u( 1) a 0, 3 u(1) a 0 3, (4.10) dy dy dy dy y1 y1 (Stokes boundary conditions): 7
du 0 dy at y = 1 and y = 1. (4.11) Now the exact solution of the boundary value problem (4.9)-(4.11), after simplifications, is obtained as, G ( m cosh mcosh ny n cosh ncosh my) uy ( ) 1 3 3 n cosh n cosh m ( m a m )sinh m m cosh m cosh n ( n a n )sinh n (4.1) where, m 1 n and a mn. a Plane Couette flow: Here, the couple stress fluid occupies the homogenous porous medium between two infinitely long horizontal parallel plates y = h and y = h. It is assumed that the pressure p is constant so that the pressure gradient is zero between parallel plates and hence the velocity is zero in the flow field. The fluid flow takes place due to the motion of the plates. The upper and lower plates are allowed to move with different constant velocities, say U 1 and U respectively. Further, it is assumed that the relative velocity between fluid and plates is proportional to the shear rate of the plates. With these, the governing equation (4.5) takes the form, 4 d u d u u 0 (4.13) 4 dy dy k with the slip boundary conditions, 3 3 du d u du d u u( h) U 3, u( h) U 3 1, dy dy dy dy yh yh (4.14) 73
(Stokes boundary conditions): du 0 dy at y = h and y = h. (4.15) where is slip constant. Introducing the following non-dimensional parameters * y, * u y u, a, = h, d = U and h U h k U1 (4.16) h the boundary value problem (4.13) (4.15), after dropping * s, becomes a d u d u (4.17) dy dy 4 u 0 4 with the slip boundary conditions, 3 3 du d u du d u u( 1) a d, 3 u(1) a 1 3, (4.18) dy dy dy dy y1 y1 (Stokes boundary conditions): du 0 dy at y = 1 and y = 1. (4.19) Now the analytical solution of this boundary value problem which is valid for the entire flow field, after incorporating the classical procedure and straightforward computations, is given by uy ( ) 3 3 n cosh n cosh m ( m a m ) sinh m m cosh m cosh n ( n a n ) sinh n d 0.5 ( n cosh ncosh my m cosh mcosh ny) d 0.5 ( n sinh nsinh my m sinh msinh ny) 3 sinh n ( n a n )cosh n (4.0) 3 n sinh n sinh m ( m a m ) cosh m m sinh m 74
Where m 1 n and a mn. a Generalized Couette flow: Here again, the couple stress fluid is bounded by two infinitely long horizontal parallel plates y = h and y = h in porous medium. Here the physical model is similar to that of the Couette flow. Additionally a constant pressure gradient is applied simultaneously in the positive x direction while the both plates are moving with different constant velocities. The resultant flow is known as generalized Couette flow or Couette- Poiseuille flow. The governing equation of this flow situation is, 4 d u d u dp u 0 4 (4.1) dy dy dx k subject to the slip boundary conditions, 3 3 du d u du d u u( h) 0, 3 u( h) U, 3 dy dy dy dy yh yh (4.) (Stokes boundary conditions): du 0 dy at y = h and y = h (4.3) where is slip constant. Making use of the following non-dimensional parameters y u h h y*, u*, p* p, a, = h U U h k and (4.4) h the boundary value problem (4.1) (4.3), after dropping * s and using dp G dx becomes 75
4 d u d u a u G (4.5) 4 dy dy subject to the slip boundary conditions, 3 3 du d u du d u u( 1) a 0, 3 u(1) a 1 3, (4.6) dy dy dy dy y1 y1 (Stokes boundary conditions): du 0 dy at y = 1 and y = 1. (4.7) Now the analytical expression of the velocity field for the problem (4.5)-(4.7) is obtained as, uy ( ) 0.5 G / ( n cosh ncosh my m cosh mcosh ny) 3 3 n cosh n cosh m ( m a m ) sinh m m cosh m cosh n ( n a n ) sinh n 0.5( n sinh nsinh my m sinh msinh ny) 3 n sinh n sinh m ( m a m ) cosh m m sinh m sinh n ( n a n G 3 )cosh n (4.8) where, m 1 n and a mn. a Results and Discussion: This section presents the graphical illustrations of the velocity field for the three unidirectional flow situations considered in the present work. For all the flow problems, the variation of velocity with diverse flow parameters is plotted for both slip ( 0 ) and no-slip ( 0 ) boundary conditions. The effects of slip parameter, couple stress 76
parameter, porosity parameter and the pressure gradient on velocity for each of the problems is described. Fig.4. Fig.4.4 show variation of velocity for various values of the slip parameter while the other parameters are fixed. It is observed from Fig.4. that, for plane Couette flow, as the slip parameter increases the velocity increases near the stationary plate while the trend is reversed near the moving plate. In the other two flow situations, it is found that the increasing of slip parameter has an increasing effect on the fluid velocity (see Fig.4.3 and Fig.4.4). That is, the more the fluid slips at the boundary the less its velocity affected by the motion of the boundary. Fig.4.5 Fig.4.7 and Fig.4.8 Fig.4.10 show the variation of velocity for Couette flow for different flow parameters with slip and no-slip boundary conditions respectively. The graphs indicate that, in Couette flow the couple stresses have no influence on the fluid velocity of for both the boundary conditions. Fig.4.6 and Fig.4.9 display the variation of fluid velocity for different values of d for Couette flow. d = 0 corresponds to the case of fixed stationary lower plate. d = 1 is for the case when both the plate are moving with the same constant velocity in the same direction while d = - 1 represents the case when both the plates are moving with the same velocity in the opposite direction. It can be seen from these figures that, the velocity increases when the lower plate moves in the same direction of the upper plate. 77
Fig.4.11 Fig.4.16 display the velocity profiles for Poiseuille flow and Fig.4.17 Fig.4. present the velocity profiles for generalized Couette flow for diverse flow parameters with slip and no-slip parameters. It is evident from these figures that, the velocity profiles for Poiseuille and generalized Couette flows are qualitatively similar to each other. From the figures, it is observed that the fluid velocity deceases with the increasing of the couple stress parameter a. In the absence of couple stresses (i.e., as 0 ), a 0 and hence decreasing of couple stress parameter shall give the classical Newtonian case. Hence, the present of couple stresses decreases the velocity of the fluid. As expected, increasing of pressure gradient G increases the fluid velocity in both the cases. In all the three problems, from the figures it is observed that, increasing of porosity parameter has a decreasing effect on the fluid velocity for both slip and no-slip boundary condition cases. 78
Fig.4.: Velocity Profiles for Plane Couette flow with various values of nondimensional slip parameter for fixed values of G and a. 79
Fig.4.3: Velocity Profiles for Plane Poiseuille flow with various values of nondimensional slip parameter for fixed values of G and a. 80
Fig.4.4: Velocity Profiles for generalized Couette flow with various values of nondimensional slip parameter for fixed values of G and a. 81
Fig.4.5: Velocity Profiles for Plane Couette flow with various values of couple stress parameter a when other flow parameters are fixed (slip boundary condition) 8
Fig.4.6: Velocity Profiles for Plane Couette flow with various values of d when other flow parameters are fixed(slip boundary condition) 83
Fig.4.7: Velocity Profiles for Plane Couette flow with various values of porosity parameter σ when other flow parameters are fixed (slip boundary condition) 84
Fig.4.8: Velocity Profiles for Plane Couette flow with various values of couple stress parameter a when other flow parameters are fixed (no-slip boundary condition) 85
Fig.4.9: Velocity Profiles for Plane Couette flow with various values of d when other flow parameters are fixed(no-slip boundary condition) 86
Fig.4.10: Velocity Profiles for Plane Couette flow with various values of porosity parameter σ when other flow parameters are fixed (no-slip boundary condition) 87
Fig.4.11: Velocity Profiles for Plane Poiseuille flow with various values of couple stress parameter a when other flow parameters are fixed (slip boundary condition) 88
Fig.4.1: Velocity Profiles for Plane Poiseuille flow with various values of pressure gradient G when other flow parameters are fixed (slip boundary condition) 89
Fig.4.13: Velocity Profiles for Plane Poiseuille flow with various values of porosity parameter σ when other flow parameters are fixed (slip boundary condition) 90
Fig.4.14: Velocity Profiles for Plane Poiseuille flow with various values of couple stress parameter a when other flow parameters are fixed (no-slip boundary condition) 91
Fig.4.15: Velocity Profiles for Plane Poiseuille flow with various values of pressure gradient G when other flow parameters are fixed(no-slip boundary condition) 9
Fig.4.16: Velocity Profiles for Plane Poiseuille flow with various values of porosity parameter σ when other flow parameters are fixed (no-slip boundary condition) 93
Fig.4.17: Velocity Profiles for Couette-Poiseuille flow with various values of couple stress parameter a when other flow parameters are fixed (slip boundary condition) 94
Fig.4.18: Velocity Profiles for Couette-Poiseuille flow with various values of pressure gradient G when other flow parameters are fixed (slip boundary condition) 95
Fig.4.19: Velocity Profiles for Couette-Poiseuille flow with various values of porosity parameter σ when other flow parameters are fixed (slip boundary condition) 96
Fig.4.0: Velocity Profiles for Couette-Poiseuille flow with various values of couple stress parameter a when other flow parameters are fixed (no-slip boundary condition) 97
Fig.4.1: Velocity Profiles for Couette-Poiseuille flow with various values of pressure gradient G when other flow parameters are fixed (no-slip boundary condition) 98
Fig.4.: Velocity Profiles for Couette-Poiseuille flow with various values of porosity parameter σ when other flow parameters are fixed (no-slip boundary condition) 99
Conclusions: Three classical unidirectional flow problems namely (i) Poiseuille flow, (ii) Couette flow and (iii) generalized Couette flow problems of an incompressible couple stress fluid through porous medium between parallel plates have been solved and studied under slip as well as no-slip boundary conditions. The analytical solutions for all the three problems have been obtained in a classical way. For all the three problems, the velocity profiles and volume flow rate are studied diverse flow parameters. The study reveals that, The presence of couple stresses decreases the fluid velocity in Poiseuille and generalized Poiseuille flow while it has no influence on the Couette flow velocity. In all the considered flow problems, the porosity parameter has a decreasing effect on the velocity of the fluid. More the fluid slips at the boundary (that is, as the slip parameter increases), less its velocity affected by the boundary. 100