CHAPTER 6 Effect of slip and heat transfer on the Peristaltic flow of a Williamson fluid in an incliped channel 6.1. Introduction Peristalsis is a well-known mechanism for pumping biological and industrial fluids. Even though it is observed in living systems for many centuries; the mathematical modeling of peristaltic transport has begun with important works by Fung and Yih (1968) using laboratory frame of reference and Shapiro et al. (1969) using wave frame of reference. Many of the contributors to the area of peristaltic pumping have either followed Shapiro or Fung. Most of the studies on peristaltic flow deal with Newtonian fluids. The complex rheology of biological fluids has motivated investigations involving different non-newtonian fluids. Peristaltic flow of non-newtonian fluids in a tube was first studied by Raju and Devanatham (1972). Mekheimer (2003) studied the peristaltic transport of MHD flow in an inclined planar channel. Hayat et al. (2008) extended the idea of Elshehawey et al. (2006) for partial slip condition. Srinivas et al. (2008) studied the Peristaltic transport in an asymmetric channel with heat transfer. Srinivas et al. (2008) studied the non-linear peristaltic transport in an inclined asymmetric channel. Vajravelu et al. (2009) analyzed peristaltic transport of a Casson fluid in contact with a Newtonian fluid in a circular tube with permeable wall. Nadeem and Akrarn (2010) discussed peristaltic flow of a Williamson fluid in an asymmetric channel. It is observed that most of the physiological fluids (for
example, blood) cannot be described by ~ewtonian model. Hence, several non- Newtonian models are being proposed by various researchers to investigate the flow behavior in physiological system of a living body. Among them Williamson model is expected to explain most of the features of a physiological fluid. Moreover, this model is nonlinear and Ntwtonian fluid model can be deduced as a special case from this model. In this chapter, the peristaltic.pumping of Williamson fluid in an inclined channel under the influence of slip and heat transfer is investigated. The peristaltic waves are assumed to propagate on the walls of the channel with speed c. Using the wave frame of reference, boundary value problem is solved. The stream function, the temperature distribution and the pressure rise are calculated. 6.2. BASIC EQUATIONS For an incompressible fluid the balance of mass and momentum are given by divv=o, (2.1) dv p-=divs+pf, dt (2.2) where p is the density, V is the velocity vector, S is the Cauchy stress tensor and f represents the specific body force and d/dt represents the material derivative. The constitutive equation for Williamson fluid is given by (Nadeem and Akram, 2010) S=-PI+r, (2.3) 5 = -[& +(/lo t A)(l-r;)-'],?.' (2.4)
in which - PI is the spherical part of the stress due to. constraint of incompressibility,, is the extra stress tensor,,& is the infinite shear rate viscosity, 4 is the zero shear rate viscosity, I- is the time constant and 7 is defined as Here li is the second invariant strain tensor. We consider the constitutive equation (2.4), the case for which &=Odry<l. The component of extra stress tensor therefore, can be written as 6.3. Mathematical Formulation Consider the peristaltic flow of a Williamson fluid in an inclined symmetric channel as shown in Figure 6.1. The channel walls are lined with non erodible porous material. The thickness of the lining is very small when compared with the width of the channel. The lower permeable wall of the channel is maintained a: temperature T1 while the upper permeable wall has temperature To. The flow is generated by sinusoidal wave trains propagating with constant speed c along the channel. The geometry of the wall surfaces is defined as
where Si is the amplitudes of the wave, d is'the mean width of the channel, h is the wave length, c is the velociti of propagation, iis the time and Xis the direction of wave propagation. Fig. 6.1 Physical Model The equations governing the motion and energy of a Williamson fluid are given by au av - ax +z'o aii -813-80 =-,-a--- ap at-- a ~ ~, (ar ax au) ax ax al p ---+U_+Vl (3.2) pg sin a (3.3) at -at I -at K' --=+u,+v--= =-v2t+v@ at ax a~ p. + pg cos a (3.4) (3.5)
where FB, Fn, Tw are components of stress, where a is the inclination of the channel with the horizontal, U, V are the velocities in X and Y directions in fixed frame, pis the density, Pis the pressure, L is the kinematic viscosity, K' is the thermal conductivity, C' is the specific heat and F is the temperature. We introduce a wave frame(x,y) moving with velocity c away from the fixed frame (X,Y) by the transformation. We define the non-dimensional quantities as follows - - Y Y i i ' F c - h A - d - X=- y== u=- v=- t=-t h== =-- T,, Tx, =-T- A' d' c' c' ' d' " p0c k c Xi - - z,,=-~-, J=-, Re=-,We=-s-, P=---P, y=- 8=- Po'oc " A Po d elpo c ' Ec = d - d pcd rc d2 -. 72 T-T, I;-q' c2 pvc' pr=-,q=- c - ) K' "0 -d2pg (3.7) Using the above non-dimensional quantities in equations (3.3), (3.4) and (3.9, the resulting equations in terms of stream function y(u = ayjay,v= -St%'/&) can be written as
where Here 6, Re, We represent the wave, Reynolds and Weissenberg numbers, respectively. Under the assumptions of long wavelength 6 << 1 and low Reynolds number, neglecting the terms of order 6 and higher, equations (3.8) and (3.9) take the form Eliminating of pressure from equations (3.1 1) and (3.12), yield The dimensionless mean flow Q is defined by Q=F+I
in which where h = I + (COS 2rx The appropriate boundary conditions for the problem in terms of stream function Yare where L is permeability parameter including slip. The partial slip conditions (3.1 8) and (3.19) are used following Nadeem et al. (2010). 6.4. Perturbation solution The equation (3.11) is a non-linear and hence its exact solution is not possible. We employ the. perturbation technique to find the solution. For perturbation solution, we expandy, Fand p as
Substituting the above expressions in eq;ations (3.1 1)-(3.12) and boundary conditions (3.16)-(3.19), we get the'fol~owin~ system of equations: System of order We0 System of order We'
0 Solution for system of order We The solution to the zeroth-order problem can be written as The axial pressure gradient is Solution for system of order we ' Substituting the zeroth-order solution (4.16) into (4.10), the solution of the first-order problem satisfying the boundary conditions takes the following form where
The axial pressure gradient is given by Summarizing the perturbation results fof small parameter We, the expression for stream function and pressure gradient can be written as The dimensionless pressure rise and frictional force per one wavelength in the wave frame are defined, respectively as and Making use of equation (4.20) the solution of the equation (3.10) using the boundary conditions 0 = 0 at y = h and 6' = 1 at JJ = -h is given by
6.5 Results and Discussion - The variation in pressure rise dp with the mean flow Q is calculated from equation (4.22) and is shown in Fig. 6.2 for different values of the slip parameter L for fixed p1, a = 4 and p0.6. It is noticed that the pumping 6 - curves intersect at a point in the first quadrant and to the left of this point, Q - decreases and to the right of this point, Q increases with an increase in L. We observe that the free pumping (A@) and co-pumping (4 < 0) Q increases with increasing L. From Eq.(4.22) we have calculated the pressure difference as a function of 0 for different values of the angle of inclination of the channel a for fixed L=0.2, PI, (=0.6 and is shown in Fig. 6.3. We observe that for a given LIP, the flux Q increases with increasing cr. We observe that for a given Q, pressure rise increases with increasing a. The variation of pressure rise with Q is calculated from equation (4.22) for different values of the gravity parameter q, for fixed L=0.2, a = E, +=0.6 6 and is shown in Fig.5.4. It is clear that the pressure rise increases with the
increase in Q. We find that for fixed^, pressure rise increases.with increasing q. Also for a given@, the increase in??increases the mean flow. The variation of pressure rise with Q is calculated from equation (4.22) for different values of the gravity parameter q, for fixed L4.2, a = 1 q=l 6 ' - and is shown in Fig. 6.5. It is found that, the flux Q increases with an increase in in both pumping and free pumping regions. But in the co-pumping region, - the pumping curves intersect at a point. After this point Q decreases with an increase in amplitude ratio 4. The variations of temperature field Qwith y are calculated fiom equation (4.24) for different values of the slip parameter L for fixed x = 1; 4 = 0.6; L = 0.1; Pr = 0.7; Ec = 0.5; We = 0.01; q = 2 is shown in Fig.6.6. It is observed that the temperature distribution 9 decreases with an increase in L. The variations of 9 with y are calculated from equation (4.24) for different values of Prandtl number Pr for fixed x = 1; 4 = 0.6; L = 0.2; Ec = 0.5; We = 0.01; q = 2 and is shown in Fig.6.7, it is found that the 0 increases with increasing Pr. The variations of 6 with y are calculated from equation (5.48) for different values of Eckert number Ec for fixed x = 1; 4 = 0.6; L = 0.2; Pr = 0.7; We = 0.01; q = 2 and is shown in Fig.6.8, it is found that the 8 increases with increasing Ec. The variations of 0 with y are calculated fiom equation (5.48) for different values of the volume flow rate q when = 1; 4 = 0.6; L = 0.2; Pr = 0.7; We = 0.01; Ec = 0.5., and are presented in Fig.6.9. It is observed that the temperature Q increases with an increase q.
-. Q Fig.6.2. The variation of @with for different values of L with r-1, a = E, p0.6 6
Fig.6.3. The variation of &with Q for different values of a with L=0.2, ~ 1 (=0.6,
- Q - Fig.6.4. The variation of @with Q for different values of q with L=0.2,
Fig.6.5. The variation of &with Q for different values of 4 with L=0.2, - Q K a = -, ~ 1. 6
Fig.6.6. The variation of 0 with y for different values of L with x = 1;
Fig.6.7. The variation of 0 with y for different values of Pr with Y
Fig.6.8. The variation of 6 with y for different values of Ec with x = 1; r,b = 0.6; L = 0.2; Pr = 0.7; We = 0.01; q = 2 Y
Fig.6.9. The variation of 0 wit11 y for different values of q with x = 1; r#~ = 0.6; L = 0.2; Pr = 0.7; We = 0.01; Ec = 0.5.