Peristaltic Transport of a Jeffrey Fluid with Variable Viscosity through a Porous Medium in an Asymmetric Channel

The peristaltic flow of a Jeffrey fluid with variable viscosity through a porous medium in an asymmetric channel is investigated. The channel asymmetric is produced by choosing the peristaltic wave train on the wall of different amplitude and phase. The governing nonlinear partial differential equations for the Jeffrey fluid model are derived in Cartesian coordinates system. Analytic solutions for stream function, velocity, pressure gradient, and pressure rise are first developed by regular perturbation method, and then the role of pertinent parameters is illustrated graphically.


Introduction
Peristalsis is a mechanism to pump the fluid by means of moving contraction on the tubes or channel walls.This process has quite useful applications in many biological systems and industry.It occurs in swallowing food through the esophagus, chyme motion in the gastrointestinal tract, the vasomotion of small blood vessels such as venules, capillaries, and arterioles, urine transport from kidney to bladder, sanitary fluid transport of corrosive fluids, a toxic liquid transport in the nuclear industry, and so forth.In view of such physiological and industrial applications, the peristaltic flows has been studied with great interest by the various researchers for viscous and non-Newtonian fluids 1-9 .
In most of the studies which deal with the peristaltic flows, the fluid viscosity is assumed to be constant.This assumption is not valid everywhere.In general the coefficients of viscosity for real fluids are functions of space coordinate, temperature, and pressure.For many liquids such as water, oils, and blood, the variation of viscosity due to space coordinate and temperature change is more dominant than other effects.Therefore, it is highly desirable Advances in Mathematical Physics to include the effect of variable viscosity instead of considering the viscosity of the fluid to be constant.Some important studies related to the variable viscosity are cited in 10-13 .
A porous medium is the matter which contains a number of small holes distributed throughout the matter.Flows through a porous medium occur in filtration of fluids.Several investigations have been published by using generalized Darcy's law where the convective acceleration and viscous stress are taken into account 14-17 .Considering the importance of non-Newtonian fluid in peristalsis and keeping in mind the sensitivity of liquid viscosity, an attempt is made to study the peristaltic transport of Jeffrey having variable viscosity through a porous medium in a two-dimensional asymmetric channel under the assumption of long wave length and the low Reynolds number approximation.A regular perturbation method is used to solve the problem, and the solutions are expanded in a power series of viscosity parameter α.The obtained expressions are utilized to discuss the influences of various emerging parameters.

Mathematical Formulation
We consider an incompressible Jeffrey fluid in an asymmetric channel of width d 1 d 2 .A sinusoidal wave propagating with constant speed c on the channel walls induces the flow.The wall surfaces are chosen of the following forms:

2.2
We assume that the flow becomes steady in the wave frame x, y moving with velocity c away from the fixed laboratory frame X, Y .The transformation between these two frames is given by where where λ 1 is the ratio of relaxation to retardation times, λ 2 is the retardation time, ρ is the density, k is the permeability of the porous medium, and ε is the porosity of the porous medium.
Introducing the following nondimensional quantities:

2.8
With the help of 2.

2.15b
where Equation 2.14 indicate that p is independent of y.Therefore, 2.10 can be written as where μ y is the viscosity variation on peristaltic flow.For the present analysis, we assume viscosity variation in the dimensionless form 10 : u y e −αy , u y 1 − αy αy 2 2 , for α ≺≺ 1.

2.17
The volume flow rate in the wave frame is given by

2.18
The instantaneous flux Q x, t in the laboratory frame is defined as

2.19
The average flux over one period T λ/c is given by

Perturbation Solution
Equation 2.16 is a nonlinear differential equation so that it is not possible to obtain a closed form solution; so we seek perturbation solution.We expand u, p and q as u u 0 αu 1 α 2 u 2 o α 3 , p p 0 αp 1 α 2 p 2 o α 3 , q q 0 αq 1 α 2 q 2 o α 3 .

3.1
Substituting these equations into 2.15a , 2.15b , 2.15c , and 2.16 , we have the following system of equations.

Zeroth-Order Equations
where

Zeroth-Order Solution
Solving 3.2 and 3.3 , we get where and the volume flow rate q 0 is given by From 3.8 , we have where

3.12
The dimensionless pressure rise at this order is 3.13

First-Order Solution
Substituting zeroth order solution 3.8 into 3.4 and then solving the resulting system along with the corresponding boundary conditions, we arrive at and the volume flow rate q 1 is given by From 3.14 , we get   The dimensionless pressure rise at this order is 3.17

Second-Order Solution
Solving 3.6 by using 3.8 and 3.14 and the boundary condition 3.5 , we obtain

3.18
and the volume flow rate q 2 is given by From 3.18 , we have

3.20
The dimensionless pressure rise at this order is

3.22
Corresponding stream functions can be defined as

Results and Discussion
We have used a regular perturbation series in term of the dimensional viscosity parameter α to obtain analytical solution of the field equations for peristaltic flow of Jeffrey fluid in an asymmetric channel.To study the behavior of solutions, numerical calculations for several values of viscosity parameter α, Daray number Da, porosity ε, amplitude ratio φ, Jeffrey fluid parameter λ 1 , a and b have been calculated numerically using MATHEMATICA software.
Figure 1 shows the variation of ΔP with flow rate θ for different values of α.It is depicted that the time-average flux θ increase with increasing the viscosity parameter α. Figure 2 represents the variation of ΔP with the flow rate θ for different values of Da.We observe that an increase in the peristaltic pumping rate pressure rises.Figures 3 and 4 are graphs of pressure rise ΔP with the flow rate θ for values of ε and λ 1 .It is observed that the

2 . 1 where b 1 condition b 2 1 b
, b 2 are amplitude of the upper and lower waves, λ is the wave length, φ is the phase difference which varies in the range 0 ≤ φ ≤ π.Furthermore, a 1 , a 2 , b 1 , b 2 , and φ should satisfy the following
and v are the velocity components in the wave frame x, y , p and P are pressure in wave and fixed frame of reference, respectively.The governing equations in the wave frame of reference are the Brinkman extended Daray equations given by 8 , 2.4 to 2.6 after dropping the bars take the form