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Advances in Mathematical Physics
Volume 2012 (2012), Article ID 169642, 15 pages
Peristaltic Transport of a Jeffrey Fluid with Variable Viscosity through a Porous Medium in an Asymmetric Channel
1Department of Mathematics & Statistics, FBAS, IIU, Islamabad, Pakistan
2Department of Mechanical Engineering, University of California Riverside, USA
Received 12 December 2011; Accepted 16 February 2012
Academic Editor: Sanith Wijesinghe
Copyright © 2012 A. Afsar Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 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 . The obtained expressions are utilized to discuss the influences of various emerging parameters.
2. Mathematical Formulation
We consider an incompressible Jeffrey fluid in an asymmetric channel of width . A sinusoidal wave propagating with constant speed on the channel walls induces the flow. The wall surfaces are chosen of the following forms: where are amplitude of the upper and lower waves, is the wave length, is the phase difference which varies in the range . Furthermore, , and should satisfy the following condition We assume that the flow becomes steady in the wave frame moving with velocity away from the fixed (laboratory) frame . The transformation between these two frames is given by where and are the velocity components in the wave frame , and 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 where
where is the ratio of relaxation to retardation times, is the retardation time, is the density, is the permeability of the porous medium, and is the porosity of the porous medium.
Using the longwave length and small Reynolds number approximation, (2.10) and (2.11) take the form The corresponding boundary conditions are whereEquation (2.14) indicate that is independent of . Therefore, (2.10) can be written as where is the viscosity variation on peristaltic flow. For the present analysis, we assume viscosity variation in the dimensionless form : The volume flow rate in the wave frame is given by The instantaneous flux in the laboratory frame is defined as The average flux over one period () is given by
3. 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 and as Substituting these equations into (2.15a), (2.15b), (2.15c), and (2.16), we have the following system of equations.
3.1. Zeroth-Order Equations
3.2. First-Order Equations
3.3. Second-Order Equations
3.4. Zeroth-Order Solution
and the volume flow rate is given by From (3.8), we have where
The dimensionless pressure rise at this order is
3.5. 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 is given by From (3.14), we get The dimensionless pressure rise at this order is
3.6. Second-Order Solution
Solving (3.6) by using (3.8) and (3.14) and the boundary condition (3.5), we obtain and the volume flow rate is given by From (3.18), we have The dimensionless pressure rise at this order is Summarizing the result obtained from (3.11), (3.16), and (3.20), we write Corresponding stream functions can be defined as
4. Results and Discussion
We have used a regular perturbation series in term of the dimensional viscosity parameter 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 , Daray number , porosity , amplitude ratio , Jeffrey fluid parameter and have been calculated numerically using MATHEMATICA software.
Figure 1 shows the variation of with flow rate for different values of . It is depicted that the time-average flux increase with increasing the viscosity parameter . Figure 2 represents the variation of with the flow rate for different values of . We observe that an increase in the peristaltic pumping rate pressure rises. Figures 3 and 4 are graphs of pressure rise with the flow rate for values of and . It is observed that the pumping rate decreases with increase of and . Figure 5 is the graph of the variation of versus the flow rate for different values of phase difference . It is observed that the pumping rate decreases with the increase of . Figures 6 and 7 plot the relation between pressure rise and flow rate for different values of and , respectively. Figure 8 represents the graph of axial velocity versus . It can be seen that an increase in decreases the magnitude of axial velocity . The effects of on the axial velocity are seen through Figure 9. It is noticed that an increase in increase the magnitude of the axial velocity. Figures 10 and 11 illustrate the effect of phase difference and Daray’s number on the axial velocity . It is observed that the magnitude of axial velocity decreases with the increasing phase difference and Daray’s number . In Figure 12 the axial velocity is graphed versus . We note that the magnitude of axial velocity increases as the channel width increases. It is worth mentioning that in the absence of porosity parameter the solutions of  can be derived as special case of the present analysis. This provides the useful check. It may be remarked that the problem for this particular model was not solved earlier even by any traditional perturbation technique. The results presented in this paper will now be available for experimental verification.
R. Ellahi thanks to United State Education Foundation Pakistan and CIES USA for honoring him by the Fulbright Scholar Award for the year 2011-2012. R. Ellahi also grateful to the Higher Education Commission and PCST of Pakistan for awarding him with the awards of NRPU and Productive Scientist, respectively.
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