Peristaltic transport in an asymmetric channel through a porous medium

doi:10.1016/j.amc.2006.03.040

Applied Mathematics and Computation

Volume 182, Issue 1, 1 November 2006, Pages 140-150

https://doi.org/10.1016/j.amc.2006.03.040 Get rights and content

Abstract

The problem of peristaltic transport of an incompressible viscous fluid in an asymmetric channel through a porous medium is analyzed. The flow is investigated in a wave frame of reference moving with velocity of the wave under the assumptions of long-wavelength and low-Reynolds number. An explicit form of the stream function is obtained by using Adomian decomposition method. The analysis showed that transport phenomena are strongly dependent on the phase shift between the two walls of the channel. It is indicated that the axial velocity component U in fixed frame increases with increasing the permeability parameter. In the case of high permeability parameter (as K → ∞), our results are in agreement with Mishra and Ramachandra Rao [M. Mishra, A. Ramachandra Rao, Peristaltic transport of a Newtonian fluid in an asymmetric channel, ZAMP 53 (2003) 532] and Eytan and Elad [O. Eytan, D. Elad, Analysis of intra-uterine fluid motion induced by uterine contractions, Bull. Math. Biol. 61 (1999) 221]. The results given in this paper may throw some light on the fluid dynamic aspects of the intra-uterine fluid flow through a porous medium.

Introduction

Peristalsis is an important mechanism for mixing and transporting fluids, which is generated by a progressive wave of contraction or expansion moving on the wall of the tube. Physiological fluids in animal and human bodies are, in general, pumped by the continuous periodic muscular oscillations of the ducts. These oscillations are presumed to be caused by the progressive transverse contraction waves that propagate along the walls of the ducts. Peristalsis is the mechanism of the fluid transport that occurs generally from a region of lower pressure to higher pressure when a progressive wave of area contraction and expansion travels along the flexible wall of the tube. Peristaltic flow occurs widely in the functioning of the ureter, food mixing and chyme movement in the intestine, movement of eggs in the fallopian tube, the transport of the spermatozoa in cervical canal, transport of bile in the bile duct, transport of cilia, and circulation of blood in small blood vessels.

In 1966, Latham [9] made an experimental study of the mechanics of peristaltic transport. The results of the experiments were found to be in good agreement with the theoretical results of Shapiro [16]. Based on the experimental work, Burns and Parkes [3] studied the peristaltic motion of a viscous fluid through a pipe and a channel by considering sinusoidal variation at the walls. In 1969, Shapiro et al. [17] analyzed peristaltic pumping with long wavelength at low Reynolds number. The small Reynolds number assumption of Shapiro et al. [17] was endorsed by Jaffrin [8] who extended the analysis by considering the higher order terms to include cases where Reynolds number was higher. Barton and Raynor [2] studied peristaltic flow in tubes using long wave approximation. Barton and Raynor also analyzed the case for low Reynolds number.

Flow through a porous medium has been of considerable interest in recent years particularly among geophysical fluid dynamicists. Many technical processes involve parallel flow of fluids of different viscosity and density through porous media. Such parallel flows exist in packed bed reactors in the chemical industry, in petroleum production engineering, and in many other processes as well. Flow through porous medium occurs in filtration of fluids and seepage of water in river beds. Movement of underground, water and oils, limestone, rye bread, wood, the human lung, bile duct, gall bladder with stones, and small blood vessels are some important examples of flow through porous medium. Another example is the see page under a dam which is very important [14]. Several works have been published by using the generalized Darcy’s law [15], where the convection acceleration and viscous-stress are taken into account [19]. El Sayed [5] studied the electrohydrodynamic instability of two superposed viscous and streaming fluids through porous medium. Varshney [18] studied the fluctuating flow of a viscous fluid through a porous medium bounded by porous and horizontal surface. Rapits et al. [12], [13] studied the steady free convection and mass transfer flow of a viscous fluid through a porous medium bounded by a vertical surface.

Recently, physiologists observed that the intra-uterine fluid flow due to myometrial contractions is peristaltic-type motion and the myometrial contractions may occur in both symmetric and asymmetric directions, De Vries et al. [4]. Eytan et al. [7] have observed that the characterization of non-pregnant woman uterine contractions is very complicated as they are composed of variable amplitudes, a range of frequencies and different wavelengths. Eytan and Elad [6] have developed a mathematical model of wall-induced peristaltic fluid flow in a two-dimensional channel with wave trains having a phase difference moving independently on the upper and lower walls to simulate intra-uterine fluid motion in a sagittal cross-section of the uterus. They have used the lubrication theory to obtain a time dependent flow solution in a fixed frame. The results obtained by Eytan and Elad [6] have been used to evaluate the fluid flow pattern in a non-pregnant uterus. The possible particle trajectories were also calculated by Eytan and Elad [6] and they have used the results to understand the embryo transport within the uterine cavity before it gets implanted at the uterine wall. More recently, Mishra and Ramachandra Rao [10] have investigated the flow in an asymmetric channel generated by peristaltic waves propagating on the walls with different amplitudes and phases.

The aim of the present paper is to investigate the peristaltic transport of an incompressible Newtonian fluid in an asymmetric channel through a porous medium. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitudes and phase due to the variation of channel width, wave amplitudes and phase differences. Peristaltic transport of intra-uterine fluid through a porous medium can be considered as an application of the present work. In this paper we have used Adomian decomposition method to solve a fourth-order linear partial differential equation with four boundary conditions. Large classes of linear and non-linear differential equations, both ordinary as well as partial, can be solved by Adomian decomposition method [1].

Section snippets

Adomian decomposition method

Let us discuss a brief outline of the Adomian decomposition method, in general. For this, let us consider an equation in the form Lu + Ru + Nu = g, where L is an easily invertible linear operator, R is the remainder of the linear operator. The remainder of the linear operator is R. The non-linear term is represented by Nu, we can write $Lu = g - Ru - Nu,$ $L^{- 1} Lu = L^{- 1} g - L^{- 1} Ru - L^{- 1} Nu .$ Now L⁻¹ is simply an n-fold integration for an nth order L. For initial-value problems we convenient define L⁻¹ for $L = \frac{d^{n}}{d t^{n}}$ as the n

Mathematical formulation and solution

We consider a two-dimensional channel (see Fig. 1) filled with an incompressible viscous fluid through a porous medium. The channel asymmetry is produced by choosing the peristaltic wave train on the walls (H₁ is the upper wall and H₂ is the lower wall) to have different amplitudes and phase due to the variation of channel width, wave amplitudes and phase differences $Y = H_{1} = d_{1} + a_{1} \cos [\frac{2 π}{λ} (X - ct)], Y = H_{2} = - d_{2} - b_{1} \cos [\frac{2 π}{λ} (X - ct) + ϕ],$ where a₁ and b₁ are the amplitudes of the waves, λ is the wave length, d₁ + d₂ is

Discussion of the results

The characteristic feature of peristaltic motion is pumping against pressure rise. From (3.18), we observe $\bar{Q} = 0 for Δ p = Δ p_{\max} = I_{1},$ and $\bar{Q} = {\bar{Q}}_{\max} = \frac{- I_{1}}{I_{2}} for Δ p = 0 (free pumping).$ When Δp > Δp_max one gets negative flux and when Δp < 0, we get $\bar{Q} > {\bar{Q}}_{\max}$ . The variation of $\bar{Q}$ as a function of $ϕ / π = \bar{ϕ}$ , normalized phase difference, is calculated from Eq. (3.18) and is presented in Fig. 2, Fig. 3 for two different cases. The first case (Fig. 2) when a = 0.7, b = 1.2, Δp = 1, d = 2 and K = 1000 (high permeability parameter) and it

Conclusions

A mathematical model to study the peristaltic transport of an incompressible viscous fluid in a generalized asymmetric channel through a porous medium is presented. The problem is studied under the assumptions of long-wavelength and low-Reynolds number. The effects of phase difference and permeability parameter on the pumping characteristics, pressure rise and velocity field are investigated. Adomian decomposition method is used for solving a fourth-order linear partial differential equation

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Peristaltic transport in an asymmetric channel through a porous medium

Abstract

Introduction

Section snippets

Adomian decomposition method

Mathematical formulation and solution

Discussion of the results

Conclusions

Am. J. Obstetrics Gynecol.

Bull. Math. Biol.

Int. J. Eng. Sci.

Solving frontier problems of physics: the decomposition method

Peristaltic flow in tubes

Bull. Math. Biophys.

Peristaltic motion

J. Fluid Mech.

Electrohydrodynamic instability of two superposed viscous streaming fluids through porous medium

Can. J. phys.

Dynamics of the intrauterine fluid-wall interface

Ann. Biomed. Eng.