Applied Bionics and Biomechanics

Volume 2015, Article ID 703574, 14 pages

http://dx.doi.org/10.1155/2015/703574

## Slip Effects on Peristaltic Transport of a Particle-Fluid Suspension in a Planar Channel

^{1}Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Giza, Egypt^{2}Department of Basic Engineering Sciences, Faculty of Engineering, Menoufia University, Shebin El-Kom, Egypt

Received 1 August 2014; Revised 16 March 2015; Accepted 16 April 2015

Academic Editor: Agnès Drochon

Copyright © 2015 Mohammed H. Kamel 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.

#### Abstract

Peristaltic pumping induced by a sinusoidal traveling wave in the walls of a two-dimensional channel filled with a viscous incompressible fluid mixed with rigid spherical particles is investigated theoretically taking the slip effect on the wall into account. A perturbation solution is obtained which satisfies the momentum equations for the case in which amplitude ratio (wave amplitude/channel half width) is small. The analysis has been carried out by duly accounting for the nonlinear convective acceleration terms and the slip condition for the fluid part on the wavy wall. The governing equations are developed up to the second order of the amplitude ratio. The zeroth-order terms yield the Poiseuille flow and the first-order terms give the Orr-Sommerfeld equation. The results show that the slip conditions have significant effect within certain range of concentration. The phenomenon of reflux (the mean flow reversal) is discussed under slip conditions. It is found that the critical reflux pressure is lower for the particle-fluid suspension than for the particle-free fluid and is affected by slip condition. A motivation of the present analysis has been the hope that such theory of two-phase flow process under slip condition is very useful in understanding the role of peristaltic muscular contraction in transporting biofluid behaving like a particle-fluid mixture. Also the theory is important to the engineering applications of pumping solid-fluid mixture by peristalsis.

#### 1. Introduction

Peristalsis is a form of a fluid transport induced by a progressive wave of area contraction or expansion along the walls of a distensible duct containing liquid. In physiology, peristaltic mechanism is involved in many biological organs such as ureter, gastrointestinal tract, ducts afferents of the male reproductive tracts, cervical canal, female fallopian tube, lymphatic vessels, and small blood vessels. In addition, peristaltic pumping occurring in many practical applications involving biomechanical systems such as roller, finger pumps, and heart-lung machine have been fabricated.

Since the first investigation of Latham [1], several theoretical and experimental studies have been conducted in the past to understand peristaltic action in different situations.

The literature on peristalsis is by now quite extensive, and a summary of most of the investigations has been presented in detail by Rath [2], L. M. Srivastava and V. P. Srivastava [3–5], Srivastava and Saxena [6], and V. P. Srivastava and L. M. Srivastava [7].

The theoretical study of the theory of particle-fluid mixture is very useful in understanding a number of diverse physical problems concerned with powder technology, fluidization, transportation of solid particles by a liquid, transportation liquid slurries in chemical and nuclear processing, and metalized liquid fuel slurries for rocketry. The sedimentation of particles in a liquid is of interest in much chemical engineering process, in medicine, where erythrocyte sedimentation has become a standard clinical test, and in oceanography as well as other fields. The particulate theory of blood has recently become the object of scientific research, Hill and Bedford [8], L. M. Srivastava and V. P. Srivastava [3–5, 9], Trowbridge [10], and Oka [11]. A number of research works on the topic, with and without peristalsis, have been reviewed by L. M. Srivastava and V. P. Srivastava [5]. Applications of the theory of particle-fluid mixture to the microcirculation and erythrocyte sedimentation included the work of Wang and Skalak [12], Bungay and Brenner [13], Skalak et al. [14], and Karino et al. [15].

Peristaltic transport of solid particle with fluid was first attempted by Hung and Brown [16]. They initiated the study of the peristaltic transport of solid particles, which included an experimental work on the particle transport in two-dimensional vertical channels having various geometries. In this connection also another paper by Brown and Hung [17] and a study by Takabatake and Ayukawa [18] are worth mentioning. Both studies have employed finite difference technique to solve two-dimensional nonlinear peristaltic flows problem. L. M. Srivastava and V. P. Srivastava [5] studied the peristaltic pumping of a particle-fluid mixture in a two-dimensional channel carried out mathematically; a perturbation solution is obtained. Mekheimer et al. [19] studied the peristaltic pumping of a particle-fluid suspension in a planar channel. El Misery et al. [20] studied the peristaltic motion of an incompressible generalized Newtonian fluid in a planar channel.

No-slip boundary conditions are convenient idealization of the behavior of viscous fluids near walls. The inadequacy of the no-slip condition is quite evident in polymer melts which often exhibit microscopic wall slip. The slip condition plays an important role in shear skin, spurt, and hysteresis effects. The boundary conditions relevant to flowing fluids are very important in predicting fluid flows in many applications. The fluids that exhibit boundary slip have important technological applications such as in polishing valves of artificial heart and internal cavities [21]. The slip effects on the peristaltic flow of a non-Newtonian Maxwellian fluid have been investigated by Eldesoky [22]. The influence of slip condition on peristaltic transport of a compressible Maxwell fluid through porous medium in a tube has been studied by Chu and Fang [23]. Many recent researches have been made in the subject of slip boundary conditions [24–33].

From the previous studies, there is no any attempt to study the effect of slip condition on the flow of a particle-fluid suspension with peristalsis. The purpose of this paper is to study the slip effects on the peristaltic pumping of a particle-fluid mixture in a two-dimensional channel. It is an application of the two-dimensional analysis of peristaltic motion of a particle-fluid mixture by L. M. Srivastava and V. P. Srivastava [5] and the two-dimensional analysis of peristaltic motion of single phase fluid by Fung and Yih [34] in the presence of slip effect. The mathematical model considers a particle-fluid mixture between infinite parallel walls with slip condition on which a sinusoidal traveling wave is imposed. A perturbation solution is obtained which satisfies the momentum equations for the case in which amplitude ratio (wave amplitude/channel half width) is small. Finally, the phenomenon of the mean flow reversal is presented and its physiological implication is discussed. Beside the engineering applications of pumping particle-fluid mixture by peristalsis, the present analysis of two-phase flow process is potentially important in regard to biofluid transport by peristalsis muscular contractions in body organs where fluids behave like particle-fluid mixtures, namely, chime in small intestine, spermatic fluid in cervical canal, urine (from a diseased kidney) in ureter, and blood suspension in arteriole.

#### 2. Formulation of the Problem

Consider a two-dimensional infinite channel of mean width (see Figure 1), filled with a mixture of small spherical rigid particles in an incompressible Newtonian viscous fluid. The walls of the channel are flexible, on which are imposed travelling, sinusoidal wave of small amplitude. The equations governing conservation of mass and linear momentum for both fluid and particle phase using a continuum approach are expressed as follows (Drew [35]; L. M. Srivastava and V. P. Srivastava [5, 9]).