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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.

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 [

The literature on peristalsis is by now quite extensive, and a summary of most of the investigations has been presented in detail by Rath [

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 [

Peristaltic transport of solid particle with fluid was first attempted by Hung and Brown [

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 [

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 [

Consider a two-dimensional infinite channel of mean width

Geometry of the problem.

In (

The concentration of the particles is considered to be so small that the field interaction between particles may be neglected. Thus, the diffusivity terms, which can model the effects of particle-particle impacts due to the Brownian motion, are neglected. It is worth mentioning here that the effect of Brownian motion was considered by others including Batchelor [

The expression for the drag coefficient for the present problem is selected as

Many empirical relations have been suggested to express the viscosity of the suspension as a function of particle concentration and viscosity of the suspending medium. Einstein was the first to obtain theoretically that the viscosity of the suspension

The boundary conditions that must be satisfied by the fluid on the walls are the slip and impermeability conditions. The walls of the channel are assumed to be flexible but extensible with a travelling sinusoidal wave, and displacement in the channel walls is in transverse direction only. Hence, boundary conditions are

We now select the following set of nondimensional variables and parameters:

Suspension Reynolds number

Wave number

Knudsen number

Amplitude ratio

Suspension parameter

Suspension parameter

Assuming the amplitude ratio

In (

Substituting (

The first set of differential equations in

Thus, the effect of the particles on the fluid velocity profile is to cause an increase in the viscosity; that is, fluid viscosity

The second and third sets of differential equations in

Thus, we obtained a set of differential equations together with the corresponding boundary conditions which are sufficient to determine the solution of the problem up to the second order in

Solutions of (

Next, in the expansion of

where

Thus, we see that one constant

The constant

The mean time average velocities may now be written as

If no-slip, that is,

Also if no-slip, that is,

A close look at (

For the sake of comparison we define mean-velocity perturbation function

It has been observed that urine, bacteria, or other materials some time pass from the

Since

For

The value of

Effect of the Knudsen number

We observed that the critical reflux pressure

Finally, in Figures

(a) Mean velocity distribution at

(a) Mean velocity distribution at

(a) Mean velocity distribution at

(a) Mean velocity distribution at

Figure

(a) Mean axial velocity (m/s) distribution at

Also we notice that the value of

Figure

Figure

Next, we return to the dimensional flow problem; the dimensional mean axial velocity ^{−2} s. The particle concentration

According to the Knudsen number, the flow regimes can be divided into various regions. These are continuum, slip, transition, and free molecular flow regimes. If

For example, for the left main coronary artery, the range of the diameter is 2.0–5.5 mm (mean 4 mm) and wavelength range

There is not any attempt to study the effect of slip conditions 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 [

Some concluding remarks are as follows.

The reversal flow increases with increasing particle concentration

Also we notice that the mean-velocity distribution increases with increasing Kn. Interpreted physiologically, this means that, under some conditions, urine in which solute particles are suspended (i.e., urine from a diseased kidney) is more susceptible to reversal flow in ureter, in comparison to pure urine without solute particles.

For example, for the left main coronary artery, the mean axial velocity

Comparing with other models for verifications of results, the present model gives the most general form of velocity expression from which the other mathematical models can easily be obtained by proper substitutions. It is of interest to note that the result of the present model includes results of different mathematical models such as the following.

The results of L. M. Srivastava and V. P. Srivastava [

The results of Fung and Yih [

The authors declare that there is no conflict of interests regarding the publication of this paper.