Peristaltic transport of a Jeffrey fluid under the effect of magnetic field in an asymmetric channel

Abstract

The peristaltic flow of a Jeffrey fluid in an asymmetric channel is studied under long wavelength and low Reynolds number assumptions. The fluid is electrically conducting by a transverse magnetic field. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitudes and phase. The flow is investigated in a wave frame of reference moving with the velocity of the wave. The expressions for stream function, axial velocity and axial pressure gradient have been obtained. The effects of various emerging parameters on the flow characteristics are shown and discussed with the help of graphs. The pumping characteristics, axial pressure gradient and trapping phenomenon have been studied. Comparison of various wave forms (namely sinusoidal, triangular, square and trapezoidal) on the flow is discussed.

Introduction

Peristaltic transport is a form of material transport induced by a progressive wave of area contraction or expansion along the length of a distensible tube, mixing and transporting the fluid in the direction of the wave propagation. This phenomenon is known as peristalsis. It plays an indispensable role in transporting many physiological fluids in the body such as urine transport from kidney to bladder, the movement of chyme in the gastrointestinal tracts, the transport of spermatozoa in the ductus efferentes of the male reproductive tract, the movement of ovum in the fallopian tubes, the swallowing of food through esophagus and the vasomotion of small blood vessels. Many modern mechanical devices have been designed on the principle of peristaltic pumping for transporting fluids without internal moving parts, for example, the blood pump in the heart–lung machine and the peristaltic transport of noxious fluid in nuclear industry. The problem of the mechanism of peristaltic transport has attracted the attention of many investigators. Since the first investigation of Latham [1], a number of analytical, numerical and experimental studies of peristaltic flow of different fluids have been reported under different conditions with reference to physiological and mechanical situations (Refs. [1], [2], [15], [27], [28], [29], [30], [31], [32] and references therein).

There are also a few studies available about the peristaltic flow of non-Newtonian fluids in asymmetric channels. Some recent interesting investigations in this direction are given in Refs. [3], [4], [5], [6], [7]. Due to the flow behaviour of non-Newtonian fluids, the governing equations become more complex to handle as additional non-linear terms appear in the equations of motion. There is also no universal constitutive model available which exhibits the characteristics of the all non-Newtonian fluids. Mention may be made to some interesting studies done previously, pertaining to non-Newtonian fluids, which may give insights into their behaviour (Refs. [8], [9], [10], [11], [12], [13] and references therein). Some recent studies [14], [16], [17] have considered the peristaltic motion of magnetohydrodynamic (MHD) Newtonian and non-Newtonian fluids in asymmetric channels. The MHD flow of a fluid in a channel with elastic, rhythmically contracting walls is of interest in connection with certain problems of the movement of conductive physiological fluids, e.g., the blood, and with the need for theoretical research on the operation of a peristaltic MHD compressor. The effect of a moving magnetic field on blood flow was studied by Stud et al. [18], and they observed that the effects of a suitable moving magnetic field accelerate the speed of blood. Srivastava and Agarwal [19] considered the blood as an electrically conducting fluid that constitutes a suspension of red cells in the plasma. Recently, Mekheimer [20] analysed the MHD flow of a conducting couple stress fluid in a slit channel with rhythmically contracting walls. Srinivas and Kothandapani [14] have analysed the MHD peristaltic flow of a viscous fluid in asymmetric channel with heat transfer. More recently, Kothandapani and Srinivas [15] have studied the influence of wall properties in the MHD peristaltic transport with heat transfer and porous medium. Hayat et al. [16] have examined the effect of heat transfer on the peristaltic flow of an electrically conducting fluid in a porous space. Literature survey bears witness to the fact that the information on MHD peristalsis of non-Newtonian fluids in an asymmetric channel is scant. To the best of our knowledge only Wang et al. [17] have studied the MHD peristaltic motion of a Sisko fluid in an asymmetric channel.

In the present analysis, the liquid considered is of Jeffrey type and is electrically conducting. The Jeffrey model is a relatively simpler linear model using time derivatives instead of convected derivatives for example the Oldroyd-B model and it represents a rheology different from the Newtonian. The main purpose of the present study is to investigate the peristaltic pumping of MHD flow of a Jeffrey fluid in a two-dimensional asymmetric channel having electrically insulated walls. The channel asymmetry is produced by choosing the peristaltic wave train on the walls which have different amplitudes and phase due to the variation in channel width, wave amplitudes and phase differences. The governing equations of fluid flow are solved subject to relevant boundary conditions. The comparison among the four waveforms is also made carefully and influence of several pertinent parameters on the stream function and pressure drop have been studied and numerical results obtained are presented. The results and discussions presented in this study may be helpful to further understand MHD peristaltic motion for non-Newtonian fluids in an asymmetric channel.