Journal of Fluids

Volume 2015, Article ID 163832, 19 pages

http://dx.doi.org/10.1155/2015/163832

## Effects of Heat and Mass Transfer on the Peristaltic Transport of MHD Couple Stress Fluid through Porous Medium in a Vertical Asymmetric Channel

Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, India

Received 30 September 2014; Accepted 10 February 2015

Academic Editor: Jamshid M. Nouri

Copyright © 2015 K. Ramesh and M. Devakar. 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

The intrauterine fluid flow due to myometrial contractions is peristaltic type motion and the myometrial contractions may occur in both symmetric and asymmetric directions. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitude, and phase due to the variation of channel width, wave amplitudes and phase differences. In this paper, we study the effects of heat and mass transfer on the peristaltic transport of magnetohydrodynamic couple stress fluid through homogeneous porous medium in a vertical asymmetric channel. The flow is investigated in the wave frame of reference moving with constant velocity with the wave. The governing equations of couple stress fluid have been simplified under the long wave length approximation. The exact solutions of the resultant governing equations have been derived for the stream function, temperature, concentration, pressure gradient, and heat transfer coefficients. The pressure difference and frictional forces at both the walls are calculated using numerical integration. The influence of diverse flow parameters on the fluid velocity, pressure gradient, temperature, concentration, pressure difference, frictional forces, heat transfer coefficients, and trapping has been discussed. The graphical results are also discussed for four different wave shapes. It is noticed that increasing of couple stresses and heat generation parameter increases the size of the trapped bolus. The heat generation parameter increases the peristaltic pumping and temperature.

#### 1. Introduction

In recent years, the flow of non-Newtonian fluids has received much attention due to the increasing industrial, medical, and technological applications. Various researchers have attempted diverse flow problems related to several non-Newtonian fluids and couple stress fluid is one of them. The theory of couple stress fluids originated by Stokes [1] has many biomedical, industrial, and scientific applications and was used to model synthetic fluids, polymer thickened oils, liquid crystals, animal blood, and synovial fluid. Some earlier developments in couple stress fluid theory with some basic flows can be found in the book by Stokes [2]. Recently, few researchers have studied some couple stress fluid flows for different flow geometries [3–8].

Nowadays, peristaltic flows have gained much attention because of their applications in physiology and industry. Peristaltic transport is a form of fluid transport induced by a progressive wave of area contraction or expansion along the length of a distensible tube/channel and transporting the fluid in the direction of the wave propagation. This phenomenon is known as peristalsis. In physiology this plays an important role in various situations such as the food movement in the digestive tract, urine transport from kidney to bladder through ureter, movement of lymphatic fluids in lymphatic vessels, bile flow from the gall bladder into the duodenum, spermatozoa in the ductus efferentes of the male reproductive tract, ovum movement in the fallopian tube, blood circulation in the small blood vessels, the movement of the chyme in the gastrointestinal tract, intrauterine fluid motion, swallowing food bolus through esophagus, and transport of cilia. Many industrial and biological instruments such as roller pumps, finger pumps, heart-lung machines, blood pump machines, and dialysis machines are engineered based on the peristaltic mechanism [9]. The intrauterine fluid flow due to myometrial contractions is peristaltic in nature and these myometrial contractions occur in both symmetric and asymmetric directions and also when embryo enters the uterus for implantation there start the asymmetric contractions. The contractions inside the nonpregnant uterus are very complicated because they are composed of variable amplitudes and different wave lengths [10]. In view of this, Pandey and Chaube [11] have investigated the peristaltic transport of a couple stress fluid in a symmetrical channel using perturbation method in terms of small amplitude ratio. Ali and Hayat [12] have studied the peristaltic motion of micropolar fluid in an asymmetric channel. Naga Rani and Sarojamma [13] have analyzed the peristaltic transport of a Casson fluid in an asymmetric channel. Hayat et al. [14] have discussed the peristaltic flow of a Johnson-Segalman fluid in an asymmetric channel. Hayat and Javed [15] have studied the peristaltic transport of power-law fluid in asymmetric channel.

The porous medium plays an important role in the study of transport process in biofluid mechanics, industrial mechanics, and engineering fields. The fluid transport through porous medium is widely applicable in the vascular beds, lungs, kidneys, tumorous vessels, bile duct, gall bladder with stones, and small blood vessels. In the pathological situations, the distribution of fatty cholesterol, artery clogging, blood clots in the lumen of coronary artery, transport of drugs and nutrients to brain cells, and functions of organs are modeled as porous medium [16]. Recently, Tripathi [17] studied the peristaltic hemodynamic flow of couple stress fluids through a porous medium. Tripathi and Bég [18] have investigated the peristaltic flow of generalized Maxwell fluid through a porous medium using homotopy perturbation method. Abd elmaboud and Mekheimer [19] have discussed peristaltic transport of a second-order fluid through a porous medium using regular perturbation method. The magnetohydrodynamic flows also gained much attention due to the widespread applications in biofluid mechanics and industry. It is the fact that many fluids like blood are conductive in nature and gave a new direction for research. The indispensable role of biomagnetic fluid dynamics in medical science has been very helpful with many problems of physiology. It has wide range of applications, such as magnetic wound or cancer tumour treatment, bleeding reduction during surgeries, provocation of occlusion of feeding vessels of cancer tumor, cell separation, transport of drugs, blood pump machines, and magnetic resonance imaging to diagnose the disease and the influence of magnetic field which may be utilized as a blood pump in carrying out cardiac operations for the blood flow in arteries with arterial disease like arterial stenosis or arteriosclerosis. Specifically, the magnetohydrodynamic flows of non-Newtonian fluids are of great interest in magnetotherapy. The noninvasive radiological tests use the magnetic field to evaluate organs in abdomen [20]. Hayat et al. [21] have studied the peristaltic transport of magnetohydrodynamic Johnson-Segalman fluid for the case of a planar channel. Wang et al. [22] have investigated the peristaltic motion of a magnetohydrodynamic generalized second-order fluid in an asymmetric channel. Nadeem and Akram [23, 24] have discussed the peristaltic transport of a couple stress fluid and Williamson fluid in an asymmetric channel with the effect of the magnetic field.

Heat transfer plays a significant role in the cooling processes of industrial and medical applications. Such consideration is very important since heat transfer in the human body is currently considered as an important area of research. In view of the thermotherapy and the human thermoregulation system, the model of bioheat transfer in tissues has been attracted by the biomedical engineers. In fact the heat transfer in human tissues involves complicated processes such as heat conduction in tissues, heat transfer due to perfusion of the arterial-venous blood through the pores of the tissue, metabolic heat generation, and external interactions such as electromagnetic radiation emitted from cell phones [25]. Heat transfer also involves many complicated processes such as evaluating skin burns, destruction of undesirable cancer tissues, dilution technique in examining blood flow, paper making, food processing, vasodilation, and radiation between surface and its environment [26]. Mustafa et al. [27] have studied the peristaltic transport of nanofluid in a channel. The heat transfer characteristics of a couple stress fluid in an asymmetric channel have been analyzed by Abd elmaboud et al. [28]. Nadeem and Akbar [29] have discussed the influence of heat transfer and magnetic field on peristaltic flow of a Johnson-Segalman fluid in a vertical symmetric channel. Some more works regarding peristaltic flows with the effect of heat transfer and magnetic field can be seen in [30–33]. Srinivas et al. [34] have studied the effects of both wall slip conditions and heat transfer on peristaltic flow of MHD Newtonian fluid in a porous channel with elastic wall properties. Mass transfer is another important phenomenon in physiology and industry. This phenomenon has great applications such as nutrients’ diffusion out from the blood to neighboring tissues, membrane separation process, reverse osmosis, distillation process, combustion process, and diffusion of chemical impurities [35]. Recently, Noreen [36] studied the problem of mixed convection peristaltic flow of third-order nanofluid with an induced magnetic field. Saleem and Haider [37] have discussed the peristaltic transport of Maxwell fluid with heat and mass transfer in an asymmetric channel. Some more relevant works on the peristaltic transport with heat and mass transfer can be seen in [38–42].

The aim of the present study is to investigate the influence of heat and mass transfer on the peristaltic flow of magnetohydrodynamic couple stress fluid through homogeneous porous medium in a vertical asymmetric channel. This paper is arranged as follows. Section 2 presents the mathematical formulation for the problem. The solution of the problem is obtained in Section 3. The four different wave forms are presented in Section 4 while the computational results are discussed in Section 5. The last section, Section 6, presents the conclusions of the present study.

#### 2. Formulation of the Problem

Let us consider magnetohydrodynamic couple stress fluid in a vertical asymmetric channel through the porous medium with the width of . The surfaces and of the asymmetric channel are maintained at constant temperatures and and the constant concentrations and , respectively (see Figure 1). The porous medium is assumed to be homogeneous. The motion is induced by sinusoidal wave trains propagating with constant speed along the channel walls as defined by the following:where and are the wave amplitudes, is the wave length, is the channel width, is the velocity of propagation, is the time, and is the direction of wave propagation. The phase difference varies in the range , in which corresponds to symmetric channel with waves out of phase and corresponds to waves in phase, and further , , , , and meet the following relation .