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.

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 [

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 [

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 [

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 [

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

Let us consider magnetohydrodynamic couple stress fluid in a vertical asymmetric channel through the porous medium with the width of

Physical model of the problem.

The continuity, momentum, energy, and concentration equations for an MHD incompressible couple stress fluid, in the absence of body couples, are [

In the fixed frame, governing equations for the peristaltic motion of an incompressible magnetohydrodynamic couple stress fluid through homogeneous porous medium in the two-dimensional vertical channel are

The coordinates, velocities, pressure, temperature, and concentration in the fixed frame

Using (

In terms of these nondimensional variables, the governing equations (

Assuming that the wave length of the peristaltic wave is very large compared to the width of the channel, the wave number

The nondimensional expressions for the four considered wave forms are given in the following.

Sinusoidal wave:

Triangular wave:

Square wave:

Trapezoidal wave:

This section is dedicated to discussion and analysis of the velocity distribution, pumping characteristics, heat and mass characteristics, and trapping phenomena for different flow parameters.

Figures

Velocity profile for (a)

Figure

Pressure gradient for (a)

Pressure gradient for various wave forms for fixed values of

Sinusoidal wave

Triangular wave

Square wave

Trapezoidal wave

The dimensionless pressure difference per unit wave length versus time mean flow rate

Pressure difference for (a)

Figure

Frictional force at the right wall for (a)

(a) Pressure difference, (b) frictional force at right wall, and (c) frictional force at left wall for different wave forms when

Figure

(a) Temperature profile, (b) concentration profile, and (c) heat transfer coefficient at the right wall for

In the wave frame, the streamlines, in general, have a shape similar to the walls as the walls are stationary. However under certain conditions some streamlines can split to enclose a bolus of fluid particles in closed streamlines. Hence some circulating regions occur. In the fixed frame of reference the fluid bolus is trapped with the wave and it moves as a whole with the wave speed. To examine the effects of

Streamlines for

Streamlines for

Streamlines for

Streamlines for various wave forms for fixed values of

Sinusoidal wave

Triangular wave

Square wave

Trapezoidal wave

Streamlines for various wave forms for fixed values of (a)

Sinusoidal wave

Triangular wave

Square wave

Trapezoidal wave

The effects of heat and mass transfer on the peristaltic flow of magnetohydrodynamic couple stress fluid through porous medium in a vertical asymmetric channel have been analyzed. The governing equations are modeled under the assumption of long wave length approximation. The exact solutions for the stream function, pressure gradient, temperature, heat transfer coefficients, and concentration are obtained. The effects of involved parameters on the velocity characteristics, pumping characteristics, heat and mass characteristics, and the trapping due to the peristalsis of the walls are discussed in detail. From the analysis the main findings can be summarized as follows:

Increasing of heat generation increases the peristaltic pumping, size of the trapped bolus, and the magnitude of heat transfer coefficient at the peristaltic walls.

Increasing of couple stresses increases the size of trapped bolus.

Increasing of heat generation increases the temperature and decreases the concentration.

The trapezoidal wave has best peristaltic pumping as compared to the other wave shapes.

The frictional forces have an opposite behaviour as compared to the pressure difference.

Wave amplitudes

Wave length

Propagation velocity

Time

Coordinates of fixed frame

Pressure in the fixed frame

Velocity vector

Darcy’s resistance in the porous medium

Density

Viscosity

Material constant associated with couple stress

Electric current density

Total magnetic field

Acceleration due to the gravity

Coefficient of thermal expansion

Coefficient of expansion with concentration

Specific heat at constant pressure

Temperature

Mass concentration

Thermal conductivity

Heat generation parameter

Coefficient of mass diffusivity

Thermal diffusion ratio

Mean temperature

Permeability parameter

Velocity components in the fixed frame

Pressure in the fixed frame

Electrical conductivity of the fluid

Uniform applied magnetic field

Dimensionless wave length

Coordinates of wave frame

Pressure in the wave frame

Velocity components in the wave frame

Reynolds number

Hartman number

Darcy number

Couple stress parameter

Local temperature Grashof number

Local concentration Grashof number

Prandtl number

Dimensionless temperature

Dimensionless concentration

Dimensionless heat generation parameter

Schmidt number

Soret number

Stream function

Time mean flow rate in the fixed frame

Time mean flow rate in the wave frame

Volume flow rate in the fixed frame

Volume flow rate in the wave frame.

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