Influence of Heat and Mass Transfer on Peristaltic Transport of Viscoplastic Fluid in Presence of Magnetic Field through Symmetric Channel with Porous Medium

In the present paper, we discussed the influence of heat and mass transfer on peristaltic transport of viscoplastic fluid in presence of magnetic field through symmetric channel with porous medium. The constitutive equation of Bingham plastic model is chosen to describe viscoplastic material. The nonlinear partial differential equations that described the motion of flow are simplified under assumptions of low Reynolds number and long wavelength. These equations are solved by mean of the regular perturbation method which is restricted to the smaller values of Bingham and Grashof numbers. Series solution for the axial velocity, temperature and concentration distribution have been computed. The flow quantities have been illustrated graphically for different interesting parameters. The pressure rise and trapping phenomena are also examined graphically. MATHEMATICA software is used to plot all figures.


Introduction
Peristalsis is well known process in which a progressive wave of contraction or expansion moves along the walls of channel causing the movement of contents of channel. This phenomena widely accurse in several applications to biological, medical and engineering; like urine motion from kidney to bladder through ureter, transportation of lymph from lymphatic vessels, heart-lung machine and many others. Many works on peristaltic flow in various geometries shown that the non-Newtonian behavior and the non-Newtonian fluid flows have many applications in engineering and medicine. So Bingham fluid is one of the non-Newtonian fluid models which is chosen for our study. Heat transfer is an important principle in biological system. There are three mechanisms of heat transfer but the convection is the most applicable heat transfer modality within the circulation of fluid in human body [1][2][3]. There are many application of MHD flows in the biomedical sciences such as cancer tumor treatment, bleeding reduction during surgeries, power generation development of magnetic devices and many others. Therefore many researcher discussed the peristaltic transport with magnetic field effects [4][5][6]. Currently, the combined effects of heat and mass transfer receive considerable attentions due to its application in the salty springs in the sea, in reservoir engineering in connection with thermal recovery, in the chemical industry etc. [7]. Srinivas and Kothandapani [8] investigated the influence of heat and ICMAICT 2020 Journal of Physics: Conference Series 1804 (2021) 012060 IOP Publishing doi: 10.1088/1742-6596/1804/1/012060 2 mass transfer on MHD peristaltic flow through porous space with compliant walls. Effect of heat and mass transfer on peristaltic flow of a Bingham fluid in the presence of inclined magnetic field and channel with different wave forms is studied by Akram et al. [9]. Ramesh [10] discussed the influence of heat and mass transfer on peristaltic flow of a couple stress fluid through porous medium in the presence of inclined magnetic field in an inclined asymmetric channel.
In this paper, the influence of heat and mass transfer on peristaltic transport of viscoplastic fluid in presence of magnetic field through symmetric channel with porous medium has been investigated. By using the perturbation method the non-linear governing equations of flow are solved analytically. The effect of all parameters on the flow regime are explained graphically.

Mathematical Formulation
Consider a peristaltic transport of an incompressible magnetohydrodynamic (MHD) viscoplastic fluid in a two dimensional symmetric channel of width 2 with porous medium as shown in figure 1. The flow is generated by propagation of wave on the channel walls train moving ahead with constant speed . The uniform magnetic field is applied in Y-direction to study the effect of it on the fluid flow. Electric field is absent. Heat and mass transfer studied through convective condition. The geometries of the channel wall is given by [1,11] , ̅ cos ̅ 1 where is the wave amplitudes, is the wavelength, ̅ is the time and , is the rectangular coordinates in the fixed frame of reference.

Basic and Constitutive Equations
Based on the above consideration, the basic governing equations that describe the flow in the present problem are given by [5,12]  in which is the velocity, is the density, is the material time derivative, is the coefficient of thermal expansion, is the Cauchy stress tensor, is the acceleration due to gravity, ̅ ( is the current density, 0, , 0 is the magnetic field, is the electrical conductivity, is the viscosity, is the permeability parameter of porous medium, ∇ is the Laplace operator, is the temperature, is the thermal conductivity, is the specific heat, ̅ is the mass concentration, is the coefficient of mass diffusion, is the thermal diffusion ratio and is the mean temperature. The term . ∇ in equation (4) can be compute from the definition of dot product of two tensor (if and are any two tensor then . ). Let and be the velocity components along the and -directions respectively in the fixed frame, the velocity vector can be written as , , ̅ , , , ̅ , 0 . 6 The Bingham plastic fluid is considered and the constitutive equations can be defined as [1,5] ̅ , 7 ̅ 2 2 , 8 in above equations, ̅ is the extra tensor, , is the identity tensor, is the pressure while the rate of deformation tensor and the tensor are defined by ∇ ∇ , √ . 9 From equations (2)-(7), the governing equation in the fixed frame are given by 0, 10 The corresponding boundary conditions are 0 , , ̅ ̅ 0 , , ̅ . 15 In view of equations (8) and (9), the components of extra stress tensor in the fixed frame becomes Peristaltic motion is unsteady phenomenon in nature but it can be assumed steady by using the transformation from the laboratory frame (fixed frame) , to the wave frame (move frame) ̅ , which defined as [1,11] ̅ where , and ̅ are the velocity components and the pressure in the wave frame, respectively. Now, we transform equations (1) and (10)-(16) in wave frame with the help of equation (17) and normalize the resulting equations by using following non-dimensional quantities [1,11]  then at the right wall and at the left wall of the channel. The non-dimension expression for pressure rise over one cycle of the wave is given ∆ , 42 which is difficult to evaluate directly, so we used MATHEMATICA software to compute it numerically.

Solution of the Problem
In above equations, we have a system of non-linear partial differential equations which is difficult to solve it exactly. So had to resort to the application of an approximation method, via the perturbation method to solve it.

Results and Discussion
To study the effect of physical parameters such as Hartman number (magnetic parameter) , permeability parameter , amplitude wave , Bingham number , Brinkman number , Grashof number , flow rate , Schmidt number , and Soret number , we have plotted the axial velocity , temperature , concentration Ω, pressure rise ∆ and trapping phenomenon in figures 2-33. MATHEMATICA software is used to plot all figures.

Velocity Distribution
Graphical results are displayed in order to see the behavior of parameters involved in the axial velocity . The effect of different values of , , , , and on the axial velocity are explained in figures 2-7. The behavior of velocity distribution is parabolic as seen in figures. Figure 2 explained the effect of on the axial velocity . This result agree with the result of Adnan and Abdulhadi [5]. It is noticed that with an increase of , the axial velocity increases at the walls of the channel, however, it decreases at the central part of the channel. Figure 3 displayed the influence of on the axial velocity. It is noticed that at the walls of the channel the axial velocity decreases slowly with an increase of , however it increases at the center of the channel. Figure 4 shown the effect of on the axial velocity . It is observed that the increase in lead to decreases at the left wall of the channel, while increasing at the middle portion and then gradually disappear as there is no effect on axial velocity near the right wall of the channel. From figure 5 noted that the axial velocity do not change at increasing in . Figures 6 and 7 displayed that the axial velocity increases with an increase in and .

Temperature Distribution
The variation in temperature profile for different values of involved parameters are displayed in figures 8-13. Figures 8 and 9 are shown that the impact of and on the temperature profile . It is noticed that the temperature distribution decreases in the central region and increases near the channel walls with increasing in and . Figures 10-13 are explained that the temperature increases by increasing in , , and . The effects of , and are consistent with the results analyzed in previous studies (Adnan and Abdulhadi [5] and Ali and Asghar [1]).

Concentration Distribution
The graphical results for concentration profile are illustrated in figures 14-21. Opposite behaviour for concentration distribution is noticed compare with the temperature distribution. Figures 14 and 15 are illustrated that the effect of and on the concentration profile Ω. It is noticed that the concentration distribution increases in the central region and decreases near the channel walls with increasing in and , but opposite behaviour is appearing with the increase in and as displayed in figures 16 and 17. Figures 18-21 are shown that the concentration decreases by increasing in , , and . Ali and Asghar [1] is also of the same opinion for the results of Brinkman number .

Pressure Rise ∆
The variation of pressure rise ∆ per wave length against the mean flow rate Θ of a symmetric channel are explained in figures 22-27 and the influence of pertinent parameters on the pressure rise ∆ are illustrated. The entire pumping region consist of three zones, which are retrograde pumping Θ 0, ∆ 0 , co-pumping Θ 0, ∆ 0 and augmented pumping Θ 0, ∆ 0 (see Adnan and Abdulhadi [5] and Misra, et al. [11]) . Figure 22 highlights the variation of ∆ for different values of Hartman number , from this figure we observed that ∆ increases by increasing in retrograde pumping while revers tend is noticed in the co-pumping and augmented regions. The opposite results are revealed for rescinding values of permeability parameter as displayed through figure 23. From figure 24 we noted that ∆ decreases with increase in Bingham number in all three region (retrograde pumping, co-pumping and augmented pumping regions). Figure 25 explained that ∆ not change by increasing of . It is visualized from figure 26 that an increase in causes increases ∆ only in augmented pumping region and the others stay in rest. Figure 27 is plotted to see the effect of on the pressure rise ∆ . It is noted that ∆ decreases in retrograde pumping and co-pumping regions and it increases in the augmented pumping region.

Conclusions
In the present paper, the influence of heat and mass transfer on peristaltic transport of viscoplastic fluid in presence of magnetic field through symmetric channel with porous medium has been investigated. The flow problem is transformed from laboratory frame to move frame by using appropriate transformation. Low Reynolds number and long wavelength are used to simplify the problem. The method of perturbation is employed to solve the governing equations of flow. In view of this study, some of the interesting outcomes are summarized as follows:  The axial velocity decreases in the central region and it increases near the boundaries of the channel with increasing but the opposite occur for increasing . Moreover, it increases over the whole cross-section with increasing and .  By increasing , the axial velocity decreases near the left wall while it increases at the center of the channel. Further has not effected on the axial velocity.  It is noted that, the temperature profile decreases over the whole cross-section of the channel while it increases at the boundaries with increasing and . Further the temperature enhances with increasing , , and .  Opposite behavior for concentration distribution is noted compared to temperature profile.
Furthermore, the concentration decreases over the whole cross-section except near the boundaries of the channel it is increases with increasing and .  The impacts of pertinent parameters on the pumping rate are different for different pumping region.  The volume and number of the trapped bolus decreases by increasing but it increases for increasing , and . Further by increasing the trapped bolus increases in number and size at the left wall of the channel but opposite behavior occur at the right wall.