Influence of Heat Transfer on MHD Oscillatory Flow for Eyring-Powell Fluid through a Porous Medium with Varying Temperature and Concentration

The aim of this research is to study the effect of heat transfer on the oscillating flow of the hydrodynamics magnetizing Eyring-Powell fluid through a porous medium under the influence of temperature and concentration for two types of engineering conditions "Poiseuille flow and Couette flow". We used the perturbation method to obtain a clear formula for fluid motion. The results obtained are illustrated by graphs.


Introduction
Many researchers have been interested in the analysis of non-Newtonian fluids during the past few decades. The main concept behind MHD is that magnetic fields can stimulate currents in a moving conductive fluid which in turn polarizes the fluid and similarly changes the magnetic field itself. MHD plays an important role in different areas of science and technology. Nigamf and Singhj [1] studied the flow between parallel plates under the influence of the transverse magnetic field and heat transfer. Raptis et al. [2] studied the hydro-magnetic free convection flow through a porous medium between two parallel plates. Hamza et al. [3] discussed the effects of the slipping state as well as the transverse magnetic field and the radiative heat transfer for the unstable flow of a thin fluid. Khudair and Alkhafajy [4] discussed the effect of heat-transfer on MHD oscillatory flow for Williamson fluid through the porous medium. Migtaa and Al-khafajy discussed the effect of heat transfer on the MHD oscillatory flow of Carreau-Yasuda fluid through a porous medium [5]. Hayat and Abdulhadi [6] discussed the peristaltic transport of MHD Eyring-Powell fluid through porous medium in a three

Al-Khafajy and Hadi
Iraqi Journal of Science, 2020, Vol. 61, No. 12, pp: 3355-3365 5533 dimensional rectangular duct. Hussain et al. analyzed the MHD flow of Powell-Eyring fluid by a stretching cylinder with Newtonian heating [7]. Begam and Deivanayaki studied the pulsatile flow of Eyring-powell nanofluid with Hall effect through a porous medium in [8]. More details about this topic are provided elsewhere [9][10][11][12][13][14][15][16][17]. Recently, a group of researchers described the effects of temperature and concentration on fluid movement. Most of these investigations agreed that the increase in temperature increases the velocity of the fluid while the fluid velocity changes in an unclear manner with the difference in concentration and according to the location of the fluid in the channel [16][17][18][19][20][21]. The present analysis aims to discuss the effects of heat transfer on the oscillating flow of the hydrodynamics of magnetizing Eyring-Power fluid through a porous medium under the influence of temperature and concentration for two types of engineering flows "Poiseuille flow and Couette flow". To our knowledge, this attempt has not yet been explored. This paper consists of six sections; section 1, which is the introduction, provides a historical overview of the studies that dealt with this topic. Section 2 includes the form of the flow channel with the formulation of the governing equations with boundaries conditions and the formula of the Eyring-Powell fluid equation. In section 3, we review the dimensionless transformations to formulate the governing equations in a way that helps in solving them. Section 4 includes problem-solving and finding the formula for temperature, concentration, and velocity for the two types of engineering flows. In sections 5 and 6, we discuss the results through illustrated graphs and review the most important observations that we reached.

Mathematical Formulation
Let us consider the flow of an Eyring-Powell fluid in a porous medium of width h under the effects of the electrically applied magnetic field and radioactive heat transfer, as illustrated in Figure-1. Suppose that the fluid has very small electromagnetic force and the electrical conductivity is small. We are considering Cartesian coordinate system such that ( ( ) ) is a velocity vector in which is the x-component of velocity and y is perpendicular to the x-axis. The continuity equation is given by: The momentum equations are: In thedirection: ( The temperature equation is given by: The concentration equation is given by: where ̅ is the axial velocity, is the density of the fluid, is the pressure, is the electrical conductivity, is the strength of the magnetic field, and is the acceleration due to gravity. In the same equations, we can define as a temperature, is specific heat at constant pressure, is the radiation heat flux, and is thermal conductivity.

Al-Khafajy and Hadi
Iraqi Journal of Science, 2020, Vol. 61, No. 12, pp: 3355-3365 5533 is heat generation, is the coefficient of mass diffusivity, ( ) is the angle between velocity field and magnetic field strength, and is the thermal diffusion ratio. The corresponding boundary conditions are given below: where is the radiation absorption. The fundamental equation for Eyring -Powell fluid is given by: where ̅ is the pressure, is the unit tensor, ̅ is the extra stress tensor, is the zero shear rate viscosity, and ̅ is the velocity gradient. We can write the component of extra stress tensor as follows:

Method of Solution
The non-dimensional governing equations are given by: where is the mean flow velocity, is the Darcy number, is the Reynolds number, is the magnetic parameter, is the Peclet number, is the radiation parameter, is the Schmidt number, is the Soret number, is the heat generation parameter, is the mean temperature, is the thermal Grashof number, and is the solutal Grashof number. Substituting equations (7) -(9) into equations (1) -(6) yields the following nondimensional equations: (10) .
With the boundary conditions By Substituting equation (15) into equation (11) after simple algebra, we have:

Solution of the Problem
This section contains the solution to the governing equations that is related to the above equations.

Solution of the Heat and Concentration Equations
To achieve this solution, we use the separating variables method, by assuming that ( ) ( ) for heat equation (13) and ( ) ( ) for the concentration equation (15), where is the frequency of oscillation with the boundary condition (18) [21]. As a result, we obtain the heat equation solution as follows: The concentration equation solution is achieved by: where √ ( ).

Solution of the Motion Equation
To solve the motion equation for two flows which are "Poiseuille flow and Couette flow", let , ( ) ( ) .
(22) where is a real constant and is the frequency of the oscillation. By substituting equation (22) into equation (19) then simplifying the result we get: We assume a small value of for the purpose of using the perturbation technique to solve the nonhomogeneous nonlinear partial differential equation (23). Accordingly, we write: ( ).

Poiseuille flow
We employ the solution of equation (25) for Poiseuille flow by using boundary condition (16) to solve the zero and first orders system. I -Zero-order system ( ) with boundary condition ( ) ( ) .

III -Zero-order solution
The solution of the zero-order equation subject to the associate boundary conditions is:

IIII -First -order solution
The solution of the first-order equation subject to the associate boundary conditions is: .

Couette flow
In this flow, the lower flake is fixed and the upper plate is moving with the velocity ℎ. The

Results and Discussion
We discuss the influence of heat transfer on MHD oscillatory flow for Eyring -Powell fluid through a porous medium with varying temperatures and concentrations for two types of engineering flows "Poiseuille flow and Couette flow" by using graphical illustrations. The temperature difference on both sides of the flow channel affects the fluid movement within the flow channel . The temperature difference depends on the parameters of R, Q, Pe and , as shown in the temperature charges. In equation (2) we notice the effects of different temperatures and concentrations, on both sides of the flow channel, on the fluid movement within the flow channel. We provide numerical assessments of analytical results and some of the graphically significant results that are presented in Figures-2-23. We used the MATHEMATICA-12 program to find numerical results and illustrations. The velocity profile of the Poiseuille flow is shown in Figures-2-9. Figure-2 shows that velocity profile decreases with increasing and . Figure-3 illustrates the influence of and on the velocity profiles on the axis. It is found that the velocity decreases with the increase of while it increases with the increase of . As illustrated in Figure-4, the velocity profile increases with the increase of and , respectively, while it decreases with the increasing the parameters and , as shown in Figure-5. Figure-6 illustrates the influence of and on the velocity profiles function on the axis. It is found that by increasing , the velocity increases, whereas it decreases with increasing . We found that the velocity increases with increasing Da, Pe, Re and Q, as demonstrated in Figures-7 and 8, respectively. Figure-9 shows that the velocity increases with the increase of A and decreases with the increase of . The velocity profile of Couette flow is shown in Figures-(10-17). It is found that the velocity increases with increasing the parameters , , , , Da, Pe, Re, Q and A, respectively, while the velocity decreases with the increase of , , , , and . Based on equation (20), Figures-(18-19) show that the temperature increases with the increase in , and Pe, while it decreases with the increase in . Based on equation (21), the concentration decreases with the increase of all parameters, (Figures-20-23).

Concluding Remarks
We discuss the influence of heat transfer on MHD oscillatory flow for Eyring-Powell fluid through a porous medium with varying temperature and concentration. Using the perturbation technique, we analyzed the velocity, temperature and concentration. We used different values to find the results of pertinent parameters, namely Darcy number, Peclet number, Grashof number, magnetic parameter, radiation parameter, Schmidt number, Soret number, heat generation parameter, frequency of the oscillation, and Reynold number. The key points are:  In the two types of flow. i.e. Poiseuille and Couette, the velocity increases with increasing the parameters , , , , Da, Pe, Re, Q and A, respectively, while the velocity decreases with increasing , , , , and .  The temperature increases with the increase in , and Pe while decreases with the increase in .  The concentration decreases with the increase of all parameters.