The Effects of Electrical Conductivity on Fluid Flow between Two Parallel Plates in a Porous Medioum

This paper deals with a mathematical model of a fluid flowing between two parallel plates in a porous medium under the influence of electromagnetic forces (EMF). The continuity, momentum, and energy equations were utilized to describe the flow. These equations were stated in their nondimensional forms and then processed numerically using the method of lines. Dimensionless velocity and temperature profiles were also investigated due to the impacts of assumed parameters in the relevant problem. Moreover, we investigated the effects of Reynolds number , Hartmann number M, magnetic Reynolds number , Prandtl number , Brinkman number F , and Bouger number ω, beside those of new physical quantities (N , ). We solved this system by creating a computer program using MATLAB.


Introduction
Because of its vital applications in sciences that influence human life, the flow of electrically directed liquids across porous parallel plates has become a major issue. This is evident in food industry, extraction of crude oil from the earth, and the movement of the blood [1].Several researchers have looked at transferring the flow of oscillator liquids ISSN: 0067-2904 Hammodat et al. Iraqi Journal of Science, 2021, Vol. 62, No. 12, pp: 4953-4963 4954 between two parallel plates in various magnetic fields. Under saturated temperature conditions, Makinde and Mhone [2] used the combined effects of a random magnetic field and thermal radiation transmission to describe the unstable flow of high-optical fluid connected through a tube filled with irregular porous walls.
For the magneto hydrodynamic (MHD) oscillatory flow of Williamson fluid across a porous plate, Khudair and Al-Khafajy [1] devised a heat transfer model for two types of flow (Couette flow and Poiseuille flow). Al-Khafajy [3] studied the effects of magneto hydrodynamic (MHD) oscillatory slip flow of Jeffrey fluid with varying viscosity, temperature, and concentration, on a porous plate. Hanvey et al. [4] studied the effects of an inclined magnetic field, as well as heat and mass transmission, on two infinite parallel plates using porous materials. The differential transformation method was used by Sheikholeslami et al. to investigate the problem of unstable nanofluid flow between parallel plates [5].
In the presence of an inclined magnetic field, , as well as radiative heat flux and heat source, Mehta et al. [6] explored oscillatory fluid flow and heat transfer via a porous material between parallel plates. Mukhopadhyay and Mandal [7] looked for numerical solutions for steady state MHD mixed convective boundary layer flow and heat transfer through a porous plate in the presence of velocity and thermal slips. Kumari and Gayal [8,9] demonstrated the effects of mass transfer, viscous suction parameter, and the dissipative impact on a twodimensional steady-state hydromagnetic viscous fluid flow between two parallel plates in the presence of heat radiation. The steady state three-dimensional MHD flow of fluid injected uniformly into the vertical channel with porous wall through one side of the channel was solved analytically by Jabr and Abdulhadi [5].Our goal here is to investigate the flow of fluids in a cross-section under the influence of an electromagnetic field (EMF) and use the method of lines to solve the partial differential equations that describe the situation. In addition, we aim to demonstrate the behavior of temperature within the cross-section and the impacts of physical quantities.

Research Method
In the following sections, we try to solve our main equations in a simple way using a dimensionless approximation, after defining the model under study and the equations that control it in addition to the boundary conditions of the problem.

Protection Equation
We consider the magnetic fluid through a porous medium moving between two horizontally parallel plates with a distance of and a cross-section length of (Figure1). The model described by the cartesian coordinate system with coordinate is parallel to the channel wall in the flow direction and the coordinate is in the channel's vertical direction. The magnetic field utilized across the channel is parallel to the -axis, with a component parallel to the y-axis is fixed and symbolized as . The component in the -axis direction is which inflows along the channel in the flow direction, while the component in the z-axis direction is equal to zero.
The heat equation is: In these equations, ( , ) are the velocity of axial motion, ( ) is the density, ( ) is the pressure, is the kinematic viscosity, is the permeability of medium, ( ) is the agnetic permeability of medium, ( ⃗ ⃗ ) is the magnetic field vector, ( ) is the magnetic field component in and directions, respectively, is a gravitational acceleration constant, is the specific heat at constant volume, thermal conductivity coefficient, is the temperature, and , are the components of radiation in the and directions, respectively. The associated boundary conditions applied on the top and the bottom of the model are:

Method of Solution
For the dimensionalization of the governing equations, we adopt the following characteristic quantities: [10] ⃗  [11], then the equation (10) becomes: But ⃗⃗ Θ [10], then the non-dimensional energy equation becomes: The non-dimensional boundary conditions become:

Solution of the Problem
In this section, we try to solve the two-dimensional motion equation (12) with the boundary equation (15) and energy equation (14) using numerical simulation for and , where and are the arbitrary lengths of the computational domain in the -direction and -direction. respectively. We consider the case where coefficients and are treated as constants at any time step of the competition [12]. The main equations of the motion and energy, (12) and (14), are solved numerically using the method of lines [13]. We discretise the domain above into points in -direction and Then, keeping the time derivative continuous, the derivatives in equation (12) were discretised as follows: We assume the condition , then we use this condition to determine the fictitious points for the main equation (12) of and .In the same way, we discretised the energy equation as:

Results and Discussion
In this section, we preesnt the numerical solutions of a magnetized fluid running through a porous medium between two horizontal plates exposed to a source of disturbance. The finite difference method (forward and backward) was applied to calculate the results of the numerical solutions that were obtained graphically. We obtained the numerical results and illustrations of ordinary differential equations, given by equations (18) and (19), using MATLAB programme (ODE solver), as shown in Figures 2-12. The energy profiles in Figures 2,5,6,7 clearly show that the greater Brinkmann number ℱ , the Bouger number , the Prandtl number , and the relative temperature , the closer we get to stability. The effect of Hartmann number M is diverging from the stability for both of the motion and energy, when M and temperature increase (Figures 3 and 10). In Figures 4 and 8, we observe that the higher the Reynolds number , the closer to stability in both equations (motion and energy). While (Figure 9) shows that the higher magnetized Reynolds number the farther from stability. Finally, Figures 11 and 12 show that the smaller the new quantities of and N in the motion equation, the further away it is from stability.

Conclusions
In this section, because of the occurrence of effects on the tested equations, we have presented some of the results obtained from the calculation done on equations (12) and (14) for different places in the region of solution. Figures 2-12 show several results. In terms of the energy equation, we can see that the higher the Brinkmann number, the Bouger number, the Prandtl number, and the relative temperature, the closer we get to stability, as shown in Figures 2, 5, 6, and 7. The effect of Hartmann number M is diverging from the stability for both of the motion and energy when M and temperature are increasing. As illustrated in Figures 4 and 8, the greater the Reynolds, the closer to stability are both equations (motion and energy). According to Figure 9, the higher the magnetic Reynolds number, the farther the system from stability. Figures 11 and 12 indicate that the closer the motion equation gets to stability, the smaller Hartmann number and the new physical quantity N.