Unsteady Couette Flow Past between Two Horizontal Riga Plates with Hall and Ion Slip Effects

Riga plate is the span wise array of electrodes and permanent magnets that creates a plane surface and produced the electromagnetic hydrodynamic fluid behavior and mostly used in industrial processes with fluid flow affairs. In cases where an external application of a magnetic or electric field is required, better flow is obtained by the involvement of the Riga plate. Riga plate acts as an agent to reduce the skin friction and enhance the heat transfer phenomena. It also diminishes the turbulent effects, so that it is possible to get an efficient flow control and it increases the performance of the machine. So the numerical investigation of the unsteady Couette flow with Hall and ion-slip current effects past between two Riga plates has been studied and the numerical solutions are acquired by using explicit finite difference method and estimated results have been gained for several values of the dimensionless parameter such as pressure gradient parameter, Hall and Ion-slip parameters, modified Hartmann number, Prandtl number, and Eckert number. In this article, the importance of the modified Hartmann number on the flow profiles is immense owing to the Riga plate. The expression of skin friction and Nusselt number has been computed and the outcomes of the relevant parameters on various distributions have been sketched and presented as well as graphically.


Introduction
The highly conducting fluids are controlled by magneto-hydrodynamics flow with the influence of an external magnetic field known as classical MHD flow control. When the fluid is too weak in electrically conducting so that Lorentz force decreases exponentially then an external electric field must be applied to achieve an efficient flow control, this is called Electro-magneto hydrodynamic (EMHD) flow. Because of the importance of magnetic fields, the Riga plate holds a key place of its tremendous applications in the field of magneto-aerodynamics, civil engineering, mechanical engineering, chemical engineering, dust or fumes in a gas, in biomechanics and rectification of groundwater, and oil. The formulation of the Riga plate is the combination of electrodes and permanent magnets create a plane surface instead of polarity and magnetization that produce a wall parallel Lorentz force to control the fluid flow that was first introduced by Gailitis and Leilausis [1]. This order minimizes the friction and pressure drag of submarines. It is also separation of the boundary layer as well as diminishing the turbulence effects and so that a better flow pattern is obtained. When the flow passes, the Lorentz force is created by gathering the electrodes and permanent magnets in the flat surface. A turbulent channel event with a small Reynolds number is utilized the Lorentz force discovered by Berger et al. [2]. Pantokratoras and Magyari [3] proposed the electro-magneto hydrodynamic free convection fluid flow with a poor conductivity along a Riga plate. Wahidunnisa et al. [4] studied the heat source of a nanofluid flow through a Riga plate with viscous dissipation. Iqbal et al. [5] explored their idea on an electrically conducted Riga plate with viscous dissipation and thermal radiation of nanofluid with melting heat and they use the Keller Box scheme to obtain the solution. Ayub et al. [6] examined the EMHD nanofluid flow along an electromagnetic actuator or Riga plate. Ahmed et al. [7] carried out the mixed convection of a nanofluid flow along a vertical Riga plate with the effect of a strong suction. The physical problems of magneto-hydrodynamic flows with Hall and Ion-slip current have practical applications as electromagnetic flow meters, electromagnetic pumps and MHD power generator, aerodynamic heating, electrostatic precipitation, geophysics, astrophysics, and many engineering and industrial processes [8]. Javeri [9] investigated the combined effect of Hall and Ion slip currents, Joule heating, and viscous dissipation on the laminar MHD channel. Eraslan [10] expressed a distribution of temperature for the MHD channels with the Hall effect. The MHD laminar flow along a porous medium has significant applications in engineering and agricultural process, groundwater flows, petroleum industry, and oil and gas purification. A lot of research work has been held on the MHD steady or unsteady flows over a vertical porous plate with Hall and ion-slip under different physical effects has been studied of their wide applications. From the point of view, the effect of Hall current and ion-slip with heat and mass convection flows of an electrically conducting fluid has been discussed by several authors such as Seddeek and Aboeldahab [11] examined an unsteady free convection fluid flow with Hall currents effect of gray gas through an infinite porous plate where a strong transverse magnetic field is imputed perpendicularly to the plate. Debnath et al. [12] have studied the effects of Hall current on unsteady hydro magnetic flow past a porous plate in a rotating fluid system. Nimr and Masoud [13] carried out an unsteady free convection flow in a porous media along a flat plate. Krishna et al. [14] have carried out an unsteady free convective magneto-hydrodynamic flow with Hall and ion slip current effects through an accelerated inclined plate with rotation which is surrounded by a porous medium with the effect of inclined angle also with the change of reference frame. They have used Laplace transform to solve these problems analytically. Angirasa and Peterson [15] presented a numerical study on heat transfer in natural transmission from an isothermal vertical surface which is a stable layered to a fluid-saturated thermally stratified porous medium. Kumar and Singh [16] analyzed the heat transfer from a vertically isothermal surface with the impact of thermal stratification in a porous medium. The influence of MHD Couette flows with Hall and Ion slip current has a great importance of experimental and theoretical applications in magnetic material processing, astrophysics, polymer technology, heating electrostatics, nuclear engineering, pumps and power generators, geophysical and industrial fields. From this point of view, an unsteady Couette flow of an incompressible fluid with Ion-slip effect and with the influence of an external uniform magnetic field which is perpendicular to the plates is discussed in [17] and the same flow with uniform suction and injection past between two parallel porous plates with heat transfer is discussed in [18] by Attia. Kumar et al. [19] examined the Couette flow in three-dimensional with heat transfer through a porous medium bounded by infinite vertical porous plates. From the great interest in unsteady laminar Couette flow of its great applications in environmental, industrial, biomedical, engineering, nuclear reactors, and oil purifications, many researchers [20,21,22] have expressed their views on its. Limthanakul and Pochai [23,25] have given an idea about groundwater flows. They discussed a two dimensional model of polluted groundwater measured around a land fill. Koroleva [24] studied a flow of Stokes-Brinkman system through an ensemble of porous cylindrical particles with different viscosity by the cell method.
Drawing motivation from the above studies, the study aims to investigate the simultaneous effects of the unsteady Couette flow past between two horizontal Riga plates with Hall and ion slip current with electromagnetic field. The explicit finite difference method has been used as a main tool to solve the problem. Also, MATLAB R2015a has been used to calculate the results. The obtained results of different parameters have been shown graphically.

Mathematical Formulation
An incompressible laminar flow of viscous fluid between two horizontal parallel Riga plates has been considered, where one of which is moving and the other is at rest. Let the lower plate be rest at 0 = y and the upper plate is moving at a distance d y = with a velocity l πυ .
Let the direction of the flow be taken along the -x axis, -y axis is perpendicular to the flow and the plates are parallel to the -z x plane. As a generalized Couette flow, the applied pressure gradient Plates are fixed at two constant temperatures 1 T for the lower plate and 2 T for the upper plate, where 1 2 T T > . The initial temperature of the fluid is assumed to be equal to the temperature of the lower plate 1 T . Due to the effect of Hall and Ion-slip current, the generalized Ohm's law may state as follows: Where, e e e β τ ω = may treat as Hall parameter. The physical model is shown in Fig.1.
is defined as magnetic forces. According to the Grinberg hypothesis this magnetic forces are defined as follows: Under the above assumptions of Couette flow, and Bousniques approximations, it is found the dimensional forms of the momentum and energy equations are: The corresponding boundary conditions are

Similarity Analysis
The following non-dimensional variables are used to make the dimensionless form of the equations (1) to (4).
The corresponding boundary conditions are is the Eckert Number.

Method of Solution
We apply the explicit finite difference method to solve the non-dimensional coupled partial differential equations (5)-(7) with the associated boundary conditions (8).
It is considered maximum length of the plate is ) 10 ( max = x and distance between the plates as the lower plate is fixed at 0 = y . This means x varies from 0 to 10 and y varies from 0 to 2. The finite difference schemes for the problems are as follows: Here, the subscripts i and j refer to x and y and the superscript k refers to time t.

Shear Stresses and Nusselt Number
The effects of pertinent parameters on the local and average shear stress from the velocity of the fluid have been investigated. The non-dimensional form of the local and average shear stress for the fluid is given by the

Steady-state Solution
The behavior of the various entities on the velocity and temperature profiles has been elaborated graphically. . There is also negligible change among these grid pairs so that any one grid pair is acceptable to find the steady-state solution. It has been seen that the same situation occurs for the other distributions. The steady-state solution has been performed for the values of

Effects of Various Parameters
To study the physical situation of the problem, it is mentioned that the figures (a), (b) , and (c) of Fig.3 to