Unsteady Couette Flow Past Between Two HorizontalRiga Plates With Hall and Ion Slip Current

In this article, we have investigated an unsteady Couette flow past between two horizontal Riga plates with the Hall and Ion-slip effects.Riga plate is the combination of electrodes and permanent magnets that creates a plane surface which produced the electromagnetic hydrodynamic fluid behavior and mostly uses in industrial processes with fluid flow affairs. It is used as an agent of reduce the skin friction and also diminish the turbulent effects. The numerical solutions are acquired by using explicit finite difference method and estimated results have been gained for various values of dimensionless pressure gradient parameter, the Hall and Ion-slip parameters, modified Hartmann number, Prandtl number, Eckert number. 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 electrically conducting fluid is controlled by magneto hydrodynamics flow with the influence of external magnetic field is known as classical MHD flow control .When the fluid is too weak in electrically conducting so that the Lorentz force decreases exponentially that's why an external electric field must be applied to achieve efficient flow control, this is called Electro-magneto hydrodynamic (EMHD) flow. There are tremendoususe of poor conductivity fluids in the field of magneto-aerodynamics, civil engineering, mechanical engineering, chemical engineering, dust or fumes in a gas, in biomechanics and purification the groundwater and oil. The combination of electrodes and permanent magnets creates a plane surface instead of polarity and magnetization is Riga plate that is introduced by Gailitis and Leilausis [1].This order minimizes the friction and pressure drag of submarines. It is also restrained the separation of boundary layer as well as diminishing the turbulence effects.When the flow passes,the Lorentz force is created by gather the electrodes and permanent magnets in the flat surface. A turbulent channel event with 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 nano fluid with melting heat and they use the Keller Box scheme to obtain the solution. Ayub et al. [6] examined the effect of EMHD nano fluid flow along an electromagnetic actuator. Ahmed et al. [7] carried out the mixed convection of a nano fluid 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 its 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 with heat transfer. Eraslan [10] expressed a distribution of temperature for the MHD channels with 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] are investigated an unsteady free convection and electrically conducting flow with the Hall currents and radiation effect of gray gas along an infinite vertical porous plate where a strong transverse magnetic field is imposed perpendicularly to the plate. Narayana et al. [12] examined the heat and mass transfer with Hall current along a vertical porous plate under the combination of buoyancy force of thermal and species diffusion in the presence of a transversely applied uniform magnetic field. Debnath et al. [13] studied the effects of Hall current on unsteady hydro magnetic flow past a porous plate in a rotating fluid system. Nimr and Masoud [14] discussed an unsteady free convection flow in a porous media along a flat plate. Krishna et al. [15] 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.Sharma et al. [16]considered a viscous, unsteady and incompressible fluid flow along a vertically infinite porous plate with Hall effect and heat source (or sink). Angirasa and Peterson [17] 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 [18] 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. These problems have been investigated by the authors such as Ghara et al. [19] studied an electrically conducting fluid of an unsteady Couette flow between two horizontal porous plates with Hall and Ion-slip effects. An unsteady Couette flow of an incompressible fluid with Ion-slip effect and an external uniform magnetic field is applied perpendicular to the plates isexamined in [20] and the same flow with uniform suction and injection is bounded by two parallel porous plates with heat transfer is discussed in [21] by Attia. Kumar et al. [22] discussed 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 [23,24,25]have expressed their views on its.
Drawing motivation from the above studies, the aim of the study is 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. Explicit finite difference methodhas 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.

Problem Formulation
An incompressible laminar flow of viscous fluid between two horizontal parallel Riga plates has been considered, where one of which is moving and other is at rest. Let the lower plate be rest at 0 = y and the upper plate is moving at a distance x direction and the fluid motion is unsteady so that the fluid may be treated as two dimensional . Therefore the continuity equation leads to ) ( y u u = . 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.

Riga Plate
A uniform magnetic force is generated by the Riga plate. The Lorentz force is defined as magnetic force. According to the Grinberg hypothesis this magnetic forces be 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 as follows: The corresponding boundary conditions are Similarity Analysis: To make the non-dimensional form of the equations (1)-(4), it is introduced the nondimensional variables are as follows: The corresponding boundary conditions are

Method of solution
The system of non-dimensional coupled partial differential equations (5)- (7) have been solved by an explicit finite difference subject to the associated boundary conditions (8). It is considered maximum length of the plate is ) 10 ( max = x and distance between the plates 2 = d i.e. 2 max = y as the lower plate is fixed at 0 = y . This means xvaries from 0 to 10 and yvaries from 0 to 2. The finite difference schemes for the problems are as follows: (1) With the boundary conditions Here, the subscripts i and j refer to x and y and the superscript k refers to time t.

3.1Shear 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

Results and Discussions
Numerical results and the graphical presentations of the fluid velocity and temperature distributions have been explained for the influence of the relevant non-dimensional parameters namely pressure gradient parameter ) (α , modified Hartmann number ),

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 seen that the same situation occur for the other distributions. The steady-state solution has performed for the values of 7 > τ . In the present analysis,

Effects of various parameters
To study the physical situation of the problem, it is mentioned that the figures(a), (b) and (c) of Fig.3to Fig.7 presented the distribution of velocity, local shear stress and average shear stress along the x-direction respectively. Figures (a), (b) and (c) ofFig.8 to Fig.11depicted  π ρυ l α − = , it means that 0 > α whenever constant pressure gradient P decreasing in the direction of motion; in that case, the velocity u has increasing effect with the increase of α over the entire width between the plates. But 0 < α influence that P increasing in the direction of motion and the velocity u has occurred back-flow, which have shown in Fig.3 Fig.7(a), it is observed that the velocity u has a cross flow where the velocity has increasing effect within the interval 1 0 < < y (approx.) thereafter it has decreasing effect, whereas xL τ and xA τ are increased with the increasing effect of 0 V .  Fig.7