MHD Flow of a Viscous Fluid with Equal Kinematic and Magnetic Viscosity Pasta Perfectly Conducting Porous Plate

The effect of MHD flow of viscous, incompressible fluid with small electrical conductivity past an infinitely long perfectly conducting porous plate started impulsively from rest in the presence of a constant transverse magnetic field fixed relative to the fluid is investigated. We also imposed a small uniform suction or injection at the plate.Skin friction at the plate, non-dimensional forms of the velocity, and the induced magnetic field have been obtained. The graphs of velocity and magnetic field are also drawn for various values of suction, and magnetic parameters are clearly explained. From the investigations, it is found that the fluid velocity decreases due to the increasing value of kinematic viscosity and constant suction velocity. It also finds that the induced magnetic field increases due to constant suction velocity.


Introduction
The branch of science that deals with the interaction of the magnetic field with electrically conducting fluid is termed MHD.Saltwater, liquid metals, plasmas, mercury, electrolytes, and molten metals are some common examples of such fluids.Noted Swiss scientist Hannes Alfven, Noble Prize winner of Physics, initiated the field of MHD.However substantial contributions in MHD made by Cowling, Shercliff, Ferraro, Plumpton, Crammer, and Pai [1][2][3][4][5] have attracted the works towards physical relevance.
There are several applications of MHD in modern technologies.In the presence of a strong electric field, the electrical conductivity is affectedby magnetic fields.Geophysical and astrophysical applications of MHD are nicely elaborated by Dormy and Nunez [6].Dynamo, motor, fusion reactors, dispersion of metals, and metallurgical industries have wide applications of MHD.Aeronautical applications of MHD were studied exclusively by Li et al. [7].Farrokhi et al. [8] focused on biomedical applications of MHD.Rana et al. [9] investigated how microbes swim in the blood flow of nano-bioconvective Williamson fluid.
The porous medium contains holes or voids that are filled with solid particles, which permit the fluid to pass through these.The mechanism of porous flow finds its applications in inkjet printing, nuclear waste disposal, electrochemistry, combustion technology, etc. Dwivedi et al. [10] studied MHD flow through the vertical channel in a porous medium, while Raju et al. [11] observed the MHD flow through a horizontal channel, taking viscous dissipation and Joule heating into account.Kuiry and Bahadur [12] studied the unsteady MHD flow of a dusty viscoelastic fluid between parallel plates with an exponentially Decreasing pressure gradient in an inclined magnetic field.Ali et al. [13] investigated the Hall effects on steady MHD Heat and mass transfer free convection flow along a stretching sheet with suction and heat generation.Islam et al. [14] studied unsteady Heat and mass transfer slip flow over an Exponentially permeable stretching sheet.Kuiry and Vishwakarma [15] discussed the Hydromagnetic flow of a viscous fluid of equal kinematic and magnetic viscosity past a non-conducting porous plate.
In this paper, the work done by the present authors [15] is extended by introducing a perfectly conducting porous plate with equal kinematic and magnetic viscosity under the influence of a transverse magnetic field.A small uniform suction or injection at the plate is imposed.The solutions to the problem under a constant transverse magnetic field fixed relative to the fluid are discussed through graphs.

Mathematical Formulation and Solutions of the Problem
The unsteady MHD flow of an incompressible viscous fluid through a perfectly conducting porous plate under the influence of a constant transverse magnetic field is considered.. Let the x-axis and y-axis be taken in the direction of fluid flow and perpendicular to the conducting wall, respectively.The z-axis is perpendicular to both the x-axis and y-axis.
Let the imposed magnetic field.⃗⃗⃗⃗ be perpendicular to the conducting wall and the suction velocity at the plate.Then, the equation of continuity at y = 0 gives everywhere over the electrically conducting infinite porous flat plate.On account of motion, let a magnetic field will be induced in the flowdirection.From the symmetry of the problem considered, all flow variables are functions of the coordinate y and time t only.Let the plate be started impulsively from rest with a constant velocity U. Then the following differential equations will govern the fluid flow [16,17] as: and where p, the fluid pressure assumed to be a constant.Boundary and initial conditions for a non-conducting plate are : at t = 0, y > 0, u = = 0 , at y = 0, t > 0, u = U, = 0, at y , u = 0, = 0.
The considered problem is to be solved after deriving first the boundary conditions to be satisfied at the wall for an arbitrary conducting wall.Then, the same is deduced for a perfectly conducting wall.From Ampere's law and Ohm's law, after taking MHD approximations into account, we have: Consequently, we have, where ⃗ ⃗ ( ) ( ) and ⃗⃗⃗ ( ) Therefore, we have Let us assume that the magnetic permeability of the medium and the induced magnetic field. in the flow direction isthe same for both the fluid and the wall.Then equation ( 4) yields that the plate started impulsively from rest with velocity U: Since the tangential component of the electric field is continuous at an interface, it yields from ( 4) and ( 5), the boundary conditions at the wall for y = 0. ( Since the wall is perfectly conducting, we have and hence we have the condition: The other boundary and initial conditions are the same as given in (3).Hence, equations ( 1) and ( 2) are to be solved subject to the boundary and initial conditions given in (3) and (7).Using equations ( 1) and ( 2), we obtained respectively as: where .
The steady-state solutions of equations ( 10) and ( 11) are respectively given as under: where A', B', C' and D' are arbitrary constants, and consequently, the solutions of equations ( 10) and ( 11) are respectively of the forms: Therefore the solutions of the problem will be given by ( ) and

√ ( )
The constants A, B, C, and D are to be determined from ( 3) and ( 7) because the first and third conditions of (3) are automatically satisfied by and .The second condition of (3) yields as: In Fig. 3 taking and the fluid velocity decreases as the suction velocity decreases.

Conclusion
The focus of the present work is to study mainly the effect of a transverse magnetic field on a viscous, incompressible fluid with equal kinematic and magnetic viscosity past through a conducting porous plate.The behavioral study of flow under the action of different parameters was carried out with the aid of graphs.The important outcomes of the present work are as follows: The magnitude of velocity decreases due to increasing values of kinematic viscosity and constant suction velocity.The induced magnetic field also increases due to constant suction velocity.The conclusions made from the above analysis are that the problem is amenable to an exact solution for the perfectly conducting plate having electrical conductivity and when the fluid has equal kinematic and magnetic viscosity.

Fig
Fig. 2. Magnetic field with no section.

Nomenclature
. 2. Magnetic field with no section.