If a magnetic field is placed before a moving conducting fluid then the motion of the fluid is changed by the influence of the magnetic field. The magnetic field is also perturbed by the motion of the fluid: one affects the other and vice versa. The motion of the conducting fluid across the magnetic field generates electric-currents, which changes the magnetic field and the action of magnetic field on these currents gives rise to mechanical forces which modify the flow of the fluid. The electromagnetic field is governed by Maxwell’s electromagnetic equations and the motion of the fluid is governed by the field equations of the fluid mechanics. In recent years, the study of MHD phenomena in liquid conductors has received considerable impetus on account of its theoretical experimental and practical applications. Schlichting  studied the problem of an incompressible viscous fluid flow problem produced by the oscillation of a plane solid wall. This problem is also known as stokes second problem. Von Keregek and Davis  performed the linear stability theory of oscillating Stokes layers. Pauton  obtained the transient solution for the flow due to the oscillation plane. Erdogan  derived the analytic solutions for the flow produced by the small oscillating wall for small and large time by Laplace transformation method. Recently Poria, Mamaloukas, layek & Magumdar  derived the solution of laminar flow of viscous conduction fluid produced by the oscillating plane wall. They solved the problem both analytically and numerically in presence of magnetic field. In this paper the main aim is to investigate the effects of a transverse magnetic field on the incompressible electrically conducting fluid flow produced by a moving plate. An attempt has been made to investigate the analytic solutions for the problem. The problem has been solved numerically using well known Crank-Nicholson Implicit scheme.
2. Formulation of the Problem
The equation of motion of the conducting fluid in presence of transverse magnetic field is
where is the fluid velocity, p the fluid pressure, n the kinetic coefficient of viscosity, the uniform magnetic field, s the electrical conductivity.
Let us consider a flat plate extended to large distances in and directions. Again we consider an incompressible viscous fluid over a half plane solid wall . Suppose the fluid is at rest at time . At the plane solid wall is suddenly set in motion in direction at constant velocity . As a result a two dimensional parallel flow will be produced near the plate. Since the fluid flows along direction and there is no velocity component along the direction perpendicular to the direction of flow, so the equation of conservation of mass reduces to
As the flow is only kept in motion by the movement of the plate, one may set
the pressure gradient . For unsteady case Equation (1) reduces
Equation (2) indicates that u is a function of and . Boundary conditions:
when for all ,
at when , (4)
at when .
We introduce the following non-dimensional quantities
where L and T represent the characteristic length and characteristic time respectively. Setting these non-dimensional quantities in Equation (3), we get
Here the number M is a non-dimensional number and is called Hartmann number.
In this case the boundary conditions may be written as
2.1. Analytic Solution
We introduce the Laplace transformation and inverse Laplace transformation as
From Equation (5), we have
With the help of boundary condition (6), we get
The Solution of Equation (11) is
Since u is finite for , we must have .
Equation (13) reduces to
Thus the Equation (13) reduces to
Taking inverse Laplace transformation of Equation (12), we have
2.2. Numerical Solution
The Equation (5) with initial and boundary conditions (6) is solved by finite difference technique. The crank-Nicholson implicit scheme is used to solve the parabolic type of equation. In this scheme, the time derivative term is represented by forward difference formula while the space derivative term is represented by the average central difference formula. To do this the temporal first derivative can be approximated by
The second derivative in space can be determined at the midpoint by the averaging the difference approximations at the beginning ( ) and at the end ( ) of the time increment as
Substituting Equation (17) and Equation (18) into Equation (5), we get
The Equation (20) may be written as
where and .
The system of algebraic equations in tri-diagonal form is solved by Thomas algorithm for each time level. In this problem some grid points have been consider for numerical computation. u is obtained at each grid point at each time interval. The Figure 1 below is drawn for various values of y when t = 0.4.
Figure 1. Velocity profile for different values of Hartmann number M.
3. Results and Discussion
Numerical results are displayed by the above figure. This figure shows that the velocity of the fluid decreases as the magnetic field increases. The velocity decreases gradually and attains almost zero velocity at a sufficient large distance from the plate.