A Study of Two Dimensional Unsteady MHD Free Convection Flow over a Vertical Plate

In this paper, unsteady free convection heat transfer flow over a vertical plate in the presence of a magnetic field is discussed in detail. The dimensionless partial differential equations of continuity, momentum along energy are analyzed with suitable transformations. For numerical calculation, an implicit finite difference method is applied to solve a set of nonlinear dimensionless partial differential equations. Dimensionless velocity and temperature profile are also investigated due to the effects of assumed parameters in the concerned problem. An explicit finite difference technique is used to compute velocity and temperature profiles. The stability conditions are also examined.


Introduction
Unsteady MHD free convection flow is created a significant effect on heat transfer over a vertical plate. It has great interest on both theoretical and practical points of view because of its enormous applications in several engineering and geophysical field. In 1952, Ostrach [1] obtained a similar solution along with a vertical plate for free convection. In 1958, Stewartson [2] observed the free convection from a horizontal plate. In 1961, Sparrow and Cuss [3] discussed the effect of the magnetic field on free convection heat transfer on the isothermal vertical plate. In 1969, Roten and Claassen [4] studied natural convection above the unconfined horizontal plate. In 1972, Soundalgekar [5] studied the viscous dis-sipation effects on unsteady free convection flow past an infinite, vertical porous plate with constant suction. In 1977, Soundalgekar and Wavre [6] investigated unsteady free convection flow past an infinite vertical plate with variable suction and mass transfer. In 1984, Hussain and Begum [7] observed the outcome of mass transfer and free convection past a vertical plate. In 1984, Afzal and Hussain [8] discoursed the mixed convection over a horizontal plate. In 1985, Tokis [9] considered a class of exact solutions of the unsteady magnetohydrodynamic free convection flows. In 1990, Huang and Chen [10] presented local similarity solutions at free convective heat transfer from a vertical plate to non-Newtonian power-layer fluids. In 1992, Hossain [11] observed the viscous and joule heating effects on MHD free convection flow with variable plate temperature. In 1992, Sattar and Alam [12] studied unsteady hydromagnetic free convection flow with hall current and mass transfer along with an accelerated porous plate. Here temperature and concentration profiles are time-dependent. In 1997, Crepeau and Clarksean [13] attained a similar solution to natural convection with internal heat generation, which decays exponentially. In 1999, Cheng [14] analyzed the consequence of a magnetic field on heat and mass transfer with natural convection along a vertical surface in porous media an integral approach. In 1999, Pop and Postelnicu [15] perceived similar solutions of free convection boundary layers over vertical and horizontal surface porous media with internal heat generation. In 2009, Cao and Baker [16] presented the slip effects on mixed convection flow and heat transfer from a vertical plate. In 2005, Chamaka and Al-Mudhaf [17] studied unsteady heat and mass transfer from rotating vertical cone with magnetic field and heat generation or absorption effect which develop our studies nearly to this field. In 2006, Alam et al. [18] investigated the numerical study of the combined free-forced convection and mass transfer flow past a vertical porous plate in a porous medium with heat generation and thermal diffusion. In 2007, Aydin and Kaya [19] scrutinized the mixed convection flow of a viscous dissipating fluid about a vertical plate. In 2008, Alam et al. [20] explored the effects of variable suction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermal radiation. In 2011, Bhattacharyya [21] [22] et al. discussed the MHD boundary layer ship flow over a flat plate and porous plate embedded in a porous medium. In 2011, Singh and Kumar [23] calculated the fluctuating heat and mass transfer on unsteady MHD free convection flow of radiating and reacting fluid past a vertical porous considered plate in the slip-flow regime. In 2013, Sarkar [24] studied the Hall effects on unsteady MHD free convective flow past an accelerated moving vertical plate with viscous and Joule dissipations. In 2013, Narahari and Debnath [25] investigated the unsteady MHD-free convection flow past an accelerated vertical plate with constant heat flux and heat source. In 2013, Alam et al. [26] premeditated the effect of heat and mass transfer in MHD free convection flow over an inclined plate with hall current. In 2017, Pandya et al. [27] observed the combined effects of Soret-Dufour, radiation, and chemical reaction on unsteady MHD flow of dusty

Mathematical Model and Governing Equations
Let us consider an electrically conducting incompressible viscous fluid of unsteady MHD free convection flow along with the vertical plate. In the Cartesian coordinate system, X-axis is along with the plate in the direction of the flow, and the Y-axis is normal to it. w T and T ∞ is the temperature of the plate and outside of the plate respectively. A uniform magnetic field is enacted normal to the plate, and the magnetic field is anticipated to be negligible while 0 B is constant which are illustrates in Figure 1.
The equations for the flow and temperature unsteady heat and mass transfer over a vertical plate in the presence of a magnetic field are given below with boundary conditions.

Continuity equation
Energy equation The appropriate boundary condition for velocity and temperature are given by where ν is the kinematic viscosity, g is the acceleration due to gravity, β is the co-efficient of volumetric expansion, T is the temperature of the fluid inside the thermal boundary layer, T ∞ is the temperature in the free stream, σ is the electric conductivity, 0 B is the magnetic field, ρ is the fluid density, κ is the kinematic viscosity, p C is the specific heat with constant pressure, 0 U is the uniform velocity of the fluid, w T is the temperature of the plate and remaining symbols have their usual meaning.

Mathematical Formulation
Applying the following usual transformations, the system of partial differential equations with boundary conditions transformed into a non-dimensional system.
( ) To apply the above transformation in Equations (1)-(3), and with corresponding boundary conditions (4), by simplification obtain the following non-linear differential equations in terms of dimensionless variables such as Continuity equation Momentum equation Energy equation with boundary conditions where,

Numerical Solution
A set of nonlinear partial differential dimensionless governing equations is Applying the explicit finite difference method into the partial Equations (5)- (7) with boundary conditions (8) we get, , , The initial and boundary conditions with the finite difference methods are as follows: ,0 ,0 ,0 , , 0, 0 Here, the subscripts i and j indicate the grid points with X and Y coordinates correspondingly and T is the temperature.

Stability and Convergence Analysis
Stability analysis is a better tool for attaining the projection of the compatibility of a real-world mathematical problem. Before doing the real computations, it's always helpful. It also represents numerical solutions that are trusted and reliable.  Applying Equation (13) and Equation (14) into Equation (10) & Equation (11), the following equations are found by simplification where, (15), Equation (16) can be written as: where, Equation (17), Equation (18) can be expressed in matrix form.
As eigenvalues of the amplification matrix T is crucial for attaining the stability condition, as a result, let Hence, matrix T is as follows: Thus, the eigenvalues of T are Here, values of 1 2 , λ λ must not surpass in modulus.
Therefore, the stability conditions are as follows: 1 e e 1 2 cos 1 The co-efficient of , , a b c is real and non-negative. Therefore, the maximum modulus of 1 , A G arise when X m α∆ =π and Analogously, the 2nd condition is as follows and convergence criteria of the method is 1 Pr ≥ .

Results and Discussion
To investigate the physical significance of the concerned problem, the several dimensionless parameters values such as magnetic Parameter M, Grashof number Gr, Prandtl number Pr, Eckert number Ec are executed with numerical computations. To illustrate the computed result, the velocity and temperature profile are plotted, the physical explanation is explained here. The velocity and temperature distributions are as follows:   Figure 3(b) reveals that the temperature profiles are lessening due to the increasing values of M. Since higher values of magnetic field parameters generate lower heat transfer, the temperature gradient decreases for reducing the rate of heat convection in the flow.

Effect of Grashof Number (Gr)
Analogously, Figure 3(c) and Figure 3(d) demonstrate the effect of Gr on the velocity and temperature profiles while other parameters are constant. Figure   3(c) demonstrates that the velocity profile is amplified for increasing values of Gr. It means to raise the velocity due to the thermal buoyancy force which generates the pressure gradient. The increasing values of Gr grow the thermal buoyancy effect. Thus, the rate of heat transfer increases, which plots in Figure   3(d).

Effect of Prandtl Number (Pr)
From Figure 3 Figure 3(g) exemplifies that the velocity profile is declined owing to rises up Ec.

Effect of Eckert Number (Ec)
Because of transferring heat to the kinetic energy of the flow, the driving force is reduced.
As frictional heating and heat energy are stored in liquid, the temperature profile is declined for rising values of Ec, which illustrate in Figure 3(h).

Conclusions
This thesis work analyzes the governing equations for two dimensional unsteady