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  obtained a similar solution along with a vertical plate for free convection. In 1958, Stewartson  observed the free convection from a horizontal plate. In 1961, Sparrow and Cuss  discussed the effect of the magnetic field on free convection heat transfer on the isothermal vertical plate. In 1969, Roten and Claassen  studied natural convection above the unconfined horizontal plate. In 1972, Soundalgekar  studied the viscous dissipation effects on unsteady free convection flow past an infinite, vertical porous plate with constant suction. In 1977, Soundalgekar and Wavre  investigated unsteady free convection flow past an infinite vertical plate with variable suction and mass transfer. In 1984, Hussain and Begum  observed the outcome of mass transfer and free convection past a vertical plate. In 1984, Afzal and Hussain  discoursed the mixed convection over a horizontal plate. In 1985, Tokis  considered a class of exact solutions of the unsteady magnetohydrodynamic free convection flows. In 1990, Huang and Chen  presented local similarity solutions at free convective heat transfer from a vertical plate to non-Newtonian power-layer fluids. In 1992, Hossain  observed the viscous and joule heating effects on MHD free convection flow with variable plate temperature. In 1992, Sattar and Alam  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  attained a similar solution to natural convection with internal heat generation, which decays exponentially. In 1999, Cheng  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  perceived similar solutions of free convection boundary layers over vertical and horizontal surface porous media with internal heat generation. In 2009, Cao and Baker  presented the slip effects on mixed convection flow and heat transfer from a vertical plate. In 2005, Chamaka and Al-Mudhaf  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.  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  scrutinized the mixed convection flow of a viscous dissipating fluid about a vertical plate. In 2008, Alam et al.  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   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  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  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  investigated the unsteady MHD-free convection flow past an accelerated vertical plate with constant heat flux and heat source. In 2013, Alam et al.  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.  observed the combined effects of Soret-Dufour, radiation, and chemical reaction on unsteady MHD flow of dusty fluid over inclined porous plate embedded in a porous medium.
The principal objective of this paper is to study the MHD free convection flow over a vertical plate. Because in recent years, free convection flow in the presence of the magnetic is attracted the attention of several researchers due to various applications in science and technology. MHD is used in a wide range of applications in engineering science, for example, MHD power generators, MHD bearings, MHD pumps etc. Also, it is used in solar physics, geophysics, meteorology, aeronautical plasma flows, electronics and chemical engineering. Therefore, it is necessary to investigate, in detail, the distributions of velocity and temperature for the flow of a viscous incompressible electrically conducting fluid along with a vertical plate across the boundary layer in the presence of a magnetic field. In this regard, the explicit finite difference techniques are used for solving non-linear partial differential equations.
2. 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. and 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 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.
Figure 1. Physical configuration and coordinates system.
The appropriate boundary condition for velocity and temperature are given by
where is the kinematic viscosity, g is the acceleration due to gravity, is the co-efficient of volumetric expansion, T is the temperature of the fluid inside the thermal boundary layer, is the temperature in the free stream, is the electric conductivity, is the magnetic field, is the fluid density, is the kinematic viscosity, is the specific heat with constant pressure, is the uniform velocity of the fluid, is the temperature of the plate and remaining symbols have their usual meaning.
3. 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
with boundary conditions
4. Numerical Solution
A set of nonlinear partial differential dimensionless governing equations is solved numerically with related boundary conditions along with an explicit finite difference method which is tentatively stable. The region of the flow is divided into a grid or mesh of lines parallel to X- and Y-axes, where X-axis indicates the plate and Y-axis is normal to the flow of the fluid which is demonstrates in Figure 2. This study measures the height of the plate Xmax (=100), i.e., X varies from 0 to 100 and assumed Ymax (=25) as taken to , i.e., Y varies from 0 to 25.
and are taken to grid spacing in the X and Y directions correspondingly and as follows and
Figure 2. Explicit finite difference system.
with the smaller time step . Let denote the values of at the end of a time-step respectively.
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:
Here, the subscripts i and j indicate the grid points with X and Y coordinates correspondingly and is the temperature.
5. 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. The general terms of the Fourier expansion for at time arbitrary say is and apart from a constant, where .
And after the time step, these terms will become
Applying Equation (13) and Equation (14) into Equation (10) & Equation (11), the following equations are found by simplification
Equation (15), Equation (16) can be written as:
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 must not surpass in modulus.
Therefore, the stability conditions are as follows:
The co-efficient of is real and non-negative. Therefore, the maximum modulus of arise when and where m and n are integers and hence are real. The values of are greatest when m and n are odd integers, which are
To satisfy the most negative permissible values are,
Thus, the stability conditions of the problem are given below.
Analogously, the 2nd condition is as follows
and convergence criteria of the method is .
6. 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:
6.1. Effect of Magnetic Parameter (M)
Figure 3(a), Figure 3(b) illustrates the variation of velocity and temperature profiles of the flow field for different values of the magnetic parameter M keeping other parameters as constant. Figure 3(a) displays that for rising values of magnetic parameter M, the velocity of the flow field falls-down. It’s the fact that the application of a transverse magnetic field to an electrically conducting fluid gives to escalate a body force accredited as Lorentz force. It reduces the speed of
Figure 3. (a-b) Effect of Magnetic parameter (M) on velocity and temperature profile where Gr = 2.5, Pr = 1, Ec = 0.1; (c-d) Effect of Grashof number (Gr) on velocity and temperature profile where M = 5.5, Pr = 1.0, Ec = 0.1; (e-f) Effect of Prandtl number (Pr) on velocity and temperature profile where Gr = 2.5, M = 5.5, Ec = 0.1; (g-h) Effect of Eckert number (Ec) on velocity and temperature profile where Gr = 2.5, M = 5.5, Pr = 1.0.
the motion of the fluid in the boundary layer. 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.
6.2. 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).
6.3. Effect of Prandtl Number (Pr)
From Figure 3(e), it is lucid that the velocity profiles are enhanced for increasing values of Pr. The thermal boundary layer thickness decreases for increasing values of Pr, in general lower average temperature within the boundary layer. Velocity and temperature in the boundary layer fall very quickly for large values of the Prandtl number.
Figure 3(f) reveals that the temperature profile rises for the rising values of Pr. It is fact that the smaller values of Pr are comparable to increase in the thermal conductivity of the fluid. Furthermore, heat is diffused away from the heated surface more rapidly for higher values of Pr.
6.4. Effect of Eckert Number (Ec)
Figure 3(g) exemplifies that the velocity profile is declined owing to rises up 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).
This thesis work analyzes the governing equations for two dimensional unsteady MHD free convection flow over a vertical plate. The governing partial differential equations and analytical explanation, graphical presentations are displayed depending on some different parameters such as Magnetic Parameter M, Grashof number Gr, Prandtl number Pr and Eckert number Ec. The effects of the mentioned parameters on the velocity and the temperature profile are analyzed significantly on the flow and heat transfer. The numerical investigations of current work are drawn in the following conclusion.
● The velocity profile falls with increasing values of M, Ec and opposite scenario for Gr, Pr.
● The temperature profile is declined with the risen up values of M, Ec and contrary consequence for Gr, Pr.
 Sparrow, E.M. and Cess, R.D. (1961) The Effect of the Magnetic Field on Free Convection Heat Transfer on the Isothermal Vertical Plate. International Journal of Heat and Mass Transfer, 3, 267-274.
 Soundalgekar, V.M. (1972) Visous Dissipation Effects on Unsteady Free Convection Flow Past an Infinite, Vertical Porous Plate with Constant Suction. International Journal of Heat and Mass Transfer, 15, 1253-1261.
 Soundalgekar, V.M. and Wavre, P.D. (1977) Unsteady Free Convection Flow past an Infinite Vertical Plate with Variable Suction and Mass Transfer. International Journal of Heat and Mass Transfer, 20, 1363-1373.
 Huang, M.J. and Chen, C.K. (1990) Local Similar Solutions at Free Convective Heat Transfer from a Vertical Plate to Non-Newtonian Power-Layer Fluids. International Journal of Heat and Mass Transfer, 33, 119-125.
 Hossain, M.A. (1992) Viscous and Joule Heating Effects on MHD Free-Convection Flow with Variable Plate Temperature. International Journal of Heat and Mass Transfer, 35, 3485-3487.
 Sattar, M.A. and Alam, M.M. (1992) Unsteady Hydromagnetic Free Convection Flow with Hall Current and Mass Transfer along an Accelerated Porous Plate with Time Dependent Temperature and Concentration. Canadian Journal of Physics, 70, 369-382.
 Cheng, C.Y. (1999) Effect of a Magnetic Field on Heat and Mass Transfer by Natural Convection from Vertical Surfaces in Porous Media an Integral Approach. International Communications in Heat and Mass Transfer, 27, 935-943.
 Postelnicu, A. and Pop, I. (1999) Similar Solutions of Free Convection Boundary Layers over Vertical and Horizontal Surface Porous Media with Internal Heat Generation. International Communications in Heat and Mass Transfer, 26, 1183-1191.
 Cao, K. and Baker, J. (2009) Slip Effects on Mixed Convection Flow and Heat Transfer from a Vertical Plate. International Journal of Heat and Mass Transfer, 52, 3829-3841.
 Chamaka, A.J. and Al-Mudhaf, A. (2005) Unsteady Heat and Mass Transfer from Rotating Vertical Cone with Magnetic Field and Heat Generation or Absorption Effect Which Are Develop Our Studies Nearly to This Field. International Journal of Thermal Sciences, 44, 267-276.
 Alam, M.S., Rahman, M.M. and Samad, M.A. (2006) 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. Nonlinear Analysis: Modeling and Control, 11, 331-343.
 Alam, M.S., Rahman, M.M. and Sattar, M.A. (2008) 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. International Journal of Thermal Sciences, 47, 758-765.
 Bhattacharyya, K., Mukhopadhyay, S. and Layek, G.C. (2011) MHD Boundary Layer Ship Flow and Heat Transfer over a Flat Plate. Chinese Physics Letters, 28, Article ID: 024701.
 Bhattacharyya, K., Mukhopadhyay, S. and Layek, G.C. (2011) Steady Boundary Layer Ship Flow over a Flat Plate and Porous Plate Embedded in a Porous Medium. Journal of Petroleum Science and Engineering, 78, 304-309.
 Singh, K.D. and Kumar, R. (2011) Fluctuating Heat and Mass Transfer on Unsteady MHD Free Convection Flow of Radiating and Reacting Fluid past a Vertical Porous Considered Plate in Slip-Flow Regime. Journal of Applied Fluid Mechanics, 4, 101-106.
 Sarkar, B.C., Das, S. and Jana, R.N. (2013) Hall Effects on Unsteady MHD Free Convective Flow past an Accelerated Moving Vertical Plate with Viscous and Joule Dissipations. International Journal of Computer Applications, 70, 19-28.
 Narahari, M. and Debnath, L. (2013) Unsteady Magnetohydrodynamic Free Convection Flow past an Accelerated Vertical Plate with Constant Heat Flux and Heat Generation or Absorption. ZAMM Zeitschrift für Angewandte Mathematik und Mechanik, 93, 38-49.
 Alam, M.S., Ali, M. and Hossain, M.D. (2013) Heat and Mass Transfer in MHD Free Convection Flow over an Inclined Plate with Hall Current. International Journal of Engineering Science (IJES), 2, 81-88.
 Pandya, N., Yadav, R.K. and Shukla, A.K. (2017) Combined Effects of Soret-Dufour, Radiation and Chemical Reaction on Unsteady MHD Flow of Dusty Fluid over Inclined Porous Plate Embedded in Porous Medium. International Journal of Advances in Applied Mathematics and Mechanics, 5, 49-58.