A Study of Two Dimensional Unsteady MHD Free Convection Flow over a Vertical Plate in the Presence of Radiation

This paper compacts with an exact analysis of radiative effects on the magnetohydrodynamic (MHD) free convection flow of an electrically conducting incompressible viscous fluid over a vertical plate. The non-dimensional continuity, momentum, and energy equations are solved using appropriate transformation. The dimensionless momentum and energy equations are solved numerically through an explicit finite difference method. The stability and convergence analysis also discussed. Finally, outcomes of the parameters on velocity and temperature profiles are displayed graphically and qualitatively.

With the consideration of radiation impacts, the energy equation has to lead to an exceptionally nonlinear partial differential equation. In 1960, Soundalgekar et al. [1] scrutinized the radiation effects on free convection flow of gas past a semi-infinite flat plate. In 1996, Hossain et al. [2] investigated the radiation effect on mixed convection along with a vertical plate and uniform surface temperature. In 1996, Das et al. [3] evaluated the radiation outcome of flow past an impulsively started infinite isothermal vertical plate, using the Laplace transform technique. In 1999, Hossain et al. [4] investigated the offshoot of radiation on free convection from a porous vertical plate. In 1999, Raptis and Perdikis [5] examined the upshot of thermal radiation and free convection flow past a moving plate. In 2001, Hossain and Pop [6] investigated the radiation effect on free convection over a vertical flat plate embedded in a porous medium with high porosity. In 2001, Chamkha [7] investigated the coupled heat and mass transfer by natural convection of Newtonian fluids in the presence of magnetic field and radiation effects. In 2001, Hossain et al. [8] studied the impact of radiation on free convection flow of fluid with variable viscosity from a porous vertical plate.
In 2003, Cookey et al. [9] well-acquainted the influence of viscous dissipation and radiation on unsteady MHD free-convection flow past an infinite heated vertical plate in a porous medium with time-dependent suction. In 2003, Chamkha et al. [10] resolved the thermal radiation effects on MHD forced convection flow adjacent to a non-isothermal wedge in the presence of a heat source or sink. In 2004, Abd EL-Naby et al. [11] calculated the finite difference solution of radiation effects on MHD unsteady free-convection flow on a vertical porous plate. In 2005, Makinde [12] studied the free-convection flow with thermal radiation and mass transfer past a moving vertical porous plate. In 2005 and 2006, Duwairi [13] and Damseh et al. [14] examined the radiation and magnetic effects on the skin friction and heat transfer for forced convection conditions. In 2008, Aydın and Kaya [15] observed the payoffs of thermal radiation taking into account the suction or injection on a steady MHD mixed convective flows of a viscous incompressible fluid about a permeable vertical plate. In 2008, Cortell [16] presented a numerical analysis for flow and heat transfer in a viscous fluid over a sheet nonlinearly stretched with effects of thermal radiation. In 2009, Deka and Neog [17] [18] reviewed the combined payoffs of thermal radiation and chemical reaction on free convection flow past a vertical plate in a porous medium and with MHD. In 2010, Roy and Hossain [19] traced the natural convection boundary layer flow of a viscous incompressible fluid along a vertical plate taking into account the conduction-radiation interaction and surface temperature oscillations. In 2011, Seth et al. [20] investigated the impacts of radiation on unsteady hydromagnetic natural convection transient flow near an impulsively moving vertical flat plate with ramped wall temperature in a porous medium. In 2012, Das [21] mastered the yielding of thermal radiation on a steady two-dimensional boundary layer flow of an incompressible electrically conducting fluid over a flat plate assuming the partial slip at the surface of the boundary, temperature-dependent fluid viscosity, variable thermal conductivity, Open Journal of Fluid Dynamics and non-uniform heat generation and absorption. In 2012, Khan et al. [22] quested the unsteady MHD free convection boundary-layer flow of a nano-fluid along a stretching sheet with thermal radiation and viscous dissipation effects. In 2012, Rao et al. [23] studied the radiation corporeity of unsteady free convection heat and mass transfer in a walters-B viscoelastic flow past an impulsively started vertical plate. In 2014, Roy et al. [24] superintendence the unsteady MHD free convection flow along with a vertical plate in the presence of radiative heat flux. In 2014, Samad and Saha [25] examined the two-dimensional heat and mass transfer free convection flow of an MHD Non-Newtonian power-law fluid along with a stretching sheet in the presence of a magnetic field and assuming simplified thermal radiation. In 2014, Fetecau et al. [26] smeared the slip effects on the unsteady radiative MHD free convection flow over a moving plate with mass diffusion and heat source. In 2015, Nagamanemma et al. [27] analyzed unsteady MHD free convective heat and mass transfer flow near a moving vertical porous plate with radiation and thermo diffusion effects. In 2015, Tomer and Kumar [28] perused the radiation effects on heat and mass transfer in a steady MHD flow over a porous vertical plate. In 2020, Tarammim et al. [29] studied two dimensional unsteady MHD free convection flow over a vertical plate.
The literature review discloses that many researchers work on steady heat transfer due to a variety of physical parameters. Although some research works are considered the unsteady problem used as simplified models, disregarding the results of the appearance of the magnetic field or thermal radiation. Analyzing this paper, we scrutinize the unsteady MHD free convection flow over a vertical plate in the presence of thermal radiation. We solve the governing equations with an explicit finite difference method. Then the upshot of the physical parameters such as the radiative parameter R, magnetic parameter M, Prandtl number Pr, Grashof number Gr, and Ekert number Ec are presented here.

Mathematical Model and Governing Equations
Let us consider unsteady MHD free convection flow with electrically conducting incompressible viscous fluid along a vertical plate in presence of radiation. The Cartesian coordinate system, the X-axis is taken along the plate in the upward direction and the Y-axis is normal to plate. w T is the temperature of the plate and T ∞ outside of the plate separately. 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 demonstrates in Figure 1. The equations for unsteady MHD heat transfer flow over a vertical plate in the presence of radiation with boundary conditions are given below: Energy equation The relevant boundary condition for velocity and temperature are given by where β is the co-efficient of volumetric expansion, ν is the kinematic viscosity, g is the acceleration due to gravity, T is the temperature of the fluid inside the thermal boundary layer, w T is the temperature of the plate, T ∞ is the temperature in the free stream, σ is the electric conductivity, 0 B is a constant magnetic field, ρ is the fluid density, κ is the kinematic viscosity, p C is the specific heat with constant pressure, 0 U is a constant indicates the uniform velocity of the fluid remaining symbols have their usual meaning.
Conferring to Rosseland approximation the radiative heat flux, The Rosseland approximation is intended for an optically thick medium, so the fluid does not absorb its particular emitted radiation, but it does absorb radiation emitted by the boundaries without self-absorption, It seems reasonable to assume that the temperature differences within the flow are expected to be small and 4 T can be articulated as a linear function of the temperature. Thus 4 T can be extended in Taylor series about T ∞ as: Neglecting the higher order terms elsewhere the first degree in ( ) By replacing the above expression into energy Equation (3) are as follows:

Mathematical Formulation
Applying the ensuing usual transformations, the system of partial differential equations with boundary conditions is changed into a dimensionless equation.
( ) Applying the above transformation in Equations (1), (2), (3), and with corresponding boundary conditions (4), after simplification we acquire the following non-linear differential equations in terms of dimensionless variables such as: Momentum equation Energy equation with boundary conditions where, Magnetic parameter,

Numerical Solution
A set of nonlinear partial differential dimensionless governing equations has been brought out numerically with the related boundary conditions along with an explicit finite difference method, which is tentatively stable. The portion of the flow is divided into a grid or mesh of lines parallel to Xand Y-axes, where X-axis indicates the plate in upward direction and Y-axis is normal to the plate.
We measure the height of plate max X (=100), i.e., X differs from 0 to 100 and Applying an explicit finite difference method into the partial Equations (6)-(8) with boundary conditions (9) we get, , , The boundary conditions with the finite difference methods are as follows: ,0 ,0 ,0 , , 0, 0, where, i and j indicate the grid points with X and Y coordinates correspondingly and the subscripts n represents a value of time. T is the temperature.

Stability and Convergence Analysis
Analysis will keep on inadequate without the stability and convergence analysis.
It's always supportive of tangible computations. Furthermore, it establishes numerical solutions (finite difference scheme) that are reliable and consistent.
Owing to the persistent of mesh sizes, the stability criteria of the problem is Open Journal of Fluid Dynamics Applying Equations (14) and (15) into Equations (11) and (12), the following equations we found by simplification. where, Equations (16), (17) can be written as: where, Equations (18) and (19) can be expressed in matrix form.
As eigenvalues of the augmentation matrix T is crucial for attaining the stability condition, as a result, let Hence, matrix T is as follows: Thus, the Eigen values of T are Here, values of 1 2 , λ λ must not surpass in modulus. Therefore, the stability conditions are: 1 e e 1 2 cos 1 The co-efficient of , , a b c are real and non-negative. Therefore, the maximum modulus of 1 , A G arise when X m α∆ =π and Y n β∆ =π where m and n are integer and therefore 1 and convergence criteria of the method is 1 Pr ≥ .     fluids, it escalates a body force prominent as Lorentz force. Furthermore, it reduces the swiftness of the movement of the fluid in the boundary layer. Figure  2(b-ii) exhibits that the temperature profiles are decreasing as a result of the ascending effect of M. As the temperature gradient is lessened for reducing the amount of heat convection in the flow which implies that larger values of magnetic field parameters produce lower heat transfer.

Conclusions
The governing equations for unsteady MHD free convection flow over a vertical plate have been analyzed in the entity of a magnetic field with thermal radiation.
Elaborate numerical computations have been brought to pass to scrutinize the outcome of the radiative parameter R, magnetic parameter M, Prandtl number Pr, Grashof number Gr, and Eckert number Ec on velocity and temperature profiles. After analysis, the following conclusions are drawn:

Conflicts of Interest
This manuscript has not been published and is not under consideration for publication elsewhere. We have no conflicts of interest to disclose.