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 dissipation 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 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.

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. T w and T ∞ is the temperature of the plate and outside of the plate respectively. A uniform magnetic field B = ( 0 , B 0 , 0 ) is enacted normal to the plate, and the magnetic field is anticipated to be negligible while B 0 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

∂ u ∂ x + ∂ v ∂ y = 0 (1)

Momentum equation

∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y = ν ∂ 2 u ∂ y 2 + g β ( T − T ∞ ) − σ B 0 2 ρ u (2)

Energy equation

∂ T ∂ t + u ∂ T ∂ x + ν ∂ T ∂ y = κ ρ C p ⋅ ∂ 2 T ∂ y 2 + ν C p ( ∂ u ∂ y ) 2 (3)

The appropriate boundary condition for velocity and temperature are given by

u = U 0 , v = 0 ,     T = T w ,       at     y = 0 u → 0 ,     T → T ∞ ,       as     y → ∞ (4)

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, B 0 is the magnetic field, ρ is the fluid density, κ is the kinematic viscosity, C p is the specific heat with constant pressure, U 0 is the uniform velocity of the fluid, T w is the temperature of the plate and remaining symbols have their usual meaning.

Applying the following usual transformations, the system of partial differential equations with boundary conditions transformed into a non-dimensional system.

u = U 0 U ,   v = V U 0 ,     Y = y U 0 ν ,   X = x U 0 ν ,     η = t U 0 2 v ,     T = T ∞ + ( T w − T ∞ ) T ¯ .

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

∂ U ∂ X + ∂ V ∂ Y = 0 (5)

Momentum equation

∂ U ∂ η + U ∂ U ∂ X + V ∂ U ∂ Y = ∂ 2 U ∂ Y 2 + G r T ¯ − M U (6)

Energy equation

∂ T ¯ ∂ η + U ∂ T ¯ ∂ X + V ∂ T ¯ ∂ Y = 1 P r ∂ 2 T ¯ ∂ Y 2 + E c ( ∂ U ∂ Y ) 2 (7)

with boundary conditions

U = 1 ,     V = 0 ,     T ¯ = 1 ,     at     Y = 0 U = 0 ,     T ¯ = 0 ,     as     Y → ∞ (8)

where,

Magnetic parameter, M = σ ν B 0 2 ρ U 0 2

Grashof number, G r = ν g β ( T w − T ∞ ) U 0 3

Prandtl number, P r = v α

Eckert number, E c = U 0 2 C p ( T w − T ∞ )

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 X_max (=100), i.e., X varies from 0 to 100 and assumed Y_max (=25) as taken to Y → ∞ , i.e., Y varies from 0 to 25.

m = 250 and n = 250 are taken to grid spacing in the X and Y directions correspondingly and as follows Δ x = 0.4 ( 0 ≤ x ≤ 100 ) and Δ Y = 0.1 ( 0 ≤ Y ≤ 25 )

with the smaller time step Δ η = 0.005 . Let U ′ , T ¯ ′ denote the values of U , T ¯ 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,

( 5 ) ⇒ U i , j − U i , j − 1 Δ X + V i , j − V i , j − 1 Δ Y = 0 (9)

( 6 ) ⇒ U ′ i , j − U i , j Δ η + U i , j U i , j − U i − 1 , j Δ X + V i , j U i , j + 1 − U i , j Δ Y                 = U i , j + 1 − 2 U i , j + U i , j − 1 ( Δ Y ) 2 + G r T ¯ i , j − M U i , j

⇒ U ′ i , j = U i , j + Δ η ( − U i , j U i , j − U i − 1 , j Δ X − V i , j U i , j + 1 − U i , j Δ Y                         + U i , j + 1 − 2 U i , j + U i , j − 1 ( Δ Y ) 2 + G r T ¯ i , j − M U i , j ) (10)

( 7 ) ⇒ T ′ ¯ i , j − T ¯ i , j Δ η + U i , j T ¯ i , j − T ¯ i − 1 , j Δ X + V i , j T ¯ i , j + 1 − T ¯ i , j Δ Y                 = 1 P r T ¯ i , j + 1 − 2 T ¯ i , j + T ¯ i , j − 1 ( Δ Y ) 2 + E c ( U i , j + 1 − U i , j Δ Y ) 2

⇒ T ′ ¯ i , j = T ¯ i , j + Δ η ( − U I , J T ¯ i , j − T ¯ i − 1 , j Δ X − V i , j T ¯ i , j + 1 − T ¯ i , j Δ Y                       + 1 P r T ¯ i , j + 1 − 2 T ¯ i , j + T ¯ i , j − 1 ( Δ Y ) 2 + E c ( U i , j + 1 − U i , j Δ Y ) 2 ) (11)

The initial and boundary conditions with the finite difference methods are as follows:

U i , 0 n = 1 ,       V i , 0 n = 0 ,       T ¯ i , 0 n = 1 (12)

U i , L n = 0 , T ¯ i , L n = 0

where L → ∞ .

Here, the subscripts i and j indicate the grid points with X and Y coordinates correspondingly and T ¯ is the temperature.

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 U , θ , φ at time arbitrary say η = 0 is e i α ¯ x ¯ and e i β ¯ y ¯ apart from a constant, where i = − 1 .

Then

U : Ψ ( η ) e i α ¯ x ¯ ⋅ e i β ¯ y ¯ T ¯ : θ ( η ) e i α ¯ x ¯ ⋅ e i β ¯ y ¯ (13)

And after the time step, these terms will become

U : Ψ ′ ( η ) e i α ¯ x ¯ ⋅ e i β ¯ y ¯ T ¯ : θ ′ ( η ) e i α ¯ x ¯ ⋅ e i β ¯ y ¯ (14)

Applying Equation (13) and Equation (14) into Equation (10) & Equation (11), the following equations are found by simplification

Ψ ′ − Ψ Δ η + U Ψ ( 1 − e − i α ¯ Δ X ) Δ X + V Ψ ( e i β ¯ Δ Y − 1 ) Δ Y = 2 Ψ ( Δ Y ⋅ cos β − 1 ) ( Δ Y ) 2 + G r θ ′ − M Ψ

⇒ Ψ ′ − Ψ + Δ η Δ X U Ψ ( 1 − e − i α ¯ Δ X ) + Δ η Δ Y V Ψ ( e i β ¯ Δ Y − 1 )         = 2 Δ η ( Δ Y ) 2 Ψ ( Δ Y cos β − 1 ) + G r Δ η θ ′ − M Ψ

⇒ Ψ ′ = Ψ { 1 − Δ η Δ X U ( 1 − e − α ¯ i Δ X ) − Δ η Δ Y V ( e i β ¯ Δ Y − 1 )                           + 2 Δ η ( Δ Y ) 2 ( Δ Y cos β − 1 ) − M Δ η } + G r Δ η θ ′

Ψ ′ = A Ψ + B θ ′ (15)

where

A = 1 − Δ η Δ x U ( 1 − e − i α ¯ Δ X ) − Δ η Δ Y V ( e i β ¯ Δ Y − 1 ) + 2 Δ η ( Δ Y ) 2 ( cos β ⋅ Δ Y − 1 ) − M Δ η

B = G r Δ η

and

θ ′ ( η ) − θ ( η ) Δ η + U θ ( η ) 1 − e − i α ¯ Δ X Δ X + V θ ( η ) e − i β ¯ Δ Y − 1 Δ Y = 1 P r 2 θ ( η ) ( cos β ⋅ Δ Y − 1 ) ( Δ Y ) 2 + E c ( U Ψ ( η ) ( e i β ¯ Δ Y − 1 ) ( Δ Y ) 2 )

⇒ θ ′ ( η ) = θ ( η ) { 1 − Δ η Δ x U ( 1 − e − i α ¯ Δ X ) − Δ η Δ Y V ( e i β ¯ Δ Y − 1 )                                 + 1 P r 2 Δ η ( Δ Y ) 2 ( cos β ⋅ Δ Y − 1 ) + E c U Δ η ( Δ Y ) 2 Ψ ( η ) ( e i β ¯ Δ Y − 1 ) }

⇒ θ ′ ( η ) = G θ + H Ψ (16)

where, G = 1 − Δ η Δ X U ( 1 − e − i α ¯ Δ X ) − Δ η Δ Y V ( e i β ¯ Δ Y − 1 ) + 1 P r 2 Δ η ( cos β ⋅ Δ Y − 1 ) ( Δ Y ) 2 and H = E c U Δ η ( Δ Y ) 2 ( e i β ¯ Δ Y − 1 )

Equation (15), Equation (16) can be written as:

Ψ ′ = A Ψ + B ( G θ + H Ψ ) = ( A + H ) Ψ + B G θ

⇒ Ψ ′ = A 1 Ψ + B 1 θ (17)

where,

A 1 = A + H

B 1 = B G

and

θ ′ = G θ + H Ψ (18)

Equation (17), Equation (18) can be expressed in matrix form.

( ψ ′ θ ′ ) = ( A 1 B 1 H G ) ( ψ θ )

i.e. η ′ = T η

where, T = ( A 1 B 1 H G ) and η = ( ψ θ )

As eigenvalues of the amplification matrix T is crucial for attaining the stability condition, as a result, let

B 1 → 0 ,       H → 0.

Hence, matrix T is as follows:

T = ( A 1 0 0 G )

Thus, the eigenvalues of T are

λ 1 = A 1 ,         λ 2 = G .

Here, values of λ 1 , λ 2 must not surpass in modulus.

Therefore, the stability conditions are as follows:

| A 1 | ≤ 1 ,         | G | ≤ 1.

Let,

a = u Δ η Δ X , b = Δ η Δ Y V , c = Δ η ( Δ Y ) 2

Hence,

A = 1 − a ( 1 − e − i α ¯ Δ X ) − b ( e i β ¯ Δ Y − 1 ) + 2 c ( cos β Δ Y − 1 ) − M Δ η

H = E c U c ( C i β ¯ Δ Y − 1 )

The co-efficient of a , b , c is real and non-negative. Therefore, the maximum modulus of A 1 , G arise when α ¯ Δ X = m π and β ¯ Δ Y = n π where m and n are integers and hence A 1 , G are real. The values of | A 1 | , | G | are greatest when m and n are odd integers, which are

A 1 = A + H = 1 − 2 a − 2 b − 4 c − 2 c E c .

To satisfy | A 1 | ≤ 1 , | G | ≤ 1 the most negative permissible values are,

A 1 = − 1 ,         G = − 1.

Thus, the stability conditions of the problem are given below.

1 − 2 a − 2 b − 4 c − 2 c E c ≤ − 1 2 ( a + b + 2 c + c E c ) ≤ 2 a + b + 2 c + c E c ≤ 1 U Δ η Δ X + | V | Δ η Δ Y + 2 Δ η ( Δ Y ) 2 + Δ η ( Δ Y ) 2 E c ≤ 1

Analogously, the 2nd condition is as follows

U Δ η Δ X + | V | Δ η Δ Y + 1 P r 2 Δ η ( Δ Y ) 2

and convergence criteria of the method is P r ≥ 1 .

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:

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Tarammim, A., Ullah, M.S. and Uddin, M.J. (2020) A Study of Two Dimensional Unsteady MHD Free Convection Flow over a Vertical Plate. Open Journal of Fluid Dynamics, 10, 342-355. https://doi.org/10.4236/ojfd.2020.104020