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.

Heat Transfer Mass Transfer MHD Vertical Plate Convection Flow
1. Introduction

Due to the plethora of applications in astronomical technology and processes entangling high temperatures, the payoffs of thermal radiation on the free convection flows have been drawing the consideration of enormous research interest. Furthermore, free convection flow in the presence of a magnetic field is crucial because of its significant impact on the boundary layer control and the execution of many engineering devices consuming electrically conducting fluids such as in MHD power generation, plasma studies, nuclear reactor using a liquid metal coolant and geothermal energy extraction. That’s why it’s a burning topic for many current researchers in the world.

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. T w is the temperature of the plate and T ∞ outside of the plate separately. 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 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:

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 − 1 ρ C p ∂ q r ∂ y (3)

The relevant 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 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, T w is the temperature of the plate, T ∞ is the temperature in the free stream, σ is the electric conductivity, B 0 is a constant magnetic field, ρ is the fluid density, κ is the kinematic viscosity, C p is the specific heat with constant pressure, U 0 is a constant indicates the uniform velocity of the fluid remaining symbols have their usual meaning.

Conferring to Rosseland approximation the radiative heat flux, q r = − 4 σ ∗ 3 κ 0 ∂ T 4 ∂ y ,

where κ 0 is the Rosseland mean absorption coefficient and σ ∗ is the Stefan-Boltzman constant.

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 T 4 can be articulated as a linear function of the temperature. Thus T 4 can be extended in Taylor series about T ∞ as:

T 4 = T ∞ 4 + 4 T ∞ 3 ( T − T ∞ ) + 6 T ∞ 2 ( T − T ∞ ) 2 + ⋯ ,

Neglecting the higher order terms elsewhere the first degree in ( T − T ∞ ) one can be found as T 4 ≈ − 3 T ∞ 4 + 4 T ∞ 3 T .

∴ q r = − 16 σ ∗ 3 κ 0 T ∞ 3 ∂ T ∂ Y .

By replacing the above expression into energy Equation (3) are as follows:

∂ T ∂ t + u ∂ T ∂ x + ν ∂ T ∂ y = κ ρ C p ∂ 2 T ∂ y 2 + ν C p ( ∂ u ∂ y ) 2 + 1 ρ C p 16 σ * 3 κ 0 T ∞ 3 ∂ 2 T ∂ y 2 (5)

2. Mathematical Formulation

Applying the ensuing usual transformations, the system of partial differential equations with boundary conditions is changed into a dimensionless equation.

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 ¯

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:

Continuity equation

∂ U ∂ X + ∂ V ∂ Y = 0 (6)

Momentum equation

∂ U ∂ η + U ∂ U ∂ X + V ∂ U ∂ Y = ∂ 2 U ∂ Y 2 + G r T ¯ − M U (7)

Energy equation

∂ T ¯ ∂ η + U ∂ T ¯ ∂ X + V ∂ T ¯ ∂ Y = 1 P r ∂ 2 T ¯ ∂ Y 2 + E c ( ∂ U ∂ Y ) 2 + R ∂ 2 T ¯ ∂ Y 2 (8)

with boundary conditions

U = 1 ,     V = 0 ,     T ¯ = 1 ,     at     Y = 0 U = 0 ,     T ¯ = 0 ,     as     Y → ∞ (9)

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 ∞ )

Radiative parameter, R = 16 σ * T ∞ 3 3 ρ C p κ 0 .

3. 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 X- and 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 X max (=100), i.e., X differs from 0 to 100 and suppose Y max (=25) as taken to Y → ∞ , it means that Y varies from 0 to 25.

Let, m = 250 and n = 250 grid spacing in 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 minor time period Δ η = 0.005 . Let U ′ , T ′ ¯ indicate the values of U , T ¯ at the terminal of a time-step separately.

Applying an explicit finite difference method into the partial Equations (6)-(8) with boundary conditions (9) we get,

( 6 ) ⇒ U i , j − U i , j − 1 Δ X + V i , j − V i , j − 1 Δ Y = 0 (10)

( 7 ) ⇒ 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 ) (11)

( 8 ) ⇒ 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 + R T ¯ i , j + 1 − 2 T ¯ i , j + T ¯ i , j − 1 ( 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                       + R T ¯ i , j + 1 − 2 T ¯ i , j + T ¯ i , j − 1 ( Δ Y ) 2 ) (12)

The 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 (13)

U i , L n = 0 ,       T ¯ i , L n = 0 ,

where

L → ∞

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.

4. 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 formulated as:

The Fourier expansion due to 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 ¯ (14)

Subsequently the time period, Equation (14) will convert

U : Ψ ′ ( η ) e i α ¯ x ¯ e i β ¯ y ¯ T ¯ : θ ′ ( η ) e i α ¯ x ¯ e i β ¯ y ¯ (15)

Applying Equations (14) and (15) into Equations (11) and (12), the following equations we 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 ′ ( η ) = A Ψ ( η ) + B θ ′ ( η ) (16)

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 ) + R 2 θ ( η ) ( cos 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 ) + R 2 ( cos β Δ Y − 1 ) ( Δ Y ) 2 }                             + E c U Δ η ( Δ Y ) 2 Ψ ( η ) ( e i β ¯ Δ Y − 1 )

⇒ θ ′ ( η ) = G θ + H Ψ (17)

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 + R 2 ( cos β Δ Y − 1 ) ( Δ Y ) 2

and H = E c U Δ η ( Δ Y ) 2 ( e i β ¯ Δ Y − 1 ) .

Equations (16), (17) 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 (18)

where, A 1 = A + H

B 1 = B G

and

θ ′ = G θ + H Ψ (19)

Equations (18) and (19) 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 augmentation 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 Eigen values 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:

| 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 are 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 integer and therefore A 1 , G are real. The values of | A 1 | , | G | are greatest when m and n are odd integers, then

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.

Therefore, the stability conditions of the problem are assumed below.

Analogously,

and convergence criteria of the method is.

5. Results and Discussion

For rummaging the problem, the outcome of numerical values of dimensionless velocity and temperature profiles within the boundary conditions have been enumerated due to several values of radiation parameter R, magnetic parameter M, Prandtl number Pr, Grashof number Gr, Eckert number Ec respectively. Furthermore, computed results of velocity and temperature profiles are portrayed and physical explanation explained here.

5.1. Effect of Radiative Parameter (R)

Figure 2(a-i) indicates that the velocity profile declines for ascending values of R. Whereas the boundary layer thickness falls down owing to rising values of R, i.e., decelerate the flow and reduce fluid velocity.

Figure 2(a-ii) demonstrates that the temperature profile increases due to rising values of R. In consequence, the increasing values of R conduct to an increase in the boundary layer thickness. As a result the heat transfer rate reduces the present thermal buoyancy force.

5.2. Effect of Magnetic Parameter (M)

Figure 2(b-i) and Figure 2(b-ii) elucidates the variation of velocity and temperature profile of the flow field owing to several values of magnetic parameter M whereas the other parameters are constant. Figure 2(b-i) demonstrates that for increasing values of magnetic parameters M, the velocity of the flow field lessens. Because when we apply a transverse magnetic field to electrically leading

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.

5.3. Effect of Prandtl Number (Pr)

Figure 2(c-i) illuminates the velocity profiles that are lessening for increasing values of Pr. Because, a higher Prandtl number creates high viscosity, consequently the velocity profiles are moving slowly. Figure 2(c-ii) displays the temperature profile which rises with the increasing of Pr. Owing to the matter, the thriving values of Pr are proportional to reduce thermal conductivity, that's why temperature profiles increases.

5.4. Effect of Grashof Number (Gr)

Figure 2(d-i) exhibits that the velocity profile decreases for rising values of Gr. Figure 2(d-ii) elucidates that the temperature profile decrease due to thriving values of Gr. It reveals that the growing of Gr decreases the rate of flow of temperature within the boundary layer.

5.5. Effect of Eckert Number (Ec)

Figure 2(e-i) demonstrates that the velocity profile declines because of the increasing values of Ec. As a result, heat transfer of the flow reduced the driving force to the kinetic energy. Since scouring heating and heat energy are stored in liquid, therefore the temperature profile increases for increasing values of Ec, which are illustrated in Figure 2(e-ii).

6. 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:

• The offshoots of R, M, Pr, Gr, and Ec on velocity have shown opposite results i.e. if the values of R, M, Pr, Gr, and Ec increase, the velocity profiles decrease.

• The thermal boundary layer has decreased on account of the various values of R, M and Gr, whereas an opposite scenario has in Pr, Ec.

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.

Cite this paper

Ullah, M.S., Tarammim, A. and Uddin, M.J. (2021) A Study of Two Dimensional Unsteady MHD Free Convection Flow over a Vertical Plate in the Presence of Radiation. Open Journal of Fluid Dynamics, 11, 20-33. https://doi.org/10.4236/ojfd.2021.111002