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The problem of transient, laminar, MHD double-diffusive free convection over a permeable vertical plate embedded in Darcy and non-Darcy porous medium is numerically investigated. Nonsimilarity solutions are obtained for constant wall temperature and concentration with a specified power law of mass flux parameter. The effects of the magnetic parameter, the inertial coefficient, Lewis number, the buoyancy ratio, and the lateral mass flux on heat and mass transfer coefficients are presented and discussed.

Recently, heat and mass transfer from different geometries embedded in fluid saturated porous medium has been studied extensively. This is due to the fact that these flows have many engineering and geophysical applications which include geothermal resources, building insulation, oil extraction, underground disposal of nuclear waste, heat salt leaching in soils, and many more. Various researchers [

There has been a renewed interest in MHD flow and heat transfer in porous and clear domains due to the important effect of magnetic field on the performance of many systems using electrically conducting fluid such as MHD power generators and the cooling of nuclear reactors. In a review article [

The aim of this work is to investigate transient MHD double-diffusive of an electrically-conducting fluid by free convection over a flat plate embedded in Darcy and non-Darcy porous medium in the presence of surface suction or blowing and magnetic field effects.

Consider the unsteady non-Darcy MHD free double-diffusive convection flow of an electrically conducting fluid over an isothermal vertical porous plate embedded in a porous medium with suction or injection. Initially the wall is at constant temperature

Introducing the following dimensionless variables and parameters:

Using the above similarity transformation the governing equations are reduced to

The physical quantities of fundamental interest of heat and mass transfer study are the heat and mass coefficients in terms of Nusselt and Sherwood numbers, respectively. The dimensionless heat and mass transfer coefficients can be expressed as

The resulting partial differential equations together with their boundary conditions have been solved numerically using an implicit finite difference technique. For the sake of brevity, the details of the solution procedure are not presented here. The results reported in this paper were validated by comparing them with those in the previously published paper. Our results show an excellent a greement with the steady state results of [

A parametric study is carried out to investigate the effects of all involved parameters on the transient velocity, temperature and concentration profiles as well as the transient local Nusselt and the local Sherwood numbers.

The effects of magnetic parameter on the transient velocity profiles, temperature distribution, and concentration distribution in the non-Darcy flow region are shown in Figures

The effect of magnetic parameter on the transient dimensionless velocity component profiles (

The effect of magnetic parameter on the transient dimensionless temperature distribution (

Figure

The effect of magnetic parameter on the transient dimensionless temperature distribution, for suction, injection, and impermeable wall cases (

The effect of the buoyancy ratio on the local Nusselt number with and without magnetic field effect is displayed in Figure

The effect of buoyancy ratio on Nusselt number with and without magnetic field effect, for suction, injection, and impermeable wall cases (

The influences of lateral mass flux on the local Nusselt number and the local Sherwood number with and without magnetic field effect and at different values of Lewis number are displayed in Figures

The effect of mass flux parameter on Nusselt number with and without magnetic field effect at different values of Lewis number (

The effect of mass flux parameter on Sherwood number with and without magnetic field effect at different values of Lewis number (

Figures

The effect of Lewis number on the Nusselt number (

The effect of Lewis number on the Sherwood number (

The effect of Lewis number on the local Sherwood number with and without the magnetic field effect in the case of fluid injection into the porous medium is plotted in Figure

The effect of Lewis number on the Sherwood number (

Figure

The effect of mass flux parameter on the Nusselt number time evolution with and without magnetic field effect, for suction, injection, and impermeable wall cases (

The time evolution of the mass transfer coefficient in the Darcy and non-Darcy flow regions at different Lewis number is illustrated in Figure

The effect of Lewis number on the Sherwood number time evolution in the Darcy and non-Darcy flow regions, for suction, injection, and impermeable wall cases (

The effects of magnetic field and lateral mass flux on transient MHD double-diffusive of an electrically conducting fluid by free convection over a flat plate embedded in Darcy and non-Darcy porous medium are numerically investigated. It was found that the presence magnetic field lowers both the Nusselt and Sherwood numbers in Darcy as well as Forchheimer flow regimes. Increasing the buoyancy ratio

Real constant

magnetic induction

concentration

specific heat of the fluid at constant pressure

concentration molecular diffusion

inertial coefficient

dimensionless, reduced stream function

suction parameter

gravitational acceleration

Hartmann number

permeability of the porous medium

effective thermal conductivity of the porous medium

Lewis number

magnetic parameter (

buoyancy ratio parameter

Nusselt number

modified Rayleigh number

Sherwood number

temperature

time

free stream temperature

wall temperature

volumetrically averaged axial velocity

volumetrically averaged lateral velocity

suction velocity

coordinates along and normal to the plate, respectively.

effective thermal diffusivity

thermal expansion coefficient

concentration expansion coefficient

similarity parameter

dimensionless inertial parameter

dimensionless temperature variable

dimensionless concentration variable

effective kinematic viscosity of the fluid

electrical conductivity of the fluid

fluid density

dimensionless time

dimensional stream function.

wall

outer edge of the boundary layer.