^{1}

This paper is concerned with analytical solution of one-dimensional unsteady laminar boundary layer MHD flow of a viscous incompressible fluid past an exponentially accelerated infinite vertical plate in presence of transverse magnetic field. The vertical plate and the medium of flow are considered to be porous. The fluid is assumed to be optically thin and the magnetic Reynolds number is considered small enough to neglect the induced hydromagnetic effects. The governing boundary layer equations are first converted to dimensionless form and then solved by Laplace transform technique. Numerical values of transient velocity, temperature, skin friction, and Nusselt number are illustrated and are presented in graphs for various sets of physical parametric values, namely, Grashof number, accelerating parameter, suction parameter, permeability parameter, radiation parameter, magnetic parameter, and time. It is found that the velocity decreases with increases of the suction parameter for both cases of cooling and heating of the porous plate whereas skin friction increases with increase of suction parameter.

The study of unsteady natural convective flow of viscous incompressible fluid past vertical bodies has wide engineering and technological applications. When free convection flows occurs at high temperature, the effects of radiation are vital important. Thermal radiation is key to many fundamental phenomena surrounding us, from solar radiation to fire incandescent lamp, and has played a major role in combustion and furnace design, design of fins, nuclear power plants, cooling of towers, gas turbines and various propulsion device for aircraft, energy utilization, temperature measurements, remote sensing for astronomy, and space exploration.

Magnetohydrodynamic (MHD) flow and heat and mass transfer processes occur in many industrial applications such as the geothermal system, aerodynamic processes, chemical catalytic reactors and processes, electromagnetic pumps, and MHD power generators. Many studies have been carried out to investigate the magnetohydrodynamic transient free convective flow. Gupta [

In recent years convective heat transfer in porous media has attracted considerable attention owing to its wide industrial and technological applications such as geothermal energy recovery, oil extraction, fibre and granular insulation, electronic system cooling, and porous material regenerative heat exchangers. Gupta et al. [

However, to the best of authors’ knowledge, MHD free convective flow past an exponentially accelerated vertical porous plate with variable temperature through a porous medium was never considered in the literature. The objective of this paper is to study magnetohydrodynamic transient heat transfer flow past an exponentially accelerated infinite vertical porous plate with variable temperature and the plate is embedded in a porous medium. The exact solutions of the dimensionless unsteady linear governing equations are obtained by Laplace transform technique.

An unsteady one-dimensional laminar free convective flow of a viscous incompressible fluid past an infinite vertical porous plate through a porous medium with variable temperature is considered. The

Physical model and coordinate system.

The local radiant for the case of an optically thin gray gas is expressed by

We assume that the temperature differences within the flow are sufficiently small such that

In order to write the governing equations, initial and boundary conditions in dimensionless form, the following nondimensional quantities are introduced:

To solve the unsteady equations (

Laplace transforms of (

Solutions of (

Knowing the velocity and temperature field, it is very interesting to study the skin friction and Nusselt number. In nondimensional form the skin friction and Nusselt number are defined, respectively, as follows:

Expression of the skin-friction

Expression of Nusselt number Nu is obtained from (

In order to get an insight into the physical solution of the problem, the numerical computations of velocity profile, temperature profile, skin friction, and Nusselt number are obtained for different values of magnetic field parameter

Effect of Gr on velocity profiles at

Effect of

Effect of

Effects of

Effects of

Effect of

Effect of

Effect of

Effect of Gr on skin friction at

Effect of

Effect of

Effects of

Effect of

The transient velocity profiles for different values of Grashof number at

The transient velocity profiles for different values of Magnetic parameter

The effects of suction parameter

The transient velocity profiles for different values of accelerating parameter

Effect of Grashof number Gr on skin friction is presented in Figure

Figure

The Nusselt number for different values radiation parameter

The analytical study on unsteady one-dimensional natural convective MHD flow of a viscous incompressible and electrically conducting fluid past an exponentially accelerated infinite vertical porous plate through a porous medium with variable temperature is presented. The exact solutions of the dimensionless governing boundary layer equations are obtained by Laplace transform technique. On the basis of the observations, results, and discussions the conclusions of the present study are as follows:

Velocity increases with increase in Gr or

Temperature increases with increase in

Skin friction increases with increase in

Rate of heat transfer increases with increase in

Accelerating parameter

Dimensionless accelerating parameter

Absorption coefficient

Specific heat at constant pressure

Transverse magnetic field strength

Grashof number

Acceleration due to gravity

Thermal conductivity of the fluid

Permeability parameter

Dimensionless permeability parameter

Magnetic field parameter

Nusselt number

Prandtl number

Radiative heat flux in the

Radiation parameter

Time

Dimensionless time

Temperature

Dimensionless temperature

Temperature of the plate

Temperature of the fluid far away from the plate

Velocity of the plate

Dimensionless velocity

Coordinate axis normal to the plate

Dimensionless coordinate axis normal to the plate.

Volumetric coefficient of thermal expansion

Suction parameter

Kinematic viscosity

Fluid density

Electrical conductivity of fluid.

The author declares that there is no conflict of interests regarding the publication of this paper.