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We have explored the influence of thermal diffusion and radiation on unsteady magnetohydrodynamic free convection flow past an infinite heated vertical plate in a porous medium. The governing boundary layer equations are written into a dimensionless form by similarity transformations. The transformed nonlinear differential equations are solved numerically with finite element methods. Numerical calculations are carried out for different values of dimensionless parameters. The results are presented graphically for velocity, temperature, and concentration profiles and show that the flow field and other quantities of physical interest are significantly influenced by these parameters.

Flow through porous medium past infinite vertical plate is common in nature and has many applications in engineering and science. A number of workers have investigated such flows and excellent literature on the properties and phenomenon may be found in the literature [

Recently, attention has been on the effects of transversely applied magnetic field and thermal perturbation on the flow of electrically conducting viscous fluids such as plasma. Various properties associated with the interplay of magnetic fields and thermal perturbation in porous medium past vertical plate find useful applications in astrophysics, geophysical fluid dynamics, and engineering. Researches in these fields have been conducted by many investigators [

The objective of the present chapter is to examine the effects of thermal diffusion and radiation on unsteady MHD free convection flow past an infinite heated vertical plate in a porous medium. The equations of continuity, linear momentum, energy, and diffusion, which govern the flow field, are solved by using Galerkin finite element method. Similarity solutions are then obtained numerically for various parameters, entering into the problem, and discussed from the physical point of view.

We consider the unsteady flow of an incompressible, viscous, and radiating hydromagnetic fluid past an infinite porous heated vertical plate with time-dependent suction in an optically thin environment. The physical model and the coordinate system are shown in Figure

The physical model and coordinate system of the problem.

At time

The boundary conditions are

Since the medium is optically thin with relatively low density and

In view of (

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

Equations (

By applying Galerkin finite element method for (

Integrating the first term in (

Now put row corresponding to the node

Here

In the previous sections, we have formulated and solved the problem of an unsteady MHD free convection flow past an infinite heated vertical plate in a porous medium with radiation. By invoking, the optically thin differential approximation for the radiative heat flux in the energy equation. In the numerical computation, the Prandtl number (

Velocity profiles for different values of

Velocity profiles for different values of

Velocity profiles for different values of

Velocity profiles for different values of

Velocity profiles for different values of

Velocity profiles for different values of

The effect of the thermal radiation parameter

Velocity profiles for different values of

Velocity profiles for different values of

Velocity profiles for different values of

Velocity profiles for different values of

Temperature profiles for different values of

Temperature profiles for different values of

Temperature profiles for different values of

Temperature profiles for different values of

Concentration profiles for different values of

Concentration profiles for different values of

In conclusion, therefore, the flow of an unsteady MHD free convection past an infinite heated vertical plate in a porous medium under the simultaneous effects of thermal diffusion and radiation is affected by the material parameters. The governing equations are approximated to a system of linear partial differential equations by using Galerkin finite element method. The results are presented graphically and we can conclude that the flow field and the quantities of physical interest are significantly influenced by these parameters.

The velocity increases as Grashof number

The fluid temperature was found to be decreasing as the heat source parameter

The fluid concentration was found to be decreasing as the Schmidt number