EFFECT OF NON-UNIFORM HEAT SOURCE AND RADIATION ON UNSTEADY MHD FREE CONVECTION FLOW PAST AN INFINITE HEATED VERTICAL PLATE IN POROUS MEDIUM

Influence of radiation and non-uniform heat source on unsteady, magneto-hydrodynamic free convection flow of viscous incompressible fluid past an infinite vertical heated plate embedded in porous medium of an optically thin environment with time dependent suction and viscous dissipation is investigated in this paper. Analytical solutions of the coupled non-linear equations are obtained for the velocity field and temperature distribution using oscillating time-dependent perturbation technique. Expressions for skin-friction and heat transfer rate are also derived. The effects of the material parameters on velocity, temperature, skin-friction, and rate of heat transfer are discussed quantitatively.


INTRODUCTION
The study of magneto hydrodynamic natural convection flow and heat transfer of an electrically conducting fluid past a heated semiinfinite vertical porous plate finds useful applications in many engineering problems such as MHD generators, nuclear reactors, geothermal extractions, heat exchanger devices and boundary layer control in the field of aeronautics and aerodynamics.
Because of numerous applications in several engineering and industrial manufacturing processes, such flows have got renewed interest among researchers. The effects of radiation are of vital importance. Recent development in hypersonic flights, rocket combustion chambers, power plants for interplanetary flights and gas cooled nuclear reactors have focused attention on thermal radiations as a mode of energy transfer, and emphasized the need for an improved understanding of radiative transfer in these processes (Cowling, 1957;Ferraro & Plumpton, 1961). Cess (1966) investigated the interaction of radiation with laminar free convection heat transfer from a vertical plate for an absorbing emitting liquid in the optically thick region and used the singular perturbation technique. Arpaci (1968) considered a similar problem in both, the optically thin and optically thick regions using the appropriate integral technique. Cheng and Ozisik (1972) investigated an absorbing, emitting and isotropically scattering fluid. Ali et. al. (1984) considered the effects of radiation on natural convective flow of a viscous fluid over a horizontal surface, while Bestman (1985) considered the same problem for the flow of non-Newtonian fluid along a vertical plate under uniform transverse magnetic field. Bestman & Adjepong (1988) also investigated free convection unsteady flow considering radiative heat transfer in a rotating fluid under the influence of uniform magnetic field. Ibrahim (1990) studied mixed convection and radiation interaction for flow of a viscous fluid considering horizontal and vertical surfaces, respectively. Hossain & Takhar (1996) studied the radiation effects on mixed convection along a vertical plate with uniform surface temperature and employed Keller Box finite Mishra, P., S. Tripathi Effect of Non-Uniform Heat Source and Radiation… 220 difference method. Hossain et. al. (2001) andHossain et. al. (1999) extended this work for free convection and radiation interaction past a porous vertical plate in the presence of constant suction and with variable viscosity effects respectively. Azzam (2002) studied a similar problem for the effects of radiation on the flow and heat transfer past a moving vertical plate in the presence of magnetic field. Hydromagnetic flow showing the effects of radiation and heat transfer over a wedge was studied by Elbashbeshy and Dimian (2002) taking into account the variable viscosity. Cookey et. al. (2003) investigated the influence of viscous dissipation and radiation in unsteady MHD free convection flow past an infinite vertical heated plate in the optically thin environment with variable suction and used radiative heat flux in differential form.
The aim of the present investigation is to study the influence of radiation, variable suction and non-uniform heat source/sink on unsteady hydromagnetic free convection flow of a viscous fluid past a heated vertical porous plate taking into account the viscous dissipation. In the analysis, we considered both the space and the temperature dependent heat source/sink followed by Abo-Eldahab and El-Aziz

MATHEMATICAL FORMULATION
We consider the unsteady free convection flow of an incompressible viscous, electrically conducting, radiating fluid past an infinite porous heated vertical plate embedded in porous medium with time dependent suction and viscous dissipation. In Cartesian coordinate system the ' x -axis is taken along the vertical porous plate in the upward direction and the ' z The boundary conditions relevant to the problem are Since the medium is optically thin with relatively low density and  , the radiative heat flux given in equation (4) such that 1   and the negative sign indicates that the suction velocity is towards the plate.
In the energy equation (3) where A* and B* space and temperature dependent internal heat generation/absorption (non-uniform heat source/sinks). We introduce following non-dimensional quantities and parameters In view of (7) and above non-dimensional quantities and parameters, the Eqs. (2) and (3) transform to: Eqs. (9) and (10) are now subject to the boundary conditions.

METHOD OF SOLUTION
The Eqs. (9) and (10) are highly non-linear equations. Since  is small ( 1   ), we can advance for analytical solution of these equations, subject to the boundary conditions (11), we use multiple parameters perturbation expansion of the form : Introducing (12) in Eqs. (9)-(10) and neglecting the coefficients of 2 o( )  , we obtain: where dashes denote differentiation with respect to z.

SKIN-FRICTION AND HEAT TRANSFER RATE
The skin-friction (  ) at the plate 0 z  is given by: The heat transfer rate ( Nu ) at the plate 0 z  is given by :

RESULTS AND DISCUSSION
In the present problem combined effect of radiation and non-uniform heat source on unsteady convection flow past an infinite heated vertical plate in porous medium with time dependent suction and viscous dissipation is studied. The velocity field and temperature distribution are evaluated in equations (28) et. al. (2007). We now proceed with the discussion and results. . It is observed that the velocity increases near the plate and after attaining a maximum value it decreases asymptotically to horizontal axis. As expected, we observe a decrease in the velocity field as the Prandtl number ( Pr ) increases. In fact, increase in Prandtl number ( Pr ) decreases the temperature buoyancy effect which also leads to a decrease in the velocity boundary layer. This observation is in good agreement with in Mbeledogu and Ogulu (2007). It is also observed that an increase in magnetic parameter ( M ) results in a decrease in the velocity field consistent with many other studies. Hence, hydromagnetic drag embodied in Eq. (9) retards the transient velocity. This is an important controlling mechanism in nuclear energy systems heat transfer, where momentum development can be reduced, for oscillatory flow regimes, by enhancing the magnetic field . We observe that velocity increases with increase in free convection parameter ( Gr ), i.e., maximum velocity corresponds to maximum free convection parameter ( 0 Gr  ). Hence, buoyancy parameter ( Gr ) has dominant effect in escalating transient velocity [25]. Also, we note that transient velocity increases with increase in permeability parameter ( K ). As expected, as K increases, the bulk porous medium is lowered, which increases the momentum development of the flow regime, thereby enhancing transient velocity (Basu, at al. (2011). It is interesting to note that in absence of buoyancy parameter, i.e., 0.0 Gr  the velocity decreases drastically but asymptotically as y increases. These observations are in good agreement of Mbeledogu and Ogulu (2007).  . It is observed that a decrease in temperature and temperature boundary layer exists with increase in Prandtl number and magnetic parameter. It is clear from the curve that the temperature of water at 60 0 C is more stable in comparison to water at 100 0 C