Radiation Effect on Natural Convection near a Vertical Plate Embedded in Porous Medium with Ramped Wall Temperature

Radiation effect on the natural convection flow of an optically thin viscous incompressible fluid near a vertical plate with ramped wall temperature in a porous medium has been studied. The exact solution of momentum and energy equations is obtained by the use of Laplace transform technique. The variations in fluid velocity and temperature are shown graphically whereas the numerical values of shear stress and the rate of heat transfer at the wall are presented in tabular form for various values of flow parameters. The results show that the fluid velocity increases with increase in Grashof number, Darcy number and time parameters whereas the fluid velocity decreases with increase in the radiation parameter and Prandtl number for ramped temperature as well as isothermal wall temperature. It is found that an increase in radiation parameter leads to rise the temperature for both ramped wall temperature as well as isothermal wall temperature. Further, it is found that an increase in Prandtl number leads to fall the temperature for both ramped wall temperature as well as isothermal wall temperature. The shear stress at the wall decreases with increases in either Prandtl number or porosity parameter while the result shows reverse in the case of radiation parameter. Finally, the rate of heat transfer is increased with increase in the radiation parameter for both ramped wall temperature as well as isothermal wall temperature.


Introduction
The phenomenon of natural convection arises in fluids when temperature changes cause density variations leading to buoyancy forces acting on the fluid particles.Such flows which are driven by temperature differences abound in nature and have been studied extensively because of its applications in engineering, geophysical and astrophysical environments.Comprehensive literature on various aspects of free convection flows and its applications could be found in Ghoshdastidar [1], Nield and Bejan [2].Ghoshdastidar gave various areas of applications of free convection flow such as those found in heat transfer from pipes and transmission lines as well as from electronic devices, heat dissipation from the coil of a refrigerator unit to the surrounding air, heat transfer from a heater to room air, heat transfer in nuclear fuel rods to the surrounding coolant, heated and cooled en-closures, quenching, wire-drawing and extrusion, atmospheric and oceanic circulation.Unsteady free convection flows in a porous medium have received much attention in recent time due to its wide applications in geothermal and oil reservoir engineering as well as other geophysical and astrophysical studies.Moreover, considerable interest has been shown in radiation interaction with convection for heat and mass transfer in fluids.This is due to the significant role of thermal radiation in the surface heat transfer when convection heat transfer is small, particularly in free convection problems involving absorbing-emitting fluids.The unsteady fluid flow past a moving plate in the presence of free convection and radiation were studied by Mansure [3], Raptis and Perdikis [4], Das et al. [5], Grief et al. [6], Ganeasan and Loganathan [7], Mbeledogu et al. [8], Makinde [9] and .All these studies have been confined to unsteady flow in a non-porous medium.have studied the influence of viscous dissipation and radiation on unsteady MHD freeconvection flow past an infinite heated vertical plate in a porous medium with time-dependent suction.Radiative and free convective effects of a MHD flow through a porous medium between infinite parallel plates with time dependent suction have been investigated by Alagoa et al. [12].have made an analysis on MHD oscillatory Couette flow of a radiating viscous fluid in a porous medium with periodic wall temperature.Sattar and Maleque [14,15] have studied the unsteady MHD Natural convection flow and mass transfer along an accelerated porous plate in a porous medium.Thermal radiation interaction with unsteady MHD flow past a vertical porous plate immersed in a porous medium has been analyzed by Samad and Rahman [16].Mahanti and Gaur [17] have studied the effects of varying viscosity and thermal conductivity on steady free convective flow and heat transfer along an isothermal vertical plate in the presence of heat sink.Transient free convection past a semi-infinite vertical plate with variable surface temperature has been investigated by Takhar et al. [18].
In this present paper, we investigate the effects of radiation on the free convection flow of an optically thin incompressible viscous fluid past an infinite vertical plate with ramped wall temperature in porous medium.The fluid considered is a gray, radiation, absorbing, emitting but non-scattering medium and the Rosseland approximation is used to describe the radiative heat transfer in the energy equation.It is seen that the velocity 1 decreases for both ramped wall temperature as well as isothermal wall temperature with an increase in either radiation parameter or Prandtl number .It is also seen that the velocity 1 increases for both ramped wall temperature as well as isothermal wall temperature with an increase in either Grashof number or time u Ra Pr Gr u  .It is found that an increase in radiation parameter leads to rise the temperature Ra  for both ramped wall temperature as well as isothermal wall temperature.Further, it is found that an increase in Prandtl number leads to fall the temperature for ramped temperature as well as isothermal case.

Formulation of the Problem and Its Solutions
Consider the unsteady free convection flow of an optically thin viscous incompressible fluid past an moving infinite vertical plate coinciding with plane 0 y  , where the flow is confined to in a porous medium.Choose a cartesian co-ordinates system with x-axis along the wall in a vertically upward direction and y-axis is normal to it into the fluid (see Figure 1).At 0 y  0 t  , the


The Boussinesq approximation is assumed to hold and for the evaluation of the gravitational body force, the density is assumed to depend on the temperature according to the equation of reference state where is the fluid temperature, T  the fluid density,   the coefficient of thermal expansion and T  and 0  being the reference temperature and the density respectively.
Using Boussinesq Approximation (1), the momentum equation in a porous medium along x-axis is where , u g ,   ,  ,  and are respectively, fluid velocity, acceleration due to gravity, coefficient of thermal expansion, kinematic viscosity, fluid density and permeability of a porous media.
where is the thermal conductivity, k p c the specific heat at constant pressure and the radiative heat flux. r The initial and boundary conditions are It has been shown by Cogley et al. [19] that in the optically thin limit for a non-gray gas near equilibrium, the following relation holds where K  is the absorption coefficient,  is the wave length, h e  is the Plank's function and subscript 0 ` indicates that all quantities have been evaluated at the temperature which is the temperature of the wall at time .Thus our study will be limited to small difference of wall temperature to the fluid temperature. where Introducing dimensionless variables Equations ( 2) and ( 6) become where the porosity parameter and the Darcy number.Da The characteristic time is defined as The corresponding initial and boundary conditions for and Taking Laplace transformation of the Equations ( 9) and ( 10), we get where The corresponding boundary conditions for 1 u and The solution of the Equations ( 14) and ( 13) subject to the boundary conditions ( 16) can be easily obtained and are given by where Taking the inverse Laplace transform of Equations ( 17) and ( 18 where where is the complementary error function and is the unit step function.

Solution in Case of Unit Prandtl Number
Prandtl number is a measure of the relative stren the viscosity and thermal conductivity of the fluid.So the where

Solution for Isothermal Case
In order to highlight the effects of the ramped temperature distribution near a vertical plate, it may be important mpare the effects of th al temperature distribution for the fluid flow.The temperature and the vecity for the fluid flow near an isothermal plate can be to co e isotherm lo expressed as When , the Solutions ( 27) and ( 28) become Pr 1 where

Results and Discussion
We u increases for both ped wall temperature as we l as isothermal wall temperature with an increase in Darcy number Da .It is seen from Figure 6 that the velocity 1 increases for both ramped wall temperature as well as isothermal wall temperature with an increase in time  .It is observed from Figure 7 that the temperature  decreases as the radiation parameter Ra increases for both ramped wall temperature as well as isothermal wall temperature.This result qualitatively agrees with expectations, since the effect of radiation is ecrease the rate of energy transport to the fluid, thereby decreasing the temperature of the fluid.It is seen from and for the isothermal wall temperature Copyright  4 creases for both ramped wall temperature as well as iso-

Figure 1 .
Figure 1.Geometry of the problem.plate and the surrounding fluid are at the same constant temperature T  .At time , the temperature of the 0 t  wall is raised or lowered to ), the solution for the fluid temperature   , are obtained and are given by


Figure 2. Velocity profiles for variations in

Figure
Figure 4. Velocity profiles for variations in when Gr Pr 0.71  , 0.04 Da  , 0.1  

Figure 6 .
Figure 6.Velocity profiles for variations in time  when Pr 0.71  , 25 Gr  , 2 Ra  and 0.04 a  .D

Figure
Figure 8. Temperature profiles for variations in when Pr 2 Ra  and 0.5  