RADIATION EFFECTS ON UNSTEADY MHD FREE CONVECTION WITH HALL CURRENT NEAR AN INFINITE VERTICAL POROUS PLATE

Radiation effect on unsteady free convection flow of an electrically conducting, gray gas near equilibrium in the optically thin limit along an infinite vertical porous plate are investigated in the presence of strong transverse magnetic field imposed perpendicularly to the plate, taking Hall currents into account. A similarly parameter length scale (h), as a function of time and the suction velocity are considered to be inversely proportional to this parameter. Similarity equations are then derived and solved numerically using the shooting method. The numerical values of skin friction and the rate of heat transfer are represented in a table. The effects of radiation parameter, Hall parameter, and magnetic field parameters are discussed and shown graphically. 2000 Mathematics Subject Classification. 76R10.


Introduction.
Free convection flows past different types of vertical bodies are studied because of their wide applications and hence it has attracted the attention of numerous investigators and scientists.Literature on unsteady MHD convection heat transfer with or without Hall currents are very extensive due to its technical importance in the scientific community.Some of the literature surveys and reviews of pertinent work in this field are documented by Schlichting [8], Soundagekhar et al. [10], Raptis et al. [7], Raptis and Perdikis [6], K. Vajravelu [11], Sacheti et al. [4], and M. A. Al-Nimr and S. Masoud [1].In all these studies the Hall current effects are not considered.
The unsteady hydromagnetic free convective flow with Hall current is studied by Singh and Raptis [9].P. C. Ram [5] studied the effects of Hall and ion-slip currents on free convective heat generating flow in a rotating fluid.The Laplace transform technique has been applied to obtain an exact solution in a closed form, when the plate is moving with a velocity which is an arbitrary function of time.In all these studies, the effect of radiation are not considered.In space technology applications and at higher operating temperatures, radiation effects can be quite significant.Since radiation is quite complicated, many aspects of its effect on free convection or combined convection have not been studied in recent years.However, Cogley et al. [2] showed that, in the optically thin limit for a gray gas near equilibrium the following relation holds: where T is the temperature, q r is the radiative heat flux, k λ is the absorption coefficient, e bλ is Plank's function, and ω is the properties of the wall.Greif et al. [3] showed that, for an optically thin limit, the fluid does not absorb its own emitted radiation, this means that there is no self-absorption, but the fluid does absorb radiation emitted by the boundaries.
In space technology and in nuclear engineering applications, such a problem is quite common.But in these fields, the presence of strong magnetic field and Hall current taking effects play an important role and these effects have not been studied in the case of free convective flow of a radiation gas under the condition mentioned above.In this paper, we investigate the solution which the buoyancy, radiation, and Hall currents act simultaneously.

Mathematical formulation.
Consider unsteady free convection flow of a viscous incompressible and electrically conducting fluid, along an infinite vertical porous plate subjected to time-dependent suction velocity.The flow is assumed to be in the xdirection which is taken along the plate in the upward direction and the y-axis perpendicular to it.A uniform strong magnetic field B 0 is assumed to be applied in the y-direction and the induced magnetic field of the flow is negligible in comparison with the applied one which corresponds to very small magnetic Reynolds number [10].On neglecting the viscous dissipation effects, the flow under consideration is governed by the following equations: ) where (u, w) are the x and z components of velocity, v is the suction velocity, T is the temperature of the fluid, u is the kinematics viscosity, σ is the electric conductivity, ρ is the density of the fluid, k is the thermal conductivity, c p is the specific heat at constant pressure, β is the volumetric coefficient of thermal expansion, g is the acceleration due to gravity, and m is the Hall parameter.
The boundary conditions are given by we now define the similarity variables as follows: where h (= h(t)) is a similarly parameter length scale and u 0 is the free stream velocity.In terms of h(t), a convenient solution of (2.1) can be given by where v 0 is a nondimensional transpiration parameter, clearly v 0 > 0 and V 0 < 0 indicates suction or injection, respectively.Accordingly (2.2), (2.3), and (2.4) take the form where the Grashof number G r = gβh 2 (T w − T ∞ )/ν 2 , the magnetic parameter M = σ µ 2 e B 2 0 h 2 /νρ, Prandtl number p r = ρνc p /K, and radiation parameter R = 4Ih 2 /ρc p ν.The boundary conditions corresponding to (2.8) are (2.9) Equations (2.8) are similar except for the term (h/ν)(∂h/∂t), where t appears explicitly.Thus, the similarity condition requires that (h/ν)(∂h/∂t) must be constant.Hence it is assumed that h ν where C is an arbitrary constant.At C = 2 and by integrating equation (2.10), one obtains h = 2 √ νt, which defines the well-established scaling parameter for unsteady boundary layer problems [8].Hence, the similarity equations are obtained as where From the velocity field, we can study the skin friction.It is given by and in view of (2.6), we have The values of F (0) and G (0) are presented in Table 2.1 for P r = 0.7 and different values of m and M.    The effects of the thermal radiation parameter R on the primary velocity and temperature profiles in the boundary layer are illustrated in Figures 3.7 and 3.8, respectively.Increasing the thermal radiation parameter R produces significant increase in the thermal condition of the fluid and its thermal boundary layer.This increase in the fluid temperature induces more flow in the boundary layer causing the velocity of the fluid there to increase.
From Table 2.1 we observe that an increase of M leads to a decrease in the value of τ x .But τ x increases with the increase of m and R. As regards τ z , we observe that the values τ z are all negative, and hence the separation in the z-direction may occur, the increase in both M and m leads to the decrease in τ z , but τ z increases with the increase of F .We can conclude a set of results corresponding to various special cases: (1) Substitute M = 0, m = 0, and R = 0 in (2.8) yields identical results to those well known in hydrodynamics [8].

Figures 3 .
Figures 3.4, 3.5, and 3.6 show the F,G, and θ profiles at different values of m, we see that F increases with increasing the parameter m, but the temperature profiles θ decrease with increasing m. Figure 3.5 shows that the secondary velocity G begins to develop as the Hall parameter m increases in the interval 0 ≤ m ≤ 1 and decreases for m > 1.The effects of the thermal radiation parameter R on the primary velocity and temperature profiles in the boundary layer are illustrated in Figures 3.7 and 3.8, respectively.Increasing the thermal radiation parameter R produces significant increase in the thermal condition of the fluid and its thermal boundary layer.This increase in the fluid temperature induces more flow in the boundary layer causing the velocity of the fluid there to increase.From Table2.1 we observe that an increase of M leads to a decrease in the value of τ x .But τ x increases with the increase of m and R. As regards τ z , we observe that the values τ z are all negative, and hence the separation in the z-direction may occur, the increase in both M and m leads to the decrease in τ z , but τ z increases with the increase of F .We can conclude a set of results corresponding to various special cases:

Figure 3 .
Figures 3.4, 3.5, and 3.6 show the F,G, and θ profiles at different values of m, we see that F increases with increasing the parameter m, but the temperature profiles θ decrease with increasing m. Figure 3.5 shows that the secondary velocity G begins to develop as the Hall parameter m increases in the interval 0 ≤ m ≤ 1 and decreases for m > 1.The effects of the thermal radiation parameter R on the primary velocity and temperature profiles in the boundary layer are illustrated in Figures 3.7 and 3.8, respectively.Increasing the thermal radiation parameter R produces significant increase in the thermal condition of the fluid and its thermal boundary layer.This increase in the fluid temperature induces more flow in the boundary layer causing the velocity of the fluid there to increase.From Table2.1 we observe that an increase of M leads to a decrease in the value of τ x .But τ x increases with the increase of m and R. As regards τ z , we observe that the values τ z are all negative, and hence the separation in the z-direction may occur, the increase in both M and m leads to the decrease in τ z , but τ z increases with the increase of F .We can conclude a set of results corresponding to various special cases:

Table 2 .
The results of the numerical computations are displayed in Figures 3.1 3. Results and conclusion.