Combined Effects of Hall Current and Radiation on MHD Free Convective Flow in a Vertical Channel with an Oscillatory Wall Temperature

The combined effects of Hall current and radiation on an unsteady MHD free convective flow of a viscous incompressible electrically conducting fluid in a vertical channel with an oscillatory wall temperature have been studied. We have considered two different cases 1) flow due to the impulsive motion of one of the channel walls and 2) flow due to the accelerated motion of one of the channel walls. The governing equations are solved analytically using the Laplace transform technique. It is found that the primary velocity and the magnitude of the secondary velocity increased with an increase in Hall parameter for the impulsive as well as the accelerated motions of one of the channel walls. An increase in either radiation parameter or frequency parameter leads to decrease in the primary velocity and the magnitude of the secondary velocity for the impulsive as well as accelerated motions of one of the channel walls. The fluid temperature decreases with an increase in radiation parameter. Further, the shear stresses at the left wall reduce with an increase in either radiation parameter or frequency parameter for the impulsive as well as the accelerated motions of one of the channel wall.


Introduction
The mechanism of conduction in ionized gases in the presence of a strong magnetic field is different from that in metallic substance.The electric current in ionized gases is generally carried by electrons, which undergos successive collisions with other charged or neutral particles.In the ionized gases, the current is not proportional to the applied potential except when the field is very weak in an ionized gas where the density is low and the magnetic field is very strong, the conductivity normal to the magnetic field is reduced due to the free spiraling of electrons and ions about the magnetic lines of force before suffering collisions and a current is induced in a direction normal to both electric and magnetic fields.This phenomenon, well known in the literature, is called the Hall effect.The study of hydromagnetic flows with Hall currents has important engineering applications in problems of magnetohydrodynamic generators and of Hall accelerators as well as in flight magnetohydrodynamics.It is well known that a number of astronomical bodies posses fluid interiors and magnetic fields.It is also important in the solar physics involved in the sunspot development, the solar cycle and the structure of magnetic stars.In space technology applications and at higher operating temperatures, radiation effects can be quite significant.The radiative convective flows are frequently encountered in many scientific and environmental processes, such as astrophysical flows, water evaporation from open reservoirs, heating and cooling of chambers, and solar power technology.The unsteady hydromagnetic flow of a viscous incompressible electrically conducting fluid through a vertical channel is of considerable interest in the technical field due to its frequent occurrence in industrial and technological applications.The Hall effects on the flow of ionized gas between parallel plates under transverse magnetic field have been studied by Sato [1].Miyatake and Fujii [2] have discussed the free convection flow between vertical plates-one plate isothermally heated and other thermally insulated.Natural convection flow between vertical parallel plates-one plate with a uniform heat flux and the other thermally insulated has been investigated by Tanaka et al. [3].Gupta and Gupta [4] have studied the radiation effect on hydromagnetic convection in a vertical channel.Hall effects on the hydromagnetic convective flow through a vertical channel with conducting walls have been investigated by Dutta and Jana [5].The unsteady hydromagnetic free convective flow with radiative heat transfer in a rotating fluid has been described by Bestman and Adjepong [6].Joshi [7] has studied the transient effects in natural convection cooling of vertical parallel plates.Singh [8] have described the natural convection in unsteady Couette motion.Singh et al. [9] have studied the unsteady free convective flow between two vertical parallel plates.The natural convection in unsteady MHD Couette flow with heat and mass transfers has been analyzed by Jha [10].Narahari et al. [11] have studied the unsteady free convective flow between long vertical parallel plates with constant heat flux at one boundary.The unsteady free convective flow in a vertical channel due to symmetric heating have been described by Jha et al. [12].Singh and Paul [13] have studied the unsteady natural convective between two vertical walls heated/cooled asymmetrically.Sanyal and Adhikari [14] have studied the effects of radiation on MHD vertical channel flow.The radiation effects on MHD Couette flow with heat transfer between two parallel plates have been examined by Mebine [15].Grosan [16] has studied the thermal radiation effect on the fully developed mixed convective flow in a vertical channel.Guria and Jana [17] have discussed Hall effects on the hydromagnetic convective flow through a rotating channel under general wall conditions.Jha and Ajibade [18] have studied the unsteady free convective Couette flow of heat generating/absorbing fluid.Effects of thermal radiation and free convection currents on the unsteady Couette flow between two vertical parallel plates with constant heat flux at one boundary have been studied by Narahari [19].Rajput and Sahu [20] have studied the unsteady free convection MHD flow between two long vertical parallel plates with constant temperature and variable mass diffusion.Das et al. [21] have studied the radiation effects on free convection MHD Couette flow started exponentially with variable wall temperature in the presence of heat generation.The effect of radiation on transient natural convection flow between two vertical walls have been described by Mandal et al. [22].Das et al. [23] have studied the radiation effects on unsteady MHD free convective Couette flow of heat generation/absorbing fluid.The effects of radiation on MHD free convective Couette flow in a rotating system have been discussed by Sarkar et al. [24].Sarkar et al. [25] have studied an oscillatory MHD free convective flow between two vertical walls in a rotating system.
The aim of the present paper is to study the combined effects of Hall current and radiation on the unsteady MHD free convective flow of a viscous incompressible electrically conducting fluid in a vertical channel with an oscillatory wall temperature of one of the channel walls.It is found that the primary velocity 1 and the magnitude of the secondary velocity decrease with an increase in either radiation parameter or frequency or Prandtl number for the impulsive as well as the accelerated motions of one of the channel walls.It is also observed that the primary velocity 1 and the magnitude of the secondary velocity 1

Formulation of the Problem and Its Solution
Consider the unsteady MHD flow of a viscous incompressible electrically conducting radiative fluid between two infinitely long vertical parallel walls separated by a distance .The flow is set up by the buoyancy force arising from the temperature gradient.Choose a Cartesian co-ordinates system with the x-axis along the channel wall at h 0 y  in the vertically upward direction, the y-axis perpendicular to the channel walls and z-axis is normal to the xy-plane (see Figure 1).Initially, at time 0 t  , the two walls and the fluid are assumed to be at the same temperatu h T and stationary.At tim > 0 , the wall 0 re e t at y  starts to move in its own plane with a velocity  

U t
and its temperature is raised to

 cos
whereas the wall at y h  is stationary and maintained at a constant temperature h , where T  is the frequency of the temperature oscil- lations.A uniform transverse magnetic field 0 is applied perpendicular to the channel walls.We assume that the flow is laminar and the pressure gradient term in the momentum equation can be neglected.It is assumed that the effect of viscous and Joule dissipations are negligible.It is also assumed that the radiative heat flux in the x-direction is negligible as compared to that in the y-direction.As the channel walls are infinitely long, the velocity field and temperature distribution are functions of y and t only.

B
Under the usual Boussinesq approximation, the flow is governed by the following Navier-Stokes equations where  is the fluid density,  the kinematic viscosity, and u  are fluid velocity components and g the acceleration due to gravity.
The energy equation is T c the fluid temperature, the thermal conductivity, p the specific heat at constant pressure and the radiative heat flux.
It has been shown by Cogley et al. [26] that in the optically thin limit for a non-gray gas near equilibrium, the following relation holds where h K  is the absorption coefficient,  is the wave length, p e  is the Planck's function and subscript ' ' indicates that all quantities have been evaluated at the temperature h T which is the temperature of the walls at time .Thus, our study is limited to small difference of wall temperatures to the fluid temperature.
On the use of the Equation ( 5), the Equation (3) becomes where The generalized Ohm's law, on taking Hall currents into account and neglecting ion-slip and thermo-electric effect, is (see Cowling [27]) where is the current density vector, j B the magnetic field vector, E the electric field vector, We shall assume that the magnetic Reynolds number for the flow is small so that the induced magnetic field can be neglected.This assumption is justified since the magnetic Reynolds number is generally very small for partially ionized gases.The solenoidal relation 0    B for the magnetic field gives everywhere in the fluid where .Further, if x y z j j j be the components of the current density , then the equation of the conservation of the current density at the walls which are electrically non-conducting.Thus z 0 j  everywhere in the flow.Since the induced magnetic field is neglected, the Maxwell's equation t . This implies that constant x E  and constant y E  everywhere in the flow.We choose this constants equal to zero, i.e. 0 x y E E   .In view of the above assumption, the Equation ( 8) gives 0 , where e e m    is the Hall parameter.Solving ( 9) and ( 10) for x j and y j , we have On the use of ( 11) and ( 12), the momentum Equations ( 1) and (2) along x-and y-directions become Introducing non-dimensional variables (15) where 6), ( 13) and ( 14) become The initial and boundary conditions for Gr , 1 Solutions of Equations ( 23) and ( 24) subject to the boundary conditions (26) are given by where The initial and boundary conditions (4) become where is the frequency parameter.
The initial and boundary conditions for   , Now, we shall considered the following cases.1) When the wall at   0 started impulsively: Then the inver- Taking the Laplace transform of Equations ( 20) and (18) and on the use of ( 19) and ( 22), we have se Laplace transforms of Equations ( 27) and (28) give the solution for the temperature distribution and the velocity field as sin π for Pr 1, 2) When the wall at   0 started acceleratedly: . Then the inverse Laplace transforms of Equations ( 27) and (28) give the solution for the temperature distribution and the velocity field as Copyright © 2013 SciRes.OJFD where 1 2 , s s and 3 s are given by (31).

Results and Discussion
We that the primary velocity 1 and the magnitude of the secondary velocity 1 v increase with an increase of Hall param ter m for the impulsive as well as accelerated motions of one of the channel walls.Figures 4 and 5 show that the primary velocity 1 u and the magnitude of the secondary velocity 1 v decrease with an increase in radia n parameter R for both the impulsive and accelerated motions of one of the channel   v decrease with an increase in Prandtl number Pr for the impulsive as well as accelerated motions of one of the channel walls.Figures 8 and 9 show that both primary velocity 1 u and the magnitude of the secondary velocity 1 v decrease with an increase in frequency parameter n for both the impulsive and accelerated motions of one of the channel walls.An increase in Grashof nu ber Gr leads to increase the primary velocity 1 u and the magnitude of the secondary velocity 1 v for both the im lsive and accelerated motions of one of e channel walls shown in  The rate of heat transfer at the channel walls 0   and 1 and are given by (see the Equations (34) and (35) below).where 1 2 , s s and 3 s are given by (31).Numerical results of the rate of heat transfer at the channel walls 0 which are presented in Tables 1-3 3 that the rate of heat transfers decrease with an increase in frequency parameter .
For the impulsive motion, the non-dimensional shear stress at the wall 0   is given by (see the Equations ( 36) and (37) below).s for Pr 1, where 1 2 , s s and 3 s are given by (31).For the accelerated motion, the non-dimensional shear stress at the wall 0 where , s s and 3 s are given by (31).

Conclusion
The combined effects of Hall current and radiation on the unsteady MHD free convective flow in a vertical channel with an oscillatory wall temperature have been studied.Radiation has a reterding influence on the fluid velocity components for both the impulsive as well as accelerated motions of one of the channel walls.Hall currents accelerates the fluid velecity components for the impulsive as well as accelerated motions of one of the channel walls.In the prence of radiation the fluid temperature  decreases.Further, the shear stress x  and the abso alue of the shear stress lute v


impulsive as well as accelerated motions.An increase in Grashof number leads to fall the fluid velocity components.An increase in the radiation parameter leads to increase the fluid temperature.Further, the shear stress Gr R x at the wall 0   due to the primary flow and the absolute value of the shear stress y  at the wall 0   due to the secondary flow decrease for the impulsive as well as accelerated motions of one of the channel walls with an increase in radiation parameter .The rate of heat transfer R with an increase in Prandtl number .Pr

Figure 1 .
Figure 1.Geometry of the problem.
e  the cyclotron frequency,  the electrical conductivity of the fluid and e  the collision time of electron.

Figures 2 - 17 .
have presented the non-dimensional velocity components and temperature distribution for several values of Hall parameter It is seen from Figures2 and 3

Figure 3 . 1 m 2
Figure 2. Primary velocity for different when and

Figure 4
Figure 4.Primary velocity for different whe u 1 R n 0.5, Pr 0.71, 2, Gr 5 m n     nd 0.2  a 

11 .
It is seen form Figures 12 and 13 that the primary velocity 1 u and the magnitude of the secondary velocity 1 v increase with an increase in time  for both the impulsiv nd accelerated motions of one of the channel w lls.It is seen from Figure 14 that the luid temperature e a a f  decreases with an increase in radiation parameter R .This result qualitatively agrees with expectations, ince the effect of radiation is to decrease the rate of energy transport to the fluid, thereby decreasing the temperature of the fluid.It is observed from Figure 15 that the fluid temperature  increases with an increase in Prandtl number.This is in agreement with the physical fact that the thermal boundary layer thickness decreases with increasing .Figure16shows that the fluid temperature Pr Pr  decreases with an increase of frequency parameter .
Figure 17 shows that the fluid temperature n  increases when time  progresses.It is seen from Figures 2-13 that the fluid velocities for the impulsive motion of one of the channel walls is always greater than the accelerated motion.

1 
 increases when time  progresses.It is seen from Table

Figure 8
Figure 8.Primary velocity for different when u 1  n

Figure 14 .
Figure 14.Temperature  for different when and R Pr 0.71, 2 n   0.2  

Figure 16 Figure 17 .
Figure 16.Temperature  for different for and 0.2  n

Figure 22
Figure 22.Shear stress x  for different n in radi n param impulsive as well as accelerated m tions of te ease atio o eter one of the channel walls.The ra of heat transfers with Constant Hea y," WSEAS Transactions on Heat sfer, Vol. 5, No. 1, 2010, pp.21-30.nd P. K. Sahu, "Transient Free Convection the Un ady Couette Flow between Two Verti Flux at One Boundar t on and Mass Tran [20] U. S. Rajput a MHD Flow between Two Long Vertical Parallel Plates with Constant Temperature and Variable Mass Diffusion," International Journal of Mathematical Analysis, number, for several values of Prandtl number