Effects of Hall Current and Ion-Slip on Unsteady MHD Couette Flow

The unsteady MHD Couette flow of an incompressible viscous electrically conducting fluid between two infinite nonconducting horizontal porous plates under the boundary layer approximations has been studied with the consideration of both Hall currents and ion-slip. An analytical solution of the governing equations describing the flow is obtained by the Laplace transform method. It is seen that the primary velocity decreases while the magnitude of secondary velocity increases with increase in Hall parameter. It is also seen that both the primary velocity and the magnitude of secondary velocity decrease with increase in ion-slip parameter. It is observed that a thin boundary layer is formed near the stationary plate for large values of squared Hartmann number and Reynolds number. The thickness of this boundary layer increases with increase in either Hall parameter or ion-slip parameter.


Introduction
The magnetohydrodynamic (MHD) flow between two parallel plates, one in uniform motion and the other held at rest known as MHD Couette flow, is a classical problem that has many applications in MHD power generators and pumps, accelerators, aerodynamic heating, electrostatic precipitation, polymer technology, petroleum industry, purification of crude oil and fluid droplets and sprays.A lot of research work concerning the MHD Couette flow has been obtained under different physical effects.In most cases, the Hall and ion slip terms were ignored in applying Ohm's law as they have no marked effect for small and moderate values of the magnetic field.However, the current trend for the application of magnetohydrodynamics is towards a strong magnetic field, so that the influence of electromagnetic force is noticeable [1].Under these conditions, the Hall currents and ion slip are important and they have a marked effect on the magnitude and direction of the current density and consequently on the magnetic force term.Soundalgekar [2] has studied the Hall and ion-slip effects in MHD Coutte flow with heat transfer.Attia [3] has studied the unsteady Couette flow with heat transfer considering ionslip.The transient Hartmann flow with heat transfer considering the ion slip has been investigated by .Attia [7] has obtained the analytical solution for flow of a dusty fluid in a circular pipe with Hall and ion slip effect.
Seddeek [8] has studied the effects of Hall and ion-slip currents on magneto-micropolar fluid and heat transfer over a non-isothermal stretching sheet with suction and blowing.The effects of Hall and ion-slip currents on free convective heat generating flow in a rotating fluid have studied by Ram [9].Mittal and Bhat [10] has discussed the forced convective heat transfer in a MHD channel with Hall and ion slip currents.Jana and Datta [11] has described the Couette flow and heat transfer in a rotating system.The Hall effect on unsteady Couette flow under boundary layer approximations has been analysed by Kanch and Jana [12].Attia [13] has studied the ion slip effect on unsteady Couette flow with heat transfer under exponential decaying pressure gradient.The combined effect of Hall and ion-slip currents on unsteady MHD Couette flows in a rotating system have been investigated by Jha and Apere [14].
The present paper is devoted to study the combined effects of Hall current and ion-slip on the unsteady MHD Couette flow between two infinite horizontal parallel porous plates under the boundary layer approximations.The upper plate is moving with a uniform velocity U while the lower plate is held at rest.The fluid is acted upon by a constant pressure gradient and a uniform suction/injection at the plates.A uniform magnetic field 0 is applied perpendicular to the plates.It is found that the primary velocity decreases while the magnitude of the secondary velocity increases with increase in Hall pa-N.GHARA ET AL. 2 rameter.It also is found that both the primary velocity and the magnitude of secondary velocity decrease with increase in ion-slip parameter.Asymptotic behavior of the solution has been analyzed for large values of squared Hartmann number and Reynolds number.It is observed that a thin boundary layer is formed near the stationary plate for large values of the magnetic parameter and Reynolds number.The thickness of this boundary layer increases with increases in either Hall parameter or ion-slip parameter.Further, it is seen that the shear stress components 0 x  and 0 z  due to the primary and sec- ondary flows at the stationary plate 0   increase with increase in Hall parameter for fixed value of squared Hartmann number and ion slip parameter.It is also seen that for fixed value of both squared Hartmann number and Hall parameter, 0 x  increases while 0 z  decreases with increase in ion-slip parameter.

Mathematical Formulation and Its Solution
Consider the viscous incompressible electrically conducting fluid bounded by two infinite horizontal parallel porous plates separated by a distance .Choose a Cartesian co-ordinate system with x-axis along the lower stationary plate in the direction of the flow, the y-axis is normal to the plates and the z-axis perpendicular to xyplane (see Figure 1).Initially, at time , both the plates are at rest.At time , the upper plate suddenly starts to move with uniform velocity along x-axis.A uniform magnetic field 0 is applied perpendicular to the plates.The velocity components are  relative to a frame of reference.Since the plates are infinitely long, all physical variables, except pressure, depend on and only.The equation of continuity then gives where , B E , , , q j  , e  , e  and i  are respectively, the magnetic field vector, the electric field vector, the fluid velocity vector, the current density vector, the conductivity of the fluid, the cyclotron frequency, the electron collision time and i  the ion-slip parameter.
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 constant everywhere in the flow where . The equation of the conservation of the charge gives y j  constant.This constant is zero since y at each plate which is electrically non-conducting.Thus 0 0 y j j   everywhere in the flow.Since the induced magnetic fields are neglected, the Maxwel's equation . This implies that x E  constant and z E  constant everywhere in the flow.In view of the above assumption, Equation (1) gives where e e     is the Hall parameter.Solving for x j and z j , we get On the use of Equations ( 4) and ( 5), the equations of motion along x-and y-directions are where  ,  and are respectively the fluid density, the kinematic coefficient of viscosity and the fluid pres sure.
The boundary conditions are cy of pressure along y-axis.Also, the fluid flow within the channel is induced due to uniform motion of the upper plate fore, using boundary condition at and (8) we get p where is the Hartmann number and Equation ( 7) shows the constan Equations ( 15) and ( 16) can be combined into the fol- where The initial and boundary conditions for On the use of ( 10) and (11), Equations ( 6) and (8), under the usual boundary layer approximations become

Introducing the non-dimensional variables
The boundary conditions (19) become 12) and ( 13) become (14) The solution of the Equation (20) subject to the boundary condition (23) is Taking inverse transform of (24) and on using (18), we get (see Equation ( 25)) where π .

e i e e i e e i e M M
s n On separating into real and imaginary parts one can easily obtain the velocity components and Equation (25).The solution given by ( 25 this case, method used by Carslaw and Jaegar [15] is used since it converges rapidly for small times.F small time  which correspond to large s , Equation (24) e use of On th (18), we have where  and  are given by (26) with Equations ( 30) and (31) show that the Hall effects become important only when terms of order )  is taken into account.

Results and Discussion
To study the effects of suction/blowing, Hall para ion-slip parameter and time on the velocity distrib we have presented the non-dimensional velocity components and against meter, utions   For small values of time, we have drawn the velocity components and on using the exact solution given by Equation (25) and the series solution given by Equations ( 30) and (31) in Figures 7, 8.It is seen that the series solution given by ( 30) and (31) converge more quickly than the exact solution given by (25) for small time.Hence we conclude that for small time, the numerical values of the velocities can be calculated from the Equations ( 30 Copyright © 2012 SciRes.OJFD N. GHARA ET AL. 6 where  and 1  are given by (26).Re .s, the s For small stress at the 0   due to the primary and the secondary flows can be obtained as where For small time, the numerical values of the shear stress components calculated from Equations (34)-(37) are given in Tables 1 and 2 for different values of (38) e  and  .It is observed that for small times the shear stresses calculated from the Equations ( 36) and (37) give better result than that calculated from Equations (34) an (35).d It is seen from Equations ( 43) and ( 44) that there exists a single-deck boundary layer of thickness of the order In this case, the velocity distribut where Equations ( 46) and (47) show the existence of singledeck boundary layer of thickness of order

Single Plate Motion
As , the velocity distribution given by (42) becomes where and ,   are given by (26).It is clear from above Equations (49) and (50) that the flow exhibits a boundary layer behavior with boundary layer thickness of order of Equations ( 52) and (53) coincide with Equat ons (36) and (37) of Gupta [16]

Figure 1 .v
Figure 1.Geometry of the problem.
Figure 6s the that the prignitude


of Equation (25).The non-dimensional shear stresses due to the primary d the secondary velocities at the stationary plate 0
these bo ary layers increases with increases in either Hall parameter or ion-slip parameter.Further, it is seen that the shear