Combined Effects of Hall Currents and Rotation on Steady Hydromagnetic Couette Flow

We study a steady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid in a rotating system between two infinitely long parallel plates in the presence of a uniform transverse magnetic field on taking Hall Current into account. The governing equations describing the flow are solved analytically. It is observed that the Hall currents accelerate the primary velocity whereas they retard the secondary velocity. The induced magnetic field is significantly affected by the Hall currents. An increase in Hall currents leads to fall in the fluid temperature. The heat transfer characteristics have also been studied. The rate of heat transfer at the lower plate decreases whereas the rate of heat transfer at the upper plate increases with an increase in Hall parameter. The asymptotic behavior of the solutions are discussed for small and large values of magnetic parameter and rotation parameter. It is interesting to note that either for strong magnetic field or for large rotation there exists a single-deck boundary layer in the region near the stationary plate. The thickness of this boundary layer first decreases, reaches a minimum and then increases with an increase in Hall parameter.


INTRODUCTION
Couette flows find widespread applications in geophysics, planetary sciences and also in many areas of industrial engineering.For many decades engineers have studied such flows with and without rotation and also for both the steady and unsteady cases.In the ionized gases, the current is not proportional to the applied potential except when the electric field is very weak.However, in the presence of strong electric field, the electrical conductivity is affected by the magnetic field.Consequently, the conductivity parallel to the electric field is reduced.Hence, the current is induced in the direction normal to both the electric and magnetic fields.This phenomenon well known in literature is known as the Hall effects.Due to these Hall Currents the electrical conductivity of the fluid becomes anisotropic and this causes secondary flow in magnetohydrodynamic primary flows.Hall Currents are of great importance in many astrophysical problems, Hall accelerator and flight MHD as well as flows of plasma in a MHD power generator.The study of the interaction of the Coriolis force with the electromagnetic force is also important.In particular, rotating MHD flows with heat transfer is one of the important current topics because of its applications in thermofluid transport modeling in magnetic geosystems and in some astrophysical problems.Hartmann and Lazarus (1937) have investigated the influence of a transverse uniform magnetic field on the flow of a viscous incompressible electrically conducting fluid between two infinite parallel stationary and insulating plates.The problem is extended in numerous ways.Closed form solutions for the velocity fields have been obtained by Tao (1960), Alpher (1961), Sutton and Sherman (1965) and Cramer and Pai (1973) under different physical effects.In the above mentioned cases the Hall term was ignored in applying Ohm's law as it has 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.Under these conditions, the Hall Currents are important and they have a significant effect on the magnitude and direction of the current density and consequently on the magnetic force.Hall effects on hydromagnetic Couette flow and heat transfer have been studied by Gupta (1972).Soundalgekar et al. (1974) have obtained the Hall effects on generalized MHD Couette flow with heat transfer.MHD Couette flow and heat transfer in a rotating system have been studied by Jana et al. (1977).Seth and Maiti (1982), Mandal et al. (1982) and Seth et al. (1982Seth et al. ( , 1985Seth et al. ( , 2009) ) have studied MHD Couette flow of a rotating system in the presence of a uniform transverse magnetic field considering different aspects of the problem.Ghosh (2002) has studied the effects of Hall current on MHD Couette flow in a rotating system with arbitrary magnetic field.Das et al. (2008) have investigated Couette flow in a viscous incompressible fluid in a rotating system in the absence of magnetic field.Seth et al. (2009) have discussed MHD Couette flow of a viscous incompressible electrically conducting fluid in the presence of a uniform magnetic field neglecting induced magnetic field.Hall effects on MHD Couette flow between two infinite horizontal parallel porous plates in a rotating system under the boundary layer approximations have been studied by Das et al. (2011).Seth et al. (2011) have obtained steady MHD Couette flow of a viscous incompressible electrically conducting fluid in a rotating system.
In the present study, we have studied the effects of Hall Current and rotation on the MHD Couette flow of a viscous incompressible electrically conducting fluid in a rotating system in the presence of a uniform transverse magnetic field on taking induced magnetic field into account.The lower plate is perfectly conducting whereas the upper plate is non-conducting.The upper plate is moving with a constant velocity U 0 while the lower plate is kept stationary.It is found that both the primary velocity u 1 and the secondary velocity v 1 decrease with an increase in magnetic parameter M 2 .It is also found that both the velocities increase with an increase in rotation parameter K 2 .It is seen that the primary velocity u 1 increases whereas the secondary velocity v 1 decreases with an increase in Hall parameter m.It is observed that both the primary and secondary induced magnetic field components b x and b y decrease with an increase in magnetic parameter M 2 whereas they increase with an increase in rotation parameter K 2 .It is also seen that the primary induced magnetic field component b x decreases while the secondary induced magnetic field component b y increases with an increase in Hall parameter m.It is found that the shear stress at the lower plate due to the primary flow increases while the shear stress at the lower plate due to the secondary flow decreases with an increase in Hall parameter.The heat transfer characteristics have also been studied on taking viscous and Joule dissipations into account.The rate of heat transfer at the lower plate decreases while the rate of heat transfer at the upper plate increases with an increase in Hall parameter.

Formulation of the problem and its solutions:
The basic equations of magnetohydrodynamics for steady flow are: together with generalized Ohm's law taking Hall current into account is (Cowling, 1957): where, , , , , Ω are respectively the velocity vector, the magnetic field vector, the electric field vector, the current density vector and angular velocity.Also , v, , ρ , p, B 0 , and are the electric conductivity, kinematic coefficient of viscosity, magnetic permeability, fluid density, modified fluid pressure including centrifugal force, applied magnetic field, cyclotron frequency and electron collision time respectively.In writing the Eq. ( 6), the ion-slip and the thermoelectric effects as well as the electron pressure gradient are neglected.
Consider the viscous incompressible electrically conducting fluid bounded by two infinite horizontal parallel plates separated by a distance d.Choose a Cartesian co-ordinates system with x -axis along the lower stationary plate in the direction of the flow, the zaxis is normal to the plates and the y-axis is perpendicular to xz-plane (Fig. 1).The lower plate is perfectly conducting whereas the upper plate is nonconducting.The upper plate is moving with a uniform velocity U 0 while the lower plate is held at rest.The plate and the fluid rotate in unison with uniform angular velocity Ω about an axis perpendicular to the plates.A uniform magnetic field B 0 is applied in the positive zdirection.Since the plates are infinitely long, all physical variables, except pressure, depend on z only.
The equation of continuity • = 0 with no-slip condition at the plates gives w = 0 everywhere in the flow where ≡ (u, v, w).The solenoidal relation • = 0 gives B z = constant = B 0 everywhere in the flow where ≡ (B x , B y , B 0 ).
x y The corresponding boundary conditions for F(η) and b(η) are: The solutions of the Eq. ( 21) and ( 22) subject to the boundary conditions ( 24) are: where, On separating into real and imaginary parts one can easily obtain the velocity components u 1 and v 1 from Eq. ( 25) and the induced magnetic field components b x and b y from Eq. ( 26).If m = 0, then the Eq. ( 25) and ( 26) are identical with the Eq. ( 23) and ( 24) of Seth et al. (2011).

RESULTS AND DISCUSSION
We have plotted the non-dimensional primary and secondary velocities u 1 and v 1 against η for several values of magnetic parameter M 2 , rotation parameter K 2 and Hall parameter m in Fig. 2 to 4. Figure 2 displays that both the primary velocity u 1 and the secondary velocity v 1 decrease with an increase in magnetic parameter M 2 which implies that magnetic field has retarding influence on both the primary and the secondary flow.Figure 3 shows that the primary velocity u 1 increases with an increase in rotation parameter K 2 whereas the secondary velocity v 1 increases in the vicinity of the lower plate and decreases away from the lower plate with an increase in rotation parameter K 2 .The rotation parameter K 2 defines the relative magnitude of the Coriolis force and the viscous force in the regime, therefore it is clear that the high magnitude Coriolis forces are counterproductive for the primary flow.It is observed from Fig. 4 that the primary velocity u 1 increases whereas the secondary velocity v 1 decreases with an increase in Hall parameter m.This implies that Hall currents accelerate the primary flow and retard the secondary flow.Also we have presented the non-dimensional primary and secondary induced magnetic field components b x and b y against η for several values of magnetic parameter M 2 , rotation parameter K 2 and Hall parameter m in Fig. 5 to 7. It is revealed from Fig. 5 that both the induced The non-dimensional shear stresses due to the primary and the secondary flows at the lower plate (η = 0) and the upper plate (η = 1) are, respectively: where λ is given by ( 27).
Numerical results of the shear stresses at the plates (η = 0) and (η = 1) are depicted in Fig. 8 to 11 against m for various values of magnetic parameter M 2 and rotation parameter K 2 .Figure 8 shows that at the lower plate (η = 0) both the shear stress τ due to the primary flow and the shear stress τ due to the secondary flow decrease with an increase in magnetic parameter M 2 .This implies that the magnetic field has tendency to reduce the primary shear stress as well as the secondary shear stress at the lower plate.It is seen from Fig. 9 that at the lower plate (η = 0) both the shear stresses τ and τ increase with an increase in rotation parameter K 2 .This means that rotation has tendency to enhance the primary as well as the secondary shear stresses at the lower plate.Further, it is observed from Fig. 10 that at the upper plate (η = 1) both the shear stress τ due to the primary flow and the shear stress τ due to the secondary flow increase with an increase in magnetic parameter M 2 which implies that the magnetic field has tendency to enhance the primary shear stress as well as the secondary shear stress at the upper plate.It is seen from Fig. 11 that at the upper plate (η = 1) both the shear stresses τ and τ decrease with an increase in rotation parameter K 2 which implies that rotation has tendency to reduce the primary as well as the secondary shear stresses at the upper plate.Further it is observed from Fig. 8 and 9 that τ increases and τ decreases with an increase in Hall parameter m. Figure 10  primary shear stress and to enhance the secondary shear stress at the upper plate.
We shall now discuss the asymptotic behavior of the solutions ( 25) and ( 26) for small and large values of M 2 and K 2 : Case I: When M 2 <<1 and K 2 <<1.In this case, the Eq. ( 21) and ( 22) become: (1 4 4 ) ... , 12 (1 4 4 ) ... .12 It is observed from Eq. ( 30) to (33) that for small values of K 2 and M 2 , the primary velocity is independent of the rotation parameter K 2 but depend on magnetic parameter and Hall parameter.On the other hand, secondary velocity, primary and secondary induced magnetic field components depend on the rotation parameter as well as magnetic parameter and Hall parameter.
Case II: When K 2 >>1 and M 2 ≈ O (1).For the boundary layer flow adjacent to the upper plate η = 1, we introduce ξ = 1 -η and then the Eq. ( 21) and ( 22) give: (1 ) 4 ( 1) where, It is seen from the Eq. ( 34) and ( 35) that there exists a single-deck boundary layer of thickness of the order О(1/α 1 ) where α 1 is given by (38).It is observed that the thickness of this boundary layer decreases with an increase in magnetic parameter M 2 .Figure 12 shows that the boundary layer thickness first decreases, reaches a minimum and then increases with an increase in Hall parameter m while the thickness of this boundary layer has oscillatory in nature with an increase in rotation parameter K 2 .
It is seen that the exponential terms in (34) to (37) damp out quickly as ξ increases.When ξ≥1/α 1 i.e., outside the boundary layer region, the velocities and induced magnetic field components become: Case III: When M 2 >>1 and K 2 ≈ O (1).In this case, the Eq. ( 21) and ( 22) give: It is seen from the Eq. ( 41) and ( 42) that there exists a single-deck boundary layer of thickness of the order О(1/α 2 ) where α 2 is given by (45).It is revealed that the thickness of this boundary layer increases with increase in Hall parameter m but it decreases with an increase in magnetic parameter M 2 .It is interesting to note that for large magnetic parameter, that is, for strong magnetic field the boundary layer thickness is independent of rotation parameter K 2 .
It is seen that the exponential terms in ( 41) to (44) damp out quickly as ξ increases.When ξ≥1/α 2 i.e., outside the boundary layer region, the velocities and induced magnetic field components reduce to: Heat transfer: The energy equation for the fully developed flow including the viscous and Joule dissipations is: where, k : The thermal conductivity μ : The dynamic viscosity The temperature boundary conditions are: where, T 0 and T 1 (T 1 T 0 ) denote the uniform temperature of the plates at z = 0 and z = d respectively.Introducing: and on the use of ( 16) and ( 23), the energy Eq. ( 48) can be written in a dimensionless form as: where, Pr is the Prandtl number, the Eckert number and C P the specific heat at constant pressure .The corresponding temperature boundary conditions for θ(η) become: (0) 0 and (1) 1.

 
Using Eq. ( 25) and ( 26), the solution of Eq. ( 51) subject to the boundary conditions ( 52) is: where, 2 2 1 2 cosh , sinh 1 (1 ) The effects of magnetic parameter M 2 , rotation parameter K 2 , Hall parameter m and Eckert number Ec on the temperature distribution have been shown in Fig. 13 to 16.It is seen from Fig. 13 to 14 that the fluid temperature θ(η) increases with an increase in either magnetic parameter M 2 or rotation parameter K 2 .Figure 15 shows that the fluid temperature θ(η) decreases with an increase in Hall parameter m.It is observed from Fig. 16 that increasing values of Eckert number Ec is to increase the fluid temperature distribution in flow region.This is due to the heat energy stored in the liquid because of the frictional heating.
The numerical values of the rate of heat transfer θ′ (0) and θ′ (1) are entered in the Tables 1 to 3 for different values of magnetic parameter M 2 , rotation parameter K 2 , Eckert number Ec and Hall parameter m.It is seen from Table 1 that the rate of heat transfer decreases both at the lower plate and the upper plate with an increase in magnetic parameter M 2 for fixed value of Hall parameter m.It is observed from Table 2 and 3 that the rate of heat transfer increases at the lower plate (η = 0) while it decreases at the upper plate (η = 1) with an increase in either rotation parameter K 2 or Eckert number Ec for fixed value of Hall parameter m.Further, it is observed from Table 1 to 3 that the rate of heat transfer decreases at the lower plate whereas it increases at the upper plate with an increase in Hall parameter m for fixed values of magnetic parameter M 2 , rotation parameter K 2 and Eckert number Ec .
Critical Eckert number at the lower plate η = 0 and at the upper plate η = 1 can be obtained from the Eq. ( 55) and (56) as: where λ is given by ( 27), A, ̅ , B, , P, Q, R, X and Y are given by (54).
The numerical values of the Critical Eckert number (Ec) η-o and (Ec) η-1 for several values of M 2 , K 2 and m are shown in Table 4 and 5 respectively.It is observed from Table 4 that the absolute value of the Critical Eckert number at the lower plate η = 0 increases while the Critical Eckert number at the upper plate η = 1 decreases on increasing magnetic parameter M 2 for fixed values of Hall parameter m.It is seen from Table 5 that the absolute value of the Critical Eckert number at the lower plate decreases and the Critical Eckert number at the upper plate also decreases with an increase in rotation parameter K 2 for fixed values of Hall parameter m.Further Table 4 and 5 show that the absolute value of the Critical Eckert number at the lower and the upper plates increases with an increase in   5: Critical eckert number at the lower and upper plates when M 2 = 10 and Pr = 3 -(Ec)η = 0 -----------------------------------------------------------------------(Ec)η = 1 --------------------------------------------------------------------- Hall parameter m for fixed values of magnetic parameter M 2 and rotation parameter K 2 .It follows from ( 55) that heat will flow from the lower plate to the fluid if Ec (Ec) η-o , while heat will start flowing from the fluid to the lower plate if Ec (Ec) η-o .Conversely, it follows from (56) that heat will flow from the fluid to the upper plate if Ec (Ec) η=1 , while heat will start flowing from upper plate to fluid if Ec (Ec) η=1 .When Eckert number Ec becomes Critical Eckert number then there is no flow of heat either from fluid to the plates or that from plates to fluid.
This reversal of heat flow may be explained on physical grounds.It is observed that in case there is significant viscous dissipation near the plate, the temperature of the fluid near the plate may exceed plate temperature and for this reason heat will flow from the fluid to the plate even if plate temperature is higher than the ambient temperature.It is noted that in our heat transfer analysis Eq. ( 48) we have considered the effects of both viscous dissipation and Joule dissipation into account and hence there is a strong reason for the flow of heat from the fluid to the plates under certain conditions.

CONCLUSION
Hall effects on MHD Couette flow between infinite horizontal parallel plates in a rotating system under boundary layer approximation have been studied.The magnetic field has retarding influence on the velocity field as well as the induced magnetic field.Rotation has accelerating influence on the velocity field as well as the induced magnetic field.Hall parameter m tends to accelerate the primary fluid flow and to decelerate the secondary flow.On the other hand, it has retarding influence on primary magnetic field and accelerating influence on secondary magnetic field.The fluid temperature increases with an increase in either magnetic parameter M 2 or rotation parameter K 2 or Eckert number Ec while it decreases on increasing Hall parameter m.In the lower plate the primary and the secondary shear stresses decrease with an increase in M 2 whereas they increase on increasing K 2 .Further, the rate of heat transfer at the lower plate decreases whereas the rate of heat transfer at the upper plate increases with an increase in Hall parameter m.There exists a single-deck boundary layer in the region near the stationary plate for large values of M 2 and K 2 .The boundary layer thickness first decreases, reaches a minimum and then increases with an increase in Hall parameter.

Fig
Fig. 11: Shear stresses and for K 2 when M 2 = 10

Table 3 :
Rate of heat transfer at the lower and upper plates when K 2 = 5, Pr = 0.71 and M 2 = 10 ′ 0 -

Table 4 :
Critical eckert number at the lower and upper plates when K 2 = 10 and Pr