COMBINED EFFECT OF FREE AND FORCED CONVECTION ON MHD FLOW IN A ROTATING POROUS CHANNEL

. This paper gives a steady linear theory of the combined effect of the free and forced convection in rotating hydromagnetic viscous fluid flows in a porous channel under the action of a uniform magnetic field. The flow is governed by the Grashof number G, the Hartmann number H, the Ekman number E, and the suction Reynolds number S. The solutions for the velocity field, temperature distribution, magnetic field, mass rate of flow and the shear stresses on the channel boundaries are obtained using a perturbation method with the small parameter S. The nature of the associated boundary layers is investigated for various values of the governing flow parameters. The velocity, the temperature, and the shear stresses are discussed numerically by drawing profiles with reference to the variations in the flow parameters.

166 D.R.V.P. RAO, D.V. KRISHNA, AND L. DEBNATH of the hydrodynamic thermal convection due to its numerous applications in geo- physics and astrophysics.Several authors including Gupta [I], Jana  [2], Nanda and   Mohanty [3], Mishra and Mudali [4], Mohan [5], Soundalgekar  [6], Yen  [7], and Gill   and Casal [8], have investigated the effects of the free and/or forced convection on hydromagnetic fluid flows confined to non-porous boundaries under various geometrical configurations.Some of these authors have also considered the effects of wall conductance on the heat transfer aspect of hydromagnetic channel flows.It has been shown that the wall conductance exerts a destabilizing influence on the flow whereas the magnetic field stabilizes the flow.
In spite of these studies, relatively less attention has been given to the simultaneous effects of the free and forced convection on the hydromagnetic rotat- ing viscous flows confined to porous boundaries.Such work seems to be important and useful partly for gaining basic understanding of such flows, and partly for possible applications to geophysical and astrophysical problems.The present paper deals with a steady linear theory of the combined effect of the free and forced convection in rotating hydromagnetic viscous fluid flows in a porous channel under the action of a uniform magnetic field.The solutions for the velocity field, temperature distribution, magnetic field, mass rate of the flow and the shear stresses on the channel boundaries are investigated using a perturbation method with the small suction Reynolds number S. The structure of the associated boundary layers is examined for various values of the governing flow parameters.The velo- city, the temperature, and the shear stresses are discussed numerically by drawing profiles with reference to the variations in the flow parameters.

MATHEMATICAL FORMULATION OF THE PROBLEM.
We consider an incompressible electrically conducting viscous steady fluid flows in a channel bounded by two porous non-conducting parallel plates at z + L.
Both the fluid and the boundary plates are in a state of solid body rotation with a uniform angular velocity about the z-axis normal to the plates.We take a Cartesian coordinate system Oxyz rotating with the angular velocity with the x- axis taken in the direction of pressure gradient and the z-axis positive upward.

FREE AND FORCED CONVECTION ON MHD FLOW 167
A uniform magnetic field of strength B parallel to the z-axis is applied to the O fluid system, and the fluid is driven by a constant horizontal pressure gradient.
The present analysis is based on the Boussinesq approximation which implies that e << I, when the density is given by 0 0 (1 e), 0 is the density at the O O temperature T is the temperature variation from T and is the coefficient of thermal expansion.The Boussinesq approximation also implies that the properties of the liquid, , the kinematic viscosity and < the thermal diffusivity, are independent of temperature and hence can be taken as constants.
With the Cartesian coordinate system and the Boussinesq approximation, the unsteady motion of a viscous conducting fluid in the presence of a magnetic field B in this rotating coordinate system is governed by the Navier-Stokes equations, the continuity equation and the energy equation with usual notations u -+ (u_ V) u__ + 2 k u__ -VP* k_g(1 ) +- + (u v)e < v2e 8t (2.3) where the velocity u (u, v, w), P is the pressure including the centrifugal force, j is the current density and g is the acceleration due to gravity.We further assume that j, B and the electric field E_ satisfy the Maxwell equations with usual notations.
The flow is assumed to be fully developed and steady so that all the physical variables except the pressure, depends only on z.It then follows from the con- tinuity equation (2.2) that w =-w a constant.Clearly, w > 0 for suction and   where P1 is the uniform pressure gradient with which the fluid is driven along the x-axis.
It is assumed that the porous plates at z + L of thickness d are cooled or heated by a constant temperature gradient A 1 along the x-axis so that the tempera- ture varies linearly along the plates.Eliminating P* from (2.4) and (2.6), we where O l(z) is an arbitrary function of z.
We next take To + AlX + @I0 and To + AlX + @II as the temperatures at the lower and upper plates respectively.
It is convenient to write down the non-dimensional form of the basic field equations (2.4) (2.9) through the non-dimensional variables (starred) defined by We next combine the non-dimensional equations (2.4) and (2.5)  where F u + iv is the complex velocity field, B B + iB is the complex magne- x y wL tic field, S o__ is the cross-flow Reynolds number, Pm Grashof number, E

L2
is the Ekman number and R P1 -)v  (2.20)   d o B1 Bxl + iByl B2 Bx2 + iBy2 l is the electrical conduct- ivity of the plate, Bxl and By are the values of Bx and By in the vacuum region d d z => + and Bx2 and By2 are the values of Bx and By in the region z =< -I ---L The magnetic field within ,the plates can be obtained using the Maxwell equation J (I E and Curl B eJ (since u_ 0 on the plates).If there is a net current through the channel in the y-direction, the induced electric field exists in the y-direction.In view of this, it is observed that the magnetic field components outside the boundary (in the vacuum) exist only in the x-direction and vanish in the y-direction.The components Bxl and Bx2 are fixed by knowing the net current in the y-direction through the channel and its return path.

SOLUTION OF THE PROBLEM.
We shall adopt a small perturbation method with S, the cross-flow Reynolds number as the perturbation parameter, in order to solve the differential equations (2.13) and (2.14).Thus we assume o.
(3.17)  Substituting F and B in (3.18) and solving the resulting differential equa- tion with the boundary conditions (3.19 ab), (z) can readily be determined.How- ever, since the solution for O(z) is quite complicated, we avoid writing the long expression for it. 4. THE VELOCITY FIELD MAGNETIC FIELD AND THE ASSOCIATED BOUNDARY LAYERS.
In order to investigate the salient feature of the boundary layers, we shall consider the following special cases related to various magnitudes of the Ekman and Hartmann numbers.
To determine the flow field, the magnetic field and the associated boundary layers on the plate z I, it is convenient to write z in (3.11) (3.12), and then take the limit of the resulting expressions as -/ 0 0 such that is finite.This leads to the following results: a a3 S H 2 u + iv a + (a +7) exp (-) (cosB-i sin) -7 (i-) + 4 Case (ii): E << i and H2O(1) In this case, the solutions are obtained from (3.11) (3.12), and have the form ----- where alr ali etc. denote the real and imaginary parts of a I respectively, and  H2Bx Sd7(l )2 + dS(l ) +_ (Bxl + Bx2) The velocity field (4.3) (4.4) shows the existence of the modified Ekman that both components of the velocity field persist, and are functions of the verti- cal coordinate z.This means the violation of the Taylor-Proudman theorem valid in an inviscid rotating fluid flows.Thus the suction or injection at the porous plates prevents the flow to reduce to two-dimensional.However, in the non-porous case the limiting flow satisfies the conditions of the Taylor-Proudman theorem.
In this case, the solutions are given by f3 f3 SG Pm 2 u fl + (f2 +7 exp (-H)-H -- -+ It is noted that the results for this case are completely independent of rotation. 5. THE NUMERICAL RESULTS AND DISCUSSION.
In order to present numerical results and their discussion, it is necessary to assume certain fixed values for Bxl +__ Bx2, R and 6.With Bxl + Bx2 3.0, Bxl Bx2 0.5, R and 0.i, the velocity, the temperature and the shear stresses are discussed numerically.The pressure gradient is taken to be positive to H.The mass flow rate Qx along the x-axis rapidly increases with increase in H for a fixed K as well as increases with increase in K for a fixed H. On the other hand,Qy, for moderate values of K retains the same nature but almost remains uniform for large K.
The non-dimensional shear stresses and at the upper and lower plates x y are plotted in Figures 6-9 for various values of the governing flow parameters S, H,G and K.It is to be noted that the reversed flow occurs whenever the shear stress on the upper and the lower plates are of the same sign.No such separation in the flow arises if the shear stress on the upper plate is opposite to that of the lower plate.Figure 6 shows that remains positive and decreases with in- x crease in H for a fixed G, although increases for increase in G for a fixed H.At the lower plate, it is negative and almost uniform for G 0. But as G in- creases it remains negative for large values of H (say greater than 4) but becomes positive as H approches to zero.This shows that for G z 0 and H 0, reversed velocities appear in the vicinity of the lower plate.However, this region of reversed velocities which grows with increase in G can be made to shrink by choos- ing sufficiently large H.The shear stress behaves similar to with occury x ence of the reversed velocities near the lower plate for small H. Figures 8-9   indicate the behavior of r and r with respect to the variation of S and K.We x y also observe that for K i no reversed flow appears with reference to either u or v for all values of S. When K is large for sufficiently small S, we find its growth near the lower plate.For a fixed S, and r at both the plates go on x y decreasing for an increase in K. Also the shear stresses increase with S for a fixed K.
The non-dimensional temperature profiles are plotted in Figures 10-12 for different sets of parameters.In all these cases the profiles are almost similar except that the concavity increases with either increase in S or G.For increase in either S,G and H it can be noted that the profiles become more and more sharp contributing to the increase in the maximum temperature attained.

2 0
solution (4.1) represents the steady hydromagnetic boundary layer flow which consists of the Ekman-Hartmann layer on the boundary plate.The non-result of Debnath[9] and also with that of Nanda and Mohanty[3] for the case E << and H (i). In particular, the Ekman-Hartmann layer reduces to the Ekman layer of thickness O(E1/2) or the Hartmann layer of thickness that the exponential terms in the above solution decays vewy EH 2 rapidly as increases and exceeds the value /(i +---In the limit the flow field and the magnetic field assumes the form u

7 S
Pm f7 (I )2 + [R 2f4 SG(I + Pm) (I-) 14) represent the steady hydromagnetic boundary -I layers flow which consists of the Hartmann Layer of thickness of the order H

Figure 5
Fig. 4: Fig. 3: and when i, Fig. 5:Mass rate of flow Qx when Substitution of (2.11) into (2.7)yields the energy equation in the non-