Effects of hall currents on MHD flow of a couple stress fluid through a porous medium in a parallel plate channel in presence of effect of incliened magnetic field.

Sarojini, M. Syamala ; Krishna, M. Veera ; Shankar, C. Uma 等

Introduction

In recently years the hydro magnetic flow in a rotating channel in the presence of an applied uniform magnetic field as well as constant pressure gradient has been considered by a number of research workers, taking into account the various aspects of the problem. The channel flow problems where the flow is maintained by torsional or non-torsional oscillations of one or both the boundaries, threw some light in finding out the growth and development of boundary layers associated with the flows occurring in geothermal phenomena. D.V. Krishna et. al [4] studied the hydro magnetic convection flow of a viscous electrically conducting fluid through a porous medium in a rotating parallel plate channel. Later M. Guria et. al [3] studied the unsteady couette flow of a viscous incompressible fluid confined between parallel plates, rotating with an uniform angular velocity about an axis normal to the plates, here the flow was induced by the motion of the upper plate and the fluid and plates rotate in unison with the same angular velocity. Claire Jacobs [1] studied the transient effects considering the small amplitude torsional oscillations of disks. This problem had been extended to the hydro magnetic case by Vidyanidhi [10], who discussed torsional oscillations of the disks maintained at different temperatures. Debnath [2] considered an unsteady hydrodynamic and hydro magnetic boundary flow in a rotating viscous fluid due to oscillations of plates including the effects of uniform pressure gradients and uniform suction. The structure of the velocity field and the associated Stokes, Ekman and Rayleigh boundary layers on the plates are determined for the resonant and non-resonant cases. Rao.D.R.V., Krishna.D.V. & Debanath, Rao et. al [6] have made an initial value investigation of the combined free and forced convection effects in an unsteady hydro magnetic viscous incompressible rotating fluid between two disks under a uniform transverse magnetic field. This analysis has been extended to porous boundaries by Sarojamma and Krishna [7], and later by Siva Prasad [8] to include the Hall current effects. Veera Krishna et. al [9] discussed the steady hydro magnetic flow of a couple stress fluid through a porous medium in a rotating parallel plate channel under the influence of a uniform transverse magnetic field making use of Brinkman's model. In this chapter we discuss the steady hydro magnetic flow of a couple stress fluid in a parallel plate channel through a porous medium under the influence of a uniform inclined magnetic field of strength [H.sub.o] inclined at an angle of inclinations [alpha] with the normal to the boundaries and taking into hall current.

Formulation and Solution of the Problem

We consider an incompressible viscous and electrically conducting couple stress fluid in a parallel plate channel bounded by a porous medium and taking hall current into account. The fluid is driven by a uniform pressure gradient parallel to the channel plates and the entire flow field is subjected to a uniform inclined magnetic field of strength [H.sub.0] inclined at an angle of inclinations [alpha] with the normal to the boundaries in the transverse xy-plane. In the equation of motion along x-direction the x-component current density--[[mu].sub.e][J.sub.x][H.sub.o] and the z-component current density [[mu].sub.e][J.sub.x][H.sub.o].

We choose a Cartesian system O(x, y, z) such that the boundary walls are at z=0 and z=l and are assumed to be parallel to xy-plane. The steady flow through porous medium is governed by Brinkman's equations. At the interface the fluid satisfies the continuity condition of velocity and stress. The boundary plates are assumed to be parallel to xy-plane and the magnetic field of strength [H.sub.o] inclined at an angle of inclinations [alpha] to the z-axis in the transverse xz-plane. This inclined magnetic field on the axial flow along the x-direction gives rise to the current density along y-direction in view of Ohm's law. Also the inclined magnetic field in the presence of current density exerts a Lorentz force with components along O(x, z) direction, The component along z-direction induces a secondary flow in that direction while its x-components changes perturbation to the axial flow. The steady hydro magnetic equations governing the couple stress fluid under the influence of a uniform inclined magnetic field of strength [H.sub.o] inclined at an angle of inclinations with reference to a frame are

[eta]/[rho] [d.sup.4]u/[dz.sup.4] = 1/p [partial derivative]p/[partial derivative]x + v [d.sup.2]u/[dz.sup.2] - [[mu].sub.e][J.sub.z][H.sub.0]Sin[alpha]/[rho] - v/k u (2.1)

[eta]/[rho] [d.sup.4]w/[dz.sup.4] = v [d.sup.2]w/[dz.sup.2] + [[mu].sub.e][J.sub.z][H.sub.0]Sin[alpha]/[rho] - v/k u (2.2)

Where, (u, w) are the velocity components along O(x, z) directions respectively. [rho] is the density of the fluid, [[mu].sub.e] is the magnetic permeability, v is the coefficient of kinematic viscosity, k is the permeability of the medium, [H.sub.o] is the applied magnetic field. When the strength of the magnetic field is very large, the generalized Ohm's law is modified to include the Hall current, so that

J + [[omega].sub.e] [[tau].sub.e]/[H.sub.0] J x H = [sigma] (E + [[mu].sub.e] q x H) (2.3)

Where, q is the velocity vector, H is the magnetic field intensity vector, E is the electric field , J is the current density vector, [[omega].sub.e] is the cyclotron frequency, [[tau].sub.e] is the electron collision time, [sigma] is the fluid conductivity and, [[mu].sub.e] is the magnetic permeability. In equation (2.3) the electron pressure gradient, the ion-slip and thermo-electric effects are neglected. We also assume that the electric field E=0 under assumptions reduces to

[J.sub.x] - m [J.sub.z] Sin [alpha] = - [sigma] [[mu].sub.e] [H.sub.0] w Sin[alpha] (2.4)

[J.sub.z] + m [J.sub.x] Sin [alpha] = - [sigma] [[mu].sub.e] [H.sub.0] u Sin[alpha] (2.5)

where m = [[omega].sub.e] [[tau].sub.e] is the Hall parameter.

On solving equations (2.3) and (2.4) we obtain

[J.sub.x] = [sigma] [[mu].sub.e] [H.sub.0] Sin[alpha] / 1 + [m.sup.2] [Sin.sup.2] [alpha] (umSin[alpha] - w) (2.6)

[J.sub.z] = [sigma] [[mu].sub.e] [H.sub.0] Sin[alpha] / 1 + [m.sup.2] [Sin.sup.2] [alpha] (u + wmSin[alpha]) (2.7)

Using the equations (2.6.) and (2.7), the equations of the motion with reference to frame are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

Let q = u + iw

Now combining the equations (2.8) and (2.9), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

The boundary conditions are

q = 0 , at z = 0 (2.11)

q = 0 , at z = l (2.12)

[d.sup.2]q / [dz.sup.2] = 0, at z = 0 (2.13)

[d.sup.2]q / [dz.sup.2] = 0, at z = l (2.14)

We introduce the non-dimensional variables

z* = z/l , q* = ql/v, [q.sub.p]* = [q.sub.p]l/v, P* = [Pl.sup.2]/[rho][v.sup.2], h* = h/l, [xi]* = [xi]/l.

Using the non-dimensional variables, the governing non-dimensional equations are (dropping asterisks)

S [d.sup.4]q / [dz.sup.4] - [d.sup.2]q / [dz.sup.2] + ([M.sup.2][Sin.sup.2][alpha](1- imSin[alpha])/1+[m.sup.2][Sin.sup.2][alpha] + [D.sup.-1]) q = P (2.15)

Where,

[M.sup.2] = [sigma][[mu].sub.e.sup.2][H.sub.0.sup.2][l.sup.2]/[rho]v is the Hartmann number,

M = [[omega].sub.e] [[tau].sub.e] is the Hall Parameter,

[D.sup.-1] = [l.sup.2]/k is the inverse Darcy parameter,

S = [eta]/[[rho]1.sup.2]v is the Couple stress parameter,

P = - [partial derivative]p / [partial derivative]x is the imposed pressure gradient.

Corresponding boundary conditions are

q = 0 , at z = 0 (2.16)

q = 0 , at z = 1 (2.17)

[d.sup.2]q / [dz.sup.2] = 0, at z = 0 (2.18)

[d.sup.2]q / [dz.sup.2] = 0, at z = 1 (2.19)

Solving the equation (2.15) making use of the boundary conditions, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.20)

The shear stresses on the upper plate and lower plate are given by

[[tau].sub.U] = [(dq/dz).sub.z=1] and [[tau].sub.L] = [(dq/dz).sub.z=0]

Where, the constants A, B, C and D are mentioned in appendix. Results and Discussion

The velocities representing the ultimate flow have been computed numerically for different sets of governing parameters namely viz. The Hartmann parameter M, the inverse Darcy parameter [D.sup.-1], couple stress parameter S and hall parameter m, and their profiles are plotted in figures (1-4) and (5-8) for the velocity components u and v respectively. For computational purpose we have assumed an angle of inclination [alpha] and the applied pressure gradient in the x-direction and are fixed. Since the thermal buoyancy balances the pressure gradient in the absence of any other applied force in the direction, the flow takes place in planes parallel to the boundary plates. However the flow is three dimensional and all the perturbed variables have been obtained using boundary layer type equations, which reduce to two coupled differential equations for a complex velocity.

We notice that the magnitude of the velocity component u reduces and v increases with increasing the intensity of the magnetic field M the other parameters being fixed, it is interesting to note that the resultant velocity experiences retardation with increasing M (Fig. 1 & 4). (Fig. 2 & 6) exhibit both the velocity components u and v reduces with increasing the inverse Darcy parameter [D.sup.-1]. Lower the permeability of the porous medium lesser the fluid speed in the entire fluid region. The resultant velocity experiences retardation with increasing the inverse Darcy parameter D-1. Here we observe that the retardation due to an increase in the porous parameter is more rapid than that due to increase in the Hartmann number M. In other words, the resistance offered by the porosity of the medium is much more than the resistance due to the magnetic lines of force. We notice that u exhibits a great enhancement in contrast to v which retards appreciably with increase in the couple stress parameter S, but the resultant velocity shows and appreciable enhancement with in S (Fig. 3 & 7). We also notice that the magnitude of the both velocity components u and v increase with increasing the hall parameter m the other parameters being fixed, it is interesting to note that the resultant velocity experiences maximum enhancement with increasing m (Fig. 4 & 8).

The shear stresses on the upper and lower plates and the discharge between the plates are calculated computationally and tabulated in the tables (I-V). The magnitude of these stresses at the upper plate is significantly high compared to the respective magnitudes at the lower plate. We notice that the magnitude of the both stresses [[tau].sub.x] and [[tau].sub.y] enhances in the upper plate and lower plates with increasing M, S and m, while on the upper plate [[tau].sub.x] and [[tau].sub.y] reduces and on the lower plate [[tau].sub.x] rapidly enhances and [[tau].sub.y] reduces with increase in the inverse Darcy parameter [D.sup.-1]. The retardation at the upper plate is significantly low compared to enhancement at the lower plate (Tables. I-IV). The discharge Q reduces in general with increase in the intensity of the magnetic field M and lower permeability of the porous medium (corresponding to an increase in [D.sup.-1]) and enhances the couple stress parameter S and m (Table. V).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Conclusions

1. The magnitude of the velocity component u reduces and v increases with increasing the intensity of the magnetic field M, also the resultant velocity experiences retardation with increasing M.

2. Both the velocity components u and v reduces with increasing the inverse Darcy parameter [D.sup.-1]. Lower the permeability of the porous medium lesser the fluid speed in the entire fluid region. The resultant velocity experiences retardation with increasing the inverse Darcy parameter [D.sup.-1].

3. We observe that the retardation due to an increase in the porous parameter is more rapid than that due to increase in the Hartmann number M. In other words, the resistance offered by the porosity of the medium is much more than the resistance due to the magnetic lines of force.

4. We observed that u exhibits a great enhancement in contrast to v which retards appreciably with increase in the couple stress parameter S, but the resultant velocity shows and appreciable enhancement with in S.

5. We observed that u and v exhibits a great enhancement appreciably with increase in the hall parameter m, likewise the resultant velocity shows enhancement with increase in m.

6. The magnitude of these stresses at the upper plate is significantly high compared to the respective magnitudes at the lower plate.

7. The discharge Q reduces in general with increase in the intensity of the magnetic field M and [D.sup.-1] and enhances with the couple stress parameter S and m.

Appendix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

References

[1] Claire Jacobs. Q. Jour. Maths. Appl. Maths., vol.24 (1971), p-221.

[2] Debnath, L. ZAMM, vol.55 (1975), p-431.

[3] Guria.M, R.N. Jana. S.K. Ghosh., Int. J. of Non- linear mechanics, vol. 91, pp. 838-843.

[4] Krishna .D.V., D.R.V.P. Rao and A. S. Rama Chandra Murthy., Journal of engg. physics and thermophysics, vol. 75 (2002), No.2, pp. 12-21.

[5] Moreau. R. Magneto hydro dynamics, Kluwer Academic publishers, Amsterdam (1990).

[6] Rao, D.R.V.P., Acta.Mech., vol. 43 (1982), p. 49.

[7] Sarojamma,G. & Krishna,D.V. Acta.Mech., vol.39 (1981), p. 277.

[8] Siva Prasad.R. Ph.D.,thesis, S.K.University, Anantapur (1985).

[9] Veera Krishna. M., S.V. Suneetha, and S.G. Malashetty, "Steady hydro magnetic flow of a couple stress fluid through a porous medium in rotating parallel plate channel". Ultra Scientist of Physical Sciences, vol. 22(2) M (2010), pp. 369-376.

[10] Vidyanidhi. Jour. Math. Phy .Sci. vol. 3 (1969), p. 193.

M. Syamala Sarojini (1), M. Veera Krishna (2) and C. Uma Shankar (3)

(1), (3)Department of OR&SQC, (2)Department of Mathematics, Rayalaseema University, Kurnool, A.P.-518002, India E-mail: cumaor@rediffmail.com, veerakrishna_maths@yahoo.com

Table I: The shear stresses ([[tau].sub.x]) on the upper plate.

 M I II III IV V

 2 0.004275 0.003865 0.003211 0.004831 1.536526
 5 0.005756 0.005211 0.004839 0.006875 1.836563
 8 0.006336 0.006008 0.005315 0.008365 2.008832
 10 0.006834 0.006331 0.005999 0.009445 5.008265

 I II III IV V

 [D.sup.-1] 1000 2000 3000 1000 1000
 S 1 1 1 2 3
 m 1 1 1 1 1

 M VI VII

 2 0.036562 0.266859
 5 0.065326 0.726652
 8 0.073365 0.855682
 10 0.096652 0.911452

 VI VII

 [D.sup.-1] 1000 1000
 S 1 1
 m 2 3

Table II: The shear stresses ([[tau].sub.y]) on the upper plate.

 M I II III IV V

 2 -0.00665 -0.00633 -0.00608 -0.00831 -0.01652
 5 -0.00746 -0.00708 -0.00633 -0.00953 -0.02365
 8 -0.00836 -0.00766 -0.00683 -0.01834 -0.06536
 10 -0.00911 -0.00833 -0.00783 -0.02008 -0.09336

 I II III IV V

[D.sup.-1] 1000 2000 3000 1000 1000
 S 1 1 1 2 3
 m 1 1 1 1 1

 M VI VII

 2 -0.03657 -0.06536
 5 -0.08326 -0.09652
 8 -0.12085 -0.12652
 10 -0.22653 -0.83115

 VI VII

[D.sup.-1] 1000 1000
 S 1 1
 m 2 3

Table III: The shear stresses ([[tau].sub.x]) on the lower plate.

 M I II III IV V

 2 0.134521 0.226532 0.383669 0.180832 -2.46575
 5 0.246532 0.300652 0.477682 0.468332 -3.66583
 8 0.383665 0.400822 0.588362 0.836522 -4.66586
 10 0.322656 0.433756 0.666832 0.999855 -4.88842

 I II III IV V

[D.sup.-1] 1000 2000 3000 1000 1000
 S 1 1 1 2 3
 m 1 1 1 1 1

 M VI VII

 2 5.465245 8.366525
 5 6.756839 9.666525
 8 8.116535 10.11532
 10 10.11586 11.20834

 VI VII

[D.sup.-1] 1000 1000
 S 1 1
 m 2 3

Table IV: The shear stresses ([[tau].sup.y]) on the lower plate.

 M I II III IV V
 2 -0.26532 -0.03636 -0.00421 -0.45226 -1.00855
 5 -0.36653 -0.04082 -0.00466 -0.83652 -1.24652
 8 -0.42208 -0.05216 -0.00621 -0.94526 -1.85765
 10 -0.48322 -0.05832 -0.00682 -0.99652 -2.32652

 I II III IV V
[D.sup.-1] 1000 2000 3000 1000 1000
 S 1 1 1 2 3
 m 1 1 1 1 1

 M VI VII
 2 -4.32652 -8.66575
 5 -5.26652 -10.0835
 8 -7.33668 -10.9995
 10 -9.44326 -12.0845

 VI VII
[D.sup.-1] 1000 1000
 S 1 1
 m 2 3

Table V: Discharge Q

 M I II III IV V

 2 1.840014 1.514985 1.355847 2.001452 2.455145
 5 1.588749 1.355466 1.144569 1.885469 2.000255
 8 1.302254 1.144541 1.000546 1.665898 1.889872
 10 1.225466 0.011451 0.002145 1.524465 1.622549

 I II III IV V

[D.sup.-1] 1000 2000 3000 1000 1000
 S 1 1 1 2 3
 m 1 1 1 1 1

 M VI VII

 2 2.825451 3.225655
 5 2.522413 2.887989
 8 2.002115 2.452244
 10 1.885474 2.000256

 VI VII

[D.sup.-1] 1000 1000
 S 1 1
 m 2 3