UNSTEADY
MHD FREE-CONVECTIVE
POROUS
MEDIUM
WITH
FLOW THROUGH
A
HALL CURRENTS
(Letter to the Editor)
P. C. R A M
Department of Mathematics, Kenyatta University, Nairobi, Kenya
(Received 18 May, 1988) Abstract. The two-dimensional unsteady free-convective flow through a porous medium bounded by an infinite vertical plate for an incompressible viscous and electrically conducting fluid is considered, when a strong magnetic field is imposed in a direction which is perpendicular to the free stream and makes an angle to the vertical direction. The effects of Hall currents on the flows are studied for various values of c~.
1. Introduction
In an ionised gas where the density is low and/or the magnetic field is very strong, the conductivity normal to the magnetic field is reduced due to the free spiralling 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 magnetohydrodynamic viscous flows with Hall currents has important engineering applications in problems of magnetohydrodynmnic generators and of Hall accelerators as well as in flight magnetohydrodynamics. Recently, Raptis and Perdikis (1981) and Raptis et al. (1985) studied the steady and unsteady flows of viscous fluid through a porous medium bounded by a porous surface subjected to a constant suction velocity by taking into account the presence of the free-convection currents. D atta and Mazumder (1976) studied the hydromagnetic freeconvective flow with Hall currents, when the imposed magnetic field is normal to the plate. Hence, the purpose of the present investigation is to study the effects of Hall currents on free-convective flow through a porous medium bounded by an infinite vertical plate, when a strong magnetic field is imposed in a direction which is perpendicular to the free stream and makes an angle e to the vertical direction. Consider the x-axis along the vertical limiting surface in the upward direction and the y-direction normal to it. A uniform magnetic field H = (0, Ho2, H o x/1 - 22), where 2 = cos c~ is imposed on the fluid. Since the plate is infinite in extent, all physical quantities, except pressure, are functions o f y ' and t' only. The equation of continuity 7 . q = 0 gives v = - Vo(Vo > 0), where q = (u, v, w). When the strength of magnetic field is very large, the generalized Ohm's law in the absence of the electric field (Cowling, 1957) is of the form weTe J x H = a J + //o2
V'pe
#eq•
Astrophysics and Space Science 149 (1988) 171-174. 9 1988 by Kluwer Academic Publishers.
en~
,
(1)
172
P . c . RAM
where a, #e, We, Te, e, ne, and Pe are the electric conductivity, the magnetic permeability, the cyclotron frequency, the electron collision time, the electric charge, the number density of the electron, and the electron pressure, respectively. Neglecting the electron pressure, thermoelectric pressure and i0nslip, from (1) we have ffHol.t e,~
Jx' -
[m(u' - U')2 - w ' ] ,
1 + m2)],2
ffHol~e2 [(u' - U) + m 2 w ' ] , 1 + m222
Jz" -
where m = we T e is the Hall parameter. We now consider further the case of a short circuit problem in which the applied electric field E = 0. Under these assumptions, the non-dimensional form of the equation of motion and energy reduce to
a Sy2]q + M ' ( q -
Ot
1
Oy
~0
O0
02 0
4 p ~ t t - p Oy
Oy2
U)
= -41-dUdt -+60,
(2) (3)
where y-
y' Vo
,
v
U-
U'
,
t pvo2 t=-, 4v
0-
Uo
T' T"-
2 2 M 2 - avpeH6 pv~ '
1,11
4vw' w=-, v~ ' T~ , T"
P=--
vpCp
,
G=
Uo vgfl(T~- T~)
k
M222(1 - irn2) 1 22m 2 +- , 1+ k
M1=
W
u=-, Uo
,(4)
Uov ~ v~k'
k-
v 2'
and q = u + iw. The corresponding boundary conditions are: q=O, q~U(t),
0=1
0=0
at
y=O,
as
y~oc.
In order to solve the differential equations (2) and (3), we assume that q = (1 - qo) + e(1
-
q l ) e iwt +
U = 1 + ~ e iwt q- O ( ~ 2 ) , 0 = 00 -Jr- ~01 e iwt +
O(e2) 9
O(/32) ,
(5) (6) (7)
UNSTEADY
MHD
FREE-CONVECTIVE
173
FLOW
Substituting Equations (5)-(7) in Equations (2) and (3), and equating the coefficients of different powers of ~, we get the solution under modified boundary conditions as qo = (1 - a l ) e - ~'Y + a 1 e
-py
(8)
,
q, = e-~2Y,
(9)
0o = e - p y ,
(10)
01 = 0 ;
(11)
where
'[1 + , f / + 4M,]
4(M, + i~],
~2 = 89 + ~/1 +
a 1 = G/(p 2-p-M1).
1.6
o( = 6 0 ~
-
~.2 o~=
45 ~
0.8
0.4
0
iI.O
t 2'0
i 3"0
I 4"0 y ~
I 5"0
i 6'0
Fig. 1. Profiles of non-dimensional velocity u.
1.0
0'8
o~ = 6 0 ~ 0"6
=
=
0.4
0"2
I
I'O
2-0
3"0
4.0
.5,0
6-0
Y
Fig. 2.
Profiles of non-dimensional velocity w.
7"0
174
P, C. RAM
2. Discussion of the Results The velocity distributions u and w are plotted against y in Figures 1 and 2 for M 2 = 5.0, G -- 5.0, k = 5.0, e = 0.01, m = 2.0, and P = 0.71, which corresponds to the air for different values of a. It is seen that both u and w increases with the increase o f a. F o r 2 = 1, the imposed magnetic field is parallel to the y-axis. In this case, neglecting the permeability term Equations (2) and (3) are the same as Equations (23) and (24) of D a t t a and M a z u m d e r (1976). Therefore, velocity and heat transfer in their problem will be the same as those of our problem. For M = 0 and m = 0, the velocity distributions are the same as Raptis and Perdikis (1985). For the steady case the velocity distributions coincide with results of Raptis et aL (1981), neglecting the Joule heating term.
References Cowling, T. G.: 1957,Magnetohydrodynamics, Interscience Publications, New York. Datta, N. and Mazumder, B. S.: 1976, J. Math. Phys. Sci. 10, 59. Raptis, A. and Perdikis, C.: 1985, Int. J. Eng. Sci. 23(1), 51. Raptis, A., Perdikis, C., and Tzivanidis, G.: 1981, J. Phys. D: Appl. Phys. 14, L99.
