H Y D R O M A G N E T I C INSTABILITY O F S T R E A M I N G F L U I D S IN P O R O U S M E D I U M

R. C. Sharma, Veena Kumar i

Department of Mathematics, Himaehal Pradesh University, Shimla-171 005, India

Received 4 June 1990

The instability of the plane interface between two uniform, superposed, electrically conduct- ing and counter-streaming fluids through a porous medium is considered in the presence of a horizontal magnetic field. In the absence of surface tension, perturbations transverse to the direction of streaming are found to be unaffected by the presence of streaming if perturbations in the direction of streaming are ignored. For perturbations in all other directions there:exists instability for a certain wavenumber range. The instability of this system is postponed by the presence of magnetic field. The magnetic field and surface tension are able to suppress this Kelvin-Helmholtz instability for small wavelength perturbations and the medium porosity reduces the stability range given in terms of a difference between the streaming veloeities and the Alfv6n velocity.

1. Introduction

The instability of the plane interface separating two uniform superposed streaming fluids, under varying assumptions of hydrodynamics and hydromagnetics, has been discussed in a treatise by Chandrasekhar [1]. The medium has been assumed to be non-porous.

The flow through a porous medium has been of considerable interest in recent years particularly among geophysical fluid dynamicists. The gross effect when the fluid slowly percolates through the pores of the rock is by Darcy's law which states that the usual viscous term in the equations of fluid motion will be replaced by the resistance term - (l~/kl)q, where # is the viscosity of the fluid, k 1 the permeabili ty of the medium (which has the dimension of length squared), and q the filter velocity of the fluid. The instability of the plane interface between two uniform superposed and streaming fluids through a porous medium has been investigated by Sharma and Spanos [3].

In many geophysical fluid dynamical problems encountered, the fluid is electrically conducting and a uniform magnetic field of the Earth pervades t he system. Therefore the aim of this paper is to study the instability of electrically conducting, streaming: fluids in a porous medium in the presence of a uniform horizontal magnetic field. The normal mode analysis and the linearized perturbation theory have been used.

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R. C. Sharma, Veena Kumari

2. Perturbation equations

The initial stationary state whose stability we wish to examine is that of an in- compressible conducting fluid in which there is a horizontal streaming in the x- direction with a velocity U(z) through a homogeneous porous medium. A uniform horizontal magnetic field pervades the system.

In flows through porous media, there are no sharp fronts and so no actual inter- facial tensions at some prescribed levels z~, as in ordinary fluid dynamics. However, there is a macroscopic interface (broad front) if viewed from a large distance, and by analogy with Laplace's formula, at each point of the macroscopic interface,

(pl - p2) . . . . = - Z ( c l + c2 ) ,

where T~ is the 'effective interfacial tension' and c l, c2 are the signed principal curva- tures of the macroscopic interface. This is the first approximation to the problem since in practice there is no 'effective interfacial tension' but in the absence of any better theory, this is being used as suggested by Chuoke et al. [2]. Let n~ denote a normal to the macroscopic interface. Then Maxwell's equations and the equations of motion, continuity and incompressibility for the fluid through a porous medium are given by

- + - ( u . v ) u = - V p + g

u+--i (VxH) xH-0g~, kl 4~

V . U = O , V . H = O ,

8O (U . = , e - - + V) 0 0 8t

81"t - -=Vx(UxH) , 8t

, , /1 s

(1)

(2)

(3)

, (4)

where U(U(z), O, 0), p, Q, 9, It, kl, e and H(H, 0, 0) denote, respectively, the fluid velocity, fluid pressure, fluid density, acceleration due to gravity, fluid viscosity, permeability of the medium, medium porosity and magnetic field. 5(z - zs) denotes Dirac's delta function and ~ = (0, 0, i).

Let u(u,v, w),h(hx, hy, hz), 8p, 80 and ~zs(x, y , t ) denote the perturbations in velocity O(U(z), 0, 0), magnetic field H, pressure p, density 0 and surface of separation z s, respectively. Then the linearized perturbation equations become

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Hydromagnetic streaming instability,..

( v0o) 0 . " (m 0hx) _Q \Or e- ~xx/ t?y k 1 4n \ Ox Oy ] e g + = - - - a p - - - v + - - , (6)

~ - + - = 8 p - - w + -

Yx b z ~ ~ \ a x a z /

+ T~ ~ x 2 + 8zs 6 ( z - z s ) - g a e , (7)

0u 0v 0w - + - + - = o , (8) ~x 0y ~z

- + v a e = - w - - , (9) 0t dz

ah~ + __Oh, + Ohm__ = O, riO) 0X 0y 0z

e - - + U h = H - - u + h z D U . (11) 0t 0x

In equation (7), 8z, can be expressed in terms of the normal component of the velocity Ws at zs since

+ V~Fx ~Zs = Ws, (,2)

where a subscript "s" distinguishes the value of the quantity at z = z~. In equation (7), use has been made of the approximations

0 2 0 2 c, ~ - - - ( S z s ) c2 ~ - - - (aZs )

0X2 ' t~y2

since k 5z~ is small. Analysing the disturbance into normal modes, we seek solutions whose dependence

on x, y and t is of the form

expi(kxx + kyy + n t ) , (13)

where n is the growth rate, k = (k 2 + k2) 1/2 is the resultant wave number and kx, ky are the horizontal wave numbers.

Eliminating u, v and 8p from equations (5 ) - (7 ) and using (8) - (12) and expression (13), we obtain

+ 4~(~n + kxU ) w - n + + w

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R. C. Sharma, Veena Kumari

ik~k~H2 D [. 4~ ie (en

- i g k ~ [- (De) - k~r~ k 9

ik~HZD[( 1 4 r e ~n + k~U)

+ i Q(en x kxU ) d ~2

where D = d/dz.

kl 4z(en + k:,u)_1 W

a(z-z 3 ~,+kx~

k. Dw - k2(De) w en + kxU

I~ ik~ H2 k 1 4n(en + kxU)

+

- - 0 , (14)

3. Two uniform streaming fluids separated by a horizontal boundary

Let two uniform fluids of densities el and ~2 be separated by a horizontal boundary at z = 0 and the density Q2 of the upper fluid be less than the density ~1 of the lower fluid so that, in the absence of streaming, the configuration is a stable one and the medium throughout is assumed to be porous. Let the two fluids be streaming with velocities U 1 and U2. Then in each region of constant ~ and U, equation (14) reduces to

(D 2 - k 2) w = 0 . (~5)

The boundary conditions to be satisfied are: (i) Since U is discontinuous at z = zs, the uniqueness of the normal displacement

of any point on the interface implies, according to (12), that

w must be continuous at the interface. (16) en + kxU

(ii) Integrating (14) between zs - e and Zs + e and passing to the limit e = 0, we obtain, in view of (16), the jump condition

Dw As[{~. ( n + k ; U ) + ~ } Dw-ik~HZ-4~ As {en + k x U } l =

= i g k 2 [A~(~) - k~9~ 1 (-en ;kxU)~' (17)

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Hydromagnetic streaming instability...

while the equation valid everywhere else (z # Zs) is

ik2H 2 w 4n(en + kxU) (D2 - k2) w = igk2(Do) ~n + kxU " (lS)

Applying the boundary condition (i) and the fact that w cannot increase exponen- tially on either side of the interface, the solutions appropriate for the two regions are

wl = A(en + kxU1) e +kz, (z < 0) (19)

w2 = A(en + kxZ2)e -kz . . (z > 0) (20)

Applying the boundary condition (17) to the solutions (19) and (20), we obtain the dispersion relation

n 3 + (~ ,v~ + ~2v2) - T, ( ~ + ~2v2) n 2

[ kx2 ( ~ , u , ~ + ~2u~) - ikx + L~ 2 ~ (~,v,U~ + ~ 2 ~ 2 u 2 ) -

{ kZT~ ~ - 2 k z v Z ] n k 3 V 2 ( u , + u 2 ) : O , (21)

where vs(=#l/O1), v2(=]~2/~2 ) are the kinematic viscosities of fluids 1 and 2, re- spectively,

(X 1 - - e l ~2 -- r and V 2 - H 2

(Or + 02)' (0, + 02) 4n(O1 + ~~ is the square of Alfv6n velocity.

Consider now the special case in which the lower and upper fluids are streaming with velocities U(=U1) and - U ( = U 2 ) , respectively.

Equation (21) reduces to

] n2 + (~ ' -- ~2) U -- k l l (0~1~I -]- ~2V2) n

2 2 { k2T" ~l = 0. (22) [- k~U2 ikx (~tv, - ~zv2) U - 2kx V~,- gk (oq - c~21 + g(o, +Q2)J] + L e 2 kt

Equation (22) yields

[ ~ ikxU )] [{2-~ )}2 in : - ~ 1 1 (~v, + ~2v2) - e (~, - :r + (a,v, + ~2v2 +

{ 4a'a2k~U2 2ikxa'a2U (v, - v2) - 2kZV2A -- ok a , - ~ 2 + 9(Ot + O2)JJ q- ~2 kl

(23)

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R. C. Sharma, Veena Kumari

Several cases of interest are now considered.

a) Absence o f s u r f a c e t e n s i o n (T s = 0) In the absence of surface tension, equation (23) becomes

in = - ~ ( 7 , V l + ~2v2) i k x U ( o : , - a2) + (~lv, + ~2v2) + 2k 1 e -

4~laEk2U 2 2ikx~tl~2 U ( v l - v2) _ 2 k 2 V 2 _ g k ( ~ t _ 62)3 1 (24) "~ e2 ka

(i) When k x = 0, equation (24) gives

.

2k~

When ~2 > a~, one of the values of in is positive which means that perturbations grow with time and so the system is unstable. When cq > c~2, both values of in are either real, negative or complex conjugates with negative real parts. The system is therefore stable. It is also clear from equation (25) that for a special case where perturbations in the direction of streaming are ignored (kx = 0), the perturbations transverse to the direction of streaming (ky # 0) are unaffected by the presence of streaming. (ii) In every other direction, instability occurs when

4~a~2kx2U 2 > g k ( a , - ~2) + 2 k 2 V 2 - (26)

g 2

The kinematic viscosities va and v2 of the two fluids here are assumed to be equal, but this simplifying assumption does not obscure any of the essential features of the problem. Thus for a given difference in velocity 2U and for a given direction of the wave vector k, instability occurs for all wavenumbers

0~2(~1 - ~2) (27) k > 2 cos z O(2~xo~zU z - V~ e z)

where 0 is the angle between the directions of k ( k x , ky, 0) and U(U, 0, 0), i.e. kx = = k cos 0. Hence for a given velocity difference 2U, instability occurs for the least wave number when k is in the direction of U and this minimum wavenumber, kmi,,, is given by

= ge2(~, _ ~2) (28) kmln 2 (2~1~2U2 - V 2 e2) "

For k > kmi., the system is unstable. It is clear from equation (28) that the presence of magnetic field increases the value of kmi n for which the system is unstable. The instability of the system is thus postponed. We thus obtain the stabilizing effect of magnetic field.

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Hydromagnetic streaming instability...

b) P r e s e n c e o f su r f ace t e n s i o n

Since the perturbations most sensitive to Kelvin-Helmholtz instability are in the direction of streaming, we put k x = k. We also assume vl = v2. When surface tension is present, equation (23) yields stability if

4~1~2U2 _ 2V 2 e2 < 982 ~ 1 -- ~2 + kTs _~ . (29) ( k g(01 + 02)J

The right-hand side (RHS) of the inequality (29) has a minimum when (d/dk) (RHS) = 0, i.e., when

g(el - c~2) _ kTs (30)

k 01 + 02

If k* denotes the value of k given by (30), we have stability if

4~le2U2_ 2VA 2~z < 2e2T~ k*. (31) 01 -~- Q2

Substituting the value of k* in accordance with (30), we obtain

4~1~2 U2 <282/(Tsg(O~l-'k-~-2)') d- 2V2 82 . (32) 'k/k, - 0 1 + 02 /

The magnetic field and surface tension therefore have stabilizing effects and completely suppress the Kelvin-Helmholtz instability for small wavelengths. The medium porosity reduces the stability range given in terms of a difference in streaming velocities and the Alfv6n velocity.

This research forms a part of the research project awarded to the first author (R.C.S.) by the University Grants Commission.

References

[1] Chandrasekhar S.: Hydrodynamic and Hydromagnetic Stability. Dover Publication, New York, 1981.

[2] Chuoke R. L., Meurs P. Van, Poel C., Van der: Trans. AIME 216 (1959) 188. [3] Sharma R. C., Spanos T. J. T.: Canadian J. Phys. 60 (1982) 1391.

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