I’ntrnat. J. Math. & Math. Sci.
Vol. 5 No. 2 (1982) 365-375
365
INSTABILITY THROUGH POROUS MEDIUM OF TWO VISCOUS
SUPERPOSED CONDUCTING FLUIDS
R.C. SHARMA and K.P. THAKUR
Department of Mathematics, Himachal Pradesh University, Simla-171005, India.
(Received July II, 1979 and in revised form December i0, 1980)
ABSTRACT.
The stability of the plane interface separating two viscous superposed
conducting fluids through porous medium is studied when the whole system is immersed
in a uniform horizontal magnetic field.
The stability analysis is carried out for
two highly viscous fluids of equal kinematic viscosities, for mathematical simpli-
city.
It is found that the stability criterion is independent of the effects of
viscosity and porosity of the medium and is dependent on the orientation and
magnitude of the magnetic field.
The magnetic field is found to stabilize a
certain wave number range of the unstable configuration.
The behaviour of growth
rates with respect to viscosity, porosity and medium permeability are examined
analytically.
KEY WORDS AND PHRASES.
Instability, porous medium, conducting
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
fluids.
76E25, 76W05
INTRODUCTION.
The instability of the plane interface between two fluids, under varying
assumptions of hydrodynamics and hydromagnetics, has been discussed by Chandrasekhar
[i].
Bhatia
[2] has studied the influence of viscosity on the stability of the
plane interface separating two incompressible superposed conducting fluids of
R.C. SHARMA AND K.P. THAKUR
366
uniform densities, when the whole system is immersed in a uniform horizontal magne-
He has carried out the stability analysis for two highly viscous fluids
tic field.
of equal kinematic viscosities and different uniform densities.
When the fluid
slowly percolates through the pores of the rock, the gross effect is represented by
Darcy’s law
which states that the usual viscous term in the equations of fluid
motion is replaced by the resistance term
fluid, k
I
(/k 1) q, where
is the viscosity of the
the permeability of the medium and q the velocity of the fluid.
[3] has experimentally observed,
Wooding
in the absence of viscous dissipation and consider-
ing only Darcy resistance, that convection sets on as a fairly regular cellular
pattern in the horizontal.
This problem in the case of a conducting fluid consider-
ing both Darcy and viscous resistances has been investigated by Prabhamani and
Rudraiah
[4].
Saville [5] has studied the stability of motions involving fluid
interfaces in porous media.
Various problems of fluid flows through porous medium
have been treated by Saffman and Taylor [6], Chouke et al [7], Scheidegger [8],
Yih
[9], Nayfeh [i0] and Rudraiah and Prabhamani [ii].
The instability of two viscous superposed conducting fluids through porous
medium may find applications in geophysics.
It is therefore the motivation of this
study to examine the effects of viscosity and medium permeability on "the stability
of the plane interface separating two incompressible superposed conducting fluids of
uniform densities, when the whole system is immersed in a uniform horizontal magnetic
field.
We examine the roles of viscosity, medium permeability and magnetic field on
the instability problem.
This aspect forms the subject matter of the present study
wherein we have carried out the stability analysis for two highly viscous fluids of
equal kinematic viscosities and different uniform densities.
2.
PERTURBATION EQUATIONS.
Consider the motion of an incompressible, infinitely conducting viscous fluid
(of variable viscosity
o(Z))
Let q(u,v,w), g0, gP and
in the presence of a uniform magnetic field
h(hx,hy,h z)
H(Hx,Hy,0).
denote the perturbations in velocity, density
pressure p and magnetic field H respectively.
Then the linearized perturbation
equations of a fluid flowing throuaporous medium when both Darcy as well as viscous
resistances are present are:
INSTABILITY OF VISCOUS CONDUCTING FLUIDS
0
q
e
t
(Vh)
Vp + g60 +
V’q
V’h
0,
09 q
1
+
367
(Vq).V+(V-V)q],
(2.1)
0,
(2.2)
V x (qXH),
e
(2.3)
p + (q.V)p
e
0.
(2.4)
Equation (4) ensures that the density of every particle remains unchanged as
we follow it with its motion,
v(=/O) denotes the kinematic viscosity of the fluid,
e is porosity (0<e<l) and g (0,0,-g) is the acceleration due to gravity,
and k
correspond to nonporous medium.
I
e
i
Analyzing the disturbances into normal
modes, we assume that the perturbed quantities have the space (x,y,z) and time (t)
dependence of the form
f (z) exp
where k
ikxx+ikyy+nt),
x’ ky are
(2.5)
2
horizontal wave numbers (k
k
x
2+ky 2), n is
the growth rate of
the harmonic disturbance and f(z) is some function of z.
For perturbations of the form (2.5), equations (2.1)-(2.4) give
H
0( +
n
n
+
v
+
kl
H
9
0( + I
(+
u
l)W
x
H
+ x
+
i
p +
i
(ik h -ik h
x y
y x
ik 6p
x
(ik h
y x
ik
-ikxhy
y
[(D2-k2)u
[( D
2
2
k )v
+ (D)(ik w+Du)]
(2.6)
+
(2.7)
H
(mhx-ikxhz)
+
D6p-g6 +
(Dh -ik h z)
Y
Y
(D)(ikyW+Dv)],
i
[(m2-k2)w+2(D)(Dw)]
(2.8)
ik u
x
+
n6p
nh
ik v
y
+
0,
ik h
xx
w(D0)/e,
(ik H
where D
xx
+
ik h
yy
+ Dh z
0,
(2.9)
(2.10)
ik H
d/dz.
+
yy
q/g,
(2 ii)
R.C. SHARMA AND K.P. THAKUR
368
Multiplying (2.6) and (2.7) by-ik and-ik respectively and adding, using (2.9)x
Y
p
(2.11) and finally, eliminating
between the resulting equation and (2.8), we
obtain the equation in w:
2
n
[k O
w
( + __)
el
i
[p(D2_k 2)2w
+
D {0
+
n
( + __)
k
1
2(Dp)(D2-k2)Dw
-gk2
Dw}]
(DO) w
(D2p)(D2+k2)w]
+
i
(k H +k H
x x
y y
2
(D2_k2)w
(2.12)
0.
TWO SUPERPOSED VISCOUS FLUIDS OF UNIFORM DENSITIES.
3.
Here we consider the case when two superposed fluids of uniform densities O
1
and
02
z
0.
and uniform viscosities
I
and
2
are separated by a horizontal boundary at
The subscripts I and 2 distinguish the lower and upper fluids respectively.
Then, in each region of constant
(D2-k 2) (D2-K2)w
01, i
and constant
02 P2’
equation (2.12) becomes
0,
(3.1)
where
K
2
k2
+
n
g
1
n-l + 4n20
{i +
(k H +k H
x x y y
)2}.
(3.2)
(in the lower fluid) and z
Since w must vanish both when z
+
(in the
upper fluid), the solutions appropriate to the two regions can be written as
w
w
I
AI e+k
2
A2e
z
+
Ble
-kz
+
B2e
-K z
where
AI, BI, A2,
K1
/
/k 2
+K z
I
+
i
2
B
2
(z<O),
(3.3)
(z>0)
3.4
are constants,
+
{i +
4n
2
(k H +k H
x
)2}
(3.5)
(k H +k H
)2}.
(3.6)
x
y y
and
/
K2
/k2
+
n
q {i +
2
+
1
4n202
x x
y y
369
INSTABILITY OF VISCOUS CONDUCTING FLUIDS
In writing the solutions (3.3) and (3.4) it is assumed that K 1 and K 2 are so
The solutions (3.3) and (3.4) must
defined that their real parts are positive.
The boundary conditions to be satisfied at
satisfy certain boundary conditions.
0 are (Chandrasekhar [i], p.432)
the interface z
(3.7)
W,
(3.8)
and
(D2+k2)w,
(3.9)
must be continuous.
-- -
Integrating (2.12) across the interface z
2
{02 (n + .)
+/-
n
{01 ( +
gk----2
ng
where w
O
Dw
2 D2_k 2 )Dw
2
2
HI
)Dw I
(02-0 l)wO
and (Dw)
O
i
+
(D2-k2)DWl +
2k2
(
2-i
i
(Dw)
O, we obtain another condition
2
(k H +k H
xx
Dw
yy
(k H +k H
y y
xx
2
2} z=O
Dw
I} z=0
(3.1o)
0
are the unique values of these quantities at z
0.
Applying the conditions (3.7)-(3.10) to the solutions (3.3) and (3.4), we
obtain
A
+ B1
I
kA
HI
+
I
A
2
KIB I
+ B2
-kA
2
(3.11)
{2k2Al + (KI2+k2)BI
-k0 2
2
1
(- + -i)A 2
(3.12)
K2B 2,
k0 I
2
i
{2k2A2 + (K22+k2)
B2}’
i
4n2gl (kxHx+kyHy)
(Z + n--l) A1
k2
2n2g (02_01) (AI+BI+A2+B2)
g
(kAl+kA2)
k
(3.13)
2
+- (HI-2)
(kAI+KIBI-kA2-K2B 2)
(3.14)
R.C. SHARMAAND K.P. THAKUR
370
Eliminating the constants
i
i
k
K
2k2I
I
R-C0.1I
nk
i
1
I
n
2(k’VA)
from (3.11)-(3.14), we obtain
AI,A2,BI,B 2
1
-i
-I
k
K
i (KI2+k2)
-2k22
[21-R-C KIk
[+C
2
+
Cll K1
-2 (K22+k
0.2
0.2u2
e
nk
i
--(k-VA)
n
klk
n
2
2
o,
(3.15)
I
t2u 2 K2.
2
nklk
where
01, 2
0.1,2
R
(.VA)
pl+P2
(c,2-0. I)
-2
n
and V
A
4(pl+P2)e
k2 Pl-P2 k2
pl+P2 ne (elV 1 (2u2)
C
ne
Alfvn velocity
is the
2
(k H +k H )
xx yy
2
vector.
Evaluating the determinant (3.15), we obtain the following characteristic
equation
(Kl-k)
+
[2k
{0.2 n
_2k[{eln
+
2(0.191-0.292
0.2v2 e
+
kl
+
(lUl e + !
+
n
{0.2 n
el
+
+(K2-k) [{0.1n
0.2)2 e
k’-=-’+
n
(k’VA)
2}
{R
(g "VA) 2
+
n (’VA)
I
(iuI+0.2)2)
e
nkl
92
(.V+A)m}{a2 (1 + n-l
+
92
I
C
{ (K2-k) + 0’2 ( + n--l)
+ C
(K2-k)
2
2
2
n2 (k’VA)
(.VA) 2}
+
n
u
1
C
g (Kl-k)
{0.1(1 + n-l)
+
n
(k’VA) 2}
-)"-)"
elle
-2 (.VA)+ 2}
T + - n (’A)2} {R le 0.1Vl+e2)2sk
I
-2k2(0.1l-0.2u2){l(
n
i
+
Ul
I)
--nk
C
(Kl-k)
+
n
(k-VA)2}]
O.
(3.16)
INSTABILITY OF VISCOUS CONDUCTING FLUIDS
Since K
and K
I
2
371
(3.16) is quite
involve square roots, the dispersion relation
We therefore carry out the stability analysis for highly viscous fluids.
complex.
Under this assumption of highly viscous fluids, we have
+__n2 + .e
k[l
K
k2kl
k
k [i
n---n-- +
+
2k
2
2
e(k H +k H
xx y,y
+
4-Ox)nk
e
2k2kl
+
1/2
2
e(Pl+02)
(.VA)2],
2
2nk 0
(3.17)
so that
n
ml-k 2Ulk +
e
(’VA) 2
e
+
2nkalV 1
(3.18)
and
n
22k +
m2-k
e
e
+
/-
(k’VA)
2nk2 2
Substituting the values of K
and K
I
2
2
(3.19)
from (3.18), (3.19) in (3.16) and putting
(the case of equal kinematic viscosities
as in Chandrasekhar
i
e
n
6
+ 3 +
U
x(l-e)+
+
+
[l]),we obtain the following dispersiom relation
2
2ala2 + (i-2 2 (i + k2u -)]n 5 + (.V-A)
e
e
i2
U {4i2+E(i-2) 2} ]n
U
(k-VA)2 {2 + 2ala2
ekk’VA
(i-2)
2
la
for mathematical simplicity
{A + (k’VA) 2 +
+
I {2(k’VA) (i +--kl
+
[e(k’VA)4{2 + 2ie 2
+
4
+
2e
2
kl
+
+ e {2A+
[q
2
e
k+
32e
+A+-k
2ala 2
(k-VA)
2
(al_a2)
2
(l-E)}
(i-2) 2}]n2
2
e
(i-2 )2}+ i2 (k’VA) {e
k
I
2
el (i-2)
e
2k2 2 (al-2) 2 (i-E) +
2la2U+ 2k I
kl
klala2
(12+2 2) + 2k2
kl
2
+ 4k
(.VA) 2 +
k
I
2x
R.C. SHARMA AND K.P. THAKUR
372
+
2k21)((Zl-2)
(l-e) }]n +
cic2
A)__
where we have written 2(k-V
(i)
2
e
2A
Cl---
+ gk
g
(k’VA)
4
(3.20)
0,
A.
__(i-2)
Stable case
For the potentially stable arrangement
i>2
we find by applying Hurwitz’
criterion to (3.20), that (as all the coefficients in (3.20) are then positive)
all the roots of n are either real and negative or there are complex roots with
negative real parts.
The system is therefore stable in each case.
The potentially
stable configuration, therefore, remains stable whether the effects of viscosity
and medium porosity are included or not.
(ii) Unstable case
For the potentially unstable arrangement
2>i
the system is unstable in the
hydrodynamic case for all wave numbers k in the presence of viscosity effects and in
the absence of porosity effects (Chandrasekhar [I]). Also the system, in the present
case, is unstable if
2(k’VA)
2
< gk
e
(a2
i)"
In the present hydromagnetic case we find, by applying Hurwitz’ criterion to (3.20)
when
2>i
that the system is stable for all wave numbers which satisfy the
inequality
2(.V+A)2 > gk (2
i.e.
2k(VlCOS@
where V
+
i ),
V2sin@)2> (2
(3.21)
el )’
the x and y directions and G
I and V 2 are the Alfvn velocities in
between k and H
(3.22)
is the angle
x
The stability criterion (3.22) is independent of the effects of viscosity and
medium porosity.
where
The magnetic field stabilizes a certain wave number range k > k*
373
INSTABILITY OF VISCOUS CONDUCTING FLUIDS
g(2-l)/2g(VlCOS@+V2sin@)2,
k*
(3.23)
of the unstable configuration even in the presence of the effects of viscosity and
medium porosity.
The critical wave number k*, above which the system is stabilized,
V
is dependent on the magnitudes
I
and V
2
of the magnetic field as well as the
orientation of the magnetic field 8.
We now examine the behaviour of growth rates with respect to viscosity,
porosity and medium permeability analytically.
(e2-l)
(3.20)
(3.20) has one positive root
is satisfied if n
o
let n
Since for
0
2>i
and 2(k’V
A)
denote the positive root.
2 <
Then
To study the behaviour of
is substituted in place of n.
growth rates with respect to viscosity and medium permeability, we examine the
dno/d
natures of
no
(6A6
3
2
no
kl
+
+
5
no
+ 5A 5
5
i
+--kl
2
and
4
dno/dkl.
+ 4A4
no
kl
22e (.A) 2
k
i2
e
(12+22)
2v2e 2A +
k
al2kl
(l-e)+
(+VA) 2
3
1
ve
+
{4ala2+e(al-a2)2}
(k’VA) (al-2)
{(k’V A) 2-A}
n
+
2
no
5
+
5A5no
4
+
4A4no
3
+
3A3no
2
o
2i292 +
2 2
Ve
k
I
}n3
o
(3.24)
o’
+
n
2
and
(6A6no
I
(l-g)} + 2(k’VA) 2 e2 (i + 2re)
k
I
kl
2
2ela2k1
dk
kl
2
2
dno
4k23g(l-e2)2(l-e)
2
k2 (el_2)2
+
kl
I
2
2A2 no + AI)
+
kl
2
e
2
62e + 4k2 2
2
(al-a 2)
k
I
3V3
(i---- (.A)2_2Age + 2
2
no
+ 3A 3
2e
i
+
3
It follows that
2A2no + AI)
dno
d
374
R.C. SHARMA AND K.P. THAKUR
2
2
2
5 [6g 8k
2
1
k
[3__ +2}] n o 4
+
+
{4i2+e(i-2)
]n
(i-2)
(2-e)(i-2)
2
(l-e)+i
el
kl -i
+
[3g22 2
2AE
+
kl
kl
kl
2k
ele2kl
2E(’V+A
+ e__+
(1-2)2 (l-E) + 4la2E
2
kl
l2kl
6k2g 2
2
2
2
kl
(el-a2)
2
22
+ 4rE__2
(k.VA)
kl
+
E
2
+
[l2k------
where A
5
(k’VA)
A
(I-E)
(k’VA)
(e12+e22) + l2kl
2
4
A
n
3
o
(i_2)
2
(l-E) +
2(-V+A 2 kl
2UE2A + E 2 (k’VA) 2 (al-e 2) 2.
2
2klela 2
kl
2
A
2
and A
1
2
/+
are the coefficients of n
respectively in (3.20) where n
and (3.25) that
2
n
o
2
4
E2A
2( l-E) ]no,
+ 2k2E
l2kl (k’VA) al 2 2 2 (k’VA) (i-2)
4
3
12
dno/dk I
it can be shown that dn
and
o
o
is substituted for n.
dno/d9 may
5
O
n
4
o
n
3
o
n
2
O
(3.25)
and n
o
It is evident from (3.24)
be both positive or negative.
/dE may be both positive or negative.
Similarly
Thus the growth
rates both increase or decrease with the increase in viscosity, porosity, and medium
permeability.
The viscosity, porosity, and medium permeability therefore have both
stabilizing as well as destabilizing tendencies on the growth rates.
REFERENCES
i.
CHANDRASEKHAR, S. Hydrodynamic and Hydromagnetic Stability.
Press, London, Chap. X, 1961.
2.
BHATIA, P.K.
3.
WOODING, R.A.
4.
PRABHAMANI, P.R. and RUDRAIAH, N.
5.
SAVILLE, D.A.
Oxford University
Rayleigh-Taylor instability of two viscous superposed conducting
fluids, Nuovo Cimento 19B, 161-168, 1974.
Rayleigh instability of a thermal boundary layer in flow through
a porous medium, J. Fluid Mech. 9, 183-192, 1960.
Stability of hydromagnetic thermoconvective
flow through porous medium, J. Appl. Mech. Ser. E. 4__0, 879-884, 1973.
Stability of motions involving fluid interfaces in porous media,
Phys. Fluids
1__2, 2438-2440, 1969.
375
INSTABILITY OF VISCOUS CONDUCTING FLUIDS
The penetration of a fluid into a porous
medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy.
Soc. A245, 321-329, 1958.
6.
SAFFMAN, P.G. and TAYLOR, G.I.
7.
CHOUKE, R.L., VAN MEURS, P. and VAN DER POEL, C. The instability of slow,
immiscible, viscous llquld-llquld displacement in permeable media,
Trans. AIME 216, 188-194, 1959.
8.
SCHEIDEGGER, A.E.
9.
YIH, C.S.
i0.
NAYFEH, A.H. Stability of liquid interfaces in porous media, Phys. Fluids i_5,
1751-1754, 1972.
ii.
RUDRAIAH, N. and PRABHAMANI, P.R.
The Physics of Flow through Porous Media.
Toronto Press, Macmillan, 1960.
Stratified Flows.
Academic Press, 1980.
Thermal diffusion and convective stability
of a two component fluid in a porous medium, 5th International Heat
Transfer Conference, Tokyo, Japan CT 3.1, 79-82, 1974.
Present Address of the first author:
Dr. R.C. Sharma
Department of Physics
University of Alberta
Edmonton, Canada
T6G 2Jl
University of