RAYLEIGH-TAYLOR I N S T A B I L I T Y OF V I S C O E L A S T I C FLUIDS W l T H S U S P E N D E D PARTICLES

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

R. C. SHARMA, A. RAJPUT

Department of Mathematics, Himachal Pradesh Universit3; Summer Hill, Shimla-171 005, hldia

Received 18 November 1991; in revised form 3 April |992

The instability of the plane interface between two viscoelastic (Ohlroydian) superposed con- ducting fluids permeated with suspended particles in porous medium is studied when the whole system is immersed in a uniform magnetic field. The dispersion relation for the Oldroydian vis- coelastic+ fluid is obtained which also yields dispersion relations for Maxwellian and Newtonian fluids in special cases, in the presence of suspended particles in porous medium in hydromag- netiy Tbe system is round tobe stable for potentially stable case. The presence of magnetic fleld stabilizes certain wave ni, tuber hand whereas the system was unstable for ail wave uumbers in the al)sente of magnctic field, for the potentially unstable configuration. The growth rates i.crease (for certain wave numbers) and decrease (for other wave numbcrs) with the inrrease in stress relaxation rime, strain retardation rime, suspended particles number density and medium permeability.

1. Introduction

The instal)ility of the plane interface between two Newtonian fluids, under vary- ing assumptions of hydrodynamics and hydromagnetics, has been discussed in detail by Chandrasekhar [4]. Bhatia and Steiner [1] bave studied the problem of thermal instability of a Maxwellian viscoclastie fluid in the presence of rotation and have round that the rotation has a destabilizing effect in contrast to the stabilizing er- fret on Newtonian fluid. Bhatia and Steiner [2] have also considered the thermal instability of a Maxwell fluid in lffdromagnetics and have found that the magnetic firld has stabilizing cffect on viscoelastie fluid just as in the case of Newtonian fiuid. The thermal instability of a viscoelastic (Oldroydian) fluid has been con- sidrrrd in the presence of magnetic field (Sharma [9]) and rotation (Sharma [10], Eltayeb [5]). Expe�9 demonstration by Toms and Strawbridge [12] has re- vcah'd that a dilute solution of methyl methacrylate in n-butyl acetate agrees well with the theorctical model of Oldroyd fluid. Chandra [3] observed in an air layer that y oecurred at mueh lower gradients than predicted if layer depth were lrss than 7 mm and called this motion "columnar instability'. However, for layers dceper than 10 mm, a B› cellular convection was observed. Thus there is a decadcs-old contradiction between the theory and the experiment. Chandra [2] added an aerosol to mark the fiow pattern. Scanlon and Segel [8] have eonsidered the effect of susi)ended particles on the onset of B› convection and round that

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R. C. Sharma, A. Rajput

the critical Rayleigh number was reduced sotely, because the heat capacity of the pure fluid was :supplement› by tbat of the particles. The effect of suspended par- ticles was thus found to destabilize the lacer. Palaniswamy and Purushotham [7] have studied the stability of shear flow of stratified fluids with fine dust and found the effect of fine dust to increase the region of instability. In all the above studies, the medium has been considered to bc non-porous.

A macroscopic equation whieh describes incompressible fiow of a Newtonian fluid of viscosity # through a macroscopically hcmogeneous and isotropic porous medium of permeability kl is the well known Darcy's equation. The usual viscous term in the equations of fluid motion is replaced by the resistance term - ( p . / k l ) g , where q is the filter velocity of the fluid. The instability of streaming fluids with fine dust in porous medium has been studied by Sharma and Sharma [11].

The object of the present paper is to study the stability of the plane interface separating two incompressible and electrically conducting superposed viscoelastic (Maxwellian and Oldroydian) fluids in the presenee of suspended particles and magnetic field in porous medium. The motivation for the present study is the fact that knowledge regarding fluid-particle mixtures is not commensurate with their seientific and industrial importance. The analysis would be relevant to the stability of some polymer solutions like a dilute solution of methyl methacrylate in n-butyl acetate and to the stabi]ity of Maxwellian viscoelastic fluids and the problem finds its usefulness in chemical technology and geophysics.

2. P e r t u r b a t i o n Equat ions

Let Tij , 71j, eij, 6ij, p, Ui, Xi, p , , )~ and A0(< )~) denote, respectively, the stress tensor, shear stress tensor, rate-of-strain tensor. Kronecker delta, scalar pressure, velocity, position vector, viscosity, stress relaxation time and strain retardation time. Then the Oldroydian viscoelastic fluid is described by the constitutive rala- tions (Oldroyd [6])

where

Tij = -P6 i j + Tij . (d ) ( a ) !+A~-~, n i = 2 # l+A0 (-�9 eq. (1)

i (o,,i o~,~~ e~j = -~ k Oz~ + Ox,: ] "

d 0 0 dt - Or + ?tj OXj

is the "convective derivative'. When A 0 = O. the fluid is t('rmed as Maxwellian viscoelastic and A = A0 = 0 yields the fluid to be Newtonian.

Consider a static state in which an in(:omprcssiblc and infinitely (electrically) conducting Maxwellian viscoelastic fluid permeated with suspended particles is

920 Czech. J. Phys. 42 (1992)

R a y l e i g h - T a y l o r i n s t a b i l l t y o f v i s c o e l a s t i c f l u i d s . . .

arranged in horizontal strata in porous medium and the pressure p and density p are fimctions of the vertical coordinate z only. A uniform horizontal magnetic field /~(H,, Hy, 0) pervades the whole system. The character of the equilibrium of this initial static state is determined, as usual, by supposing that the system is slightly disturbed and then by following its further evolution.

Let ff(u, v, w), h(hx, hy, h~), 6p and 6p denote respectively the perturbations in fluid velocity (0, 0, 0), magnetic field/I(Hx, H~, 0), density p and pressure p; g(~, t) and N(~,t) denote the velocity and number density of the suspended particles respectively, g = (0, 0 , -g ) and ~ = (x, y, z). Then the linearized hydromagnetic perturbation equations of viscoelastic fluid through porous medium are

[ #e ( V x f ~ ) x / t + K N x - v @ + ~~p + ~ E

( o), - I+Ao~ ~1 ~,

(~- ~)]

(2)

v . ~ = o, (3)

v . ~ = o, (4)

o�9 ~ = v x (~ x ri), (~)

0 b~ ~p + ( f v )p : 0, (6)

where c is the medium porosity and K = 6~r#7/, 7/being the particle radius, is the Stokes drag coefficient. Assuming uniform particles size, spherical shape and small relative velocities between the fluid and particles, the presence of particles adds an extra force term in the equations of motion (2), proportional to the velocity difference between particles and fluid.

Since the force exerted by the fluid on the pa�9 is equal and opposite to that exerted by the particles on the fiuid, there must be an extra force term, equal in magnitude but opposite in sign, in the equations of motion for the particles. The buoyancy force on the particles is neglected. Interparticle reactions are also ignored for w,'~ assume that the distances between particles are quite large compared with their diameters. If mN is the mass of particles per unit volume, the linearized perturbed equations of motion and continuity for the particles, under the above a.ssumptions, are

OM or + v . y = o, (8)

Czech, J. Phys. 42 (1992) 921

R. C. Sharma, A. Rajput

where M = ~ N/No and No, N stand for initial uniform number density and per- turbation in number density, respeetively. Equation (6) ensures that the density of every particle remains unchanged as we follow it with its motion.

Analyzing the disturbances into normal modes, we seek solutions whose depen- dence on x, y, z and t i s given by

l (z) exp(ik~x + ik~y + rit), (9)

where k,, k v are horizontal wave numbers (k 2 = k~ + k~), n is the growth rate of harmonic disturbance and f(z) is some function of z.

For perturbations of the form (9), Eqs. (2)--(7), after eliminating ~, give

(1 § An) [ . ~ (1 § rn) § KNv ] " u+ (1+ ~.)(1+ ~ o . ) V .

[ tl'eH:l (ikyhz --ikzhy)], = ( l + A n ) - i k~@+ (10)

(l + An) [P_~~ (l + ru) + K N v ] P - - n e v + (l + ru)( l + )ton)-~i v

[ I ' y ' = (1 + An.) -ik,@ + (11)

[pu K N r ] p (l+.~,t) _-7 ( l+ rn )+ ~ ,t w+(l+vn)( l+)~on)~w

I'y {H~,(ikflt: - Dh~) + H™ - D/%)}], (12) = (1 + .Xn) [-D(Sp - g@ + -~~ , �9

ik~u + ikvv + Dw = O,

ik~hx + ikvh ~ + Dhz = 0,

~n.h = (ik~H~ T ik,H,),7,

~nlfp = - w Dp,

wherc r = m / K and D = d/dz.

(13) (14)

(15) (16)

Multiplying Eqs. (10) and (11) by - ik , , - i k u respectively and adding, using Eqs. (13)-(16) in it and finally eliminating 6p between the resulting equation and Eq. (12), we obtain the equation in w:

(1 + ~n)(1 + rn) n [D(pDw) - k2pw] + (1 + An) rn [D(KNDw) - k~I™

4 (1 + Aon)(1 + rn) [D(pvDw) - k2pvw] + gk---~-2 (1 + ~n)(1 + rn)(Dp)w k ] ch

+(1 + ~n)(1 + rn) lt---s (k~H~ + kyH,j)2(D 2 - k2)w = 0. (17) 4un_~

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R a y l e i g h - T a y l o r i n s t a b i l i t y of v i s c o e l a s t i c f l u i d s . . .

3. T w o uniform viscoelast ic fluids separated by a horizontal b o u n d a r y

Here we consider the case of two uniform viscoelastic fluids of densities, viscosi- tics pi, jt~ (lower fluid) and P2, P2 (upper fluid) separated by a horizontal boundary z = 0. Then, in each region of constant p and constant #, Eq. (17) becomes

(D 2 - k2)w = 0. (18)

Since w must vanish both when z ~ o0 (in the upper fluid) and z --* -oo (in the lower fluid), the general solution of Eq. (18) can be written as

wl = Ae ~z , ( z < 0 ) (19)

w2 = A e -~~ , ( z > 0) (20)

where the saine constant A has been chosen to ensure the continuity of w at z = 0. Integrating Eq. (17) across the interface z = 0, we obtain the boundary condition

n(1 + An) n (1 + An)(1 + Tri) Ao(pDw) + Ao(mNDw) e

1 + ~ (1 + Tri)(1 + Aon) Ao(ItDw) + gk---~-2en (1 + An)(1 + rn) Ao{p)w

+ Pe (1 + An)(1 + r,t)(krH. + kvH,) 2 A0(Dw) = 0. (21) 4*rny

Applying the boundary condition (21) to the solutions (19) and (20), we obtain

[ ArK(NI + N2) + Aore ] A v n 4 + A + r + Pl + P2 ~ (alva + a2v2) n s

+ [1 + e(Ao + r) rK(N1 + N2) k~ (y + {x2v2) + pi + p2

T V, (y + y ~

+ 2{~. r ~ - T (y - y = 0, (22)

w h e r c

and

(~. 13^) 2 = p,,(k,H~, + k,Hv) ~ 4~e(pl + P2)

I t t ,2 V] ,2 :

PI,2

, 0(1,2 Pi,2

Pi +P2

Czech. J. Phys. 42 (1992) 923

R. C. Sharma, A. Rajput

For the problem under consideration, Eq, (22) is the dispersion relation for O1- droydian viscoelastic fluid and A0 = 0 yields the dispersion relation for Maxwellian visy fluid.

( i) S t a b l e ca se

For the potentially stable arrangement y < OZl, we find, by applying Hurwitz criterion to Eq. (22), that all the coefficients in (22) are positive, and so all the roots of n are either rem and negative; or there are complex roots with negative rem parts. The system is therefore stable in each case. The potentially stable con- figuration, therefore, remains stable whether the fluid is viscoelastic (Oldroydian or Maxwellian) permeated with suspended particles in porous medium in hydro- magnetics or hot.

( i ) U n s t a b l e c a s e

For the potentially unstable arrangement c~2 > c~1, the system is unstable in the hydrodynalnic case for M1 wave numbers k in the presence of viscoelasticity, suspended particles and porosity effects. Also, the system, in the present case, is unstable if

gk 2(]~" VA) 2 < - - (OE2 -- OL1).

In the present hydromagnetic case we find, by applying Hurwitz criterion to Eq. (22) when c~2 > c~1, tha t the system is stable for all wave numbers which satisfy the inequality

gk 2 ( k . V~) ~ > - - (y - ~~) . (23)

i.e.,

2k(V1 cos0 + V~ sin0) 2 > -g (y - OZl), (24) c

where VA and V2 are the Alfv› velocities in the x and y directions and 0 is the angle between/~ and H=: The stability criterion (24) is independent of the effects of viscoelasticity, suspended particles and medium porosity. The magnetic fietd stabilizes a certain wave number range k > k* where

k* -~ g(o~2 - o~1) (25) 2c(I~ cos 0 + V2 sin 0) 2 '

of the unstable configuration even in the presence of the effects of viscoetasticity, suspended particles and medium porosity. The critieal wave number k*, above which the system is stabilized, is dependent on the magnitudes of magnetic field, fluid densities as well as the orientation 0 of the magnetic field.

™ Since for c~2 > oz1 and 2(f~. I)A) 2 < (c~2 a l ) , Eq. (22) has one positive root,

let no denote the positive root. Then (22) is satisfied if no is substi tuted in place of n. To study the behaviour of growth rates with respect to strain retardation time,

924 Czech. J. Phys. 42 (1992)

R a y l e i g h - T a y l o r i n s t a b i l i t y o f v i s c o e l a s t i c f l u i d s . . .

stress relaxation time parameters, suspended particles number density and medium pcrineability, we examine the natures of (dn0)/(d)t0), (dn0)/(d$), (dno)/(dN), and (dno)/(dkl). Equation (22) yields

dn0 = _ k l (alt'l + a2v2)(1 + Tno)n¤

dA0 P (26)

where

[ ~0re ] ATK(N1 + N2) + (celUi + n n¤ P=4ATn03+3 ( ; ~ + r ) + P i+P2

+2 [1 + e(Ao + r) TK(N1 + N2) kl (alUl + a2u~) + Pi + P~

+~r 2(�9 ~~)~ - T (y - y ,~o

~ (~1~i + y �9 (27)

It is evident from Eq. (22) that (dno)/(dA0) may be positive or negative. A similar result holds Cor (d~o)/(dA), (d~0)/(dN) and (d~0)/(dkl). We thery r162 that the growth rates inr (for certain wave numbers) and decrease (for other wave numbers) with the increase in stress relaxation time, strain retardation time, suspended particles number density and medium permeability.

4. Conclusion

A detailed account of Rayleigh-Taylor instability of Newtonian, viscous fluid in hydromagnetics through non-porous madium has been given by Chandrasekhar [4]. A dilute solution of methyl methacrylate in n-butyl acetate agrees well with the theoretical model of Oldroydian viscoelastic fluid. The knowledge concerning fluid-particle mixtures is not commensurate with their scientific and industrial im- portance. The presence of magnetic field (e.g. Earth's magnetic field) and porous medium are relevant and important in geophysics. Keeping in mind the relevance to the stability of some polymer solutions like a dilute solution of methyl methacrylate in n-butyl acetate and to the stability of Maxwellian viscoelastic fluids and the use- fulness in chemical technology and geophysics,the present paper attempts to study the stability of the plane interface separating two incompressible and electrically conducting superposed Oldroydian viscoelastic fluids in the presence of suspended particles and magnetic field in porous medium.

For the potentially stable configuration (lightcr density fluid overlying heav- ier one), the systcm is stable whether the fluid is viscoelastic (Oldroydian or Maxwellian) permeated with suspended particles in porous medium in hydromag- netics or not.

Czech. J. Phys. 42 (1992) 925

R. C. Sharma , A. Ra jpu t : Rayle igh-Taylor i n s tab i l i t y . . .

For the potentially unstable arrangement (heavier density fluid overlying lighter one), the system is unstable in the hydrodynamic case for ail wave numbers k in the presence Gf viscoelasticity, suspended particles and porosity effects whereas the system becomes stable in the hydromagnetic case for a certain wave number range. The magnetic field stabilizes a certain wave number range (which was unstable in the hydrodynamic case) even in the presence of the effects of viscoelasticity, suspended particles and medium porosity. The stability criterion is independent of the effects of viscoelasticit); suspended particles and medium porosity but depends on the magnitudes of magnetic field, fluid densities as well as the orientation 0 of the magnetic field. Moreover, the growth rates increase (for certain w~we numbers) and decrease (for other wave numbers) with the increase in stress relaxation time, strain retardation time, suspended particles number density and medium permeability.

References

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York, 1981. [5] Eltayeb I. A.: Z. Angew. Math. Mech. 52 (1975) 599. [61 Oldroyd J. G.: Proc. Roy. Soc. (London) A 245 (1958) 278. [7] Palaniswamy V. I., Purushotham C. M.: Phys. Fluids 24 (1981) 1224. [8] Scanlon J. W., Sege[ L. A.: Phys. Fluids 16 (1973) 1573.

[9] Sharma R. C.: Acta Phys. Hung. 38 (1975) 293. [I0] Sharma R. C.: Acta Phys. Hung. 40 (1976)11. [11] Sharma R. C., Sharma N. D.: Czech. J. Phys. 42 (1992) 907. [12] Toms B. A., Strawbridge D. J.: Trans. Faraday Soc. 49 (1953) 1225.

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