T H E I N S T A B I L I T Y OF S T R E A M I N G FLUI DS W I T H F I N E D U S T IN P O R O U S M E D I U M

Il,. C. SlIARMA

Department of Mathematics, Himachal Pradesh University,

Summer Hill, Shimla - 171 005, India

N. D. SHARMA

Department of Mathematics, Govt. College,

Dhararnshala, Himachal Pradesh, Indla

Received 18 November 1991; in revised form 3 April 1992

The instability of the plane interface between two uniform, superposed, and streaming fluids permeated with suspended particles through porous medium is considered. The effect of a uni- form horizontal magnetic field on the problem is Mso studied. 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, whereas for perturbations in ail other directions there exists instability for a certain wavenumber range. The instability of the system is postponed by the presence of magnetic field. The magnetlc 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 in streaming velocities and the Alfv› velocity. The suspended particles do not affect the above results.

l . I n t r o d u c t i o n

The instability of the plane interface separating two uniform superposed stream- ing ttuids, under varying assumptions of hydrodynamics and hydromagnetics, has been discussed in a treatise by Chandrasekhar [1].

The effect of suspended particles on the stability of superposed streaming fluids might be of industrial and chemical engineering importance. Further motivation for this study is the fact that knowledge concerning fluid-particle mixtures is not commensurate with their industrial and scientific importance. Scanlon and Segel [4] bave considercd the effect of suspended particles on the onset of B› convection and hmnd that the critical Rayleigh number was reduced solely because the heat ca pacity of the pure gas was supplemented by that of the particles. The effect of suspcnded particles was thus found to destabilize the layer. Palaniswamy and Purushotham [5] haveconsidered the stability of shear flow of stratified fluids with fine dust and have found the effect of fine dust (suspended particles) to increase thc region of instability.

The medium has been considered to be non-porous in all the above studies. The flow through porous medium has been of considerable interest in recent years particularly y geophysical fluid dynamicists. The gross effect, as the fluid slowly percolates through the maeroseopically homogeneous and isotropic porous

Czechoslovak Journal of Physics, Vol. 42 (1992), No. 9 907

R. C. Slmrma, N. D. Sharma

medium, is represented by Darcy's law which states that the usual viscous terre

in the equations of fluid motion will be replaced by the resistance terre - # q,

where # is the viscosity of the fluid, k: the permeability of the medium and q the filter velocity of the fluid. The instability of the plane interface between two uniform superposed and streaming fluids through porous medium has been investigated by Sharma and Spanos [5].

The present paper deals with the effect of suspended particles on the stability of superposed streaming fluids in porous medium. The problem finds its usefnlness in chemical engineering, paper and pulp technology and several geophysicM situations. In many geophysicM fluid dynamical problems encountercd, the fluid is electrically conducting and a uniform magnetic field of the Ear th pervades the system. A study has, therefore, also been ruade of the instability of electrically conducting, streaming fluids permeated with suspended particles in porous medium in the presenee of a uniform horizontal magnetic field. These aspects form the subject mat ter of the present papœ

2. Formulation of the problem and perturbation equations

The initial s tat ionary state whose stabili~y we wish to examine is that of an incompressible fluid embœ by suspended particles (fine dust) in which there is horizontal streaming in the x-direction with a velocity U(z) through a homogeneous porous medium. The character of the equilibrium of this initial state is determined by supposing that the system is slightly disturbed and then following its furtlmr evolution.

Let p, p., p and U ( u ( z ) , 0, 0) denote respectively the density, viscosity, pressure

and velocity of the pure fluid; I2(2, t) and N(~, t) denote the velocity and number density of the particles respectively. K = 67rtuT, where r/ is the particle radius, is the Stokes drag coefficient, :~ = (x, y, z) and ~ = (0, 0, 1). Let .q, kj ans : stand for acceleration due to gravity, medium permeability and medium porosity, respec- tively. Suppose that at some presy level z.~ density may change discontinuously and bring into play ettbcts due to 'effective interfacial tension' T.~, and let i�9 de- note the normal to the interface. Then the equations of nmtion, continuity and incompressibility for the fluid,particle mixture through porous medium are

, [ . , 1 ] . 5 7 + (~7 v ) 0 = - v v - f , S - ~ ~7

+ ™ ~ + ~ y ~ ( : - : ~ ) : ~ + y ( r 2 3 9 (1)

v C = o. (.,)

or, c oW + (0. v)z = o, (a)

w]mrc ~�9 - ,:s) denotes Dirac's delta funy

908 Czech J. Phys 42 (1992)

Thy ins tabi l i t y o f streaming tl~lids it) porous m e d i a . . ,

If m N is the mass of particles per unit volume, then the cquations of motion and COlginuity for the particles are

1 e N ( 0 - fO - ~' r = o, (4) y g

O N - œ + v . ( x ~) = o. (5)

The presence of paxtlcles adds an extra force terre, proportional t,o the velocity dit[\',rcnce bct, ween particlcs and fluid which appears in equations of motion (1). Since the force exerted by the fluid on the particles is equal and opposite to that exerted by the particles on the fluid, there nmst be an extra force terre, equal in magnitude but opt)ositc in sign, in the equations of motion (4) fur the particles. The buoyancy force on the particles is neglectcd. Interparticle reactions are also hot y for we assume that the distances between particles are quite large COmlmred with their diameter. These a.ssumptions have been used in writing the equations of motion for the particles. Since the nniform numbcr density of particles is aSSUlll(?d t(> l)0 smatl, the change in pardcle concentration is ignored and so Eq. (5) l>ecomes rcdundant in view of (2) and (4) and h™ not uscd hercafter.

L(ql I‡ ll : U. lU), {S /)™ [51) , g ( l�87 f, ,5) and ~ zs ( 3?�87 ~]�87 z ) denote the t)erturbations in fluid velo('ity U(U(z), 0, 0), fluid density, thfid pressure, part ides vclocity I �9 0, 0) and surfaces of separation respectively. Thon thc linearized prer turbat ion equations of the ttuid-parti(:le las, er become

LOt+™ (~.v)O t' .l™ < ('V -

: - - v ap - .y - ~ ~ + y <y

[(o, :5) ] V �9 i; = 0,

[ O ] dp y ~ } (U" [ ) (5 {, = l ' ~ ( 1 ] "

(G)

(7)

(s )

l™ V = /~r iL (9)

K A T +

hl E(l. (6). b,z, tan be exi)ressed in terlns of the normal colnponent of tlhe velocity ~~'~ ai zs Sill('e (o o) wherc ~he sul)s('ript s distinguishes the value of the quantiiy at z = %.

Czech. J. Phys. 4:2 (1992) 909

R. C. Sharma, N. D. Sharma

Analysing the disturbance into normM modes, we seek solutions whose depen- dence on x, y and t i s of the form

exp[i(k.x + kvy +nt)], (11)

t - -

where n is the growth rate, k = %/k 2 + k~ is the resultant wave number and k,,k v are horizontal wave numbers.

Eliminating ~7 and substituting for 5p and ~z~, Eqs. (6) and (8)-(10) with the help of expression (11) yield

[~-~2 # #KN ] g + ~Tw DU �8 (en + k~U) + ~ + klKN + Ire

= -VSp - ig w(Dp) ,~ + ik2T~ ws 6(z - z~)ff~, en + k~U " en + k~U

(12)

where �8 is the unit vector in the x-direction and D = d/dz. Writing the three component equations of (12) and eliminating u, v and 5p with

the help of (7), we obtain

{[iP (en+k~U)+ # KNlt ] ik, p } D -fi ~ + k l I ( N + t t e D w - 7 ( D U ) w

_k2[ip(en+k~U)+~ ,~'N,, ] -'fi ~ + kl KN + te w

=igk 2 ( D p ) - g T s S ( z - z s ) en+k,U" (13)

3. T w o u n i f o r m s t r e a m i n g f luids s e p a r a t e d by a ho r i zon ta l b o u n d a r y

Let two uniform fluids of densities Pi,P2 permeated with suspended particles number densities N1, N2 be separated by a horizontal boundary at z = 0 and the density P2 of'the upper fluid "be less than the density Pi of the lower fluid so that , in the absence of streaming, the configuration is stable, and the porous medimn throughout is assumed t o b e isotropic and homogeneous. Let the two fluids be streaming with velocities U1 and U2. Then in cach region of constant p, #, N and U, Eq. (13) reduces to

(D 2 - k2)w = 0. (1-4)

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, according to (12), implies that

W

(ch + k,~U) (15)

910 Czech. J. Phys. 42 (1992)

Tbe instability o f s treaming ttuids in porous media...

must be continuous at an interface. (ii) Integrating (13) between z~ - 7/and zs + ~/and passing to the limit 7/= 0, we obtain, in view of (15), the jump condition

�9 tt ttKNtts] D w - ~ ( D U ) w } A~ { [~(en + k~U) + -'s + klKN +

[ ~__~�9 ( W ) (for z = z s ) =igk2 As(p) - En+k=U s '

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

(16)

�9 . e_u_~ ] . D { [ ~ ( r kzU)+ ~ +klKN+Fe] Dw ~~- (DU)w}

Iie , , , , - ~ 1 ~ -k = [ e~ (En + k~:U) + ~'1 + klZ™ #eJ w = igk=(Dp) Eu + k,U" (17)

Here A�87 = .f(zs + 0)- f(zs - 0) is the jump which a quantity experiences at the interface z = z~; and the subscript s distinguishes the value a quantity, known to be continuous at an interface, takes at z = zs.

Since w/(` + kzU) must be continuous on the surface z = 0 and w cannot increase exponentially on either side of the interface, the solutions appropriate for the two regions are

w~ = A(~~ + k.U~) e+% (z < 0) (18)

w2=A(en+kxU2) e -kz. ( z > 0 ) (19)

Applying the boundary condition (16) to the solutions (18) and (19), we obtain the dispersion relation

{ ie Tt 2 -{.. ~-(oL1U 1 -[- OE2U2) - -~�9 (oElVl + oe2v2)

i6 [ Olivl O~2V2 } kl 1 + PlVle + P2P2e n

k,KN------~ 1+ klK-----~~ {~~ + ~ ( ~ , U ~ +y 2 ) - ik , . -~-1 (y + y

k2T ]

PlVl e + p2V2 e ' = O,

kl 1 + klKN~ 1 + klKN2 (20)

Czech. J. Phys. 42 (1992) 911

R. C. Sharma, N. D. Sharnla

where ~~, = p,/(p~ + [�87 ~2 = P2/(P, + P2); ul (= I t , /P~) and u2(= It2/p,e) y the kinemat ic viscosities of fluids 1 and 2, respectively. E(lu~tti(m (20) yields

c i,,. = - 9 7 . (™171 + y - (™ + y

{ [ ~ ]2 i/~,a:oE](_t2 ((/ , ipl--(~2oe2)( U1 - U2) -[- (` la ] "q- OE2/J2 ̀ a: 1

C,. t (.,.2/~;2 , - [ ii:2 T ] } ' / 2 - } - ~ ( U I -- U2) 2 -- g]r (0~1 -- (s ~- ,(](Pl -}- P2)j ' (21)

w h e r e

a, ll (|

1 a l = l + fll ut ™

1 + kj K N I

1 a 2 = l + D2P2 s

1 + kj KN~

W h e n the fluid is pure (i.e. absence of suspended imrticles), al -+ 1~ a2 -+ 1 and Eq. (21) r edHces to t h e r e s u l t of Sh&l'in& a, nd Sp&ilOS [~], E( l. (2~) . S e v e r a l cy of interest are now considered.

(a) A b s e n c e of surface t e n s i o n ( T = O)

In the. J ) s e n c e of surface tension, Eq. (21) bey

lit = -- O~l//lal -~ G!2L'2a2) -- ~�9 1 q-o~2U 2 .b

:::[: ~ (oElv'lal -{- 0:2//26/2) /gl

a l a:2k 2 ]I/~ + ~ ( u , - u=) = - j k ( ~ , - y j, , (22)

(i) W h e n k,,: = 0, Eq. (22) yields

s in = - (c~luj o.1 + a'~u2a2)

2kl

{ _ ~ ]1/~ (23)

W h e n y > ce1, one of the values of in is posi t ive which means tha t the pe r tu r - ba t ions grow with rime and so the sys tem is unsty

912 Czech. J. Phys. 42 (1992)

The instability of streaming fluids in porous m e d i a . . ,

When c~2 < o/1, both the values of in are either real, negative or complex con- jugates with negative real parts and so the system is stable. It is also clear from Eq. (23) that for the special case when perturbations in the direction of streaming are ignored (k~ = 0), the perturbations transverse to the direction of streaming (k.v r 0) are unaffeeted by the presence of streaming. (ii) In every other direction, instability occurs when

OZlOZ2k2 (U1 -- U2) 2 ) ..qk(Ozl -- OZ2), g2 (24)

i.e. for a given difference in velocity U1 - U2 and for a given direction of the wave vector/~, instability occurs for ail wave numbers

k > c~1o~2(U1 - U2) 2 cos 2 O '

where kx = k cos 0, 0 being the angle between the directions of f~(k~, ky,O) and U(U, 0, 0). Hence for a given velocity difference U1 - [72, instability occurs for the least wave number when fo is in the direction of U and this minimum wavenumber, kmin, is given by

g~2 (c~~ - o~2) (26) ~mi~, = o ~ 1 o ~ 2 ( U 1 - U 2 ) 2 "

For k > kmin, t h e system is u n s t a b l e .

(b) Presence of surface tension

In the presence of surface tension, Eq. (21) yields stability if

OE1OE2~g 2 [ k2 T ] ~2 (U1 - U2)2 < qk (0: l - 0!2) -t- g/--~--2"~Dl�8 )J (27)

Since the perturbations most sensitive to Kelvin-Helmholtz instability are in the direction of streaming, we put k~ = k and then the condition of stability beeomes

[OE1 -- OE2 ~~~2(U~ - u~) 2 < S - - + ~(pl + p2) " (2 �8

The right hand side (RHS) of the inequality (28) has a minimum when (d/dk) (RHS) = 0, i.e., when

g(~~ - c~2) k T - (29)

k /91 ~ D2

If k* denotes the value of k giv• by (29), we have stability if

2y (U', - U2) 2 < k*. (30)

c ~ l a z ( m + p2)

Czech, J. Phys. 42 (1992) 913

R. C. Sharma, N. D. Sharma

Substituting the value of k* in accordance with (29), we obtain

2e 2 ,/T.q(al_.~ o~2) (31) (u~ - u2) 2 <

y171 V (pi +p2)

This is the same result as obtMned by Sharma and Spanos [5], Eq. (34), for the instability of streaming fluids ili porous medium in the absence of suspended parti- clcs. The surface tension therefore has stabilizing effect and completely suppresses the Kelvin-Helmholtz instability for small wavelengths. The medium porosity re- duces the stability range given in terres of a difference in streanfing velocities.

4. Effect o f magne t i c field

Here the problem and the configuration is assumed to be the same except that the fluid is electrically conducting (and ignoring interfacial tension effect) and a uniform horizonty magnetic field pervades the system. Then the equations of motion and the Maxwell's equations are

-67 + 7 (57" v)57 = - v p - pgY,

i, 57+KN.I~ 1 ( V x H ) x H , k, - 7 - ( - 57) + (32)

v . ~ = o, (33)

0 ~ e - - ~ = V x (57 x / t ) , (34)

where/ t (H, 0, 0) is the uniform horizontal magnetic field. Equations (2)-(5) remain unaltered.

Let ft(h˜ denote the pcrtnrbation in magnetic field / t . Then the lin- ea�9 perturbation equations~ using expression (11), become

n+ ~ + ~ + k lKN+p,e u

+ P-~ (DU)w = -ik~ 5p, (35)

U

/teH = -ikv(Sp + -~~ (iL'~-h v - ik.J~=), (36)

914 Czecb. J. Phys. 42 (1992)

The instability of streaming//uids in porous media.. .

[™ ) ] u + ky + p KNp, -~1 + k l K N + Ire w

peH (ikJtz - Dh~), = -D6p - g6p + (37)

ikrhx + ikuhu + Dhz = O,

i(r + k~U)hz = ik~Hu + hzDU,

i(en + k~U)hu = ikxHv,

i(r + kxU)hz = ik~Hw,

(38)

(39)

(40)

(41) together with Eqs. (7)-(10). Eliminating h~, hv, hz, u, v and 619 using the relations

ik2u= - ( k~Dw+ky™ (42)

and ik~ kylteH 2 (DU)w D

ik�87 ~ (DU)w - 4~'(en + k~U) 2 ™

�9 # K N p ik~p~H 2 , (43) lpe2 (~,z + kzU) + ~1. + k l K N + l t e 41r(en+kzU)

™ = ik~v - ikyu being the z-component of vorticity, wc obtain

"ikxk~IteH2(DU)w ] D(ADw)- ik~D [œ (DU)w I - D 4zr(en.+ k~U) z B - k2Aw

4~r (en + kzU) k�87 eu + k,~U

ik2/D,,, f P I'eH~k~ ] "1 )

"t il,. k ~ U ) - - - ~ ' - - - ~ - - --~~IteH2 -1

ik~lteH2k 2 i.qk2(Dp)w + w - = 0,

4Tf(en + kxU) en, + k~U (44)

where

attd

B =

A=i---P~ e / + + k~, U ~ Il,

e

K N p kl K N + tre'

•t K N It + kl k I K N +,pe,,

-- n + + + ~ kl K N + I�87

ik2xiteH 2 4~r(~n + h~U)

(45)

(46)

Czech. J. Phys. 42 (1992) 015

t"t. C. Sharma, N. D. Sharma

Here also we consider the case that two uniform, streaming fluids permeated with suspended particles are separated by ~ horizontal boundary z = 0. Then in eack region of constant p, l�87 N z.tld U, Eq. (44) reduces t.o

(D 2 - k2)w = 01 (47)

The boundary conditions t o b e satisfied are then (15) and

{ [ i p ~ K N ~ ] D w Z~o )z(~~ + k.u) + s + k~KW + ~~

l k , # y Ao 4--;- ~,~ u

-ig,2 ~0(p)(~,~™231 (4s)

Applying the boundary condition (48) to the appropriate solutions (18) and (19), we obtain the dispersion relation

s -- k l (~ nu c~2v2a2) n

+ 7 (~~u~ + ~~u~) - ik~ (y + ~2.~a2U2)

- [gk(~~ - ~~) + 2k~v2] } = 0, (49)

where 1 1

ai = 1 + PlUie , a2 = 1 + , p2™ 1 + k~ K-----~-I 1 + 77X-%

Equation (49) yields

s�99238 - P ' H 2

47r(p~ + p2)

in ~ [ ~ I~ +~176 ~~'I~176 ]

{ [ ~ 1 ~ ~~1o~~: ~~~ ~~~~ 22~ ~ (OEll]I (Il Jl- OE2u2a2) + g2

ik~ozl a:2 ~ 1/2 (alvl - a2v2)(U1 - U2) - bk(~l - a~) + 2k~V21 ~ �9 (50)

kl

(i) VChen k~ = 0, Eq. (.50) implies that the system is stable ,a, hcn a2 < cci and unstable when a:2 > o , . Equation (50) also implies that perturbations transverse

916 Czech. J. Phys. 42 (1992)

The instability of streaming fluids in porous m e d i a . . .

to the direction of streaming (k u # 0) are unaffected by the prcsence of streaming for the speeial case when perturbations in the direction of streaming are ignored. (ii) In every other direction, instability occurs when

cqoz2k~ (U1 - U2) 2 > gk (a l c~2) + 2k~V~. g2 (51)

Thus for a given difference in velocity [71 - U2 and for a given direction of the wave vector k, instability occurs for all wavenumbers

k > (52) cos~ o[o21o2~(u~ - u~ )2 - 2c2v~] '

whe�9 0 is the angle between the directions of ]~(k,, ky, 0) and LT(U, 0, 0), i.e., k, = k cos0. Hence for a given velocity differenee U1 - U2, instability oecurs for the least wave mimber when fo is in the direction of U and this minimum wavenumber, kmi~,, is given by

kmin ---- gE'2(o21 -- O22) OE1OE2(U 1 _ U2)2 _ 2 s 2 . ( 5 3 )

For k > kmin, the system is unstable. It is clear from Eq. (53) that the presence of magnetic field increases the value of kmin for which the system is unstable. Thus the instability of the system is postponed. We thus obtain the stabilizing effect of magnetic field. (iii) Since the perturbations most sensitive to Kelvin-Helmholtz instability are in the direction of streaming, we put kz = k.

When magnetic field is present. Equation (50) yi• stability if

o21(~2 (U1 - Un < g [c~1 - er2 + #~H2 ] (54) c ~ k 2~~(pl + e~)

The right hand side (RHS) of the above iuequality has a minimum when (d/dk) (RHS) = 0, i.e., when k --+ oe. Therefore we shall have stability if

2e2V�9 ( u ~ - u 2 ) 2 < - - ( 5 5 )

O21 ~

The magnetic field therefore has stabilizing effect and completely suppresses the Kelvin-Helmholtz instability for small wavelengths. The mœ porosity reduces the stability range given in terms of a differœ in streaming velocities and the Alfv› vclocity.

5. Conc lus ion

The instability of the plane interface between two uniform superposed and streaming fluids through porous medium has been investig~~ted by Sharma and

Czech. Ji Phys. 42 (1992) 917

R . C. S h a r w a , N . D. S h a r m a : The i n s t a b i l l t y o f s t r e a m i n g t t , i d s . . .

Spanos [5]. Since the knowledge concerning fluid-particle mixtures is not commen- surate with their scientific and industrial importance and the effect of suspended particles on the stability of superposed streaming fluids might be of industrial and chemical engineering importance, here a study has been ruade of the instability of streaming fluids permeated with suspended particles through porous medium. In many geophysical fluid dynamical problems, the fluid is electrically conducting and a uniform magnetic field of the Earth pe.rvades the system; the effect of a uniform horizontal magnetic field on the problem is also considered.

The magnetic field and surface tension are round to have stabilizing effects and are able to suppress the Kelvin-Helmholtz instability for small wavelength per- turbations. The medium porosity reduces the stability range given in terres of a difference in streaming velocities and the Alfv› velocity. The suspended particles do not affect the above results.

Appendix: Interfacial tension term in porous media

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

(p~ - p2)~=~. = -T. (c~ + c2),

wherc T~ fs an 'effective interfaciM tension' and cl, c2 are the signed principal curwttures of the macroscopic interface. This fs the first approximation to the problem as iii actual practice there fs no 'effective interfacial tension' but in the absence ofany better theory, this fs being used as suggested by Chuoke et al. [2]. Let 6Zs(X, Y, t) denote the perturbation in surface of separation zs and Os stands for the normal to the interface. In equation (6), use has been ruade of the approximations

0 2 (9 2 c ~ ~ Ox z(~z~), c 2 ~ 0y ~ ( 6 z d .

since k(bz~) fs small.

R cfi ,r( ,nccs

[I] Chandrasekhar S.: IIjdrodynamic and Ilydromagnetic Slnbilily. I)over Publications, New

York, 1981.

[2] Chuoke R. L., Me,fs van P., Poel van dy C.: Trnns. AIME 211; (1959) 188.

[3} Palaniswamy V. 1., Purushothanl C. M.: Phys. Fluids 24 (1981) 1224.

[4] Sy J. W., Scgel L. A.: Phys. Fluids 16 (197:1) 1573.

[5] Sharma R. C., Spanos T. J. T.: Cana dian J. Phys. 60 (1982) 1391.

918 Czech. J. Phys. 42 (1992)