stability of two superposed viscous-viscoelastic fluids · stability of two superposed...

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STABILITY OF TWO SUPERPOSED VISCOUS-VISCOELASTIC FLUIDS by Pardeep KUMAR and Roshan LAL Original scientific paper UDC: 532.516 BIBLID: 0354-9836, 9 (2005), 2, 87-95 The Rayleigh-Taylor instability of a Newtonian viscous fluid overlying a Rivlin-Ericksen viscoelastic fluid is considered. Upon application of nor- mal mode technique, the dispersion relation is obtained. As in both Newto- nian viscous-viscous fluids the system is stable in the potentially stable case and unstable in the potentially unstable case, this holds for the present problem also. The behaviour of growth rates with respect to kinematic vis- cosity and kinematic viscoelasticity parameters are examined numerically and it is found that both kinematic viscosity and kinematic viscoelasticity have stabilizing effect. Key words: Rayleigh-Taylor instability, normal mode technique, linearized theory, Rivlin-Ericksen viscoelastic fluid Introduction The instability of the plane interface separating two Newtonian fluids when one is accelerated towards the other or when one is superposed over the other has been stud- ied by several authors, and Chandrasekhar [1] has given a detailed account of these inves- tigations. The influence of viscosity on the stability of the plane interface separating two incompressible superposed fluids of uniform densities, when the whole system is im- mersed in a uniform horizontal magnetic field, is studied by Bhatia [2]. He has carried out the stability analysis for two fluids of equal kinematic viscosities and different uniform densities. A good account of hydrodynamic stability problems has also been given by Drazin and Reid [3], and Joseph [4]. The fluids have been considered to be Newtonian in all the above studies. Sharma and Sharma [5] have studied the stability of the plane interface separating two viscoelastic (Oldroydian) superposed fluids of uniform densities. In another study, Sharma [6] has studied the instability of the plane interface between two Oldroydian viscoelastic superposed conducting fluids in the presence of a uniform magnetic field. Fredricksen [7] has given a good review of non-Newtonian fluids. Molten plastics, petro- leum oil additives and whipped cream are examples of incompressible viscoelastic fluids. There are many non-Newtonian fluids that cannot be characterized by Oldroyd’s [8] con- stitutive relations. The Rivlin-Ericksen elastico-viscous fluid is one such fluid. 87

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Page 1: STABILITY OF TWO SUPERPOSED VISCOUS-VISCOELASTIC FLUIDS · STABILITY OF TWO SUPERPOSED VISCOUS-VISCOELASTIC FLUIDS by Pardeep KUMAR and Roshan LAL Original scientific paper UDC: 532.516

STABILITY OF TWO SUPERPOSED VISCOUS-VISCOELASTIC

FLUIDS

by

Pardeep KUMAR and Roshan LAL

Original scientific paperUDC: 532.516

BIBLID: 0354-9836, 9 (2005), 2, 87-95

The Ray leigh-Tay lor in sta bil ity of a New to nian vis cous fluid over ly ing aRivlin-Ericksen viscoelastic fluid is con sid ered. Upon ap pli ca tion of nor -mal mode tech nique, the dis per sion re la tion is ob tained. As in both New to -nian vis cous-vis cous flu ids the sys tem is sta ble in the po ten tially sta ble caseand un sta ble in the po ten tially un sta ble case, this holds for the pres entprob lem also. The be hav iour of growth rates with re spect to ki ne matic vis -cos ity and ki ne matic viscoelasticity pa ram e ters are ex am ined nu mer i callyand it is found that both ki ne matic vis cos ity and ki ne matic viscoelasticityhave sta bi liz ing ef fect.

Key words: Ray leigh-Tay lor in sta bil ity, nor mal mode tech nique, linearized the ory, Rivlin-Ericksen viscoelastic fluid

Introduction

The in sta bil ity of the plane in ter face sep a rat ing two New to nian flu ids when oneis ac cel er ated to wards the other or when one is superposed over the other has been stud -ied by sev eral au thors, and Chandrasekhar [1] has given a de tailed ac count of these in ves -ti ga tions. The in flu ence of vis cos ity on the sta bil ity of the plane in ter face sep a rat ing twoin com press ible superposed flu ids of uni form den si ties, when the whole sys tem is im -mersed in a uni form hor i zon tal mag netic field, is stud ied by Bhatia [2]. He has car ried out the sta bil ity anal y sis for two flu ids of equal ki ne matic vis cos i ties and dif fer ent uni formden si ties. A good ac count of hy dro dy namic sta bil ity prob lems has also been given byDrazin and Reid [3], and Jo seph [4].

The flu ids have been con sid ered to be New to nian in all the above stud ies.Sharma and Sharma [5] have stud ied the sta bil ity of the plane in ter face sep a rat ing twoviscoelastic (Oldroydian) superposed flu ids of uni form den si ties. In an other study,Sharma [6] has stud ied the in sta bil ity of the plane in ter face be tween two Oldroydianviscoelastic superposed con duct ing flu ids in the pres ence of a uni form mag netic field.Fredricksen [7] has given a good re view of non-New to nian flu ids. Mol ten plas tics, pe tro -leum oil ad di tives and whipped cream are ex am ples of in com press ible viscoelastic flu ids. There are many non-New to nian flu ids that can not be char ac ter ized by Oldroyd’s [8] con -sti tu tive re la tions. The Rivlin-Ericksen elastico-vis cous fluid is one such fluid.

87

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Srivastava and Singh [9] have stud ied the un steady flow of a dusty elastico-vis cousRivlin-Ericksen fluid through chan nels of dif fer ent cross-sec tions in the pres ence of atime de pend ent pres sure gra di ent. In an other study Garg et al. [10] have stud ied the rec ti -lin ear os cil la tions of a sphere along its di am e ter in a con duct ing dusty Rivlin-Ericksenfluid in the pres ence of a uni form mag netic field. Ther mal in sta bil ity in Rivlin-Ericksenelastico-vis cous fluid in pres ence of mag netic field and ro ta tion, sep a rately, has been in -ves ti gated by Sharma and Kumar [11, 12]. In an other study, Sharma and Kumar [13] have stud ied the hydromagnetic sta bil ity of two Rivlin-Ericksen elastico-vis cous superposedcon duct ing flu ids and the anal y sis has been car ried out, for two highly vis cous flu ids ofequal ki ne matic vis cos i ties and equal ki ne matic viscoelasticities.

It is this class of elastico-vis cous flu ids we are in ter ested in, par tic u larly to studythe sta bil ity of the plane in ter face be tween vis cous and viscoelastic (Rivlin-Ericksen)flu ids.

The sta bil ity of the plane in ter face be tween vis cous (New to nian) andviscoelastic (Rivlin-Ericksen) flu ids may find ap pli ca tions in geo phys ics, chem i cal tech -nol ogy, and bio-me chan ics and is there fore, stud ied in the present paper.

Formulation of the problemand perturbation equations

Con sider a static state in which an in com press ible Rivlin-Ericksen viscoelasticfluid is ar ranged in hor i zon tal strata and the pres sure p and the den sity r are func tions ofthe ver ti cal co-or di nate z only. The char ac ter of the equi lib rium of this ini tial static stateis de ter mined, as usual, by sup pos ing that the sys tem is slightly dis turbed and then by fol -low ing its fur ther evo lu tion.

Let rn (u, n, w), dr, and dp de note the per tur ba tion in fluid ve loc ity (0, 0, 0), den -

sity r and pres sure p, re spec tively. Then the linearized per tur ba tion equa tions rel e vant tothe prob lem are:

r r r n nm m¶

¶d d

rr rv

tp g

tv

z t z= -Ñ + - + ¢

æ

èç

ö

ø÷Ñ + +

¢æ2 d

d

d

dèç

ö

ø÷ +æ

èç

ö

ø÷

w

x

v

z

r

(1)

Ñ × =rv 0 (2)

¶twDdr r= - (3)

where n m r( / )= and ¢ = ¢n m r( / ) denote the kinematic viscosity and the kinematicviscoelasticity of the fluid,

rg g( , , )0 0 - is the acceleration due to gravity, x x y z= ( , , ) and

D z=d d/ . Equation (3) ensures that the density of every particle remains unchanged aswe follow it with its motion.

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THERMAL SCIENCE: Vol. 9 (2005), No. 2, pp. 87-95

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An a lyz ing the dis tur bances into nor mal modes, we seek so lu tions whose de -pend ence on x, y, and t is given by:

exp(ikx x + iky y + nt) (4)

where kx, and ky are horizontal wave numbers, k k k2 2 2= +x y , and n is a complex constant.For per tur ba tions of the form (4), eqs. (1) – (3) give:

r r n n m mnu ik p n D k u ik w Du D nD= - + + ¢ - + + + ¢x xd ( )( ) ( )( )2 2 (5)

r n r n n n n m mn ik p n D k ik w D D nD= - + + ¢ - + + + ¢y yd ( )( ) ( )( )2 2 (6)

r r r n n m mnw D p n D k w Dw D nD= - - + + ¢ - + + ¢d dg ( )( ) ( )2 2 2 (7)

ik u ik Dwx y+ + =n 0 (8)

ndr = –wDr (9)

Elim i nat ing dp be tween eqs. (5)-(7) and us ing eqs. (8) and (9), we ob tain:

n D Dw k w D n D k Dw

k n D

[ ( ) ] { [ ( )( ) ]

( )(

r r r n n

r n n

- - + ¢ - -

- + ¢

2 2 2

2 2 22

2 2 22

- + -

- + ¢ + - +

k wk

nD w

D D nD D k w k D nD

) } ( )

{ [( )( ) ] (

gr

m m m ¢ =m )( )}Dw 0 (10)

Two uniform viscous and viscoelastic (Rivlin-Ericksen)fluids separated by a horizontal boundary

Con sider the case of two uni form flu ids of den si ties, vis cos i ties; r2, m2 (up perNew to nian fluid) and r1, m1 (lower Rivlin-Ericksen viscoelastic fluid) sep a rated by a hor -i zon tal bound ary at z = 0. Then, in each re gion of con stant r and con stant m, m', eq. (10)be comes:

(D2 – k2)(D2 – q2)w = 0 (11)

where q k n n2 2= + + ¢/ ( )n n .Since w must van ish both when z ® +4 (for the up per fluid) and z ® –4 (for the

lower fluid), the gen eral so lu tion of eq. (11) can be writ ten as:

w zkz q z1 1 2

1 0= + <+ +A e A e ( ) (12)

89

Kumar, P., Lal, R.: Stability of Two Superposed Viscous-Viscoelastic Fluids

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w kz q z2 3 4

2= +- -A e A e (13)

where A1, A2, A3, and A4 are constants of integration,

q kn

nq k

n1

2

1 12

2

2

= ++ ¢

= +n n n

and (14)

In writ ing the so lu tions (12) and (13), it is as sumed that q1 and q2 are so de finedthat their real parts are pos i tive.

Boundary conditions

The so lu tions (12) and (13) must sat isfy cer tain bound ary con di tions. Clearly,all three com po nents of ve loc ity and tan gen tial vis cous stresses must be con tin u ous. Thecon ti nu ity of Dw fol lows from eq. (8) and the con ti nu ity of u and n. Since:

t m m m mxz xz xe= + ¢æ

èç

ö

ø÷ = + ¢ +2 2

¶tn Du ik w( )( )

andt m m m myz yz ye= + ¢

æ

èç

ö

ø÷ = + ¢ +2 2

¶tn Dv ik w( )( )

are continuous,ik ik n D k wx xz y yzt t m m+ = - + ¢ +( )( )2 2

is continuous across an interface between the two fluids. Hence, the boundary conditionsto be satisfied at the interface z = 0 are that:

w (15)

Dw (16)and

( )( )m m+ ¢ +n D k w2 2 (17)

must be continuous. Integrating eq. (10) across the interface z = 0, we obtain anothercondition:

[ ] ( ) ( )( )r r m m m2 2 1 1 0 22 2

2 1 12 21 1

Dw Dwn

D k Dwn

n D kz- - - - + ¢ -= Dw

k

nw

k

nn Dw

z1

02

2 2 1 0

2

2 1 12

é

ëêù

ûú=

= - - - - - ¢

=

g[ ] [ ]( )r r m m m 0 (18)

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THERMAL SCIENCE: Vol. 9 (2005), No. 2, pp. 87-95

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where w0, (Dw)0 are the common values of w1, w2 and Dw1, Dw2 respectively, at z = 0.

Dispersion relation and discussion

Ap ply ing the bound ary con di tions (15)-(18) to the so lu tions (12) and (13), andelim i nat ing the con stants A1, A2, A3, and A4 from re sult ing equa tions, we ob tain:

det(aij) = 0 (19)

where i, j = 1, 2, 3, 4 and

a a a a a a k a q a q a11 12 13 14 21 23 22 1 24 2 311 1 2= = = = - = = = = =, , , , , k n

a n q k a k a

21 1

32 1 1 12 2

332

2 342

( )

( )( ), ,

m m

m m m m

+ ¢

= + ¢ + = - = - 2 22 2

41 1

2

2 2 1 1 1 1 422 2

( )

( ),

q k

aR k

nn a

R k

+

= - + + - - ¢ = +a n a n a n an

n q

aR k

nn

( ) ,

(

n a n a n a

a n a n a n

2 2 1 1 1 1 1

43 2

2

2 2 1 12

- - ¢

= - + - - - ¢1 1 44 2 2 1 1 1 1 2

1 21 2

1 2

2a n a n a n a

ar

r r

), ( ) ,

,,

aR k

nn q= - - - ¢

=+

andg

Rk

n= -

2 2 1( )a a

(20)

Eq. (19) yields the fol low ing char ac ter is tic equa tion:

( ) ( ) ( )(q k k nk

nn1

22 2 1 1 1 1 2 2 1 1 1 12- - - - ¢ - - - ¢n a n a n a n a n a n a q k

R q k k

2 2

2 2 22 21 2

- +é

ëêù

ûú+

ìíï

îï

+ - -üýï

þï-

)

( )( )( ) (

a

n a n1 1 1 1 12

2 2 1 1 1 1 2

a n a

n a n a n a

+ ¢ - ×ìíï

îï

× - - - ¢ -

n q k

k

nn q

)( )

( )( k

q kk

nn q

)

( )( ) ( )(

ëêù

ûú+

+ - - - ¢

a

n a n a n a n a

2

2 2 22 2

2 2 1 1 1 1 1 - +é

ëêù

ûúüýþ

+

+ - + ¢ - × -

k

q k n q k R

)

( ) ( )( ) ( )

a

n a n a

1

2 1 1 1 1 12 2 1 +

ìíï

îï

+ - - ¢ - - ¢2 22 2 1 1 1 1 2 2 1 1 1 1k n

k

nn q( ) ( )(n a n a n a n a n a n a 1 1 0- +

é

ëêù

ûúüýþ

=k ) a (21)

91

Kumar, P., Lal, R.: Stability of Two Superposed Viscous-Viscoelastic Fluids

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The dis per sion re la tion (21) is quite com pli cated, as the val ues of q1 and q2 in -volve square roots. We, there fore, make the as sump tion that the flu ids are of high vis cos -ity and high viscoelasticity. Un der this as sump tion, we have:

q kn

k nk

n

k nk

n

k n= +

+ ¢= +

+ ¢

é

ëê

ù

ûú = +

+ ¢1 1

1

2 22 2( ) ( ) ( )n n n n n n(22)

so thatq k

n

k nq k

n

k1

12

22 2- =

+ ¢- =

( )n n nand (23)

Sub sti tut ing the val ues of q1 – k and q2 – k from eqs. (22) and (23) in eq. (21), weob tain the dis per sion re la tion:

a n a n a n a n a n1 12

1 13

1 1 2 22

1 122 1 1 4¢ ¢ + + + + ¢ +

+

[ ] [( )( )]

[

k n k n

4 221 2 1 2

212

12

22

22

1 1 2 1

k k

k n

n n a a a n a n

a n a a

+ + -

- ¢ - -

( )

( )]g gk ( )( )a a a n a n2 1 1 1 2 2 0- + = (24)

As sum ing n1 = n2 = n ([1], p. 443), as this sim pli fy ing as sump tion does not ob -scure any of the es sen tial fea tures of the prob lem, the dis per sion re la tion (24) be comes:

a n a n n a n

n a n

1 12

1 13 2

1 12

2 21

2 1 1 4

2

¢ ¢ + + + ¢ +

+ - ¢

[ ] [ ( )]

[

k n k n

k 1 2 1 2 1 0g gk n k( )] ( )a a n a a- - - = (25)

Stable case

For the po ten tially sta ble ar range ment (a2 < a1), all the co ef fi cients of eq. (25)are pos i tive. So, all the roots of eq. (25) are ei ther real and neg a tive or there are com plexroots (which oc cur in pairs) with neg a tive real parts and the rest neg a tive real roots. Thesys tem is, there fore, sta ble in each case. Hence the po ten tially sta ble ar range ment re -mains sta ble for the sta bil ity of two superposed vis cous-viscoelastic (Rivlin-Ericksen)flu ids. This re sult is also true when the flu ids are vis cous [1].

Unstable case

For the po ten tially un sta ble ar range ment (a2 > a1), the con stant term in eq. (25)is neg a tive. The eq. (25), there fore, al lows at least one change of sign and so has at leastone pos i tive real root. The oc cur rence of pos i tive root im plies that sys tem is un sta ble fordis tur bances of all wave num bers. The sys tem is, there fore, un sta ble for po ten tially un -sta ble case.

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THERMAL SCIENCE: Vol. 9 (2005), No. 2, pp. 87-95

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We now ex am ine the be hav iour of growth rates with re spect to ki ne matic vis -cos ity and ki ne matic viscoelasticity nu mer i cally. We have plot ted the growth rate n (pos -i tive real value) ver sus the wave num ber k for sev eral val ues of the ki ne matic vis cos ity and the ki ne matic viscoelasticity in fig. 1 and fig. 2, re spec tively.

In fig. 1, the growth rate n is plot ted against wave num ber k, for fixed value of ¢ =n1 2, a1= 0.38, a2 = 0.62 and for n = 1, 2, 5. The growth rate de creases with in crease in

ki ne matic vis cos ity show ing its sta bi liz ing ef fect on the sys tem. In fig. 2, growth rate n is

93

Kumar, P., Lal, R.: Stability of Two Superposed Viscous-Viscoelastic Fluids

Figure 1. The variation of the growth rate n (positive real value) with the wave number k for the kinematic viscosities n = 1, 2, 5 when a1 = 0.38, a2 = 0.62, and ¢n1 = 2

Figure 2. The variation of the growth rate n (positive real value) with the wave number k for the kinematic viscoelasticities ¢n1= 3, 5, 10 when a1 = 0.38, a2 = 0.62, and n = 2

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plot ted against wave num ber for fixed n = 2, a1= 0.38, a2 = 0.62 and for ¢ =n1 3, 5, 10. It isseen that for the same wave num ber k, the growth rate n de creases as the ki ne maticviscoeasticity ¢n1 in creases, show ing the sta bi liz ing char ac ter of ki ne matic viscoelas-ticity.

Conclusions

A de tailed ac count of sta bil ity of superposed New to nian flu ids, un der vary ingas sump tions of hy dro dy nam ics, was given by Chandrasekhar [1]. With the grow ing im -por tance of non-New to nian flu ids in chem i cal en gi neer ing, mod ern tech nol ogy and in -dus try, the in ves ti ga tions on such flu ids are de sir able. The Rivlin-Ericksen fluid is onesuch im por tant non-New to nian (viscoelastic) fluid. The Ray leigh-Tay lor in sta bil ity of aNew to nian vis cous fluid over ly ing Rivlin-Ericksen viscoelastic fluid is con sid ered in thepres ent pa per. Fol low ing the linearized per tur ba tion the ory and nor mal mode anal y sis,the dis per sion re la tion is ob tained. The sta bil ity anal y sis has been car ried out, for math e -mat i cal sim plic ity, for two highly vis cous and viscoelastic flu ids of equal ki ne matic vis -cos i ties. As in both New to nian vis cous-vis cous flu ids, the sys tem is found to be sta ble for sta ble con fig u ra tion and un sta ble for un sta ble con fig u ra tion for the pres ent prob lem also.The dis per sion re la tion is also solved nu mer i cally and it is found that both ki ne matic vis -cos ity and ki ne matic viscoelasticity have sta bi liz ing ef fect on the growth rate.

Acknowledgments

The au thors are grate ful to the ref eree for his crit i cal com ments, which led to asig nif i cant im prove ment of the pa per.

Nomenclature

g – acceleration due to gravity, [m/s2]rg – gravity field, [m/s2]k – wave-number, [1/m]kx, ky – horizontal wave-numbers, [1/m]n – growth rate, [1/s]p – fluid pressure, [Pa]t – time, [s]rv – fluid velocity, [m/s]

Greek letters

m – dynamic viscosity [kg/ms]r – density, [kg/m3]n – kinematic viscosity, [m2/s]n’ – kinematic viscoelasticity, [m2/s]

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References

[1] Chandrasekhar, S., Hy dro dy namic and Hydromagnetic Sta bil ity, Do ver Pub li ca tions, NewYork, 1981

[2] Bhatia, P. K., Ray leigh-Tay lor In sta bil ity of Two Vis cous Superposed Con duct ing Flu ids,Nuov. Cim., B19 (1974), pp. 161-168

[3] Drazin, P. G., Reid, W. H., Hy dro dy namic Sta bil ity, University Press, Cam bridge, 1981[4] Jo seph, D. D., Sta bil ity of Fluid Mo tions II, Springer-Verlag, NewYork, 1976[5] Sharma, R. C., Sharma, K. C., Ray leigh-Tay lor In sta bil ity of Two Viscoelastic Superposed

Flu ids, Acta Phys. Hungar., 45 (1978), 3, pp. 213-220[6] Sharma, R. C., In sta bil ity of the Plane In ter face be tween Two Viscoelastic Superposed Con -

duct ing Flu ids, J. Math. Phys. Sci., 12 (1978), 6, pp. 603-613[7] Fredricksen, A. G., Prin ci ples and Ap pli ca tions of Rhe ol ogy, Prentice-Hall Inc., New Jer sey,

USA, 1964[8] Oldroyd, J. G., Non-New to nian Ef fects in Steady Mo tion of Some Ide al ized Elastico-Vis cous

Liq uids, Proc. Roy. Soc. Lon don, A 245 (1958), pp. 278-297[9] Srivastava, R. K., Singh, K. K., Drag on a Sphere Os cil lat ing in a Con duct ing Dusty Vis cous

Fluid in Pres ence of Uni form Mag netic Field, Bull. Cal. Math. Soc., 80 (1988), 4, pp. 286-291[10] Garg, A., Srivastava, R. K., Singh, K. K., Drag on a Sphere Os cil lat ing in Con duct ing Dusty

Rivlin-Ericksen Elastico-Vis cous Fluid, Proc. Nat. Acad. Aci. In dia, 64A (1994), III, pp.355-361

[11] Sharma, R. C., Kumar, P., Ef fect of Ro ta tion on Ther mal In sta bil ity in Rivlin-EricksenElastico-Vis cous Fluid, Z. Naturforsch., 51a (1996), pp. 821-824

[12] Sharma, R. C., Kumar, P., Ther mal In sta bil ity in Rivlin-Ericksen Elastico-Vis cous Fluid inHydromagnetics, Z. Naturforsch., 52a (1997), pp. 369-371

[13] Sharma, R. C., Kumar, P., Hydromagnetics Sta bil ity of Rivlin-Ericksen Elastico-Vis cousSuperposed Con duct ing Flu ids, Z. Naturforsch., 52a (1997), pp. 528-532

Author’s address:

P. Kumar, R. LalDepartment of MathematicsICDEOL, Himachal Pradesh UniversitySummer-Hill, Shimla-171005, India

Corresponding author (P. Kumar):E-mail: [email protected]

Paper submitted: March 5, 2005Paper revised: June 25, 2005Paper accepted: August 18, 2005

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Kumar, P., Lal, R.: Stability of Two Superposed Viscous-Viscoelastic Fluids