Electroviscoelastic Rayleigh-Taylor instability of Kelvin fluids. Effect of a constant tangential electric field

FLUIDDYNAMICS RESEARCH Fluid Dynamics Research 19 (1997) 327-341

ELSEVIER

Electroviscoelastic Rayleigh-Taylor instability of Kelvin fluids. Effect of a constant tangential electric field A b o u E1 Magd A. M o h a m e d , Elsayed F.A. Elshehawey, Yusry O. E1-Dib* Department of Mathematics, Facul~ of Education, Ain Shams University, Heliopolis, Cairo, Egypt Received 23 October 1995; revised 30 April 1996; accepted 7 October 1996

Abstract

By using the method of multiple scales, an investigation of the Rayleigh-Taylor problem of interfacial stability in a two-layer system of electroviscoelastic Kelvin fluids is performed. Examination of the effects on the stability of the interface by applying a constant tangential electric field is made. Through the linear perturbation analysis a fourth-order partial differential equation which governs the motion of rheological fluids is obtained. The scheme reported here depends on the idea that the flow of a slightly non-Newtonian fluid is about the same as that for a Newtonian fluid. The contribution of elasticity is included in the first-order problem. A solvability condition is obtained in this analysis. A first-order differential equation which controls the surface deflection is obtained and solved. Also, stability conditions are introduced theoretically. Some graphs are drawn to indicate the stability regions. The case of large viscosity is considered for numerical calculations. It is found that the elasticity parameter plays a destabilizing role under the effect of a tangential electric field, while the viscosity having a damping nature in Newtonian fluids plays a dual role in non-Newtonian fluids. It is shown that the electric field plays a dual role in stability criteria.

1. Introduction In most materials under appropriate circumstances, effects of both elasticity and viscosity are noticeable. If these effects are not further complicated by behavior that is neither elasticity nor viscosity, we call the material viscoelastic. The term viscoelastic will be used to describe the properties of materials which, under appropriate conditions, are able to store both energy in elastic deformation and dissipate energy as heat. The stability of viscoelastic fluids has received increasing interest due to its technological applications such as in petroleum industries, coextrusion of fluids, fiber and many other industries.

*Present address: Department of Mathematics, Faculty of Applied Science Umm-Qura University, P.O. Box 3711, Makkah, Saudi Arabia. 0169-5983 / 97 / $17.00 © 1997 The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved. PII S0 1 6 9 - 5 9 8 3 ( 9 6 ) 0 0 0 5 2 - 4

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Stability used to rupture biological and artificial membranes and thin films is involved in the mechanisms of a wide variety of biological and engineering processes and is important for practical applications. Typical examples are: membrane fusion, membrane rupture and mechanical destruction, cell division and fusion, colloid stability, etc. (Evans and Hochmuth, 1976). Recently, several experimental studies have been made on the problem of interfacial viscoelastic stability. Lee and White (1974) studied the deformation of an interface between two viscoelastic fluids and showed the existence of instability in these flows. Han et al. (1984) examined instability of two-polymer melt flows. In this investigation the region of interfacial stability was delineated as a function of viscosity ratio and layer-depth ratio. Wilson and Khomami (1992, 1993a, b) examined experimentally an instability of a multilayer flow of viscoelastic fluids, where stability and growth rate are determined as a function of disturbance wave number and layer-depth ratio. Rheological studies have indicated that the biological cell membrane exhibits elastic as well as viscous behavior during mechanical deformation. This is particularly true for the red blood cell (Pohl, 1978). In fact, research efforts attempted by Skalak (1973) to model red blood cell membrane deformation have focused primarily on the elastic nature of the membrane rather than its viscous property. This viscoelastic behavior was suggested by micropipet experiments, which is interpreted in terms of a two-dimensional linear Kelvin model (1982, 1983). Application of an electric field to the fluids introduces a new aspect to the stability problem. Electrohydrodynamic instability of a viscoelastic liquid layer with simultaneous applications of a vertical electric field and a vertical temperature gradient has been studied by Takashima and Ghosh (1979). An expanding field of applications is observed in molecular biology. Viscoelastic films were treated by Maldarelli and Jain (1982a, b), who extended their studies to include small biological cell membranes. The above authors (1982a) dealt with internal electric forces in their studies. However, external electric forces and stresses are important because they can exert a relatively large net force (Zelazo and Melcher, 1974). Mohamed et al. (1994, 1995) studied an electro-viscoelastic Rayleigh-Taylor instability of Maxwell fluids under the influence of a tangential electric field. In their study they found that the increase in the relaxation time has a destabilizing influence. Therefore, the kinematic viscosity has changed its role when the fluid has a Maxwellian nature. The analysis presented in this work is concerned with the linear instability of two viscoelastic fluids with respect to the onset of surface waves under the influence of gravity and a tangential electric field. This problem has been treated by Chandrasekhar (1961) for viscous fluids in the absence of elasticity or electric field. Instability of a viscous fluid with a constant electric field has been treated by Melcher and Schwarz (1968). Kelvin fluids are considered here because of an experimental evidence demonstrated by Evans (1976) indicating that the mechanical behavior of a biological cell membrane (specifically the red blood cell membrane) may be described by the use of this model. One of the most difficult problems in mechanics is the determination of the parameters concerned with the shape of the interface between two viscoelastic fluids. Owing to the mathematic complexity the use of perturbation technique is quite useful. A perturbation scheme for slightly non-Newtonian fluid was first successfully applied to the Rayleigh-Taylor instability by Rivlin and Ericksen (1955) and Mohamed et al. (1994). A perturbation scheme with multiple scales has also been applied in terms of electric models (see M o h a m e d et al., 1994, 1995; EI-Dib, 1994, 1995 and others).

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2. Formulation of the problem We shall study two-dimensional progressive waves at the interface of the two semi-infinite Kelvin-type viscoelastic fluids. These fluids are regarded as dielectric, homogeneous, isotropic and incompressible. The Cartesian coordinate system (x, y) is introduced (See Fig. 1), where the interface is at y = 0 in the absence of the wave. The upper and the lower densities of the fluids are p(a) and p(2) (p(1) > p(2)), respectively. The gravity g acts in the negative y-direction. The fluids are subjected to an external electric field Eo in x-direction. The basic state is motionless, and the interfacial perturbation produces the velocity vector v(x, y, t) and the displacement vector u(x, y, t), which are related by the following equation:

8 v ==u

C[

+ (v.V)u.

(1)

There are two surface forces that must be accounted for by the stress tensor o-u. One results from the effect of the viscoelastic force given by a[j s = -- P(Si; +

#'~ + G

(2)

LCXj + 8 x i J '

where P is the hydrostatic pressure,/t is the coefficient of viscosity and G is the elastic moduli. The other surface force is due to the electric surface force. Thus the Maxwell's stress tensor (Melcher (1963)) is

qie)"e

=

eEiEi

E2(Sq,

- - ~1 ~,

(3)

Y

Fluld (1) P

(1)

, c

(1)

, ~

(1)

,

G(1)

o g

~

)E

) X

p

(2)

, c

(2)

' /~

(2)

'

G(2)

)E )E

0

o

l g

Fluid (2)

Fig. 1. S k e t c h o f t h e s y s t e m u n d e r c o n s i d e r a t i o n . T h e y axis is t a k e n vertically u p w a r d s . T h e x axis is t a k e n h o r i z o n t a l l y at t h e fiat interface. Eo is t h e t a n g e n t i a l electric field in t h e fluid l a y e r s . / 1 ) a n d / 2 ) a r e t h e fluid's densities, E~1~ a n d c ~2) a r e t h e dielectric constants,/~(1) a n d f 2 ) a r e t h e v i s c o s i t y coefficients, G I1) a n d G (2) a r e t h e elastic p a r a m e t e r s a n d ~ is t h e d e f l e c t i o n o f t h e interface.

A.E.M.A. Mohamed et al. /Fluid Dynamics Research 19 (1997) 327-341

330

where e is the dielectric constant. The electric field is a superposition of the constant field in the basic state and the perturbation. Hence, the total stress tensor is

Since there is no electric volume force density here because e is constant in the fluid and there is no volume charge in the bulk of the fluid, the electric forces only act on the interface. Accordingly, the motion in the bulk of each fluid phase is governed by

PLot+(v'v)v

=-VP+V

2 G+It~

u+pg.

(4)

The continuity equation is described by exx + eyy + ezz = 0,

(5)

v.u

(6)

or

where eij

= 0,

(i,j

= x, y, z) are the components of the strain tensor given by

(7)

eu = 2 L ~Xj + c?xi J "

The equilibrium solution of Eq. (4) is P(o')(y, t) = - p(r)gy + C(or)(t) (r = 1, 2),

(8)

where the suffix r denotes the upper or the lower medium. C(o~) is the time-dependent constant of integration. The balance of the normal stress tensor at the interface leads to (P(J) - P(o2)) = - ½(e(') - e(2))E2.

(9)

Thus, in the equilibrium state, the pressure Po will be discontinuous across an interface, while it will be continuous in the absence of electric field. In formulating Maxwell's relation for the system, we assume that the quasi-static approximation is valid for the problem (Melcher, 1963). With a quasi-static model, it is recognized that relevant time rates of change are sufficiently low such that contributions due to a particular dynamic process are negligible. The objective in electrified fluids is concerned with phenomena in which the electric field far exceeds the magnetic field and where the propagation times of electromagnetic waves are short compared to those of interest to us. Then under the quasi-electrostatic approximation Maxwell's equations are reduced to V.(eE)=0

and

VxE=0

or

E=-Vq~,

where ~b is the electrostatic potential. The boundary conditions are applied at infinity far from the interface and at the interface. The former requires that the electric field and the velocity vector tend to zero at infinity. The conditions at the interface are as follows: (i) The tangential and normal components of the electric field are continuous. (ii) The normal and tangential components of the velocity are continuous.

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331

(iii) The tangential stress c o m p o n e n t is continuous. (iv) The j u m p of the normal stress balances with the surface tension. These b o u n d a r y conditions are the same as those given by Chandrasekhar (1961) and Melcher (1963).

3. Perturbation equations The location of the interface is denoted by

y = ~(x, t) = 7(t)e ikx,

(10)

where the wave n u m b e r k is assumed to be real and positive. If the perturbation is assumed infinitesimal, the unit normal vector to the interface, can be written as: n

=

-

ik¢ex + ey,

(11)

where e~ and e~, are the unit vectors in the X-, and Y-directions, respectively. The linearized equations for the flow and the electric field are 02//1 P 0T -

gP1 +/~ ~ + r V 2 Ul

(12)

V ' u l = 0,

(13)

V2 ~)1 = 0,

(14)

where z = G/l~ represents the retardation time. The subscript 1 refers to the perturbed quantity. Taking the divergence of Eq. (12), and using Eq. (13), we get V2p, = 0.

(15)

On the other hand, operating V 2 on Eq. (12) and using Eq. (15), we get

V 2 V IJ ~ + ~

-p~7 ~ ul(x,y,t)=O.

(16)

The linearized b o u n d a r y conditions at y = 0 mentioned above are ~--'~ [(~(11) -- (~)i2)] = 0,

(17)

0 ik(e (1) - e(2)) ~Eo + ~yy (e(1) qS(x1) - e (2) qS]2)) = 0,

(18)

0 u]l ? = 0 .~2~= 0 & ~ u,,, ~ 4,

(19)

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A.E.M.A. M o h a m e d et al. / F l u i d D y n a m i c s Research 19 (1997) 3 2 7 - 3 4 l

a [ -~° u(,1,!- ~° u]~,!] ay

o'

(20)

(~2 (~2) [~(1' ( ~@~t-[- "C'1') u(ll)?-- //(2)(Stt -I- "C(2))b/(12'1=0, (1) - - 0.(2) 0-yy yy -_ _ TV 2

(21) (22)

where T is the surface tension of the interface. With the conditions at infinity for lyl ~ oc such that O]~(x, _+oc, t) = 0

and

u]~)(x, +_~, t) = O.

(23)

4. Method of solution

The perturbations of the velocity, pressure and electric field are assumed to have the following forms:

Ul(X, y, t) = ~(y, t)e ikx,

(24)

Pl(X, y, t) = P(y, t)e ikx ,

(25)

0~(x, y, t) = 0(y, t)e ikx .

(26)

Then, substituting Eq. (26) into Eq. (14), and solving it with the boundary conditions (17) and (18) and the conditions at infinity, we have 0~l)(x, y, t) = iEo (c/1) + e/z)) 7(t)e ikx-ky

for y > 0,

(27)

012)(x, y, t) = iEo (e(l) + e~2)) 7(t)e ikx+ky for y < 0.

(28)

On the other hand, substituting Eqs. (24) and (25) into the y-component of Eq. (11), we have

Oy p~,~ (y, t) = __a

~

- k~

+ r('~ IJ("-P~') Oz ]o('~(Y't)

c~dJ

(r=1,2)

(29)

where U(x,y, t ) = O(y,t)e ik~ is the y-component of the displacement vector u(x,y, t). Also, substituting Eq. (24) into Eq. (16), we have adJ

O~'~(Y'0

= o.

(30)

This last equation may be solved by the method of separation of variables assuming timewiseharmonic oscillation. But it will lead to a very complicated transcendental dispersion relation, and the interpretation of the resulting relation is rather difficult. Therefore, stability conditions cannot be obtained.

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In order to carry out the stability analysis we use the method of multiple time scales (cf. Nayfeh, 1979). The method is based on the assumption that the retardation time r is small, i.e. the material is slightly elastic. Then the first step of perturbation deals with the state without elasticity, and the second step is concerned with the effect of elasticity. Let ?. denote a smallness parameter defined as r(1) = ~(1)

and

./7(2)~- ~.~(2),

where ~(1),(2) are finite. We introduce two time-scales, To=t

and

Tl=~t.

Then the differential operator d / d t can be expressed by the derivative expansions: d d - t = D o + g D 1 + "";

D. - OT " •

Assume that the unknown quantities are represented by the following expansions: y(t, ~;) = 7 o ( T o , T1) + ~ l ( T o ,

T1) +

(31)

"",

U(r)(y, t, ~) = 0(or)(y, To, T1) -}- gU(lr)(y, To, T1) + " " ,

(32)

p(r)(y, t, ~) = P(or)(y, To, T1) + ~p~r)(y, To, T1) + "'" •

(33)

Substituting the above expansions into Eqs. (29) and (30), and boundary conditions (19)-(22), we obtain a series of equations in powers of 5. 4.1. The zero-order solution

As zeroth-order solutions we assume the following forms: ]2o(To, T1) ---- 7 o o ( T 1 ) e o~°T° ,

(34)

~)(y,

(35)

TO , T1 ) ~_ ~ ( r ) ( y , T1)eO~oTo,

P(or)(y, To, T1) = P(~)o(Y,

T1) e°)°T°

(36)

where COois the growth rate and the unknown function 700 will be determined later in the firstorder solution by the solvability condition. By substituting the above expansions into the zerothorder equations, the solution of the resulting differential is straightforward but lengthy and will not be included here. The details are outlined by Chandrasekhar (1961) and Melcher and Schwarz (1968). However, the electrified viscous fluid dispersion relation is obtained in the following form: k 4 ( m l - 1) (m2 - 1) (/t (1) - ]/(2))2 _ 4k 2 COO (~(1) __ ]A(2)) Wp(1)(m2 -- l) -- p(2)(ml -- 1)] -- 4p(1)p(2)co 2 -- [/)(1)(m 2 -- 1) + p(2)(m I -- 1)] [co02(p (1) -k- t0 (2)) +

kiTE]

O,

(37)

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A.E.M.A. Mohamed et al. / Fluid Dynamics Research 19 (1997) 327 341

where p(J) (.0 o

m2 = 1 + k2//u) ,

(38)

rre = k 2 T _ (p(a) _ e* -

p(Z))g +

(39)

ke* E 2 ,

(40)

8 (1) -I- g(2) •

The above dispersion relation (37) is obtained by Melcher and Schwarz (1968) and by M o h a m e d et al. (1994) for the case with electric field and by Chandrasekhar (1961) without it. Melcher and Schwarz discussed the stability near the marginal states while Chandrasekhar examined some special cases. M o h a m e d et al. (1995) obtained a similar dispersion relation for Maxwell fluids and discussed a special case where the two fluids have the same kinematic viscosity. 4.2. A n u m e r i c a l illustration a n d s t a b i l i t y a n a l y s i s o f the s t a t e o f the v i s c o u s s o l u t i o n

The interpretation of Eq. (37) is rather difficult because the growth rate Cno appears as an implicit quantity. In order to carry out further calculations we will have to be concerned with highly viscous fluids (i.e. small Reynolds number). In order to interpret the physical implications of the stability conditions we shall select a model for numerical c o m p u t a t i o n and we select the case of large viscosity. Thus, we use the binomial theorem to expand (38) in the following form:

8k4//(j) 2 + 16k6ffj)3 + .-..

mj = 1 + 2k2//~j)

(41)

Substituting (41) into (37) and neglecting all powers of (//- 1) greater than or equal two, we obtain the following dispersion relation in approximate form: Aco 2 + Booo + aE = 0,

(42)

where A = (//(2) __ //(1)) [p(2)//(1)(2//(2)

+ //(1)) _ p(1)//(2)(2//(1)

+//(2))]/2k//(1)//(2)(fl(1)

+//(2)).

B = 2k(//(1) + / / ( 2 ) ) .

In view of the Hurwitz criteria for stability, the stability occurs when A>0,

B>0

and

aE>0.

The first condition is satisfied if either //(2) > //(1) and

p(2) > p(1)//(2)(2//(1) q-//(2)) //(1)(2//(2) +//(1)),

i/{1) > //(2) and

p{1) >

(43)

or

p(2)/1(1)(2/l(2) -I- //(1))

//(2)(212(1) + //(2))"

(44)

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335

The second condition is trivially satisfied. The third condition is satisfied if E2 >

(p(l) _ p(2)) g _ k2 T

(45)

ka*

This condition is also valid for inviscid fluids (cf. Melcher, 1963). Combining conditions (43) with (45) and condition (44) with (45) we obtain the stability condition as E2>7-g

-kzr,

p{l) 9 1

/*<1,

(46)

/*>1,

(47)

and E2>~e*l ~P{2)g[-#(/*+( [_2-fi+2)1

ll-k2T

}'

where/* -/*,)//*(2) is used. Note that: ifp (2) > p~l), then condition (45) is trivially satisfied regardless of the value of the field. This is because the system is statically stable in the absence of the field. This follows from the fact that the right-hand side of inequality (46) is always negative for/* < 1. But when/* > 1, the stability occurs for all values of the wave number satisfying the following relation k2

> p(~g V/*(/* -+- 2)

L TI

11

(48) "

The corresponding relation in the inviscid case is k2 > @

(p - 1),

p = p,)/p(2).

(49)

Since p >/*(/* + 2)/(2/, + 1) (which follows from relation (44)) is the condition for stability when /* > 1, it follows from the comparison between (48) and (49) that the cutoff wave number in the inviscid case is larger than the cutoff wave number in the case of viscous fluid. This means that the viscosity produces a larger stable region than that for the inviscid case. The added region is due to the presence of viscosity which results in damping modes suppressing the vibrating system. 4.3. The first-order solution

To carry out the solution in the first-order problem we substitute the solutions, which are governed by the zeroth-order state, into the governing equations for the first-order quantities. The resulting set of equations governing the fluid motion in the upper and lower regions along with the boundary conditions gives the following solvability condition: ( R D , + Q/~)7oo(t) = O.

In terms of the original variable t, we obtain

(Rd )

= 0,

(50)

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where the original variable t is used and the constant coefficients R and Q are, in general complex, and given in the approximate case of a large viscosity by R = F109 2 + F2o9 o -Jr- F3,

(51)

Q = Ogo(F~coo + Fs),

(52)

where El = 4p(1)p(2)]/(D]/(2) E4(]/(1) +//(2))(/3(1) -/- p(2))2 + (p(1) _ p(2))(p(l)]/(2) _+_ p(2)]/(l))] _ 3p(1)p(2)(]/(1) _+_ ]/(2)) []/(1)2/./(2)2(/)(1) _ p(2))2 ~_ (]/(1) 2 _.1_]/(2)2)(p(112]/(2) 2 _1_ p(z)-']/(1)z)] ..1_ ]/(1)]/(2)(]/(1) ..1_ /2(2))[2(p(1)2 ]/(2) _~ p(2)2]/(1))2 _

_ p(1)p(2)(p(X)]/(2)

_

2p(1)p(2)]/(l)]/(2)(p(1)

__ p(2I)2

p(2)]/(1))2 + 4p(1)2p(2)2]/(1)]/(2)],

F2 = 8k2]/(1)2]/(2)2(]/(1) + ]/(2))[7(p(1) + p(2))(]/(~) + ]/(2)) + 2p(a)p(2)(p(1)]/(1) + p(2~]/~2)) + (p(1) _ p(2)) (pl2)2]/(~) _ p~1)-'~2))], 1:3 = 48k4 p(1)p(2)]/(1)2]/(2)2 (]/(1) + ]/(2))3 F4 = ]/m]/(2~[(p(l~]/121~ - p~2~]/~1~)(p(Z~]/~r~ _ pl~]/(2~T~2~) + 2(]/~l~r 11~ _ ]/(z~r~2~)(p~2~]/~ ~ _ p(l~]/12~~) + ]/(1)]/(2)(p(1)]/(2) + p(2)]/(1))(p(2).r(1) + p(1).c(2) ) + p(1)p(2)]/(1)]/(2)(]/(1) + ]/(2))(z.(l) @ 27(2))

_ 2(]/~ + ]/~2~)(p{~)]/~2~z(1~ + p~2)]/~r~2~)], F5 = 4k2p(1)p(2)]/~a]/(e)e(]/(~) + ]/(2)) [(z (~) + "r(2))(p(1)]/(2) + p(2)]/(1)) + 2(p (a) - p(2)) (]/(1).c(1) -- //(2)-'/7(2))] ' The solution of Eq. (50) will impose the stability condition of the problem. The system is stable if 7(t) is b o u n d e d as t --+ c~. Since Eq. (50) is a first-order differential equation its solution has the form 7(t) = e x p ( - Ot), where O = Q/R. Thus, stability occurs for positive values of ~2 which can be satisfied when the real part of RQ is positive. Suppose that the dispersion relation (42) has pure real roots, i.e. its discriminant has positive values, then the stability criteria can be sought in the following form: Coo(F4co 0 q- F5) (Faco 2 + Fzco 0 -I- F3) > 0,

(53)

provided that B 2 --

4Aae >>-O.

(54)

The transition curves separating the stable region from the unstable region correspond to B 2 --

4Aa~ = 0,

(55)

O0o = 0,

(56)

F4e)o + Fs = 0,

(57)

Fie) 2 + F2o9o + F3 = 0.

(58)

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337

Using relation (42), the above transition curves (55)-(58) can be written in terms of E 2, respectively, as follows: E~

(p~l) _ p ( 2 ) ) g _ k2 T =

ke,*

B2

(59)

+ 4Ak~----g ,

Eo2 = (p(1) _ p ( 2 ) ) g _ /(2 T ke* '

(60)

E2

(61)

=

(p~a) _ p ( 2 ) ) 9 _ / ( 2 T Fs(BF4 - AFs) ke* + ke* F 2 '

a2Eo* + ax E2o + ao = 0,

(62)

where the a are constant coefficients given by a 2 = / ( 2 g * Z A F 2, aa = 2 k ~ * A F Z ( k 2 T

-

(p(~) -

ao = A 3 F ~ + A B F 3 ( B F 1 + (AF2 -

P~2))9) + k e * ( A B F a F2 - 2AZFa F3 - B 2 F 2 + ( A F 2 - B E t ) Z ) ,

- AF2) + (k2T

BF1) 2) + A F 2 ( k Z T

-

-

(p~) - p~z))9)(ABF~F2

-

2AZF1F3

-

BZF 2

(p~l) _ p~2))9)2"

5. A numerical estimation Calculations for the stability conditions (53) and (54) are displayed in the following graphs for the plane (E 2 - k). Five transition curves have been calculated for different wave numbers k and are displayed in the plane (E 2 - k). These curves are indicated by the symbols I-III, IV1 and IV2. The curve I represents the transition curve (59) which depends on viscosity and is independent of the elasticity parameter. This curve partitions the plane (E 2 - k) into two regions relative to the nature of the roots of the dispersion relation (42). Thus, the region labeled by the symbol U* is out of the scope of the stability condition (53) where condition (54) is not satisfied, while the behavior of the real roots appears in the region which lies on the right-hand side of curve I. The curve labeled by the symbol II represents the transition curve (60) which is independent of viscosity and elasticity parameters. The curve m a r k e d by the symbol III indicates the transition curve (61) which depends on the elasticity parameters. The curves IV1 and IV2 denote the two curves of (62). With those curves the plane (E 2 - k) has been partitioned into stable and unstable regions. We may note that condition (53) depends on COo, which is determined by Eq. (42). Since Eq. (42) has two roots, it is necessary that every root should satisfy condition (53) independently. The region, in the (E 2 - k) plane, labeled by Sj represents situations where every point in the region satisfies inequality (53) for two values of ~Oo. Fig. 2 represents a sample case when p ( t ) = 0.99823 g/cm 3, p(2)= 0.879 g/cm 3, ~ t ) = 80.08, e(2) = 2.29, 1./(1) = 3 g c m - 1 s - 1 #(2) = 2 g c m - ~s - 1, T = 35 dynes/cm and 9 = 980.665 c m / s 2. In the above case the system is statically unstable in the absence of the electric field. Three stable regions

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A.E.M.A. Mohamed et al. /Fluid Dynamics Research 19 (1997) 327-341 10

/

Ill

$3 /IV 2

9

//111

/,

//

E2 o

U

2

S2

T I

2-

'/

H*

IV1

0 0.1

0.3

0,5

0,7

0.9

1.1

1,3

....... ~ . , . . . . . . . . . . . . . . . . . 1,5 1.7 1.9 2.1 2.3

2.5

2.7

2.9

r-- k Fig. 2. Variation of Eo: with the wave n u m b e r k as given by relations (59) (62) for a system having p(1) = 0.99823 g/cm 3, p(Z)=0.879g/cm s, e , ) _ 8 0 . 0 8 , e(2)=2.29, p ( 1 ) = 3 g c m - l s - 1 , p(2)=2gcm-ls 1, T = 3 5 d y n e s / c m and g = 980.665 cm/s 2. The curve marked by • refers to the case of z = 0, while the symbol o refers to the case of z = 0.5. The symbols (Sj) refer to stable regions and (Uj) denotes unstable regions. The symbols I - I I I refer to the transition curves (59)-(61), respectively. While the transition curves of (62) are represented by IV1 and IV2.

$1, $2 and $3 appear in the graph. The Sl-region lies between the curves II and I, the S2-region lies between the curves IV~ and III and the S3-region is bounded by the curve IV2. On the other hand, there are three unstable regions which are displayed in the plane (E 2 - k). U1 which lies between the curves II and IV1, U2 lies between the curves III and IV2 and U*-region which appears on the left-hand side of the curve I. It can be noted that the two unstable regions U1 and U2 increase as the electric field increases, while the unstable region U* decreases. Thus, increasing the electric fields plays a dual role; a destabilizing effect in the regions U1 and U2 and a stabilizing effect in the U*-region. Two different cases for the ratio of the elasticity parameter z (~ - "c(1)/z(2)) are considered in Fig. 2. The curve which is marked by the symbol • refers to the value of-c = 0, while the curve which is marked by the symbol o refers to the case of z = 0.5. It is clear that the transition curve III is affected by the variation in elasticity ratio: an increase in the ratio -c increases the width of the unstable region U2. This shows that increasing the ratio T plays a destabilizing role in the presence of the electric field. Fig. 3 represents the case of increasing the viscosity of the upper fluid to the value of ~/(1) = 3.4 g c m - ~ s-1 for a specified elasticity ratio z = 0.5. A small increase in kt(~) yields a small shift to the right-hand side for the curve I and a small increase relative to the curves IVa and III is associated with a large increase for the curve IV2. Thus, small increases in the upper viscosity play a destabilizing effect in the presence of the electric field and elasticity effects. When the viscosity of the upper fluid is increased to the value p(1) = 4 g c m - 1 s - 1 (the results are displayed in Fig. 4) the unstable region UI increases while the unstable region U2 decreases in width. Also the curve IV2 moves up so that the stable region $3 disappears from the plane (Eoz - k). A comparison between Figs. 2 4 shows that the unstable region Ux increases while the unstable region U2 decreases and the stable region $3 decreases in width. Thus, a destabilizing mode is observed for a large increase in the viscosity of the upper fluid.

A.E.M.A. Mohamed et al. /Fluid Dynamics Research 19 (1997) 327-341

339

10

10

9-

E 2 o

u

8-

8

2

Eo

6 5

S 2

L

i

4

/

2

765-

4$2

IV

3-

3

2-

1

U*

~

o,

10

0.1

0,3

0.5

0,7

0.9

1.1

1,3

1.5

1.7

1.9

2.1

~-

2.3

2.5

2.7

0.1

2.9

0.3

0.5

0.7

0.9

1.1

Fig. 4. The same system considered in Fig. 2 except that # m __ 4 g c m

--

1t/\

/

/

U2

! S2

E

"13

03

0.5

0.7

0.9

1,1

1.3

1.5

1.7

1.9

--D..

2.1

1.9

2,1

2.3

2.3

2.5

2.7

2.9

k

s - ~ a n d z = 0.5.

a s -1 a n d r = 0.5.

/7 Oa

/

9 8 7

0.1

1.7

Iv2

20

S 7

1.5

--q,-

Fig. 3. The same system considered in Fig. 2 except that p(1) = 3.4 g e m - 1

10

1.3

k

2.5

2.7

2.9

~1/

//

5 432 1 0 ................................................................. 0.1 0.3 0.5 0.7 0.9 1.1 1,3 1.5 1,7

k

Fig. 5. The same system considered in Fig. 2 except that

s2

H..................................... ~ 1.9

2,1

2.3

--~,,. //(2) =

2.3 g e m

2.5

2.7

2.9

k

x s - 1 a n d r = 0.5.

Fig. 6. The same system considered in Fig. 2 except that/~(zl = 2.8 g c m - 1 s - 1 a n d z = 0.5.

If the viscosity of the upper fluid is held at a fixed value where #(1) = 3 g e m -1 s -1, then the viscosity of the lower fluid has changed to the value of/t (2) = 2.3 g c m - 1 s - 1 in Fig. 5 and the value of/~12) = 2.8 g c m - a s - 1 in Fig. 6. It appears that curve I has shifted to the left-hand side as the viscosity of the lower fluid has increased so that the unstable region U* decreases and leaves a largest stable region $1. A m a x i m u m shifting to the left-hand side occurs and then the curve I disappears. But, the continuous increase i n / t (2) makes this curve appear again and lie in the S3-region as shown in Fig. 6. However, a comparison between Figs. 5 and 6 shows that an increase

340

A.E.M.A. Mohamed et al. /Fluid Dynamics Research 19 (1997) 327-341

in the Sl-region is associated with a small decrease in $2- and S3-regions as the viscosity of the lower fluid is increased. Thus, a dual role for the increasing viscosity #12) is clear in the stability criteria.

6. Conclusions

We have presented the problem of the gravitational stability of electrorheological fluids of the Rayleigh-Taylor type. The motivation for such studies stems from the need to understand and control the location and motion of the rheological fluids. Rheological studies have indicated that the biological cell membrane exhibits elastic as well as viscous behavior during mechanical deformation. Our analysis was undertaken principally to clarify the coupling between the effect of electric field and viscous and elastic effects on the stability of an interface between two Kelvin fluids. By using the method of multiple scales, we have performed an analysis based on the idea that the flow is slightly elastic. The stability of the system was analytically discussed and the results were confirmed numerically. We come to the following conclusion: (i) In the presentation of the problem, the effect of elasticity is revealed through the parameter r (retardation time). It is shown that increasing the ratio z ( = 27(1)/z -(2)) leads to a destabilizing role in the presence of the electric field. (ii) In pure viscous analysis, it is found that viscosity possesses a damping nature. In the presence of a small elasticity parameter z, another role is observed. It plays the same destabilizing role as the elasticity in some cases. In other cases it plays a dual role similar to that of elasticity. (iii) It appears that increasing the electric field plays an important role in the stability criteria. For a given specific wave number the electric field gives a variation from a stable mode to unstable mode. This shows that the field plays a dual role in the stability criteria. This phenomenon has been demonstrated by M o h a m e d and Elshehawey (1983) in the non-linear electrohydrodynamic Rayleigh Taylor instability for inviscid fluids.

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Nayfeh, A.H. and D.T. Mook (1979) Non-linear Oscillations (Wiley, Virginia). Pohl, H.A. (1978) Dielectrophoresis (Cambridge University Press, Cambridge). Rivlin, R.S. and J.L. Ericksen (1955) J. Ratl. Mech. Anal. 4, 323. Skalak, R., A. Tozeren, R.P. Zarda and S. Chien (1973) Biophys. J. 13, 245. Steinchen, A., D. Gallez and A. Sanfeld (1982) J. Colloid. Interface Sci. 85, 5. Takashima, M and A.K. Ghost (1979) J. Phys. Soc. Japan 47, 1717. Wilson, G.M. and B. Khomami (1992) J. Non-Newtonian Fluid Mech. 45, 355. Wilson, G.M. and B. Khomami (1993a) J. Rheol. 37, 315. Wilson, G.M. and B. Khomami (1993b) J. Rheol. 37, 341. Zelazo, R.E. and J.R. Melcher (1974) Phys. Fluids 17, 61.

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