Electroviscoelastic Instability of a Kelvin Fluid Layer Influenced by a Periodic Electric Force

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

207, 54 – 69 (1998)

CS985710

Electroviscoelastic Instability of a Kelvin Fluid Layer Influenced by a Periodic Electric Force Abou El Magd A. Mohamed, Elsayed F. A. Elshehawey, and Yusry O. El-Dib1 Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis, Cairo, Egypt Received December 15, 1997; accepted June 19, 1998

small-scale biological cell membrane deformation processes can be induced by physicochemical events that cause the membrane to become unstable to fluctuations in its interface. The assumption that the cell membrane is isotropically viscous, as modeled by these authors (13–17), warrants refinement. Rheological studies have indicated that the cell membrane exhibits elastic as well as viscous behavior during mechanical deformations. The hydrodynamic stability of biomembranes has been studied by several authors, using a viscoelastic rheology (18, 19). This viscoelastic behavior was suggested by micropipet experiments which were interpreted using a two-dimensional linear Kelvin model (20). The hydrodynamic instability to rupture of biological and artificial membranes and thin films is involved in the mechanisms of a wide variety of biological and engineering processes and can be important for applications to practice. Typical examples are membrane fusion, membrane rupture and mechanical destruction, cell division and fusion, nerve excitation, and colloid stability. The viscoelastic films were treated by Maldarelli and Jain (18), who extended their studies to include small cell membranes. These authors dealt with internal electric forces in their studies. Mohamed et al. (21) have analyzed the influences of an external steady electric forces on the interfacial stability of thin viscoelastic fluid phase between two different viscoelastic fluids. The object of the present study is to provide an extension to the Mohamed et al. model (21) in order to discuss the influence of an unsteady electric force on interfacial stability of viscoelastic thin layer of Kelvin type. Application of an electric field to the fluids uncovers new aspects of stability parameters. Electrohydrodynamic instability in a viscoelastic liquid layer under the simultaneous action of a vertical electric field and vertical temperature gradient is studied by Takashima and Ghost (22). The stability of capillary-gravity waves for an electroviscoelastic of Maxwell-type and Kelvin-fluid was studied by Mohamed et al. (23, 24). The gravitational stability of the interface between two electrorhological fluids of Kelvin type has been examined by El-Dib (25). He demonstrated that the vertical electric field retards the stabilizing influence of both the viscosity and elasticity parameters, whereas the presence of a tangential field suppresses the

The electroviscoelastic stability of a Kelvin fluid layer is discussed in the presence of the field periodicity. The surface elevations are governed by two transcendental coupled equations of Mathieu type which have not been attempted before. Analysis for the surface waves in axisymmetric modes and antisymmetric deformation which are governed by a single transcendental Mathieu equation is considered. The method of multiple scales expansion is applied to the stability analysis. The solution and the characteristic curves are obtained analytically. It is shown that the region between the two branches of the characteristic curves is unstable, whereas all points which lie outside the characteristic curves are stable. The special case of large viscosity is introduced for numerical calculations. It is found that the increase of kinematic viscosity, field frequency, and the elasticity parameter possesses a dual role in a damping nature. The phenomena of the coupled resonance is observed. The resonance region and the resonance points are functions of viscosity, elasticity, and field frequency, with nonlinear relations in the wavenumber. © 1998 Academic Press

1. INTRODUCTION

At the base of the investigation pertaining to the interfacial stability of thin liquid films is the desire to understand the dynamics of partial coalescence in continuous liquid phase. Indeed, the type of film system examined in this investigation underscores this intention. Felderhof (1), Ivanov et al. (2), Sche (3), Joosten (4), Pre`vost and Gallez (5), Li and Tankin (6), and Rangel and Sirignano (7) have analyzed the interfacial stability of inviscid and viscous films bounded by gas phases. Such investigations are instructive in detailing the fluid mechanics of bubble coalescence in a dispersion. A host of efforts (8 –12) have been devoted to the stability of viscous films surrounded by liquid phases. These studies are applicable to the attachment of droplets in an emulsion. A significantly different potential application of thin film stability theory was introduced by Sanfeld and Steinchen (13, 14), Bisch and Sanfeld (15), Wendel et al. (16), Gallez et al. (17), and others. The essential idea in their studies is the concept that 1 To whom correspondence should be addressed. Present address: UmmQura University, Department of Mathematics, Faculty of Applied Science, P.O. Box 6503/75, Makkah, Saudi Arabia.

0021-9797/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.

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55

ELECTROVISCOELASTIC INSTABILITY

destabilizing influence of the stratified density. An expanding field of applications is observed in molecular biology. The electrohydrodynamic stability of liquid films is used to study the behavior of hydrocarbon films (26, 27) and lipid films (16, 28). It is known that if an electric field is applied to adjacent biological cells, a cell membrane is formed which can be treated as a dielectric insulating fluid layer separating two fluids (29). The electrohydrodynamic stability of an inviscid fluid layer bounded from above and below by two different inviscid fluids have been studied by Mohamed et al. (30 –32). The dielectric fluids are stressed by gravity force and vertical constant electric field (30) which play a destabilizing role when there are no surface charges present at the interfaces. In the presence of the surface charges on the interface, the field is still destabilizing, but the effect is partially shielded in some situations. In Ref. 31, the nonlinear deformations of fluid layer is influenced by a tangential electric field. These inviscid models cannot, in general describe real physical problems. In Ref. 32, the weakly nonlinear electrohydrodynamic stability of a fluid layer sandwiched between two semi-infinite fluids was investigated. The nonlinear theory of perturbation is applied for axisymmetric and antisymmetric modes. 2. ELECTROVISCOELASTIC MODEL AND THE FUNDAMENTAL EQUATIONS

A surface wave is essentially a two-dimensional deformation of an interface separating two media. Melcher studied the electrohydrodynamics of a generalized surface wave by linearizing the electrohydrodynamic equations of motion and the boundary conditions and the associated electric, velocity, pressure, and surface deformation fields (33, 34). Consistent with the first-order linear approximation, he used perturbation theory to derive the surface wave solutions for the electric and velocity potentials and the surface deformations for different wave modes. Following Melcher and Schwartz (34) and applying the electrohydrodynamic capillary wave theory derived by Miskovsky et al. (35) to the planar model including viscosity, the dispersion relation was obtained within the harmonic approximation from the set of electrohydrodynamic equations, surface coupling, and other boundary conditions (35). Recently, Mohamed et al. (21, 24) and El-Dib (25) applying Melcher’s and Miskovsky’s model to the planar model including the viscoselasticity of the Kelvin type. In this section, we briefly review the application of the linearized electrohydrodynamic equations to the planer viscoelastic model. We suppose the viscoelastic fluid layer has thickness 2h, density r(2), viscosity m(2), and elastic moduli G (2) . The layer is embedded between two semi-infinite viscoelastic fluids. The upper medium has density r(1), viscosity m(1), and elastic moduli G (1) . The lower fluid has density r(3), viscosity m(3), and elastic moduli G (3) . Both of the fluids are incompressible and isotropic. The system is assumed to be stressed by a

tangential periodic electric field (E 0 cos v t) in the X direction. E 0 is the amplitude of the periodic electric force, and v is the frequency. The volume charge density of the viscoelastic layer and the surrounding phases is assumed to be zero. There is no surface charge density at the interfaces of the layer. The middle plane of the layer is taken to be y 5 0. For the viscoelastic problem, a complete, nonlinear constitutive relation need not be formulated for the dynamic component of the stress tensor because only the linearized form of this relation is necessary for the linear stability analysis. As an appropriate linear viscoelastic formulation, the Kelvin model is adopted here because of the experimental evidence (20) which indicates that the mechanical behavior of a biological cell membrane (specifically the red blood cell membrane) may be described by this model. The following linear constitutive relations can be derived for the Kelvin model: The velocity vector V( x, y, t) is implicitly related to the displacement vector u( x, y, t) by V5

­u 1 ~V z ¹!u. ­t

[2.1]

Two forces must be accounted for by the stress tensor s ij . One is the surface force which results from the effect of the viscoelastic force of the Kelvin type as given by

FS

s vis ij 5 2P d ij 1 2 m

D G

­ 1 V z ¹ 1 G e ij, ­t

[2.2]

where P is the hydrostatic pressure, m is the coefficient of viscosity, G is the elastic, moduli, and e ij (i, j 5 x, y, z) are the components of the strain tensor given by e ij 5

F

G

1 ­u i ­u j ­u k­u k 1 2 . 2 ­ x j ­ x i ­ x i­ x j

[2.3]

The other one is the body force which is caused by the electrical forces as given by F 5 rf E 2

S

D

1 1 ­e E ? E¹ e 1 ¹ E ? E r , 2 2 ­r

[2.4]

where rf is the free charge density, e is the dielectric constant, and r is the mass density of the fluid (36). The first term, called the Coulomb forces, is the force per unit volume on a medium containing free electric charge. It is the strongest electro-fluid-dynamic term and usually dominates when the dc electric field is present. The second term, called the dielectric force, is caused by the force exerted on a dielectric fluid by a nonuniform electric field. It is usually weaker than the Coulomb force and only dominates when the ac electric field of high enough frequency is imposed.

56

MOHAMED, ELSHEHAWEY, AND EL-DIB

The third term, called the electrostriction term, being the gradient of a scalar is treated as a modification to the compressible fluid pressure and should be omitted from this analysis because incompressible flow is presented here (36). Manipulation of Eq. [2.4] that incorporates the irrotational nature of the electric field intensity shows that the stress tensor representation of combined free charge and polarization force densities is

s ele ij 5 e E iE j 2 s ij 5 s

vis ij

F

G

F

5 2¹P 1 ¹ 2 G 1 m

1s ,

­ 1Vz¹ ­t

S

D

­V 1 ­ 5 2¹P 1 1 m 1 t ¹ 2u 1, ­t ­t

[2.12]

¹ z V 1 5 0,

[2.13]

¹ 2f 1 5 0,

[2.14]

[2.5]

ele ij

S

r

and

1 e E 2d ij, 2

[2.6]

as discussed in Ref. 36. Because there are no electrical volume force density terms here because e is constant in the system and there is no volume charge in the layer and in both upper and lower fluids. Thus electrical forces only act on interfaces. Its contribution passes through the normal component stress term in the boundary condition at the surface of separation. However, in the bulk, the volume equation has 2¹[(1/2)eE20] 5 0, and so no contribution; it is an impulse at the boundaries, which is treated as a modification to an interfacial pressure P* 5 P 2 (1/2)eE20 (37). Accordingly the motion in the bulk of each fluid phases is described by ­V r 1 ~V z ¹!V ­t

where c (r) 0 are the time-dependent constants of integration. The linearzation equation in the bulk of the viscoelastic layer and the surrounding fluids are described by

DG

u 2 r ge y.

[2.7]

[2.8]

where ex and ey are the unite vectors in the x and y direction. We assume that the quasi-electrostatic approximation is valid, then Maxwell’s equations are reduced to ¹ z ~ e E! 5 0 and ¹ 3 E 5 0 or E 5 2¹ f ,

j j~ x, t! 5 g j~t!e ikx, j 5 1, 2;

g1(t) and g2(t) are arbitrary functions of time which determine the behavior of the amplitude of the disturbance of the interface; and k is the wavenumber which is assumed to be positive. According to the linear perturbation theory, the unit normal vectors to the interfaces are defined as n j 5 2ik j je x 1 e y.

¹ f 5 0.

[2.9]

[2.10]

The equilibrium solution of the problem is ~r! ~r! P ~r! 0 ~ y, t! 5 2r gy 1 c 0 ~t!, r 5 1, 2, 3,

[2.16]

Two types of boundary conditions suffice to constrain the field equations properly: conditions at an infinite (perpendicular) distance from the fluid layer and conditions at the dividing surfaces. The former conditions express the following requirements:

­ ~ f ~ j! 2 f ~1j11!! 5 0, ­x 1

j 5 1, 2,

ik~ e ~ j! 2 e ~ j11!! j jE 0cos v t 1

­ ~ e ~ j!f ~1j! 2 e ~ j11!f ~1j11!! 5 0, ­y y 5 ~21! j11h.

[2.18]

3. The continuity of the normal component of the velocity vector across the interfaces y 5 6h. This requires that V ~y j!~ x, y, t! 5 V ~y j11!~ x, y, t! 5

[2.11]

y 5 ~21! j11h. @2.17#

2. The continuity of the Maxwell displacement in the absence of surface charges at the surface of separation. This leads to

where f is the electrostatic potential, which satisfies Laplace’s equation 2

[2.15]

1. The jump in the tangential component of the electric field is zero across the interfaces y 5 6h. This implies that

The continuity equation is described by e xx 1 e yy 1 e zz 5 0,

where t 5 G/m is the retardation rate of time. The deformed interfaces of the film are identified by j1 and j2 variables where

­ j ~ x, t!, ­t j

y 5 ~21! j11h. [2.19]

57

ELECTROVISCOELASTIC INSTABILITY

4. The tangential component of the velocity vector across the interfaces y 5 6h is zero. This implies that ­ ~V ~ j!~ x, y, t! 2 V ~y j11!~ x, y, t!! 5 0, ­y y

y 5 ~21! j11h. [2.20]

5. The discontinuity of the normal component of the stress s ij at the two interfaces y 5 6h. This leads to n y~ s ~iyj! 2 s ~iyj11!! 5 2n iS j¹ 2j j,

y 5 ~21! j11h,

@2.21#

~2! ~2! ky 2ky ikx f ~2! !e , 1 ~ x, y, t! 5 ~C 1 ~t!e 1 C 2 ~t!e ~3! ky1ikx f ~3! , 1 ~ x, y, t! 5 C 1 ~t!e

uyu , h,

y , 2h,

[3.7] [3.8]

where Cs are arbitrary time-dependent functions to be determined by the appropriate boundary conditions. Substituting from [3.4] and [3.5] into the y component of Eq. [2.12], we get ­ Pˆ ~ y, t! 5 ­y

FS

­2 2 k2 ­ y2

DS

D

G

­ ­2 ˆ ~ y, t!, 1t m2r 2 U ­t ­t [3.9]

where S 1 and S 2 are the surfaces tension through the surfaces separating fluids and

s ij 5 2P d ij 1 e E iE j 2

S

­ 1 e E 2d ij 1 m 1t 2 ­t

DS

D

­u i ­u j 1 . ­ xj ­ xi [2.22]

6. The continuity of the tangential stress across the interfaces y 5 6h requires that n x~ s ~xxj! 2 s ~xxj11!! 1 n y~ s ~yxj! 2 s ~yxj11!! 5 0,

y 5 ~21! ~ j11!h. [2.23]

3. METHOD OF SOLUTION AND THE CHARACTERISTIC EQUATIONS

Taking the divergence of Eq. [2.12] and using Eq. [2.13] to eliminate the displacement vector u 1 , we get ¹ 2P 1 5 0.

[3.1]

On the other hand, operating ¹2 on Eq. [2.12] and using Laplace’s equation [3.1], to eliminate the pressure P 1 , we get

F S

¹ 2 ¹ 2m

D

G

­2 ­ 1 t 2 r 2 u 1~ x, y, t! 5 0. ­t ­t

[3.2]

In order to obtain a traveling wave equation, we may assume the following dependence:

f 1~ x, y, t! 5 fˆ ~ y, t!e ikx,

[3.3]

u 1~ x, y, t! 5 uˆ ~ y, t!e ikx,

[3.4]

P 1~ x, y, t! 5 Pˆ ~ y, t!e ikx.

[3.5]

Substituting Eq. [3.3] into Eq. [2.14], the solutions of the resulting differential equation are ~1! 2ky1ikx f ~1! , 1 ~ x, y, t! 5 C 2 ~t!e

y . h,

[3.6]

ˆ ( y, t) e ikx is the y component of the where U( x, y, t) 5 U displacement vector u( x, y, t). Also substituting [3.4] into of the y component of Eq. [3.2], we get

S

­2 2 k2 ­ y2

DFS

DS

­2 2 k2 ­ y2

D

G

­ 1 ­2 ˆ ~ y, t! 5 0, U 1t 2 ­t n ­t 2 [3.10]

where n 5 m/r is the kinematic viscosity parameter. Although solution of Eqs. [3.9] and the fourth order partial differential equation [3.10] are variable separable, the separation leads to solutions of the form f( y)exp( b t), where b is some constant and f( y) is a function of y. The time dependence given by this solution is not appropriate for the periodic boundary conditions imposed on the problem. Its application limits g j (t) to the exponential form, which leads to inconsistency of various relations obtained from the boundary conditions. Moreover, including the viscoelastic effect makes a more complex problem and creates a difficulty to in treating and analyzing the problem. Because of the complexity of the system, we prefer to consider the Fourier series approach. Accordingly, Eq. [3.10] has the following solution (the details are given in the appendix): ˆ ~ y, t! 5 A 0~t!e ky 1 B 0~t!e 2ky U 1 ~e ky Î11u!A 1~t! 1 ~e 2ky Î11u!B 1~t!,

[3.11]

where u is a linear differential operator and is given in the appendix. A 0 (t), A 1 (t), B 0 (t), and B 1 (t) are arbitrary timedependent functions to be determined from the appropriate boundary conditions. According to [3.11], the pressure Pˆ ( y, t) is expressed in the exponential form such that

r d2 ~A ~t!e ky 1 B 0~t!e 2ky!. Pˆ ~ y, t! 5 2 k dt 2 0

[3.12]

Because the displacement and the pressure must be finite as y ¡ 6`, we take A0 5 0 in the upper fluid and B0 5 0 in the

58

MOHAMED, ELSHEHAWEY, AND EL-DIB

lower fluid, so that the solutions of Eqs. [3.9] and [3.10] become ~1! 2ky ikx u ~1! 1 ~e 2ky Î11u1!B ~1! y ~ x, y, t! 5 $B 0 ~t!e 1 ~t!%e , y . h,

[3.13] ~2! ~2! ky 2ky u ~2! 1 ~e ky Î11u2!A ~2! y ~ x, y, t! 5 $A 0 ~t!e 1 B 0 ~t!e 1 ~t! ikx 1 ~e 2ky Î11u2!B ~2! 1 ~t!%e , uyu , h, [3.14] ~3! ky ky Î11u3 ikx u ~3! !A ~3! y ~ x, y, t! 5 $A 0 ~t!e 1 ~e 1 ~t!%e , y , 2h,

[3.15] P ~1! 1 ~ x, y, t! 5

r ~1! d 2 ~1! B ~t!e 2ky1ikx, y . h, k dt 2 0

P ~2! 1 ~ x, y, t! 5 2

[3.16]

r~2! d2 ~2! ky 2ky ikx ~A~2! !e , uyu , h, 0 ~t!e 2 B0 ~t!e k dt2 [3.17]

r ~3! d 2 ~3! P ~ x, y, t! 5 2 A ~t!e ky1ikx, y , 2h. k dt 2 0 ~3! 1

[3.18]

shall confine the stability analysis to consider axisymmetric and antisymmetric deformations. Therefore, the variables g1 and g2 may be related by

g2 5 Jg1

[3.21]

where J 5 21 denotes the axisymmetric deformation and J 5 1 refers to the antisymmetric mode. Dropping the suffix 1, for simplicity, from g1 and eliminating D 2 g between the preceding equations, we obtain @F~D! 1 ~a 1 E 20b cos2v t!# g ~t! 5 0,

[3.22]

which is a single transcendental Mathieu equation governing the behavior of axisymmetric ( J 5 21) and antisymmetric ( J 5 1) deformations. The constant coefficients a and b and the function of the operator D follow: a 5 a 4 2 a 2 1 J~a 3 2 a 1!,

[3.23]

b 5 b 4 2 b 2 1 J~b 3 2 b 1!,

[3.24]

F~D! 5 F 21 0 ~D!@F 4~D! 2 F 2~D! 1 JF 3~D! 2 JF 1~D!#, [3.25]

In order to derive the characteristic equations in dimensionless form, we introduce the characteristic length of order (h), the characteristic time of order (h/g) 1/ 2 , and the characteristic mass of order (S 2 h/g). Other dimensionless quantities are given by

r ~ j! 5 r˜ ~ j!~S 2/gh 2!, n ~ j! 5 n˜ ~ j!~ gh 3! 1/ 2, t ~ j! 5 t˜ ~ j!~ g/h! 1/ 2, k 5 k˜ h 21, S 1 5 S˜ S 2, v 5 v˜ ~ g/h! 1/ 2, and E 20 5 E˜ 20~S 2/h!. The characteristic equations are the result of the solutions given by [3.6]–[3.8] and [3.13]–[3.18] of basic equations governing the electrified fluid motion [2.14], [3.9], and [3.10] and the boundary conditions [2.17]–[2.23]. These nondimensional characteristic equations are found to be

[3.19]

$F 0~D!@D 2 1 ~a 3 1 E 20b 3cos2v t!# 1 F 3~D!% g 2~t! 1 $F 0~D!@a 4 1 E 20b 4cos2v t# 1 F 4~D!% g 1~t! 5 0,

4. MODULATION AND STABILITY ANALYSIS BY USING MULTIPLE-SCALES PERTURBATION

To employ the method of multiple scales expansion (38), we shall introduce a dimensionless (smallness) parameter e˜ such that E 20 5 e˜ E 200 . Thus, Eq. [3.22] in absence of e˜ reduces to @F~D! 1 a# g ~t! 5 0.

$F 0~D!@D 2 1 ~a 1 1 E 20b 1cos2v t!# 1 F 1~D!% g 1~t! 1 $F 0~D!@a 2 1 E 20b 2cos2v t# 1 F 2~D!% g 2~t! 5 0,

where F 21 0 (D) is the inverse of F 0 (D). As the operator function F(D) reduces to D 2 , the ordinary Mathieu equation, which has been studied extensively, arises, even though a transcendental Mathieu equation [3.22] has not been analyzed before. In what follows, we shall use a perturbation technique to investigate and analyze Eq. [3.22]. Employing of the multiple scales method is very useful.

Let its solution having the form

g ~t! 5 h exp~iV!t, [3.20]

where the superposed ; will be omitted for simplicity and the constants as and bs and the operator functions Fs are given in the appendix. It is clear that these equations are very complicated. They are of the transcendental coupled Mathieu equation type. Such equations have not been treated before except in the case of a constant field (21). We

[4.1]

[4.2]

where V is, in general, a complex constant (V 5 V r 1 iV i , with real V r and V i ) given by the characteristic equation F~iV! 1 a 5 0.

[4.3]

Because the function F(iV) has a complex nature, its imaginary parts will be zero near the marginal state.

59

ELECTROVISCOELASTIC INSTABILITY

Making a small modulation such that T n 5 e˜ n t, n 5 0, 1, 2, the differential operator D can be expanded as the derivative expansion D > D 0 1 e˜ D 1 1 e˜ 2D 2 1 · · · , D n ;

­ . ­T n

[4.4]

Because the solution reduces to [4.2] when e˜ vanishes, we assume that the set of solutions of Eq. [3.22] can be expressed in the form

[4.5]

where h1 and h2 are unknowns to be determined. Inserting Eqs. [4.4] and [4.5] into [4.1] and expanding it to include the periodic term, we get @F~iV 1 e˜ D1 1 e˜ D2 1 · · ·! 1 ~a 1 e˜ E b cos vt!#@h0exp~iVT0! 2 00

2

2

1 e˜ h1 1 e˜ 2h2 1 · · ·# 5 0. [4.6] The function F can be expanded using Taylor expansion in the form F~D! 5 F~D 0! 1 F9~D 0! D 1e˜

F

G

1 1 F9~D 0! D 2 1 F0~D 0! D 21 e˜ 2 1 · · · , 2

[4.7]

where (9) denotes the first derivative with respect to D 0 . Inserting the expansion [4.7] into [4.6] and collects coefficients of each power of e˜ and equating it to zero. These equations must hold independently because sequences of e˜ are linearly independent. The resulting equations can be solved successively. Thus we have

[4.8]

e˜ 2:@F~D 0! 1 a# h 2 5 @iF9~iV! D 1 2 E 200b cos2v T 0# h 1

F

1 iF9~iV! D 2 1

G

1 F0~iV! D 21 h 0~T 1, T 2! 2 3 exp~iVT 0! 1 C.C.,

iF9~iV! D 1 2

D

1 2 E b h 0~T 1, T 2! 5 0, 2 00

[4.10]

with the following uniform valid expansion:

H

exp~iV 1 2i v !T 0 1 exp~iV 2 2i v !T 0 5 2 bE 200 1 2 F~iV 1 2i v ! 1 a F~iV 2 2i v ! 1 a 3 h 0~T 1, T 2! 1 C.C.

[4.9]

where C.C. represents complex conjugate of the preceding terms. It is clear that Eq. [4.8] contains nonhomogeneous terms. The uniform solution is required to eliminate the secular terms. This elimination introduces the solvability condition corresponding to the terms containing the factor exp(iV r T 0 ). Thus, in order to analyze the solution of Eq. [4.8], we need to distinguish between two cases. The first is the nonresonance

J [4.11]

The corresponding solvability condition in the resonance case is formulated in the form iF9~iV! D 1h 0 2

1 2 E b@2 h 0 1 h# 0exp~2i s 1T 1!# 5 0, 4 00

[4.12]

where h# 0 is the complex conjugate of h0 and the nearness of the frequency v to V r is expressed by introducing the detuning parameter s1 according to

v 5 V r 1 e˜ s 1.

[4.13]

On the order of (e˜ 2), the solvability conditions corresponding to the resonance case of v near V r , the resonance case of 2v approaching V r , and the nonresonance case are, respectively, @D 2 1 i a 1b 2E 400# h 0~T 1, T 2! 1 ibE 200@ b 1bE 200 1 e˜ s 1b 0# 3 h# 0~T 1, T 2!exp~2i s 1T 1! 5 0,

e˜ :@F~D 0! 1 a# h 1 5 @iF9~iV! D 1 2 E 200b cos2v T 0# 3 h 0~T 1, T 2!exp~iVT 0! 1 C.C.,

S

h 1~T 0, T 1, T 2!

g ~t; e˜ ! 5 h 0~T 1, T 2!exp~i v 0T 0! 1 e˜ h 1~T 0, T 1, T 2! 1 e˜ 2h 2~T 0, T 1, T 2! 1 · · · ,

case, when the field frequency v is not near the disturbance frequency V r . The second case is the resonance case arising when the frequency v approaches V r . Therefore, the solvability condition corresponding the first case is found as

[4.14]

@D 2 1 i a 2b 2E 400# h 0~T 1, T 2! 1 i b 2b 2E 400h# 0~T 1, T 2!exp~2i s 2T 2! 5 0,

[4.15]

@D 2 1 i a 2b 2E 400# h 0~T 1, T 2! 5 0,

[4.16]

where the nearness of the frequency 2v to V r is expressed by introducing the detuning parameter s2 according to 2 v 5 V r 1 e˜ 2s 2.

[4.17]

It can easily be verified that the first-order solvability conditions [4.10] and [4.12] and the second-order solvability conditions [4.14]–[4.16] are the first two terms in the multiple scales expansions of

60

MOHAMED, ELSHEHAWEY, AND EL-DIB

dh 1 iq~ a 1q 1 2 a 0! h 1 iq@ b 1q 1 a 0 1 ~ v 2 V r! b 0# dt 3 h# exp@2i~ v 2 V r!t# 5 0,

[4.18]

dh 1 iq~ a 2q 1 2 a 0! h 1 iq 2b# 2h# exp@2i~2 v 2 V r!t# 5 0, dt [4.19] dh 1 iq~ a 2q 1 2 a 0! h 5 0. dt

The interpretation of the characteristic equation [4.3] is rather difficult because of the implicit form of the growth rate V in the transcendental function F(iV). Moreover the roots of Eq. [4.3] and, hence, values of the quantity V r are not available because of the complexity of the transcendental equation. Because of the very complex nature of Eq. [4.3] and hence the solvability conditions [4.21]–[4.26], as the function F(iV) involve a transcendental of an implicit the growth rate V, a mathematical simplification is useful here.

[4.20] 5. NUMERICAL ESTIMATION AND CONCLUSIONS

where the parameters e˜ s1 and e˜ 2s2 are eliminated using Eqs. [4.13] and [4.17], respectively, and E 20 5 e˜ E 200 , q 5 bE 20 are used. The coefficients as and bs are given in the appendix. It is easy to show that the stability criteria at the resonance case of v approaches V r is presented in the form Im@q~ a 1q 1 2 a 0!# , 0

[4.21]

and $3 a 0a# 0 1 ~ v 2 V r!@ a 1 1 a# 1 2 ~ a 0b# 0 1 a# 0b 0!#

In spite of the complexity of the mathematical problem, we can discuss the stability of the system in a special case. The case of large kinematic viscosities is considered (which corresponds to small Reynolds numbers) such that the order of the operator u greater than or equal to 2 will be neglected. For the purpose of simplicity, the elastic parameters are assumed to be the same; then we take u1 5 u2 5 u3 5 u [the case of equal kinematic viscosity as in Chandrasekhar (39)]. Consequently, the function F(D) will be simplified in the form

2 ~ v 2 V r! 2b 0b# 0%q 2 1 2~ v 2 V r!~ a 0 1 a# 0!q 1 ~ v 2 V r! . 0, 2

[4.22]

where all terms that include powers of e˜ greater than 2 are neglected. The notion (Im) refer to the imaginary parts. At the resonance case of 2v approaches V r the stability conditions, in a good approximation for small finite q, is formulated in the form Im~q~ a 2q 1 2 a 0!! , 0,

~iV! 2M 1 ~iV! n N 1 nt N 1 a 5 0,

Vi 5 [4.24]

@3 a 20 1 2~ v 2 V r!~ a 1 2 a 0b 0! 2 ~ v 2 V r! 2b 20#q 2 [4.25]

@4 a 20 1 2 a 2~2 v 2 V r!#q 2 1 4 a 0~2 v 2 V r!q 1 ~2 v 2 V r! 2 . 0,

where M and N are given in the appendix. The zeroth-order characteristic equation F(iV) 1 a 5 0 leads to [5.2]

which is a quadratic equation in (iV) satisfied by

It is noted that the stability criterion at the nonresonance case obeys condition [4.23] only. It is convenient to observe that the stability behavior near the marginal state can be governed by the conditions

1 4 a 0~ v 2 V r!q 1 ~ v 2 V r! 2 . 0,

[5.1]

[4.23]

@4 a 0a# 0 1 ~2 v 2 V r!~ a 2 1 a# 2!#q 2 1 2~2 v 2 V r!~ a 0 1 a# 0!q 1 ~2 v 2 V r! 2 . 0.

F~D! 5 D 2M 1 n ~D 1 t !N,

[4.26]

which are held in the resonance cases of v near V r and 2v near V r , respectively. In the nonresonance case, the requirement for stability is that the quantity ( a 2 q 1 2 a 0 ) has a zero-imaginary part.

nN 2M

and

V 2r 5

4M~ nt N 1 a! 2 n 2N 2 . 4M 2

@5.3#

Thus the stability conditions in the zeroth order (i.e., in the absence of the electric field) are 4M~ nt N 1 a! 2 n 2N 2 $ 0

and

MN . 0.

[5.4]

It is clear at the exact resonance v 5 V r or 2 v 5 V r that the field frequency v represents a function of n, and t with a nonlinear relation in the wavenumber k. The stability criterion at the resonant case of v near V r and 2v near V r are, respectively, 1 2 ~ v ~7V r 2 v ! 2 V 2r ~V r 1 3 v !!q 2 2 32MV 3r ~ v 2 V r!q v 1 64M 2V 4r ~ v 2 V r! 2 . 0,

[5.5]

ELECTROVISCOELASTIC INSTABILITY

61

FIG. 1. The stability diagram for a system having r(1) 5 1.5, r(2) 5 0.9, r(3) 5 1.45, e(1) 5 29, e(2) 5 15, e(3) 5 20, S 5 70, t 5 2.7, n 5 24, and v 5 20.3. The calculations that are displayed in this diagram indicate the transition curves E *1 and E *2 for the resonance case of v 3 V r in the case of the axisymmetric deformation ( J 5 21). Curves marked by the symbol 0 refer to the transition curves E *1 , whereas curves market by the symbol * refer to the curve E *2 .

S

D

6 v V 2r 2 V 3r 2 4 v 3 2 q 2 16MV r~2 v 2 V r!q V r~V 2r 2 v 2! 1 32M 2V 2r ~2 v 2 V r! 2 . 0.

[5.6]

In terms of the electric field E 20 , these stability conditions are, respectively, ~E 20 2 E *1!~E 20 2 E *2! . 0

[5.7]

2 ~E 20 2 E ** 1 !~E 0 2 E ** 2 ! . 0,

[5.8]

and

where the transition curves E *1,2 and E ** 1,2 are E *1,2 5

8 v V 2r ~ v 2 V r!M b~ v 2~7V r 2 v ! 2 V 2r ~V r 1 3 v !!

H

3 2V r 6 E ** 1,2 5

F

1 ~V 3r 1 v 3 1 7 v V r~V r 2 v !! v

4V 2r M~V 2r 2 v 2!~2 v 2 V r! b~6 v V 2r 2 V 3r 2 4 v 3!

H F

3 26

6V 3r 1 8 v 3 2 4 v V r~ v 1 3V r! V r~V 2r 2 v 2!

GJ 1/ 2

GJ

,

[5.9]

1/ 2

.

[5.10]

It is noted that the instability region lies between the transition

curves E *1 and E *2 and between the curves E ** 1 and E ** 2 . The points which lie outside the transition curves are stable. The numerical illustration for the transition curves E *1 and E *2 for the resonance case of v approaches V r is made for the case of the axisymmetric deformation ( J 5 21). The results are shown in Figs. 1– 4 for a system having r(1) 5 1.5, r(2) 5 0.9, r(3) 5 1.45, e(1) 5 29, e(2) 5 15, e(3) 5 20, S 5 70, and different values of n, v, and t. The region labeled by the symbol S refer to stable region. The region labeled by the symbol U represents an unstable region (resonance region). Curves marked by the symbol O refer to the transition curves E *1 , whereas curves marked by the symbol * refer to the curve E *2 . In these figures every unstable region is bounded by two branches E *1 and E *2 given by Eq. [5.9]. These branches are the characteristic boundary of the region at which the transition from stability to instability occurs. It is observed that a stable zone has appeared in between two unstable region and the phenomenon of coupled resonance arises in this investigation. The presence of a coupled resonance has been observed before by El-Dib (40) for Newtonian fluids having large Reynolds numbers (small viscosity approximations). It is shown, in this paper, that this phenomenon occurs even for a small Reynolds number (large kinematic viscosity). No instability occurs at the sharp resonance. At the exact resonance, we found that the field frequency v is related to both the viscosity parameter n and the elasticity parameter t by the relation

62

MOHAMED, ELSHEHAWEY, AND EL-DIB

FIG. 2. The same system represented in Fig. 1 for two cases of kinematic viscosity. The upper transition curves refer to n 5 30, and the lower curves represent the case of n 5 20.

n 2 2 4M tn /N 1 4M~ v 2M 2 a! 5 0. At a specific resonance point, the increase of both the kinematic viscosity and the elasticity parameter leads to decreases in the value of the field frequency v. This shows that small values of the frequency are needed to produce the parametric resonance in the presence of the viscoelastic effects.

In Fig. 1 the characteristic curves E *1 and E *2 have been calculated for different wavenumbers k against the parameter E 20 of the electric field. In this calculations, the kinematic viscosity n 5 24, the elasticity parameter t 5 2.7, and the field frequency v 5 20.3. The characteristic curves E *1 and E *2 have intersected at k 5 1.523, which represents a common resonance point. It is observed that the width of the unstable region

FIG. 3. The same system as that represented in Fig. 1 with two different cases of the field frequency v. The upper transition curves refer to v 5 20.6, whereas the lower curves correspond to v 5 20.

ELECTROVISCOELASTIC INSTABILITY

63

FIG. 4. The same system as that considered in Fig. 1 where the upper curves represent the case of t 5 3.3 and the lower curves refer to the case of t 5 2.

is increased as E 20 is increased, whereas in the limiting case as E 20 3 0 the system is stable except at the resonance point. The more destabilizing influence for the field appears corresponding to wavenumbers smaller than the resonance point. The stabilizing role that occurs at the exact resonance has increased as E 20 is increased. Thus a dual role for the electric field arises in the resonance case. It appears in Figs. 2–5 that the resonance regions and the resonance points are functions of the kinematic viscosity n, the elasticity parameter t, and the field frequency v with nonlinear relations in the wavenumbers. The decrease in the parameters n, v, and t leads to move the transition curves E *1 and E *2 down. The curve E *1 moves down more rapidly than the curve E *2 which yields two separated resonance points and produces an increased in the stable zone lies between the unstable regions. The increase of these parameters leads to move the transition curves up and leaving the k axis. Thus, no resonance points lie on the k axis at this stage. The curve E *1 moves up rapidly than the curve E *2 and possesses a dual role in a damping nature for n, v, and t. The system exhibits more damping in the stable zone (bounded by the curve E *1 ) than damping in the unstable regions as E *1 and E *2 are, respectively, moved up. The stability diagram shown in Fig. 2 represents the same system considered in Fig. 1 except that n has decreased to the value n 5 20 one time and increased to the value n 5 30 another time. As n is decreased, there two different wavenumbers occur at the resonance points. They lie on the k axis at k 5 1.38889 and at k 5 1.636101. The width of the unstable region has decreased, while the width of the stable zone has

increased. This shows the stabilizing effect for decreasing the kinematic viscosity. As n is increased, the transition curve E *1 has moved up more rapidly than the curve E *2 . This causes the stable zone surrounded by the unstable regions to contract. Thus, the influence for increasing the kinematic viscosity has larger damping effect in the stable zone than the damping effect in the unstable zone. Increasing n leaves a larger stable region, and the viscous damping effect arises. Consequently, the stabilizing effect for a small electric field has increased, and the dual role for relativity large field arises. In Fig. 3, the effect of the field frequency v is examined. Because the system is as shown in Fig. 1, except that v slightly decreased to v 5 20 once and slightly increased to v 5 20.6 another time. The comparison between Fig. 3 and Fig. 2 shows that the field frequency behaves as the kinematic viscosity in the stability criterion. As v is slightly decreased, the unstable regions decrease. The resonance point is separated into two points and are located at k 5 1.345908 and at k 5 1.706501. For small increases in v, the two unstable regions are connected and increased in its width. On the other side, the stable zone bounded by E *1 has decreased. The dual role in the damping effect for the field frequency v has not observed before in inviscid or viscous fluids. In small viscous fluids with the presence of surface charges, the system exhibits a destabilizing influence as the field frequency is increased (40). A dual role for the field frequency is observed in the nonlinear stability for inviscid fluids (41). Even in the linear stability for small viscous fluids, a dual role for the filed frequency v is observed in a rigidly rotating fluid column (42). The stabilizing influence

64

MOHAMED, ELSHEHAWEY, AND EL-DIB

FIG. 5. Comparison of the axisymmetric case and antisymmetric case for the same system as in Fig. 1. The transition curves with a common resonance point appears in the axisymmetric case, whereas the curves with two different resonance refer to the antisymmetric deformation.

of v is noted in the case of a periodic rotating of an inviscid fluid column (43). In the linear theory of surface waves for non-Newtonian fluids of Maxwell type, Mohamed et al. (44) demonstrated that the increase of the field frequency v has a stabilizing effect. They found that the field frequency plays a destabilizing role in the absence of the elasticity parameter (i.e., pure viscous effects). We can now conclude that the field frequency plays against the kinematic viscosity in pure viscous fluids as well as in Maxwell fluids and behaves like the kinematic viscosity in Kelvin fluids. The calculations displayed in Fig. 4 are as in Fig. 1 where all parameters are fixed except the elasticity parameter t which has two different values t 5 2 and t 5 3.3. As the parameter t is changed from the value 2.7 (in Fig. 1) to the value t 5 2 (in Fig. 4), the transition curves E *1 and E *2 have moved down and have intersected the k axis at k 5 1.41 and at k 5 1.626. The comparison between this case at hand with Fig. 1 shows that the stabilizing influence for deceasing the elasticity parameter t. Comparing Fig. 4 with Figs. 2 and 3 shows that the behavior of the parameter t in stability criteria is similar to the behavior of the parameter n. This conclusion is similar to that observed in the absence of v (21). The calculations for the case of J 5 1 i.e. in the case of antisymmetric deformation are displayed in Figs. 5 and 6 for the same system as considered in Fig. 1. The graph of Fig. 5 represents the comparison between the case of J 5 21) and the case of J 5 1. Because, in the case of J 5 21, the coupled resonance point is located at k 5 1.535, we observe, in the case of J 5 1, that the resonance point has separated into two

resonance points. Both points are shifted to new locations on the k axis at k 5 1.1369 and at k 5 1.797008. This shifting leads to an increase in the stable zone. Thus, the comparison shows that the case of J 5 1 (the antisymmetric mode) is stabilizing than the case of J 5 21 (the axisymmetric mode) for the same system. Figure 6 shows that the coupled resonance holds for n having the value 66.1 at the resonance point k 5 1.42, whereas the other parameters are fixed. The stability diagram shows that the resonance case of v approaches V r occurring for relatively large n compared with the case of J 5 21. A wide stable zone between in unstable regions is observed. The field is more stabilizing than in the case of J 5 21. The behavior of both n, v, h, and t still holds in the case at hand similar to the case of J 5 21. APPENDIX—FOURIER SERIES APPROACH TO SOLVE EQ. [3.10]

Let the solution of Eq. [3.10], in each phase, take the form

O F ~t! ~ky!n! . `

ˆ ~ y, t! 5 U

n

[A.1]

n

n50

Applying the operator

S

­2 2 k2 ­ y2

DFS

DS

­2 2 k2 ­ y2

t1

D

­ 1 ­2 2 ­t n ­t 2

G

65

ELECTROVISCOELASTIC INSTABILITY

FIG. 6. Stability diagram for the case of antisymmetric deformation with common resonance point for the same system in Fig. 1 except that n 5 66.1.

on both sides of [A.1], using Eq. [3.10], and equating coefficients of like powers of (ky) to zero, we obtain

S

k2 t 1

S

k2 t 1 5

D

S

k2 t 1

D

d 1 d2 B m11~t! 5 B ~t!, dt n dt 2 m

m 5 1, 2, 3, . . . . [A.7]

d @F˜ 2n14~t! 2 2F˜ 2n12~t! 1 F˜ 2n~t!# dt 1 d2 @F˜ ~t! 2 F˜ 2n~t!#, 5 n dt 2 2n12

D

[A.2]

d @F˜ 2n15~t! 2 2F˜ 2n13~t! 1 F˜ 2n11~t!# dt

Operating on both sides of [A.6] and [A.7] by ( t 1 d/dt) 21 , if we assume the initial condition to be zero, we obtain

A m11~t! 5 u ~D! A m~t!,

B m11~t! 5 u ~D! B m~t!,

D;

1 d2 @F˜ ~t! 2 F˜ 2n11~t!#, n 5 0, 1, 2, 3, . . . . @A.3# n dt 2 2n13

d , dt [A.8]

where the linear differential operator u (D) (viscoelastic operator) is given by

It is useful to define

O m!~nn!2 m!! B ~t!,

[A.4]

O m!~nn!2 m!! A ~t!.

[A.5]

n

F˜ 2n~t! 5

m

m50

u j~D! 5

1 kn

2 ~ j!

~ t ~ j! 1 D! 21D 2,

j 5 1, 2, 3.

[A.9]

n

F˜ 2n11~t! 5

m

One can easily show that

m50

Therefore, Eqs. [A.2] and [A.3] can be described by the notations

S

D

d 1 d2 k2 t 1 A m11~t! 5 A ~t!, dt n dt 2 m

[A.6]

A n11~t! 5 u n~D! A 1~t!,

B n11~t! 5 u n~D! B 1~t!, n 5 1, 2, 3, . . . ,

u n~D! 5 u ~D! z u ~D! z u ~D! · · · n times.

[A.10]

66

MOHAMED, ELSHEHAWEY, AND EL-DIB

Now, substituting from [A.4] and [A.5] into [A.1], we get

From the properties of modified Bessel functions, we have

ˆ ~ y, t! U

S

O `

5 A 0~t!

d d~ky!

O `

n50

n 1 1 A 1~t! 1! n51

`

n52

FS D ky 2

D

O n~n 2 1!~n 2 2!r!· · · ~n 2 r 1 1! ~2n~ky!1 1!! `

1 Ar~t!

n5r

S

`

`

`

1

n51

n52

D

O

2n! 5 G~2n 1 1! 5

Îp

G

FS D ky 2

and

r11/ 2

I r21/ 2~ky!

1

2 2nG~n 1 1/ 2!G~n 1 1!,

Îp

`

`

r11/ 2

r

r50

n50

O r!1 B ~t! ÎpS ky2 D `

2n1r11/ 2

1

O n!1 A ~t! S ky2 D `

G

n11/ 2

n

I n11/ 2~ky!

n51

O n!1 B ~t!S ky2 D `

n

n11/ 2

I n21/ 2~ky!,

2 2n11G~n 1 3/ 2!G~n 1 1!,

O r!1 A ~t! ÎpS ky2 D O n!G~n 11r 1 3/ 2! O

I r11/ 2~ky!

1 B 0~t!cosh ky 1 Îp

then we get

n50

I r21/ 2~ky!;

n51

1

~2n 1 1!! 5 G~2n 1 2! 5

3

r11/ 2

5 A 0~t! sinh ky 1 Îp

We make use of the properties of the gamma function

`

r11/ 2

are independent. Therefore, the solution [A.13] is the complete solution. Note that for n 5 0 the arbitrary functions A 0 (t) and B 0 (t) are not defined by relation [A.10]. Therefore, [A.13] requires the arrangement

[A.11]

S D

ky 2

ˆ ~ y, t! U

n~n 2 1!~n 2 2! . . . ~n 2 r 1 1! ~ky!2n . 1 Br~t! r! 2n! n5r

ky 3 2

I r11/ 2~ky! 5

2

n50 `

ˆ ~ y, t! 5 U

G S D

r11/ 2

2n11

O 1 1 B ~t! O 1!n 1 B ~t! O n~n2!2 1! 1 · · ·

1 B0~t!

ky 2

hence, the two functions

O n~n2!2 1! 1 · · ·

1 A 2~t!

FS D

r11/ 2

r

[A.14] where

Î

p ky I ~ky! ; sinh ky and 2 1/ 2

Î

p ky I ~ky! ; cosh ky 2 21/ 2

are used. It is necessary to derive an explicit expression for the pressure P 1 ( x, y, t). This can be obtained by substituting from [A.14] into Eq. [3.9] such that

r50

S D

1 ky n!G~n 1 r 1 1/ 2! 2

2n1r21/ 2

.

[A.12]

­ d2 Pˆ ~ y, t! 5 2r 2 @A 0~t!sinh ky 1 B 0~t!cosh ky#. ­y dt Integrating both sides with respect to ( y), we obtain

According to the definition of the modified Bessel functions (45), we obtain ˆ ~ y, t! 5 Îp U

O `

r50

S D

1 ky A r~t! r! 2

1 Îp

r

r50

[A.15]

r11/ 2

I r11/ 2~ky!

O r!1 B ~t!S ky2 D `

r d2 @A ~t!cosh ky 1 B 0~t!sinh ky#, Pˆ ~ y, t! 5 2 k dt 2 0

r11/ 2

I r21/ 2~ky!.

[A.13]

where the constant of integration is included into the arbitrary constants A 0 (t) and B 0 (t). Clearly, Eq. [A.15] satisfies Laplace’s equation [3.1]. Because the hyperbolic functions can be expressed in terms of Bessel functions (45) such that

67

ELECTROVISCOELASTIC INSTABILITY

cosh~ky Î1 1 u ! ; cosh ky 1 Îp

O n!1 u S ky2 D `

b4 5 n11/ 2

I n21/ 2~ky!,

n

k2 $ r ~2!~ e ~1! 2 e ~2!! 2~ e ~2!cosh 2k 1 e ~3!sinh 2k! r*e* 1 e~2!~e~1! 2 e~2!!~e~2! 2 e~3!!~r~2!cosh 2k 1 r~1!sinh 2k!},

[A.16]

n51

r * 5 r ~2!~ r ~1! 1 r ~3!!cosh 2k 1 ~ r ~2! 1 r ~1!r ~3!!sinh 2k, 2

sinh~ky Î1 1 u ! ; sinh ky 1 Îp

e * 5 e ~2!~ e ~1! 1 e ~3!!cosh 2k 1 ~ e ~2! 1 e ~1!e ~3!!sinh 2k, 2

O n!1 u S ky2 D `

n11/ 2

n

I n11/ 2~ky!,

[A.17]

n51

F 0~D! 5 u 1u 22u 3$ r ~1!r ~3!l 1~D!@1 1 Î1 1 u 1#@1 1 Î1 1 u 3# 1 r ~2!l 2~D!~ r ~1!u 3@1 1 Î1 1 u 1# 1 r ~3!u 1

Eq. [A.14] may have the equivalent form

3 @1 1 Î1 1 u 3! 1 r ~2! u 1u 3sinh 2k

ˆ ~ y, t! 5 A *0~t!cosh ky 1 ~cosh@ky Î1 1 u #! A *1~t! U

3 @sinh2k Î1 1 u 2#},

1 B *0~t!sinh ky 1 ~sinh@ky Î1 1 u #! B *1~t!.

2

[A.18]

If we put [A.18] in the exponential form, we obtain Eq. [3.11]. The values of the coefficients and the operators that appear in the Eq. [3.19] and [3.20] are

a1 5 a2 5

k ~2! 2 r @k 2 ~ r ~2! 2 r ~3!!#, r* k ~ r ~2!cosh 2k 1 r ~1!sinh 2k!@k 2 2 ~ r ~2! 2 r ~3!!#, r*

a4 5

k ~2! 2 r @k S 2 ~ r ~1! 2 r ~2!!#, r* 2

D2 @ d ~D!~ r ~2!cosh 2k 1 r ~3!sinh 2k! r *sinh2 2k 1 3 sinh 2k 1 r ~2! d 0~D!], 2

F 2~D! 5

k ~ r ~2!cosh 2k 1 r ~3!sinh 2k!@k 2S 2 ~ r ~1! 2 r ~2!!#, r*

a3 5

b1 5

F 1~D! 5

r ~2!D 2 @ d ~D!~ r ~2!cosh 2k 1 r ~3!sinh 2k! r *sinh2 2k 0 1 d 2~D!sinh 2k],

F 3~D! 5

D2 @ d ~D!~ r ~2!cosh 2k 1 r ~1!sinh 2k! r *sinh2 2k 2 3 sinh 2k 1 r ~2! d 0~D!], 2

F 4~D! 5

r ~2!D 2 @ d ~D!~ r ~2!cosh 2k 1 r ~1!sinh 2k! r *sinh2 2k 0 1 d 1~D!sinh 2k],

d i~D! 5 4 u 522i~ r ~2i21!u 2 2 r ~2!u 2i21!@ r ~2!u 2i21

k $~ e ~1! 2 e ~2!! 2~ e ~2!cosh 2k 1 e ~3!sinh 2k! r*e*

1 r ~2i21!u 2 Î11u 2i21] 3 $ r ~2!u 522il 2~D! 1 r ~522i!l 1

3 ~ r ~2!cosh 2k 1 r ~3!sinh 2k!

3 ~D!@1 1 Î11u 522i#} 1 2 r ~2!r ~522i! 3 u 1u 2u 3

1 r ~2!e ~2!~ e ~1! 2 e ~2!!~ e ~2! 2 e ~3!!}, k2 b2 5 $ r ~2!~ e ~2! 2 e ~3!! 2~ e ~2!cosh 2k 1 e ~1!sinh 2k! r*e*

3 @1 1 Î11u 512i#$ r ~2i21!u 2@1 1 Î1 1 u 2i21# 3 e 2k 2 2~ r ~2i21!u 2 2 r ~2!u 2i21!sinh 2k} l 3~D! 1 r ~2!2u 1u 22u 3$ r ~2i21!u 522i@1 1 Î1 1 u 2i21#e 2k

1 e ~2!~ e ~1! 2 e ~2!!~ e ~2! 2 e ~3!!~ r ~2!cosh 2k

1 r 522iu 2i21@1 1 Î1 1 u 522i#e 22k}

1 r ~3!sinh 2k)},

3 e 2k@sinh 2k Î1 1 u 2#,

k2 b3 5 $~ e ~2! 2 e ~3!! 2~ e ~2!cosh 2k 1 e ~1!sinh 2k! r*e*

d 0~D! 5 r ~1!u 1u 2u 3@1 1 Î1 1 u 1#$ r ~3!u 2 Î1 1 u 3#cosh 2k 2 2~ r ~3!u 2 2 r ~2!u 3!sinh 2k}e 2kl 4~D! 1 r ~3!u 1u 2u 3

3 ~ r ~2!cosh 2k 1 r ~1!sinh 2k!

3 @1 1 Î1 1 u 1#$ r ~1!u 2@1 1 Î1 1 u 1#cosh 2k

1 r ~2!e ~2!~ e ~1! 2 e ~2!!~ e ~2! 2 e ~3!!},

2 2~ r ~1!u 2 2 r ~2!u 1!sinh 2k}e 2kl 4~D!

68

MOHAMED, ELSHEHAWEY, AND EL-DIB

1 4 u 1u 3~ r ~1!u 2 2 r ~2!u 1!~ r ~3!u 2 2 r ~2!u 3! 3 sinh2 2k l 4~D! 2 u 1u 2u 3$ r ~1!u 2@1 1 Î1 1 u 1#e 2k 2 2~ r ~1!u 2 2 r ~2!u 1!sinh 2k}$ r ~3!l 2 3 ~D!@1 1 Î1 1 u 2# 1 r ~2!u 3@sinh 2k Î1 1 u 2# 3 sinh 2k} 2 u 1u 2u 3$ r u 2@1 1 Î1 1 u 3#e ~3!

l 1~D! 5 ~2 1 u 2!@sinh 2k Î1 1 u 2#sinh 2k 2 2 Î1 1 u 2 3 @cosh 2k Î1 1 u 2#cosh 2k 1 2 Î1 1 u 2,

m 00 5

2 r *@6k cosh 2k 1 4k sinh 2k 1 sinh 2k# 1

2 @sinh 2k Î1 1 u 2#sinh 2k 2 Î1 1 u 2,

1 ~ r ~2!cosh 2k 1 r ~522i!sinh 2k! 2k 2

1 2 r ~522i!~cosh2 2k 2 4k 2!] 1 ~ r ~2i21! 2 r ~2! ! 2

2

3 @ r ~522i!~cosh2 2k 2 4k 2!

The formula for the coefficients which appears in Eqs. [4.14]–[4.20] are

2 2 r ~2!~sinh 4k 2 k cosh2 2k!] 1 2 r ~2!r ~522i!~ r ~2i21!

D

3 cosh 2k 1 r ~2!sinh 2k)~4k 2 2 2k sinh 4k!

1 F9~iV! F0~iV! 4 2 # 2 2F9 2~iV! 32F9 3~iV! F 9~iV!

1 2r~1! r~2! r~3! e2k sinh2 2k 1 r~2! ~4k cosh 2k 1 sinh 2k! 2

3 ~ r ~2i21!e 2k 1 r ~522i!e 22k!e 2ksinh 2k},

3 @F~iV 2 2i v ! 1 a# 21 ,

S

J

3 $ r ~2i21!~ r ~2i21! 2 r ~2!!@ r ~2!~sinh 4k 2 4k!

l 4~D! 5 Î1 1 u 2sinh 2k 2 @sinh 2k Î1 1 u 2#.

F0~iV! F9~iV! 12 # , 16F9 3~iV! F 9~iV!

H

r ~2!2 1 ~2! ~1! r ~ r 1 r ~3!!@~3 2 2k!sinh 4k 1 4k# 2k 2 2

1 2 r ~1!r ~3!@~1 2 2k!cosh22k 2 2k#

l 3~D! 5 Î1 1 u 2@cosh 2k Î1 1 u 2#cosh 2k

b1 5

r *sinh 2k @ r *sinh 2k 2 2k r ~2!~ r ~1! 1 r ~3!! k2

m i1 5 m 00 1

2 @sinh 2k Î1 1 u 2#cosh 2k,

J D

1 @~n 22 2 n 12! 2 J~n 11 2 n 21!#, m 00

1 r ~1!r ~3!~1 2 4k 2!]

l 2~D! 5 Î1 1 u 2@cosh 2k Î1 1 u 2#sinh 2k

S

N5 where

3 ~D!@1 1 Î1 1 u1# 1 r~2!u1@sinh 2k Î1 1 u2#sinh 2k},

H

1 @~m 22 2 m 12! 2 J~m 11 2 m 21!#, m 00

2k

2 2~ r ~3!u 2 2 r ~2!u 3!sinh 2k}$ r ~1!l 2

a1 5

M5

n i1 5 ~ r ~2!cosh 2k 1 r ~522i!sinh 2k!$~ r ~2i21! 2 r ~2! ! 2

2

3 @ r ~2!~sinh 4k 2 4k! 1 2 r ~522i!~cosh2 2k 2 4k 2!# 1 4 r ~2!r ~522i!~ r ~2i21!cosh 2k 1 r ~2!sinh 2k!sinh2 2k

a 0 5 1/4F9~iV!,

1 2 r ~2! ~ r ~2i21!e 2k 1 r ~522i!e 22k!e 2ksinh 2k} 2

F0~iV! b0 5 2 , 4F9 2~iV!

a2 5

b2 5

1 $2F0~iV! 2 F9 2~iV!~@F~iV 1 2i v ! 1 a# 21 16F9 3~iV!

m i2 5

r ~2! ~2! ~ r cosh 2k 1 r ~522i!sinh 2k!$ r ~2!~ r ~1! 1 r ~3!! 2k 2 3 @~3 2 2k!cosh 2k sinh 2k 1 2k#

1 @F~iV 2 2i v ! 1 a# 21)}

1 2 r ~1!r ~3!@cosh2 2k 2 2k~cosh2 2k 1 1!#

1 @F~iV 2 2i v ! 1 a# 21. # 16F 9~iV!

2 r *@6k cosh 2k 2 4k sinh 2k 1 sinh 2k#}

The formula for the coefficients which appear in Eq. [5.1] are

1

r ~2! ~522i! ~522i! $r ~r 2 r ~2!!@ r ~2!~sinh 4k 2 4k! 2k 2

1 2 r ~2i21!~cosh2 2k 2 4k 2!] 1 ~ r ~522i! 2 r ~2! ! 2

2

ELECTROVISCOELASTIC INSTABILITY

3 @ r ~2i21!~cosh2 2k 2 4k 2! 2 2 r ~2! 3 ~sinh 4k 2 k cosh2 2k!] 1 2 r ~2!r ~2i21! 3 ~ r ~522i!cosh 2k 1 r ~2!sinh 2k!~4k 2 2 2k sinh 4k! 1 2 r ~1!r ~2!r ~3!e 2ksinh2 2k 1 r ~2! ~4k cosh 2k 1 sinh 2k! 2

3 ~ r ~522i!e 2k 1 r ~2i21!e 22k!e 2ksinh 2k}, n i2 5 2 r ~2!~ r ~2!cosh 2k 1 r ~522i!sinh 2k!@2k r ~2!~ r ~1! 1 r ~3!! 2 r *~2k cosh 2k 1 sinh 2k!] 1 r ~2!$~ r ~522i! 2 r ~2! ! 2

2

3 @ r ~2!~sinh 4k 2 4k 2!# 1 4 r ~2!r ~2i21!~ r ~522i!cosh 2k 1 r ~2!sinh 2k)sinh22k 1 2 r ~2! ~ r ~522i!e 2k 2

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