# Electrohydrodynamic linear stability of two immiscible fluids in channel flow

**ABSTRACT** The electrohydrodynamic instability of the interface between two viscous fluids with different electrical properties in plane Poiseuille flow has recently found applications in mixing and droplet formation in microfluidic devices. In this paper, we perform the stability analysis in the case where the fluids are assumed to be leaky dielectrics. The two-layer system is subjected to an electric field normal to the interface between the two fluids. We make no assumption on the magnitude of the ratio of fluid to electric time scales, and thus solve the full conservation equation for the interfacial charge. The electric field is found to be either stabilizing or destabilizing, and the influence of the various parameters of the problem on the interface stability is thoroughly analyzed.

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**ABSTRACT:**A numerical model for electrokinetic flow of multiphase systems with deformable interfaces is presented, based on a combined level set-volume of fluid technique. A new feature is a multiphase formulation of the Nernst-Planck transport equation for advection, diffusion and conduction of individual charge carrier species that ensures their conservation in each fluid phase. The numerical model is validated against the analytical results of Zholkovskij et al. (2002) [1], and results for the problem of two drops coalescing in the presence of mobile charge carriers are presented. The time taken for two drops containing ions to coalesce decreases with increasing ion concentration.Journal of Computational Physics 10/2013; · 2.49 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we have discussed the linear stability analysis of the electrified surface separating two coaxial Oldroyd-B fluid layers confined between two impermeable rigid cylinders in the presence of both interfacial insoluble surfactant and surface charge through porous media. The case of long waves interfacial stability has been studied. The dispersion relation is solved numerically and hence the effects of various parameters are illustrated graphically. Our results reveal that the influence of the physicochemical parameter β is to shrink the instability region of the surface and reduce the growth rate of the unstable normal modes. Such important effects of the surfactant on the shape of interfacial structures are more sensitive to the variation of the β corresponding to non-Newtonian fluids-model compared with the Newtonian fluids model. In the case of long wave limit, it is demonstrated that increasing β, has a dual role in-fluence (de-stabilizing effects) depending on the viscosity of the core fluid. It has a destabilizing effect at the large values of the core fluid viscosity coefficient, while this role is exchanged to a regularly stabilizing influence at small values of such coefficient.Acta Mechanica Sinica 28(6). · 0.62 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we study the linear stability of the interface between an Upper Convective Maxwell fluid and a hydrodynamically passive fluid subject to an electric field applied either parallel or normal to the flat interface between the two fluids. The fluids are leaky-dielectric and we apply surface-coupled model. We solve the model equations analytically and study the dispersion and neutral curves for various parameters representing the applied potential, the fluid’s elasticity, the physical and the electrical properties of the fluids, and the heights of the fluids in the presence of both normal and parallel electric fields. It is found that the critical wavenumber is independent of the Weissenberg number. However, increasing the Weissenberg number increases the maximum growth rate for both the normal and the parallel fields. The critical wavenumber increases with the dimensionless applied voltage for the normal field. Lastly for the normal field, for some values of the dimensionless parameters, the growth rate reached very large values representing some type of singularity as has been observed in the literature. However, for the same values of the parameters no singularity is observed for the parallel field.The European Physical Journal Special Topics 03/2013; 219(1). · 1.76 Impact Factor

Page 1

Electrochimica Acta 51 (2006) 5316–5323

Electrohydrodynamic linear stability of two immiscible

fluids in channel flow

O. Ozena,b,∗, N. Aubrya,b, D.T. Papageorgioua,b, P.G. Petropoulosa

aDepartment of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology,

University Heights, Newark, NJ 07102, USA

bDepartment of Mechanical Engineering, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA

Received 7 November 2005; received in revised form 25 January 2006; accepted 1 February 2006

Available online 10 March 2006

Abstract

The electrohydrodynamic instability of the interface between two viscous fluids with different electrical properties in plane Poiseuille flow has

recently found applications in mixing and droplet formation in microfluidic devices. In this paper, we perform the stability analysis in the case

where the fluids are assumed to be leaky dielectrics. The two-layer system is subjected to an electric field normal to the interface between the two

fluids. We make no assumption on the magnitude of the ratio of fluid to electric time scales, and thus solve the full conservation equation for the

interfacial charge. The electric field is found to be either stabilizing or destabilizing, and the influence of the various parameters of the problem on

the interface stability is thoroughly analyzed.

© 2006 Elsevier Ltd. All rights reserved.

PACS: 47.15.Fe; 47.15.Gf; 47.20.Ma; 47.65.+a; 85.85,+j

Keywords: Electrohydrodynamics; Two-fluid Poiseuille flow; Linear instability; Pattern formation; Microfluidics

1. Introduction

Recently, the instability of flows in microchannels has been

a primary focus of research due to the rapid development in the

area of microfluidics. In many microsystems rapid mixing is

highly desired, which is achieved by using complex geometries

or external fields [1,2]. However, not much has been accom-

plished in the domain of immiscible fluids, which is of impor-

tance as the instability of the liquid–liquid interface may lead

to rupture and eventually to a liquid-in-liquid droplet formation.

The linear stability tells us what the necessary conditions are

for the instability of the interface. Knowing the wavenumber for

the most unstable perturbation may, in turn, give information

on what the size of the resulting droplets is. One can generate

monodisperse droplets using the model described in this paper

[3], which are important in the reproducibility of microsystems

and the uniformity of microstructures [4] and in drug delivery

∗Corresponding author.

E-mail addresses: ozgur.ozen@njit.edu (O. Ozen), Aubry@adm.njit.edu

(N. Aubry), depapa@aphrodite.njit.edu (D.T. Papageorgiou),

peterp@ouzo.njit.edu (P.G. Petropoulos).

devices [5–7]. Thus, understanding the physics underlying the

interfacial instability of a two-fluid flow should be of consider-

able interest.

Electrohydrodynamics is the study of the relation between

the electric field and fluid mechanics. One important problem in

electrohydrodynamics is the impact of the electric field on the

stability of a two-fluid system. The discontinuity of the electri-

cal properties of the fluids across the interface affects the force

balance at the fluid–fluid interface, which may either stabilize

or destabilize the interface in question. There are two common

approaches, which we now describe. The first one, the bulk

coupled model, assumes a conductivity gradient in a thin diffu-

sion layer between the two fluids, resulting in an electrical body

force on the fluids. Hoburg and Melcher used such a model to

study the stability of two fluids stressed by a tangential electric

field with a conductivity gradient in a diffusive layer [8]. Recent

applications of this approach to microchannels have been pre-

sented by Lin et al. [9] who studied the electrokinetic flow in a

microchannel with a conductivity gradient. Lin et al. performed

the linear stability analysis and nonlinear simulations following

the framework of Hoburg and Melcher [8], and also considered

the diffusion of conductivity as in ref. [10]. Storey et al. [11]

0013-4686/$ – see front matter © 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.electacta.2006.02.002

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O. Ozen et al. / Electrochimica Acta 51 (2006) 5316–5323

5317

addressed the nonlinear stability using asymptotics, and Chen

et al. [12] considered the absolute and convective instability for

the same problem. Tardu [13] investigated the linear stability of

aPoiseuilleflowundertheeffectofanelectrostaticdoublelayer

in a microchannel.

The second approach, the surface coupled model, considers

a jump in electrical conductivity at the interface of the two flu-

ids. Each fluid has a constant electrical conductivity, and there

are no electric body forces present in the fluids. Melcher and

Schwarz used this second viewpoint to study the stability of

two fluids under the influence of a tangential electric field [14]

as well as a normal electric field [15], both in an unbounded

domain. There is, however, no base flow present in their

analyses.

Previously, Abdella and Rasmussen [16] analyzed Couette

flow in an unbounded domain subjected to a normal electric

field. They considered a system of two viscous fluids with

different fluid and electrical properties, and used Airy func-

tions and Airy integrals to derive a generic dispersion rela-

tion. They analyzed two special cases in detail: (i) the electro-

hydrodynamic free-charge case [17] and (ii) the electrohydro-

dynamic polarization charge configuration [17]. Tilley et al.

[18] and Savettaseranee et al. [19] addressed the rupture of a

thin film under the stress of an electric field. More recently,

Craster and Matar [20] derived a coupled system of evolution

equations to investigate the stability of the interface between

two thin leaky dielectric fluid layers. Thaokar and Kumaran

[21] studied the stability of the interface between two dielec-

tric fluids confined between moving parallel plates subjected

to a normal electric field in the zero Reynolds number limit

and for fast relaxation times, using linear and weakly nonlinear

analysis.

In this paper, we study the interfacial stability of a two-fluid

flow in a channel using the surface coupled model. In our

study, the electric body force vanishes, and the electric field

and fluid dynamics are coupled only at the interface through

the tangential and normal interfacial stress balance equations.

We perform a linear stability analysis of two superposed

viscous flows in a channel stressed by a normal electric

field. The interface is not perfectly conducting and admits

free charge due to the fact that the two fluids have finite

conductivities. Finally, we present numerical results using the

Chebyshev spectral Method, and study the influence of the

various parameters of the problem on the interface stability

property.

2. The physical and mathematical model

The physical model, depicted in Fig. 1, consists of a viscous

conducting liquid layer of depth, h, in contact with another vis-

cous liquid of depth, h*, in a channel of infinite length in the

horizontal direction. The equations that model the physics are

given by the momentum balance in each layer, i.e.,

ρ

?∂? υ

∂t+ ? υ · ∇? υ

?

= −∇P + ρ? g + µ∇2? υ

(1)

Fig. 1. The physical model.

and

ρ∗

?∂? υ∗

∂t

+ ? υ∗· ∇? υ∗

?

= −∇P∗+ ρ∗? g + µ∗∇2? υ∗,

(2)

the Laplace equation in each layer, i.e.,

∇2V = 0

and

(3)

∇2V∗= 0,

and also the continuity equation in each layer, assuming that

both fluids are incompressible. In the preceding equations, ? υ, P

and V refer to the velocity, pressure and voltage potential fields,

respectively, and the asterisk denotes the fluid in the region of

depth, h*. It is assumed that the voltages at the plates are held

constant and the plate at z=h*is grounded, i.e., V(−h)=Vband

V*(h*)=0. In the equations above, ρ and µ denote the density

and the viscosity of the fluids. Hereafter, the gravitational accel-

eration, ? g, is dropped from the equations since we assume that

the gravity points in the direction into Fig. 1.

The no-slip boundary condition, applied along both plates

of the channel, gives rise to υx(−h)=υz(−h)=0 and υ∗

υ∗

located at z=Z(x,t), leads to two kinematic conditions because

the fluids are immiscible and no mass transfer occurs across the

interface. The kinematic conditions are given by

(4)

x(h∗) =

z(h∗) = 0, while the mass balance equation at the interface,

(? υ − ? u) · ? n = 0

and

(5)

(? υ∗− ? u) · ? n = 0,

where the unit outward normal is defined by

(6)

? n =

−(∂Z/∂x)?i +?k

(1 + (∂Z/∂x)2)1/2

and the interface speed is

? u · ? n =

(∂Z/∂t)

(1 + (∂Z/∂x)2)1/2.

At the interface, the tangential components of the velocities

of the two fluids are equal to each other, and thus ? υ ·? t = ? υ∗·? t

holds. Here, the unit tangent vector is given by

? t =

?i + (∂Z/∂x)?k

(1 + (∂Z/∂x)2)1/2.

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O. Ozen et al. / Electrochimica Acta 51 (2006) 5316–5323

The interfacial tension and the coupling between the fluid

dynamics and the electric fields enter the problem through the

balance of forces at the interface. By taking the dot product of

the force balance with the unit normal and unit tangent vectors

separately, we obtain the normal and tangential stress balance

equations. The force balance is expressed by the equation

??T

where??T is the stress tensor, γ the interfacial tension and 2H is

the surface mean curvature given by

∗

· ? n −??T · ? n + γ2H? n = 0,

(7)

2H =

∂2Z/∂x2

(1 + (∂Z/∂x)2)3/2.

The stress tensor,??T, is composed of a fluid component,??T

an electrical component,??T

??T

and

?

where? E is the electric field given by? E = −∇V,??I the identity

tensor, and ε0and ε are the electrical permittivity of the vacuum

andtherelativeelectricalpermittivityofthecorrespondingfluid.

The superscript ‘t’ indicates the transpose of the corresponding

tensor.Thecontinuityofthetangentialcomponentoftheelectric

field across the interface implies that

F, and

E, whose expressions are

F

= −P??I + µ(∇? υ + ∇? υt)

??T

E

= ε0ε

? E? E −1

2|? E|2??I

?

,

? E ·? t =? E∗·? t.

The completion of the mathematical description of the prob-

lemrequirestwoadditionalinterfacialconditionsforthesurface

charge density, q. The first one comes from Gauss’ Law, i.e.,

(8)

−q = ε0ε∗? E∗· ? n − ε0ε? E · ? n,

while the second one depends on the model considered. Using

the leaky dielectric model, which allows for charge distribution

on interfaces due to current fluxes from the bulk, we describe

the physics of the electric-field in both fluids by writing

(9)

qt− ? u · ∇sq + ∇s· (q? us) + qγ? u · ? n = σ? E · ? n − σ∗? E∗· ? n.

(10)

Here the subscript s denotes surface quantities and σ is the elec-

trical conductivity of the corresponding fluid. The nonlinear

equations are brought into dimensionless form using the fol-

lowing scales:

(x,z) = (hx?,hz?),t =h

Ut?,Z(x,t) = hZ?(x?,t?),

V = Vh∗V?,

For plane Poiseuille flow, the velocity scale is the horizontal

velocity, U, at the interface when the interface is flat. Although

? υ = U? υ?,P =µU

h

P?,q =ε0Vb

h

.

the dimensionless quantities are primed, the primes are dropped

from the final equations. The Navier–Stokes equations then

become

?∂? υ

and

?∂? υ∗

where Re=ρUh/µ denotes the Reynolds number, while the

equations for the voltage potentials in dimensionless form are

Re

∂t+ ? υ · ∇? υ

?

= −∇P + ∇2? υ

(11)

ρ∗

ρRe

∂t

+ ? υ∗· ∇? υ∗

?

= −∇P∗+µ∗

µ∇2? υ∗,

(12)

∇2V = 0

and

(13)

∇2V = 0

The tangential and normal stress balance equations take the fol-

lowing dimensionless form:

(14)

[(∇? υ + ∇? υT) + εEb? E? E] : ? n? t

?µ∗

and

?

?µ∗

=

µ(∇? υ∗+ ∇? υ∗T) + ε∗Eb? E∗? E∗

?

: ? n? t

(15)

P −

(∇? υ + ∇? υT) + εEb

?

? E? E −1

2|? E|2??I

??

: ? n? n +

1

Ca2H

= P∗−

µ(∇? υ∗+ ∇? υ∗T)

?

b/µUh is the electric Weber number, and

Ca=µU/? is the Capillary number. The condition describing

the conservation of interfacial charge becomes

+ε∗Eb

? E∗? E∗−1

2|? E∗|2??I

??

: ? n? n,

(16)

where Eb= ε0V2

qt− ? u · ∇sq + ∇s· (q? us) + qγ? u · ? n = S

?

? E · ? n −σ∗

σ

? E∗· ? n

?

(17)

,

where S=hσ/Uε0is a new dimensionless parameter. We can

rewrite S as (h/U)/(ε0/σ), so the new parameter S is actually the

ratio of fluid to electric time-scales.

Wenowaddressthelinearstabilitypropertyoftheproblemby

applying arbitrary small disturbances to the interface and deter-

mining the time growth (or decay) rate of such disturbances.

We thus linearize the above equations about a known base state

and investigate the onset of the interfacial instability from the

perturbed model. Hereafter, the subscript ‘0’ refers to the vari-

ables of the base state and the subscript ‘1’ to the variables of

the perturbed state. The perturbed voltage can then be expanded

in the following form:

?

V = V0+ δV1+dV0

dz0z1

?

+ O(δ2),

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O. Ozen et al. / Electrochimica Acta 51 (2006) 5316–5323

5319

where δ is a small perturbation parameter representing the devi-

ation from the base state and z1is the mapping of the perturbed

configuration in (x, z) onto the reference configuration in (x0,

z0). The meaning of such a mapping can be found in Johns and

Narayanan [22] and at the interface it simply reduces to the per-

turbation of the surface deflection to first order, Z1, a variable

which will be determined during the course of our calculation.

Assuming a two-dimensional variation in x0and z0, we can fur-

therexpandV1andothersubscript‘one’variablesusinganormal

mode expansion. For example,

V1=ˆV1(z0)eωt0eikx0+ c.c.

Here ω denotes the inverse time constant, k the wavenumber

associated with a given perturbation, and c.c. denotes the com-

plex conjugate. A wavenumber arises because the system is

infinite in the lateral direction. The same expansion is used for

the two components of velocity in each fluid, the pressure in

both fluids, and the surface deflection, Z.

3. The base state solution and the perturbed equations

In the base state, the velocity field and the voltage potential

are decoupled from each other, as can be seen by considering

the interfacial conditions where hydrodynamic and electrical

variables appear together. When the system is unperturbed, the

fluidflowisassumedtobefullydevelopedinbothregionsinthe

lateral direction. The vertical component of the velocities and

its derivatives are equal to zero, and the interface is flat, which

implies that only the electrical part of the stress tensor survives

in the normal stress balance. In the tangential stress balance, we

notethatthereisnocontributionfromtheelectricalstresstensor

since the electric field is unidirectional in the unperturbed state,

i.e., normal to the flat interface, in both regions. This allows us

to solve for the velocities and voltage potentials independently

of each other. The details of the base state velocity calculations

for plane Poiseuille flow have been given by other authors [23].

For this reason, we simply present the final expressions of the

horizontal component of the velocity in the two fluids:

?

1

d∗(1 + d∗)

and

?

?

In the previous equations, d*is the scaled depth of the fluid

denotedbyanasterisk.Theexpressionsforthevoltagepotentials

in the base state take the form

σ∗/σ

(d∗+ σ∗/σ)z0+

υx0=

1

d∗(1 + d∗)

d∗+µ∗

?

µ

?

z2

0

?

+

d∗2−µ∗

µ

z0+ 1(18)

u∗

x0= −µ

µ∗

1

d∗(1 + d∗)

d∗2−µ∗

d∗+µ∗

µ

?

z2

0+

µ

µ∗

1

d∗(1 + d∗)

×

µ

?

z0+ 1.

(19)

V0= −

d∗

(d∗+ σ∗/σ)

(20)

and

V∗

0= −

1

(d∗+ σ∗/σ)z0+

Using Gauss’ Law, one can easily obtain the following expres-

sion for the surface charge density:

d∗

(d∗+ σ∗/σ).

(21)

q0=ε∗− ε(σ∗/σ)

(σ∗/σ) + d∗.

After perturbing the domain and the boundary equations in

the manner outlined in the previous section and omitting the hat

of the z-dependent part of the perturbed variables, we derive the

following momentum equations in dimensionless form:

?

?

0

(22)

Re ωυx1+ ikυx0υx1+ υz1dυx0

dz0

?

= −ikP1+

d2υx1

dz2

− k2υx1

?

(23)

Re[ωυz1+ ikυx0υz1] = −dP1

dz0

+

?

d2υz1

dz2

0

− k2υz1

?

(24)

ρ∗

ρRe

?

ωυ∗

x1+ ikυ∗

x0υ∗

?

x1+ υ∗

z1

dυ∗

dz0

x0

?

?

= −ikP∗

1+µ∗

µ

d2υ∗

dz2

x1

0

− k2υ∗

x1

(25)

and

ρ∗

ρRe[ωυ∗

z1+ ikυ∗

x0υ∗

z1] = −dP∗

1

dz0

+µ∗

µ

?

d2υ∗

dz2

z1

0

− k2υ∗

z1

?

(26)

,

while the perturbed voltage potentials are described by the per-

turbed dimensionless Laplace equations given by

d2V1

dz2

0

− k2V1= 0(27)

and

d2V∗

dz2

1

0

− k2V∗

1= 0.

(28)

These equations are subjected to the following boundary condi-

tions at both plates:

υx1= 0,

υ∗

υz1= 0,

υ∗

V1= 0

V∗

atz = −1

atz = d∗.

x1= 0,

As mentioned above, the dimensionless interfacial conditions

are also perturbed, which leads to the following kinematic con-

ditions and no-slip condition:

z1= 0,

1= 0

(29)

υz1− ikυx0Z1= ωZ1,

υ∗

x0Z1= ωZ1

(30)

z1− ikυ∗

(31)

Page 5

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O. Ozen et al. / Electrochimica Acta 51 (2006) 5316–5323

and

υx1+dυx0

dz0

Z1= υ∗

x1+dυ∗

x0

dz0

Z1.

(32)

Thetangentialandnormalstressbalanceequationsthenbecome

?

dz0

0

?

dz0

0

+ε∗EbdV∗

dz0

dz0

dυx1

+d2υx0

dz2

Z1+ ikυz1

?

+ εEbdV0

dz0

?

?

ikV1+ ikdV0

dz0Z1

?

=µ∗

µ

dυ∗

x1

+d2υ∗

?

x0

dz2

Z1+ ikυ∗

z1

0

ikV∗

1+ ikdV∗

0

Z1

?

(33)

and

P1−

?

2dυz1

dz0

− ikdυx0

?

dz0

Z1

?

− εEbdV0

dz0

?

dV1

dz0

−k2

CaZ1

= P∗

1−µ∗

µ

2dυ∗

dz0

z1

− ikdυ∗

x0

dz0

Z1

− ε∗EbdV∗

0

dz0

dV∗

dz0

1

.

(34)

In the Eqs. (33) and (34), note that

dυx0

dz0

=µ∗

µ

dυ∗

dz0

x0

(35)

d2υx0

dz2

0

=µ∗

µ

d2υ∗

dz2

x0

0

.

(36)

Applying a similar technique to the rest of the perturbed inter-

facial conditions, we obtain

V1+dV0

dz0Z1= V∗

1+dV∗

0

dz0

Z1

(37)

ε∗dV∗

dz0

1

− εdV1

dz0

= −q1

(38)

and

ωq1+ ikυx0q1+ ikq0υx1= S

?σ∗

σ

dV∗

dz0

1

−dV1

dz0

?

.

(39)

Beforeweturntothenumericalresults,muchcanbededuced

about the instability conditions of the problem for the limiting

case S?1 by simply determining the voltage profiles in both

fluid domains as a function of the base state voltage gradients

and the surface deflection, Z1. Since the parameter S measures

the ratio of fluid to charge relaxation time-scales, assuming that

Sislargeimpliesthatthechargerelaxationtimeisveryfast.The

equations for the perturbed voltage in both layers are given in

Eqs. (27) and (28). Thus, the voltage profiles in the non-asterisk

and the asterisk fluids are given by

V1= Acosh(kz) + Bsinh(kz)

and

(40)

V∗

1= A∗cosh(kz) + B∗sinh(kz).

(41)

Using the boundary conditions V(−1)=0 and V*(d*)=0 along

with the interfacial conditions (37) and (39), and assuming that

S is very large, we find that

dV∗

dz0

Moreover, we know that

?σ∗

and

dV1

dz0

σ

1

=

1 − (σ∗/σ)

(σ∗/σ)tanh(k) + tanh(kd∗)

dV∗

dz0

0

Z1.

(42)

dV0

dz0

=

σ

?dV∗

0

dz0

(43)

=

?σ∗

?dV∗

1

dz0

(44)

whenSisverylarge.Now,returningtothenormalstressbalance

and rewriting the terms due to the presence of the electric field

in terms of the gradient of the base state voltage potential in the

asterisk fluid and the surface deflection, Z1, we end up with the

normal stress balance equation in the following form:

?

?dV∗

?

In Eq. (45), the contributions of the Maxwell stress tensor and

the surface curvature are on the right-hand side. Recall that the

surface tension always plays a stabilizing role on the instability

ofthisproblem.Inthisnewformofthenormalstressbalance,we

knowthecontributionoftheelectricfieldstressesintermsofthe

surfacedeflection;accordingly,wecandeterminetheconditions

under which these electric stress contributions are in the same

direction as the surface tension effects. In the equation above,

we also know that ((σ*/σ)tanh(k)+tanh(kd*))>0. Independent

of the sign of Z1, we thus learn that when (σ*/σ)2>(ε*/ε) and

σ*/σ <1(or(σ*/σ)2<(ε*/ε)andσ*/σ >1),theelectricfieldplays

a stabilizing role in this interfacial instability problem. Follow-

ing the same path, we also deduce that when (σ*/σ)2>(ε*/ε)

and σ*/σ >1 (or (σ*/σ)2<(ε*/ε) and σ*/σ <1), the electric field

plays a destabilizing role. We recall here that these conclusions

are drawn based on the assumption that S is very large, namely

for fast charge relaxation times.

P1−

2dυz1

dz0

− ikdυx0

?2??σ∗

1 − (σ∗/σ)

(σ∗/σ)tanh(k) + tanh(kd∗)

dz0

Z1

?

?2

− P∗

1+µ∗

?ε∗

µ

??

?

?

2dυ∗

dz0

z1

− ikdυ∗

x0

dz0

Z1

?

= εEb

0

dz0

σ

−

ε

×

Z1+k2

CaZ1.

(45)

4. Results of linear stability calculations

Numerical calculations were performed to determine the

growth rate at different wavenumbers for various initial con-

ditions and fluid properties. The Chebyshev spectral tau method

[24] is used to solve the eigenvalue problem resulting from the

linearization of the problem around the base state. For each cal-

culation, the number of terms in the expansion is increased until

the convergence of the numerical results is ensured.

Since we have already predicted the role of the electric field

when S is large, we start by numerically testing our conclusions

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5321

Fig. 2. Numerical validation of the stability criteria determined from our theoretical analysis (see text in Section 3) when S is large, (a) σ*/σ =0.5, ε*=2 and ε=10,

(b) σ*/σ =2, ε*=5 and ε=1, (c) σ*/σ =2, ε*=3 and ε=1, (d) σ*/σ =0.5, ε*=5 and ε=10, demonstrating that the electric field is stabilizing in (a) and (b), and

destabilizing in (c) and (d) as expected from the four conditions deduced when S is large.

in this limit. The input parameters for the fluids are chosen to

be Re=1, µ*/µ=1, ρ*/ρ=1, Ca=1, S=108and h*/h=1, while

those related to the electrical properties of the fluids are given

in the caption of Fig. 2. In each case, we determine the effect of

the electric field applied across the fluid layers compared to the

case with no electric field and as the value of Ebis increased.

Theresultsofourcalculations,asdisplayedinFig.2athroughd,

agreewithourtheoreticalpredictionsforthestabilitycriteria.In

Fig.2aandbtheelectricfieldisstabilizingsince(σ*/σ)2>(ε*/ε)

andσ*/σ <1(a)and(σ*/σ)2<(ε*/ε)andσ*/σ >1(b),anddesta-

bilizing in Fig. 2c and d since (σ*/σ)2>(ε*/ε) and σ*/σ >1 (c)

and (σ*/σ)2<(ε*/ε) and σ*/σ <1 (d).

Recall that in the present work we do not assume that S is

necessarilylarge,i.e.,S?1,andthatwesolvefortheinterfacial

charge. We will first present our results for cases which differ

fromeachotheronlyinthevalueofS,inordertounderstandthe

effectoftherelaxationofotherresearchers’assumptionthatSis

large. The other input parameters are chosen as follows: Eb=1,

Re=1, µ*/µ=1, ρ*/ρ=1, σ*/σ =0.1, ε*=3, ε=4, Ca=1 and

h*/h=1. The value of the input parameter S was varied from

10−4to 104by factors of 10.

We have plotted the real part of the dimensionless growth

rate of the perturbation against the dimensionless wavenumber

for some of the cases only. The reason for this selection is that

the difference between the real part of the growth rates cannot

be discerned when the values of S are either too large or too

small. We observe in Fig. 3 that the smaller S the higher the crit-

icalwavenumber,whilethemaximumgrowthratedoesnotvary

muchforthissetofinputparameters.Additionalinterestingfea-

turesintroducedbytheinclusionofSinthemathematicalmodel

Fig.3. EffectofdifferentvaluesofthedimensionlessparameterSonthegrowth

rate and critical wavenumber. The input parameters are Eb=1, Re=1, µ*/µ=1,

ρ*/ρ=1, σ*/σ =0.1, ε*=3, ε=4, Ca=1 and h*/h=1.

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Fig. 4. Effect of the dimensionless parameter S on the growth rate over a broad

range of wavenumbers, so that the effect at large wavenumbers can be clearly

observed. The input parameters are the same as in Fig. 3.

emerge when we compare the growth rates at higher wavenum-

bers for different values of the dimensionless parameter, S. For

instance, for small values of S the stabilizing role of the surface

tension at higher wavenumbers is no longer dominant, as Fig. 4

shows,incontrasttowhathappensinmanyinterfacialinstability

problems.

When we fix all the input parameters except the ratio of the

viscosities µ*/µ, we find that the higher the viscosity ratio the

more stable the interface is. In Fig. 5, we have plotted the real

partofthegrowthrateagainstthewavenumberinthecasewhere

Eb=1, Re=1, ρ*/ρ=1, σ*/σ =0.1, ε*=3, ε=4, Ca=1, S=103

and h*/h=1 for different values of the viscosity ratio, µ*/µ.

Increasing the viscosity ratio results in slightly increasing the

critical wavenumber and decreasing the maximum growth rate.

It thus has a stabilizing effect for the selected values of input

parameters due to the increased viscous dissipation.

Fig. 5. Effect of the viscosity ratio on the growth rate and critical wavenumber.

Fig. 6. Effect of the ratio of the fluid depths on the growth rate and critical

wavenumber.

In addition, we have investigated the effect of the ratio of

the fluid layers depths on the instability of the interface. In

our sample calculations, we have taken Eb=1, Re=1, ρ*/ρ=1,

σ*/σ =0.1, ε*=1, ε=2, Ca=1, S=103and µ*/µ=0.1, and var-

ied the depth ratio h*/h. For the previous input parameters, the

critical wavenumber and maximum growth rate decreased as

the ratio of the fluid depths increased from 0.3 to 1 (see Fig. 6),

thus showing that a decrease in the depths mismatch between

the two layers has a stabilizing effect for the selected input

parameters. The calculations presented thus far enable us to

understandtheroleofthefluiddynamicsontheinstabilityofthe

problem.

However, in order to understand the effect of the electri-

cal properties on the instability, we now fix both the depths

of the fluid layers and the fluid mechanical properties, and vary

the ratios of the fluids’ electrical conductivities and permittivi-

ties. First, we present calculations for a two-fluid system where

Eb=1, Re=1, p*/p=1, ε*=1, ε=2, Ca=1, S=103, h*/h=0.5

and µ*/µ=0.1, and a ratio of the electrical conductivities vary-

ingfrom0.1to0.9.Thereasonforcarryingoutthesecalculations

in the interval 0<σ*/σ <1 is that, in practice, highly conducting

fluidsalsoexhibithighelectricalpermittivity,andsincewehave

taken the permittivity of the fluid denoted by an asterisk to be

lower, its electrical conductivity has to be lower as well. Fig. 7

displays the results of our calculations, showing that decreasing

the ratio of electrical conductivities σ*/σ from 0.9 to 0.1 (and

thus increasing the conductivity mismatch between the two flu-

ids) increases both the critical wavelength and the maximum

growth rate, thus having a destabilizing effect. Turning to the

study of the effect of the electrical permittivity ratio, we now fix

Eb=1, Re=1, ρ*/ρ=1, σ*/σ =0.1, Ca=1, S=103, h*/h=0.5

and µ*/µ=0.1, and vary the ratio ε*/ε by simply fixing ε*=1

and varying ε from ε=2 to 10. It was found that both the maxi-

mum growth rate and the critical wavenumber increased as the

ratio of the electric permittivities was raised from 0.1 to 0.5

(Fig. 8).

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Fig. 7. The effect of the ratio of the electrical conductivities on the growth rate

and the critical wavenumber.

Fig. 8. Effect of the ratio of the electrical permittivities on the growth rate and

critical wavenumber.

5. Conclusion

We have studied the linear stability of a two-fluid flow in

a channel where the fluids are assumed to be leaky dielectrics

with different electric properties (conductivities and permittivi-

ties) and subjected to an electric field normal to their interface.

For this purpose, we have derived and then linearized the equa-

tionsofmotionwheretheinteractionbetweenthehydrodynamic

and electric problems occurs through the stress balance at the

fluid interface. The growth rate of the perturbation was then

computed for various wavenumbers by using the Chebyshev

spectral tau method and its variation studied as a function of the

dimensionless parameter S=hσ/Uε0, as well as viscosity, fluid

depth,electricalconductivityandpermittivityratios.Whiletwo-

layer flows in channels of small dimensions are rather stable,

the instability of the fluid–fluid interface is highly desirable in

certain cases, particularly for microfluidic applications where

the mixing of reagents or the formation of drops are crucial

steps in the process. However, in systems of larger scale, the

instability of the fluid–fluid interface in a channel is often an

undesiredphysicalphenomemon.Insuchsituations,controlling

the flow requires the stabilization of the interface. In searching

for a method capable of either stabilizing a potentially unsta-

ble interface or destabilizing a potentially stable one, we have

investigatedtheroleoftheelectricfieldonthetwo-layerchannel

flow problem, demonstrated that either destabilization or stabi-

lization can be obtained and presented growth rates in situations

wheretheelectricfieldisstabilizingordestabilizingoverabroad

range of wavenumbers. Not only the dimensionless parameter

S was found to affect the critical wavenumber, but also all the

ratios of electrical and mechanical properties of the two fluids

(mentioned above) were observed to have a significant influ-

enceonboththemaximumgrowthrateandcriticalwavenumber,

and should thus play a critical role in applications, particularly

microfluidic ones, where either an unstable interface (e.g., for

mixing or droplet formation) or a stable one (e.g., for material

deposition) is desired.

Acknowledgments

This work was partially supported by the New Jersey Com-

mission on Science and Technology through the Center for

Micro-Flow Control.

References

[1] A.O.E. Moctar, N. Aubry, J. Batton, Lab. Chip 3 (4) (2003) 273.

[2] I. Glasgow, J. Batton, N. Aubry, Lab. Chip 4 (6) (2004) 558.

[3] O. Ozen, N. Aubry, D. Papageorgiou, P. Petropoulos, Phys. Rev. Lett.,

in press.

[4] Q. Xu, M. Nakajima, Appl. Phys. Lett. 85 (17) (2004) 3726.

[5] T. Thorsen, R.W. Roberts, F.H. Arnold, S.R. Quake, Phys. Rev. Lett. 86

(18) (2001) 4163.

[6] S.L. Anna, N. Bontoux, H.A. Stone, Appl. Phys. Lett. 82 (3) (2003)

364.

[7] T. Cubaud, C. Ho, Phys. Fluids 16 (12) (2004) 4575.

[8] J.F. Hoburg, J.R. Melcher, J. Fluid Mech. 73 (1976) 333.

[9] H. Lin, B.D. Storey, M.H. Oddy, C. Chen, J.G. Santiago, Phys. Fluids

16 (6) (2004) 1922.

[10] J.C. Baygents, F. Baldessari, Phys. Fluids 10 (1) (1998) 301.

[11] B.D. Storey, B.S. Tilley, H. Lin, J.G. Santiago, Phys. Fluids 17 (12)

(2005) 018103-1.

[12] C.-H. Chen, H. Lin, S.K. Lele, J.G. Santiago, J. Fluid Mech. 524 (2005)

263.

[13] S. Tardu, Tran. ASME 126 (1) (2004) 10.

[14] J.R. Melcher, W.J. Schwarz, Phys. Fluids 11 (12) (1968) 2604.

[15] J.R. Melcher, C.V. Smith, Phys. Fluids 12 (4.) (1969) 778.

[16] K. Abdella, H. Rasmussen, J. Comp. Appl. Math. 78 (1997) 33.

[17] J.R. Melcher, Field Coupled Surface Waves, MIT Press, Cambridge,

MA, 1963.

[18] B.S. Tilley, P.G. Petropoulos, D.T. Papageorgiou, Phys. Fluids 13 (12)

(2001) 3547.

[19] K. Savettaseranee, D.T. Papageorgiou, P.G. Petropoulos, B.S. Tilley,

Phys. Fluids 15 (3) (2003) 641.

[20] R.V. Craster, O.K. Matar, Phys. Fluids 17 (2005) 32104-1.

[21] R.M. Thaokar, V. Kumaran, Phys. Fluids 17 (8) (2005) 084104-1.

[22] L.E. Johns, R. Narayanan, Interfacial Instability, Springer-Verlag, NY,

2002.

[23] S.G. Yiantsios, B.G. Higgins, Phys. Fluids 31 (11) (1988) 3225.

[24] J.P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications,

Inc., Mineola, NY, 2001.

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