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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
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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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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)
Reωυx1+ 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)
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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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O. Ozen et al. / Electrochimica Acta 51 (2006) 5316–5323
5323
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.
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