Nc

OD

h

�

�

�

�

a

ARRAA

KEEDOB

1

udpctpd

(

0h

Colloids and Surfaces A: Physicochem. Eng. Aspects 423 (2013) 27– 34

Contents lists available at SciVerse ScienceDirect

Colloids and Surfaces A: Physicochemical andEngineering Aspects

jo ur nal homep a ge: www.elsev ier .com/ locate /co lsur fa

umerical simulation of electrically deformed droplets lessonductive than ambient fluid

. Ghazian ∗, K. Adamiak, G.S.P. Castleepartment of Electrical and Computer Engineering, Western University, London, ON, Canada

i g h l i g h t s

3-D deformation of a droplet sus-pended in a uniform dc electric fieldis studied.Three different regimes wereobserved for droplet deformation.Droplets experienced an oscillatorymotion with the major axis becomestilted.The breakup starts with creation of ahole in the middle of the droplet.

g r a p h i c a l a b s t r a c t

r t i c l e i n f o

rticle history:eceived 12 December 2012eceived in revised form 24 January 2013ccepted 27 January 2013vailable online 4 February 2013

a b s t r a c t

In this paper, the 3-D deformation of an initially uncharged and spherical droplet suspended in anotherimmiscible fluid under dc uniform electric field is numerically investigated. Both the droplet and theambient fluids are considered as incompressible Newtonian fluids. In all the cases both fluids are slightlyconductive (“leaky” dielectrics) with the ambient phase more conductive than the droplet. Three regimeswere observed for droplet deformation: (1) oblate deformation (which can be predicted from the small

eywords:lectrohydrodynamicslectric fieldropletscillation

perturbation theory), (2) oscillatory oblate-prolate deformation and (3) breakup of the droplet. It wasfound that an increased electric field causes the prolate deformation to be decreased in the oscillationregime. Further increase of the electric field leads to breakup, which creates a toroidal shape. It wasshown that the threshold electric field strength depends on the viscosity ratio of both fluids. The criticalelectric capillary number beyond which the droplet eventually breaks up in a symmetric manner has

reakup been determined.

. Introduction

A fluid droplet suspended in another immiscible fluid deformsnder the effect of the sufficiently strong electric field. It mayeform to either an oblate or prolate shape depending on thehysical properties of the droplet and ambient fluids [1]. Theoupling between fluid motion and electric fields has many applica-

ions including electrosprays, electrohydrodynamic pumps, ink-jetrinting, etc. If the fluids are perfect dielectrics, the electric stressiscontinuity just has a normal component that can be balanced

∗ Corresponding author. Tel.: +1 5196612111.E-mail addresses: oghazian@uwo.ca (O. Ghazian), kadamiak@eng.uwo.ca

K. Adamiak), gcastle@uwo.ca (G.S.P. Castle).

927-7757/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.colsurfa.2013.01.048

© 2013 Elsevier B.V. All rights reserved.

by the interfacial tension at steady state and the droplet alwaysdeforms into the prolate shape. Assuming slightly conductingmaterials creates a tangential electric stress at the interface dueto the mobile charges, the effect of electric field on these surfacecharges will drag the fluid into motion. The direction of the flowdepends on the charge distribution on the surface and the dropletcan deform into the oblate shape.

Taylor [2] introduced small deformation theory based on theleaky dielectric model to describe the deformation of the dropletsin weak electric fields theoretically. To characterize the dropletdeformation, the degree of deformation is conventionally defined

by:

D = l1 − l2l1 + l2

(1)

2 : Phy

wlrarupfi

D

wtaa

R

wi

fl

R

w

t

diaiulcufsmtstpmcdd

utlhoa[aMitpiofdh

8 O. Ghazian et al. / Colloids and Surfaces A

here l1 and l2 are the axes of the droplet in the direction paral-el and perpendicular to the electric field, respectively. A positive Depresents a deformation of the droplet that has an increased lengthlong the direction of the electric field (prolate), while a negative Depresents a deformation of the droplet, which deforms perpendic-larly to the direction of the electric field (oblate). Taylor’s analysisredicted relationship between droplet deformation D and electriceld strength as

= 9aεexE2

16�S(2 + R)2×[

S(

R2 + 1)

+ 3 (RS − 1)2� + 32� + 5

− 2]

(2)

here a is the radius of the undeformed droplet and � is the surfaceension. The subscript “in” and “ex” denote the values for the dropletnd the ambient fluid, respectively. The ratios of physical propertiesre defined as:

= �in

�ex, S = εex

εin, � = �in

�ex, (3)

here � is electrical conductivity, ε is permittivity, and � is viscos-ty.

According to Taylor’s model, the conduction response of theuids can be characterized by the product of the R and S

S = tc,ex

tc,in,

here

c.in = εin

�inand tc.ex = εex

�ex

If RS > 1, the interface charge distribution is dominated by theroplet and the fluid motion is from the equator to the pole while

f RS < 1, the charge distribution is dominated by the exterior fluidnd the fluid motion is from the poles to the equator [3,4]. Var-ous breakup modes and steady state shapes were investigatedsing numerical simulations. In most cases, the numerical simu-

ations of droplet deformation have been limited to axisymmetricases [5–8]. Lac and Homsy [4] considered a naturally buoyant andncharged droplet suspended in another liquid subjected to a uni-orm electric field assuming creeping flow conditions and axialymmetry of the problem. They presented a review for variousodes of droplet deformation, assuming a wide range of conduc-

ivity and permittivity ratios. Various breakup modes and steadytate shapes were investigated. A correlation on the drop deforma-ion between the simulation and the theoretical predictions is alsoroposed by Hua et al. [7] considering three different electric fieldodels: a leaky dielectric model for droplets with finite electrical

onductivity, a perfect dielectric model for electrically insulatingroplets and a simplified constant surface charge model for chargedroplets.

In the studies mentioned above, the oblate deformation sim-lations should be valid only in small electric capillary numberso avoid the electrorotation and symmetry breaking of the prob-em. The Taylor model does not fully capture all phenomena thatave been observed in experiments. The deformation and burstingf liquid droplets suspended in liquid dielectrics and exposed ton electric field were measured experimentally by several authors9–12]. The bursting mode was found to show considerable vari-tion with the electrical properties of the systems by Allan andason [9]. They also found that when the permittivity of the droplet

s smaller than that of the continuous phase, the droplets were flat-ened into a sheet, which then turned over until it was no longerarallel to the electrodes. Vizika and Saville [11] experimentally

nvestigated the deformation of droplets in ac and dc fields. They

bserved that silicone oil droplets in castor oil change their shaperom oblate to prolate under sufficient dc electric field but theyid not describe this behaviour. Ha and Yang [13] did a compre-ensive study for droplets more conductive than ambient fluid and

sicochem. Eng. Aspects 423 (2013) 27– 34

considered experimentally the deformation and breakup of New-tonian and non-Newtonian droplets. Three different cases of highlyconducting droplets, conducting droplets and slightly conductingdroplets were examined. They found that the electrohydrostatictheory is satisfactory when the ratio of resistivities for the dropletand continuous-phase is less than 10−5.

Recent experimental studies have discovered non-axisymmetric shapes for the droplets [14–17]. It was foundthat the symmetry breaking happens only for droplets with RS < 1due to the reverse dipole created inside the droplet. The rotationalflow is created and the major axis of the droplet makes an obliqueangle with respect to the electric field. The effect of ac and dcfields on continuous electrorotation of droplets with the rotationof the symmetric axis of each droplet was investigated by Krauseand Chandratreya [14]. It was found that the rotation velocitiesin the dc field were in agreement with a theoretical treatmentfor electrorotation of solid spheres in dc fields (See the Introduc-tion of Ref. [18]). Ha and Yang [15] experimentally investigatedthe electrorotation of the less conducting droplet suspended inmore conducting liquids and its effect on deformation and burstbehaviour of the droplet. It was stated that in the case of highlyviscous droplets, the deviation from the rigid body theory is small.They also found the threshold electric field strength beyond whichthe droplet breaks up. The deformation and break up of a droplet ina steady and uniform electric field was investigated experimentallyby Sato et al. [16]. Three different modes were reported for siliconeoil suspended in more conducting castor oil. A peculiar oscillatorymotion of the droplets accompanied by cyclic oblate-to-prolateand prolate-to-oblate variations in shape was found. Salipante andVlahovska [17] have presented a systematic experimental study ofthe non-axisymmetric rotational flow in strong fields. The criticalelectric field, droplet inclination angle and the rate of rotation weremeasured for small and high viscosity droplets. It was found thatthe droplet inclination angle increases with an increase in field.

It is difficult to measure the conductivities of the oily fluidsused in these experiments with conventional methods due to theirextremely low conductivities. It can be concluded that some of thediscrepancies noted between experiments can be the results of dif-ferences in measuring the conductivities of the fluids. On the otherhand, it is clear that the rotation of the droplet with an obliqueaxis to the applied electric field will destroy the axisymmetric con-ditions. In order to have a comprehensive understanding of theproblem, the full 3D simulation is needed.

In the present study, we numerically investigate the defor-mation, oscillation and breakup of a weakly conducting dropletsuspended in an ambient medium with higher conductivity. It isthe first time that the deformation of such droplet is investigatednumerically in a 3D configuration. There has been also relativelylittle work on the breakup of a less conducting droplet via theoblate-type deformation.

2. Problem statements

Fig. 1 illustrates the 3D model considered in this study. A smallliquid droplet is suspended in another immiscible fluid and isexposed to a uniform electric field parallel to the Y-axis. This fieldis generated by applying different electric potentials to the parallelplates in the X–Z planes.

The center of the droplet is placed in the middle of the rectangu-lar cube. The liquids are assumed to be incompressible Newtonianwith the same density, �, so that the drop is under neutrally buoy-ant condition. The interface separating the two fluids is assumed to

have a constant interfacial tension coefficient. The size of the com-putational domain is 10 times the droplet radius in the directionof the electric field and 8 times the droplet radius in the directionsnormal to the electric field. At the initial stage, the shape of the

O. Ghazian et al. / Colloids and Surfaces A: Phys

dam

adbcies

wflttb

mlnTosbs(sattibevu

a

Fig. 1. Model of a suspended droplet in an electric field.

roplet is assumed to be spherical, the center of droplet is locatedt the middle of the parallel-plate capacitor and both fluids areotionless.In order to investigate the dynamics of droplet deformation in

n electric field it is necessary to solve the Navier-Stokes equations,escribing the fluid motion, as well as track the interfaces betweenoth fluids. The laminar two-phase flow system studied here isoupled with the applied electric field and electric charges on thenterface. Additional body forces are added to the Navier-Stokesquations for considering the surface tension (Fst) and electrictress (Fes).

�∂u

∂t+ � (u · ∇)u = ∇ ·

[−PI + �

(∇u + (∇u)T)]

+ F st + Fes

∇ · u = 0

(4)

here u denotes fluid velocity, I is the identity matrix, � is theuid density, � is the dynamic viscosity and P is the pressure. Inhe current simulation, no-slip boundary conditions are applied forhe electrodes and pressure outlet conditions are applied for otheroundaries.

To describe the evolution of the droplet shape, the level setethod [19], suitable for free boundary problems, is applied. In the

evel set method, the interface is considered to have a finite thick-ess of the same order as the mesh size instead of zero thickness.he physical property changes smoothly from the value on one sidef the interface to the value on the other side in the interfacial tran-itional zone. The method describes the evolution of the interfaceetween the two fluids tracing an iso-potential curve of the levelet function (). In general, in droplet ( = 1) and in ambient fluid = 0). The interface is represented by the 0.5 contour of the levelet function ( = 0.5). The movement of the interface is governed by

differential equation for this function. To keep the level set func-ion a distance function, a reinitialization process is needed. Ideally,he interface should not change its position during this reinitial-zation procedure, but in many applications the zero level set canecome distorted by parasitic numerical inaccuracies if the gradi-nts in the neighbourhood of the interface are either very large or

ery small. For this reason, an improved reinitialization method issed.

Level set methods automatically deal with topological changesnd it is in general easy to obtain high order of accuracy. The time

icochem. Eng. Aspects 423 (2013) 27– 34 29

evolution of the interface is modelled via transport of the levelset function due to the underlying physical velocity field. Thefunction is governed by the Eq. (5).

∂�

∂t+ u · ∇� = ˛∇ ·

(εls∇� − �

(1 − �

) ∇�∣∣∇�∣∣)

(5)

In Eq. (5), εls is the parameter controlling the interface thicknessand � is the reinitialization parameter. A suitable value for � is themaximum velocity magnitude occurring in the model. The densityand viscosity, which are different for oil and water, are automati-cally calculated from the level set variable , as well as the surfacetension force.

The surface tension force is given by

F st = ∇ · T

T = �(I −(

nnT)

)ı(6)

n is the interface normal and ı is the Dirac-delta function thatis nonzero only at the fluid interface. The interface normal is calcu-lated from

n = ∇�∣∣∇�∣∣ (7)

The use of a Dirac-delta function will ideally create a sharp inter-face in the mathematical formulation. However, to implement thisin the numerical simulation, the Dirac-delta function should beapproximated by

ı = 6∣∣�(1 − �)

∣∣ ∣∣∇�∣∣ (8)

The electric forces cause the deformation and they can be cal-culated from the electric field distribution, which depends on theposition and shape of the droplet. In the absence of any time-varying magnetic field, the curl of the electric field is zero (∇ × E =0) and the electric field can be expressed in terms of the electricpotential (V).

E = −∇V (9)

The charge conservation can be expressed in each medium asfollows

∇ · (�∇V) = 0 (10)

where � is the electric conductivity of the medium.In a two-fluid system, assuming that the electric relaxation time

is less than the time scale of the fluid motion, the electrical conduc-tivity is constant within each fluid and Eq. (10) for electric potential(V) can be reduced to Laplace equation in each medium

∇2V = 0 (11)

It is assumed that there is no space charge in the fluids except thesurface charge on the interface, created by the difference betweenpermittivities and conductivities of both fluids. At the interfacebetween the two fluid media, the continuities of electric potentialand electric current are preserved.

||V || = 0 and ||�∇V · n|| = 0 (12)

where ||V|| represents a jump across the interface. The aboveboundary conditions at the interface between two fluids can beembedded in the governing equation Eq. (10) for electric potential

with variable electric conductivity � in the difference fluid regionsof the system.

With the solution of Eq. (10), the electric potential can beobtained, and then the electric field strength can be calculated using

30 O. Ghazian et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 423 (2013) 27– 34

umerical results (Right). The applied electric field strength is 4.5 kV cm−1.

Ec

tAbsNc

T

fo

F

fflect

Vt

Table 1List of experimental parameters.

System R = �in�ex

S = εexεin

, � = �in�ex

�(N/m)

Fig. 2. Breakup of NN6 system [13]: Experimental data (Left) and n

q. (9). The current density ( J) and the electric displacement ( D)an also be found by

J = �E + ∂D

∂t

D = ε0εrE

(13)

�0 is the permittivity of vacuum, and εr is the relative permit-ivity (ratio of the permittivity of a substance to that of vacuum).ssuming that the fluids are incompressible, the electric stress cane calculated by taking the divergence of the Maxwell stress ten-or, which couples electrostatic and hydrodynamic phenomena.eglecting the effect of magnetic field, the Maxwell stress tensoran be defined as follows:

Mij= εrε0EiEj − 1

2(εrε0E2)�ij (14)

The momentum equation is modified by inserting the electricorce, Fes, which can be determined by calculating the divergencef the Maxwell stress tensor ( TM):

es = ∇ · TM (15)

The conductivity and relative permittivity are constant, but dif-erent, for each fluid. The volume fraction changes from zero in oneuid to one in the other one. In order to have all the physical prop-rties in the interface, the two phase relative permittivity (εr) andonductivity (�r) can be defined based on the volume fraction ofhe phases:

εr = εinVfin + εexVfex

�r = �inVfin + �exVfex

(16)

fin and Vfex are the volume fractions of the droplet and the con-inuous phase, respectively. Using Eq. (16), the physical properties

Fig. 3. Breakup of NN21 system [13]: Experimental data (Left) and numer

NN6 >105 0.05 0.043 0.0054NN21 �10 0.73 0.874 0.0033

change smoothly from the value on one side to the value on theother side. The governing equations for two-phase flow and electricfield have been solved with the software COMSOL MULTIPHYSICSÔusing the Finite Element Method (FEM) [20].

3. Numerical simulation validation

In order to validate the numerical simulations, two differentbreakup modes of the prolate type deformation are compared withthe available experimental data of Ha and Yang [13]. There is alarge set of parameters to consider and in the results that followwe assume that the drop phase and the suspending fluid havethe same parameters as measured in [13].Two different systems(denoted NN6 and NN21) were chosen from their experimental sys-tem. According to their measurements, the relevant parameters areas follows (Table 1):

For a highly conducting droplet (NN6), the droplet first deformsinto an elongated ellipsoid. After reaching the critical deformation,the drop begins to stretch rapidly and the blobs at each ends moveaway leaving two daughter droplets at the ends and a main bodyin the middle (Fig. 2).

For a slightly conducting drop (NN21) having the viscosity com-

parable to the ambient phase, the droplet elongates into the thinthread with sharply pointed ends rather than relaxes into an ellip-soid like the highly conducting drop (Fig. 3). Although the drop endsbecome pointed, they did not refer this mode to tip streaming.

ical results (Right). The applied electric field strength is 3.2 kV cm−1.

: Physicochem. Eng. Aspects 423 (2013) 27– 34 31

itasstta

sci1crd

4

caoari

C

Tpd

m

m

b

4

m

F�

O. Ghazian et al. / Colloids and Surfaces A

In the range of electric capillary numbers covered in the exper-ments, the measured deformations and break-up match very wellhe numerical results. For a highly conducting droplet (NN6),ccording to the details of the experiments, the droplet begins totretch rapidly on frame 140 (the frame rate is 15 frames/s) whicheems to agree perfectly with our numerical results (t = 8.5 s). Forhe NN21 system, experimental figures illustrate the forming ofhe pinching necks on frame 210 (t = 14 s) which is also in goodgreement with our numerical results (t = 12 s).

To check the sensitivity of the numerical simulations on theize of the computational domain, the electrodes distance has beenhecked for three different values of 8 a, 10 a and 12 a (where as the radius of the undeformed droplet). The drop shapes for the0 a and 12 a are almost identical, but differences are evident whenompared to the 8 a solution. Since time-dependent highly accu-ate solutions are computationally expensive we employ the 10 aistance for all the computations reported in what follows.

. Results and discussion

The simulations were done for different conductivity and vis-osity ratios. The degree of droplet deformation can be plotted as

function of the electric capillary (CaE) number for given ratiosf the viscosities, permittivities, and conductivities of the dropletnd continuous phases. The electric capillary number describes theelative strength of the electric force with respect to the capillarynterfacial force.

aE = aεexE2

�(17)

he problem depends on a single dynamic parameter, CaE , and threeroperty ratios, R, S and � [4]. Three different modes of the dropletistortion can be distinguished:

(A) At very low electric fields the droplet deforms into a sym-etric oblate shape.(B) Increasing electric field causes the oscillatory oblate-prolate

otion.(C) The droplet turns into the torus shape, which leads to a

reakup, when the electric field intensity is further increased.

.1. Small deformation

In order to verify the accuracy of the numerical model, the defor-ations of a droplet with a = 1.3 mm at low electric field calculated

ig. 4. Droplet deformation versus electric capillary number for R = 0.15, S = 1.38, = 1.31.

Fig. 5. Sequence of images showing the oscillation of the 1.3 mm droplet forE = 4 kV/cm. (a) Observation normal to the applied electric field. (b) Observationparallel to the applied electric field.

3 : Physicochem. Eng. Aspects 423 (2013) 27– 34

nsatdlsbstito

4

ocroiEo

�

bmvmsftiw

oetsti

2 O. Ghazian et al. / Colloids and Surfaces A

umerically have been compared with the analytical results of themall deformation theory [2]. In this case the droplet deforms inton oblate shape with the axis of symmetry parallel to the direc-ion of the electric field. Fig. 4 compares the numerical results forroplet deformation with those obtained from the asymptotic Tay-

or theory for R = 0.15, S = 1.38 and � = 1.31. The numerical resultshow good agreement with Taylor’s theory up to D ≈ −0.05. This isecause Taylor’s theoretical analysis is based on the assumption ofmall droplet deformation. When the droplet deformation is small,he deformation increases almost linearly with the electrical cap-llary number, which agrees well with the prediction by Taylor’sheory. For stronger electric fields (CaE > 0.27), the droplet starts toscillate and the numerical results deviate from the Taylor’s theory.

.2. Oscillatory oblate-prolate motion

This phenomenon is very interesting from a mechanical pointf view, because using a dc electric field with a constant strengthauses elongation and contraction of droplets and the droplet expe-iences highly nonlinear oscillatory motion when the magnitudef the electric field is increased. Fig. 5 shows the image sequencellustrating the oscillatory motion of a droplet with a = 1.3 mm in

= 4 kV/cm (CaE = 1.9) from two directions. The physical propertiesf the droplet and the surrounding fluid are:

= 1.31 R = 0.15 S = 1.38

Sufficiently strong electric field causes the droplet to oscillateetween the oblate and prolate shape. The major axis of the dropletakes an oblique angle comparing to the first regime and this angle

aries during the oscillation between the oblate and prolate defor-ation. The droplet shape changes from an oblate shape with tilted

ymmetric axis in to the prolate shape with horizontal axis and thenrom prolate shape to oblate shape. While the field is constant inhis regime, the droplets maintained a cyclic motion without break-ng up. This is a steady-state oscillation with the constant amplitude

hile the electric field strength is kept constant.It is obvious that the major axis of the droplet tilts, if it is

bserved from the direction parallel to the electric field. The dropletlongates in the direction which is normal to the electric field until

= 0.3 s, then it starts coming back to its original position. At t = 1.2 the major axis of the droplet makes a tilt angle, then at t = 1.5 she major axis of the droplet loses the tilt angle and a new cycles again repeated. Increasing the electric field in this mode causes

Fig. 7. Sequence of images showing the breaku

Fig. 6. Frequency of droplet oscillation (f) versus electric capillary number (CaE) forR = 0.15, S = 1.38, � = 1.31.

decrease of the amplitude of oscillation in the direction parallel tothe electric field.

The nature of oblate–prolate droplet oscillation is different fromthe ordinary resonant oscillation of the droplets. In Fig. 6, the fre-quency of the oscillation is plotted versus CaE for R = 0.15, S = 1.38and � = 1.31. By increasing the electric field, the frequency of theoscillation will increase while the amplitude of the oscillation alongthe electric field (prolate deformation) will decrease.

The frequencies were calculated based on the time intervalbetween two maximum prolate deformations. For a sufficientlyhigh electric field (CaE ≥ 2.5), no oscillations are observed and thebreak up will happen.

4.3. Break up

Further increasing of the electric field causes the droplet breakup. This break up occurs in a symmetric manner. The droplet will

first deform into a torus shape and this happens very fast compar-ing with the oscillation mode. This is the first time that this kindof break up is captured numerically. Fig. 7 illustrates the electro-hydrodynamic burst of the droplet with the radius of a = 1.3 mm in

p of the 1.3 mm droplet for E = 7 kV/cm.

: Physicochem. Eng. Aspects 423 (2013) 27– 34 33

Esotl

esfftafiufq

4

tctfettecsvtm

O. Ghazian et al. / Colloids and Surfaces A

= 7 kV/cm and with the same physical properties as in the previousections. The breakup starts with creation of the hole in the middlef the droplet. This hole grows and changes the droplet shape to theorus. The hole inside the droplet continues to grow and eventuallyeads to the droplet breakup.

Charges carried by conduction accumulate at the interface. Thelectric field acting on these surface charges creates a tangentialtress which drags the fluid into motion and forces the fluid awayrom the center. If RS < 1, the conduction in the ambient fluid isaster than the droplet and charges at the poles are repelled fromhe electrodes, pushing the droplet into an oblate shape caused by

hydrodynamic shear force acting on the interface. If the electriceld strength exceeds a critical value, the droplet becomes flattennder the action of high electric field and the torus like shape isormed. The torus expands in time to a fluid ring, which subse-uently disintegrates due to the capillary instabilities.

.4. Effect of viscosity and conductivity ratios on droplet breakup

It was observed by Ha and Yang [21], [22] that for the prolate-ype deformation of a more conducting droplet, the critical electricapillary number weakly depends on the viscosity ratio. This is dueo the fact that for a more conducting droplet the flow that resultsrom the electrical stress is too weak to induce hydrodynamicffect on the droplet deformation and breakup. On the other handhe viscosity ratio can have significant effect on the breakup inhis case. Fig. 8 illustrates the effect of viscosity on the criticallectric capillary number of this kind of breakup for two differentonductivity ratios (R). It is obvious that for highly viscous droplets,

tronger electric field is required for the droplet breakup. For a lessiscous droplet, however, the electric energy will be transformedo viscous dissipation associated with the strong internal fluid

otions. As a result, the higher electric energy is required to break

Fig. 9. Droplet shape for different con

Fig. 10. Flow patterns of the dropl

Fig. 8. Critical electric capillary number as a function of the viscosity ratio forS = 1.38.

up a less viscous droplet. It can be understood from Fig. 6 thateven though there is no variation in the electric field strength, thebehaviour of the droplet may be controlled by changing the ratio ofviscosity of the fluids in the system. It is clear that the conductivityratio (R) has a significant effect on the critical electric capillarynumber of the breakup and may change the behaviour of thedroplet. By decreasing the conductivity ratio (R = 0.05) and keepingthe same viscosity ratio (� = 1.31) and permittivity ratio (S = 1.38),the breakup starts at lower electric capillary numbers (CaE = 1.9).For R = 0.15, the numerical results have also been compared withthe available experimental data of Ha and Yang [15]. As mentionedbefore, the difference between numerical and experimental resultscan be attributed to the difference in measuring the conductivities

of the oily fluids. The electrical conductivity of insulating oils isa material parameter which strongly depends on temperatureand electric field strength and can also vary with ageing. These

ductivity ratios (R) at t = 0.05 s.

et for E = 7 kV/cm. (XY view).

3 : Phy

di

utDflsa

tdts

spfiattwa

tabitl

5

pemd

–

–

–

[

[

[

[

[

[

[

[

[[

[[21] J.W. Ha, S.M. Yang, Effects of surfactant on the deformation and stability of a

4 O. Ghazian et al. / Colloids and Surfaces A

ifferences can reach easily one or even several orders of magnituden different experimental conditions.

Increasing the conductivity of the continuous phase will speedp the breakup and the growing of the hole inside the droplet, sohat the breakup will happen at a lower electric capillary number.ecreasing the conductivity ratio (R) causes the induced electricalow to become more powerful, resulting in faster breakup. Fig. 9hows the growing rate of the hole development inside the droplett t = 0.05 s for different values of R.

It is obvious that for R = 0.2 and R = 0.1, there is no hole at = 0.05 s, but for R = 0.05 the hole has already been formed inside theroplet. Generally, a larger difference in the electrical conductivi-ies will result in larger electric charge accumulation on the dropleturface, which induces larger deformation.

Fig. 10 illustrates schematically the flow patterns inside and out-ide the droplet from XY view. When a droplet reaches a steadyrolate or oblate shape under the influence of an external electriceld, circulating flow patterns of fluid can be formed both insidend outside of the droplet. The Taylor circulations are visible at

= 0 and the fluid motion is from pole to equator. As time proceeds,he developed circulating flow patterns diminish, the flow patternill change and the droplet will not keep a steady circulating flow

s it was predicted by Taylor’s theory.Because the fluid interface cannot support tangential stresses,

he high tangential electric stress at the droplet interface acceler-tes the fluid in the breakup mode very fast and the droplet wille squeezed into a plane under the effect of the high electric field

ntensities. This changing of the flow pattern will cause the dropleto be stretched in a plane parallel to the electrodes which finallyead to the axisymmetric breakup of the droplet.

. Conclusions

We have studied numerically the deformation of a droplet sus-ended in a uniform dc electric field. We observed that dropletsxperienced an oblate-prolate oscillatory motion with the tiltedajor axis. We have determined three types of behaviour for the

roplets, which are less conducting than ambient fluid:

Small deformation perpendicular to the electric field in accor-dance with Taylor theory, which happens at small electriccapillary numbers.

Oscillatory motion between the oblate and prolate deformationobserved for moderate capillary numbers. The frequency of theoscillation for constant physical properties was also investigated.

Further increasing of the electric field causes the droplet to breakup passing through a torus shape. The effect of viscosity ratio onthe droplet break up shows that there exists a minimum for thecritical electric capillary number of breakup. It was also observed

[

sicochem. Eng. Aspects 423 (2013) 27– 34

that the onset of breakup will move to lower electric capillarynumbers by decreasing the conductivity ratio (R). We also charac-terized a specific kind of deformation (torus shape) which leads tothe breakup and for the first time the effect of conductivity ratio(R) on the breakup of less conducting droplet was investigated.

It was also revealed that the differences in experimental con-ditions such as measuring the conductivity ratio may change thecritical electric capillary number and the droplet behaviour.

References

[1] J.R. Melcher, G.I. Taylor, Electrohydrodynamics–a review of role of interfacialshear stress, Annu. Rev. Fluid Mech. 1 (1969) 111.

[2] G.I. Taylor, Studies in electrohydrodynamics. I. Circulation produced in a dropby an electric field, Proc. R. Soc. London, Ser. A 291 (1966) 159.

[3] J.Q. Feng, T.C. Scott, A computational analysis of electrohydrodynamics of aleaky dielectric drop in an electric field, J. Fluid Mech. 311 (1996) 289.

[4] E. Lac, G.M. Homsy, Axisymmetric deformation and stability of a viscous dropin a steady electric field, J. Fluid Mech. 590 (2007) 239.

[5] J.Q. Feng, Electrohydrodynamic behaviour of a drop subjected to a steady uni-form electric field at finite electric Reynolds number, Proc. R. Soc. London, Ser.A 455 (1999) 2245.

[6] N. Dubash, A.J. Mestel, Behaviour of a conducting drop in a highly viscous fluidsubject to an electric field, J. Fluid Mech. 581 (2007) 469–493.

[7] J. Hua, L.K. Lim, C. Wang, Numerical simulation of deformation/motion of a dropsuspended in viscous liquids under influence of steady electric fields, Phys.Fluids 20 (2008) 113302.

[8] A.M. Benselama, J.L. Achard, P. Pham, Numerical simulation of an unchargeddroplet in a uniform electric field, J. Electrostat. 64 (2006) 562–568.

[9] R.S. Allan, S.G. Mason, Particle behavior in shear and electric fields. I. Deforma-tion and burst of fluid drops, Proc. R. Soc. London, Ser. A 267 (1962) 45.

10] S. Torza, R.G. Cox, S.G. Mason, Electrohydrodynamic deformation and burst ofliquid drops, Philos. Trans. R. Soc. London, Ser. A 269 (1971) 295.

11] O. Vizika, D.A. Saville, The electrohydrodynamic deformation of drops sus-pended in liquids in steady and oscillatory electric fields, J. Fluid Mech. 239(1992) 1.

12] N. Bentenitis, S. Krause, Droplet deformation in dc electric fields: the extendedleaky dielectric model, Langmuir 21 (2005) 6194–6209.

13] J.W. Ha, S.M. Yang, Deformation and breakup of Newtonian and non-Newtonianconducting drops in an electric field, J. Fluid Mech. 405 (2000) 131–156.

14] S. Krause, P. Chandratreya, Electrorotation of deformable fluid droplets, J. Col-loid Interface Sci. 206 (1998) 10.

15] J.W. Ha, S.M. Yang, Electrohydrodynamics and electrorotation of a drop withfluid less conductive than that of the ambient fluid, Phys. Fluids 12 (2000)764.

16] H. Sato, N. Kaji, T. Mochizuki, Y.H. Mori, Behavior of oblately deformed dropletsin an immiscible dielectric liquid under a steady and uniform electric field,Phys. Fluids 18 (2006) 127101.

17] P.F. Salipante, P.M. Vlahovska, Electrohydrodynamics of drops in strong uni-form dc electric fields, Phys. Fluids 22 (2010) 112110.

18] T. B. Jones, Quincke rotation of spheres, IEEE Trans. Ind. Appl. IA-20,845 (1984).19] E. Olsson, G. Kreiss, A conservative level set method for two phase flow, J.

Comput. Phys. 210 (2005) 225–246.20] Comsol Multiphysics User’s Guide, (COMSOL 4.2).

drop in a viscous fluid in an electric field, J. Colloid Interface Sci. 175 (1995)369.

22] J.W. Ha, S.M. Yang, Breakup of a multiple emulsion drop in a uniform electricfield, J. Colloid Interface Sci. 213 (1999) 92.