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Journal of Electrostatics 71 (2013) 471e477

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Journal of Electrostatics

journal homepage: www.elsevier .com/locate/elstat

Investigation on dynamics of double emulsion droplet in a uniformelectric field

Purushottam Soni*, Vinay A. Juvekar, Vijay M. NaikDepartment of Chemical Engineering, IIT Bombay, Mumbai 400076, India

a r t i c l e i n f o

Article history:Received 10 September 2012Accepted 4 December 2012Available online 25 December 2012

Keywords:Finite elementPhase fieldElectro-hydrodynamicsLeaky dielectricDouble emulsion

* Corresponding author. Indian Institute of TechnoChemical Engineering, Organic Process Lab, Room No400076, India. Tel.: þ91 (0)22 25764225.

E-mail address: puru85@iitb.ac.in (P. Soni).

0304-3886/$ e see front matter � 2012 Elsevier B.V.http://dx.doi.org/10.1016/j.elstat.2012.12.006

a b s t r a c t

Phase field method based on CahneHilliard free energy formulation is adopted for predicting thebehavior of double emulsion droplet suspended in a continuous phase under the influence of a uniformelectric field. The role played by the inner droplet on the electric-field-driven fluid flow, and also ondeformation of the outer droplet is predicted by present numerical simulation. Three different kind ofdeformation type of outer and inner droplet (prolateeoblate, prolateeprolate and oblateeprolate) hasbeen observed. With increase in the volume fraction of inner drop, transition in the deformation of outerdrop from prolate to oblate occurs at lower value of fluid permittivity ratio.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

The behavior of droplet in presence of electric field is a veryimportant problem of electro-hydrodynamics and has been studiedwidely over the past five decades because of its industrial relevance[1e3] in the fields such as metal extraction, liquideliquid extrac-tion, drug delivery, direct contact heat exchangers, emulsification,electrocoalescence, photography, leather processing etc. Thedynamics and stability of single drop suspended in an immiscibleliquid under the influence of applied electric field is well studied[4e14]. But, the effect of an electric field on the dynamics of doubleemulsion droplets has not been explored in detail.

Usually, double emulsions (also called multiple emulsions) areformed by dispersing a primary emulsion in another immiscibleliquid. The examples of such emulsions are: watereoilewater (W/O/W), oilewatereoil (O/W/O) or oileoileoil (O0/O/O0) emulsions.The behavior of multiple emulsions with large number of internaldroplets in electric field has been studied experimentally by Ha andYang [15]. The effect of electric field and also of shear on interactionof two immiscible single drops of different fluid suspended ina third immiscible fluid is investigated and by thermodynamiccalculations, a condition for the engulfment of one drop into

logy Bombay, Department of. 227, Mumbai, Maharashtra

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another fluid drop was predicted by Torza and Mason [16]. Thedeformation and stability of emulsion drops in electric field areconsiderably influenced by the presence of inner drops. Gouz andSadhal [17] did the stability analysis of a compound multiphasedrop formed by a fluid sphere completely covered by anotherimmiscible fluid drop in an infinite fluid under the influence ofelectric field. A semianalytical solution was obtained for both theelectric and the flow fields by using the bipolar coordinate system.A finite element based computational technique is developed forthe dynamics of composite drop in uniform electric field by Tsu-kada et al. and a good qualitatively agreement between experimentand simulationwas achieved [18]. However, their study is limited toa narrow range of physicochemical properties of fluids. An inves-tigation on fluid dynamics of double emulsion droplet in uniformelectric field is carried out by Ha and Yang [19]. An analytic solutionis given by using domain perturbation method in small deforma-tion limit. Fluid flow pattern induced by the applied electric fieldare characterized in all the phases. Direction of flow and itsintensity depends on the charge induced on the interface. Wheninner and outer drops have the same charge secondary circulatingflow will be generated in the annular region.

In present work, we focus on the concentric double emulsion,consisting of leaky dielectric fluid phases, in a steady as well asalternating electric field. Due to the electrohydrodynamic stressesfluid flow (uniaxial or biaxial, depending on the electric properties ofthe fluids) is generated in all three phases. At lower electrical capil-lary number, inner and outer drops deforms into a prolate or oblatespheroid of definite shape. The presence of inner drop alters thefluid

P. Soni et al. / Journal of Electrostatics 71 (2013) 471e477472

motion and deformation of outer drop. Therefore, the effect ofvolume fraction of inner drop, and, its electric properties is investi-gated in great detail here,whichwere not exploredbefore. This studyis also very important in understanding the stability of liquideliquidinterface for designing the liquid emulsion membranes.

In this paperwe have adopted a phase fieldmethod based on theCahneHilliard free energy formulation coupledwith NaviereStokesequation to simulate the deformation of a concentric doubleemulsion in presence of uniform electric field (DC/AC). The systemconsists of a composite drop suspended in a continuous phase. Thiscomposite drop consists of a smaller inner drop suspended withinthe larger outer drop. For the simplicity it is assumed that the innerdrop and the continuous fluid are made of same fluid, while theouter drop consists of another immiscible liquid. The entire systemis contained in a cylinder with insulated curved surface. Highintensity steady or alternating axial field is imposed on the system.In this paper,wehave studieddeformationof the inner and theouterdrops and compared the deformation of the outer drop in thepresence/absence of the inner drop. This study is the first steptowards understanding the dynamics of a concentric double emul-sion numerically in steady as well alternating electric field.

2. Problem description and formulation

As shown in Fig. 1a, a neutrally buoyant composite sphericaldroplet of radius awith innerdroplet of radiusba is suspended in thecontinuous immiscible fluid, where b is the volume fraction of theinner drop. In present study, all fluids are considered to be incom-pressible and Newtonian. The whole system is subject to an exter-nally applied uniform electric field E

!. In the presence of imposed

electric field and depending on the fluid properties, outer and innerdrops may deform into any of the following four combinations:(outereinner) prolateeprolate, prolateeoblate, oblateeprolate andoblateeoblate spheroids. We denote the fluid density by r, viscosityof the fluid by h, the fluid dielectric constant by ε and the electricalconductivity of the fluid by k. To characterize the system, we definethe following fluid properties ratios; mass density ratio M12 ¼ 1/M23 ¼ r2/r1, viscosity ratio l12 ¼ 1/l23 ¼ h2/h1, ratio of dielectricconstants Q12 ¼ 1/Q23 ¼ ε2/ε1, ratio of electrical conductivitiesR12 ¼ 1/R23 ¼ k1/k2. The interfacial tension is denoted by g12 ¼ g23.

Fig. 1. (a) Schematic representation of the composite droplet suspended in the continuougeometry used in numerical simulation with specified boundary conditions.

The problem is formulated in a 2-dimensional axisymmetriccylindrical coordinate system (Fig. 1b). To avoid the effect of walls onthe deformation of drop, the radius of the cylinder is taken to be 10times larger than the radius of the composite drop. A steady oralternating current electric field is applied between the top andbottom electrodes. Under the influence of the electric field, both thedrops deform. In thepresent study, due to the limitations of themodelit is assumes that the inner drop and the continuous medium aremade of same fluid. The densities of all the fluids are assumed to beequal sothat theeffectof gravitycanbeneglected.Alsoviscosity rationis taken to be unity l12 ¼ l23 ¼ 1, and interfacial tension g12 ¼ g23.

2.1. Governing equation for two phase flow

It is assumed that the fluids are immiscible, incompressibleNewtonian fluids and fluid flow is considered to be in laminarregime. Therefore the momentum balance and continuity equationfor the fluid flow in all phases can be written as:

r

�v u!vt

þ ð u!$VÞ u!�

¼V$h� pI þ h

�V u!þ ðV u!ÞT

�iþ F!

g þ F!

st þ F!

El

(1)

D$ u! ¼ 0 (2)

Where r is the fluid density, p is the pressure of the fluid, h is thefluid viscosity, and u! is the velocity vector. F

!g is the force due to

gravity F!

g ¼ r g!, where g! is the gravity vector. F!

st is the bodyforce due to surface tension, and F

!El is the volumetric electrical

force due to electrical stress (Maxwell stress). The surface tensionforce can be defined as F

!st ¼ GVf, where G is the chemical

potential (J/m3), expressed as:

G ¼ l

"� V2fþ

f�f2 � 1

�a2

#(3)

Where l is the mixing energy density (N) which is given byl ¼ 3ag=2

ffiffiffi2

p. Here, g is the interfacial tension (N/m). f is the

s medium between two parallel plate electrodes and (b) 2-dimensional axisymmetric

Table 1Boundary conditions used with reference to Fig. 1b.

Boundary Abbreviation Boundary condition Mathematical formulation

Fluid flowSymmetry

boundaryAD Axial symmetric r ¼ 0

Walls AB, BC, CD No slip u! ¼ 0Interface EFG, KLM Initial fluid interface ½�pI þ hðV u!þ ðV u!ÞT Þ�n ¼ 0

Electric fieldSymmetry

boundaryAD Axial symmetric r ¼ 0

Electrode AB Positive potential V ¼ Vþ ¼ V0=2cos utElectrode BC Negative potential V ¼ V� ¼ �V0=2cos utWall CD Electrical insulation �εrε0vV/vn ¼ 0

P. Soni et al. / Journal of Electrostatics 71 (2013) 471e477 473

dimensionless phase field function. f has value �1 and 1 for thetwo types of fluid, respectively, and 0 in a very thin region on thedrop/medium interface.

The evolution of interface is governed by phase field method,which is based on a CahneHilliard free energy equation written as:

vf

vtþ u!$Vf ¼ Vs$VG (4)

where the velocity of fluid is u!, s is the mobility coefficient (m3 s/kg); which is given by s ¼ ca2, where c is the mobility tuningparameter (m s/kg) and a is the interfacial thickness (m). The termu!$Vf accounts for the convective flux produced by viscous flow,and the term Vs$VG denotes the diffusion flux. In the Phase Fieldapplication method CahneHilliard equation is decomposed intotwo second-order PDEs.

vf

vtþ u!$Vf ¼ V$

sl

a2Vj (5)

j ¼ �V$a2Vfþ�f2 � 1

�f (6)

The volume fraction of the two fluids are defined asVf1 ¼ (1 � f)/2 and Vf1 ¼ (1 þ f)/2. The mass density and viscosityin the drop and the surrounding medium are defined as a contin-uous scalar, i.e. r ¼ r1Vf1 þ r2Vf2, h ¼ h1Vf1 þ h2Vf2.

2.2. Governing equation for electric field

The charge conservation equation can bewritten in all the phaseas follows:

vQf

vtþ V$J ¼ 0 (7)

Where Qf is the volume charge density in the bulk fluids and J iscurrent density. The current density J is defined as J ¼ k E

!(Ohmic

conduction law). Here, E!

is the electric field and k is the electricalconductivity. From the gauss law for the free charge densityV$ðε0εr E

!Þ ¼ Qf , we obtain the following equation:

V$

ε0εr

v E!vt

þ k E!!

¼ 0 (8)

In the absence of any time-varying magnetic field, fromMaxwell’s equations, the curl of the electric field has to be zero i.e.V� E

! ¼ 0. Therefore, the electric field can be expressed asa gradient of the electric potential V; E

! ¼ �VV . Hence, Eq. (8) canbe rewritten as

V$

�ε0εr

vVVvt

þ kVV�

¼ 0 (9)

The first term in the parentheses corresponds to the displace-ment current and second term corresponds to the Ohmic current.Where, ε0 is the permittivity of the free space and εr is the dielectricconstant of the fluid. Permittivity and the conductivity in the dropand the surrounding medium are defined as ε ¼ ε1Vf1 þ ε2Vf2 andk ¼ k1Vf1 þ k2Vf2.

Electrical force can be calculated by taking the divergence of theMaxwell’s stress tensor sE:

F!

El ¼ V$sE (10)

sE ¼ ε0εr

�E!

E!� 1

2E2I�

(11)

I is the identity matrix.The boundary conditions used in the numerical simulation for

the fluid flow and electrodynamics problems, are summarized inTable 1.

In this paper, the system is numerically investigated usingCOMSOL Multiphysics 3.5a, which is based on the finite elementmethod. The full NaviereStokes equations are solved in all the fluidphases. To reduce the computational efforts, the electric field isassumed to be axially symmetric so that only half of the domain isconsidered in the analysis. Triangular mesh is used in all thedomain. To get better mass conservation the density of the elementdefined at the interface is taken to be high. Up to 1% tolerance in themass conservation is permissible.

3. Results and discussions

In Fig. 2, aspect ratio l/b (l is semi major axis and b is semi minoraxis) of (a) outer drop (b) inner drop is plotted against dimension-less electric field E0¼ E/(g12/4pε1de); where de is the diameter of theouter drop. Where black circle represents the experimental resultsfor b ¼ 0.5 to 0.6 from Tsukada et al. 1997 for vegetable oil/siliconeoil/vegetable oil system under the influence of uniform steadyelectric field. Red (b ¼ 0.5) and blue (b ¼ 0.6) line shows oursimulation results for the two limiting conditions. Simulationresults are in good qualitatively agreement with the experimentalone. The deformation of the interface increases with E0. Withincrease in the size of inner dropdeformation of outerdrop becomesmore pronounced. It is because; as the volume fraction of inner dropb increases themagnitude of the electric field-driven-fluid flow andMaxwell stress at the interface increases significantly.

In Fig. 3, the degree of deformation of the outer drop, inner dropand of single drop is plotted with dimensionless electrical capillarynumber CaE ¼ ε0ε1E

2a/g12 for the two different kind of deformationtype, i.e. prolateeoblate (outereinner) and prolateeprolate (outereinner). As we can see from the graph in the small deformation limitdegree of deformation of both the outer and inner dropletsincreases linearly with the electrical capillary number CaE. Degreeof deformation D, is defined as D ¼ (l � b)/(l þ b). For prolatedeformation D > 0 and for oblate deformation D < 0. In both thecases, inner drop is less conducting in comparison to the annularphase. The degree of deformation of the outer drop is less than thatof the single drop. Since R12 is less than unity, therefore inducedsurface charge at the interface would be small. So that, the viscousflow generated due to the mismatch in tangential componentwould be weak. Also the Maxwell stress acting on the interfacebecomes reduced. Therefore, we can say if outer drop deforms inprolate spheroid the presence of inner drop reduces the

Fig. 3. Degree of deformation as a function of the electrical capillary number at M12 ¼ M23 ¼ 1, l12 ¼ l23 ¼ 1, R12 ¼ 0.1 and b ¼ 0.6. (a) Prolateeoblate Q12 ¼ 1 (b) prolateeprolateQ12 ¼ 10 Points: black for outer droplet, violet for inner droplet, red for single droplet. Solid lines are the best fit to numerical data. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

Fig. 2. Effect of electric field strength on drop aspect ratio. (a) Outer droplet (b) inner droplet. Points: black for experimental data b ¼ 0.5 to 0.6 (Tsukada et al., 1997), red presentnumerical simulation for b ¼ 0.5 and blue present numerical simulation for b ¼ 0.6. Lines are best data fit for numerical simulation. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

Fig. 4. Degree of deformation as a function of the electrical capillary number foroblateeprolate deformation at M12 ¼ M23 ¼ 1, l12 ¼ l23 ¼ 1, Q12 ¼ Q23 ¼ 1, R12 ¼ 10 andb ¼ 0.6. Points: black for outer droplet, violet for inner droplet, red for single droplet.Solid lines are the best fit to numerical data. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

P. Soni et al. / Journal of Electrostatics 71 (2013) 471e477474

deformation of outer drop. The deformation of inner drop is usuallyvery small in comparison to the outer drop. It is due to the Maxwellstress acting on the interface 23 (inner/outer droplet) are muchsmaller than which is acting on the interface 23 (continuousmedium/outer droplet) and also restoring force would be moresignificant for the inner droplets due to their smaller curvature.

Fig. 4 shows the casewhere outer droplet deforms into an oblatespheroid. Usually in the presence of inner drop, deformation of theouter droplet is more in comparison to the absence of inner drop.When the continuous phase is more conducting in comparison toannular region the deformation of the inner drop exceeds to that ofouter drop. In this case Maxwell stress acting on the interface 23 ismuch higher in comparison to that of acting on the interface 12. Theincrease in the capillary number does not change the direction offluid flow, but the intensity of the velocity field will increase,therefore drop will deform more.

In Fig. 5 the effect of volume fraction of inner drop (b) on defor-mation of outer droplet is explored for three different cases, i.e.prolateeprolate, prolateeoblate and oblateeprolate at lower elec-trical capillary number CaE¼ 0.1. In presence of an electric field therewould be a charge buildup at the interfaces, due to which electro-hydrodynamic force acts on the interface. This force creates fluidmotion in all the phases. For the prolateeoblate case, as shown inFig. 6 direction of the electric field-driven-fluid flow is from equator

Fig. 6. Steady state fluid flow pattern induced by externally applied electric field and electric field profile for prolateeoblate deformation at CaE ¼ 0.1. (a) volume fraction of innerdrop b ¼ 0.2 (b) volume fraction of inner drop b ¼ 0.8.

Fig. 5. Effect of volume fraction of inner droplet on the deformation of outer droplet at M12 ¼ M23 ¼ 1, l12 ¼ l23 ¼ 1, CaE ¼ 0.1. (a) Prolateeoblate Q12 ¼ 0.1, R12 ¼ 0.1 (b) prolateeprolate Q12 ¼ 10 R12 ¼ 0.1 (c) oblateeprolate Q12 ¼ 1 R12 ¼ 6 Points: black for outer drop, violet for inner drop and line are best fit to numerical data. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. Steady state fluid flow pattern induced by externally applied electric field and electric field profile for prolateeprolate deformation at CaE ¼ 0.1. (a) volume fraction of innerdrop b ¼ 0.2 (b) volume fraction of inner drop b ¼ 0.8.

P. Soni et al. / Journal of Electrostatics 71 (2013) 471e477 475

Fig. 8. Steady state fluid flow pattern induced by externally applied electric field and electric field profile for oblateeprolate deformation at CaE ¼ 0.1. (a) volume fraction of innerdrop b ¼ 0.2 (b) volume fraction of inner drop b ¼ 0.7.

Fig. 9. Effect of the fluid permittivity ratio Q12 on the degree of deformation of outerdrop atM12 ¼M23 ¼ 1, l12 ¼ l23 ¼ 1, R12 ¼ 0.1 and CaE ¼ 0.1. Points: red e Ajayi’s theoryfor single drop, black e b ¼ 0.6, violet e b ¼ 0.7, green e b ¼ 0.8. Solid lines are best fitto numerical data. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)

P. Soni et al. / Journal of Electrostatics 71 (2013) 471e477476

to pole in the continuous medium, and in opposite manner in theannular region. Toroidal circulation patterns are developed in all thephases, same as predicted for the single drop [11]. It is seen fromFig. 5a increase in the size of inner droplet doesn’t affect the defor-mation of the outer droplet for prolateeoblate case. But, the defor-mation of inner droplet increaseswithb, which is straight forward. Inthis case, annular region is more conducting in comparison to theinner drop. Therefore, the electric field in the annular region wouldbe less as compared to inner drop. Lines of electric field can’t pene-trate into the inner drop and they tend to accumulate in annularregion at equator as b increases. Change in themagnitude of velocityfield and electric field is very small with increase in b. Therefore,change in the deformation of the outer drop is negligible.

For the prolateeprolate case, the deformation of the outer dropdecreases, and of inner drop increases with increase in the volumefraction of inner drop in Fig. 5b. Also, this case is same as theprolateeoblate case, electric field accumulate in the annular regionas shown in Fig. 7. In this case intensity of the velocity field is veryweak. Both the interface are charged in opposite sense. Therefore,as b increases the gap between two interface decreases; as a resultof that both the interface approach to each other. Hence, defor-mation of inner drop increases, and deformation of outer dropletdecreases.

On the other hand, for the oblateeprolate case, the direction offluid flow developed by electrohydrodynamic stresses is from poleto equator in continuous phase, and equator to the pole in annularphase as shown in Fig. 8. The circulating flow in the annular regioninduces a flow in the inner drop, with an opposite direction ofcirculation. The presence of inner droplet, increases the deforma-tion of outer droplet for the oblateeprolate case. In this case,continuous medium is more conducting in comparison to theannular region. Therefore, equal but opposite charge will bedeveloped at the interfaces. As b increases, both interfacesapproaches to each other. In this process fluid will move from poleto equator in continuous medium; which will also help inincreasing deformation of outer drop. Hence, deformation of both(inner as well as outer droplet) increases with increase in thevolume fraction of inner droplet.

In Fig. 9 effect of fluid permittivity ratio at constant fluidconductivity ratio is explored in the small deformation limit. As it isseen from the figure, as the ratio Q12 increases the degree defor-mation of single drop decreases. And transition from prolate tooblate occurs at particular value of Q12 as explained for single drop

[5]. As volume fraction of inner drop increases, transition occurs atthe lower value of Q12 as shown in Fig. 9; this is due to the followingreason: as volume fraction of inner droplet increases the lines ofelectric field accumulate at the equator in annular region. This willtry to push the outer drop towards equator direction. Also, this willcreate the intense fluid flow in annular region. This helps to reducethe deformation of outer drop. Therefore, an oblate deformation ofouter drop is obtained at lower value of Q12.

Fig. 10 shows the degree of deformation of drops as a Function oftime, in presence and absence of the inner drop. It is seen from thesefigures that the drop oscillates in AC electric field. The frequency ofoscillation is twice of the applied electric field as explained by Torza[4] for single drop. For the oblateeprolate case distortion of thecomposite drop becomes more, on the other hand for prolateeoblate case deformation of outer drop becomes less, same asexplained for the DC field for the two cases. But, the deformation in

Fig. 10. Effect of frequency on the degree of deformation at M12 ¼ M23 ¼ 1, l12 ¼ l23 ¼ 1, b ¼ 0.6 and CaE ¼ 0.2. (a) Oblateeprolate deformation Q12 ¼ 1; R12 ¼ 6. (b) Prolateeoblatedeformation R12 ¼ 0.1, Q12 ¼ 0.1. Line color: black e outer drop, violet e inner drop and red e single drop. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

P. Soni et al. / Journal of Electrostatics 71 (2013) 471e477 477

ACfield is less as compared to that of steadyelectricfield at the samecapillary number. However, the deformed droplets are not restoredto their original shapes when the electric field becomes zero. Someresidual deformation is retained. As a result, their deformationincreases with every cycle, till a steady oscillatory deformation isattained. The oscillatory steady state for single drop is regainedfaster as compared to that achieved for outer drop in presence ofinner droplet. It is due to the retardation effect produced by innerdroplet. Due to the difference in the electrical properties of fluidsa phase lag is obtained in inner and outer droplet. Hence, as a resultof phase lag a retardation effect is produced.

4. Conclusions

In the present study, we have investigated the dynamics ofconcentric double emulsion in the presence of steady as well asalternating electric field. Fluid flow and deformation of outer andinner drops arewell predicted by present numerical simulation. Theeffect of electric field and volume fraction of inner drop is alsoexamined. It is observed that the presence of inner drop alters thedeformation. In some cases, deformation of outer drop increases,and in some cases decreaseswith increase in size of inner droplet. Inpresence of inner droplet transition of outer droplet from prolatedeformation to oblate occurs at the lower value of fluid permittivityratio.

Acknowledgments

Authors would like to thanks Prof. Rochish Thaokar for hisvaluable suggestions, and also to Unilever Industries limited for thefunding.

References

[1] P. Atten, Electrocoalescence of water droplets in an insulating liquid, Journalof Electrostatics 30 (1993) 259e269.

[2] P.J. Bailes, S.K.L. Larkai, Experimental investigation into the use of high voltaged. c. fields for liquid phase separation, Transactions of the Institution ofChemical Engineers 59 (1981) 229e237.

[3] D.O. Thompson, A.S. Taylor, D.E. Graham, Emulsification and demulsificationrelated to crude oil production, Colloids and Surfaces 15 (1985) 175e189.

[4] S. Torza, R.G. Cox, S.G. Mason, Electrohydrodynamic deformation and burst ofliquid drops, Philosophical Transactions of the Royal Society A: Mathematical,Physical and Engineering Sciences 269 (1971) 295e310.

[5] O.O. Ajayi, A note on Taylor’s electrohydrodynamic theory, Proceedings of theRoyal Society of London. Series A. Mathematical and Physical Sciences 364(1978) 499e507.

[6] D.A. Saville, Electrohydrodynamics: the Taylor-Melcher leaky dielectric model29 (1997) 27e64.

[7] D.A. Saville, Electrohydrodynamic oscillation and stability of a charged drop,Physics of Fluids 17 (1974) 54e60.

[8] O. Vizika, D.A. Saville, The electrohydrodynamic deformation of drops sus-pended in liquids in steady and oscillatory electric fields, Journal of FluidMechanics 239 (1992) 1e21.

[9] J.D. Sherwood, The deformation of a fluid drop in an electric field: a slender-body analysis, Journal of Physics A: Mathematical and General 24 (1991)4047e4053.

[10] G. Taylor, Disintegration of water drops in an electric field, Proceedings of theRoyal Society of London. Series A. Mathematical and Physical Sciences 280(1964) 383e397.

[11] G. Taylor, Studies in electrohydrodynamics. I. The circulation produced ina drop by electrical field, Proceedings of the Royal Society of London. Series A.Mathematical and Physical Sciences 291 (1966) 159e166.

[12] T. Tsukada, T. Katayama, Y. Ito, M. Hozawa, Theoretical and experimentalstudies of circulations inside and outside a deformed drop under a uniformelectric field, Journal of Chemical Engineering of Japan 26 (1993) 698e703.

[13] C.T. O’Konski, J. Thacher, The distortion of aerosol droplets by an electric field,Journal of Physical Chemistry 57 (1953) 955e958.

[14] J.S. Eow, M. Ghadiri, A. Sharif, Deformation and break-up of aqueous drops indielectric liquids in high electric fields, Journal of Electrostatics 51e52 (2001)463e469.

[15] J.W. Ha, S.M. Yang, Breakup of a multiple emulsion drop in a uniform electricfield, Journal of Colloid and Interface Science 213 (1999) 92e100.

[16] S. Torza, S.G. Mason, Three-phase interactions in shear and electrical fields,Journal of Colloid And Interface Science 33 (1970) 67e83.

[17] H.N. Gouz, S.S. Sadhal, Fluid dynamics and stability analysis of a compounddroplet in an electric field, Quarterly Journal of Mechanics and AppliedMathematics 42 (1989) 65e83.

[18] T. Tsukada, J. Mayama, M. Sato, M. Hozawa, Theoretical and experimentalstudies on the behavior of a compound drop under a uniform DC electric field,Journal of Chemical Engineering of Japan 30 (1997) 215e222.

[19] J.W. Ha, S.M. Yang, Fluid dynamics of a double emulsion droplet in an electricfield, Physics of Fluids 11 (1999) 1029e1041.