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Journal of Electrostatics 73 (2015) 89e96

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Journal of Electrostatics

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

Head-on collision of electrically charged droplets

O. Ghazian*, K. Adamiak, G.S.P. CastleDepartment of Electrical and Computer Engineering, University of Western Ontario, London, ON, Canada N6A 5B9

a r t i c l e i n f o

Article history:Received 7 May 2014Received in revised form4 September 2014Accepted 27 October 2014Available online 18 November 2014

Keywords:DropletsCollisionElectrohydrodynamicsElectric force

* Corresponding author. Tel.: þ1 5196612111.E-mail address: oghazian@uwo.ca (O. Ghazian).

http://dx.doi.org/10.1016/j.elstat.2014.10.0090304-3886/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

Interaction between two identical charged droplets is investigated numerically. In the first part, themechanism of Coulomb attraction between two conducting droplets is investigated. Numerical simu-lation shows that two conducting droplets carrying charges of the same polarity under some conditionsmay be electrically attracted.

The second part of the study concerns the collision dynamics of two identical dielectric chargeddroplets. At lowWeber numbers, two droplets carrying charges of the same polarity can repel each other.As Weber number increases, the drop collision leads to their coalescence. With Weber number furtherincreased, satellite droplets are formed.

© 2014 Elsevier B.V. All rights reserved.

Introduction

Charged droplets play a significant role in a wide range of ap-plications, such as electrospray atomization [1e4], fuel injectionand formation of clouds [5,6].

The electric force can also be used to enhance the separation andcoalescence of small droplets. The interaction between electric fieldand single charged droplets has already been extensively studied[7e10], but droplet collision is still not well understood. Park [11]produced collisions between streams of water droplets travelingin still air and showed pictorially that near head-on collision be-tween pairs of equally sized droplets resulted in stable coalescence.Ashgriz and Poo [12] developedmodels for predicting the boundarybetween the coalescence and separation regimes. In general, theoutcome of the drop collision can be categorized into four differenttypes: bouncing, coalescence, separation, and shattering collisions.At higher Weber number for head-on or near head-on cases, re-flexive separation may happen resulting in formation of satellites.“Shattering” occurs at extremely high Weber numbers, which isbeyond the scope of the conventional application.

As reported by Qian and Law [13], for head-on collisions of waterdroplets at atmospheric pressure bouncing is not observed; for thesame conditions however, the collision between hydrocarbondroplets may result in bouncing. The collision behavior of fueldroplets was found to vary significantly from those of water

droplets. The most noticeable difference is the bouncing phenom-ena. Estrade et al. [14] published information about the number ofsatellite droplets, their sizes and velocities produced by bouncingcollisions. Brenn et al. [15] produced a nomogram for the variouscollision regimes and for the number of satellite droplets formedduring droplet collision depending on the Weber number andimpact parameter, which agreed quite well with the experimentalresults of Ashgriz and Poo [12].

There are a few studies on the numerical simulation of thedroplet collision. Nobari et al. [16] used the front tracking methodin axi-symmetric formulation for the central collision; the methodwas able to capture the features of bouncing, coalescence and re-flexive separation with up to one satellite droplet formed.Mashayek et al. [17] studied the coalescence collision of twodroplets in axi-symmetric geometry, using a Galerkin FiniteElement Method. Recently, Pan and Suga [18] using the implicitcontinuous-fluid Eulerian method coupled with the level setmethodology for a single phase in a fixed uniform mesh system,simulated the three major regimes of binary collision (bouncing,coalescence and separation), both for water and hydrocarbondroplets. Their numerical results suggest that the mechanism ofbouncing collision is governed by the macroscopic dynamics, whilethe mechanism of coalescence is related to the microscopic dy-namics. Tanguy and Berlemont [19] performed simulation using aLevel-Set Method. Results of coalescence, reflexive separation andstretching separation were found in good agreement with experi-ments. Nikolopoulos et al. [20,21] conducted numerical investiga-tion of both head-on and off-center droplet collision based on thevolume of fluid (VOF) method. Their results provided a detailed

Nomenclature

d0 initial distance of dropletsD0 initial droplet diameterE electric fieldFes electric forceFst surface tension forceI identity matrixn interface normalP pressureq droplet chargeqRay Rayleigh limitr droplet radiusRe Reynolds numberT capillary pressure tensorTM Maxwell stress tensort nondimensional time

t0 time scaleu velocityU0 impact velocityWe Weber number

Greek symbolsa reinitialization parameter3r relative permittivity3ls parameter controlling the interface thickness30 permittivity of vacuumf level-set functionrair air densityrdrop droplet densitymair air dynamic viscositymdrop droplet viscositys surface tension

O. Ghazian et al. / Journal of Electrostatics 73 (2015) 89e9690

picture of the collision process, the ligament formation and di-mensions, the pinch-off mechanism, as well as the creation of thesatellite droplet. They further investigated the effect of gas, liquidproperties and droplet size ratio on the central collision betweentwo unequal-size droplets in the reflexive regime [22], results ofwhich show that the droplet size ratio, rather than the Reynoldsnumber based on the gas properties, is an important parameteraffecting the collision outcome. Chen et al. [23] analyzed energyand mass transfer during binary droplet collision based on the VOFsimulation. The mass transfer process was studied in detail,whereas the energy transfer process was only investigated with theoverall energy balance. Estrade et al. [14] for the first time includedinformation about the number of the satellite droplets in separa-tion collisions.

Literature on the investigation of interactions between twocharged droplets is extremely scarce [24]. The feasibility of coa-lescence of two perfectly conducting, electrically charged dropletswas studied from a thermodynamic point of view by Gallil et al.[25]. A suitable expression was developed for the electrical energyof the two droplets whichmake the initial contact. It was proven byLekner [26] that two charged conducting spheres will almost al-ways attract each other at a close distance. Surprisingly, this is trueeven when they have like charges. The one exception is when thetwo spheres have a charge ratio which would result from dropletsmaking mechanical contact. Phase Doppler anemometry mea-surements and flow visualizations were used to measure thestructures of electrostatically atomized hydrocarbon fuel sprays byShrimpton and Yule [27].

The purpose of the present study is to investigate the electro-static interaction of charged droplets considering different physicalparameters and impact velocities with the aim of providing broaderandmore in depth insight into the collision of charged droplets anddifferent outcome regimes. The NaviereStokes equations with thevolumetric forces due to surface tension and electric charges aresolved numerically by the finite element methodology.

There are three aspects of the present investigation. First, inorder to validate the model the numerical results are comparedin detail with the images of the simulated liquid droplet collisionobtained by Pan and Suga [18]. Secondly, the mechanism ofCoulomb attraction between two like charged conducting drop-lets is investigated. The third aspect of the study concerns thecollision dynamics of two charged droplets and satellite dropletformation.

The mathematical model

The flow is considered as axi-symmetric, incompressible andlaminar. The main parameters affecting the process are grouped intwo dimensionless numbers: Reynolds Re ¼ rdropD0(2U0)/mdrop andWeber We ¼ rdropD0(2U0)2/s, where D0 is the initial drop diameter,U0 is the impact velocity, and rdrop, mdrop and s are the liquid density,viscosity and surface tension, respectively.

In order to investigate the dynamics of droplet deformation inan electric field it is necessary to solve the NaviereStokes equationsgoverning the fluid motion, as well as track the interface betweenboth fluids. The laminar two-phase flow studied here is coupledwith the applied electric field and electric charges on the interface.Additional body forces are added to the NaviereStokes equationsfor considering the surface tension (Fst) and electric stress (Fes).

rvuvt

þ rðu$VÞu ¼ V$h� PI þ m

�Vuþ ðVuÞT

�iþ Fst þ Fes

V$u ¼ 0(1)

where u denotes fluid velocity, I is the 3� 3 identity matrix and p isthe pressure.

To represent the free boundaries of the droplet, the Level-SetMethod has been incorporated into the simulations. The methoddescribes the evolution of the interface between the two fluidstracing an iso-potential curve of the level set function (f). In gen-eral, inside the droplet f equals to one (f ¼ 1) and in ambient fluidf equals to zero (f ¼ 0). The interface is represented by the 0.5contour of the level set function (f ¼ 0.5). The function f is gov-erned by

vf

vtþ V$ðfuÞ ¼ aV$

�3lsVf� fð1� fÞ Vf

jVfj�

(2)

where 3ls is the parameter controlling the interface thickness and a

is the reinitialization parameter. For identifying each phase sepa-rately a volume fraction is introduced. The values of density r andviscosity m are calculated using linear interpolation between thevalues of the two phases.

It is assumed that there is no space charge in the fluids exceptthe surface charge on the interface. Assuming that the fluids areincompressible, the electric stress can be calculated by taking thedivergence of the Maxwell stress tensor, which couples

O. Ghazian et al. / Journal of Electrostatics 73 (2015) 89e96 91

electrostatic and hydrodynamic phenomena. Neglecting the effectof magnetic field, the Maxwell stress tensor can be defined as:

TMij ¼ 3r30EiEj �12

�3r30E

2�dij (3)

where 30 is the permittivity of vacuum, 3r is the relative permittivityand E is the electric field.

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

Fes ¼ V$TM (4)

The surface tension force can be computed as the divergence ofthe capillary pressure tensor T

Fst ¼ V$TT ¼ V$

�s�I � nnT�d� (5)

where n is the interface normal, d is a smoothed Dirac delta func-tion centered at the 0.5 contour of f and s is the surface tensioncoefficient.

The computational domain is shown in Fig.1. Inmost of previousnumerical and experimental studies, the droplets were assumed tohave equal, but opposite velocities before collision. At the begin-ning of the simulation, t ¼ 0, the centers of the droplets are 2D0apart approaching each other with a relative velocity 2U0, whilstthe surrounding gas has zero velocity. Due to the symmetry onlyone droplet, shown in the figures, needs to be considered, becausethe other droplet is identical and the top edge of the domain is thesymmetry line.

In order to investigate the grid dependency of the results, threegrids with 20, 40, and 100 cells per initial droplet diameter wereconsidered. It was found that the calculated parameters of dropletshapes did not vary significantly, if the element size is smaller thanD0/40. Therefore, all further simulations were performed usingsuch a mesh.

The radius and height of the cylindrical computational domainsare 3D0 and 5D0, respectively. The increase in size of the chosendomain has been found to have a negligibly small effect on thesolution. The computational domain is therefore large enough toneglect the effects of the domain truncation.

Fig. 1. Initial configuration used in numerical simulation of droplet impact.

The behavior of the binary fluid system is governed by the level-set equation. In the present study, the discretization and calculationprocedure follow that described by Ghazian et al. [28e30]. The levelset function moves with the fluid at velocity u as a passive scalarvariable. The density and the viscosity are calculated throughoutthe computational domain depending on the value of f. The fluidproperties continuously change across the interfacial region. Notethat in the experiments, the droplet diameters were in the range of100e500 mm. In such cases, the gravity effects are negligible sincethe Bond number is always small.

Verification of numerical model

Fig. 2 shows a sequence of results from the present simulationfor two uncharged colliding droplets compared with the simula-tions of Pan and Suga [18].

As the droplets approach each other, high pressure is built up inthe gap; the droplets are flattened, conversion of the droplet kineticenergy into surface tension energy takes place and gas is squeezedout in a form of a jet sheet.

The merged mass continues to deform into a donut shape. Aftercoalescence, it is retracted into a cylindrical rod, which laterstretches longer and thinner, until it eventually breaks into twoprimary droplets with one secondary drop. During most of thecollision process, consisting of the stretching filament, its disinte-gration between bulbous ends, and the further breakup, the com-parison shows very reasonable agreement.

Results and discussion

In this section the model is extended to study the Coulombattraction between two stationary conducting droplets. In thesecond part, the collision dynamics of two identical chargeddroplets is reported. Some interesting features of the chargeddroplet disintegration and satellite droplet formation are illustratedvia various examples.

Two stationary conducting droplets

The presented results of simulation illustrate how the surfacesof the two droplets are deformed due to the electrostatic in-teractions between them. In particular, modification of the elec-trostatic interactions caused by the droplet deformations will beinvestigated. This aspect has not been discussed elsewhere.

As shown below, surprisingly Coulomb attraction may existbetween two conducting droplets carrying the same sign charges.

In the first example, we consider the case of two identicalspherical droplets separated from each other by an initial distanceof 2.5 radii between their centers. The maximum electric chargethat a single isolated droplet can theoretically carry can be evalu-ated from the Rayleigh limit:

qRay ¼ 8p�30sr

3�1=2

where 30 is the vacuum permittivity and r is the radius of thedroplet. The simulations were performed for the following pa-rameters: rdrop ¼ 1000 kg/m3, rair ¼ 1 � 10�3 kg/m3,mdrop ¼ 0.001 Pa s, mair ¼ 2 � 10�5 Pa s, s ¼ 0.073 N/m, r ¼ 100 mm.The maximum Rayleigh charge is calculated to be qRay ¼ 20 pC.

As it can be expected, droplets with the samemagnitude and signof charge (Fig. 3A) repel each other. However, in the case where thedroplets have different charge magnitudes (Fig. 3B), the dropletsattract each other, even though theycarrycharge of the samepolarity.The two droplets will eventuallymove towards and collidewith each

Fig. 2. Pan and Suga simulation [18] (left) and our simulated (right) snapshots of head-on collision of two uncharged water droplets in air. The initial droplet diameter is 300 mmand the relative velocity of collision is 0.28 m/s.

Fig. 3. Interaction between two identical spherical droplets with the same (A) and different charges (B) at an initial distance of 2.5 radii between centers.

O. Ghazian et al. / Journal of Electrostatics 73 (2015) 89e9692

other after forming conical protrusions at the contact point. It can bededuced that the lower part of the upper droplet carries negativecharge, while the upper part of it carries the opposite charge.

By keeping the same value for charge (Fig. 3B) and increasingthe distance between the droplets, the droplets will repel eachother (Fig. 4). This example indicates that the distance plays animportant role in Coulomb attraction for the droplets charged withthe same polarity. It also proves that conducting droplets willalmost always attract each other at close proximity, evenwhen theyhave like charges.

Contrary to common sense, it seems that two conductingdroplets carrying charges of the same polarity can attract eachother in some situations. This apparent paradox can be easily un-derstood by noting that the charge on a conductor is not fixed butfree to move on the surface of the droplets under the action of anelectric field. When the two conducting droplets move towardseach other, the surface charges will redistribute and a net attraction

Fig. 4. Two identical spherical droplets with different charges at a larger distance of2.7 radii between the two centers, q1/qRay ¼ 0.1 (bottom), q2/qRay ¼ 0.25 (top).

force will be created between them due to the unsymmetricalcharge distribution. In addition, the electric force will also result indeformation of the two droplets close to the facing sides, which inturnwill enhance the charge redistribution since charges will moveto locations of high surface curvature. Obviously, this destabilizingmechanism will not happen in the case of rigid spheres.

Fig. 5 illustrates the limit between the attraction and repulsionregimes for two identical droplets, where d0 is the initial distancebetween the droplet centers. It is clear that in order to get theattraction regime, the charge ratio should increase when the

Fig. 5. The limit between attraction and repulsion regimes for two identical droplets.

Fig. 6. Shape evolution of the head-on collision of two equal-size droplets for We ¼ 6.7, Re ¼ 66.7 and q/qRay ¼ 0.5.

O. Ghazian et al. / Journal of Electrostatics 73 (2015) 89e96 93

distance between the droplets increases. There is a maximumdistance (d0 ¼ 2.8 r), beyond which the repulsion is always domi-nant. It can be concluded that the attraction occurs at a shorterdistance when droplets have closer charges.

Impacting dielectric droplets

In this section the collision of two charged dielectric droplets isconsidered. The simulations were performed for the following pa-rameters: rdrop ¼ 1000 kg/m3, rair ¼ 1 � 10�3 kg/m3,mdrop ¼ 0.003 Pa s, mair ¼ 2 � 10�5 Pa s, s ¼ 0.03 N/m, D0 ¼ 200 mm.

Initial distance between both droplets is equal to two dropletdiameters. At the beginning of the simulation, t ¼ 0, a uniformvelocity U0 is imposed on each of the two liquid droplets inopposite but approaching directions, while the surrounding gas isstationary.

Grid dependence tests have been already performed usingdifferent discretization densities and it has been confirmed that thepresented results are reasonably grid independent.

Fig. 8. Shape evolution of the head-on collision of two equa

Fig. 7. Shape evolution of the head-on collision of two equa

Effect of Weber numberTwo regimes can be clearly identified at low Weber numbers in

this process. In the first one, the droplets initially start to movetowards each other, but the kinetic energy is not sufficiently high toovercome the electrostatic repulsion. Since the droplet velocity istoo small, the droplets repel each other due to the electrostaticforce and do not even touch (Fig. 6). To nondimensionalize time wehave the choice to use the advection time (t0 ¼ D0/U0) of thedroplets before impact. Our results are presented using theadvective time scale.

Fig. 7 illustrates the case of the droplets with the same value ofthe charge, but for higher Weber number. It is clear that thedroplets collide and form a larger one, which oscillates until itreaches steady-state. Simulations are continued until oscillations ofthe combined droplet completely decay.

The simulationwas also carried out for increased droplet chargeand keeping the velocity constant (Fig. 8); it is clear that here thedroplets are repelling each other even though the magnitude of theWeber number is the same as in Fig. 7.

l-size droplets for We ¼ 13, Re ¼ 94.3 and q/qRay ¼ 0.7.

l-size droplets for We ¼ 13, Re ¼ 94.3 and q/qRay ¼ 0.5.

Fig. 9. The limit between coalescence and repulsion regimes for two identicalimpacting droplets with the initial distance of d0 ¼ 4r.

Fig. 13. Computational domain for simulating collision of droplets with different size.

O. Ghazian et al. / Journal of Electrostatics 73 (2015) 89e9694

This behavior can be explained by considering the fact that sincetheWeber number and the initial impact velocity are kept constant,the electrostatic force becomes dominant and prevents dropletsfrom colliding.

A map can be drawn to show the limit between the repulsionand coalescence regimes. The graph in Fig. 9 shows the boundarybetween the repulsion and coalescence for the fixed initial distance

Fig. 10. Shape evolution of the head-on collision of two neutral eq

Fig. 12. Shape evolution of the head-on collision of two equal-s

Fig. 11. Shape evolution of the head-on collision of two equal-s

of d0 ¼ 4r. At higher droplet charge the coalescence occurs, buthigher Weber number is necessary. Theoretically, the dropletcharge can be increased up to the Rayleigh limit, before the dropletbecomes unstable. In the investigated models the droplet charge

ual-size droplets for We ¼ 106.7, Re ¼ 266.67 and q/qRay ¼ 0.

ize droplets for We ¼ 106.7, Re ¼ 266.67 and q/qRay ¼ 0.5.

ize droplets for We ¼ 106.7, Re ¼ 266.67 and q/qRay ¼ 0.3.

Fig. 14. Snapshots of droplet motion of larger (upper) and smaller (lower) neutral spherical droplets with R2/R1 ¼ 1.5 at We ¼ 160 and Re ¼ 400.

Fig. 15. Snapshots of droplet motions between a larger (upper) and a smaller (lower) neutral spherical droplets with R2/R1 ¼ 1.5 at We ¼ 360 and Re ¼ 600.

O. Ghazian et al. / Journal of Electrostatics 73 (2015) 89e96 95

was increased up to 80 percent of the Rayleigh limit withoutobserving any instability.

Effect of the droplet charge in the break-up regimeThe effect of the droplet charge on the collision between two

identical dielectric droplets has also been considered. The dropletshave been impacted at much higher velocities and both dropletshave been charged up to 50% of the Rayleigh limit, which isqRay ¼ 13 pC for the 100 mm size droplets considered in thissection.

The charge was placed in the small finite interface surroundingthe surface of the drop using the level-set variable distribution. Thetotal charge was kept constant and calculated by integration overthe surface of the droplet. The total surface area has been calculatedand a uniform charge distribution was considered on the dropletsin a way that the total charge is still kept constant.

Fig. 10 shows the head-on collision of two identical neutraldroplets.

The droplets first coalesce and then reach the maximumdeformation in the radial direction forming a thin disc with atoroidal rim. This shape then contracts under the effects of surfacetension attempting to recover the spherical shape. However, due toinertia a liquid cylinder is formed and it continuously stretchesuntil it eventually breaks, if it has sufficient energy. It can be seenthat the ligament stretches leaving a daughter droplet between thebulbous ends.

A relatively small amount of gas is trapped in the coalescedmassof droplets. Suchmicro-bubble entrapment, due to the formation ofcurved interfaces on the approaching sides of the droplets, has beenobserved experimentally by Ashgriz and Poo [12].

In addition, a recent high-resolution computation of twoapproaching drops using the VOF method also showed that a smallportion of ambient fluid gets trapped during coalescence [31]. Theformation of amicro-bubble at lowWe reported in our paper is thusapparently a physical macroscopic effect and not a numericalartifact.

If the droplets are charged up to 30% of the Rayleigh limit(Fig. 11), there is no obvious change in the collision pattern, if theWeber number is kept at the same level. By further increasing theinitial droplet charge up to 50% (Fig. 12), the daughter dropletsbreak into smaller ones and this continues until the radius of thedroplet gets close to the mesh size.

Since the surface tension of the stretched droplet cannot holdthe large amount of kinetic energy, even with the stabilization ef-fect of viscous dissipation, the merged droplet eventually separatesand a smaller, satellite droplets are continuously formed. It was

found that by increasing the charge, the droplets will disintegrateinto a larger number of daughter droplets after collisions.

Unequal-size droplets

In practical atomization processes, unequal-size droplet colli-sions are more frequent than equal-size cases; however, they havebeen less examined both experimentally and numerically. Equal-size droplet collisions have minimal mixing after droplet coales-cence due to the symmetrywith respect to the collision plane. Thus,breaking the symmetry through the unequal droplet sizes mayresult in improved mixing upon coalescence. In the present study,simulations were carried out to investigate the dynamics ofunequal-size droplets collision. Before the droplets collide, there isa squeezed gas between the droplet interfaces which producessome localized excess energy and needs to be ejected. A smallerrepulsive force from the compressed gas flow between the dropletscan be assumed for the smaller droplet due to the smaller frontalarea which promotes coalescence. For the larger droplet, moreenergy can be dissipated by internal motion and the coalesceddroplet can be stabilized.

The charge ratio on droplets is equal to the surface ratio of thedroplets, which assumes identical surface charge density. Thesymmetry boundary condition cannot be used in this part due todifference in size of the droplets. The schematic of the problem isshown in Fig. 13.

The impact of droplets with radii of 100 mm and 150 mm isshown in Fig. 14. The initial velocity of U0 ¼ 2 m/s is assumed forboth droplets. The Weber number and nondimensional time arecalculated based on the diameter of the larger droplet.

For unequal-size droplet collision, the droplet deformation ismore complex because of the loss of the symmetric shape. Twodroplets impinge head-on and spread outwardly in the radial di-rection to form a flying-saucer-like shape upon merging. Byincreasing the Weber number (Fig. 15), the saucer-like shape be-comes more stretched in the radial direction and a ring at the edgeof the combined droplet becomes smaller in size leading to ejectionof a small droplet.

Conclusions

The head-on collision of two liquid drops is studied using theFEM. The effects of the impact velocity, drop size ratio and electriccharge on the behavior of the combined droplet are investigated.The present study proves the feasibility of applying the Level-Set

O. Ghazian et al. / Journal of Electrostatics 73 (2015) 89e9696

Method as a numerical technique to investigate the collision dy-namics of electrically charged droplets.

It is demonstrated that the presented numerical method is ableto capture the droplet collision in the presence of the electriccharges on the surface of the droplets. It was also shown that twoconducting droplets carrying charges of the same polarity undersome conditions may be electrically attracted.

The formation of charged daughter droplets has been investi-gated and it was found that the number of the satellite dropletsafter collision appears to increase with an increase in the dropletcharge.

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