Droplet dynamics in confinement

ContentslistsavailableatScienceDirect

Journal

of

Computational

Science

j o u r n al ho me p a g e :w w w . e l s e v i e r . c o m / l o c a t e / j o c s

Droplet

dynamics

in

confinement

N.

Ioannou

_,

_H.

_Liu

_,

_Y.H.

_Zhang

a_{JamesWeirFluidsLaboratory,DepartmentofMechanical&AerospaceEngineering,UniversityofStrathclyde,GlasgowG11XJ,UK}

b_{SchoolofEnergyandPowerEngineering,Xi’anJiaotongUniversity,Xi’an710049,China}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received26October2015

Receivedinrevisedform9March2016 Accepted12March2016

Availableonlinexxx

Keywords:

Dropletdynamics LatticeBoltzmannmethod Deformation

Breakup Confinement

a

b

s

t

r

a

c

t

Thisstudyistounderstandconfinementeffectonthedynamicalbehaviourofadropletimmersedinan immiscibleliquidsubjectedtoasimpleshearflow.ThelatticeBoltzmannmethod,whichusesaforcing termandarecolouringalgorithmtorealizetheinterfacialtensioneffectandphaseseparation respec-tively,isadoptedtosystematicallystudydropletdeformationandbreakupinconfinedconditions.The effectsofcapillarynumber,viscosityratioofthedroplettothecarrierliquid,andconfinementratio arestudied.Thesimulationresultsarecomparedagainstthetheoreticalpredictions,experimentaland numericaldataavailableinliterature.Wefindthatincreasingconfinementratiowillenhance deforma-tion,andthemaximumdeformationoccursattheviscosityratioofunity.Thedropletisfoundtoorient moretowardstheflowdirectionwithincreasingviscosityratioorconfinementratio.Also,itisnoticed thatthewalleffectbecomesmoresignificantfortheconfinementratioslargerthan0.4.Finally,the crit-icalcapillarynumber,abovewhichthedropletbreakupoccurs,isfoundtobemildlyaffectedbythe confinementfortheviscosityratioofunity.Uponincreasingtheconfinementratio,thecriticalcapillary numberincreasesfortheviscosityratioslessthanunity,butdecreasesfortheviscosityratiosmorethan unity.

©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Emulsionsconsistofimmisciblefluidscommonlyfoundin pro-ductionprocessesinfood,chemical,andpharmaceuticalindustries. Sincethedropletsizeandshape determineimportantemulsion propertiessuchasstability,rheology,andparticlemorphology,it isimportant tounderstandthemechanismof droplet deforma-tionandbreakupduringemulsification.Inaddition,thestudyof dropletdeformation and breakupcan providevaluable insights intoimmisciblefluiddisplacementinporousmedia,whichplays animportantroleinenhancedoilrecovery,geologicCO2 seques-tration,andremediationofnonaqueous-phaseliquids.Inrecent years,thedropletdeformationandbreakuphavereceivedmore attentionbecauseofthegrowinginterestinmicrofluidic technolo-gies,wheredropletsarecirculatedinchannelsandtheirsizeisoften comparablewithorevensmallerthanchanneldimension.A signif-icantnumberoftheoretical,experimentalandnumericalstudies havebeenreportedregardingdropletdeformationandbreakupin ashearflowsincethepioneeringworkofTaylor[1].Themain fea-tureofshearflowisitsrelativesimplicity,whileitcontainsrich physics[2].

∗Correspondingauthor.

E-mailaddress:[email protected](Y.H.Zhang).

AssketchedinFig.1,asphericaldropletwithradiusRisinitially placedhalfwaybetweentwoparallelplatesthatareseparatedby adistanceH.Shearflowisintroducedtothissystembythe move-mentoftheplateswithequalbutoppositevelocitiesU,andthe resultingshearrateis˙ =2U/H.Thedropletandthecarrierfluid areassumedtohaveequaldensities,aconstantinterfacial ten-sion,andtheirdynamicviscositiesaredandm.Inthesimple

shearflow,twomostimportantforcesexertingonthedropletare: theviscousforces(mR˙ 2),leadingthedroplettodeformandturn

towardstheflowdirection,andthecapillaryforces(R),resisting thedropletdeformationandretainingthesphericalshapeofthe droplet.Consequently,thedropletundergoesanelongationinthe directionoftheL-axis,andacompressioninthedirectionofthe

B-axis(seetherightpanelofFig.1).Thecapillarynumber,which isdefinedasCa=R˙ m/,isusedtomeasuretherelative

mag-nitudeofviscousandcapillaryforces.Theinertialforcecanalso influencethedynamicalbehaviourofthedroplet,andits impor-tanceisdescribedbytheReynoldsnumber,definedastheratioof inertialtoviscousforcesi.e.Re=R˙ 2_/

m.Inadditiontothe

capil-laryandReynoldsnumbers,theviscosityratio(=d/m)andthe

proximityofdroplettothewallsarefoundtoaffectstronglythe deformationandbreakupofthedroplet.Theproximityisevaluated bytheconfinementratio,definedas2R/H.Underconsiderationof lowcapillarynumberandconfinementratio,thedropletwill even-tuallybecomeellipsoidalintheStokesflowregime(Re<1).The http://dx.doi.org/10.1016/j.jocs.2016.03.009

Fig.1. Schematicillustrationofdropletdeformationinasimpleshearflow.

deformationparameterDcanbedefinedbythelengthsofthemajor (L)andminor(B)axesofthedeformeddroplet,i.e.

D=L−B

L+B. (1)

WithincreasingCa,thedropletcanreachasteadystatebut devi-atefromtheellipsoidalshape.Thiscausesdifficultyinobtainingthe correctvaluesofdeformationparameterandorientationangle(), whichisdefinedastheanglebetweenthedropletmajoraxisand thehorizontalplane.Asthecapillarynumberisfurtherincreased abovethecriticalvalue,thecapillaryforcescannolongerretain theshapeofdroplet;thedominantviscousforcesleadthedroplet toformalongthin neckand finallybreakupintotwo ormore fragments.Thecriticalcapillary number(Cacr),abovewhichthe

dropletbreakupwilloccur,isinfluencedbyboththeviscosityand confinementratios[3].

Theinterestindropletdeformationdatesbacktotheworkof Taylor(1934)[1],whoderivedatheoreticalexpressiontodescribe smalldeformationsinthebulkshearflowintermsoftheviscosity ratioandthecapillarynumber:

DT=

19+16

16+16Ca. (2)

Thisexpressionhasbeendemonstratedtopredictexperimental resultswellinavarietyofcases,wherethevalueofconfinement ratio was around 0.2. However, the influence of wall confine-mentis not taken intoaccount in Eq.(2).It wasreported that thepresenceofwallshasanegligiblecontributiontothedroplet deformationfortheconfinementratio2R/H≤0.4[4,5].Whenthe confinementratioishigherthan0.4,thedeformationcannotbe predictedaccuratelybyEq.(2).Moreover,Eq. (2)is notableto describethe dropletdeformation for very largeviscosityratios

[1,6].Morediscussionaboutthisequationcanbefoundinthese

reviewpapers[7–10].Taylorderivedhismodelusingsmall defor-mationperturbationproceduretothefirstorder,withCaasthe expansionparameter,soheobtainedaconstantorientationangle of45◦.Theperturbationprocedurewaslater tobeextendedto thesecondorderinCatoyieldanexpressionfortheorientation angle[11–14]:=(/4)−((16+19)(3+2)/80(1+))Ca[15].The orientationanglewasalsoformulateddifferentlyinthe phenomen-ologicalmodelsofMaffetoneandMinale[16]andMinale[17].

Toaddressthewallconfinementeffectondropletbehaviour, ShapiraandHaber[18]solvedtheStokesflowsaroundadroplet usingthemethodofreflection,whichtakesintoaccounttherelative

positionofthedroplettothewall.TheresultingSHmodelcombines theTaylordeformationandanadditionaltermaccountingforthe influenceofwallsonthedeformation:

DSH=DT

1₊2.5 1₊

3

, (3)

whereCSisreferredtoasashapeparameteranditsvaluedepends

ontherelativepositionofthedroplettothewalls.Foradroplet positionedhalfwaybetweenthetwowallsCSistakenas5.6996.

Topredictthedropletdeformationundertransientconditions, MaffettoneandMinale[16]proposedaphenomenologicalmodel, i.e.MMmodel,inwhichthedropletshapeisassumedtoremain ellipsoidal.AsecondordertensorS,whoseeigenvaluesreflectthe squaresofthesemi-axesofanellipsoid,isusedtodescribethe dropletshape.Basedonthisassumption,theyderivedanevolution equationforthetensorS,whichconsistsofaco-rotational deriva-tive,thecontributionsoftheviscousstressandthecapillaryforce. Thevaluesofthesemi-axesforseveraltypicalflowswerepredicted, includingthesimpleshearflow,theuniaxialextensionalflowand theplanarhyperbolicflow.Inparticular,forthesimpleshearflow, thedeformationparameterinasteadystateiscalculatedas

DMM=

1+Ca 2₋

m2

1+(1−m22)Ca 2

m2Ca ,

(4) where

m1=

40(+1)

(2+3)(19+16), (5)

and m2=

5 2+3+

3Ca2

2+6Ca2. (6)

significantatlargecapillarynumbers.Inanattempttomatchthe experimentalresultsregardinglargedeformationatlargecapillary numbers(e.g.,Ca=0.3)withananalyticalexpression,two mod-elswereproposed.OneistheorientationanglescaledSHmodel. However,itsuffersfromthedifficultyofestimatingtheorientation angleathighconfinementratios[7].Theotherisacombination oftheaforementionedtwomodels(i.e.,SHandMM).Specifically, thetermDTintheSHmodelisreplacedbyDMMintheMMmodel,

leadingtoanewmodel,calledastheMMSHmodel:

DMMSH=DMM

1+2.5 1+

3

. (7)

SincetheMMSHmodeldoesnotneedtoestimatethedroplet ori-entationangleanditspredictionsagreewellwiththeexperimental data[6],itwillbeusedinSection3forvalidationofournumerical results.Inaddition,Minale[17]refinedthephenomenologicalMM model,byforcingthemodeltorecovertheanalyticallimitsofthe SHmodelformoderateconfinementratios,i.e.2R/H≤0.5.Forlarge confinementratios,theexperimentaldatafromSibilloetal.[19]

andVananroyeetal.[6],wereusedtore-evaluatethecoefficients

m1andm2intheMMmodel.

Complementingtheoreticalandexperimentalstudies, numer-ical simulations have been extensively used to investigate the droplet deformation and breakup in a simple shear flow

[3,4,20–24]. Among these works, Janssen and his co-workers

[3,23,24]usedtheboundaryintegralmethod(BIM)tostudythe

effectofviscosityandconfinementratiosondropletdeformation andthecriticalcapillarynumber.Itwasfoundthatthesimulated shape ofthedroplet andthecritical capillary numberare both inexcellentagreementwiththeexperimentalresults.However, theBIMisusuallyfor thesolutionof Stokesflow, inwhichthe fluidinertiaistotallyneglected.Also,itsuffersfromthe difficul-tiesofaccessingthelowviscosityratioregime[3],andofdealing withtopologicalchangessuchasbreakupandcoalescenceofthe interfaces[7,25,26].Thesedisadvantagescanbeovercomebythe latticeBoltzmannmethod(LBM),whichtracksevolutionofthe dis-tributionfunctionofanassemblyofmoleculesandisbuiltupon microscopicmodelsand mesoscopickineticequations.TheLBM cancapturemultiphaseinterfaceanddescribeinterfacial dynam-icssuchasphasesegregationandinterfacialtensionthroughthe incorporationofintermolecularinteractions.Anumberof multi-phase andmulticomponent modelshave beenproposed, which canbedividedintofourmaintypes:colourgradientmodel[27], free-energy model [28], interparticle-potential model [29], and mean-fieldtheorymodel[30].Althoughthedropletdynamicsina shearflowwassuccessfullysimulatedbytheformerthreemodels ortheirvariants[31–36],noneofthemhassystematically exam-inedtheimpactofviscosityandconfinementratiosonthedroplet deformationandbreakup,whichisthefocusofthiswork.In partic-ular,thecolourgradientLBMisusedherebecauseofitsadvantages suchaslowspuriouscurrents,highnumericalaccuracyandstrict massconservationforeachfluidwhichhavebeendemonstratedin ourrecentworks[37,38].

2. Numericalmethod

Weconsiderisothermalmotionoftwoimmisciblefluidsand adoptacontinuumsurfaceforce(CSF)model[39]todescribethe system.TheCSFformulationallowsustoconsiderthetwophases as a singlefluid with space-dependentproperties and replaces thejumpconditionattheinterfacewithanadditionalforcethat actsonlyintheinterfaceregion.Unlikeatwo-fluidformulation, itdoesnotneedtosolveamoving boundaryproblem, whichis computationallyexpensiveandextremelydifficulttoimplement especially when large topological changes occur around the

interface.IntheCSFformulation,theflowoftwoimmisciblefluids isgovernedbyasinglesetofNavier–Stokesequations(NSEs):

∇

∂

∂

∇

uu

∇

∇

∇

∇

wheretisthetime,uisthefluidvelocity,isthetotaldensity,p

isthepressure,andisthedynamicviscosity.ThelastterminEq.

(9)representstheinterfacialtensionforceactingattheinterface betweenthetwofluidsandisdefinedas

Fs=nı, (10)

whereistheinterfacialtension;,thelocalinterfacecurvature;

n,theunitnormaltotheinterface;andı,theDiracdeltafunction

usedtolocalizetheforceexplicitlyattheinterface,whichshould satisfy

∞

−∞

ıdz=1, (11)

inordertorecoverproperlythestressjumpconditioninthe sharp-interfacelimit.Herezisthespatiallocationnormaltotheinterface. AlthoughEqs.(8) and(9)allowcomputingthevelocityfield, an advection equation has to be additionally solved to cap-turetheevolution ofinterfaceintraditional multiphasesolvers, e.g.,thevolume-of-fluid(VOF)andlevel-setmethods,wherethe sophisticatedinterfacereconstructionalgorithmorunphysical re-initialization process is often required. In order toavoid these issues,weusethelatticeBoltzmanncolourgradientmodel,first proposedbyGunstensenetal.[27]andlaterimprovedby Halli-dayandhiscoworkers[40],tosimulatethedynamicalbehaviour ofa dropletsubjectedtoa simpleshearflow. Thecolour gradi-entmodelpossessesmany advantagesinsimulatingimmiscible two-phaseflows,includingtheabilityofcapturinginterface auto-matically,low spurious velocities, highnumericalaccuracy and strictmassconservationforeachfluid.Inthismodel,twosetsof distributionfunctionsfR

i and fiB areintroducedtorepresentthe

“red”and“blue”fluids.Thetotaldistributionfunctionisdefined byfi=f_iR+f_iB,whichundergoesaBhatnagar–Gross–Krook(BGK)

collisionstepas f_i†(x, t)=fi(x,t)−

1

(fi(x, t)−f

eq

i (x,t))+i, (12)

wherefi(x,t)isthetotaldistributionfunctionintheithvelocity

directionatthepositionxandtimet,f_ieqistheequilibrium distri-butionfunctionoffi,f_i†isthepost-collisiondistributionfunction,

isthedimensionlessrelaxationtime,andiistheforcingterm.

Theequilibriumdistributionfunctionisobtainedbya second-orderTaylorexpansionofMaxwell–Boltzmanndistributionwith respecttothelocalvelocityu:

f_ieq=wi

c2

s

+(ei· u)2

2c4

s

− u2 2c2

s

, (13)

whereiscalculatedby=R+Bwiththesubscripts‘R’and‘B’

referring totheredand bluefluidsrespectively, cs isthespeed

ofsound,ei isthelatticevelocityintheith direction,and wiis

theweightfactor.Forthethree-dimensional19-velocity(D3Q19) model,thelatticevelocityeiandtheweightfactorsaregivenby

ei=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

(0,0,0)c, i=0;

(±1,0,0)c,(0,±1,0)c,(0,0,±1)c, i=1,2,...,6; (±1,±1,0)c,(0,±1,±1)c,(±1,0,±1)c, i=7,8,...,18,

Fig.2.Thedeformationparameter(a),andtheorientationangle(b)asafunctionofthecapillarynumberfor=1.

wi=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

1/3, i=0;

1/18, i=1,2,...,6;

1/36, i=7,8,...,18, (15)

andthespeedofsoundcs=c/√3=ıx/√3ıtwithıxandıtbeing

thelatticelengthandtimestep,respectively.

Thespatialdistributionofthetwofluidsisdescribedusinga colourfunction(orphase-fieldfunction),whichisdefinedas

N(x,t)=R(x, t)−B(x,t)

R(x, t)+B(x,t), −

1≤N≤1. (16)

Bydefiningn=−

∇

_∇

=|

∇

fur-therwrittenas

Fs=−

1 2

∇

N_{, (17)}

wherethelocalinterfacecurvatureisrelatedtotheunitnormal totheinterfaceby

=−

∇

∇

Fig.3.Deformationparameterasafunctionofthecapillarynumberfor=1and 2R/H=0.69.

andIisthesecond-orderidentitytensor.Withtheinterfacial ten-sionforcegivenbyEq.(17),theforcingtermithatisappliedto

realizetheinterfacialtensioneffect,readsas[41]

i=

wi

c2

s

+ei· u

c4

s

ei

· Fsıt. (19)

Accordingto Guo et al. [42], the local fluid velocity shouldbe defined to incorporate the spatially varying interfacial tension force,i.e.,

u(x,t)=

i

fi(x,t)ei+

1

2Fs(x, t)ıt. (20) UsingtheChapman–Enskogmultiscaleexpansion,Eq.(12)can reducetotheNSEsinthelowfrequency,longwavelengthlimitwith thepressureandthefluidviscositydefinedby

p=cs2, (21)

=cs2

−1₂

Toallowforunequalviscositiesofthetwofluids,wedetermine theviscosityofthefluidmixturebyaharmonicmean,i.e.

1

1+N

2R +

1−N

2B ,

(23)

wherek(k=RorB)isthedynamicviscosityoffluidk.Ithasbeen

shownthatthechoiceofEq.(23)canensureaconstantviscosity stressacrosstheinterface,whichprovideshigheraccuracythan otherchoices[43].

The partial derivatives required for the curvature and nor-malvectorcalculationsareobtainedusingthe19-pointcompact

finite-differencestencil[44].Forexample,foravariable ,its par-tialderivativescanbecalculatedby

∂

∂

1 c2

s

i

wi (x+ eiıt)eia. (24)

Althoughtheforcingtermgeneratesaninterfacialtension, it doesnotguarantee theimmiscibilityofbothfluids.Topromote phasesegregationandmaintaina reasonableinterface,the seg-regation(recolouring)algorithmofLatva-KokkoandRothmanis used[45].Itcanovercomethelatticepinningproblemandcreates asymmetricdistributionofparticlesaroundtheinterfacesothat unphysicalspuriouscurrentscanbeeffectivelyreduced. Follow-ingLatva-KokkoandRothman,thepost-segregation(recoloured) distributionfunctionsoftheredandbluefluidsare

fR

i(x,t)=

R

f

†

i(x,t)+ˇ

RB

wicos(ϕi)|ei|, fB

i(x,t)=

B

f

†

i(x,t)−ˇ

RB

wicos(ϕi)|ei|,

(25)

whereˇisthesegregationparameterandissettobe0.7for numer-icalstabilityandmodelaccuracy[46];ϕiistheanglebetweenthe

colourgradientandthelatticevectorei,whichisdefinedby

cos(ϕi)= ei·

∇

|ei||

∇

(26) Aftertherecolouringstep,theredandbluedistribution func-tions propagate to the neighbouring lattice nodes, known as propagationorstreamingstep:

fk

i(x+ eiıt,t+ıt)=f_ik(x,t), k=RorB, (27)

andtheresultingdistributionfunctionsarethenusedtocalculate thedensitiesofbothfluidsi.e.k₌

ifik.

3. Resultsanddiscussion

Inallthecasesdiscussedbelow,asphericaldropletwiththe radiusR=20latticesisinitiallyplacedinthecentreofthe compu-tationaldomain,asillustratedinFig.1.Thecomputationaldomain hasa lengthof 71 latticesin they-direction.The lengthinthe

x-directionischosenas101inthestudyofdropletdeformation, whereas indroplet break-up this lengthcouldbe increasedup to401in somecasesinorder toavoidtheoccurrence thatthe droplettouchesitselfviatheperiodicboundaryconditionsbefore thebreakupoccurs.Inaddition,theheightofthecomputational

domainisvariedintherangeof51≤H≤223toobtaindifferent confinementratios2R/H.Periodicboundaryconditionsareapplied inthex-andy-directions,whilethehalf-waybounce-backscheme isusedonthetopandbottomwalls[47]:

fi(x,t+ıt)=f_ops† (i)(x,t)+6wi ei· uw

c , (28)

whereuwisthemovingvelocityofthesolidwallwithuw=(U,0,0)

forthetopwallanduw=(−U,0,0)forthebottomwall,andops(i)

standsforthedistributionfunctionwhichhasoppositedirection toei.Forthesakeofsimplicity,bothfluidsareassumedtohave

equaldensities.TheReynoldsnumberisfixedat0.1,whichissmall enoughtocharacterisetheflowasStokesflow.

3.1. Dropletdeformation

Thenumericalmethodisfirsttestedbysimulatingthedroplet deformation in a weakly confined shear flow. Simulations are conductedat=1andtheconfinementratio2R/H=0.18forthe capillarynumbersrangingfrom0.1to0.3.Fig.2(a)plotsthedroplet deformationparameterasafunctionofthecapillarynumber,in whichourLBM resultsarecomparedwiththeprevious experi-mentaldata[6,19],thenumericalresultsobtainedbytheboundary integralmethod(BIM)[4,48]andthevolumeoffluid(VOF)method

[20], as wellas thepredictionsfrom Taylor theoryand MMSH model.Notethatthedegreeofconfinementisnotexplicitly avail-ableforsomeofthedatasources,inwhichtheflowisconsidered tobeunbounded.Itcanbeseenfromthecomparison,theLBM resultsagreewellwiththeseexperimentaldata,numericalresults andmodelpredictions;andgoodagreementamongthemsuggests thatthewallshaveanegligibleinfluenceonthedropletbehaviour forthepresentconfinementratio,i.e.2R/H=0.18,consistentwith thepreviousnumericalandexperimentalobservations[4,8].

Inadditiontothedeformationparameter,wealsocalculatethe orientationanglesofthedeformeddroplet(),definedastheangle betweenthelongestaxisofthedropletandthex-axis,atvarious capillarynumbers,andcomparethemtothosereportedinRefs.

[4,20,24,48](seeFig.2(b)).Clearly,thereisfairlygoodagreement

especiallybetweenourresultsandthoseobtainedbythe exper-iment and theBIM at anexactly matched confinementratio, i.e.

2R/H=0.18.

Next,thesimulationswereconductedforahighconfinementof 0.69,whichisachievedbyadjustingtheheightofcomputational domainwhilstkeeping otherparameters thesame asin Fig.2.

Fig.3showsthecomparisonbetweenthesimulateddeformation

Fig.7. VelocityvectorsandstreamlinesaroundthedropletforCa=0.2:thecolumnsareforthesameconfinementratio(left:0.4,right:0.7);andtherowsareforthesame viscosityratio(top:=0.3,middle:=1,bottom:=5).

parameterandtheTaylortheory,theMMSHmodelandthe experi-mentalresultsofVananroyeetal.[6].Consistentwiththeprevious findings,theMMSHmodelcanprovideexperimently-matched pre-dictionsofthedropletdeformation,butthepredictionsfromthe Taylortheoryarenolongervalidforhighconfinementratios.Also, oursimulationresultsagainmatchquitewellwiththepredictions fromtheMMSHmodelandtheexperimentaldata,verifyingthatthe presentLBMcansuccessfullypredictdeformationofahighly con-fineddroplet.Fig.4showsthefinalshapeofthedeformeddroplet inthemiddlex–zplaneforCa=0.2,0.25and0.3,inwhichthe con-tourswithN_{=−0.5,0,and0.5aredisplayed.Itisobservedthat}

thedropletdeviatesfromtheellipsoidalshapetosomeextentat highcapillarynumbers,e.g.Ca=0.25and0.3.Therefore,the con-finementratioandthecapillarynumberarerestrictedto2R/H≤0.7 andCa≤0.3,andboth ofthemwillnottakelargevaluesatthe sametimeintherestofthesimulationsinordertoensurethatthe deformeddropletmaintainsanellipsoidalshape.

Then, the effectof viscosity ratio ondroplet deformation is studiedattwomoderateconfinementratios,0.4and0.5.Foreach confinementratio,threedifferentviscosityratiosareconsidered, i.e.=0.3,1,and3.Thevariationofdeformationparameterwith thecapillarynumberispresentedinFig.5.Fortheviscosityratio of0.3,ourLBMresultsagreewellwiththeMMSHmodelatlowCa, butgraduallydeviatefromtheMMSHmodelwithincreasingCa.

Specifically,ourLBMresultsliebetweentheTaylortheoryandthe MMSHmodel.Thereasonforthedeviationremainsanopen ques-tionandneedsfurtherstudy.Notethatsimilarobservationswere reportedexperimentallybyVananroyeetal.[6].Forthe viscos-ityratioofunity,ourLBMresultsagreequitewellwiththeMMSH model,andtheexperimentaldata[19]wherethedatawereonlyfor

theconfinementratioof0.5.Fortheviscosityratioof3,theobtained resultsofdropletdeformationarealittlelowerthanthepredictions fromtheMMSHmodelfortheconfinementratioof0.4,whereas fortheconfinementratioof0.5onecanclearlyobserveexcellent agreementbetweenthesimulationsandtheMMSHmodel. More-over,ourresultsdeviatesignificantlyfromtheTaylortheoryfor both2R/H=0.4and2R/H=0.5.Thus,itcanbeconcludedthatthe Taylortheorycannotdescribedropletdeformationinconfinement withtheconfinementratioassmallas0.4,whichissomehow differ-entfromthepreviousfindingi.e.thewalleffectscanbeneglected when2R/H≤0.4[4].

Theeffect of theconfinementand viscosityratios isfurther investigatedatafixedcapillarynumberof0.2.Theconfinement

ratiorangesfrom0.18to0.7.Fig.6(a)showsthevariationofthe deformationparameterfor=0.3,0.5,1,3,and5.Itisobserved thatthedeformationincreaseswithincreasingviscosityratioupto unity.Uponafurtherincreaseoftheviscosityratio,the deforma-tiondecreases.Inaddition,weobservethatincreasingconfinement ratioleadstobiggerdeformationofthedropletforallthe viscos-ityratios.Interestingly,thedropletdeformationdoesnotpresent alinearcorrelationwiththeproximityof thewalls.Itincreases almostlinearly,withdifferentrates,beforeandafter2R/H=0.4. Uptothisvaluethedeformationexhibitsaslowincrease,which becomesmorerapiduponhighervalues, especiallyforthehigh viscousdroplets.Thiscanbeexplainedtogetherwiththe corre-spondingorientationangles. Thevariationof thecorresponding

orientation angle () with the confinement ratio is shown in

Fig.6(b).Increasingviscosityratiocausesthedroplettoalignmore withtheshearflow,thusresultinginasmallerorientationangle. For≥1,boththedeformationparameter(seeFig.6(a))andthe orientationangle(Fig.6(b))revealmoresignificantvariationswith theconfinementratio2R/H≥0.4.Thissuggeststhatthewall con-finementhasa strongereffectondropletdeformationformore

viscousdroplets.For=3,5,theorientationanglesdonotreduce furtheronce2R/H≥0.3.Therefore,thecarrierfluidhasanarrower flowpassagetopassthedropletwhichleadstobuilt-upshearstress aroundthedroplet.Consequently,thedeformationrateincreases morerapidlyfor=3,5withincreasingconfinement.

Inordertoillustratetheeffectoftheviscosityratioandthe con-finementratioontheflowfieldinsideandaroundthedroplet,Fig.7

depictsthevelocity contoursandstreamlinesinthemiddlex–z

planefor2R/H={0.4,0.7}and={0.3,1,5}atCa=0.2.Notethatwe herelimitourinvestigationto2R/H=0.4and0.7sincetheimpact ofwallbecomesmoreevidentwiththeconfinementratiolarger than0.4.Inthisfigure,thethicksolidlinesrepresentthedroplet interfacewhere N_{=0,whiletheblacksolidlinesrepresentthe}

streamlines,witharrowsindicatingthedirectionofthelocalflow. Itisconfirmedthattheeffectoftheviscosityratioandthe confine-mentratioarenotonlyonthedropletshapeandorientationangle, butalsoontheflowfieldpatternsinsideandaroundthedroplet. Increasingviscosityratio leadstothestreamline patterninside thedropletvaryingfromadoublevortextoasinglevortex struc-ture.Interestingly,ananalogousvariationinthestreamlinepattern wasobservedinthepreviousnumericalstudies[20,21,34,35]when theReynoldsnumberisdecreasedfromarelativelyhighvalue.In addition,astheviscosityratioincreases,theinclinationangleof theinternalvortices(streamlines)andtheorientationangleofthe dropletbothdecrease,andtheirdifferencealsoreduces.Fortwo dif-ferentconfinementratios,thestreamlinepatternsarequalitatively similarbutquantitativelydifferent,e.g.theinclinationof stream-linesinsidethedropletissmallerattheconfinementratioof0.7, whichcorrespondstoalargerdeformationandsmallerorientation angleofthedroplet.

3.2. Dropletbreakup

Inthissection,weinvestigatethecriticalcapillarynumberof thedropletbreakupatvariousviscosityandconfinementratios. Thesimulationsareconductedfortheconfinementratiosranging from0.37to0.8atthreetypicalviscosityratios,i.e.=0.3,1and 5.AlthoughtheLBMhasbeenusedextensively tosimulatethe dropletbreakup,e.g.,Refs.[32–35],tothebestofourknowledge,no studiestodatehaveconsideredsuchabroadrangeofparameters. Thisispartiallybecauseoftheunsolvedissuesoftheexisting mod-elsthemselves,suchasartificiallyenlargedinterfacethicknessand case-dependentmobilityinthephase-field-basedmodel,high spu-riouscurrents,numericalinstabilityintheinterparticle-potential model.

Fig.8showstheeffectoftheconfinementratioonthedroplet breakupattheviscosityratiosof0.3,1,and5,whicharerepresented bycircles, squares, and diamonds,respectively. Empty symbols areusedin thecaseswhere nodropletbreakupisobserved.To distinguishdifferentmodesof breakup,we usehalf-filled sym-bols to denote the binary breakup, and full-filled symbols to denotethemultiplebreakup,wherethedropletsplitsintothree ormoredaughterdroplets. Inaddition,thebreakupisdepicted at the moments before and after it occurs, for a number of cases.Followingtheaforementioneddefinition,thecritical cap-illarynumber,Cacr,liesbetweentheemptyand filled symbols.

Theconfinementratiohasdifferenteffectsonthecriticalcapillary number, depending on the viscosity ratio. For the low viscos-ityratio,i.e.=0.3,thecriticalcapillarynumberisvariedfrom 0.4≤Cacr≤0.5 at2R/H=0.37,to0.5≤Cacr≤0.6at2R/H=0.5 and

finallyto0.55≤Cacr≤0.625at2R/H=0.625,suggestingthat Cacr

increaseswithincreasingconfinement.Contrarytothelow vis-cosityratiocase,Cacr decreaseswithincreasingconfinementfor

thehighviscosityratio,i.e.=5.Specifically,thecriticalcapillary numberdecreasesfrom0.5≤Cacr≤0.6to0.3≤Cacr≤0.35 when

oneincreasestheconfinementratiofrom0.55to0.8.Fig.9shows thesnapshots ofthedropletbreakup processforCa=0.35,=5 and2R/H=0.8.Thedropletdeformscontinuouslyandeventually breaks up into two equal-sized daughter droplets, as opposed to the unbounded case where the high viscosity ratio droplet (>4)doesnotbreakregardlessofthecapillarynumber.Also,it isinterestinglyobservedinFig.8thattheconfinementratiohas smalleffectonthecritical capillarynumber for=1,which is

around0.4.Thesefindingsagreewellwiththeprevious experi-mentalandnumericalstudies[3,24].However,inthesereported works[3,24]throughalargenumberofexperimentsand/or simu-lations,itwasfoundthatfor=1thatthecriticalcapillarynumber firstdecreasesandthenincreaseswiththeconfinementratio.The minimumcriticalcapillary numberoccursat2R/H=0.5.To cap-ture this subtle variation of the critical capillary number with theconfinementratio,alargenumberofsimulationsarerequired whichwillbestudiedseparatelybyexploitingmassivelyparallel computing.

Eventhoughtheoverallagreementisquitesatisfactorybetween thepresentresultsandthoseobtainedbyJanssenetal.[3],some quantitative differences are also noticed. For example, Janssen etal.[3]experimentallyfoundthatbinarybreakupisexpectedto occurforadropletwith=1.0atCa=0.45and0.37≤2R/H≤0.45, whereasinoursimulationsternarybreakupisobserved.Fig.10

showsthesnapshotsofthedropletbreakupprocessforCa=0.45, =1 and 2R/H=0.45. Asthe droplet is stretched by the exter-nalshear flow,it first deforms toanellipsoidal shape (t˙ =4). Themajoraxislengthens while theminoraxisshrinks,leading totheformationofawaistnearthecentreofdropletatt˙ =16. Thedropletisthen elongatedtoa ‘dumbbell’shape, andatthe sametimethebulbsattheendofdropletachieveastableshape (t˙ =20).Att˙ =25,twovisiblenecksareformedbetweenthe centralportionof thedropletand bulbs.The dropletcontinues to thin and eventually breaks up into three parts (i.e., ternary breakup) with two daughter droplets having much larger size (t˙ =27).

Inaddition,thepreviousexperimentalstudyindicatedthatthe lowestCacrforallofviscosityratios(0.3≤≤5)andconfinement

ratios(0.1≤2R/H≤0.9)isaround0.4[3],butinoursimulations thelowestCacrisfoundbetween0.3and0.35,whichcanbeclearly

seeninFig.8forthecaseof=5and2R/H=0.8.Thereasonforthese differencesremainsunclear,andislikelytobetheresultofa com-binationoffactors,includingtheexperimentalerrorsandmodel accuracyindescribingtheinterfacialdynamics.

4. Conclusions

The influence of capillary number, viscosity ratio and con-finement ratio have been systematically studied for droplet deformationandbreakupinasimpleshearflow.Increasing con-finementleadstoanincreaseindropletdeformationforawide rangeofviscosityratios,andthedropletwiththesameviscosityas thecarrierfluidpresentsthelargestdeformationatallconfinement ratios.Theorientationangleofthedeformeddropletdecreaseswith increasingviscosityratio.Itisobservedthatthepresenceofwalls hasamore significantimpactonbothdroplet deformationand orientationanglewhentheconfinementratioincreasesto0.4or higher.Theflowfieldinsideandaroundthedropletshowsthat con-finementandviscosityratioscanleadtoaperplexpatternwhich mayaffectdropletbehaviour.Finally,thecriticalcapillarynumber, abovewhichthedropletbreakupoccurs,isinvestigatedforabroad rangeofflowconditions.Dependingonthevalueofviscosityratio, theconfinementratiohasdifferenteffectsonthecriticalcapillary number:astheconfinementincreases,thecriticalcapillarynumber increasesfor<1butdecreasesfor>1;andfor=1thecritical capillarynumberiskeptatavalueofaround0.4,regardlessofthe confinementratio.

Acknowledgements

TheresultswereobtainedusingtheUKEPSRCfunded ARCHIE-WeSt High Performance Computer (http://www.archie-west.

supported by EPSRC under grant no. EP/M021475/1 and EP/L00030X/1. H.L. gratefully acknowledges the financial sup-ports from the National Natural Science Foundation of China (Grant No. 51506168) and the Thousand Talents Program for DistinguishedYoungScholars.

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NikolaosIoannouobtaineda5-yearDiploma,equivalent toMasterofSciencedegree,inMechanicalengineering fromtheSchoolofMechanicalEngineering,National Uni-versityofAthensin2010.In2014,heobtainedaMasterin ComputationalMechanicsfromthesameinstitution.He isaPh.DcandidateunderthesupervisionofProf. Yong-haoZhangintheMechanicalandAerospacedepartmentof theUniversityofStrathclyde,andamemberoftheJames WeirFluidsLaboratorysinceDecember2012.Hisresearch involvesinterfacialdynamicsatthemicroscaleand non-Newtonianfluids.

HaihuLiuisanAssociateProfessoratXi’anJiaotong Uni-versity,China.HereceivedhisBEngandMEngdegreesin FluidMachineryandEngineeringfromXi’anJiaotong Uni-versityin2004and2007,respectively.Hethenjoinedthe JamesWeirFluidsLaboratoryasaPhDstudentin2007. AfterawardedhisPhDin2011,heworkedasa post-doctoralresearchassociateintheDepartmentofCivil& EnvironmentalEngineeringintheUniversityofIllinoisat Urbana-Champaign,USA.InSeptember2013,hejoined UniversityofStrathclydeasaLecturerandthenbecame anAssociateProfessoratXi’anJiaotongUniversityChina in2014.