Jet

surface

LaurentDuchemin,St´ephanePopinet,ChristopheJosserand

andSt´ephaneZaleski

Abstract

Westudynumericallybubblesburstingatafreesurfaceandthesub-sequentjetformation.TheNavier-Stokesequationswithafreesurface

andsurfacetensionaresolvedusingamarker-chainapproach.Differenti-ationandboundaryconditionsnearthefreesurfacearesatisfiedusingleast-squaresmethods.Initialconditionsinvolveabubbleconnectedtotheoutsideatmospherebyapreexistingopeninginathinliquidlayer.Theevolutionofthebubbleisstudiedasafunctionofbubbleradius.Ajetformswithorwith-outtheformationofatinyairbubbleatthebaseofthejet.Theradiusofthedropletformedatthetipofthejetisfoundtobeaboutonetenthoftheinitialbubbleradius.Aseriesofcriticalradiusesexist,forwhichatransitionfromadynamicswithorwithoutbubblesexist.Forsomeparametervalues,thejetformationisclosetoasingularflow,withaconicalcavityshapeandalargecurvatureorcuspatthebottom.ThisiscomparedtosimilarsingularitiesinvestigatedinothercontextssuchasFaradaywaves.

1Introduction

Bubblesburstingatthewatersurfaceareafamiliareverydayoccurrence.Theyalsotakepartinimportantprocessesoftransportandexchangeacrossliquid/gasinterfaces,causedbytheejectionofjetsandvariouskindsofsmalldroplets.Theseareinvolvedinthetransferofheat,massandvariouscontaminantsbetweentheoceansandtheatmosphere[1].Indeed,breakingwavescausetheformationofalargenumberofbubblesbeneaththewaterlevel.Theefficiencyoftheresultingmasstransfer,includingthetransferofCOdependsontheinitialpropertiesoftheejecteddroplets(size,initialvelocity).

1

Thephenomenaproducingaerosolsduringtheburstingofabubbleareoftwokinds:thefirstistheruptureofthefilmseparatingthebubblefromtheatmo-sphere.Thisfilmatomizationcanproduceseveralhundreddropletsofaroundamicrometerindiameterwhichprobablyrepresentalargefractionofthetransfers[1].Sincethescalesinvolvedduringthisruptureareoftheorderof100nm,aphysicaldescriptionisoutsidethescopeofcontinuumfluidmechanics.Indeed,long-rangemolecularforcessuchasVanderWaalsforcesorelectrostaticrepul-sionmustbetakenintoaccount[2].

Thesmallcavityremainingafterthefilmrupturecollapsesundertheeffectofbothsurface-tensionandbuoyancy.Thiscollapsegivesbirthtoanarrowverticaljetwhicheventuallybreaksintooneorseveraldroplets(seeFig.1).Thisphe-nomenonconstitutesthesecondaerosolproductionprocessandistheprincipaltopicofthispaper.

Theseaerosolsareofadifferentkind:theyareejectedvertically—whichisnotthecaseforfilmaerosols—andtheirdiameterisaboutonetenthofthesizeoftheinitialcavity,i.e.about100mforatypicalbubbleradiusofonemillimeter.Dependingontheirmassandinitialvelocity,thedropletswilleitherfallbackintowaterorevaporate.

Thetopicofthispaperistheinvestigationofthebubbleevolutionaftertheinitialfilmrupturing,includingthejetformation.AnumericalmethodsolvingtheNavier-Stokesequationsanddescribingthefreesurfacewithhighprecisionisused.Previousnumericalstudiesofthesephenomenahavebeenmadepostulatingmostlyinviscidfluids;however,amodifiedboundaryelementmethodtakingintoaccountsmallviscouseffectswasalsoused[2,3].ANavier-Stokessimulationwasshownin[4],withaVOF-typemethodinaregimewherethebubbleisverydeformed.

Inmostpreviousstudiestheeffectoffilmatomizationonjetbirthwasassumedtobenegligible.Fewcomparisonsweremadewithexperimentaldata.Someex-perimentalstudieswerealsoconductedtomeasurequantitiessuchasjetvelocity[5,6,7],sizeofthefirstejecteddroplet,heightatwhichthedropletdetachesfromthejet,orheightreachedbythedroplet.Theseexperimentsarefairlydif-ficulttoconduct,becauseofsurfacecontaminationwhichmodifiessignificantlythefree-surfaceboundaryconditionandthesurfacetensioncoefficient.

Asournumericalresultswilldemonstrate,thejetformationisinmanycasessingularandself-similar.Singularjetsformingatafreesurfacehavealreadybeenobservedandstudiedindifferentcontexts.Indeed,inthebubble-burstingprob-lemaswellasinseveralotherfree-surfaceflows,oneobservestheformationofaconicalcavity,withaveryhighcurvatureorcuspatitsbase.Insomecasesa

2

Figure1:Jetproducedbythecollapseofasphericalcavity.TheenddropletwilleventuallydetachduetotheSavart-Plateau-Rayleighinstability.

3

smallbubbleistrappedatthebottomofthecavity.Athinnarrowjetsubsequentlyformsinaself-similarmanner.ThisphenomenonwasobservedexperimentallyinFaradaywavesbyLonguet-Higgins[8]andLathrop[9],inthedevelopmentofthejetinsideabubblecontainingasinkflowinthenumericalstudyof[10].Forburstingbubblestheconicalcavitymaybeseenintheexperimentsandsimula-tionsbutthesingularcharacterofthejetformationhasnotbeeninvestigatedtoourknowledge.Thephenomenonmayalsobeseeninthecavityformedbyfallingraindrops[11,12,13,14].

TheevolutionoftheconicalcavityhasbeenstudiedbyLonguet-Higgins[12]asaspecialcaseofafamilyofhyperbolicsurfaces:conicalsurfaceswereshowntobeaspecialcaseofthehyperbolicsurfacesofRef.[8].Theseconicalsurfacesarepreservedbythevelocitypotential

(1)

whereisthesphericalradialdistanceandnorthpole,whichyieldsthevelocityfield

thepolaranglemeasuredfromthe

(2)

whereisanarbitraryfunction.Indeedanyconicalfreesurfaceinthisflowremainsconical.Forpositivethecavityopensintimeasintheexperiment.Ofcoursetheactualflowisnotexactlyconical.Thebottomoftheconeisrounded,andoscillatesinshapeascapillarywavesconvergetowardsthebottomoftheconicalsurface.Atsomeinstantintimethebottommaydevelopacusp,fol-lowedbyjetformation.Thisprocessisobviouslysingularatleastforsomevaluesoftheparameters,butthereisnoagreementamongtheabovecitedpublicationsontheexactnatureofthesingularity.

Indeedonemayinquireintothespecificscalingformofthesingularity.TheEulerequationswithoutsurfacetensionandgravitywillinprincipleadmitself-similarsolutionsoftheform

(3)

whereisanarbitraryscalingfunctionandisthesingularitytime.Thesolutionmaybevalidbeforeand/orafterthesingularitytime.Theexponentsshouldsatisfy

.IndeedwiththisconditionallthetermsintheBernoulliequation

balance.Howeverwhensurfacetensionisadded,theonlywaytoformaself-similarsolutionthatbalancesinertiaandsurfacetensionisbyselecting.

4

Thisisbecausetheonlylengthscalethatcanbebuiltiswhereisthesurfacetensionandthedensity.Thenthesimilarityvariableisandtheflowvelocitydivergeslikenearthesingularityforafixedvalueofthesimilarityvariable.ThisideaisatthebasisofexponentsfoundforinstancebyMiksisandKeller[15].Thistypeofscaling,wasappliedbyZeffetal.[9]toobservationsofjetformationinFaradaywaves.Theleadingordertermforthevelocitypotentialisoftheform

(4)

However,aseriesofalternatetheoriesforsingularfree-surfaceflowsandinparticulartheconicalcavityandjetformationwasproposedbyLonguet-Higgins.HehasshownthatthetypeofflowdescribedbyEq.(2)hadadivergentvelocity

thusadivergenceforafixedvalueoftherealwith

(unscaled)distance[12].Inthissolutionthescalingisnotfixedbyabalancewithsurfacetension.Instead,surfacetensionisaddedasaperturbationtotheconicalsolution,intheformofasinkflow[12].TheLonguet-Higginssolutionyieldsananglefortheconicalcavityof5,ingoodagreementwiththenumericalobservationsof[11].Anotherself-similarsolutionforjetformationwasfoundnumericallyby[10]obtainingyetanotherscaling,forthecaseofjetformationinsideabubble.Thepotentialisthenapproximatedby

(5)

Thisarticleisorganizedasfollows.Wefirstdescribethegeneralcontextofthisstudy,thenon-dimensionalnumberscontrollingtheproblemandthescalinglawsdeducedfromdimensionalanalysis.Wethenbrieflyintroducethenumericalmethodweuseanditsmainadvantages.Afirstcomparisonwithexperimentalprofilesispresented.Finally,adetailedparametricstudyisconductedusingasimpleinitialshapeforthecavityandneglectinggravity.Wemeasurethevolumeofthefirstejecteddroplet,thevelocityofthejetandthemaximumpressureen-counteredontheaxisofsymmetryanddiscusstheresults.Insomecircumstances,atinybubbleisformedatthebaseofthejet.Theself-similarflowoccurringwhentheconicalcavityandthecuspformisinvestigated.

2Initialconditionsandexpectedscalinglaws

Giventhesmallsizeofthebubblesweareinterestedin(diameterisaroundonemillimeter),someassumptionscanbemaderegardingtheparametersgoverning

5

jetbirth.Thefirstideaistosupposethatthecavityismotionlessattheinitialtime.Experimentshaveshownthat,evenintheabsenceofsurfactants,thebubblecanstayatthefreesurfaceinaquasistaticequilibriumforafewseconds[16].Thebubbleisthenseparatedfromtheatmospherebyathinliquidfilm,thecavitybeingsubjecttosurfacetensionandbuoyancyforces.Amodelforthisstaticconfigurationisamoreorlessdeformedbubbleadjacentoverpartofitssurfacetoafilmofnegligiblethickness.Thisconfigurationmaybecomputed,orobtainedfromtheexperimentaldataasinthecasereportedin[7].

Whenthefilmreachesacriticalthickness(about100nm)afterdrainingslowly,itbreaksmoreorlessrapidly(dependingonthepresenceofsurfacecontami-nants).Itisthenpossibletorunsimulationsbytakingthecurrent,staticconfigu-rationandremovingthethinfilm.Whilewedothisinonecase,thedrawbackisthatasharpcornerexistsattherimoftheneckorjuncturebetweenthefilmandthebulkliquid.Thesmalllengthscalesinvolvedmaycreatenumericalconver-genceproblems.Moreover,asweshowbelow,smalllengthscalesaregeneratedindependentlyofinitialconditionsbythesteepeningofcapillarywavesandjetformation.Keepingthesmalllengthscalesintheinitialconditionsmakesitmoredifficulttoobservetheintrinsicallygeneratedsmallscales.Wethusdecidedtodrasticallysmooththerimoftheneck.Inmostcalculations,theinitialshapewasdefinedasfollows.Asphericalcavityisseparatedfromtheatmospherebyacircularhole,theborderoftheholebeingacircularrim(seeFig.2).

Thecollapsebehaviordependsonlyonfourphysicalparameters:thekine-maticviscosity,,andtheaccelerationduetogravity.Outofthefourphysicalparametersonlytwolengthscalescanbedefined,thecapillarylength

andthetheviscous-capillarylength.Inpurewa-termmandmrespectively.Iftheradiusofthebubble

,capillaryeffectsarepredominantcomparedwiththegravityeffects;if,viscouseffectsareexpectedtobenegligiblecomparedtothecapillary

ones.Forthephenomenonisdominatedbysurfacetensionandinertia.

Wealsodecidedtoneglecttheeffectofgravitywhichisacorrectapproxima-tionfor.Therefore,onlytheOhnesorgenumbergovernsthephenomenonanddimensionalanalysisgivesvelocityintheform

Figure2:Theinitialconfigurationinthe“largerim”case.ThegridisaCartesiangrid.

7

Whenever,wealsoexpectthatviscosityplaysnorole.Theonlywaytoeliminateviscosityistosupposethatthefunctionhasafinite,non-zerolimitwhengoestozero.Thenon-dimensionalvelocityofthejetthenbehaveslike

(7)

where.

Similarargumentsleadtoascalinglawforthenon-dimensionalpressureoftheform

(8)

where

.

3Numericalmethod

Thechoiceofthenumericalmethodisconditionedbythetermsweneedtosolveaccurately.Inourproblem,thefirsttermofinterestissurfacetension:beingthemaindrivingforceintheparameterrangeweconsider,itisimportanttomodelitcorrectly.Giventhelargedensityratiobetweenwaterandairwecanmoreoverassumethattheinfluenceofthegasphaseisnegligible.

Accordingtothesetwoassumptions,weusedanumericalmethodwhichsolvesthefullaxisymmetricNavier-Stokesequationsinafluidboundedbyafreesurfacewhileallowinganaccuratedescriptionoftheinterfacialtermssuchassur-facetension.Thismethodhasbeendocumentedelsewhere[17,18]andhasbeenshowntoproduceaccuratequalitativeandquantitativeresultswhencomparedwithboththeoreticalandexperimentaldata.

Inshort,aregularCartesianfixedgridisused.Masslessparticles(markers)advectedbytheflowdefinethepositionoftheinterface.Linkedbycubicsplines,theydescribeaccuratelythegeometryofthefreesurface.Forcellswhicharenotcutbythefreesurface,aclassicalfinite-volumeschemeisapplied.Forthecellsinthevicinityoftheinterface,finitedifferencescannotbecomputedsincevelocitiesarenotdefinedinthe“gas”phase.Therefore,anextrapolationofthevelocityfieldnearthefreesurfaceontheothersideisnecessary.Thisextrapolationmusttakeintoaccounttheboundaryconditionsonthefreesurface(inparticularlythenullityofthetangentialstress).Thisisdonebyusingaleast-mean-squareprocedureconstrainedbytheconditionofvanishingtangentialstress.Comparisonswith

8

theoreticalresultsshowthatthisapproachgivesanaccuratedescriptionoftheviscousdissipativetermsassociatedwiththeboundaryconditions.

Thepressureontheboundaryisobtainedasfollows.Thelocalcurvatureisestimatedfromthesplinereconstruction.Thelocalnormalviscousstressisestimatedfromtheaboveleast-squaresprocedure.Thenthepressureisobtainedfromthenormal-stressboundarycondition.ThepressureontheboundaryservesasaboundaryconditionforthePoissonequationforthepressure.Thisequationisinturnsolvedusingamultigridalgorithm.

grid,exceptthecomparisonMostcomputationshavebeenmadeona

withtheexperimentalprofilesfromMacIntyre,whichhasbeenmadeonagridandsomeselectedcomputationswhichwererefinedtogrids.

4Comparisonofthenumericalresultswithexperi-mentalprofiles

Wehavefirstinitializedthecalculationwitharealisticshape,andtakenintoac-countallthephysicalparameters,i.e.capillarity,viscosityandgravity.ThegoalwastocomparetheresultswithaseriesofshapespublishedbyMacIntyre[7].TheinitialshapeofthefreesurfacehasbeenobtainedfromthefirstimagegivenbyMacIntyre,justafterthefilmrupture.

Fig.3illustratestheexperimentalandthecomputationalresults.Thenumer-m/s,kg/s,kg/mandtheicalparametersare:

volumeofthebubbleisthesameastheonegivenbyMacIntyre:l.Thecomputationaltimeisaboutonedayonthegrid.Theoverallagreementisverysatisfactory.Inparticularcapillarywavesarewelldescribed,incontrasttotheearlierpublishedresultsusingboundaryintegralmethods[3,2].Webelievethatthislackofcapillarywavesisduetothestrongsmoothingneededtoavoidnumericalinstabilitiesinboundaryintegraltechniques(andprobablyalsotoaninsufficientspatialresolution,whichisalsolimitedbynumericalstability).Inourmethod,real,molecularviscosityispresentandthefinegridweuseallowsinprincipletosolvethesmallspatialscalesofthecapillarywaves.

ThetimeintervalbetweenimagesisthesameastheonegivenbyMacIntyre,i.e..Adifferenceintimebetweenprofilescanbeseen,eveniftheshapeisverysimilar.ApossibleexplanationisthepresenceofsurfacecontaminantsintheMacIntyreexperiment.Thesecontaminantscouldchangethesurfacetension,evenmodifyitsvaluelocally,thereforechangingthebehaviorofthefreesurface

9

Figure3:Timesequenceofthejetformationinabubbleburstingatafreesurface.Top:experimental[7]andbottom:computationalresults.Profilesare

apart.

10

throughgenerationofMarangonicurrents.Theycouldalsomaketheinterfacepartiallyorentirelyrigid,changingthefree-surfaceboundarycondition.

Figure4:Vorticityisolinesduringthecollapseofthebubbleforthesamecondi-tionsasinFig.3.

11

Figure5:Sameaspreviousfigure.

Figure6:Sameaspreviousfigure.

12

Figure7:Sameaspreviousfigure.

13

SolvingthefullNavier-Stokesequations,wehaveaccesstovorticitywhichcan,aswewillseelater,haveanimportanteffectevenonverysmallstructuresinalowviscosityfluidsuchaswater.Fig.4showsthevorticityisolinesduringthecollapseofthecavity.Thevorticityisconfinedtoathinboundarylayerbeforethejetbirth.Lateronhowever,avorticityconeisentrainedbelowthejetandtheshearstressthereiscomparabletothatinthenarrowjet.Thisdetachmentofvorticityillustratestheformationofadownwardjet,alreadyobservedbyBoulton-StoneandBlakewiththeirmodifiedboundaryintegralmethod[3,2].

5Resultsoftheparametricstudy

Asetofcomputationshavebeenmadeforradiibetweenand(

)withtheinitialshapedescribedabove.

TheevolutionoftheprofilesisverysimilartothatshownonFig.3.Aconicalcavityformswithatrainofcapillarywavesconvergingtotheaxis.ThenumberofcapillarywavesdependsstronglyontheOhnesorgenumber:thehigherthisnumber,thehigherthenumberofcapillarywavesconvergingtothebaseofthecavity.Fig.8showsalarge-radiuscasewithalargenumberofwaves(seealsoFig.11).Insomecases,especiallynearthejetbecameverythin(Fig.9)andthelocalradiusofcurvaturesmallerthanthegridsize.Thecalculationthenbecomesinaccurateandhastobestopped.

Forsomeparametervaluesweobserveantendencytotrapabubbleontheaxisofsymmetryjustbeforetheformationofthejet.Wehavesearchedsystem-aticallyforbubbleentrapment.Therearetwocompetingchangesofshape:thejetformationisheraldedbyachangeofcurvatureatthebaseofthecavity,whilethebubblepinchingisprecededbytheformationofanoverhangintheinterface,i.e.theheightoftheinterfacebecomesmulti-valued.Thusourcriterionforincipientbubbleformationisasfollows:(a)Theheightbecomessteep,thenmulti-valued,and(b)thecurvatureatthebaseremainspositive.ThisisonlyanindicationthatabubblewillbetrappedbeforethejetformsasshownonFig.10,butweneedsuchacrudecriterionbecausethebubblesareverysmallforthekindsofgridswehave.Wefoundafirstbubbleentrapmentregionfor

,thesecondonebetween.Other

suchregionsathighervaluesofarelikely,butdifficulttoobservenumer-ically.Oneindicationistheexistenceoflargetrainsofcapillarywavesatlarge

asshownonFig.11.

Thetopologyoftheinterfacechangeswhenabubbleistrapped.Thispinching

14

0.5

0.4

0.3

0.2

0.1−0.2

Figure8:Capillarywavesfor

.As

15

1.0

0.0−0.5

isasingulareventakintothepinchingofagascylinderbytheSavart-Plateau-Rayleighinstability.Weshallcallitapinchingsingularitytodistinguishitfromotherfreesurfacesingularities.Topursuethecalculationnumericallybeyondapinchingsingularity,oneshouldinprincipleperformsurgeryonthemarkerchainandcontinuethesimulation.Thisishoweverdifficultbecausetheproblemslightlychangesinnature:thepressureinsidethesmalltrappedbubblecannotbesettoatmosphericpressurebutshouldinprincipledependonthebubblevolumethroughsomeequationofstate.Thischangesmarkedlythenatureofthecalcu-lation.Moreoverthetrappedbubblesareextremelysmallandverydifficulttoresolvewithoutmeshadaptation.Thusinmostcaseswecontinuedthesimulationwithoutmarkersurgery.Whenthetrappedbubbleisverysmall,themarkerchainreorganizesitselfspontaneouslyandthecalculationproceeds.Insomecases,asintherightmostbubbleentrapmentregion,itseemsthattheeffectonthedynam-icsissmall.Inothercases,asintheleftmostentrapmentregion,thecalculationhastobestoppedorprovidesunreliableresultswhichwereremovedfromthequantitativeanalysesbelow.

Wehaveredoneallthecalculationsforadifferentinitialcondition.Theover-allconfigurationisthesameasonFig.2buttherimthicknessishalved.Alltheabovequalitativeresultsareidentical.Inparticular,wedonotobserveanysteepeningofthecapillarywavesorthinnerjetsaswereducetherimsize.Thisisaclearindicationthatthesmalllengthscalesweobserveformspontaneously,independentlyofinitialconditions.

5.1Jetvelocity

Afirstquantityofinterestisthevelocityofthejet,ortheejectionspeedofthefirstdrop.Fig.12showsthenon-dimensionalvelocityofthejet,measuredwhenthetopofthejetreachesthemeanwaterlevel.CirclesymbolscorrespondtothelargerrimthicknessasonFig.2whilesquaresymbolscorrespondtothinnerrims.Apartfromaverticalshift,themeasuredvelocitiesareverysimilar.Thisshiftmayinpartbeexplainedbythefactthatwemeasurethejetvelocityatthemeanwaterlevelforbothcases,whichisatadifferentdistancefromthebaseofthetwocavities.

andtimestheviscous-capillaryForalargerangeofradii(between

length),thenumericalresultsareingoodagreementwiththeinviscidscaling.Forsmallradiithevelocitystartsdecreasingasdecreases.Forthesmallestradiiwehaveinvestigatedthecavityrelaxestoaflatsurfaceshapewithoutjetformation.

17

Figure10:Beginningoftheentrapmentofabubblebythecollapsingcavity,for

(abubble).

18

10

5velocity on the axis0−50

Figure11:Thevelocityoftheinterfaceontheaxisfor.Theoscillationscorrespondtothearrivalofatrainofcapillarywaves.Forthislarge

capillarywavesarenumerousandofshortwavelength.Theveryvalueof

largeexcursioninvelocitymaybeduetotheexistenceofafurtherbubbleentrap-mentregion,howevertheverysmallscalesinvolvedmakenumericalresolutiondifficult.

19

10

1

10

0

10

−1

10

−2

10

Figure12:Non-dimensionaljetvelocityasafunctionofthenon-dimensionalbubbleradius.Thetworegionsbetweenverticalstraightlinescorrespondtotheradiiforwhichabubbleistrappedatthebaseofthejet.

20

TheregionswherebubblesformatthebaseofthejetareindicatedasverticallinesonFig.12.Intheleftmostregion,around,forthereasonsdiscussedabove,thereisagapindatapoints.Itisthuspossiblethatmuchhigherjetvelocitiesmaybereachedinthatregion.

5.2Maximumpressureontheaxisofsymmetry

Wehavecomputedthemaximumpressureontheaxisofsymmetrywhenthejetreachesthemeanwaterlevel.Fig.13showsthispressureandafitin.Oncemore,thenumericalresultisingoodagreementwiththescalinglawfor

andtimestheviscous-capillarylength.Wealsoremarkaradiibetween

smalljitteraboutthestraightlineontherighthandsideofthecurve,perhapsasaresultofthesingularbehaviorinthebubbleentrapmentregion.Noteagainthatinthelefthandsideofthecurvewecouldnotreliablycalculatepressure.

5.3Radiusofthefirstejecteddrop

ExperimentaldataobtainedbySpieletal.[6]tendtoshowthattheradiusofthefirstejecteddropisaboutone-tenththeradiusoftheinitialbubble.

Wehaveobtainedthisradiusfromthenumericalsimulationsasfollows.Thecomputationstopswhenthejetthicknessreachesthesizeofonecomputationalcell.Thejetrupturewilloccursoonthereafter.Thevolumeenclosedbythefree-surfacebetweenthispointofminimumthicknessandthetipofthejetisthenagoodapproximationofthevolumeoftheejecteddroplet.Theequivalentradiusisdefinedastheradiusofasphericaldropletwiththesamevolume.Fig.14shows.Forlargeweobtainalinearfitwhichisconsistentwiththeexperimentallyobservedvalueof.Thislinearbehaviorisconsistentwiththeviscosity-independentregimeofEqs.(7,8)inwhichtheonlylengthscaleis.Ontheotherhand,thereisalargefractionofthedatawherethisregimedoesnotholdandtheejecteddropradiusismuchsmallerthan.Noticeagainthegapinvaluesaround.Therethejetwastoothintobewell-resolvednumerically,andtheactualdropletsizemaybemuchsmaller.Varyingtheinitialconditionhaslittleeffect,exceptatsmallradiiwherethethinnerrimleadstoalargerdroplet.

21

10

0

10

−1

10

−2

10

−3

10

−4

10

−5

10

−6

10

1

0.1

0.01

10

6Singularjetformationbycurvaturereversal

Theformationofathin,high-velocityjetinandaroundthefirstbubbleentrapmentregionleadstosuspecttheexistenceofasingularity.Thescaling(3)discussedaboveyields

(9)

(10)

thedistancetotheaxisofsym-whereisthesurfaceelevationand

metry.Werescaledtheradialandverticalcoordinatesofthesurfacepointsby

for.Wedeterminedbyfittingtwooftherescaled

profilesontooneanother.TheresultsareshownonFig.15.

Alltheprofileshavebeentranslatedverticallyinorderforthepointontheaxisofsymmetrytobeatthesameverticalcoordinate.Therescaledprofilessuperimposewellatsmallvaluesofthesimilarityvariable.Theshapeoftheprofilescloselyresemblestheexperimentalandnumericalprofilesinothertypesofflow[11,14,9].Howeveratalargedistancefromthesingularitytheconeangleisabout.ThisshouldbecomparedwiththeangleofthecavityseenintheMcIntyredatashownonFig.3.There,onprofile6wemeasureanangleof,asmalldifferencewithourcalculations.Incontrast,theotherphysicalprocessesdiscussedintheintroductionyieldrelativelylargerangles.

Thefiniteviscosityshouldalsointroduceadiscrepancywiththetheoreticalsimilaritysolution.Itseemshoweverthatitseffectsaresmallinthatcase.

Inreference[9]itwasshownthatforFaradaywavestherewasaconnectionbetweenbubbleentrapmentandsingularities.Inourcasethepictureseemsdif-ferent.Theself-similarsolution(10)isobservedintheentirefirstbubbleentrap-mentband.Ontheotherhand,thissolutionisnotseeninoraroundthesecondbandofbubbleentrapment,whereweshouldinprinciplealsohaveasingularity.Howevertheshapeoftheinterfaceisverydifferentinthatcase(Fig.16)andasuperpositionusingtheaboverescaledvariablescouldnotbefound.ApossibleexplanationisthattheconicalcavityassociatedwiththesingularityispresenthereonlyonthelargescaleasseenonFig.10.Theconvergenceofsmall-scalecap-illarywavesisnotablebyitselftogenerateaself-similarconicalflow.Thustheconicalflowwouldbeatleastasimportantasbubbleformationinproducingthesurface-tensiondrivenself-similarscaling.

Atantalizingpossibilityistheexistenceoffurtherbandsofbubbleentrapmentandsingularitiestotherightofthesecondband.Whilebubbleentrapmentisobservedinsomecases,detailsofthedynamicsarenotwell-enoughresolved.It

24

0.3

0.2

0.1

0

−0.1

−0.2

Figure15:Comparisonbetweensuccessiveprofilesmadenon-dimensionalusingthescalingdescribedinthetext.Left:unrescaledprofiles.Right:rescaledprofiles.Theretheopeningangleofthecavityisaround.Sincethenumericalmethodusesadaptivetime-stepping,theprofilesarenotseparatedbyequaltimeintervals.

25

0.3

0.28

0.26

0.24

0.22

0.2−0.05

−0.03−0.010.01

Figure16:Shapeoftheinterfaceontheedgeofthesecondbubbleformationregionat.InthatcasearescalingofthetypeshownonFig15couldnotbefound.AsinthepreviousFigureprofilesarenotseparatedbyuniformtimeintervals.

26

islikelythatbubbleentrapmentandcuspsingularitiesarerelatedtotheamplitudeoftheconvergingcapillarywaves.Asviscosityisreduced,evermorecapillarywavesareobservedtoconvergeontheaxis.Forverylargevaluesof,waveshavingbothshortwavelengthandsmallamplitudeareformed.

27

7Conclusions

Wehavepresentedanumericalstudyoftheburstingprocessofbubblesatafreesurface.Theschemeusedwasbasedonanaccuratedescriptionofthefreesurfacewiththehelpofamarkerschain.Thismethodhasshowngoodcapabilitiestoresolvesmallcapillarywaves.Thelargescalefeaturesofthedynamics,thepres-sureandfinaldropletradiusmaybepredictedwithaccuracy,exceptnearthefirstbubbleentrapmentregionnear.Thepredictionsarequantitativelyinagreementwithexperiment:theangleofopeningofthecavityissimilartotheangleobservedintheexperimentsofMacIntyreandthesizeofthedropletatthetipofthejetisclosetotheexperimentallyreportedsize.Themeasurementsofjetvelocitynearshowasurprisinglylargevelocity.Theinterfaceshapescaleswithacharacteristiclength

predictedbythebalanceofsurfacetensionandinertia.Theshapeoftheinterfaceresemblesshapesfoundinotherjet-formingflowsandcuspsingularities,buthasquantitativedifferencessuchastheopeningangleoftheconicalcavity.

Theconnectionofthisscalingwithbubbleentrapmentislessclear.Wefoundthescalinginawideregion.Theoccurrenceofselfsimilarflowandanapprox-imatesingularityisnotconnectedtotheexactboundaryofabubbleentrapmentband.Wealsofoundbubbleentrapmenttransitionswhichwerenotassociatedtothescaling.Finallytheangleoftheconicalcavityagreeswiththeexper-imentaldataforburstingbubbles,butnotwiththeanglesseenorpredictedinotherflows.Thisindicatesthatothertypesofsingularities,correspondingtodifferenttopologiesorinitialconditions,maybeobserved.Furtherworkshouldexploreindetailthenatureofthesesingularitiesusingforinstancemeshrefinement.

Alsoofinterestwouldbeastudyoftheinfluenceoftheinitialshapeofthebubble.Wehaveshownthatafactoroftwochangeintherimthicknesshadnoqualitativeeffect,andverylittlequantitativeeffectonthecollapseprocess.Howeverotherchangesintheinitialconditionmaycauseachangeinthepositionofthevarioussingularities.Inotherwords,foragivenradius,itwouldbepossibletoreachasingularitybychangingtheshapeofthebubble.

28

References

[1]M.Coantic.Masstransfertacrosstheocean-airinterface:smallscalehydro-dynamicandaerodynamicmechanisms.PhysicoChemicalHydrodynamics,1:249–279,1980.[2]J.M.Boulton-StoneandJ.R.Blake.Gasbubblesburstingatafreesurface.

J.FluidMech.,254:437–466,1993.[3]J.M.Boulton-Stone.Theeffectofsurfactantsonburstinggasbubbles.J.

FluidMech.,302:231–257,1995.[4]M.Rudman.Avolume-trackingmethodforincompressiblemulti-fluidflows

withlargedensityvariations.Int.J.Numer.Meth.Fluids,28:357–378,1998.[5]D.C.Blanchard.Thesizeandheighttowhichdropsareejectedfromburst-ingbubblesinseawater.J.Geophys.Res.,94(C8):10999–11002,19.[6]D.E.Spiel.Onthebirthsofjetdropsfrombubblesburstingonwatersur-faces.J.Geophys.Res.,100(C3):4995–5006,1995.[7]F.MacIntyre.Flowpatternsinbreakingbubbles.

77(27):5211–5228,1972.

J.Geophys.Res.,

[8]M.S.Longuet-Higgins.Bubbles,breakingwavesandhyperbolicjetsata

freesurface.J.FluidMech.,127:103–121,1983.[9]B.W.Zeff,B.Kleber,J.Fineberg,andD.P.Lathrop.Singularitydynamics

incurvaturecollapseandjeteruptiononafluidsurface.LetterstoNature,403:401–404,2000.[10]M.S.Longuet-HigginsandH.Oguz.Criticalmicrojetsincollapsingcavi-ties.J.FluidMech.,290:183–201,1995.[11]H.N.OguzandA.Prosperetti.Bubbleentrainmentbytheimpactofdrops

onliquidsurfaces.J.FluidMech.,203:143–179,1990.[12]M.S.Longuet-Higgins.Ananalyticmodelofsoundproductionbyrain-drops.J.FluidMech.,214:395–410,1990.[13]M.Rein.Thetransitionalregimebetweencoalescingandsplashingdrops.

JournalofFluidMechanics,306:145–65,1996.

29

[14]D.Morton,LiowJong-Leng,andM.Rudman.Aninvestigationoftheflow

regimesresultingfromsplashingdrops.PhysicsofFluids,12(4):747–63,2000.[15]J.B.KellerandM.J.Miksis.Surfacetensiondrivenflows.SIAMJournalon

AppliedMathematics,43(2):268–77,1983.[16]D.C.BlanchardandL.D.Syzdek.Filmdropproductionasafunctionof

bubblesize.J.Geophys.Res.,93(C4):39–3654,1988.[17]S.PopinetandS.Zaleski.Afronttrackingalgorithmfortheaccuraterepre-sentationofsurfacetension.Int.J.Numer.Meth.Fluids,30:775–793,1999.[18]S.PopinetandS.Zaleski.Bubblecollapsenearasolidboundary:anumer-icalstudyoftheinfluenceofviscosity.SubmittedtoJFM.

30