Jetformationin bubblesburstingata freesurface

LaurentDuchemin,St́ephanePopinet,ChristopheJosserandandSt́ephaneZaleski

Abstract

We study numericallybubblesbursting at a free surfaceand the sub-sequentjet formation. The Navier-Stokes equationswith a free surfaceandsurfacetensionaresolved usinga marker-chainapproach.Differenti-ationandboundaryconditionsnearthefreesurfacearesatisfiedusingleast-squaresmethods.Initial conditionsinvolveabubbleconnectedto theoutsideatmosphereby apreexisting openingin a thin liquid layer. Theevolution ofthebubbleis studiedasafunctionof bubbleradius.A jet formswith or with-out theformationof a tiny air bubbleat thebaseof thejet. Theradiusof thedropletformedat thetip of thejet is foundto beaboutonetenthof theinitialbubbleradius.A seriesof critical radiusesexist, for whicha transitionfromadynamicswith or withoutbubblesexist. For someparametervalues,thejetformationis closeto asingularflow, with aconicalcavity shapeanda largecurvatureor cuspat the bottom. This is comparedto similar singularitiesinvestigatedin othercontexts suchasFaradaywaves.

1 Intr oduction

Bubblesburstingat the watersurfacearea familiar everydayoccurrence.Theyalsotake part in importantprocessesof transportandexchangeacrossliquid/gasinterfaces,causedby theejectionof jetsandvariouskindsof smalldroplets.Theseareinvolved in the transferof heat,massandvariouscontaminantsbetweentheoceansandtheatmosphere[1]. Indeed,breakingwavescausetheformationof alarge numberof bubblesbeneaththewaterlevel. Theefficiency of the resultingmasstransfer, including the transferof CO� dependson the initial propertiesoftheejecteddroplets(size,initial velocity).

1

Thephenomenaproducingaerosolsduringtheburstingof abubbleareof twokinds: the first is the ruptureof the film separatingthe bubble from the atmo-sphere.This film atomizationcanproduceseveralhundreddropletsof aroundamicrometerin diameterwhich probablyrepresenta largefractionof thetransfers[1]. Sincethe scalesinvolved during this ruptureareof the orderof 100 nm, aphysicaldescriptionis outsidethe scopeof continuumfluid mechanics.Indeed,long-rangemolecularforcessuchasVanderWaalsforcesor electrostaticrepul-sionmustbetakeninto account[2].

Thesmallcavity remainingafterthefilm rupturecollapsesundertheeffect ofbothsurface-tensionandbuoyancy. This collapsegivesbirth to a narrow verticaljet which eventuallybreaksinto oneor several droplets(seeFig. 1). This phe-nomenonconstitutesthe secondaerosolproductionprocessand is the principaltopicof this paper.

Theseaerosolsareof a differentkind: they areejectedvertically — which isnot thecasefor film aerosols— andtheirdiameteris aboutonetenthof thesizeoftheinitial cavity, i.e. about100 � m for a typical bubbleradiusof onemillimeter.Dependingon theirmassandinitial velocity, thedropletswill eitherfall backintowateror evaporate.

The topic of this paperis the investigationof the bubbleevolution after theinitial film rupturing, including the jet formation. A numericalmethodsolvingtheNavier-Stokesequationsanddescribingthefreesurfacewith highprecisionisused.Previousnumericalstudiesof thesephenomenahavebeenmadepostulatingmostlyinviscid fluids;however, amodifiedboundaryelementmethodtakingintoaccountsmall viscouseffectswasalsoused[2, 3]. A Navier-Stokessimulationwasshown in [4], with a VOF-typemethodin a regimewherethebubbleis verydeformed.

In mostpreviousstudiestheeffectof film atomizationonjet birthwasassumedto benegligible. Few comparisonsweremadewith experimentaldata.Someex-perimentalstudieswerealsoconductedto measurequantitiessuchasjet velocity[5, 6, 7], size of the first ejecteddroplet, height at which the droplet detachesfrom the jet, or heightreachedby the droplet. Theseexperimentsarefairly dif-ficult to conduct,becauseof surfacecontaminationwhich modifiessignificantlythefree-surfaceboundaryconditionandthesurfacetensioncoefficient.

As our numericalresultswill demonstrate,thejet formationis in many casessingularandself-similar. Singularjetsformingata freesurfacehavealreadybeenobserved andstudiedin differentcontexts. Indeed,in the bubble-burstingprob-lem aswell asin severalotherfree-surfaceflows, oneobservesthe formationofa conicalcavity, with a very high curvatureor cuspat its base.In somecasesa

2

Figure1: Jetproducedby thecollapseof a sphericalcavity. Theenddropletwilleventuallydetachdueto theSavart-Plateau-Rayleighinstability.

3

smallbubbleis trappedat thebottomof thecavity. A thin narrow jet subsequentlyforms in a self-similarmanner. This phenomenonwasobserved experimentallyin Faradaywavesby Longuet-Higgins[8] andLathrop[9], in thedevelopmentofthe jet insidea bubblecontaininga sink flow in thenumericalstudyof [10]. Forburstingbubblesthe conicalcavity may be seenin the experimentsandsimula-tions but the singularcharacterof the jet formationhasnot beeninvestigatedtoourknowledge.Thephenomenonmayalsobeseenin thecavity formedby fallingraindrops[11, 12, 13,14].

Theevolutionof theconicalcavity hasbeenstudiedby Longuet-Higgins[12]asaspecialcaseof a family of hyperbolicsurfaces: conicalsurfaceswereshownto bea specialcaseof thehyperbolicsurfacesof Ref. [8]. Theseconicalsurfacesarepreservedby thevelocitypotential���������� ��� � ����������

(1)

where

is thesphericalradialdistanceand�

thepolaranglemeasuredfrom thenorthpole,which yieldsthevelocity field�������������� �"!$#%�'&(#*)�+,�-#

(2)

where�����

is anarbitraryfunction. Indeedany conicalfreesurfacein this flowremainsconical.For positive

�thecavity opensin timeasin theexperiment.

Of coursethe actualflow is not exactly conical. The bottomof the coneisrounded,andoscillatesin shapeascapillarywavesconverge towardsthebottomof theconicalsurface.At someinstantin timethebottommaydevelopacusp,fol-lowedby jet formation.Thisprocessis obviouslysingularat leastfor somevaluesof theparameters,but thereis no agreementamongtheabove citedpublicationson theexactnatureof thesingularity.

Indeedonemay inquire into thespecificscalingform of thesingularity. TheEuler equationswithout surfacetensionandgravity will in principle admit self-similar solutionsof theform�.��/0#1��2�43 5�6�7�3 8:9;�/:3 <�6�7�3>=@?��

(3)

where9

is anarbitraryscalingfunctionand�7

is thesingularitytime. Thesolutionmaybevalid beforeand/orafterthesingularitytime. TheexponentsshouldsatisfyA �B)DCE�GF

. Indeedwith this conditionall the termsin theBernoulli equationbalance.However whensurfacetensionis added,the only way to form a self-similar solutionthatbalancesinertiaandsurfacetensionis by selecting

CH�I)�J�K.

4

Thisis becausetheonly lengthscalethatcanbebuilt is �LM�N�OP � JRQS��TVUXW

whereO

isthesurfacetensionand

Qthedensity. Thenthesimilarity variableis Y ��/$JR %L

andthe flow velocity divergeslike

= TVUXWnearthe singularityfor a fixed valueof the

similarity variableY . This ideais at thebasisof)�JRK

exponentsfoundfor instanceby Miksis andKeller [15]. This typeof scaling,wasappliedby Zeff et al. [9] toobservationsof jet formationin Faradaywaves. The leadingorderterm for thevelocitypotentialis of theform���[Z\3 5�]�7�3 TVUXW TVU � � TVU � �^�_�������a`

(4)

However, a seriesof alternatetheoriesfor singularfree-surfaceflows andinparticulartheconicalcavity andjet formationwasproposedby Longuet-Higgins.He hasshown that thetypeof flow describedby Eq. (2) hada divergentvelocitywith

�b���Hc 3 '�G�7�3 = TVUXWthus a

= TVUXWdivergencefor a fixed value of the real

(unscaled)distance

[12]. In this solutionthe scalingis not fixed by a balancewith surfacetension. Instead,surfacetensionis addedasa perturbationto theconicalsolution,in the form of a sink flow [12]. The Longuet-Higginssolutionyieldsananglefor theconicalcavity of

)R�d�4Ffe�g�h5, in goodagreementwith the

numericalobservationsof [11]. Anotherself-similar solution for jet formationwasfound numericallyby [10] obtainingyet anotherscaling,for the caseof jetformationinsideabubble.Thepotentialis thenapproximatedby���������� TVUji � TVUji �^�_�������a`

(5)

This article is organizedasfollows. We first describethe generalcontext ofthis study, thenon-dimensionalnumberscontrolling theproblemandthescalinglawsdeducedfrom dimensionalanalysis.Wethenbriefly introducethenumericalmethodwe useand its main advantages.A first comparisonwith experimentalprofiles is presented.Finally, a detailedparametricstudy is conductedusingasimpleinitial shapefor thecavity andneglectinggravity. Wemeasurethevolumeof thefirst ejecteddroplet,thevelocity of the jet andthemaximumpressureen-counteredontheaxisof symmetryanddiscusstheresults.In somecircumstances,atiny bubbleis formedat thebaseof thejet. Theself-similarflow occurringwhentheconicalcavity andthecuspform is investigated.

2 Initial conditionsand expectedscalinglaws

Giventhesmall sizeof thebubbleswe areinterestedin (diameteris aroundonemillimeter), someassumptionscanbemaderegardingtheparametersgoverning

5

jet birth. The first idea is to supposethat the cavity is motionlessat the initialtime. Experimentshaveshown that,evenin theabsenceof surfactants,thebubblecanstayat the free surfacein a quasistaticequilibrium for a few seconds[16].Thebubbleis thenseparatedfrom theatmosphereby a thin liquid film, thecavitybeing subjectto surfacetensionand buoyancy forces. A model for this staticconfigurationis a moreor lessdeformedbubbleadjacentover partof its surfaceto afilm of negligible thickness.Thisconfigurationmaybecomputed,or obtainedfrom theexperimentaldataasin thecasereportedin [7].

Whenthefilm reachesacritical thickness(about100nm)afterdrainingslowly, it breaksmoreor lessrapidly (dependingon the presenceof surfacecontami-nants).It is thenpossibleto run simulationsby takingthecurrent,staticconfigu-rationandremoving thethin film. While we do this in onecase,thedrawbackisthata sharpcornerexistsat therim of theneckor juncturebetweenthefilm andthe bulk liquid. The small lengthscalesinvolved may createnumericalconver-genceproblems.Moreover, aswe show below, small lengthscalesaregeneratedindependentlyof initial conditionsby the steepeningof capillary wavesand jetformation.Keepingthesmalllengthscalesin theinitial conditionsmakesit moredifficult to observe the intrinsically generatedsmall scales.We thusdecidedtodrasticallysmooththe rim of the neck. In most calculations,the initial shapewasdefinedasfollows. A sphericalcavity is separatedfrom theatmosphereby acircularhole,theborderof theholebeingacircularrim (seeFig.2).

The collapsebehavior dependsonly on four physicalparameters:the kine-matic viscosity k ,

O,Q

and the accelerationdue to gravity l . Out of the fourphysicalparametersonly two lengthscalescanbe defined,the capillary lengthmnLo�p�O$JRQ l � TVU � andthe the viscous-capillarylength

mnq\�rQ k � JRO . In purewa-ter

msLd�t)u`wvmm and

mnqE�xey`weuF%z � m respectively. If the radiusof the bubblem4{ mnL, capillaryeffectsarepredominantcomparedwith thegravity effects; ifm}| mnq, viscouseffectsareexpectedto benegligible comparedto thecapillary

ones.Formnqd{ m4{ mnL

thephenomenonis dominatedby surfacetensionandinertia.

We alsodecidedto neglecttheeffectof gravity which is a correctapproxima-tion for

m~{ msL. Therefore,only theOhnesorgenumber � ��Q k � JRO�m

governsthephenomenonanddimensionalanalysisgivesvelocity in theform

��� Q@mO ����� Q k �O�mE� #(6)

where�

is anunknown function.

6

Figure2: The initial configurationin the “large rim” case. The grid is a � F�) �Cartesiangrid.

7

Whenevermnq�{ mB{ mnL

, we alsoexpectthat viscosityplaysno role. Theonly way to eliminateviscosityis to supposethatthefunction

�hasafinite, non-

zerolimitZ

when k goesto zero. The non-dimensionalvelocity of the jet thenbehaveslike ���q�� Z}� mmnqS� = TVU �

(7)

where��qs��O�JRQ k .

Similar argumentsleadto a scalinglaw for the non-dimensionalpressureoftheform �� q cGZ�� � mmnq � = T

(8)

where� qs��O � JDQ k � .

3 Numerical method

Thechoiceof thenumericalmethodis conditionedby thetermsweneedto solveaccurately. In our problem,thefirst termof interestis surfacetension:beingthemaindriving forcein theparameterrangewe consider, it is importantto modelitcorrectly. Given the large densityratio betweenwaterandair we canmoreoverassumethattheinfluenceof thegasphaseis negligible.

According to thesetwo assumptions,we useda numericalmethodwhichsolvesthefull axisymmetricNavier-Stokesequationsin afluid boundedby a freesurfacewhile allowing anaccuratedescriptionof theinterfacialtermssuchassur-facetension.This methodhasbeendocumentedelsewhere[17, 18] andhasbeenshown to produceaccuratequalitative and quantitative resultswhen comparedwith boththeoreticalandexperimentaldata.

In short,a regular Cartesianfixed grid is used. Masslessparticles(markers)advectedby theflow definethepositionof theinterface.Linkedby cubicsplines,they describeaccuratelythegeometryof thefreesurface.For cellswhich arenotcutby thefreesurface,aclassicalfinite-volumeschemeis applied.For thecellsinthevicinity of theinterface,finite differencescannotbecomputedsincevelocitiesarenotdefinedin the“gas” phase.Therefore,anextrapolationof thevelocityfieldnearthefreesurfaceon theothersideis necessary. This extrapolationmusttakeinto accounttheboundaryconditionsonthefreesurface(in particularlythenullityof the tangentialstress). This is doneby using a least-mean-squareprocedureconstrainedby the condition of vanishingtangentialstress. Comparisonswith

8

theoreticalresultsshow that this approachgivesan accuratedescriptionof theviscousdissipativetermsassociatedwith theboundaryconditions.

The pressureon the boundaryis obtainedas follows. The local curvatureis estimatedfrom the spline reconstruction.The local normalviscousstressisestimatedfrom theabove least-squaresprocedure.Thenthepressureis obtainedfrom thenormal-stressboundarycondition.Thepressureon theboundaryservesasa boundaryconditionfor thePoissonequationfor thepressure.This equationis in turnsolvedusingamultigrid algorithm.

Most computationshave beenmadeon a � F�) � grid, except the comparisonwith theexperimentalprofilesfrom MacIntyre,which hasbeenmadeon a

Ffe,)Rz �grid andsomeselectedcomputationswhichwererefinedto

Ffe,)Rz �grids.

4 Comparison of the numerical resultswith experi-mental profiles

We have first initialized thecalculationwith a realisticshape,andtaken into ac-countall thephysicalparameters,i.e. capillarity, viscosityandgravity. Thegoalwasto comparethe resultswith a seriesof shapespublishedby MacIntyre [7].Theinitial shapeof thefreesurfacehasbeenobtainedfrom thefirst imagegivenby MacIntyre,just afterthefilm rupture.

Fig. 3 illustratestheexperimentalandthecomputationalresults.Thenumer-ical parametersare: k ��Ffe =S�

m�/s,

O���ey`�e,v�)kg/s

�,Q���Ffe�e�e

kg/mW

andthevolumeof the bubbleis the sameasthe onegiven by MacIntyre:

)u` � v � l. Thecomputationaltime is aboutonedayon the

Ffe,)Dz �grid. Theoverall agreementis

very satisfactory. In particularcapillarywavesarewell described,in contrasttotheearlierpublishedresultsusingboundaryintegral methods[3, 2]. We believethat this lack of capillary wavesis dueto the strongsmoothingneededto avoidnumericalinstabilitiesin boundaryintegral techniques(andprobablyalsoto aninsufficientspatialresolution,which is alsolimited by numericalstability). In ourmethod,real, molecularviscosity is presentand the fine grid we useallows inprincipleto solve thesmallspatialscalesof thecapillarywaves.

Thetime interval betweenimagesis thesameastheonegivenby MacIntyre,i.e.

F�J���e�e�e,�. A differencein timebetweenprofilescanbeseen,evenif theshape

is very similar. A possibleexplanationis thepresenceof surfacecontaminantsintheMacIntyreexperiment.Thesecontaminantscouldchangethesurfacetension,evenmodify its valuelocally, thereforechangingthebehavior of thefreesurface

9

Figure3: Timesequenceof thejet formationin a)u` � v ��� bubbleburstingat a free

surface. Top: experimental[7] andbottom: computationalresults. ProfilesareF�JR��e�e�e,��`apart.

10

throughgenerationof Marangonicurrents. They could alsomake the interfacepartiallyor entirelyrigid, changingthefree-surfaceboundarycondition.

Figure4: Vorticity isolinesduringthecollapseof thebubblefor thesamecondi-tionsasin Fig. 3.

11

Figure5: Sameaspreviousfigure.

Figure6: Sameaspreviousfigure.

12

Figure7: Sameaspreviousfigure.

13

Solving the full Navier-Stokesequations,we have accessto vorticity whichcan,aswewill seelater, haveanimportanteffectevenonverysmallstructuresina low viscosityfluid suchaswater. Fig. 4 shows thevorticity isolinesduringthecollapseof thecavity. Thevorticity is confinedto athin boundarylayerbeforethejet birth. Lateronhowever, avorticity coneis entrainedbelow thejet andtheshearstressthereis comparableto that in thenarrow jet. This detachmentof vorticityillustratesthe formationof a downward jet, alreadyobserved by Boulton-StoneandBlakewith theirmodifiedboundaryintegralmethod[3, 2].

5 Resultsof the parametric study

A set of computationshave beenmadefor radii betweenF%e =S� A and

Ffe = � A (Ffe �'� m�J�mnq � Ffe �) with theinitial shapedescribedabove.

Theevolutionof theprofilesis verysimilar to thatshown onFig. 3. A conicalcavity formswith a train of capillarywavesconverging to theaxis. Thenumberof capillary wavesdependsstronglyon the Ohnesorge number: the higher thisnumber, the higherthe numberof capillary wavesconverging to the baseof thecavity. Fig. 8showsalarge-radiuscasewith alargenumberof waves(seealsoFig.11). In somecases,especiallynear

m�JRmnqs��Ffe Wthejet becamevery thin (Fig. 9)

andthe local radiusof curvaturesmallerthanthegrid size. Thecalculationthenbecomesinaccurateandhasto bestopped.

For someparametervalueswe observe an tendency to trap a bubbleon theaxisof symmetryjust beforetheformationof the jet. We have searchedsystem-atically for bubbleentrapment.Therearetwo competingchangesof shape:thejet formationis heraldedby a changeof curvatureat thebaseof thecavity, whilethebubblepinchingis precededby theformationof anoverhangin theinterface,i.e. the height � � ��

of the interfacebecomesmulti-valued. Thus our criterionfor incipientbubbleformationis asfollows: (a) Theheight � �^ ��

becomessteep,then multi-valued,and (b) the curvatureat the baseremainspositive. This isonly an indicationthata bubblewill betrappedbeforethejet formsasshown onFig. 10, but we needsucha crudecriterion becausethe bubblesarevery smallfor the kinds of grids we have. We found a first bubbleentrapmentregion for� vR� � m�J�msq � )�euFf�

, thesecondonebetween� v���e�e � m�J�mnq � )�����e�e�e. Other

suchregionsat highervaluesofm�J�mnq

arelikely, but difficult to observe numer-ically. Oneindicationis the existenceof large trainsof capillary wavesat largem�J�msq

asshown on Fig. 11.Thetopologyof theinterfacechangeswhenabubbleis trapped.Thispinching

14

−0.2� −0.1� 0� 0.1 0.20.1

0.2

0.3

0.4

0.5

Figure8: Capillarywavesform�JRmnqs��F%z�z�e�e

. Asm�J�mnq

15

−0.5  0.50.0

1.0

PS

fragreplacements mG�IFfe � AmG��) A\A

Figure9: The initial phaseof jet formationasseenin two simulations.For thelarge bubble (dashedline,

m�J�msq¡� F%z�z�e�e) the jet is relatively wide and well

resolvednumerically. For smallerbubbles(solid line,m�JRmnqd�4v�)�e

) the jet maybecomeextremelythin.

16

is a singularevent akin to the pinchingof a gascylinder by the Savart-Plateau-Rayleighinstability. We shall call it a pinchingsingularityto distinguishit fromother free surfacesingularities. To pursuethe calculationnumericallybeyonda pinching singularity, one should in principle perform surgery on the markerchainandcontinuethesimulation.This is however difficult becausetheproblemslightly changesin nature:thepressureinsidethesmalltrappedbubblecannotbesetto atmosphericpressurebut shouldin principledependon thebubblevolumethroughsomeequationof state.This changesmarkedly the natureof the calcu-lation. Moreover the trappedbubblesareextremelysmall andvery difficult toresolvewithoutmeshadaptation.Thusin mostcaseswecontinuedthesimulationwithoutmarker surgery. Whenthetrappedbubbleis verysmall,themarkerchainreorganizesitself spontaneouslyandthecalculationproceeds.In somecases,asin therightmostbubbleentrapmentregion, it seemsthattheeffect on thedynam-ics is small. In othercases,asin the leftmostentrapmentregion, thecalculationhasto be stoppedor provides unreliableresultswhich were removed from thequantitativeanalysesbelow.

We have redoneall thecalculationsfor a differentinitial condition.Theover-all configurationis the sameason Fig. 2 but the rim thicknessis halved. Allthe above qualitative resultsare identical. In particular, we do not observe anysteepeningof thecapillarywavesor thinnerjetsaswe reducetherim size. Thisis a clearindicationthat thesmall lengthscaleswe observe form spontaneously,independentlyof initial conditions.

5.1 Jet velocity

A first quantityof interestis the velocity of the jet, or the ejectionspeedof thefirst drop. Fig. 12 shows thenon-dimensionalvelocity of thejet, measuredwhenthe top of the jet reachesthe meanwater level. Circle symbolscorrespondtothelargerrim thicknessason Fig. 2 while squaresymbolscorrespondto thinnerrims. Apart from a vertical shift, the measuredvelocitiesarevery similar. Thisshift may in partbeexplainedby the fact thatwe measurethe jet velocity at themeanwaterlevel for bothcases,which is at a differentdistancefrom thebaseofthetwo cavities.

For alargerangeof radii (between)'¢£F%eRi

andFfe �

timestheviscous-capillarylength

msq), thenumericalresultsarein goodagreementwith theinviscid scaling.

For small radii the velocity startsdecreasingasm

decreases.For the smallestradii we have investigatedthe cavity relaxes to a flat surfaceshapewithout jetformation.

17

Figure10: Beginningof theentrapmentof a bubbleby thecollapsingcavity, form�J�msqn�IFfe�¤(a

F�`>z A\A bubble).

18

0¥ 0.005¥ 0.01¥ 0.015¥ 0.02¥time

−5

0

5

10ve

loci

ty o

n th

e ax

is

Figure11: The velocity of the interfaceon the axis form�JRmnq£�¦)u`w���"F%e ¤

. Theoscillationscorrespondto thearrival of a train of capillarywaves. For this largevalueof

m�J�mnqcapillarywavesarenumerousandof shortwavelength.Thevery

largeexcursionin velocitymaybedueto theexistenceof a furtherbubbleentrap-mentregion, however the very small scalesinvolvedmake numericalresolutiondifficult.

19

10§ 210§ 3

10§ 410§ 5

10§ 610§ 710

−2

10−1

100

101

PS

fragreplacements

¨�©ª«­¬

®°¯'±_²�³ �

Wide rimThin rim´�µ6¶�·>¸f¹ ±fº »

Figure 12: Non-dimensionaljet velocity as a function of the non-dimensionalbubbleradius. The two regionsbetweenverticalstraightlines correspondto theradii for which abubbleis trappedat thebaseof thejet .

20

Theregionswherebubblesform at thebaseof thejet areindicatedasverticallines on Fig. 12. In the leftmost region, around

m�J�mnq��¼Ffe�W, for the reasons

discussedabove,thereis a gapin datapoints.It is thuspossiblethatmuchhigherjet velocitiesmaybereachedin thatregion.

5.2 Maximum pressureon the axisof symmetry

We have computedthemaximumpressureon theaxisof symmetrywhenthe jetreachesthemeanwaterlevel. Fig. 13 shows this pressureanda fit in

�Vm�JRmnqf� = T.

Oncemore, the numericalresult is in goodagreementwith the scalinglaw forradii between

)u`½FfeRiand

Ffe �timestheviscous-capillarylength.We alsoremarka

smalljitter aboutthestraightline on theright handsideof thecurve,perhapsasaresultof thesingularbehavior in thebubbleentrapmentregion. Noteagainthatintheleft handsideof thecurvewecouldnot reliablycalculatepressure.

5.3 Radiusof the first ejecteddrop

Experimentaldataobtainedby Spielet al. [6] tendto show that theradiusof thefirst ejecteddropis aboutone-tenththeradiusof theinitial bubble.

We have obtainedthis radiusfrom thenumericalsimulationsasfollows. Thecomputationstopswhenthe jet thicknessreachesthe sizeof onecomputationalcell. Thejet rupturewill occursoonthereafter. Thevolumeenclosedby thefree-surfacebetweenthis point of minimum thicknessandthe tip of the jet is thenagoodapproximationof thevolumeof theejecteddroplet. The equivalentradiusmn¾

is definedastheradiusof asphericaldropletwith thesamevolume.Fig. 14 shows

mn¾_J�m. For large

m�J�mnqwe obtaina linear fit

mn¾H¿ eu`ÀFfK�mwhich is consistentwith theexperimentallyobservedvalueof

m�JuFfe. This linear

behavior is consistentwith the viscosity-independentregime of Eqs. (7, 8) inwhichtheonly lengthscaleis

m. Ontheotherhand,thereis a largefractionof the

datawherethis regimedoesnot hold andtheejecteddropradiusis muchsmallerthan

m�JuF%e. Noticeagainthegapin valuesaround

m�J�mnqs��Ffe W. Therethejet was

too thin to bewell-resolvednumerically, andtheactualdropletsizemaybemuchsmaller. Varyingtheinitial conditionhaslittle effect, exceptat small radii wherethethinnerrim leadsto a largerdroplet.

21

10§ 210§ 3

10§ 410§ 5

10§ 610§ 710

−6

10−5

10−4

10−3

10−2

10−1

100

PS

fragreplacements

Bubbleradius( ®°¯'±_²�³ � )

Pre

ssur

e(

ÁÃÂÄÅ Æ^ÇÅ )

Figure13: Maximumpressureon theaxisof symmetrywhenthe jet reachesthemeanwater level. As in the previous figure the bubbleentrapmentregionsaremarked.

22

10¥ 210¥ 3

10¥ 410¥ 5

10¥ 610¥ 70.01

0.1

1

PS

fragreplacements

Wide rimThin rim

È ÉÊ È

Ë�Ì�ËnÍ

ËsÎ�ÌRË[ÏGÐyѽÒfÓ

Figure14: Ratioof theradiusof thefirst ejecteddropandtheradiusof theinitialbubbleasa functionof

m�J�mnq.

23

6 Singular jet formation by curvaturereversal

Theformationof athin,high-velocityjet in andaroundthefirst bubbleentrapmentregion leadsto suspectthe existenceof a singularity. The scaling(3) discussedaboveyields � �^Ô$#1��2�N���7Õ�Ö�� � UXW_× �^Ô5���72�6�� = � UXW �-#

(9)�.�^Ô$#*+y#���Ø����7Ø�6�� TVUXW 9:�Ô5��70�6��*= � UXW #*+Ù��7Ø�]��*= � UXW �-#(10)

where � is thesurfaceelevationandÔÖ�~ 5�������

thedistanceto theaxisof sym-metry. We rescaledthe radial andvertical coordinatesof the surfacepointsby���7.���� � UXW

form�JRmnq6�¼v�)�e

. We determined�7

by fitting two of the rescaledprofilesontooneanother. Theresultsareshown on Fig. 15.

All the profileshave beentranslatedvertically in order for the point on theaxis of symmetryto be at the samevertical coordinate. The rescaledprofilessuperimposewell at small valuesof the similarity variable Y . The shapeof theprofilescloselyresemblestheexperimentalandnumericalprofilesin othertypesof flow [11, 14, 9]. However at a large distancefrom the singularity the coneangleis about

v�K�h. This shouldbe comparedwith the angleof the cavity seen

in theMcIntyre datashown on Fig. 3. There,on profile 6 we measureanangleof

����h, a small differencewith our calculations. In contrast,the otherphysical

processesdiscussedin theintroductionyield relatively largerangles.The finite viscosityshouldalso introducea discrepancy with the theoretical

similarity solution.It seemshowever thatits effectsaresmall in thatcase.In reference[9] it wasshown that for Faradaywavestherewasa connection

betweenbubbleentrapmentandsingularities.In our casethe pictureseemsdif-ferent.Theself-similarsolution(10) is observedin theentirefirst bubbleentrap-mentband. On theotherhand,this solutionis not seenin or aroundthe secondbandof bubbleentrapment,wherewe shouldin principlealsohave a singularity.However theshapeof the interfaceis very differentin that case(Fig. 16) andasuperpositionusingtheabove rescaledvariablescouldnot be found. A possibleexplanationis thattheconicalcavity associatedwith thesingularityis presenthereonly on the largescaleasseenon Fig. 10. Theconvergenceof small-scalecap-illary wavesis not ableby itself to generatea self-similarconicalflow. Thustheconicalflow would beat leastasimportantasbubbleformationin producingthesurface-tensiondrivenself-similarscaling.

A tantalizingpossibilityis theexistenceof furtherbandsof bubbleentrapmentand singularitiesto the right of the secondband. While bubble entrapmentisobservedin somecases,detailsof thedynamicsarenot well-enoughresolved. It

24

−0.2Ú −0.1Ú 0Ú 0.1 0.2−0.1

0

0.1

0.2

0.3

Figure15: Comparisonbetweensuccessiveprofilesmadenon-dimensionalusingthescalingY ��!(J���7R�n�� � UXW

describedin thetext. Left: unrescaledprofiles.Right:rescaledprofiles. Theretheopeningangleof thecavity is around

v�K�h. Sincethe

numericalmethodusesadaptive time-stepping,the profilesarenot separatedbyequaltime intervals.

25

−0.05 −0.03 −0.01 0.01Û 0.03Û 0.05Û0.2

0.22

0.24

0.26

0.28

0.3

Figure 16: Shapeof the interfaceon the edgeof the secondbubble formationregion at

mN� � v���e�e . In thatcasea rescalingof thetypeshown on Fig 15 couldnotbefound.As in thepreviousFigureprofilesarenotseparatedby uniformtimeintervals.

26

is likely thatbubbleentrapmentandcuspsingularitiesarerelatedto theamplitudeof the converging capillary waves. As viscosity is reduced,ever morecapillarywavesareobservedto convergeontheaxis.For verylargevaluesof

m�JRmnq, waves

having bothshortwavelengthandsmallamplitudeareformed.

27

7 Conclusions

We have presenteda numericalstudyof theburstingprocessof bubblesat a freesurface.Theschemeusedwasbasedonanaccuratedescriptionof thefreesurfacewith the help of a markerschain. This methodhasshown goodcapabilitiestoresolvesmallcapillarywaves.Thelargescalefeaturesof thedynamics,thepres-sureandfinal dropletradiusmaybepredictedwith accuracy, exceptnearthefirstbubbleentrapmentregion near

m�J�mnq£�xFfe�W. The predictionsarequantitatively

in agreementwith experiment:theangleof openingof thecavity is similar to theangleobservedin theexperimentsof MacIntyreandthesizeof thedropletat thetip of thejet is closeto theexperimentallyreportedsize.

Themeasurementsof jet velocitynearm�J�mnqn�}Ffe W

show a surprisinglylargevelocity. The interfaceshapescaleswith a characteristiclength

�c 3 :��7�3 � UXWpredictedby thebalanceof surfacetensionandinertia.Theshapeof theinterfaceresemblesshapesfoundin otherjet-formingflows andcuspsingularities,but hasquantitativedifferencessuchastheopeningangleof theconicalcavity.

Theconnectionof thisscalingwith bubbleentrapmentis lessclear. We foundthescalingin a wide region. Theoccurrenceof self similar flow andanapprox-imatesingularityis not connectedto theexactboundaryof a bubbleentrapmentband.We alsofoundbubbleentrapmenttransitionswhich werenot associatedtothe

3 ��b�7R3 � UXWscaling.Finally theangleof theconicalcavity agreeswith theexper-

imentaldatafor burstingbubbles,but notwith theanglesseenor predictedin otherflows. This indicatesthatothertypesof singularities,correspondingto differenttopologiesor initial conditions,maybeobserved.Furtherwork shouldexploreindetailthenatureof thesesingularitiesusingfor instancemeshrefinement.

Also of interestwould be a studyof the influenceof the initial shapeof thebubble. We have shown that a factor of two changein the rim thicknesshadno qualitative effect, andvery little quantitative effect on the collapseprocess.Howeverotherchangesin theinitial conditionmaycauseachangein thepositionof thevarioussingularities.In otherwords,for agivenradius,it wouldbepossibleto reachasingularityby changingtheshapeof thebubble.

28

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