TechnologyResearchUSDepartmentofEnergyunderContractNoW31109Eng - PDF document

1 TechnologyResearch,U.S.DepartmentofEnerg
TechnologyResearch,U.S.DepartmentofEnergy,underContractNo.W-31-109-Eng-38.M.B.acknowledgesanNSFpostdoctoralfellowship.J.Eggers,``Universalpinchingof3Daxisymmetricfree-surface¯ow,''Phys.Rev.Lett.,,3458J.EggersandT.F.Dupont,``Dropformationinaone-dimensionalap-proximationoftheNavier-Stokesequation,''J.FluidMech.,205M.P.Brenner,X.D.Shi,andS.R.Nagel,``Iteratedinstabilitiesduringdroplet®ssion,''Phys.Rev.Lett.,,3391X.D.Shi,M.P.Brenner,andS.R.Nagel,``Acascadeofstructureinadropfallingfromafaucet,''Science,219R.M.Kerr,``Evidenceforasingularityofthethree-dimensional,incom-pressibleEulerequations,''Phys.FluidsA,1725A.PumirandE.D.Siggia,``DevelopmentofsingularsolutionstotheaxisymmetricEulerequations,''Phys.Rev.Lett.,1511A.J.Majda,``Vorticityandthemathematicaltheoryofincompressible¯uid¯ow,''Commun.PureAppl.Math.,5187R.E.Ca¯ischandG.C.Papanicolaou,SingularitiesinFluids,Plasmas,andOptics,Vol.C404ofNATOASISeriesKluwerAcademic,Dor-drecht,1993P.ConstantinandM.Pugh,``GlobalsolutionsforsmalldatatotheHele-Shawproblem,''Nonlinearity,393A.L.BertozziandM.Pugh,``Thelubricationapproximationforthinvis-cous®lms:regularityandlongtimebehaviorofweaksolutions,''Com-mun.PureAppl.Math.,85P.Constantin,T.F.Dupont,R.E.Goldstein,L.P.Kadanoff,M.Shelley,andS.-M.Zhou,``DropletbreakupinamodeloftheHele-Shawcell,''Phys.Rev.E,4169T.Dupont,R.Goldstein,L.Kadanoff,andS.-M.Zhou,``Finite-timesin-gularityformationinHele-Shawsystems,''Phys.Rev.E,4182A.L.Bertozzi,M.P.Brenner,T.F.Dupont,andL.P.Kadanoff,``Sin-gularitiesandsimilaritiesininterface¯ows,''TrendsandPerspectivesinAppliedMathematics,editedbyL.SirovichSpringer-Verlag,Berlin,,pp.155±208.A.L.Bertozzi,``Symmetricsingularityformationinlubrication-typeequationsforinterfacemotion,''SIAMJ.AppliedMath.R.E.Goldstein,A.I.Pesci,andM.J.Shelley,``Topologytransitionsandsingularitiesinviscous¯ows,''Phys.Rev.Lett.,3043D.Bensimon,L.P.Kadanoff,S.Liang,B.I.Shraiman,andC.Tang,``Viscous¯owsintwodimensions,''Rev.Mod.Phys.,977D.A.Kessler,J.Koplik,andH.Levine,``Patternselectionin®ngeredgrowthphenomena,''Adv.Phys.,255R.Almgren,``SingularityformationinHele-Shawbubbles,''Phys.Fluids,344M.A.Lewis,``Spatialcouplingofplantandherbivoredynamics:thecontributionofherbivoredispersaltotransientandpersistent`waves'ofdamage,''Theor.Popul.Biol.P.W.Voorhees,G.B.McFadden,R.F.Boisvert,andD.I.Meiron,``NumericalsimulationofmorphologicaldevelopmentduringOstwaldripening,''ActaMetal.,207G.I.Barenblatt,Similarity,Self-Similiarity,andIntermediateAsymptoticsConsultant'sBureau,Plenum,NewYork,1979R.E.Goldstein,A.I.Pesci,andM.J.Shelley,``Attractingmanifoldforviscoustopologytransition,''Phys.Rev.Lett.,3665S.Boatto,L.P.Kadanoff,andP.Olla,``Traveling-wavesolutionstothin-®lmequations,''Phys.Rev.E,4423R.E.Goldstein,T.G.Mason,andE.Shyamsunderprivatecommunica-S.Akamatsu,O.Bouloussa,K.To,andF.Rondelez,``Two-dimensionaldendriticgrowthinLangmuirmonolayersofD-myristoylalanine,''Phys.Rev.A,R45041370Phys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner ofwhichappearinpreviousworkontheforcedHele-Shawcell.Wesupporttheevidencepresentedinthenumericalsimulationsbyconstructingself-similarlocalsolutionst

2 othegoverningequationsthatagreewiththenu
othegoverningequationsthatagreewiththenumerics;takento-gethertheseresultsprovideconvincingevidencethatthereareseveraldifferentmechanismsfortheformationof®nitetimesingularities.Weemphasizethattheobservationofsolutionsdependingonthedetailsoftheinitialconditionsandboundaryconditionsisquitedifferentfromotherphysi-calsituations.AllofthesimilaritysolutionsintheHele-Shawproblemhavescalingexponentsunrelatedtodimen-sionalanalysis;thisisinstarkcontrasttothree-dimensionaldropletbreakup,wherethereisasinglesimilaritysolutionwithdimensionalexponents.Theexistenceofsomanydif-ferentsimilaritysolutionsintheHele±Shawcellisinterest-ingbecauseitmeansthatinthiscaseargumentsaboutuni-versalityofthesingularitybecomevacuous:slightchangesininitialconditionsandboundaryconditionsthesingularbehavior,eventhoughthesingularityhappensonatimescalearbitrarilyfasterthantheboundaryforcing.ThereasonforthisstrikingdifferencebetweentheHele±Shawproblemandthethree-dimensionaldropletbreakupmightberelatedtotheimportanceof¯uidinertiainthelatterAparticularlyintriguingresultofthepresentworkistheobservationthatsimilaritysolutionscanapparentlydestabi-lizeatarbitrarilysmallthickness,withthethicknessatwhichinstabilityoccursdependingontheinitialcondition.Beforetheinstabilitysetsin,arbitrarilymanydecadesofscalingcanoccur.Itisimportanttonotethatlesswellresolvednumericswouldmisstheinstabilitiesandthereforeprovideaninaccu-ratedescriptionofthesingularitybehavior.Instabilitieshavebeenpreviouslyobservedinsimilaritysolutionscharacteriz-ingthreedimensionalaxisymmetricdropletbreakup;everinthatcasetheinstabilitiesaremanifestationsoftheRayleighinstability,whichisabsentintheourtwodimen-sionalsystem.Theinstabilitiesobservedherearemoresubtle,andacompleteunderstandingwouldrequirestabilityanalysisofthedifferentmatchingregions.Theexistenceofunstablesimilaritysolutionshighlightsourcurrentlackofunderstandingoftheselectionofscalingsolutionsnearsingularities.Whatcausestheselectionofaparticularsingularbehavior?Doestheselectiondependonboundaryconditions?Althoughstabilityanalysisofasinglesimilaritysolutionwithawellde®nedasymptoticbehaviorissee,forexample,Ref.3,itisunknownhowtoperformstabilityanalysiswhenthereareseveralmatchingregionswithdifferenttimedependences.Evenmoreimpor-tant,thetimedependencesarealwaysdeterminedbyassum-ingacertainasymptoticbehaviorawayfromthesingularity;wedonotknowwhatdeterminestheparticularasymptoticbehaviorselected.Withoutunderstandingtheseissues,wecannotruleoutthatofthesingularbehaviorsdescribedtodateintheHele-Shawcellorforthatmatterinanysys-temwheresingularitiesformmaybeinherentlyunstable.Forexample,wedonotobserveinstabilitiesoftheconstantvelocityexplodingsingularity;however,thisapparentstabil-itycouldbearemnantofour®nitenumericalresolution.Manycomputationalstudiesaddressscalingpropertiesofsingularities.Inmorecomplexequationssuchas3DEu-ler,onecanonlyresolveafewdecadesofscalingwithcur-rentcomputationaltechnology.Itmaybethatsuchdestabi-lizationsalsooccurinsystemslikethisyetthenumericaltoolsarenotre®nedenoughtoobservethem.Ourstudyshowsthatnumericalcalculationsof®nitetimesingularities,evenwhenmanydec

3 adesofscalingarepresent,maynotshowthetru
adesofscalingarepresent,maynotshowthetrueendstateofthesystem.Weemphasizethatthisiseventruewhenthecalculationsareresolvedwellbelowallimportantphysicallengthscales,asinstabilitiescanoccuronscalesdeterminedbyintermediatematchingregions,whichcanbearbitrarilysmall.Anotherfundamentalquestionreiteratedbythisstudyiswhyarethesingularitiesofthelubricationapproximationnotdescribedbyexactsimilaritysolutions,predictedbydi-mensionalanalysis?Inthreedimensionaldropletbreakup,suchdimensionalsimilaritysolutionsarerelevantforun-derstandingrupture.However,althoughwehaveuncoveredatleastsixdifferentsimilaritysolutionsobservedtransientlyorasymptoticallyinthesimulations,noneofthemareexactsimilaritysolutions.Finally,weendwithapleatoexperimentalists:OtherthantheinitialexperimentsthatpromptedstudiesofdropletbreakupinaHele-Shawcell,nosystematicexperimentalinvestigationshavebeencarriedout.Themainreasonisthatsuchexperimentsaredif®cultbecauseoftheneedtoresolvealargerangeofscalestotestthedetailedscalinglaws.InatypicalHele-Shawexperiment,theplatespacingisaround1mm,whichinthebestofcircumstancescouldgiveonlyadecadeofscaling,hardlysuf®cienttotesttheoreticalpredic-tions.Langmuirmonolayer®lmshavebeenusedtotesttheo-riesoftwodimensionaldendriticgrowth,andcouldpossi-blybemodi®edforHele-Shawexperimentsaswell,allowingalargerrangeofscales.Inanycase,thesubtledetailsofselectingsingularsolutionswillbedif®culttoobservewithonlyalimitedrangeofscales;however,thepresentresultssuggestthattherearealsopredictionsthatwouldbebothworthwhileandpossibletotestexperimentally:itshouldbeexperimentallyfeasibletotunebetweenthediffer-entsimilaritysolutions.Sincethedifferentsingularbehav-iorshavedifferentqualitativefeaturese.g.,®nitetimever-susin®nitetime,stationarypinchpointversusmovingpinchpoint,symmetricversusasymmetric,implodingversusex-.Theirqualitativefeaturesshouldbeeasilydistin-guishable,evenbythenumberofsatellitedropsthatareleftbythismostinterestingtopologicaltransition.WethankLeoKadanoff,ToddDupont,andPeterCon-stantinfordiscussionsandencouragement.ThisresearchwaspartiallysupportedbytheMRSECProgramoftheNationalScienceFoundationunderAwardNo.DMR-9400379.Inad-dition,R.A.ispartiallysupportedbyanAlfredP.SloanFoundationResearchFellowship,andA.B.issupportedbyOf®ceofNavalResearchGrantNo.N00014-95-1-0752andtheMathematical,Information,andComputationalSciencesDivisionsubprogramoftheOf®ceofComputationalandPhys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner seemstohaveaslightlysteeperslope.Theinstabilityoccursexactlywhenthecharacteristicscaleofthepinchregionisequaltothecharacteristicscaleoftheintermediateregion,Theinstabilitycanalsobepredictedbystudyingthepropertiesofthesolutioninthepinchregion.Intheaboveanalysisofthepinchregion,wecomputedthe®rsttwotermsinanexpansionandshowedthatthesecondtermhassizeInorderforthistermtobelowerorderthanthe®rstterm,of,theratiomustbesmallcomparedwith1.Figure23showsforthestablesymmetricsin-Ontheotherhand,inthecasewherethesingularityde-initiallydecreasesbutthenstartstoincreaseveryslowlyFigure24.Arapidriseinthenprecedesthedestabilizationrightbeforebifurcationoftheminimumoc-Figure

4 25Thefactthattheunstablesymmetricsingula
25Thefactthattheunstablesymmetricsingularityandthestablesymmetricsingularityhavedifferenttimedependencesanddifferentintermediateregionsinthetwocasesin-dicatesthatweareobservingtwodifferentsingularsolutionsofthePDE.Thusfarwehavenotbeenabletoconstructanentirelyconvincingsimilaritysolutionthatrecoversthescal-ingpropertiesofeitherofthetwocases.VIII.CONCLUSIONSThispaperdescribesasetofcomplexphenomenaasso-ciatedwithscalingandsingularityformationinathinneckintheunforcedHele-Shawcell.Weshowseveraltypesofpossiblebehaviorassociatedwithsingularityformation,two FIG.22.Intermediatelengthscaledashedlineandpinchlengthscalesolidlineplottedagainstminimumthickness FIG.23.Plotofasafunctionofforthestablesymmetricsingularity0.085.Theratiodecreasesmonotonicallyas FIG.24.Plotofasafunctionoffortheunstablesymmetricsingu-larityat0.08.Inthiscasetheratiogrowsveryslowlyas0,againwithalogarithmictimedependence. FIG.25.Plotofasafunctionoffortheunstablesymmetricsingu-larityat0.08.The®gureshowsacloseupofthesharpincreaserightbeforebifurcationoccurs.1368Phys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner Figure18showsexcellentagreementofthenumericswiththissimilaritysolution.Therescaleddatahasfrom10to10.Therescalingusesthemaximumofasarescalingparameter.Althoughwehavefoundthecorrectfunctionalformforthespatialdependenceinthepinchregion,wehavenotde-terminedthetimedependenceofthesolution.Asintheex-plodingsingularityaboveitisnaturaltoimaginethatthetimedependencemightoccurasamatchingconditiontoanintermediateregion.B.IntermediateregionIndeed,thesymmetricsingularityfor0.85alsohasstructureonanintermediatelengthscale,whichismuchlargerthan.Theexistenceofthisintermediatescalewas®rstpointedoutinRef.14fortheforcedHele-ShawcellasthelengthscaleoverwhichFigure19showstheminimumthicknessasafunctionofboththisintermediatelengthscaleandthepinchregionlengthscalede®nedasaboveasthelengthscaleoverwhich,fortheinitialcondition0.085shownabove.ThedatasuggeststhescalinglawsThereisacleardifferenceinsizebetweentheintermediatescaleandthepinchscale,withtheintermediatescalealwaysmuchlargerthanthepinchscale.MeasuringtheintermediatescaleasafunctionoftimegivesapowerlawconsistentwithFigure20C.DestabilizationofthesolutionThusfarwehavepresentednumericalevidencefortheexistenceofalocallysymmetricsingularity.Thesimulationdescribedaboveshowsovertendecadesofscalinginthecharacteristicwidthofthesolution.However,forslightlylowervaluesoftheparameter,thesymmetricsingularitybecomesunstable.Asinthecaseofthe``exploding''singu-larityinstabilitydescribedintheprecedingsection,thisin-stabilitycansetinatanarbitrarilysmallthickness,depend-ingontheinitialdata.WeillustratetheinstabilityusinginitialdatawithseeFigure21Atearlytimestheuppermostcurvethesolutionfallsbytenordersofmagnitude,andseemstoapproachthesym-metricsingularity.However,thesolutioneventuallybifur-catesintoanexplodingsingularity,withtwominima.Evidenceoftheimpendingbreakdownofthesymmetricscalingstructurecanbefoundwellbeforeitoccursinboththeintermediateregionandthepinchregion.Plottingtheminimumthicknessasafunctionoftheintermediatelengthscaleandthepinchscaletellsadramaticallydiffere

5 ntstoryfromthestablesymmetricsingularity
ntstoryfromthestablesymmetricsingularitydiscussedabove.Figure22showsthattheintermediatescaleandthepinchscalehaveessentiallyidenticalscalinglawsuptopos-siblelogarithmiccorrections.Infact,theintermediatescale FIG.19.Minimumthicknessasafunctionofboththeintermediatelengthscaleandthepinchlengthscale,measuredasdescribedinthe FIG.20.Thecharacteristiclengthscaleintheintermediateregionasafunctionofthetimetothesingularity,.Thedottedlineshowsthescalinglaw FIG.21.Destabilizationofthesymmetricsingularityfor0.08.Theuppermostcurvecorrespondstotheearliesttime,andthelower-mostcurvecorrespondstothelatesttime.Althoughthethicknessattheorigininitiallyfallsbyalmosttenordersofmagnitude,this``symmetricsingularity''even-tuallybifurcatesintoanexplodingsingularity,withtwominima.Phys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner Nearthecriticalvalueof0.0664,thecriticalseemstoapproachzero.NearzeroexhibitsapowerlawasshowninFigure16.A®ttothepowerlawgivesthescaling3.5.Anexplanationofthispowerlawbehavioriscurrentlylacking.VII.SYMMETRICSINGULARITYThethirdtypeofsingularbehaviorresultingfromthesimpleinitialdataisthesymmetricsingularity.Thistypeofscalingbehaviorresultsfrominitialdatawithlargerthanthatproducingtheexplodingsingularity;anex-ampleofthissolutionfor0.085isshowninFigure17.Theinterfacebreaksattheoriginafterthe®nitetime0.002547266.Thissingularitymechanismwas®rstdiscoveredinRef.13.Forequation1,theyconstructedasimilaritysolutionthatwelldescribessolutionsfromnumericalsimu-lations.However,theiranalysisbreaksdownforthecaseofpresentinterestwhere1;moreover,itwaspointedoutinRef.14thatthereisasecondrelevantlengthscalegoverningthesingularity,neglectedintheanalysisofRef.13.Thepresentresultsclarifythesituationconsiderably,al-thoughwestilllackacompletetheoryforthistypeofsin-gularity.Weshowbelowthattherearebothstableandun-stablesymmetricsingularities.Thepinchregionforbothofthesesolutionshasaspatialdependencesimilartothatpro-posedinRef.13forequation1,althoughthetimedependenceismorecomplicated.Thesingularityisdemonstratedtohaveanintermediatelengthscale,withanontrivialscalinglaw.Thesingularity,ontheotherhand,appearstobegovernedbyasinglelengthscalethroughout.Weshowthatthesebehaviorsintheintermediateregionleadtodestabilizationofthepinchregionscalingfollowedbybreakupofthissingularitystruc-A.ThepinchregionThesolutioninthepinchregionisoftheform )isacharacteristiclengthscale,whichgoestozeroasthesingularityoccurs,and)denotesthelocalmini-mumofthesolution.Thevariableisthesimilarityspacevariable.Theexpansioninisinpowersof;wehaveneglectedtermsoforder,whicharesmallerthantheoneswrittenfortherangeofinterest jD1 2h21g8G2g jhGh1G5g j4FSdS111 0sinceisdecreasing.Wenowcomputethefunc-),whichgivestheleadingordercorrectiontotheparabolicshape,forsmall.Todothis,wemustidentifythedominanttermsoneachsideofForEq.suchsymmetricsingularitiesoccurwith,forwhich0.Forlogarithmic-typecorrectionsappear:ournumericssuggest(log),wherehasatmostpolynomialgrowthatin®nity.Then(log),whichissmallerbyafactoroflog.Wethereforearguethatthedominanttermistheleadingone,,andwelookforaconsistentbalanceunderthisassumption.Ontheright-handside

6 ,theleadingtermisthe®rstone, .Balancing
,theleadingtermisthe®rstone, .Balancingthedominantterms,andseparatingvariablesforthetwodependenceson,wedetermineboth 111 2h2gj4jÇd8 d. FIG.17.Symmetricsingularityformationfor0.085.Theneckof¯uidbreaksinthecenter,symmetricallyaboutthepinchpoint. FIG.18.Rescaledthirdderivativeforapproximately15decadesinthecharacteristicwidth1366Phys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner Theonlyapriorirequirementforthescalingexponentisthatthelengthscaleintheintermediateregionbemuchsmallerthanthelengthscalegoverningthepinchregion;thisimplies2.Thesolution5/2hasthescalinglaws,and;thisisconsistentwiththemeasuredscalinglawsfortheacceleratingintermediateregiondiscussedinboththissectionandtheprecedingsec-tion.Inthedeceleratingregime,Figure11showsthattheouterintermediatelengthscaleobeystheapproximatescal-inglaw,suggesting4and.However,asobservedabove,Figure14showsasharperdecreasethanthisinthedeceleratingregion,andmightnotevenobeyastrictpowerlaw.Thisdiscrepancysuggestsapossiblemechanismforin-stabilityofthe``exploding''similaritysolutionsolution:theinnerintermediateregionsobeysascalinglawthandotheouteronesseeFigure11.Denotingtheouterin-termediatelengthscaleandtheinnerintermediatelengthscale,wehaveinthedeceleratingregion.Intheacceleratingregion,,anddecreasesmuchmoreslowlywithdecreas-Recallthattheintermediateregiondictatesthetimede-pendenceofthesingularity,aswellasthevelocityofthepinchpoint.Thefactthattherearedifferenttimedepen-dencesonthetwosidesofthepinchpointmeansthatthesingularityis``frustrated:''shoulditmoveaccordingtothedirectionsofitslefthandoritsright?Aheuristicwayofestimatingwhentheinstabilitywillsetinisasfollows:Wefocusontheinstabilitythatoccurswhenthesingularityisdecelerating;asimilarargumentap-pliestotheacceleratingphase.Thevelocityofthepinchpointdictatedbythedynamicsintheouterintermediatere-gionisthevelocityofthepinchpointdictatedbythedynamicsintheinnerintermediateregionis.When,thedynamicsoftheouter¯owpullstheminimaoutward.However,eventuallythereisatransitioninwhichtheinner¯owhasastrongerin¯uenceonthepinchpoints.Atthispoint,theinner¯owbecomesmoreimportant,andthesingularitybecomesunstable.ThisheuristicargumentsuggeststheinstabilitycriterionAtthetimeoftheinstabilityshownabove,sothecriterionpredictsthatthetransitionshouldoccurwhen.Thisagreesquitewellwiththeactualvalueatthetransition,.Asimilarargumentappliedtotheinstabilityoftheacceleratingsingularitygivesthesamelevelofpredictability.Insummary,weconstructaone-parameterfamilyofad-ditionalsolutionsgoverningtheintermediateregion;atspe-cialparametervaluesthesolutionsareobservedinnumericalsimulations.Theselectionmechanismofthesespecialpa-rametersisnotunderstood;moreover,itisapparentthattheobservedsolutionshavebothhavestableandunstabledirec-tions,asevidencedbyinitialconvergencetothesolutionsfollowedbyaninstability.However,theconstantvelocityintermediateregionappearstoberobust,sincewehaveneverobservedatransitionawayfromtheconstantvelocitysolution.Thestabilityandinstabilityoftheseintermediateregionscanbeheuristicallyexplainedasacompetitionbe-tweenthe¯uidintheinnerandouterintermediateregions.Beforeproceedingtothe

7 nextsection,weaddresshowtheinstabilityth
nextsection,weaddresshowtheinstabilitythresholddependsontheparameterintheinitialconditions.Astheparameterisincreasedtoward0.0664,theintermediateregionconvergestothecon-stantvelocitysolutionandastableexplodingsingularity.Forbelowthisthreshold,thereisalwaysaninstability.Theminimumthicknessatwhichtheexplodingsimilaritysufferstheinstabilitydependsontheinitialcondition.Figure15showshowtheturnaroundthicknessdependson FIG.15.Thethicknessoftheinterfacewhentheexplodingsingularitygoesunstableasafunctionoftheparametercharacterizingtheinitialcondi- FIG.16.ThethicknessoftheinterfacewhentheexplodingsingularitygoesunstableasafunctionfromthecriticalparameterPhys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner solutionfallsslightlyinthecenterwithtwominimathatpropagateoutward,attemptingtobreakatthepointsThesolutioninthepinchregionfollowstheexplodingbehaviorformanydecades:Figure10showthescalinglawsforthecharacteristiclengthscaleinthepinchregion,de®nedasthelengthscaleoverwhichthethicknessdoubles.Thereisalargerangeofscaleswherethescalingbehaviorcoincideswiththatoftheexplodingsingularity.However,thedataintheintermediateregiontellsadif-ferentstory.Figure11shows)versusforthepresentcase.Thesolidlinecorrespondstothemeasurementoftheinsideofthepinchpoints.Thedottedlinecorrespondstoameasurementofontheoutsideofthepinchpoints.Theupperlong-dashedstraightlinecorrespondstothelaw;thelowerdashedstraightlinecorrespondstothescalinglaw.Thisplothasseveralimportantfea-tures:At,theoutsideintermediatescalebeginstofollowthesamelawnotedaboveforearlytimes.However,inthisregime,theinsideintermediatescaledoesnotsatisfyanynoticeablescalinglaw.Atalengthscaleofapproximately10,thereisatransitioninboththeinnerandouterintermediateregions.Beyondthistransition,theouterintermediateregionseemstoobeythescalinglaw;however,theinnerintermediateregiondisplaysasharperdependence,consistentwithThisbehaviorintheintermediateregionhasanimpor-tantconsequence,asnowshown:continuingthesimulationbeyondthelasttimeshownintheprevious®gures),thebehaviorchangesdramatically.Figure12showstheminimumthicknessimmediatelyafterFigure9.Thetemporalbehaviorofthesolutionisdemonstratedbyplottingthelogarithmofversusthelogarithmof,thedistancefromtheoriginofthelocationoftheFigure13.Beyondthecriticalthicknesstheminimaturnaroundandpropagatetowardtheorigin.Afterturningaround,thesolutionapproachesthenewsingularitymechanism,theimplodingsingularity,discussedintheprecedingsection.Apotentialsourceofthedeviationinintermediateregionfromthescalingsolutiondiscussedintheprecedingsectionliesinthevelocityofthepinchpoints.Figure14showsthisvelocityasafunctionoftheminimumthickness,AtearlytimescorrespondingtowheninFiguretheminimaare;atlatertimesingtowhen)theminimaare.AsshowninFigure14,thescalinglawsforthevelocitiesintheacceleratingregimearegivenbyLetusnowreexamineequationgoverningthedy-namicsintheintermediateregion.Solutionstoresultbyassumingthatisnotconstant.Thesesolutionsobeythescalinglaws log~l/j!. FIG.12.Transitionbehaviorfor0.0662,fortimesfollowingthoseofFigure9:solidline,0.00243;dashedline,0.0032.Thenatureofthesolutionhaschangeddramatically,evenatthis

8 extremelyshorttimebefore FIG.13.Theminim
extremelyshorttimebefore FIG.13.Theminimumthicknessasafunctionoftheminimalocations,fortheunstableexploding``singularity''ofFigures9and12.Atthesolutiontransitionstotheimplodingsingularity. FIG.14.Minimumthicknessasafunctionofthevelocityofthepinchpoint,fortheunstableexploding``singularity''ofFigures9and12.Notethepresenceofanacceleratingandadeceleratingregime.Thedottedlinerepresentsthescalinglaw1364Phys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner Thus,thedecayoftheintermediateregiontoaparabolade-terminesthetimedependentscalesinthepinchregion.Notethatanimplicitassumptionofthesescalinglawsisthattheintermediateregionsonbothsidesofeachminimahavethesametimedependences.ThisistestedandshowntobetrueinFigure8.However,wewillseeinthenextsectionthatthisassumptionsometimesbreaksdown,causinginstabilitiesinthesimilaritysolution.Forthesimulationshownabove,wecanindeedseethenontrivialscalingoftheintermediateregion.Figure8showsthelengthscaleoftheintermediateregionasafunctionoftheminimumthickness.Noticethatbeforetheasymp-toticbehaviorsetsin,thereisatransientbehaviorinwhichonlytheinnerside)ofthepinchre-gionssatis®esthescalinglaw.Atthethicknessthereisacrossoverinwhichboththeouter)intermediateregionsandtheinnerintermediateregionsobeythesamescalinglaw,andthescalinglawagreeswiththatproposedinRef.12.Thebehavioroccurringbeforethecrossoverisinterestingandisdiscussedinthenextsection.NotethatFigure7showingthedependenceofshowsnoevidenceofthiscrossover.VI.INSTABILITYOFTHEEXPLODINGSINGULARITYDifferentinitialconditionsfromthoseoftheprecedingsectionleadtoanexplodingsingularitythatdestabilizes,evenafterarbitrarilymanydecadesofscalingtheprecisenumberdependingoninitialconditions.ToillustratethiswhichwasneverobservedinsimulationsoftheforcedHele-Shawcell,orinFigure8,weconditiontheinitialconditionwith0.0662.Atearlytimesthesolutionmimicsthebehaviorshowninthepreviousexample:the FIG.8.Intermediatelengthscaleasafunctionofminimumthickness,forthestableexplodingsingularityofFigure6:solidline,measuredontheoutersideofthepinchpoints;dottedline,measuredontheinnersideofthepinchpoints.Thedashedlineshowsthetheo-reticalprediction FIG.9.Approachtoexplodingsimilaritysolutionfor0.0662atearlytimes.Thesolid,dotted,dashed,anddot-dashedlinescorrespondtotimes2.3,2.42,2.427,2.43,respectively,seeminglyleadingtoa®nitetimesingularityat FIG.10.Pinchlengthscaleasafunctionofminimumthickness,fortheunstableexplodingsingularityofFigure9.Thedottedlineshowsthetheoreticalprediction FIG.11.Intermediatelengthscaleasafunctionof,fortheunstableexplodingsolutionofFigure9:dottedcurve,measuredontheinsideofthesingularity;solidcurve,measuredontheoutside.Theupperdashedlinecorrespondsto,thelowerdashedlinetoPhys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner Thesescalinglawsagreequitewellwiththeevidencefromthenumerics.V.EXPLODINGSINGULARITYForlargervaluesof,solutionsapproachasingularityinwhichtherearetwominimumpointsthatmoveeachotherasthesingularityapproaches.AnexampleofanexplodingsingularityisshowninFigure6fortheinitialAtearlytimestheminimumthicknessisat0.Thissingleminimumthenbifurcatesintotwominima,whichpropagateawayfromeac

9 hother,formingsimultaneoussin-gularities
hother,formingsimultaneoussin-gularitiesat.Severaldifferentscalinglawsareassoci-atedwiththissingularity.Figure7showsthecharacteristiclengthscale,de®nedasthedistanceoverwhichthethick-nessoftheinterfacedoubles,asafunctionoftheminimumThistypeofsingularitywas®rstdiscussedinRef.12,inthecontextofathinnecksqueezedbyexternalpressure.Itisalsoreadilyobservedwhen¯uidisdrainedfromtheneckataconstantrateorwhenthethinneckisforcedbyRayleigh-Taylorinstability.Fordetailsthereaderisreferredtotheabovereferences.Atheoryforthistypeofsingularitywas®rstproposedinRef.12.Belowwesummarizethemajorfeaturesoftheso-lution;ourderivationisslightlysimplerandcorrespondinglylessrigorousthanthatofRef.12.Ourpurposeistoexposethemainfeaturesofthetheoreticalsolutioninordertoex-tendthesolutiontotreatnewphenomenainthenextsection.Threeseparatescalingregionsarepresent,thepinchre-localminimum,theintermediateregion,andtheouterregion.Inthepinchregionthesolutionobeyssothatveryclosetothesingularitythecurrentdoesnotdependonspace.Taking .Here)isthepositionofthepinchpoint.Asinthecaseofthepreviousexampleofthe``implod-ing''singularity,thesolutioninthepinchregiondoesnotdeterminethetimedependenceof.Thistimedependenceresultsfromamatchtoanintermediateregion.Thesolution)toorequivalently.Sincethisbehaviorisindependentof,itcannotdeterminethetimedependenceof.Thuswemustanalyzelower-orderterms.Alittlealgebrashowsthat Thesecond-largesttermintheexpansionmustbematchedtoanintermediateregion.Theintermediateregionhastheleadingorderasymptoticbehavior(ofthepinchregion;thatis,intheinterme-diateregion,hastheformForthesingularitytooccurin®nitetime,mustvanishin®nitetime.We®ndthetimedependenceofbylookingforasimi-laritysolution )isanintermediatelengthscale,andthesimilarityvariableis.PluggingintotheoriginalequationWedeterminetheasymptoticbehaviorofboth)and)by®rstidentifyingleadingtermsin,andthensepa-ThebalancechosenbyRef.12takes3,andconstantandThentheleading-ordertermsof1,thatis,for,givetheordinarydifferentialequationfor SolutionstoforsmallAsymptoticmatchingrequiresthatthesmall-oftheintermediatesolutionagreewiththelarge-ofthepinchsolution.Thatis,wemusthave log~l/j!;~tc2t! log~tc2t!. FIG.7.Minimumthicknessasafunctionofpinchlengthscale,forthestableexplodingsingularityofFigure6.Thedottedlineshowsthetheoreticalprediction1362Phys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner Ifweconsiderallsuchsolutionsto,whenislarge,thesolutioniscompactlysupportedon)with0.Thesesolu-tionsareundesirablebecausetheycannotbematchedtotheotherregions.However,thereisauniquecriticalvalue)suchthatthesolutiontouchesdownlike.Itiseasytoconstructsuchasolutionnumericallyforanybyusingashootingmethod.Noticethat,sincehasa®nitepositivelimitat,anonzero¯uxof¯uidleavesthecentralregion.2.PinchregionInthepinchregionaround,ournumericalevi-denceindicatesthatthecurrent)isconstant,sothatwecanwriteThisisconsistentwiththenumericalandasymptoticstudiesofthein®nite-timeandthe®nite-timesingularitiesintheforcedgeometry.Asdescribedindetailinthosereferences,thissuggeststhatthepinchregionmaybedescribedbyascalingsolutionthattoleadingorder,satis®estheconstant¯uxequatio

10 nthatis,thetimederivativeislowerordertha
nthatis,thetimederivativeislowerorderthanthe¯uxterm(inthePDE.Welookforsolu-tionswiththeself-similarform).Withthespecialchoice1/(2wehavethetime-independentequationforthesimilarityspatialpro®lehasspecialsolutionswith 2h2,h!`,A 8 isanarbitraryconstant.Thetimedependence),orequivalentlythecurrent),aredeterminedbymatchingthissolutiontothecentralandoutersolutions.3.OuterregionIntheouterregion,thesolutionisoftheform)),whereisthelocationoftheminimum.PluggingintoequationTherelevantsolutionhasthepinchregionmovingatcon-stantvelocity,andalso0independentoftime.Thesolutionis arearbitraryconstants.4.MatchingconditionsTocompletethesolutionsitisnecessarytomatchthethreeregionstogether.Therearematchingconditionscon-nectingthepinchregiontoboththeouterregionandthecentralregion.Forthepinchsolutiontomatchontotheoutersolution,werequireThismatchingconditionrequiresthatboththetimedepen-dencesandthespatialdependencesbalance.Thisbalanceisachievedbytaking1/2,forthepinchsolution,andtaking0fortheoutersolution.Also,thepinchsolutionmustmoveatthevelocitydictatedbytheoutersolution.Matchingthepinchsolutionontothecentralsolutionrequiresthat Thisconditioncanbesatis®edbytaking.Also,forthetimedependencestomatch,the¯uxof¯uidleavingthecentralregionmustenterthepinchregion.The¯uxleavingthecentralregionisoforderThe¯uxenteringthepinchregionisoforder.ThuswehavethescalinglawFinally,notethattheminimumpointmovestowardtheori-0withvelocity(.Demandingthattheouteredgeofthecentralregionmoveswiththesamevelocityastheinneredgeoftheouterregion®xesPuttingtheseconditionstogethergivethescalinglaws FIG.6.Explodingsingularityat0.07.Thesolid,dotted,anddot-dashedcurvescorrespondtotimes0,2,2.4,2.422,respectively,corre-spondingtoasingularityatPhys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner Theinitialcondition,withasinglelocalminimum,bi-furcatesintotwominima,whichthenpropagatetowardtheorigin,mergingatthesingulartime.Severalnontrivialscal-inglawsareassociatedwiththissingularbehavior.Figure4showstheheightatthepinchpoints,),asafunctionoftheirposition).Asthethicknessapproacheszero,0,thepinchpointspropagatetowardtheorigin,0.ThedataisconsistentwiththescalinglawAnotherrelevantquantityis(0,),thethicknessoftheinterfaceatthecenter0.Asthepinchpointspropa-gatetowardtheorigin,mass¯owsawayfromtheneighbor-hoodoftheorigin,sothat(0)decreases.Figure4also(0)asafunctionofthepinchlocation).ThedataisconsistentwiththescalinglawThe®nalrelevantscalinglawisthecharacteristicwidtharoundthepinchpointasafunctionof.Aswede-terminebytheexplicitconstructionbelow,thesingularityhasalocallyparabolicstructure,sothatasveri®edinFigure5.B.ConstructionofsimilaritysolutionNowweconstructaone-parameterfamilyofsimilaritysolutionstoequation.Ataspecialvalueofthisparameterthesimilaritysolutionwelldescribesthebehaviorpresentedabove,bothinexplainingthescalinglawsandinpredictingtheshapeoftheinterfaceclosetothesingulartime.Thesingularityhasthreedifferentself-similarscalingregions,whichwecallthecentralregion,thepinchregion,andtheouterregion.Thescalingexponents,andhencethetimede-pendence,aredeterminedbyrequiringthatthesolutionsinthethreeregionsmatch.Thec

11 onstructionissimilarinspirittothedescrip
onstructionissimilarinspirittothedescriptionofthein®nite-timesingularityintheforcedHele-Shawcell.Weproceedby®rstdescribingthesolutionineachregionseparately,andthenmatchingthemtogethertodeterminethescalingexponents.1.CentralregionInthecentralregion,welookforanexactsimilaritysolutionoftheform inwhichthetimedependence),thepro®lefunction),andtheexponentaretobedetermined;heredenotesthesimilarityvariable.Substituting,andthenseparatingthedependences,givesthetwoequationsimpliesthat1/4.Thisisanexampleofanexactsimilaritysolutiontotheequation.Althoughitdescribeswellthestruc-tureinthecenteroftheimplodingsingularitypro®le,suchanexactsolutionhasnotbeenobservedinthepinchregionofasingularityofthelubricationequationforany.ItisnotknownwhethersuchasolutionexistsforanyqSolutionstothatdescribethecentralregionareknowntoexistandarecompactlysupportedin.Thesesolutionssatisfythesymmetryconditions FIG.4.Minimumpositionvsminimumthickness,andthicknessatorigin(0)fortheimplodingsingularityofFigure3.Theuppermostdottedlinerepresentsthescalinglaw,whilethelowermostdottedlinerepresentsthescalinglaw FIG.5.Pinchlengthscalevsminimumthickness,fortheimplodingsingularityofFigure3.Thedottedlinerepresentsthescalinglaw1360Phys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner sistentmatchbetweenthesetworegionsisnecessaryforthesingularitytopersist.Wearguethatthebreakdownofthismatchisresponsibleforitsdestabiliation.Symmetricsingularity.Forvaluesofgreaterthanbutlessthan0.09,asymmetricsingularityforms:thesolutionissymmetricaboutasingleminimumat0,and(0,0.Thissolutionwas®rstdiscussedinRef.13andlaterinRef.14.ThisbehavioralsopossessesanintermediateNosingularity.Iftheinitialheightisnotsmall,thentheinitialblobbynecksimplyrelaxestoaconstant¯atneck,asexpectedbasedontheoreticalresults.behaviorissummarizedinthephasediagramofFigure2.Thetransitionsbetweendifferentsingularitybehaviorsaremorecomplicatedthanthosefoundinpreviousstudies.Ononesideofeachcriticalparametervalue,thesolutionappearstoexhibittheneighboringbehavior,showingclearself-similarscalingovermanydecades.However,atsomecriticalneckthickness,thesolutionchangesbehaviorintothetruebehaviorforthatparametervalue.Theneckthicknessatwhichthistransitionoccursbecomesarbitrarilysmall,andthenumberofdecadesofdeceptiveself-similarityincreasesasthevalueofmovesclosertothecriticalvalue.Thus,asillustratedinFigure2,whenisjustsmaller,thesolutioninitiallyexhibitsthebehaviorcharac-teristicofthe``explodingsingularity''untiltheminimumreachesaverysmallvalue,atwhichthebehaviorchangestotheimplodingsingularity.Thischangeoverisseendramaticallyinthefactthatthedistancebetweenthetwominimaceasestoincreaseandstartstode-crease.Similarly,forjustsmallerthan,thesymmetricsingularitybehaviorisobservedformanydecadesbeforetheminimumbifurcatesintotwoexplodingsingularities.Forjustlargerthan,thesolutioninitiallyexhibitsthefea-turesofthesymmetricsingularity,beforereversingitselfandrisingtowardtheuniformstate.Theevidenceforourargumentsisbasedonlyonnu-mericalsimulation;hencewecannotexcludethepossibilitythatfurthersurprisesoccurateventhinnerneckwidths.However,amajorpointofthispaperisidenti®ca

12 tionofthebetweenthedeceptiveself-similar
tionofthebetweenthedeceptiveself-similarbehaviorandtheultimatesingularitystructurethatemergesuponen-hancedresolution.Inmanyinstances,theonlyobserveddif-ferencebetweenanapparentlystablestructureandanun-stableoneisthatofdifferentbehaviorintheintermediatematchingregions.Aremainingmajorchallengeistocon-structageneralexplanationofwhythesedifferentscalingsultimatelydestabilize.Inthefollowingthreesections,wediscusseachofthe®nite-timesingularitymechanisms.InSectionIVwepresentnumericalevidenceforthe``imploding''singularitymecha-nismandderiveasimilaritysolutionthatreproducesitsscal-ingproperties.Wehaveneverobservedinstabilityofthissolutionandtransitiontoadifferentbehavior.InSectionV,wediscussthe``exploding''singularityandgeneralizetheknownsimilaritysolutiontoaone-parameterfamilyofsolutionswithdifferentscalingexpo-nents.Threemembersofthisfamilyappearinourcalcula-thesolutionofRef.12,withpinchpointsmovingapartataconstantvelocity;asolutionwithacceleratingpinchpoints;asolutionwithdeceleratingpinchpoints.Thelattertwocasesapparentlysufferaninstabilityata®nitebutarbitrarilysmall,dependingoninitialdataneckthick-ness.Weprovideapartialexplanationoftheinstabilityofthenewmembersofthisfamily,basedonamismatchinvelocitiesbetweentheintermediateregionsonthetwosides.InSectionVIwediscussthesymmetricsingularitymechanism.Herewedistinguishbetweentwotypesofsolu-tion,onlyoneofwhichappearstobestable,andweprovideapartialexplanationofthedestabilizationandtransitiontotheexplodingsingularity.IV.IMPLODINGSINGULARITYInthissectionwediscussnumericalobservationsandascalinganalysisfortheimplodingsingularity.A.NumericalobservationandscalingrelationshipsWhentheinitialminimumthicknessisverysmall,thesolutionapproachesasingularityinwhichtwopinchpointspropagatetowardeachother,mergingatthesingulartime.Thisisanewtypeofsingularity;ithasnotbeenseenintheforcedgeometryandisreportedhereforthe®rsttime.Figure3illustratesthisbehaviorfor FIG.2.Phasediagramofsingularityformationforsolutionsofinitialdata.Theshadedbarsindicate``instabilities'':rangesofvaluesofinwhichthesolutionexhibitsmanydecadesofscalingcharacteristicoftheneighboringsingularity,beforechangingtothetruesingularitybehavior. FIG.3.Implodingsingularityat0.01.Thesolid,dotted,dashedandlongdashedcurvescorrespondtotimes0,3.3,3.4,3.9,respec-Phys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner Theapproximationsthatpermitreductionfromtheorigi-nalproblemtothetwo-dimensional,andthentotheone-dimensional,modelsystemsbreakdownastheneckbe-comesverythinandthesingularityisformed.Henceitisnaturaltoquestiontherelevanceofthesingularitiesstudiedheretotheoriginalproblem.Basedbothonourasymptoticsolutionsnearthesingularityandnumericalcomputationsofthefullequations,thelubricationapproximation,thatis,thereductionofthetwo-dimensionalHele-Shawsystemtotheone-dimensionallubricationmodel,appearstoremainvalidaspinchoffoccurs.However,theconstructionoftheHele-Shawmodelitselfbreaksdownwhentheneckwidthbecomessmallerthantheplatespacingduetotheeffectofthesecondcomponentofcurvature.Wedonotaddressthisquestioninthispaper,exceptforthefewremarksinSectionVIII.Thelubricationequationisnatu

13 rallygeneralizedtoinwhich1isthethin-neck
rallygeneralizedtoinwhich1isthethin-necklimitofHele-Shaw¯ow,andthecase3isobtainedfromthedynamicsofathinliquidlayeronarigidsurface.Varyingvariesthenatureofthedegeneracyas0.Thedynamicsofsolutionsmaythenbeexploredasafunctionofanditappearsthat1isanespeciallydif®cultborderlinecase.III.SIMPLEINITIALDATAThispaperexaminesthebehaviorofsolutionstoequa-withaoneparameterfamilyofinitialconditions1.5cos0.6cos20.1cos3Figure1.Thisdataisperiodicin,andthusthe)remainsperiodic.Alternatively,wecouldim-pose``neutral''boundaryconditions0attheendsofa®nitedomain;intheHele-Shawsystemtheseconditionscorrespondtonotransportofmassinoroutofthedomainandtonormalcontactangle.Wechosetheinitialdatatobecompletelysmooth,withasingleminimumofheight,atwhichthe®rstfourspatialderivativesofvanish.Theseconditionsproduceanarrayofbulgeswithavery¯atinterconnectingregionsof.Thecurvaturetakesapositivemaximumvalue(1/)cosAsaconsequenceoftheaboveconditions,0forneartobutgreaterthanzero;hence,underthedynamics0onthesameregion.Thus,atleastforshorttimes,thefourth-orderdynamicswilldrivetheneckthicknesstodecreasebelowitsinitialvalue.Itisthismechanismthatpermitssingularityformation,andthatwouldbeabsentinasecond-orderequation.Webelievethatthephenomenaweobserveinthispaperforinitialdataaregeneric,inthesensethattheywouldbeobservedforavarietyofinitialconditions.Itisnecessarythattheinitialdatabeextremely¯at;previousexplorationswithsimilardatacontainingonlytwoFouriermodesexhib-itednosingularity.Ournumericalsimulationsusea®nite-differencewithadynamicallyevolvingadaptiveThecodesarewelltestedandcanfullyresolvemanydecadesofbehaviorintheapproachtothesingularity.Althoughthesingularitieshavestructuresthatlocallyhaveasimpleself-similarform),thetimedependentquantitiescannotbedeterminedbydimensionalanalysis.Moreover,unlikethetypicalscenarioinsuch``secondtype''scaling,theanaly-sisofthesimilaritysolutiondoesnotinvolvesolvinganon-lineareigenvalueproblembutinsteadinvolvessolvingmatchingconditionsbetweendifferentregionsofthesolu-tioneachofwhichhasitsownsimilarityscaling.Itisthisorlackthereofthatwefocusoninanalyzingtheinstabilitiesdescribedinthefollowingsections.Asisvar-ied,thesimulationsshowseveraldifferentpossiblebehav-iorsofthesolution,threeofwhichleadto®nitetimesingu-laritieswiththesekindofwell-de®nedself-similarscalingImplodingsingularity.Forinitialconditionswith0.0665,thesolutiondevelopstwolocalminimathataremirrorimagesofeachother.Theymovetowardseachotheratroughlyconstantspeed,coalescingat0atthesingulartime.Thissingularityisreportedonforthe®rsttimehere.Theexampleexhibitsthreedifferentscalingre-gions,theregionaroundalocalminimum,thecentralregionbetweenthetwopinchpoints,andtheouterregionawayfromthesingularity.WepresentamatchingargumentinthenextsectiontopredictthetimedependencesofthethreeExplodingsingularity.Forvaluesofgreaterthanbutlessthan0.0825,thesolutiondevelopestwominimathatmoveapartwithroughlyconstantspeed,withthepinchoff0occurringattwolocations.Thisbehaviorwas®rstobservedinRef.12forpressureboundaryconditions.Asdiscussedinthatpaper,thissingularityhasaninnerself-similarregionandanintermediateregion.Acon- FIG.1.Thein

14 itialconditionforparametervalue0.1.Ifnos
itialconditionforparametervalue0.1.Ifnosingu-larityformsin®nitetime,thesolutionmustrelaxtoitsaverage1358Phys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner Hele-Shawcellforcedbyexternalpressuresorimposed¯ow,severaldifferenttypesoflocalsimilaritysolutionswithdifferentscalingpropertiescanoccur:an``in®nite-timesingularity'';a®nite-timesingularityinwhichthepinchpointmoveswithconstantvelocity;anda®nite-timesingularityinwhichthepinchpointisstationary.ingtheinitialandboundaryconditionsleadstothedifferentsingularbehaviors.Moreover,thesamearrayofsingularitystructuresisobservedinatwo-dimensionallayerforcedbydensitystrati®cationingravity.ThispaperaddressesruptureofathinneckinaHele-Shawcellintheabsenceofforcing.Basedonthegeneralprinciplesofuniversalityoutlinedabove,onewouldexpectthatthelocalnatureofsingularitiesshouldbenodifferentintheunforcedcasethanintheforcedcase.However,ourstudyrevealssomequalitativefeaturesofthesingularitiesthathavenotpreviouslybeenobservedintheforcedcase.Moststrikingly,wediscoverthroughnumericalsimulationsthattwoofthethreesimilaritysolutionscanatanarbitrarilysmalltimedistancefromthesingularityforappro-priateinitialconditions.Aftertheinstabilitythesolutionde-velopsasingularityatalatertime,byadifferentsimilaritysolution.Wealso®ndanewsimilaritysolutionthathasneverbeenobservedintheforcedHele-Shawcell.Thismechanismappearstobemorestablethantheothertwo.OurinvestigationoftheunforcedHele-Shawcellusesaoneparameterfamilyofinitialconditionswheretheparam-correspondstothewidthofthe``thread''connectingthelargerdropsintheinitialinterfaceshape.Forsuf®cientlythe``blobby''neckrelaxestoa¯atneckwithoutbreaking.However,forsmallervaluesofthenecktriestobreakupbyacomplexsequenceofsimilaritysolutions.Inthispaper,wediscussthescalingandstabilityofeachofthesesolutionsthroughhighlyresolvednumericalsimula-tions;manyfeaturesofthesimulationsareexplainedthroughasymptoticanalysis.II.REVIEWOFGOVERNINGEQUATIONSInaHele-Shawcell,athinlayerofviscous¯uidmovesbetweentwonarrowlyspacedglassplates.Typically,the¯uiddoesnot®lltheentiregap;theremainingspaceis®lledbya¯uidofnegligibleviscositysuchasair.The¯uidandtheinterfacemoveunderthein¯uenceofsurfacetensionandapossibleexternalforcing;viscosityinthe¯uidresistsmo-tionviatheno-slipconditionontheplatesurfaces.Thefullsystemisdescribedbythethree-dimensionalincompressibleNavier-Stokesequationswithinthe¯uidlayer,togetherwiththeLaplacepressureconditionatthe¯uid/airinterface,cou-plingthemeancurvatureoftheinterfacesurfacewiththepressuredropacrosstheinterface.Iftheplateseparationismuchsmallerthananytrans-versedimension,thenthewell-knownHele-Shawtwo-dimensionalmodelsystemisaverygoodapproximation.ByDarcy'slaw,thedepth-averaged¯uidvelocity)isthegradientofanondimensionalizedvelocitypotentialproportionaltonegativepressureandde®nedinthetwo-dimensional¯uidregion.Byincompressibility,ishar-monicinateachtime:0.TheLaplacepressureconditionbecomes,inwhichisthetwo-dimensionalcurvatureofthe¯uid/airinterfacecurve.ThisDirichletproblemissolvedateachinstantoftime,andma-terialconsistencyrequiresthattheinterfacenormalvelocity.Formoredetails,

15 seereviewarticlessuchasRefs.16and17.Wear
seereviewarticlessuchasRefs.16and17.Weareinterestedinchangesoftopologyinthesystem.Forexample,wetakea¯uiddropletof®nitesize,surroundedbyair,andaskwhetherforsomeinitialshapesthedropletcanbreakintotwoormoredroplets.Ithasbeenthatiftheinitialshapeisclosetoacircle,thenthedropletshaperelaxestoacircleinin®nitetime.Recentnu-mericalworkindicatesthatiftheinitialdroplethastheformofadumbbellwithathinneck,itcanbreakapartsim-plyasaresultofsurfacetension.Intheregionofthethinneck,alubricationapproxima-tionreducesthetwo-dimensionalHele-Shawsystemtoaone-dimensionalmodel.Wedenoteby)thehalf-widthoftheneck;thenassumingthat1,andthat1sothatthepressure),yieldsthe``lubricationequation''Thisequationalsofollowsfromasystematicasymptoticex-pansioninWeareinterestedintheruptureofthinnecks,when0inata®nitetime.Inthelubrica-tionmodelthefourth-ordertermisandplaysaninterestingroleintheformationofsingularities.Simpleshowsthatsuchvanishingofrequiresthatatleastthefourthspatialderivativeofmustbecomein®nite;thuswearejusti®edincallingsucheventssingularities.Forcirculardroplets,thelubricationequationisnotuniformlyvalidoverthewholeliquidregion;itbreaksdownwheretheendscloseoff.InRef.18thelubricationmodelwasusedinthecenteroftheneck,withboundaryconditionstakenfromanoutersolution.Themodelcanbeextendedtohandleclosedends,asinRef.2forthree-dimensionalliquidcolumns.Alternatively,specialphysicalboundaryconditionscanforcetheentirenecktobethinand¯at,asinRefs.11and12.Toavoidthesecomplications,weconsiderometry,ratherthanaclosed®nitedrop.Thatis,inplaceofaweconsideraninitialcon®gurationinwhichtheliquidformsanin®nitesequenceofbulgesseparatedbythinnecks.Inorderforthelubricationapproximationtoholdthroughoutthebreakingoftheneck,itmustremainthinand¯atforalltime.Webelieve,andthenumericalresultsofRef.18providepartialcon®rmation,thatthelocaldynamicsofthethinneckarethesameinthisgeometryasforacloseddroplet.Inperiodicgeometry,theneckofconstantthicknessisaglobalattractorforthelubricationapproximation.BesidestheintrinsiceleganceandsimplicityoftheHele-Shawmodel,partofitsappealcomesfromthefactthatthemathematicalformulationdescribesseveraldifferentphysi-calproblems,includingsolidi®cationintheone-sidedlow-undercoolinglimitandpopulationdensityinherbivore/plantdynamics.Thus,forexample,thesingularitiesstudiedheredescribenotonlythepinchingofa¯uidneckinaHele-Shawcell,butalsothesingularitiesproducedbetweentwoparticlesgrowingtogetherinOstwaldripening.Phys.Fluids,Vol.8,No.6,June1996Almgren,Bertozzi,andBrenner Electronic-mail:almgren@math.uchicago.eduElectronic-mail:bertozzi@math.duke.eduTowhomcorrespondenceshouldbeaddressed;Electronic-mail:1356Phys.Fluids(6),June19961070-6631/96/8(6)/1356/15/$10.001996AmericanInstituteofPhysics StableandunstablesingularitiesintheunforcedHele-ShawcellRobertAlmgrenTheUniversityofChicago,DepartmentofMathematics,Chicago,Illinois60637AndreaBertozziMathematicsandComputerScienceDivision,ArgonneNationalLaboratory,Argonne,Illinois60439,andDukeUniversity,DepartmentofMathematics,Durham,NorthCarolina27708MichaelP.BrennerDepartmentofMathematics,MassachusettsInstituteofTechnology,Cambridge,Massach