ouslydifferentiableintheirargumentsWedenotethematerialderivativeofthevelocity ID: 385132
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Authortowhomcorrespondenceshouldbeaddressed.Electronicmail:PHYSICSOFFLUIDS,017102/017102/7/$23.00©2008AmericanInstituteofPhysics,017102-1 ouslydifferentiableintheirarguments.Wedenotethemate-rialderivativeofthevelocityÞeldby u denotesthegradientoperatorwithrespecttothespatialvariabledenotethepathofasphericalparticleofden-immersedintheßuid.Lettheparticlebesphericalwithradius1,letdenoteacharacteristiclengthscaleintheßow,letRedenotetheReynoldsnumber,andletbetheLagrangianvelocityofthesphericalparticle.TheparticlesatisÞestheMaxeyÐRileyequationofmotioncf.,e.g.,Benczik,Toroczkai,andTlorBabianoetal. vu,1where=1 =2 3St,St= 9a Re.Notethatthelargertheinertiaparameter,thelesssigniÞ-canttheeffectofinertia;inthelimit,Eq.themotionofapassiveidealtracerparticle.HallerandSapsisprovedthatfor0smallenough,admitsagloballyattractinginvariantslowmanifold.Forneutrallybuoyantparticles,ithastheformfornon-neutrally-buoyantparticles,isgivenbyagraph,ThedynamicsonisgovernedbythereducedMaxeyÐRileyequationsinertialequationi.e.,bytheequationofmotionforinÞnitesimalßuidUsinganobservationofBabianoetal.wecancon-cludethattheinvariantmanifoldandthecorrespondingreducedequationexistforallvaluesofintheneutrallybuoyantcase.SpeciÞcally,Eq.isequivalenttoAspointedoutbyBabianoetal.thislastequationcanberecastintheformor,equivalently, vux,tux,t+Iux,t4d ThisshowsthatdeÞnedinEq.isaninvariantmani-foldforanyIII.GLOBALATTRACTIVITYOFTHESLOWexistsforany0,itisnotguaranteedtobegloballyattractingforlargervaluesofi.e.,forsmallervaluesof.HerewegiveasufÞcientconditionunderwhichtheglobalattractivityofisguaranteed.Theorem1.AssumethatforsomeÞxedthesmall-esteigenvalueÞeldofthesymmetrictensorÞeldisuniformlypositiveforallandtThentheinvariantmanifoldMisgloballyattracting,i.e.,allneutrallybuoyantparticlemotionssynchronizeexponentiallyfastwithinÞnitesimalLagrangianßuidtrajectoriesProof:ApplyingthechangeofcoordinatestoEq.,weobtainthesystemNotethattheinvariantsubspaceofthisequationcor-respondstotheinvariantmanifoldforany.WeÞxasolutionofEq.,substitutethesolutioninto,andmultiplythecomponentoftheresultingequa-tionbytoobtain 2d Herewehaveintroducedtherate-of-straintensor andusedthenotationtorefertothemaximaleigen-valueofatensor.IntegrationofEq.referringtotheminimaleigenvalueoftheappro-priatetensor.Weconcludethatthe=0subspaceisgloballyattractingifforall.ThisiscertainlysatisÞedifforsomepositiveconstantandforallFortwo-dimensionalincompressibleßows,isthesmallerrootofthecharacteristicequation=0.Therefore,thetwo-dimensionalversionofthesufÞcientcon-ditioninTheorem1requiresthat detor,equivalently,017102-2T.SapsisandG.HallerPhys.Fluids,017102 holdsuniformlyforallIV.LOCALDIVERGENCEALONGTHESLOWEveniftheabovesufÞcientconditionfailstohold,canstillbegloballyattractive,althoughtrajectoriesmaytem-porarilydivergefromwhiletheypassbyregionsofdi-vergenceon.Asthefollowingresultshows,theseregionsoflocaldivergencearepreciselythedomainswherethesuf-ÞcientconditionofTheorem1isviolated.Theorem2.ForanytheinvariantmanifoldMrepelsallcloseenoughtrajectoriesaslongastheysatisfyProof:ForaÞxedinitialcondition,letnotethefundamentalmatrixsolutionoftheÞrstequationof.BydeÞnition,wethenhaveWesuppressthedependenceoffornotationalsimplicity.Observingthat,weÞndthatperturbationstothemanifoldwillgrowordecaydependingontheeigenvalueconÞgurationoftheCauchyÐGreenstraintensor.SpeciÞcally,ifhasaneigenvaluelargerthan1foragiven,typi-perturbationstowillincreaseinnormovertheOurinterestisÞndinglocationsofinstantaneousgrowthinthedirectionatpointsofthemanifold.Byinstanta-neousgrowth,wemeangrowthinthelimitof.Notethatinthatlimit,alleigenvaluesoftendto1,thusinstantaneousgrowthisgovernedbyhowtheuniteigenval-uesofperturbawayfromunityasisincreased.Toseethis,wewillcalculatetheinstantaneoussta-bilityindicator=lim whichisaLyapunov-exponent-typequantity,exceptthatthelimitistakenat,asopposedtoSincewehaveassumedthatistwicecontinuouslydifferentiable,standardregularityresultsforordinarydiffer-entialequationsguaranteethatisalsotwicecontinuouslydifferentiable.Thus,wecanexpandwithrespecttotimeintoaTaylorseriesoftheform.UsingEq.,weobtainFortheCauchyÐGreenstraintensorweobtaintheexpression=1,...denotetherealeigenvaluesandthecorrespondingeigenvectors,respectively,ofthesym-metrictensor.Becauseissemisimple,itsandeigenvectorswillbedifferentiablewithrespecttotimeatsee,e.g.,Lancaster,.Thus,byexpandingtheeigenvalueprobleminaTaylorseriesin,weobtaintermsinthislastequation,weobtain FIG.1.ColoronlineExactdomainofneutrallybuoyantparticlesÕdivergenceyellowonlineanditslowerestimateencircledbyblackbythecriterionofBabianoetal.fortwodifferenttimes.017102-3InstabilitiesinthedynamicsofneutrallybuoyantparticlesPhys.Fluids,017102 Thisthree-parameterfamilyofspatiallyperiodicßowspro-videsasimplesteady-statesolutionofEulerÕsequations,asshownbyArnold.Fortheßowparameters,wechoose=12,=15,=10;forourinertiaparameter,weselect=0.075.Inourcomputationofinertialparticletrajec-tories,weagainuseafourth-orderRungeÐKuttaalgorithmwithabsoluteintegrationtolerance10InFig.,weshowtheunstableregionsoftheßowsat-isfyingEq..Thecoloringinsidethesedomainsillustrateshowthestabilityindicatorvariesinsidetheunstablevolumes.InthesameÞgure,thearrowsshowthevelocityvectorsInFig.,wepresenttwosnapshotsofinertialparticledynamicsviewedinthespaceá,á,,whereistheinstantaneouscoordinateoftheparticle.Weshowtheslowmanifoldcomputedforeachtimeattheinstantaneousverticalparticleposition;regionsen-circledbytheblacksolidcurvesredregionsonlinethedomainoflocaldivergenceontheslowmanifold.ThesmallersubplotsintheÞgureshowthedistanceoftheparticlesfromtheslowmanifold.Notehowparticlesarere-pelledbytheslowmanifoldoverthedomainoflocaldiver-gence,whileparticlesconvergetotheslowmanifoldovertheregionsthatareoutsideoftheblackcurvesblueregionsVI.CONCLUSIONSWehavederivedananalyticcriterionforregionsofdi-vergenceontheslowmanifoldthatgovernstheasymptoticmotionofneutrallybuoyantparticlesinageneralunsteadyßuidßow.WehavealsoshownthatanearlierformulafortheseunstableregionsbyBabianoetal.alwaysgivesalowerestimate.Furthermore,wehavederivedasufÞcientcriterionfortheglobalattractivityoftheslowmanifold.Un-derthiscondition,allinertialparticlemotionssynchronizewithLagrangianßuidmotion.Weillustratedourresultsoninertialparticledynamicsinatwo-dimensionalmodelofthevonKrmnvortexstreetinthewakeofacylinderbyJung,Tl,andZiemniakandintheclassicthree-dimensionalsteadyABCßow.Studyingthetopologyofregionsofdivergenceontheslowmanifoldforthree-dimensionalßowsrequiresfurtherwork.Anapplicationofthepresentresultstoidentifyingregionsofinaccuraciesforparticleimagevelocimetrywillbegivenelsewhere.WethankHarrySwinneyandTamsTlforusefuldis-cussionsonthesubjectofthispaper.WealsothankJohnWeissforsuggestingtheABCßowasanexample.ThisresearchwassupportedbyNSFGrantNo.DMS-04-04845,AFOSRGrantNo.AFOSRFA9550-06-0092,andaGeorgeandMarieVergottisFellowshipatMIT.M.R.MaxeyandJ.J.Riley,ÒEquationofmotionforasmallrigidsphereinanonuniformßow,ÓPhys.Fluids,883E.E.Michaelides,ÒThetransientequationofmotionforparticles,bubbles,anddroplets,ÓJ.FluidsEng.119,233J.Rubin,C.K.R.T.Jones,andM.Maxey,ÒSettlingandasymptoticmotionofaerosolparticlesinacellularßowÞeld,ÓJ.NonlinearSci.T.J.Burns,R.W.Davis,andE.F.Moore,ÒAperturbationstudyofparticledynamicsinaplanewakeßow,ÓJ.FluidMech.E.MograbiandE.Bar-Ziv,ÒOntheasymptoticsolutionoftheMaxeyÐRileyequation,ÓPhys.Fluids,051704G.HallerandT.Sapsis,ÒWheredoinertialparticlesgoinßuidßows?ÓPhysicaDtobepublishedA.Babiano,J.H.E.Cartwright,O.Piro,andA.Provenzale,ÒDynamicsofasmallneutrallybuoyantsphereinaßuidandtargetinginHamiltoniansystems,ÓPhys.Rev.Lett.,5764R.D.Vilela,A.P.S.deMoura,andC.Grebogi,ÒFinite-sizeeffectsonopenchaoticadvection,ÓPhys.Rev.E,026302A.J.Szeri,S.Wiggins,andL.G.Leal,ÒOnthedynamicsofsuspendedmicrostructureinunsteady,spatiallyinhomogeneous,two-dimensionalßuidßows,ÓJ.FluidMech.,207R.T.PierrehumbertandH.Yang,ÒGlobalchaoticmixingonisentropicsurfaces,ÓJ.Atmos.Sci.,2462G.Boffetta,G.Lacorata,G.Redaelli,andA.Vulpiani,ÒDetectingbarrierstotransport:Areviewofdifferenttechniques,ÓPhysicaD,58C.Jung,T.Tl,andE.Ziemniak,ÒApplicationofscatteringchaostoparticletransportinahydrodynamicalßow,ÓChaos,555I.J.Benczik,Z.Toroczkai,andT.Tl,ÒSelectivesensitivityofopenchaoticßowsoninertialtraceradvection:Catchingparticleswithastick,ÓPhys.Rev.Lett.,164501P.Lancaster,ÒOneigenvaluesofmatricesdependentonaparameter,ÓNumer.Math.,377V.I.Arnold,ÒSurlatopologiedesecoulementsstationnairesdesßuidesparfaits,ÓC.R.Hebd.SeancesAcad.Sci.,17017102-7InstabilitiesinthedynamicsofneutrallybuoyantparticlesPhys.Fluids,017102