NONLINEAR DYNAMICS OF ELECTRIFIED THIN LIQUID FILM

SIAMJ.APPL.MATH.c2007SocietyforIndustrialandAppliedMathematicsVol.67,No.5,pp.1310–1329NONLINEARDYNAMICSOFELECTRIFIEDTHINLIQUIDFILMS∗DMITRITSELUIKO†ANDDEMETRIOST.PAPAGEORGIOU†Abstract.Westudyanonlinearnonlocalevolutionequationdescribingthehydrodynamicsofthinfilmsinthepresenceofnormalelectricfields.Theliquidfilmisassumedtobeperfectlyconductingandtocompletelywettheupperorlowersurfaceofahorizontalflatplate.Theflatplateisheldatconstantvoltage,andaverticalelectricfieldisgeneratedbyasecondparallelelectrodekeptatadifferentconstantvoltageandplacedatalargeverticaldistancefromthebottomplate.Thefluidisviscous,andgravityandsurfacetensionact.Theequationisderivedusinglubricationtheoryandcontainsanadditionalnonlinearnonlocaltermrepresentingtheelectricfield.Theelectricfieldislinearlydestabilizingandisparticularlyimportantinproducingnontrivialdynamicsinthecasewhenthefilmrestsontheuppersideoftheplate.Wegiverigorousresultsontheglobalboundednessofpositiveperiodicsmoothsolutions,usinganappropriateenergyfunctional.Wealsoimplementafullyimplicitnumericalschemeandperformextensivenumericalexperiments.Throughacombinationofanalysisandnumericalexperimentswepresentevidencefortheglobalexistenceofpositivesmoothsolutions.Thismeans,inturn,thatthefilmdoesnottouchthewallinfinitetimebutasymptoticallyatinfinitetime.Numericalsolutionsarepresentedtosupportsuchphenomena,whicharealsoobservedinhangingfilmswhenelectricfieldsareabsent.Keywords.thinfilm,electrohydrodynamics,nonlocalevolutionequationAMSsubjectclassifications.76D03,76D08,76D27,76E17DOI.10.1137/0606635321.Introduction.Aviscousliquidfilmwettingtheuppersurfaceofaflathor-izontalsubstrateisexpectedtobestable,undernormalconditions,andeventuallyreturnstoitsuniformundisturbedvalueofthefilmthickness.Ifthefilmishanging(i.e.,itwetstheundersideofthesubstrate),thengravityisdestabilizing.Thispaperisconcernedwiththeadditionofelectricfieldsnormaltothesubstrate.Usingexper-imentsandlineartheory,TaylorandMcEwan[26]observedthatasufficientlystrongfieldcanovercomeviscousforcesinoverlyingfilmsandinducewavyperturbations.Weaimtomodelandanalyzethenonlinearstagesofthisphenomenonforoverlyingandhangingfilms.IntheabsenceofelectricfieldstheproblemwasstudiedbyEhrhardandDavis[9],whoconsideredspreadingofviscousdropsonsmoothhorizontalsurfaceswhichareuniformlyheatedorcooled.Theirisothermalevolutionequationfortheinterfaceco-incideswithourswhenthereisnoelectricfield.YiantsiosandHiggins[30]consideredthebehaviorofaviscousfilmboundedbelowbyawallandabovebyanunboundedsecondheavierimmisciblefluid.ForthecasewhentheBondnumberB1(itmea-surestheratiobetweengravitationalandinterfacialforces)andtheviscosityratiom=μ1/μ2isO(1),whereμ2isthefilmviscosity,theyobtainedthesameevolutionequationfortheinterfaceasdidEhrhardandDavis[9].Ehrhard[8]usedthemodelderivedbyErhardandDavis[9]todescribethequasi-steadyevolutionofaviscousdrophangingontheearth-facingsideofasmoothhorizontalplate,whichiseitheruni-∗ReceivedbytheeditorsJune22,2006;acceptedforpublication(inrevisedform)March15,2007;publishedelectronicallyJune21,2007.†DepartmentofMathematicalSciencesandCenterforAppliedMathematicsandStatistics,NewJerseyInstituteofTechnology,Newark,NJ07102([e-mail],[e-mail]).ThefirstauthoracknowledgessupportfromNJITthroughitsPresidentialResearchInitiative.TheworkofthesecondauthorwassupportedbyNationalScienceFoundationgrantnumberDMS-0072228.1310NONLINEARDYNAMICSOFELECTRIFIEDTHINLIQUIDFILMS1311formlyheatedorcooled.Also,othersimilarequationsarisinginthemodelingofthinliquidfilmshavebeenderivedandstudiedbyBertozzi[1],Dussan[7],Greenspan[13],HaleyandMiksis[14],Hocking[16],Myers[18],andOron,Davis,andBankoff[19].Hereweconsidertheproblemofaperfectlyconductingliquidfilmonahorizontalplanewiththeupperelectrodeplacedfarfromthegroundedsubstrate.Therehasbeenconsiderableinterestinelectricallyinducedinstabilitiesandtheiruseinpatternformationandtransferinphotolithographicapplications.DemonstrationsofsuchinstabilitiesinthiscontexthavebeenmadebySchafferetal.[23],[24]andLinetal.[17],forexample.Theoreticalworkshavefocusedonlineartheory,asinPeaseandRussel[21]andreferencestherein,aswellaslongwavetheoriesinthethinfilmandsmallgapapproximation(i.e.,thesecondelectrodeisplacedclosetothegroundedbottomplate)inShankarandSharma[25],andthemorerecentleakydielectricstudyofCrasterandMatar[5].Thepresentworkisrelatedtobutdifferentfromthosecitedabove,butweexpectthatthemathematicaltoolsdevelopedherecanbeusedinthoseproblemsalso.Alongwavetheoryleadstoanonlocalnonlinearevolutionequationatthelead-ingorder,whichbyachangeofthesignofthegravitationalparameteralsodescribeshangingfilms.(NonlocalequationshavealsobeenderivedinrelatedproblemsbyPa-pageorgiouandVanden-Broeck[20],Savettaseraneeetal.[22],andTilley,Petropoulos,andPapageorgiou[27].)IfH(x,t)denotesthescaledinterfacialposition,thentheequationtakestheformHt+13H31CHxxx−GHx+2WeH[Hxx]x=0,(x,t)∈R×R+,(1.1)H(x,t)=H(x+2L,t),whereC0,We0,andGcanbepositiveornegativeforoverlyingorhangingfilms,respectively.Intheabsenceofanelectricfieldthefilmislinearl