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TUMFakult atf urMathematikNumericalandAsymptoticResultsontheLinearStabilityofaThinFilmSpreadingdownaSlopeofSmallInclinationWagner,B.A.;M unch,A.ABCDEFGHIJKLMNOTUM-M9705M arz97TechnischeUniversit atM unchenCatalogingData:Wagner,B.A.;M unch,A.:NumericalandAsymptoticResultsontheLinearStabilityofaThinFilmSpreadingdownaSlopeofSmallInclination;Techn.Univ.M unchen,Fak.f.Math,ReportTUMM9705(97)MathematicsSubjectClassi cation:76D08Editor:H.W ahling(waehling@mathematik.tu-muenchen.de),Fakult atf urMathematikderTechnischenUniversit atM unchen,D-80290M unchen,Germany.FortheElectronicVersionsee: XCVIIFakult atf urMathematikundFakult atf urInformatikderTechnischenUniversit atM unchen.D-80290M unchen.Allrightsreserved.PrintedinGermany.NumericalandAsymptoticResultsontheLinearStabilityofaThinFilmSpreadingdownaSlopeofSmallInclinationA.M unchandB.A.WagnerInstitutf urAngewandteMathematikundStatistikLehrstuhlf urAngewandteMathematik(DAH)TechnischeUniversit atM unchenPostfach20103280010M unchenGermanyAbstractForathinliquid lmmovingdownaslope,thelubricationapproximationwithasmallNavier-Slipisusedasamodel.Thetravellingwavesolutionisderivedforsmallinclinationangle ,usingsingularperturbationmethods,andcomparedtothenumericalsolution.Forthelinearstabilityanalysiswecombinenumericalmethodswiththelong-waveapproximationand ndasmallbut nitecritical belowwhichthe owremainslinearlystable.Thisiscontrastedwiththevanishingofthehumpofthetravellingwavesolution.Finally,thisprevailingoflinearstabilityofthetravellingwaveatsmallinclinationanglesiscomparedwithrecentrelatedresultsusingaprecursormodelinstead.Herethough,adependencyonthemagnitudeofthecontactangleisfound,which,wethink,hasnotbeenobservedbefore.1IntroductionThecontrolofthebehaviourofthethree-phasecontactlineofathinspreading uidsheetismostimportantinmanyindustrialcoatingprocesses.Yet,themodellingofthethree-phasecontactlinestillremainsphysicallynotcompletelyunderstood,andtherefore,alsothemechanismsunderlyingtheeventualdevelopmentofinstabilities,suchas ngerorsawtoothpatterns.Oneofthe rsttoexperimentallyobservethesephenomenawasHuppert[18],andmanyresearchershaveinvestigatedthesesince.AscanbeseenfromfurtherexperimentalworkbySilviandDussan[29],JerrettanddeBruyn[19],deBruyn[3],SchwartzandTejada[27],Johnsonetal.[23],accuratemeasurementsof uidpropertiessuchas uiddepth,dynamic1contactangle,contactlinevelocity,ofamoving uidsheetalsopresentdi culties,inparticularduringtheonsetoftheinstabilityaswellasforsmallinclinationangles.Qualitatively,justbeforetheonsetoftheinstabilityahumpclosetothecontactlineisobservedand,dependingifthesizeofthecontactangleissmallorlarge,eithersawtoothpatternsor ngerpatternsevolve.Themaindi erencebetweenthesepatternsarethatsawtoothpatternseventuallyleadtoacompletecoatingwhile ngerpatternsleavedryoruncoatedregions,thepredictionandsimulationofwhichmaybeimportantforindustrialapplications.Oneofthe rsttomodelthespreadingofathin uid,drivenbygravitywasGreenspan[12],whousedthethinnessofthe lmtoderivethemoresimpli edyetstillnonlinearlubricationapproximationastheleadingorderproblemforCa 1,andintroducesaslipfunctionsuchthatnosingularityinthegoverningequationfortheheightofthe lmoccursatthecontactline.Healsousesacontactanglemodelthatyields,forsmallfrontvelocities,alinearrelationshipbetweentheslipvelocityandthecontactangle.Furtherstudies,byslightlyvaryingthecontactanglemodelandslipfunctionweredonebyHocking[16],Cazabatetal.[5],GoodwinandHomsey[10],HockingandMiksis[17],Miksisetal.[21].Ontheotherhand,goingbacktodeGennes[6],precursormodels,wherethenarrowregionconnectingaverythin lminfrontoftherelativelythicker uidsheetreplacesthecontactline,havebeenproposedbyTroianetal.[33],BertozziandBrenner[1].Bothmodelspredictedapreferredwavelengthofthelinearstabilitytospanwiseperturbationsoftheleadingfrontandfoundgoodqualitativeagreementincomparisontoexperiments.Asthedi erenceofthe owpatterns(sawtoothor ngers)isimportantforapplications,somaybe,wethink,thepossibleprevailingoflinearstabilityofthetravellingwaveatsmallenoughcontactangles.Weaddressthisquestionhere.Inouranalysisweuseaslipmodelandcomparetorecentpredictionsof[1].There,acriticalinclinationangle wasfoundbelowwhichthe owremainedlinearlystable.However, wasaboveinclinationanglesforwhichinstabilitiesstilloccurinexperiments[1].Inthelinearstabilityanalysisofourmodelacriticalanglewasfoundaswell,howeveritdependedstronglyonthevalueofthecontactangleandbecamesmallerforlargercontactangles.Similary,thereisacritical ,alsodependentonthecontactangle,atwhichthehumpofthe uidsheetvanishes.Bothfunctionsarethencomparedanddiscussed.Forthese(typically)smallinclinationangles,numericalsimulationbecomeincreasinglydi cult.Ontheotherhand,thefastestgrowingmodebecomesverysmallinthatrange,sothatitbecomesusefultoperformalong-waveanalysisoftheresultingeigenvalueproblem.Byimprovingournumericalcodetoadegreeofaccuracysothatthegrowthratesoverlapwiththosepredictedbythelong-waveanalysis,weobtainalsointhisextendedrangeofsmall thegrow