Faraday Waves in Small Cylinders and the Sidewall Non-Ideality

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Faraday Waves in Small Cylinders and the Sidewall Non-Ideality
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1 online resource (143 p.)
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english
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Batson, William R
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University of Florida
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Doctorate ( Ph.D.)
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University of Florida
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Chemical Engineering
Committee Chair:
Narayanan, Ranganatha
Committee Members:
Kopelevich, Dmitry I
Weaver, Jason F
Pilyugin, Sergei S
Zoueshtiagh, Farzam

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Subjects / Keywords:
immiscible -- instability -- interface -- parametric -- resonance -- waves
Chemical Engineering -- Dissertations, Academic -- UF
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Chemical Engineering thesis, Ph.D.
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This work is the result of a scientific inquiry into the current understanding of experimental single-mode Faraday waves, from the perspective of the linear stability theory. Given an electromechanical shaker capable of imposing vibrations of several centimeters at frequencies of up to 15 Hz, experiments were directed toward laterally ”small” systems in which the cell modes are discretized and the excited wavelength was of the order of the lateral dimension. In this regime, the theoretically tractable boundary condition for the sidewalls is a stress free condition, which is a challenge to produce experimentally. In reality, the no-slip behavior of the fluid along the sidewalls and interfacial contact line effects such as capillary hysteresis introduce sidewall stresses. In the interest of comparing an experiment to the theory, it was therefore necessary to develop an experiment which respected this assumption. This marks the first attempt to match the single mode Faraday experiment to a linear theory that rigorously treats viscosity. Past experiments comparing the observed threshold to a theory have found agreement by phenomenologically accounting for the overall system damping. While remarkable that such a match can be made, the observed thresholds of past experiments in general do not agree with the predictions of the viscous linear stability theory, thereby motivating this work.
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by William R Batson.
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Thesis (Ph.D.)--University of Florida, 2013.
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Adviser: Narayanan, Ranganatha.
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FARADAYWAVESINSMALLCYLINDERSANDTHESIDEWALLNON-IDEALITY By WILLIAMR.BATSONIII ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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c r 2013WilliamR.BatsonIII 2

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Tomyfamilyandfriends 3

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ACKNOWLEDGMENTS ThismanifestationmarksanaccomplishmentforwhichIammo stproudof,asI believeIhavegivenasmuchofmyselftoitasIpossiblycould ,justiedbytheimmense amountofsatisfactionithasgivenme. IwouldliketothankmyparentsBillyandSusanforgivingmethe irunconditional loveandsupportthroughalltheseyears.Theyhavesettheex amplewhichIwillforever trytofollow.Iamequallygratefultobeabletosharesomuch withmybrotherThomas andsisterGrayce. Beyondfamily,Irstandforemostwouldliketothankallwhot each.Iamhere becauseofanoutstandinglineofteachers,andtheopportun itiestheyhaveprovided me.Learningismymostprizedhobbyandthesearethefacilit ators. RangaNarayanan,aftermyfamily,willstandasoneofthemos tinuentialpeople inmylife.Thepursuitofscienceismydream,andregardless ofwhathappensinthe future,IcanalreadysaythatIhavedonethat,becauseofhim .Heisadearfriend whoIhaveanutmostrespectfor,givenhistrueloveforwhath edoes,whichIhopehe continuestodoforalongtime. AwarmreectionofRanga'spersonality,myFrenchadvisor, FarzamZoueshtiagh, isdevotedtohisworkandhisfamily,andstandsasrolemodel forapersonIstriveto become.AlongsidehiminLilleIwasabletoworkwithanincred ibleleveloffocusand accomplishnearlyeverygoalIset,becauseherepeatedlyex tendedhimselftoensure mycomfortandunlockresources. IamalsoveryappreciativeofthestaffpersonsofbothIEMNin Lilleandthe UniversityofFloridaDepartmentofChemicalEngineering.H el eneDelsarte,thebusiest personatIEMN,wasespeciallyhelpfulinmydealingswithadm inistration(andtolerable ofmyFrench!).CarolynMiller,whonavigatedmytravelpape rworkatUF,wasalways concernedformybestinterest. 4

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ThesatisfactionIderivefromworkisintimatelyconnected withmyfriendships,and oftentimesitseemslikeonewouldnotexistwithouttheothe r.ThereforeImustsaythe spreadingofmyselfthroughGainesvillewasaspecialpathl adenwithgreatmemories. JosephC.RevelliIIIandChristopherP.Muzzilloaretwocar ingguyswhoseopinions andthoughtsIwillforevervalue. IthankthePartnerUniversityFundandtheEmbassyofFrancef oraChateaubriand FellowshipduringSpring2011andtheUniversityofFloridaf orthepart-alumni fellowshipforamajorpartofmyDoctoralwork.TheNational ScienceFoundation providedfundingviagrantNSFOISE0968313.Iamgratefultoal loftheminnosmall measure. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 15 LiteratureReview ................................... 21 SpatiotemporalChaos ............................. 21 ExperimentalFaradayWavePatternFormation ............... 22 LinearTheoryandComparisontoExperiments ............... 24 NonlinearTheoreticalAdvancement ..................... 29 FaradayWavesinFerrouids ......................... 31 FaradayWavesinMicrouidics ........................ 32 2MODELANDLINEARSTABILITYANALYSIS ................... 34 GoverningEquations ................................. 34 PerturbedEquations ................................. 37 LinearStabilityAnalysis ............................... 39 SpatiallyInniteSystemResultsandDiscussion .............. 42 SpatiallyFiniteSystems ............................ 47 ACaseStudy:theEffectofGravity ...................... 51 TheCaseofDouble-FrequencyParametricForcing ................ 53 Double-FrequencyPredictionsforHorizontallyInniteSyst ems ...... 54 Double-FrequencyPredictionsforCylindricalContainers .......... 59 3PREVIOUSEXPERIMENTSONSINGLE-MODEEXCITATION ......... 61 4EXPERIMENTALMETHOD ............................. 67 ChoiceofLiquids ................................... 67 CellDesign ...................................... 70 ElectromechanicalShaker,ImageCaptureandProcessing ........... 72 SingleandDouble-FrequencyAmplitudeDetermination ............. 76 AnalysisoftheOutputCellMotionQuality ..................... 78 ExperimentalRepeatability ............................. 79 5SINGLE-FREQUENCYEXPERIMENTSANDDISCUSSION .......... 82 SidewallMeniscusandFilmBehavior ....................... 82 ExperimentalThresholdDependenceUpontheUpperPhaseViscos ity .... 85 NonlinearGrowthandSaturation .......................... 88 FC70and1.5cStSiliconeOilInstabilityThresholds ............... 90 6

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SaturatedInterfaceAmplitudes ........................... 93 HigherOrderNonlinearPhenomenaandthePathtoTurbulence ........ 94 SystemDampingStudy ............................... 100 6DOUBLE-FREQUENCYEXPERIMENTS ..................... 102 LinearBehavior .................................... 102 NonlinearBehaviorandInteraction ......................... 104 ExperimentalConclusion .............................. 106 7FINALDISCUSSIONANDCONCLUDINGREMARKS .............. 111 ExperimentalMethod ................................ 111 RealizationoftheStress-FreeBoundaryCondition ............. 111 ExperimentalRepeatability .......................... 112 ExperimentalInitialCondition ......................... 113 DataAnalysisTechniques ........................... 113 ExperimentalResults ................................ 114 MeniscusWaveInhomogeneity ........................ 116 Double-FrequencyPhenomena ........................ 117 NonlinearBehavior ............................... 118 APPENDIX AINVISCIDRESULTANDCOMMENTS ....................... 120 BLINEARSTABILITYMATLAB R r CALCULATION .................. 123 C B M 1 AND B M 2 ..................................... 130 DOUTREACHSUMMARYOFWORK ........................ 132 REFERENCES ....................................... 137 BIOGRAPHICALSKETCH ................................ 143 7

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LISTOFTABLES Table page 6-1Parametricconditionsandwaveheightmeasurements .............. 110 8

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LISTOFFIGURES Figure page 1-1MeasurementoftheinstabilitythresholdbyBechhoefer etal .......... 19 2-1Fixed-frequencylinearstabilitythresholdforahoriz ontallyinnitebilayer. ... 43 2-2Fixed-frequencystabilitydependenceuponvarioussys temparameters .... 44 2-3Minimumthresholddependenceuponfrequencyforhorizo ntallyinnitebilayers 46 2-4Cylindricalcellmodesandwavenumberssatisfyingthes tress-freecondition .. 48 2-5Fixed-frequencylinearstabilityofsystemswithcylin dricalmodediscretization 49 2-6Minimumthresholddependenceuponfrequencyforacylin dricalsystem .... 50 2-7Fixed-frequencylinearstabilityinearth-basedandze ro-gravityenvironments 52 2-8Minimumthresholdinearth-basedandzero-gravityenvi ronments ........ 52 2-9Fixed-frequencystabilityofinnitesystemswithdoub le-frequencyexcitation 55 2-10Minimumthresholddependenceuponbasicfrequencyfor aninnitesystem .. 57 2-11Minimumthresholddependenceupontheratio foraninnitesystem ..... 57 2-12Minimumthresholddependenceuponbasicfrequencyfor acylinder ...... 58 2-13Minimumthresholddependenceupontheratio foracylinder. ......... 59 3-1Pastexperimentalthresholdsincylinders ..................... 62 4-1Schematicdiagramofthecellandtheelectromechanicals haker ........ 71 4-2Samplecameraimagedepictingexcitationofa(0,1)mode. ........... 73 4-3Time-spacedataforasaturated(0,1) sh mode ................... 74 4-4ExperimentalcellmotionsignalsandFFTanalysis ................ 75 4-5SampleFourierspectradepictingtheappearanceofundes iredfrequencies .. 77 4-6Outputcellmotionsignalqualitydependenceonparamet ricconditions. .... 78 4-7Experimentalrepeatabilityissues .......................... 81 5-1Experimentalvisualizationofthemeniscusdynamics ............... 83 5-2Experimentalvisualizationoftheexcitationofa(0,1) sh mode .......... 84 5-3FilmdynamicsinaFC70and50cStsiliconeoilsystem ............. 86 9

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5-4FilmdynamicsinaFC70and1.5cStsiliconeoilsystem ............. 87 5-5Dependenceoftheupperphaseviscosityontheoreticala greement ...... 88 5-6Experimentalvisualizationofaco-dimension2point ............... 89 5-7Theoreticalcomparisonforthelargeheightexperiment alsystem ........ 90 5-8Theoreticalcomparisonforthesmallheightexperiment alsystem ........ 92 5-9Saturatedinterfacedeectionowingtomeniscusandpara metricexcitation .. 93 5-10Shearinstabilitiesona(0,1) sh mode. ........................ 95 5-11Precessionofa(0,1) sh mode. ............................ 96 5-12Shearinstabilitiesona(1,1) sh mode ........................ 97 5-13Shearinstabilitiesona(2,1) sh mode ........................ 97 5-14Breakupofa(2,1) sh mode .............................. 98 5-15Orderedbreakupofa(0,1) sh mode. ......................... 99 6-1Thresholddataandpredictionswith( M 1 M 2 )=(3,4)excitation .......... 103 6-2Saturatednonlineardataforaharmonicmodeexcitedwith asinglefrequency. 107 6-3Saturatednonlineardataforharmonicmodesexcitedwith twofrequencies .. 108 6-4Saturatednonlineardataforsubharmonicmodesexcitedw ithtwofrequencies 109 A-1ThestabilitydiagramofBenjamin&Ursell. .................... 122 A-2SampleMathieuequationsolutions. ........................ 122 10

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy FARADAYWAVESINSMALLCYLINDERSANDTHESIDEWALLNON-IDEALITY By WilliamR.BatsonIII May2013 Chair:RanganathanNarayananMajor:ChemicalEngineering Thisworkistheresultofascienticinquiryintothecurren tunderstandingof experimentalsingle-modeFaradaywaves,fromtheperspect iveofthelinearstability theory.Givenanelectromechanicalshakercapableofimpos ingvibrationsofseveral centimetersatfrequenciesofupto15Hz,experimentswered irectedtowardlaterally “small”systemsinwhichthecellmodesarediscretizedandt heexcitedwavelength wasoftheorderofthelateraldimension.Inthisregime,the theoreticallytractable boundaryconditionforthesidewallsisastressfreecondit ion,whichisachallengeto produceexperimentally.Inreality,theno-slipbehavioro ftheuidalongthesidewalls andinterfacialcontactlineeffectssuchascapillaryhyst eresisintroducesidewall stresses.Intheinterestofcomparinganexperimenttothet heory,itwastherefore necessarytodevelopanexperimentwhichrespectedthisass umption.Thismarks therstattempttomatchthesinglemodeFaradayexperiment toalineartheorythat rigorouslytreatsviscosity.Pastexperimentscomparingt heobservedthresholdtoa theoryhavefoundagreementbyphenomenologicallyaccount ingfortheoverallsystem damping.Whileremarkablethatsuchamatchcanbemade,theob servedthresholds ofpastexperimentsingeneraldonotagreewiththepredicti onsoftheviscouslinear stabilitytheory,therebymotivatingthiswork. Instrumentaltotheproductionofasystemthatrespectsthe stress-freecondition isthechoiceofsystemliquids.Intheseexperimentsitwasf oundthattheliquidsFC70 11

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andsiliconeoil,which,otherthanaslightmeniscus,produ cedaatinterfacethatmoved withlittleeffortoncethecontainerwastilted.Thisbehav iorstandsincontrasttothatof awater-airoranimmisciblesiliconeoil-waterinterfacei naglassbeaker,whichsuffer frompinningtothesidewall.CloserinspectionoftheFC70oilinterfaceatthesidewalls showsformationofatinyoillminatiltedcellwherebulkFC 70haddisplacedbulk siliconeoil.Itisbelievedthislmisinstrumentalinredu cingthestressesassociatedwith thesidewall,anditsdynamicsalonearearemarkableresult worthyofcontinuedstudy. Wavedecayexperimentsconrmthedominanceofthebulkvisc ouscontributiontothe sidewalldampingeffectswhenmeasuringtheoverallsystem damping,supportingthe casethatthestress-freeconditionhasbeenapproximated. Inproceedingtointerprettheexperiments,rsttheviscou slinearstabilitytheory ofKumar&Tuckerman[ 52 ]ispresented,andmodiedwiththestressfreeboundary conditiontoaccountformodediscretization.Thelinearth eoryiscapableofpredicting thethresholdamplitude,abovewhichtheatinterfaceisun stableanddeection occurs.Inhorizontallyinnitesystemsthewell-knownres ultisthattheinstabilityis subharmonicallyexcited–withafrequencyhalfthatofthef orcingfrequency.Themain implicationofmodediscretizationisthesystemnolongern ecessarilyhasaccessto amodewhosefrequencyishalftheimposedfrequency.Instea dthecontinuumof availablemodesisdiscretized,andeachavailablemodecan beexcitedinsideofitsown frequencyband.Thecorrespondingthresholdamplitudesfo reachbanddescendtoa minimumamplitudenearthenaturalfrequencyofthemode,an dthepointsatwhichthe thresholdsofneighboringmodesintersectareco-dimensio n2points,conditionswhere twomodesareneutrallystable.Anotherimportantimplicati onofanitesystemisthe abilitytoaccessinstabilitytongueswithharmonicandsup erharmonicresponses. Thetheoreticalconceptsofthecriticalthresholdandfreq uencybandsatwhich modesappeararestudiedextensivelywiththeFC70andsilic oneoilsystem.Ina systemoscillatingbelowtheinstabilitythreshold,tiny owperturbationsareseendue 12

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totheemissionofwavesfromtheoscillatingmeniscus,anun avoidablenon-ideality. Imposingvibrationsabovethethresholdamplituderesults inthegradual(orsudden) deectionandgrowthoftheinterfacetosomenewstate.Exper imentalthresholdsare thereforemarkedbyperformingaseriesoftrialsatdiffere ntamplitudes,andthelowest amplitudeatwhichtheinstabilityisobservedismarkedast hethreshold.Datasetsare builtbymeasuringthethresholdfortheexperimentalfrequ encyband,andboundedwith theco-dimension2points.Agreementbetweenthesedatasets isseentoimprovewhen lowerviscositysiliconeoilsareused,presumablyduetoth edecreaseinthesidewall lmthicknessandshearingstresses. Twocompletedatasetsofseveralsubharmonic,harmonic,an dsuperharmonic modesarepresentedforFC70and1.5cStsiliconeoilsystemso fdifferentlayerheights, andarecomparedtothepredictionsofthelineartheory.Whil eslightlyhigherthan predictedthresholdsareobservednearthemodenaturalfre quencies,theagreement isquitegoodasonemovestowardtheco-dimension2points.T hedeviationnearthe naturalfrequenciesappearstobegreaterformodeswithgre aternumberofazimuthal nodes,suggestinganassociatedincreaseinwalldamping.C onsiderabledeviationis seeninthelower-than-predictedthresholdsfortheharmon icmodes,mostnoticeablyfor themodeshowingthesameazimuthaluniformityasthesidewa llmeniscus,suggesting aninteractionbetweentheharmonicmodeswiththemeniscus waves,whichare alsoharmonicwiththecellmotion.Similardeviationisseen fortheotherharmonic modes,albeitlessduetothespatialmismatchoftheinstabi litywiththemeniscus wave.Thisisanotherkeyndingofthiswork,asitcouldpote ntiallybemodeledasan inhomogeneousinteractionwiththeinstability. Thepowerofthelineartheorycomesinbeingabletopredictt hemodeofinstability andthecriticalthresholdatwhichitappears,whereasgrow thbeyondtheinnitesimal stateanonlineartheoryisrequired.Anonlineartheorywas notcompletedinthiswork andthereforetheexperimentwasusedprimarilyforcompari sontothelineartheory. 13

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Qualitativenonlinearobservations,however,werenumero us,andmanyarepresented intheinterestofbothgivingabroadersenseoftheexperime ntandinsightintothe parameterspacesaweaklynonlineartheorymightshowagree ment.Ingeneralitis observedthatexcitationoftheinstabilityatanamplitude closetothethresholdoften resultsinamodethatgrowsandsaturatestoaniteamplitud ethatcloselyresembles thelinearspatialformofthemode.Continuedincreaseofth eimposedamplitude eventuallyresultsintheappearanceofsecondaryinstabil itieswhichcaneitherbea sourceofnonlineardampingorapathwaytomodebreakup.Addi tionally,behavior suggestsasubcriticalbifurcationforfrequenciesbelowt henaturalfrequencyand supercriticalabove,consistentwithpastexperiments. Thenalresultpresentedisforwhentheinstabilityisexci tedwithtwofrequencies, anoveltyforsinglemodeexperiments.Thetheorycanbewrit tenandpredictionsmade forthecasewhenthesetwofrequenciesareintegermultiple sofabasicfrequencyand abasicamplitude,andtheresultsofthismethodarepresent edanddiscussed.An importantimplicationoftheadditionofasecondfrequency componentistheaddition ofthreedegreesoffreedom:theamplitude,frequency,andp haseshiftofthesecond component.Alargerparameterspaceisthereforenecessary toconductathorough investigation.Whileinitialexperimentsondoublefrequen cyexcitationofsinglemodes isshowntoagreewiththetheoryforafewselectedmodes,the thresholdbehavior oftheharmonicmodethatwasobservedtointeractwiththeme niscuswavesinthe singlefrequencycaseisseentoproduceevenmoreirregular behaviorwhenthesecond frequencycomponentisadded. 14

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CHAPTER1 INTRODUCTION Thisworkconcernsaninvestigationintothewavephenomeno noriginallyreported byFaraday[ 30 ]in1831,appearingatthesurfaceofaliquidlayerthatwasm adeto vibrateupanddowninanoscillatoryfashion.Henceforthbe aringhisname,Faraday wavesaremostsimplycharacterizedbythedeection,growt handsaturationofthe normallyatuidinterfacetoastandingwavewhentheampli tudeofvibrationsurpasses acriticalthreshold.Faradayalsoobservedthewavefreque ncytobeonehalfthatof theimposedvibrations,i.e.subharmonicresonance,which isnowunderstoodtobe characteristicofparametricexcitation.Subharmonicreso nancewasalsoobservedin theexperimentofMelde[ 57 ],whoinducedsuchbehaviorinastringbyperiodically adjustingitstensionatafrequencyequaltotwicethefrequ encyofthestring'sdominant mode.MotivatedbytheexperimentsofMeldeandFaraday,and followinganalysisof thelunarperigeebyHill[ 39 ],Rayleigh[ 68 ]putforthamathematicalargumentforthe existenceofsubharmonicsolutionsappearingaboveacriti calparametricthresholdin singledegreeoffreedommechanicalsystems. Classicalexamplesofsuchsystems,includingsimplependu lumsoramass attachedtoaspring,exhibitnonlinearitiesduetothedepe ndenceofthesystemforce onthestateofthesystem.Intheinvestigationofhisownexp erimentslikethatofMelde, Raman[ 67 ]extendedRayleigh'sanalysisbyinclusionofcubicnonlin earityinthespring tensionwithrespecttoitsdisplacement.Nonlinearbehavi orinFaraday'swavesis alsopresent,aswaverestorativeforcessuchasbuoyancyan dsurfacetensionand dissipativeforcessuchasviscositydependupontheampli cationofthewaveandthe uidvelocities.Thereforecommonanalysesofsuchsystems tendtorstfocusonthe linearizedbehaviorofthesystem,inthecaseofRayleigh's analysistheparametric thresholdrepresentsthedepartureofthesystemfromaquie scentstate.Inthecaseof Faraday'swaves,Benjamin&Ursell[ 11 ]in1954successfullydescribedthedeparture 15

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fromtheatstatebyreducingtheinvisciduidequationsof motiontotheMathieu equation,viz., @ tt m +[ p m +2 q m cos T ] m =0 TheclassicalformoftheMathieuequation,shownabove,dep ictstheparametric excitationofalinearharmonicoscillator.Thescalingunc overedbyBenjamin& Ursellshowsthatthesquareofthenaturalwavefrequencyof aFaradaysystem, determinedbytheuiddensity,surfacetension,andwavele ngth,toberepresentedby theparameter p m ,theparametricamplitudeby q m andthewaveresponseamplitude bythedependentvariable m .Whatcanbeshowneitherbynumericalintegrationor advancedanalysis,inaccordancewithobservationsofFara dayandRayleigh,isthatfor certainvaluesof p m and q m theresponse m growswithoutboundwithhalf-integer( 2 and 3 2 )frequencieswithrespecttotheparametricfrequency.Suf cientreduction oftheamplitude q m resultsinaboundedsolutionforlongtimeindicatingsyste m stability,andtheexistenceofaneutrallystable q m .FundamentalphysicsoftheFaraday phenomenonaregleanedfromthismodelasthedenitionsof p m and q m provide excellentapproximationfortheexcitedwavenumber. However,theMathieuequationisinsufcientforpredictin gthethresholdamplitudes inanexperimentsuchasthatofFaraday.Thereasonistwo-fo ld,andhighlightsthe fundamentaldifferencesbetweenhigh-frequencyexperime nts,andlow-frequency experimentssuchasthoseBenjamin&Ursellperformedandcom paredtheirmodelto. First,dissipation,aneffectpresentinallrealsystems,i sabsentinthismodel,anditwill beshownthattheeffectofuidviscositycannotsimplybein corporatedbyinsertion oflineardampingintotheMathieuequation.Thesecondreas onisanalogoustothe quantummechanicalparticleinabox,andtheboundarycondi tionsitmustsatisfy,which placesrestrictionsontheallowedwavefunctionsandenerg ies.Atlowenergylevels thespacingbetweenallowedwavefunctionenergiesisnite ,formingadiscreteset, 16

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whileatsufcientlyhighenergytheselevelsformacontinu ousset.Thesameistrueof theFaradaywavesystem,wherethereexistsaregime,access edbysufcientlyhigh parametricfrequencies,wherethespacingbetweentheallo wedwavenumbersisvery smallandthesetformsacontinuum.Theregimeofinterestto thisdissertation,islow frequencyexcitation,wherethewavenumbersallowedbythe systemformadiscretized set,andarestronglydependentupontheboundarycondition sandthelateralgeometry ofthecontainer. DuetotheabsenceofdissipationintheMathieuequation,pe rfectresonance isattainedforinnitesimalforcingamplitudeswhenevert heimposedfrequencyisa half-integermultipleofawave'snaturalfrequency.There foreahigh-frequencysystem havingaccesstoacontinuumofmodeswouldinvariablybeuns tablebecauseone ofthesemodesisboundtobeinperfectresonancewiththepar ametricfrequency. DevelopmentoftheviscouslinearstabilitymodelbyKumar& Tuckerman[ 52 ]has allowedforthepredictionoftheexperimentalthresholdin ahigh-frequencysystem. Ontheotherhand,theMathieuequationhasbeenshowntobesu fcientinthe low-frequencydiscretizedsystem,asexperimentswithlow -viscosityuidscanrelegate thesourcesofdissipationtothesidewalls,whichexperime ntallyhavebeenshown tobelinear.Discretizedexperimentsthereforehavenegle ctedanalysisfromthe perspectiveofauidmechanicalsystem,butratherasapara metricallyexcitedsingle degreeoffreedomsystemwithlineardamping.Whileavalidap proach,whatwillbe discussedisthatthepredictionofthisdissipationisextr emelydifcult,requiringits measurementdirectlyfromtheexperiment.Insomewaysthis phenomenologicalnature isunsatisfying,andwhatwillbeshowninthisworkisthatan experimentaleffortcanbe madetoapproximatetheappropriateboundaryconditions,a llowingcloserconnection totheKumar&Tuckermanmodel.Indoingsonotonlyhasitbeco mepossibleto accuratelypredictselectedmodesandthresholdamplitude s,butalsotopinpointthe causesfordifferencewhichlendthemselvestoamyriadofri chphenomena. 17

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Itwasinvestigationofthepatternformingbehavior,origi nallyobservedbythe high-frequencyexperimentsofFaraday,thatwasseenasapa thwaytodisorderand chaosandledtothedevelopmentoftheviscouslinearstabil ityanalysisofKumar& Tuckerman.OtherhydrodynamicinstabilitiessuchasRayle igh-B enardconvection andTaylor-Couetteowhadexhibitedtransitionsfromregu larpatternstochaos andturbulence,andsuchtransitionsweresimilarlydispla yedintheFaradaywave experimentsofKeolian etal. [ 47 ],Gollub&Meyer[ 33 ]andEzerskii etal. [ 29 ]as theparametricamplitudewasincreasedwellpastthecritic alpoint.Faradaywave patternssymmetricandasymmetricwiththecontainergeome trywerestudiedforthe instabilitynearerthethresholdbyDouady&Fauve[ 26 ].Ofnoteisthattheylledtheir squarecontainertothebrim,therebypinningtheinterface witha90 contactangleand eliminatingtheparasiticwavesemittedbyameniscussubje ctedtovibrations.Douady [ 25 ]studiedtheeffectofthemeniscusonmodediscretizationa tmoderatefrequencies bymeasuringthelinearthresholdforseveralmodesinabrim fulsystemandonewith aprominentmeniscus.However,thesethresholdswerenotco mparedtoatheory, andDouadyacknowledgedthatthecellmodesinabrimfulsyst emdonotrespect theproperboundaryconditionsandthatamodeldidnotexist thatcorrectlytreated viscosity.Theimportanceofviscositytothelinearthresh oldinahigh-frequencysystem wasfurtherhighlightedbythehigh-frequencyexperiments ofFauve etal. [ 31 ],who observedsystematicallyhigherthresholdsthanthosepred ictedbyamodelincluding linearviscousdissipation. Followingclassicalhydrodynamicstabilityanalysislike thatofChandrasekhar[ 16 ], Kumar&Tuckerman[ 52 ]studiedthestabilityofahorizontallyinnitebilayersu bjected toparametricforcing,whilerespectingtheroleofviscosi typresentintheNavier-Stokes equations.TheydidthisbymeansofaFourier-Floquetanaly siswhichcastthelinear stabilityofaspatialwavenumber k intoaninnitesetequationsfortheperturbed interfacialdeection,coupledbyvirtueoftheparametric forcingtermcos t.Setting 18

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Figure1-1.MeasurementoftheinstabilitythresholdbyBech hoefer etal. [ 8 ]. thegrowthrateoftheFloquetexponenttozero,theneutrall y-stableamplitudesAwere numericallycalculatedastheeigenvaluesofthetruncated system.Repeatedsolutionof thisproblemforallvaluesof k producedadiagramoftheneutralamplitudesAoutlining severalregionsforwhichasolutionwouldgrowintime,simi lartothediagramdepicting thestabilityofsolutionstotheMathieuequation.Differe nt,howeverwasthedampingof thethresholdtonguestoaniteamplitude,indicatingthep resenceofviscouseffects. Theglobalminimumofthisdiagramtherebyestablishedthet hresholdamplitudeat whichtheinterfacebelongingtoahorizontallyinnitesys temwoulddestabilize. Thiscalculationrepresentsthemaintheoreticaltoolused inthisworkandwillbe presentedindetailinChapter 2 .Thepredictedamplitudesandwavenumberswere successfullyreproducedinanexperimentbyBechhoefer etal. [ 8 ],whoseresultsare presentedinFigure 1-1 .Thefocusedregimeofthisdissertationaretheoscillatio ns inthethresholdsoccurringatlowfrequencies,whichtheya cknowledgeisduetothe discretizationofcellmodes.Datapointscorrespondingto thesethresholdswere characterizedbydistinctpatterns,whereasathigherfreq uenciestheyobservedwell 19

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orderedsquares.Thesedistinctpatterns,orcellmodesast heywillbereferredto throughoutthiswork,aredependentnotonlyuponthecontai nergeometrybutalso uponthesidewallcontactbehavioroftheuidinterface.Ma thematically,thetractable conditionwhichallowstheseparationofthehorizontalspa tialdependenceofeachcell modeistheso-calledstress-freecondition,andwasusedby Benjamin&Urselltoadapt theMathieuequationpredictionstotheirsingle-modeexpe riment. Thishistoryanddiscussionformsthemotivationforthiswo rk,because,todate, adaptationoftheKumar&Tuckermanmodelwiththestress-fr eeconditionhasnotbeen made,norhasadiscretizedFaradaywavesystembeendesigne dwhichreplicates thiscondition,allowingfortheaprioripredictionofthei nstabilitythreshold.Thework thatwillbepresentedrepresentsthecourseofstudytakent oputforthsuchaneffort, andisnotonlycomposedofthedesignandresultsoftheexper iment,butalsoan amountoftheoreticaldetailsufcienttoreproducethelin earstabilitypredictionsto whichtheexperimentwascompared.Anadditionalexperiment alcaseconsidered wasthatwheretheparametricforcingwascomposedoftwofre quencies,atheoretical extensiontreatedbyBesson etal. [ 12 ],whichisalsopresentedalongwithadaptationof thestress-freeconditioninChapter 2 .Anespeciallyinsightfulpartofthiswork,providing furtherhistoricalcontext,istheretrospectivecomparis oninChapter 3 oftheviscous lineartheorytothepreviousexperimentsthatmeasuredsin glemodethresholds, highlightingthatpreviousexperimentshavebeencontroll edbysidewallorinterfacial effects.Experimentaldesign,repeatability,andmeasurem entanalysisaredetailedin Chapter 4 andhighlightstheconsiderationstakentowardsproducing anidealsystem. Chapter 5 presentsthegeneralexperimentalbehavior,theagreement anddisparities withthetheory,andtheinvestigationintohigherordernon linearphenomena.The inuenceofthedouble-frequencyparametricforcingonthe experimentalagreement withthetheoryisinvestigatedinChapter 6 .Here,quantitativenonlineardataisalso presentedwhichrepresentsthefurthestextentoftheexper imentalinvestigation.Finally, 20

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discussion,conclusions,andavenuesforfuturestudyarep resentedinChapter 7 Beforeproceeding,however,amoredetailedlookattheprevi ousworkonparametric resonanceandFaradaywaveswillbegiven. LiteratureReview Inthefollowingliteraturereviewtheexperimentalandthe oreticaladvancement ofFaradaywaveswillbepresented,alongwithanuancedscie nticapplicationanda technologicalapplication.First,theresurgenceoftheph enomenoninthe1980swillbe discussedinthecontextofexperimentswhichprobedthedif ferenttransitionsmadefrom regularwavestodisorderedbehaviorandchaos.Followinga retheexperimentswhich movedclosertotheinstabilitythreshold,whichpromotedt hephenomenontothestatus ofbeingthepremiertestbedfornonlinearpatternformatio n.Thediscoveryofthisrich behaviorbeggedstrongertheoreticalexplanation,whichw asmadebydevelopmentof linearandweaklynonlinearstabilityanalyses,whichwill alsobediscussedatlength. SpatiotemporalChaos InterestintheFaradayproblembegantobuildintheearly19 80swiththeexperiments ofKeolian etal. [ 47 ]whofollowedapursuitforchaosandturbulencearisingfro m hydrodynamicinstabilities.Theyparametricallyexcited awaterlayerinathincylindrical annulus,thereforewithlimitedspatialbehavior,andobse rvedthatanincreased parametricamplituderesultedinsequencedincreasesinth eresponsefrequency fromtheinitialsubharmonicresponse.Gollub&Meyer[ 33 ]introducedaspatialvariation intothisnonlinearbehaviorbystudyingthetransitionsma deinalargecylindricallayer ofwater.Relativetothecurrentwork,theynotedandmeasur edathresholdwhichdid notmonotonicallydecreasewithfrequency,undoubtedlydu etomodediscretization. Theirinteresthoweverwasspatiotemporalvariation,andt heyobservedsuchtransitions fromsubharmonicmodeswithspatialaxisymmetry,followed bymodeswithperiodic modulationstotheaxisymmetry,thenquasiperiodicmodula tionswithazimuthalspatial modulations,andnallytocompletespatiotemporaldisord er.Atlowerfrequencies 21

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Ciliberto&Gollub[ 20 ]reportedsuchtransitionsintheinteractionbetweentwoc ell modesneartheco-dimension2point,andatlargefrequencie s,Ezerskii etal. [ 29 ] reportedthesetransitionsforthedisorderingofregulars quarepatterns.Theeffectsof thelossofmodedegeneracyinslightlyrectangularcellson theinteractionbetween discretizedmodeswasstudiedbySimonelli&Gollub[ 71 ]andevenqualitatively reproducedthephasespacepredictedbytheweaklynonlinea rtheoryofGu etal. [ 34 ]. Bycarefuladjustmentofthelayerheight,Henderson&Miles[ 37 ]broughtsubharmonic modesintointernalresonancebothwiththeirsubharmonica ndsuperharmonicmodes, andcomprehensivelymappedaphasespaceforthedifferentr esultingspatiotemporal behaviors,inadditiontoattemptingtheoreticalcomparis onwiththatofBecker&Miles [ 9 ].AuniquemethodwasthatofCiliberto etal. [ 19 ],whodetectchaotictransitionsin ahigh-frequencysystembytheadjustmentintherequireddr iveaccelerationresulting fromthechangingmodedissipation.ExperimentalFaradayWavePatternFormation AnotheraspectofFaraday'swaveswhichdrewtheattentionof everyonein the1990swerethepatternsFaradaydeemedsobeautifulhims elfinhisoriginal experiments.Comprehensivelyreviewedfornumerousphysi calsystemsbyCross& Hohenberg[ 22 ],includinghydrodynamicsystems,thepatternformingnat ureofthe Faradaysystemisanattractivepatternformingsystemduet otheshorttimescales andtheeaseofproductionoflargeaspectratiosystems.The regimeinwhichthis behaviorisobservedisthereforeoppositeofthatforthisd issertation:highfrequencies wheremodediscretizationispreferablyabsent.However,i twaslearnedthattheuid mechanicaltreatmentofviscositywasnecessarytopredict thesepatternsandtherefore theKumar&Tuckermanlinearanalysis,whichisalsousedint hiswork,formsthe basisforthecurrentmostcompletepatterntheory.Addition ally,thepatternworkon parametricexcitationwithdoublefrequenciesservesasth eonlyworktowhichthe 22

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discretizedresultsofthisworkcanbecomparedto,assuche xperimentshavenotbeen performedinthediscretizedregime. Higher-ordertransitionsfromregularpatternstomoredis orderedstateshave beenobserved.Earlyexperimentsstudyingthepatternforma tionnearthethreshold excitedbysingleparametricfrequencieswereconductedby Douady&Fauve[ 26 ], howevertheseexperimentswereperformedatmoderatefrequ encieswhereeffectsof discretizationpersisted,andtheyinterpretedtheirpatt ernsinthecontextofindividual superpositionsofdiscretewaves,oneofwhichwashexagona l.Tullaro etal. [ 78 ] followedtheexperimentsofEzerskii etal. [ 29 ],byconductingexperimentsathigh-frequency suchthattheexcitedwavelengthwassufcientlysmallerth anthecontainersize,and bothobservedsquarepatternsjustabovetheinstabilityth reshold.Christiansen etal. [ 18 ]furtherincreasedthesystemaspectratioandgeneratedap hasediagramofregions inwhichpatternsformedby3,4(quasicrystalline),6and8p lanewavesappeared. Fauve etal. [ 31 ]studiedtheinstabilityneartheliquid-vaporcriticalpo intofcarbon dioxide,andobservedatransitionfromthenormalsquarepa tternstoastripedpattern asthedifferenceintemperaturefromthethermodynamiccri ticalpointwasdecreased. Notablytheyalsoobservedsystematicallyhigherthreshol dsfortheirlargeaspectratio systemwhencomparedtopredictionsassuminglineardampin g.Kudrolli&Gollub[ 49 ] extensivelystudiedthepatternsformedbysingle-frequen cyforcingandestablished thedependenceofatransitionfromsquarestostripesonthe uidviscosity,andfurther reportthehigherordertransitionsfromhexagonsandstrip estospatiotemporalchaos. M ¨ uller[ 61 ]wasthersttoreportanequilateraltriangularpattern,a ccessedby excitationwithtwofrequenciesofratio1:2andexplainedq ualitativelybyamplitude equationsdescribingtheinteractionofthreemodes.Edward s&Fauve[ 28 ]extended thetwofrequencyworkofM ¨ ullertothatofaneven-oddcaseofratio4:5,thereby breakingthesubharmonic-harmonictimesymmetryoftheins tability.Indoingsothey establisheddependenceofthepatternsonthemixingratioa ndphaseanglebetween 23

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thetwofrequencies and .Squares,stripes,andhexagonswereobserved,aswell asnovelquasi-patternsnearvaluesof producingbicriticalitywithrespecttoboth frequencies,andresultingfromtheinteractionof12separ atemodes.Nearthebicritical pointin4:5and6:7parametricforcingandobtainedbyvaryi ngthephaseangle Kudrolli etal. [ 50 ]observedtheappearanceofso-calledsuperlattice-Iands uperlattice-II patterns,andwasexplainedbythemasbeingtheinteraction oftwoseparatehexagonal lattices.SuchsuperlatticesweregeneralizedbyArbell&Fin eberg[ 2 ]tothatof2-mode superlatticesresultingfromtriadwaveinteractions,and additionallyobservedthe appearanceofnewso-calledsubharmonicsuperlatticestat esawayfromthebicritical point.Theseaspectsoftwo-frequencypatternformationin theFaradayexperiment wereextensivelystudiedfor18differentfrequencyratios byArbell&Fineberg[ 3 ].This myriadofpatternformationbehaviorprovidesinsightinto phenomenaandinteractions thatcouldpotentiallybeobservedinfutureexperimentsdo uble-frequencyexcitationin discretizedexperiments.LinearTheoryandComparisontoExperiments Giventhenonlinearnatureoftheuidequationsofmotion,t hetheoreticalstudyof theFaradayinstabilityistypicallyapproachedperturbat ivelywhereanalyticalsolutions canbederivedgivenvariousassumptions.Assumingtheparam eteraboutwhichthe equationsareperturbedisverysmall,thebasicstateofthe systemcanbedescribed asaquiescentlayeredsystemofzerouidvelocities,aati nterface,andanoscillating verticalpressuregradient.Translationofthissolutiont otheequationdenedbythe termswhichareofrstorderintheperturbationproducesap roblemwhoseanalysis constituteslinearstabilitytheory,becausethegrowthor decayofthesolutioninlong timeingeneraldeterminesthestabilityofthebasicstate. Anecessaryassumption thathastobemadeinthisanalysisisthespatialextentofth esystem,forwhich ahorizontallyinnitesystemisthemostsimplebecausethi sallowstherstorder solutionstobewrittenasbeingspatiallyperiodicofwaven umber k .Theoreticallythe 24

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horizontallyinnitesystemhasaccesstoaninnitecontin uumofthesewavenumbers, andfortheconditionsofinstabilitytobedeterminedthene utralconditionsatwhich thesewavenumbersneitherdecaynorgrowaredetermined.In thecaseofthe Faradayinstabilitywhereitisobservedthereexistsacrit icalthresholdamplitudefor anygivenfrequencyatwhichtheinterfacedestabilizes,th ewavenumberwiththelowest associatedthresholdwouldrepresenttherstunstablemod einanexperimentthatwas conductedatsuccessivelyhigheramplitudes. ThisanalysiswasrstperformedbyBenjamin&Ursell[ 11 ]forahorizontally inniteinvisciduidwithapassiveupperlayer.Indoingso theyhighlightedthatevery wavenumber k possessedanunforced,naturalfrequency,andthattheinst ability resultedfromaresonancethatoccursbetweenthisandthepa rametricfrequency.The absenceofdissipationintheirmodel,however,meantthatp erfectresonancecould occurforawavenumberwhenexcitedwithafrequencyequalto ahalf-integermultiple ofitsown,andthus,inahorizontallyinnitesystemwhichh asaccesstoaninnite numberofmodes,thefreesurfaceisunstabletoallparametr icexcitations.Benjamin& Ursellthemselvescomparedtheirtheorytoadiscretizedex perimentwhichdidnothave accesstoaninnitenumberofmodes,andthiswillbediscuss edfurtheronasitisthe regimeoffocusforthisdissertation. ThemodelusedinthisdissertationwasdevelopedbyKumar&T uckerman[ 52 ], astheyperformedthelinearstabilityoftheanalysisofthe nonlinearsystemwithout makingtheinviscidassumption.Inclusionofviscositypre ventsthereductionofthe linearproblemtoaMathieuequationdescribingeachwavenu mber,butratherthe instabilityofeachwavenumberisdenedbyaninnitesetof coupledequationsby Fourier-Floquetanalysiswhichmustbesolvednumerically forthethresholds.In doingsothewavenumbercorrespondingtotheminimumthresh oldistherstexcited inanexperiment,whichwasconrmedbyBechhoefer etal. [ 8 ]forawiderangeof 25

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frequencies(seeFigure 1-1 ).Thethresholdforsystemsexcitedbytwofrequencieswas theoreticallypresentedandexperimentallyconrmedbyBes son etal. [ 12 ]. Anideawhichisrepeatedseveraltimesthroughoutthisworki stheinherent differencesbetweenFaradayexcitationathighandlowfreq uencies.Themain assumptionsofthetheoriesofKumar&TuckermanandBesson etal whichrequire careinexperimentarethatthesystemmustproduceabasesta teofaatinterface andnoow,andmusthaveaccesstothewavenumbercorrespond ingtotheminimum predictedthreshold.Thenormalcauseforviolationofthe rstconditionistheproduction ofameniscusbytheinterfaceatthecontainersidewalls.Mo dulationofgravitybycell vibrationcausesthealterationofthedesiredprolemenis cus,whichinevitablyemits atravelingwavetotheinterioroftheinterface.Continued cellvibrationresultsina ripplepatternontheinterfacewhichtakesthegeometryoft hecell.Thisnon-ideality hasadverseeffectsbothontheexperimentalthresholdando nthenonlinearpattern formation,andthereforehigh-frequencyexperimentsempl oydifferentsidewall techniqueswhicheliminatetheeffect.Thebrimfultechniq ueofDouady&Fauve [ 26 ]isonesuchexamplewhichpinstheinterfaceata90 angleatthesidewall,thereby minimizingtheparasiticwaves.Thiswasoneofthemethodsu sedbyBechhoefer etal andmanyofotherexperimentsstudyingpatternformation. Equallyimportanttothemeniscuswhichmustbeconsideredin experimentsisthat thesystemmusthaveaccesstothewavenumberofthepredicte dminimumthreshold. Atsufcientlylowfrequencythishoweverisnotguaranteed, becauseinthisregime aboundaryconditionmustbeappliedtospecifythediscreti zationofthemodes.Two mathematicallyattractiveboundaryconditionsarethatei thertheinterfacecontact positionremainsxed,orthattheslopeoftheinterfacenor maltothewallremains90 whilemovingfreelyinthevertical.Whiletheformerpinnedc onditionisrealizedby thebrimfulconditionofDouady&Fauve,theresultingeigen functionsfortheinterface aregenerallynotcompatiblewiththelinearstabilityanal ysesofBenjamin&Ursellor 26

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Kumar&Tuckerman.Theoryforsuchwaves,includingapproxi mationoftheirnatural frequencieshasbeenputforthbyBenjamin&Scott[ 10 ],butitsincorporationinto Faradaywavetheoryiseeting.Thelatterstress-freecond itioniscompatiblewiththe linearstabilitytheories,allowingdecouplingofthesyst emswavenumbers,andisthe conditionBenjamin&Urselltoadapttheirtheorytotheirdis cretizedexperiments.It shouldbenotedhowever,thatinnitecontainerssuchastha tofBenjamin&Ursell, stressesfromthesidewallsinevitablyappearduetoviscou sStokesboundarylayers, capillaryhysteresisandinterfacialdissipativeeffects .Theirsystemusedwater,a sufcientlylowviscosityuid,suchthatthesystemdissip ationwasinfactsolelydened bythesestresses,aswillbeshowninChapter 3 Knowledgeoftheoverallsystemdissipationiscrucialinbei ngabletopredictthe thresholdamplitudeatwhichtheinstabilityappearsinany regimeoftheinstability.While carefulexperimentaldesignandlargeaspectratiosathigh frequencieshasconstrained theseeffectstothebulkuids,enablingconnectiontotheK umar&Tuckermantheory, whatwillbeshownisthatforallpreviousexperimentsinthe discretizedregimeisthat thedissipativeeffectshavebeenconstrainedtotheseside walleffects.Predictionofthis behaviorisenormouslycomplex,andexhibitsaremarkablec onnectionbetweenthe molecular,boundarylayerandcontainerlengthscales.The dampingoflongwavelength waterwavesinrectangularbasinswasstudiedbyKeulegan[ 48 ],whoalsoestablished thatsidewallsurfacewettingwasakeyfactorasidenticale xperimentsonwavedecay producedfargreaterdampingratesofthewaterwavesinluci tecontainersthanin glasscontainers.Loweringofsurfacetensionbytheadditi onofaerosolbroughtthe dampingratesinthetwocontainersintoagreement.Miles[ 58 ]developedamodelfor thedampingofwavesforslightlyviscousuidsthatseparat elytreatedthecontributions duetowallboundarylayers,surfacelmsproducedbycontam ination,andcapillary hysteresis.Applicationofthemodeltothewavedampingresu ltsofKeuleganandCase &Parkinson[ 15 ]yieldedqualitativeexplanationsfortheirobserveddamp ingrates 27

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butnotquantitative.Hockingpresentsamodelwhichinclud estheeffectofcontact hysteresisbymeansof“wetting”boundaryconditionwhichd escribesalinearrelation betweentheinterfacecontactangleandvelocity,andheuse sthismodeltoanalyze therelativeimportanceofcapillarytoviscouseffectsint hedampingratesobservedby Benjamin&Ursell,Case&ParkinsonandKeulegan. InapplyingthemodelofMiles[ 59 ]tothesingle-modeFaradayexperimentsof Henderson&Miles[ 36 ],itwasseenthattherateswerecorrectlypredictedprovid ed thatthewaterviscositywastakentobe3cSt.Uponusingthisd ampingratethestability modelusedbyHenderson&Mileswasabletoaccuratelypredic ttheFaradayinstability thresholdforthefundamentalaxisymmetricmodeintheircy lindricalcell.Tipton&Mullin [ 77 ]andDas&Hopnger[ 23 ]bothwereabletomatchtheirexperimentalthresholds withcurvesgeneratedusingdampingratesmeasuredfromthe experiment.Whilethis approachisinformative,italsomotivatesthisworktodeve lopanexperimentwhose thresholdscanbepredictedapriori. Neverthelesstheseworksalsostudiednonlinearphenomena .Thediscretized regimeisinherentlycharacterizedbyaseriesof“oscillat ions”intheinstabilitythreshold asfrequencyisincreased,correspondingtoeachallowedce llmode.Eachofthese oscillationsdescendstoaminimumthresholdlocatedneart henaturalfrequencyofthe mode,andthemodesexcitedatlowerfrequenciesandhighert hresholdsarereferred toas“detuned”modeswhilethoseathigherfrequenciesandt hresholdsarereferred toas“tuned”modes.Anobservationmadebyprevioussingle-m odeexperimentssuch asHenderson&Miles,Tipton&MullinandDas&Hopngerwasth atthebifurcation issubcriticalfordetunedmodesandsupercriticalfortune dmodes,consistentwith parametricallyexcitedsingledegreeoffreedomsystems.H ysteresiswasalsoreported wherethethresholdatwhichanexciteddetunedmodedecayed waslessthanthe onsetthreshold.Mullin&Tiptonprobedthebifurcationstr uctureofthefundamental axisymmetricmodeanditsdependenceupontheliquidlayerh eights.Das&Hopnger 28

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observedregimesinwhichperioddoublingandperiodtripli ngtothemainsubharmonic responseoccurred.NonlinearTheoreticalAdvancement Inthelinearstabilityanalysis,oneisabletodetermineth econditionsatwhich atinyperiodicperturbationisneutrallystable,thatis,i tneithergrowsnordecaysin time.Experimentalprecisionrestrictsonefrombeingablet ondthispointdirectly, instead,itisnecessarytointerpolatetheneutralthresho ldliesbetweentheconditions ofonestableandanotherunstablestate.Theunstablestate naturallydepartsfrom thebasicstate,astinyperturbationspresentintheexperi mentgrowintime.While thelinearstabilityanalysispredictsthegrowthconstant sforthedeparturefromthe basicstate,theassumptionsmadeareonlyvalidasmalldist ancefromthecritical pointandtheassociatedlongtimegrowthisexponential.Vis ualizationofaowor interfacialdeectionintheexperimentoftenimpliesthat nonlineareffectshavealready becomeimportant.However,inregionssufcientlycloseto theonsetofinstability,itis oftenseenthatthelinearformoftheperturbationhasmanif esteditselfinthenonlinear experiment,anditsaturatestoaniteamplitude.Suchbehav iorisapotentialcandidate foraweaklynonlinearanalysisinwhichthetermsofhighero rderintheperturbation expansionmaybeconsidered.Uponusingacorrectlyscaledp erturbationparameter ofniteamplitude,whatisseenisthatsolvabilityconditi onsariseinthehigherorder problemsforthesolutionsgrowingoutoftherstorderprob lem.Theconditionmanifests itselfasadifferentialequationdeterminingthetemporal growthoftheamplitudeof therstordersolutions,inwhichthetimederivativeofthe amplitudeisbalancedby alineartermwhosecoefcientcorrespondstothegrowthrat eoflinearstability.The nonlinearitythenappearsastermseitherquadraticorcubi cintheamplitude,whose coefcientsgovernthedeparturefromlineargrowth.Thise quationiscommonlyreferred totheGinzburg-Landauequation,afterthosewhorstinves tigatedit.Itsappearancein 29

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hydrodynamicinstabilitiesiscoveredbyDrazin&Reid[ 27 ],whonoteitsresemblanceto theequationforlogisticalpopulationgrowth. TheimportanceofsuchtheorytoFaradaywavesisbasedonthe observation thatanunstablesystemdoesnotgrowinnitely,andthatnon lineardampingresultsin saturationtoaniteamplitudestandingwave.Inthecaseof single-modeexcitation, weaklynonlinearanalysishasnotbeenextendedfromthevis cousKumar&Tuckerman lineartheory,andviscosityhasbeenincorporatedeitherb ylineardampingorviscous boundarylayers.Suchanalysisresemblesthatofnonlinears ingledegreeoffreedom systems,coveredbyNayfeh&Mook[ 63 ].Miles[ 59 ]developedtheamplitudeevolution equationsbyaLagrangianformulationforinviscidFaraday wavesinsmallcylinders subjecttolineardamping,andisthetheorytowhichHenders on&Miles[ 36 ]compared theiramplituderesponses,seeingreasonableagreement.G u etal. [ 34 ]derivethe amplitudeequationfromtheinterfaceconditionforaCarte siangeometry,again includingdissipationvialineardamping,andtheresultso fVirnig etal. [ 79 ]are comparedtothistheory.Hill[ 38 ]presentsauniquetheoryforlinearlydampedFaraday resonanceinacartesiangeometrywherecorrectionsduetos idewallandinterfacial boundarylayersareincorporatedintothesolvabilitycond itions. Thevarietyofpatternsobservedintheregimeofhighsingle anddouble-frequency excitationhasbeenaprimaryinterestofweaklynonlineara nalysis.Ingeneralthese patternsaretheresultofexcitationoftheinstabilityata nitedistanceabovethe thresholdsuchthatabandofwavenumbersisexcitedsimulta neously,andthenonlinear interactionofthesemodesgivesrisetotheobservedpatter n.Milner[ 60 ]madetherst attempttodescribethisbehavior,andbyminimizationofaL yupanovfunctionalwritten fortheweaklynonlineargrowthofarbitrarysetsofwavenum berstheyndthatsquares wereastablepattern,inaccordancewiththeobservationof Tullaro etal. [ 78 ].This wasdonefortheinviscidequations,andakeystepwasthatda mpingwasincludedby substitutionoftheexpandedvelocitypotentialintotheex pressionforviscousdissipation. 30

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AnobservationwhoseimportancewasoverlookedbyMilnerwas thedivergenceofthe nonlineardampingcoefcientswhentwoexcitedwavenumber sseparatedbytheright anglesummedtoathirdwavewithwhichtheyresonatedwithat secondorder.Termed triadresonances,theirimportancetopatternformationwa snotedbyZhang&Vi nals [ 82 ]andincludedintheirweaklynonlineartheory.Theyassume dviscouseffectswere constrainedtoasmalllayerneartheinterfaceandpotentia lowinthebulk,aso-called quasipotentialapproximation.Inmakingthisassumptiona ndaccountingforthetriad interactionstheypredictthesquarepatterninthepurelyc apillarywaveregime,withthe possibilityofhexagonalandquasipatternsnearthelowlim itoftriadinteractioninthe mixedgravity-capillaryregime.Chen&Vi nals[ 17 ]writetheweaklynonlineartheory fromthefullyviscousNavier-Stokesequationsandproduceq uantitativeagreement betweentheirpredictionsandthepatterntransitionsobse rvedinthephasespaceof Kudrolli&Gollub[ 49 ].FinallySkeldon&Guidoboni[ 72 ]offerrigoroustreatmentof thefullyviscousproblemsubjecttodouble-frequencyforc ingandndgoodmatching withthe“regular”patternsofKudrolli etal. [ 50 ]andqualitativeexplanationoftheir superlattice-IandIIpatterns.FaradayWavesinFerrouids Ferrouids,inventedinthe1960s,arecolloidalsuspensio nsofmagneticparticles whoseBrownianmotionpreventssettlingundergravityandsu rfactantcoatingprevents agglomeration,seeRosensweig[ 70 ].Resultingisamagnetizableuidwhichhasseen manycommercialapplicationssuchasliquido-ringsinseal sorbearinglubricants, andcurrentinterestliesinapplicationtomicro-andnanoscaledevicesorbiological applicationssuchasdrugdelivery.ThesubharmonicFarada yinstabilityonaferrouid interfaceresultingfromanoscillatingmagneticeldwaso bservedbyPerry&Jones[ 64 ] inathinchannelandthebehaviorwasconnectedtothatofthe Hillequation.Bashtovoi &Rosensweig[ 7 ]observedtheinstabilityinasmallcylindricalcontainer andreported effectsofmodediscretization,inadditiontoatheoretica lconnectiontotheMathieu 31

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equation.TheRosensweiginstabilityofaferrouidinterf aceisobservedwhenasystem issubjectedtoanormalmagneticeldofcriticalmagnitude ,asreportedbyCowley& Rosensweig[ 21 ],andexperimentsbyP etr elis etal. [ 65 ]showedthisinstabilitycouldbe stabilizedbyparametricoscillationofthemagneticeld. Anespeciallypeculiarbehavior offerrouidsisthe“negativeviscosity”effect,rstshow nbyBacri etal. [ 4 ],which resultsfromaresonanceofthemagneticparticledipolewit hanoscillatingmagnetic eld,therebytranslatingparticleangularmomentumtothe surroundingcolloidaluid. LinearstabilitytheoriesfortheFaradayinstabilityinma gneticuidsemployingmethods analogoustothatofKumar&Tuckermanhavebeenpresentedby Bajaj&Malik[ 6 ]for single-frequencyandBajaj[ 5 ]fordouble-frequencyexcitation.Areviewofferrouidsi s givenbyRinaldi etal. [ 69 ]. FaradayWavesinMicrouidics Whilethepattern-formingnatureoftheFaradaysystemmaybe themostscientically attractiveaspectofthephenomena,therecurrentlyexists interestinFaradaywavesin microuidicdesign.Duetothelargewaveresponseobserved insingle-modeexcitation, thediscretizedregimeserveswellasameanstoachievebulk uidmixing.Excitation oftheinstabilitywellabovethethresholdinevitablyresu ltsinhighlynonlinearsloshing, aeldcomprehensivelyreviewedbyIbrahim[ 42 ]withconsiderationofFaradaywaves. Suchmixingwouldbeenhancedforimmisciblesystemswithden sitystratication, asstudiedbyZoueshtiagh etal. [ 83 ].Intheinterestofdesigningsmallerandsmaller systems,theinstabilitywasexcitedbyultrasoundingeome trieswithwidthassmall as100micronsbyXu&Attinger[ 81 ].Atomizationofdropletssubjectedtoparametric vibrationwasstudiedbyJames etal. [ 45 ],andthismechanismhasbeenposedbyQi etal. [ 66 ]asameanstodesignportableultrasonicnebulizers,inrep lacementoftypical aerosoldesigns.Ultrahigh-frequencyactuationoftheins tabilityatfrequenciesover100 kHzandtheoreticaldeviationduetomicroscaleturbulence hasrecentlybeenreported 32

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byBlamey etal. [ 13 ].TheapplicationoftheFaradayinstabilitytomicrouidi csystems hasbeenreviewedbyFriend&Yeo[ 32 ]. 33

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CHAPTER2 MODELANDLINEARSTABILITYANALYSIS Inthischapteralinearstabilitymodelwillbedevelopedto whichtheexperiments canbecompared.Previousexperimentsonsingle-modeFarada yexcitationhave focusedusingsystemswithverylowviscosityuids,andhav emeasuredthesystem dissipationdirectlyfromtheexperiments.Indoingsothel inearandnonlinearbehavior canbeadequatelydescribedasasingledegreeoffreedomsys temwithlineardamping, anotherexamplebeingasimplependulum.Uniquetothiswork isthatthelinearstability modelforsingle-modeexcitationinacylindricalsystemwi llbederiveddirectlyfromthe Navier-Stokesequations,respectingtheuidmechanicalna tureoftheinstabilityand allowingananalyticaldescriptionofthebulkviscouseffe cts. Thenonlinearequationsaregivenrst,andintheinteresto fanalyzingthe departureofthesystemfromabasiclinearstate,thesystem isthenlinearized.The stabilityofthebasicstateisthendetermined,wheretheco nditionsofneutralstabilityof wavenumbers k aresolvedfor.ThisisdonebyaFourier-Floquetmethodresu ltinginan eigenvalueproblemandissolvedbyacheapnumericalcalcul ationlikethatperformed byKumar&Tuckerman[ 52 ].Theseresultsdescribethestabilityofhorizontallyat interfaceseparatingtwoimmiscibleuid,andthedependen ceoftheseresultsonthe modelparametersarediscussed.Nextthestress-freebound aryconditionrequiredby horizontallynitesystemsisappliedtotheseresults,whi chplacesconstraintsonthe wavenumbersallowedbythesystem.Thenalloftheseresults areappliedtoacase studyoftheeffectofgravity.Finally,thelinearanalysis isextendedtosystemssubjected twodouble-frequencyexcitation,followingthatofBesson etal. [ 12 ],inadditiontothe adaptationoftheirmodeltothecaseofanitecell. GoverningEquations Toanalyzethestabilityofaatimmiscibleinterfacesubje ctedtoanoscillatory verticaldisplacementwithstress-freesidewallswerstt akeahorizontallyinniteuid 34

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bilayerofdepths h 1 and h 2 withtheinterfacelocatedat z =0.Wethenobservethatthe caseofstress-freesidewallscanbeadaptedfromthehorizo ntallyinniteuidanalysis. TheequationsofmotiongoverningowofNewtonian,incompr essibleuidsinthebulk domainsarewrittenas, j ( @ t V j + V j r V j )= r P j + j r 2 V j + j g k and r V j =0 where j =1 indicatesthelowerlayerand j =2 theupperone.Thevector V isthe velocityeldoftheuidswithpressureeld P ,ofdensity andofviscosity .The verticaldisplacementandimposedacceleration d ( )= A cos (2–1) a ( )= A 2 cos areaccountedforintheequationsofmotionbyadoptionofam ovingreferenceframe viathevelocitytransformation V 0 = V + @ t d ( t ) k wheretheprimeindicatesthemovingframe.Substitutionint otheequationsofmotion, anddenotingtheconvectivederivative, @ 0 t ,by @ 0 t = @ t + @ t d ( t ) @ z theresultisthegroupingoftheforcingacceleration a ( t ) withthegravitationalacceleration, g .Removingprimes,wehave, j ( @ t V j + V j r V j )= r P j + j r 2 V j + j g + A 2 cos t k 35

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and r V j =0 No-slipconditionsapplyatthetopandbottomwallsandcont inuityofvelocityisapplied attheinterface.Thekinematicconditionresultsfromthej umpmassbalance(see Slattery[ 74 ]),i.e., [[ ( V U ) n ]]=0 at z = Z ( x y t ) (2–2) where U n isthespeedoftheinterface, n istheunitsurfacenormalpointinginto thelightuid,and Z = Z ( x y t ) isthepositionoftheinterface.Thesurfacenormal n = r f jr f j 1 ,where f =0= z Z ( x y t ) ,is n = k Z x i Z y j 1+ Z 2 x + Z 2 y 1 = 2 = k Z x i Z y j jr Z j (2–3) In 2–2 thebracesrepresentajumpquantity,evaluatedas [[ Q ]]= Q 2 Q 1 .Expandingthe jumpin 2–2 wehave ( 2 1 )( V U ) n =0 V n = U n V n = U n n V n = U (2–4) Writingthetotaldifferentialforthemotionoftheinterfac einaninnitesimaltime t we write Z ( x y t + t ) Z ( x y t )= @ Z @ t + @ Z @ x @ x @ t + @ Z @ y @ y @ t + @ Z @ z @ z @ t 0= @ Z @ t + U r Z 0= @ Z @ t + U n n jr Z j 0= @ Z @ t + U jr Z j 36

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andmakinguseof 2–4 followedby 2–3 wearriveat @ Z @ t = V n jr Z j @ t Z = V z V x @ x Z V y @ y Z at z = Z ( x y t ) (2–5) notingthatthesubscriptsonthevelocityindicatethecart esiancomponentwhilethose onthesurfaceelevationindicatepartialderivatives.The stressbalanceattheinterface, [[ P I + ( r V +( r V ) | )]] n = r 2 H n at z = Z ( x y t ) statesthepressureandviscoustangentialstressdifferen cesbetweenthetwophases arebalancedbytheforcesarisingfromsurfacecurvature,w here r istheinterfacial tensionand 2 H istwicethemeansurfacecurvature.Themeancurvature,isd enedas 2 H = r r Z [1+ jr Z j 2 ] 1 = 2 = r 2 Z [1+ jr Z j 2 ] 1 = 2 + r Z 1 2 rjr Z j 2 [1+ jr Z j 2 ] 3 = 2 = r 2 Z (1+ jr Z j 2 ) 1 2 r Z rjr Z j 2 [1+ jr Z j 2 ] 3 = 2 (2–6) PerturbedEquations Observingtheinstabilitytooccurasatransitionfromaqui escentstateofnoowor nointerfacialdeectiontoastateofow,andtakingthistr ansitiontobeaninnitesimal perturbationoforder ofthisquiescentstate,thus V = v P = p 0 + p and Z = (2–7) wherethe O (0) velocityandinterfacialdeectionarezero.Substitutiono f 2–7 intothe governingEquations 2–2 2–5 and 2–16 ,andcollectingthetermsofcommonpowersin 37

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,revealstheno-owbasestatetobesimply @ z p 0 j = j g + A 2 cos t (2–8) whichstatesthatthegravitationalmodulationisbalanced byaverticalpressure gradient.Thestressbalancegivesaconditionofpressurec ontinuityattheinterface where p 01 = p 02 .Theexpansionofdependentvariableslocatedat z = Z ( x y t ) is attainedwith u = u 0 + ( u 1 + @ z u 0 )+ 1 2 O 2 inordertomapthevariablesfromtheunknowninterfaceposi tionto z =0 (seeJohns& Narayanan[ 46 ]).Collectionofthe O ( ) termsgeneratesthelinearizeddomainproblem j @ t v j = r p j + j r 2 v j (2–9) andthekinematicconditionandstressbalanceat z =0,i.e., @ t = w (2–10) and [[ ( p j + @ z p 0 ) I + j ( r v j +( r v j ) | )]] n = r r 2 n (2–11) Thetwotangentialcomponentsofthestressbalancecanbeco mbinedwithcontinuityto obtainthecondition 1 @ zz r 2H w 1 = 2 @ zz r 2H w 2 at z =0 (2–12) where r H = i @ x + j @ y isthehorizontaldivergence.Thenormalcomponentofthest ress balancebecomes [[ p j + @ z p 0 +2 j @ z w j ]]= r r 2H at z =0 (2–13) 38

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andapplying r 2H tothisresultsin [[ r 2H p j + r 2H @ z p 0 +2 j r 2H @ z w j ]]= r r 4H at z =0. (2–14) Anexpressionfor r 2H p j canbeevaluatedbyapplying r H totheperturbeddomain Equations 2–9 r 2H p j = j @ t r 2 @ z w j (2–15) notingthatviacontinuitywehave r H v j = @ z w j .Substitutionof 2–15 andthebase pressuregradient 2–8 into 2–14 yields [[ j @ t + j r 2 +2 j r 2H @ z w j + j g + A 2 cos t r 2H ]]= r r 4H at z =0. (2–16) Thetemporalevolutionofthisequationforagivenimposeda mplitude A andfrequency ultimatelydeterminesthestabilityofthelinearsystem.I tmaybenotedthatthe kinematiccondition 2–10 relates w intermsof therebyyieldingahomogeneous problemin LinearStabilityAnalysis Theperturbedsystemisanalyzedbyconsideringhorizontal lyspatialmodeswith wavenumber k .Atermwithtwoderivativesintimearisesfromtheperturbe dpressure eld,buttheappearanceof cos t viathebasepressuregradientpreventsusfrom expressingthestatevariablesinpureexponentialtimemod eswithgrowthrates, Insteadtheperiodicityofthesystemmustbeaccountedforb yincludingaFloquet exponent ,possiblycomplex,intheinniteFourierseries.TheFouri erseriesiswritten inmodesofthebasicfrequency, ,forconvenience.Accountingforboththehorizontal spatialandtemporaldependenceofthesystem,foreverydep endentvariable, ,we write =e i k r 1 X n = 1 e [ + i ( + n )] ^ n ( z ) (2–17) 39

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Toevaluatethe z -derivativesof w in 2–16 ,thedomainperturbedvelocityprolemust becalculated.Applying rr tothedomainEquation 2–9 ,thepressureeldis eliminated,yielding j @ t j r 2 r 2 v j =0. (2–18) Substitutionoftheexpansion 2–17 intothe z -component,thefourth-orderordinary differentialequationgoverningeachFouriermode, n ,for w is + i ( + n ) j @ zz k 2 w jn =0, (2–19) towhichthesolutionsare w jn = a jn e kz + b jn e kz + c jn e q jn z + d jn e q jn z (2–20) with q 2 jn = k 2 + + i ( + n ) j where = isthekinematicviscosity.Inthecaseof + i ( + n )=0 thefunctions containing q jn arereplacedwith z e kz .Theboundaryconditionsdeterminethecoefcients in 2–20 .Theno-owandno-slipboundaryconditionsonthebottoman dtopsurfaces are, w 1 n = @ z w 1 n =0 at z = h 1 and w 2 n = @ z w 2 n =0 at z = h 2 40

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andattheinterface z =0, w 1 n = w 2 n @ z w 1 n = @ z w 2 n 1 @ zz w 1 n + k 2 = 2 @ zz w 2 n + k 2 and w 1 n = w 2 n =( + i ( + n )) n hold.Thepressure p in 2–16 canbereplacedwithanexpressionincludingthevertical velocitycomponent w usingthe x and y componentsoftheperturbedequationsof motionalongwithcontinuity,andthenalformofthenormal componentofthestress balanceforeachFouriermode n isthen [[ j ( + i ( + n ))+3 j k 2 @ z w n j @ zzz w n ]]+ g r k 2 k 2 n = A 2 k 2 2 ( n +1 + n 1 ). (2–21) Theidentity cos t = 1 2 e i t +e i t hasbeenused,resultinginthecouplingofthe n th Fouriermode n tothe n +1 and n 1 modes.Thelinearstabilityofthesystem issolelygovernedbytheinniteseriesofequationsgivenb y 2–21 ,andsubstitution ofthederivativescalculatedfrom 2–20 givesaseriesofcoupledlinearequations homogeneousin n .Truncatedtoanitenumberofmodes N ,withthegrowthconstant settozero,thestabilityofthelinearizedproblemcanbeca stasaneigenvalue problemforsolutionsofneutralstabilityandtheircorres pondingeigenvaluesbeingthe amplitudes A atwhichtheyoccur,expressedas D = A B (2–22) Herethematrix D isgeneratedbythel.h.s.of 2–21 andoperatesontheeigenvector .Theresponsefrequency issettozeroin 2–21 forharmonicsolutionsandto 1 2 for subharmonicsolutions.Thematrix B isadoublebandedmatrixoflargely1'sand0's usedtoselectthecoupledmodes ,butdiffersslightlyforthe n =0 mode.Truncation 41

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from n =0to N resultsintheinclusionofthe 1 coefcientinthematrix B forthe n =0 modeandisreplacedwiththeconjugate 1 for =0 and 0 for = 2 ,byvirtueofthe timesymmetryoftheexpansion 2–17 .For =0 2–22 becomes 2666666666666666664 D r 0 D i 0 0000 D i 0 D r 0 0000 00 D r 1 D i 1 00 00 D i 1 D r 1 00 0000 D r 2 D i 2 0000 D i 2 D r 2 3777777777777777775 2666666666666666664 r 0 i 0 r 1 i 1 r 2 i 2 ... 3777777777777777775 = A 2666666666666666664 002000000000100010010001001000000100 3777777777777777775 2666666666666666664 r 0 i 0 r 1 i 1 r 2 i 2 ... 3777777777777777775 andfor = != 2 2666666666666666664 D r 0 D i 0 0000 D i 0 D r 0 0000 00 D r 1 D i 1 00 00 D i 1 D r 1 00 0000 D r 2 D i 2 0000 D i 2 D r 2 3777777777777777775 2666666666666666664 r 0 i 0 r 1 i 1 r 2 i 2 ... 3777777777777777775 = A 2666666666666666664 1010000 10100 100010010001001000000100 3777777777777777775 2666666666666666664 r 0 i 0 r 1 i 1 r 2 i 2 ... 3777777777777777775 ThesolutionofEquation 2–22 yieldsasetofamplitudeeigenvalues A ,forwhichthe lowestrealeigenvaluecorrespondstotherstexcitedmode inaphysicalsystemwhen theforcingamplitudeisgraduallyincreasedfromzeroupwa rd(cf.Kumar&Tuckerman [ 52 ]). SpatiallyInniteSystemResultsandDiscussion Solvingthelinearizedproblemforallwavenumbers k isasetoftonguesof instabilitysimilartothensproducedbytheMathieuequat ion[ 11 ][ 63 ](see A ,but withthetipssmoothedbyviscosityandnotdescendingtozer oamplitude,cf.Figure 42

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0 400 800 1200 1600 2000 0 0.4 0.8 1.2 1.6 2 wavenumber k ,1/mvibrationalamplitude A ,cm minimuminstabilitythreshold 2 3 2 2 Figure2-1.Fixed-frequencylinearstabilitythresholdfo rahorizontallyinnitebilayer. SystemuidsareFC70(1880kgm 3 ,12cSt)andsiliconeoil(846kgm 3 1.5cSt)with h 1 = h 2 =0.5cm,andfrequency f = != 2 =9Hz.Interfacialtension estimatedtobe 7 dyncm 1 [ 75 ]. 2-1 .Forthesecalculations N wastakentobe12.Thelinearsystemresponsearising fromthesetonguesalternatesbetweensubharmonic( = 2 )andharmonic( =0), wherethersttongueissubharmonic.Tomakeaclarication thatisimportanttoan experiment,werefertothersttongueassubharmonic,thes econdasharmonic,and thethirdassuperharmonic,asthewavesexcitedinthesereg ionsexecuteone-half, one,andone-and-a-halfperiodspercellperiod,respectiv ely.Inanexperimental systemthatapproachesthelaterallyinnitelimitliketha tofBechhoefer etal [ 8 ],the wavenumberwiththelowestthresholdamplitudewouldbeexc itedrstinaseriesof trialsofincreasingvibrationalamplitudes.InFigure 2-1 theminimumthresholdoccursat A =0.091cmwithawavenumber k of259.5m 1 .AsampleMATLAB R r codegenerating thisthresholdispresentedinAppendix B Figures 2-2 presentsseveralexamplesofhowtonguebehaviorisaltered by commonsystemparameters.PicturedinFigure 2-2 (a)istheeffectofincreasingthe parametricfrequency.Itisclearthatincreasingfrequenc ycausesashiftofalltongues towardshighwavenumbers,whichisaresultofthesystemreq uiringawavenumberwith 43

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0 500 1000 1500 0 0.2 0.4 0.6 0.8 1 wavenumber k ,1/mvibrationalamplitudeA,cm (a) increasingfrequency 0 500 1000 1500 0 0.3 0.6 0.9 1.2 1.5 wavenumber k ,1/mvibrationalamplitudeA,cm (b) increasingviscosity 0 500 1000 1500 0 0.3 0.6 0.9 1.2 1.5 wavenumber k ,1/mvibrationalamplitudeA,cm (c) decreasinglayerheight 0 200 400 600 800 0 1 2 3 4 5 6 7 8 wavenumber k ,1/mvibrationalamplitudeA,cm (d) bicritical Figure2-2.Fixed-frequencystabilitydependenceuponvar ioussystemparameters. PredictionsareforFC70and1.5cStsiliconeoilbilayersfor( a) h 1 = h 2 =5mm atf=9,11,13Hz;(b) h 1 = h 2 =5mmand f =9Hzwith 2 =1.5,20,50cSt;(c) h 1 = h 2 =5,3,1mmat f =9Hzand(d) h 1 = h 2 =1.5,1.3,1.1mmatf=5Hz. PhysicalparametersarethesameasinFigure 2-1 ahighernaturalfrequencytomeettheresonantcondition.T hisoccursbyselectionof higherwavenumberswithgreatercapillarycontribution.T hisselectioncanbepredicted ingeneralusingtheinvisciddispersionrelationforsingl e[ 11 ]anddoublelayer[ 52 ] systems.Whileheretheminimumthresholddecreases,itwill beshownlaterand explainedthatincreaseswithrespecttoincreasingfreque ncyarepossibleaswell. TheeffectofincreasinguidviscosityisshowninFigure 2-2 (b),wheretheupper layersiliconeoilviscosityisincreasedfrom1.5to20to50 centiStokes.Itisseen thattheincreaseinviscositycausesadampingofthetongue s,raisingtheminimum thresholds.Tonguedampingisgreaterforthehigherharmon ictongues,duetothe 44

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increasedviscouseffectofhigherwavenumber,choppierwa ves.Additionally,slight shiftstowardhigherwavenumbersareobserved. Figures 2-2 (c)and(d)presentthecomplexeffectoflayerheight.There exists adeeplayerlimitbeyondwhichlargerheightsdonotshowany effectontheonset threshold.Itiswellexplainedmathematicallybythe tanh kH terminthesingleuid inviscidtheory(seeAppendix A ).Figure 2-2 (c)showsthatloweringthelayerheights h 1 = h 2 =5to4andto3mmcausesshiftoftherstsubharmonictonguet owardhigher wavenumbers,andthresholdamplitudes.Observingthatres onanceisobtainedon accountofagravitationalaswellascapillarycontributio n,itisthereforeapparentthata reducedgravitationaleffectleadstoagreatercapillaryc ontribution.Theeffectofheight saturatesaswavenumberisincreasedandthestabilitydiag ramscoincidewitheach otheratsufcientlyhighwavenumbers. Loweringthelayerheightstoextremevaluesespeciallycau sesinterestingbehavior, wheretheharmonictonguecanbecomemoreunstablethanthe rstsubharmonic tongue.ThisisseeninFigure 2-2 wherethelayerheights h 1 = h 2 aredecreasedfrom 1.5to1.3to1.1mm.Uponperformingacalculationforalower frequencyof5Hz whereinertialeffectsshouldbecontrolling,itisseen,fo rthe1.1mmcase,thatthe harmonictongueistheleaststable.Inthe1.3mmcasethesub harmonicandharmonic minimaarepositionedatnearlyidenticalamplitudes,givi ngrisetoaco-dimension 2point.Thisbehavior,duetodiminishedinertialeffectsa ndincreasedviscouslm effects,wasrstpredictedbyKumar[ 51 ].Theharmonicresponsewithathinlayerwas conrmedexperimentallybyM ¨ uller etal. [ 62 ],andanonlinearpatternformingstudyof theco-dimension2pointwasdonebyWagner etal. [ 80 ]. ThetheoreticalcurvematchedbyBechhoefer etal [ 8 ]canbegeneratedby calculatingtheminimumpointinFigure 2-1 forarangeoffrequencies,notingthat theircurvesdepictforcingacceleration, A 2 ,insteadofforcingamplitude, A .Anexample ofthiscalculationisgiveninFigure 2-3 (a)forseveraldifferentlayerheights,whilethe 45

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0 5 10 15 20 25 30 35 0.04 0.06 0.08 0.1 0.12 frequency f ,HzvibrationalamplitudeA,cm 0 10 20 30 0 0.2 0.4 0.6 0.8 1 (a) increasingheight 0 5 10 15 0 100 200 300 400 500 frequency f ,Hzexcitedwavenumber k ,1 =m 0 10 20 30 0 500 1000 1500 (b) increasingheight Figure2-3.Minimumthresholddependenceuponfrequencyfo rhorizontallyinnite bilayers.Shownisthedependenceofthe(a)thresholdamplit udeand(b) wavenumberselectionforFC70andsiliconeoilbilayersof h 1 = h 2 =0.75,1,1.5 and2.5cm.PhysicalparametersarethesameasinFigure 2-1 .Circled pointdenotestherstexcitedmodefromFigure 2-1 correspondingwavenumberselectionisshownin 2-3 (b).Itcanbeseenthatthegeneral behaviorofthesecurvesdependsuponthelayerheight.Atlow forcingfrequencythe minimumthresholdincreaseswithoutboundasfrequencyisd ecreased,agreeingwith intuitionthatalightoverheavysystemisstable.Increasi ngtheforcingfrequencycauses asharpdropintheminimumthresholdamplitude.Forsystems oflargelayerheight thethresholdcandroptoaminimumvaluebeforerisingandth endroppingagain.The wavenumberselectioninFigure 2-3 (b)showsthatforlowerlayerheightswavenumber selectionishigher.Theshifttowardhigherwavenumbersar isesfromthediminished inertialeffectduetothedensityjumpacrosstheinterface fromwhichresonanceis achievedresultinginagreatercapillarycontribution.Hi gherwavenumbersfeelgreater viscouseffectsandthisisseenintherisesinthethreshold presentinthesystemsof h 1 = h 2 =1,1.5,and2.5cminFigure 2-3 (a).Forthe h 1 = h 2 =0.75cmcase,theheightis sufcientlylowsuchthattheviscouseffectsareimportant atmuchlowerfrequencies thanintheothercases,andalocalminimuminthethresholdc urveisthereforebarely seen. 46

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SpatiallyFiniteSystems Extensionofthismodeltothecaseofacellwithnitelateral dimensionpermitting separationofhorizontalspatialdimensionwasoutlinedby Benjamin&Ursell[ 11 ]for bothrectangularandcylindricalcross-sections.Theyapp liedtheseconditionstotheir stabilityresultsgeneratedbytheMathieuequation,andth eapplicationtotheviscous modelismuchthesame.Forasystemwitharectangularcrosssectionofwidth W and breadth L ,thesurfacewavesmustsatisfytheboundaryconditions @ @ x =0 and @ @ y =0 atthe x =0 and W and y =0 and L sidewalls,respectively.Theseconstraintsdiscretize thewavenumberselectiontothosewhichform90 anglesatthesidewalls.Functions thatfollowthisconstrainttaketheformoftrigonometricf unctionsofthehorizontal directions,viz., ( x y )= ^ cos k x x cos k y y with k x = n W and k y = m L (2–23) where n and m increasefromzerotoinnity.Dening k 2 = k 2 x + k 2 y ,andspecifyinga squarecross-sectioni.e., W = L ,aseriesofdimensionlesswavenumberscanbewritten forincreasing n and m ,viz. k nm W = ( n 2 + m 2 ) 1 2 ,producing k 0,1 W =3.14, k 1,1 W =4.443, k 0,2 W =6.283, k 1,2 W =7.025, k 2,2 W =8.886,and k 0,3 R =9.425andsoon.Thisseries speciesthewavenumbersallowedinFigure 2-1 foranitecellofsquarecross-section. Forthecylindricalcase,theallowedeigenfunctions m = ( r ) aregivenbythe functionssatisfying @ 2 @ r 2 + 1 r @ @ r + 1 r 2 @ @ + k 2 m m =0 47

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-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 ^ 1 ; 1 ^ 0 ; 1 ^ 2 ; 1 ^ 3 ; 1 ^ 4 ; 1 ^ 1 ; 2 Figure2-4.Cylindricalcellmodesandwavenumberssatisfy ingthestress-freecondition. whichcanbedecomposedas ^ l m = J l ( k l m r )sin l (2–24) where k l m isthe m thzeroof J 0 l ( k l m R ) R beingthecontainerradius.Theindices l and m ,indicatethenumberofazimuthalandradialnodes,respect ively.Thespatial 48

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0 100 200 300 400 500 0 1 2 3 4 5 wavenumber k ,1/mthresholdamplitude A ,cm (a) k 1 ; 1 k 4 ; 1 k 0 ; 1 k 3 ; 1 k 2 ; 1 minimuminstabilitythreshold 0 100 200 300 400 500 0 0.4 0.8 1.2 1.6 2 wavenumber k ,1/mthresholdamplitude A ,cm (b) k 0 ; 1 k 2 ; 1 k 1 ; 1 k 3 ; 1 k 4 ; 1 minimuminstabilitythreshold Figure2-5.Fixed-frequencylinearstabilityofsystemswi thcylindricalmode discretization.PredictionsareforaFC70andsiliconeoilb ilayerof h 1 =3.1 cmand h 2 =3.3cmoscillatedatfrequenciesof(a)3.3Hzand(b)7.045H z. Circledthresholdscorrespondtotheallowedsetofdiscret ewavenumbers foracylindricalcontainerof R =2.55cm.Physicalparametersarethesame asinFigure 2-1 formofthevariouscylindricallinearmodes l m arepresentedinFigure 2-4 ,andthe non-dimensionalizedvaluesof k l m R forthesemodesare k 1,1 R =1.841, k 2,1 R =3.054, k 0,1 R =3.832, k 3,1 R =4.201, k 4,1 R =5.318,and k 1,2 R =5.331[ 1 ]. Thecriticalthresholdscorrespondingtotheseallowedmod es(inadditiontohigher indexmodes)havebeenlabelledonstabilitydiagramsoftwo differentfrequenciesin Figures 2-5 (a)and(b).Hereitisapparentthatthewavenumbercorrespo ndingtothe subharmonicminimumthresholdisexcitedonlyinthecasewh ereitcoincideswithone oftheallowedmodes.Harmonic,andevermore,superharmoni cexcitationisuncommon athighfrequencies,butatlowfrequenciesinadiscretized systemonecanpotentially skiptherstsubharmonictongueandexciteharmonicandsup erharmonicmodes,asin Figure 2-5 (a),wherethe(2,1) h modewouldbeexcitedrstatanamplitudeof0.92cm. ThestabilitydiagramofFigure 2-5 (b)wascalculatedforacarefullychosenfrequencyto highlighttheappearanceofaco-dimension2point,asetofc onditionswheretwomodes 49

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2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 3.5 4 frequency f ,Hzthresholdamplitude, A ,cm 5 5.5 6 6.5 7 7.5 8 0 0.1 0.2 0.3 0.4 0.5 co-dimension2points (1 ; 1) sh (0 ; 1) sh (1 ; 1) h (2 ; 1) h (0 ; 1) h (3 ; 1) h (1 ; 2) h (1 ; 1) sh (2 ; 1) sh (0 ; 1) sh (1 ; 2) su (2 ; 1) sh Figure2-6.Minimumthresholddependenceuponfrequencyfo racylindricalsystem. PredictionsareforabilayerofFC70( 1916 kgm 3 ,12cSt, h 1 =3.1cm)and siliconeoil( 846 kgm 3 ,1.5cSt, h 2 =3.3cm)inbothaninnitesystem (dashedline)andcylindrical R =2.55cmsystem.Subscripts sh h and su representsubharmonic,harmonic,andsuperharmonicmodes arebothneutrallystableforthesameamplitudeandfrequen cy.Here,both(2,1) sh and (0,1) sh modesaresimultaneouslyexcitedatanamplitudeof0.123cm CombinedinFigure 2-6 areallofthediscussedaspectsofdiscretization,where theminimumamplitudesofinstabilityfromcalculationssu chasFigure 2-5 are plottedversusamultitudeoffrequenciesforacylindrical system.Presentarethe overlappingdipsofinstabilityforeachmode,descendingt ominimumthresholds.Each dipcorrespondstoasinglemode l m speciedbyEquation 2–24 .Whenpositionedat oneofthelocalminima,slightadjustmenttoalowerorhighe rfrequencycausesthe thresholdtoriseastheresonanttransferofenergytothewa vebecomeslessefcient. Themodesexcitedatfrequencieshigherthantheminimaaret ermedtunedmodes,and thoseatlowerfrequenciesasdetunedmodes.Thisdistincti onisimportantbecause manyincludingBenjamin&Ursellhavenoteddifferentnonlin earbehaviornearthe thresholdfortunedanddetunedmodes,whichwillbediscuss edlaterinthecontext 50

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ofdeterminingtheexperimentalthreshold.Furtherdepart urefromalocalminimum resultsintheintersectionwithanadjacentmode.Theseint ersections,orcusps,arethe co-dimension2pointshighlightedbyFigure 2-5 (b).Thegeneralorderingofthemodes isdictatedbythewavenumber,whereatlowfrequenciesharm onicandsuperharmonic modesareprevalent,followedbysubharmonicmodesatthehi gherrange.Thephysical parametersusedtocalculateFigure 2-6 arethesameasinoneoftheexperiments ofthisstudy,andoneresultofthisistheviscousdampingof sharpresonancesatlow frequenciesenteringfromthehigherharmonictongues.One ofthebenetsofusing aviscoussystemisaseriesofwell-spacedmodes,resulting fromthedampingof resonancesofhigherharmonictongues.Itwillbeshowninth enextsectionthattheuse oflessviscoussystems,typicalofpreviouswork,makesind ividualmodesmoredifcult todiscernduetotheinclusionofthesehigherharmonics.In cludedforcomparisonin Figure 2-6 istheinnitesystemthreshold,muchlikeFigure 2-3 (a).Atlargefrequencies thenitesystemthresholdisverynearlyidenticaltothein nitethresholdandcontinued increaseinfrequencycausesthemergingofthetwo.ACaseStudy:theEffectofGravity Anexampledemonstratingthepredictivecapabilitiesofthe modeliscomparison ofitsapplicationtoidenticalsystemsbasedonearthandmi crogravityenvironments, highlightingtheroleofgravityontheinstabilitythresho ld.Arelevantapplicationisthe propellantinarocketfueltank,whereuponvibrationsduri ngtake-offareknownto induceFaradayinitializedsloshingIbrahim[ 42 ]. InthecontextoftheRayleigh-Taylorinstability,alsodri venbyadensitycontrast acrosstwophases,oneintuitivelyknowsthatgravityhasad estabilizingeffectonthe system'sstability.ThisstandsincontrastwiththeFarada ysystem,whereitisfoundthat gravitynotonlypossessesadestabilizingeffectakintoit sroleintheRayleigh-Taylor instability,butitscontributiontothenaturalfrequency ofamoderesultsinashifttoward higherwavenumberswhenitisremoved,asseeninFigure 2-7 (a).Here,thesystemis 51

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0 100 200 300 400 500 0 1 2 3 4 5 6 wavenumber k ,1/mamplitude A ,cm g =9.81m/2 2 g =0 minimumthresholds (a) 0 200 400 600 800 1000 1200 0 0.2 0.4 0.6 0.8 1 wavenumber k ,1/mamplitude A ,cm g =9.81m/s 2 g =0 minimumthresholds (b) Figure2-7.Fixed-frequencylinearstabilityinearth-bas edandzero-gravity environments.SystemsareinnitebilayersofFC70and1.5cSt siliconeoil with h 1 = h 2 =3cmoscillatingat f =(a)2Hzand(b)8Hz. 0 5 10 15 20 25 30 35 40 0 0.04 0.08 0.12 0.16 0.2 frequency f ,Hzamplitude A ,cm g=0 g=9.81 m=s 2 (a) 2 3 4 5 6 7 8 0 0.04 0.08 0.12 0.16 0.2 frequency f ,Hzamplitude A ,cm g =9.81m/s 2 g=0 (4,1) sh (1,1) sh (2,1) sh (0,1) sh (3,1) sh (b) Figure2-8.Minimumthresholdinearth-basedandzero-grav ityenvironments. PredictionsareforFC70and1.5cStsiliconeoilbilayersas(a )innite systemsand(b)a R =4cmcylindricalcell.Systemparametersareidentical tothoseofFigure 2-7 oscillatedatafrequencywherezerog minimumthresholdislowerthanitsearth-based value,followingtheRayleigh-Taylorjustication.Howev ertheoppositeisseeninFigure 2-7 (b),wherethesamesystemisoscillatedatahigherfrequenc y,andtheshifttowards ahigherwavenumbersisaccompaniedbyanincreaseinthemin imumthreshold.Itis thereforeapparentthereexistsafrequencywherethezerog ravitysystemtransitions frombeinglesstomorestablethantheearth-basedsystem. ThiscrossoverintheminimumthresholdisdepictedinFigur e 2-8 (b)andresults fromincreasedviscouseffectsduetohigherwavenumbersel ectioninzerog systems. 52

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Atlargefrequenciesthezerog systemwillcontinuetoselecthigherwavelengths,at thereforegreaterthresholdsaswell.Giventhatthecrosso vertypicallyoccursatlow frequencieswherediscretizationaresignicant,thethre sholdsforasmallcylindrical cellaredepictedinFigure 2-8 .Inthediscretizedsystemthenotionofthecrossover frequencybecomesmoreambiguousasmultiplecellmodesint heearth-basedsystem becomelessandthenmorestablethanthezerog systembeforethezerog system becomesconsistentlymorestable.Itisalsoworthnotingth atinthezerog cellthe effectsofdiscretizationareminimized,asthewavenumber selectionhasbeenshifted intoaregionofthespectrumwithsmallerspacing. TheCaseofDouble-FrequencyParametricForcing Theinstabilitymayalsobeexcitedwithparametricforcing composedoftwo separatefrequencies.Followingthesingle-frequencyana lysis,predictionscanbemade forbothhorizontallyinnitesystemsandalsothoseinsmal lcontainerswhensubjected toamotionwithtwofrequencycomponents.Theadditionofas econdfrequency componentnaturallyaddsthreedegreesoffreedomtothepro blem:anamplitude, frequency,andphaselagofthesecondcomponent.Firstpres entedtheoreticallyby Besson etal. [ 12 ],thesedegreesoffreedomwereaccountedforbywritingthe forcing functionas d ( t )= A [cos cos( M 1 )+sin cos( M 2 + )] (2–25) a ( t )= A 2 M 2 1 cos cos( M 1 )+ M 2 2 sin cos( M 2 + ) intheirtheory,whichreplaces 2–2 inthesingle-frequencyanalysis.Thetwocomponents arewrittenintermsofabasicfrequency andaredenotedbytheintegers M 1 and M 2 andtheangle speciesthephaselag.Thefrequencyratio dividestheoverallforcing amplitude A betweenthetwocomponents,andincreasingfrom =0 to90 represents asweepfrompure M 1 topure M 2 excitation.Differentcomparativestudiescould beconductedbyadjustmentofthewaytheforcingfunctionis written,forexample, 53

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replacementofthemixingangle andoverallamplitude A withtwoindependent amplitudes A 1 and A 2 .However,whenthetwofrequenciesarewrittenasinteger multiplesofabasicfrequency ,Floquetanalysisidenticaltothesinglefrequencycase ispossible.TherebyinsteadofeachFouriermode n couplingtothemodes n 1 and n +1 ,thedouble-frequencycaseseeseachmode n beingcoupledtofourmodes, n M 1 n + M 1 n M 2 and n + M 2 .Inclusionofthenewfunctionintotheanalysisleadingto 2–21 givesthenewformofeachFouriercomponentofthenormalcom ponentofthestress balancetobe [[ j ( + i ( + n ))+3 j k 2 @ z w n j @ zzz w n ]]+ g r k 2 k 2 n = A 2 k 2 2 sin( )( n + M 1 + n M 1 )+cos( ) e i n + M 2 +e i n M 2 (2–26) Therealityconditionson n remainunchangedfromthesinglefrequencyanalysis.The B matrixof 2–22 nowtakesaform B =sin( ) B M 1 +cos( ) B M 2 ,withtheintegers M 1 and M 2 specifyingthebandspacing.Thematrices B M 1 and B M 2 aregiveninAppendix C forboththeharmonicandsubharmoniccases.Again,thethres holdsofneutralstability aredeterminedfromthesetofeigenvaluesobtainedfrom 2–22 ,withthenewformof the B matrix.Repeatedsolutionforharmonicandsubharmonicres ponseforarangeof wavenumbers k providesthefamiliarstabilitydiagramssimilarto 2-1 Double-FrequencyPredictionsforHorizontallyInniteSyst ems Withatotalofvedegreesoffreedom,alargenumberofcompar ativestudies mustbemadetouncovertheeffectofadditionofasecondfreq uencycomponent. Thegeneraleffectsofdensity,viscosity,andlayerheight remainunchangedfromthe singlefrequencycase,howevertheadditionofthethesecon dcomponent'sdegreesof freedompresentmanynewphenomena.Figures 2-9 (a)-(f)depicttheappearanceand growthofnewtonguesofinstabilityduetotheadditionofth esecondcomponent.When =0 ,thesystemisforcedwithasinglefrequency M 1 andthestabilityisidenticalto thesinglefrequencyresultofFigure 2-1 .Increasing to10 inFigure 2-9 (b),anisland 54

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0 500 1000 1500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 wavenumber k ,1/mamplitude A ,cm =0 (a) 0 500 1000 1500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 wavenumber k ,1/mamplitude A ,cm (b) =10 0 500 1000 1500 0 0.2 0.4 0.6 0.8 1 1.2 wavenumber k ,1/mamplitude A ,cm (c) =30 0 500 1000 1500 0 0.2 0.4 0.6 0.8 1 wavenumber k ,1/mamplitude A ,cm (d) =60 approximatebicriticalthreshold 0 500 1000 1500 0 0.2 0.4 0.6 0.8 1 wavenumber k ,1/mamplitude A ,cm (e) =70 0 500 1000 1500 0 0.2 0.4 0.6 0.8 1 1.2 wavenumber k ,1/mamplitude A ,cm (e) =90 Figure2-9.Fixed-frequencystabilityofinnitesystemsw ithdouble-frequencyexcitation. Systematvariousangles (a)-(f)isaa h 1 = h 2 =3cmbilayerofFC70 ( 1 =1888kgm 3 ,12cSt)and1.5cStsiliconeoil( 2 =846kgm 3 ).Basic frequency f = != 2 =is3Hzwith ( M 1 M 2 ) =(3,4). 55

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regionofinstabilityappearsbetweenthersttwooriginal tongues,andanothertongue hasappearedbetweentheoriginalsecondandthirdtongues. Sharpresonanttongues alsoappearatlowwavenumbers.At =30 thenewtonguescontinuetoappearand grow,andsomeoftheoriginaltonguesfromthesinglefreque ncyresulthavenoticeably shrunk.At =60 theprimaryharmonictongueappearingduetothesecondforc ing componenthasgrownsuchthatitsminimumthresholdisequal lyunstablewiththe originalprimarysubharmonictongue.Thistypeofco-dimen sion2pointmanifestsitself atacertainfrequencyratio inanydoublefrequencysystem,andtheapproximate angleatwhichisappearscanbedeterminedbysettingthetwo accelerationsofdifferent frequencyequal,thatis, A 1 ( M 1 ) 2 = A 2 M 2 2 2 A cos M 2 1 = A sin M 2 2 tan = M 2 M 1 2 For ( M 1 M 2 ) = (3,4) forcing,bicriticalityshouldappearat =60 ,butduetohigher dampingofthemain M 2 tongue,theviscouscontributiongivesaslightlyhigheran gle. Continuedincreaseof pastthetheco-dimension2pointinthefurtherlessening andnaldisappearanceofthetonguesowingtothe M 1 forcing.Alltongueswith =90 excitationareobtainedwith =0,implyingaharmonicresponse,andthisisaresult ofthesecondfrequencybeinganeven-integermultipleofth ebasicfrequency.The responseinthersttongueisharmonicwithrespecttothefr equency ,butremains subharmonicwithrespectto M 2 Followingtheanalysisforhorizontallyinnitesystemsin thesinglefrequencycase, themovementoftheminimalthresholdandwavenumberselect ionindoublefrequency systemsisdisplayedinFigure 2-10 forvariousvaluesof ,andresemblesthebehavior ofsinglefrequencysystems.Thecurvesfor =0and90 collapseontoeachotherwhen thefrequenciesarerescaledtakingintoaccountthevalues of ( M 1 M 2 ) .Figure 2-10 can 56

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1 2 3 4 5 6 7 8 9 10 0.025 0.03 0.035 0.04 0.045 0.05 0.055 frequency f ,Hzamplitude A ,cm 60 70 10 30 =0 90 (a) 1 2 3 4 5 6 7 8 9 10 0 200 400 600 800 1000 1200 1400 1600 1800 frequency f ,Hzwavenumber k ,1/m (b) Figure2-10.Minimumthresholddependenceuponbasicfrequ encyforaninnite system.Shownisthe(a)minimumthresholdand(b)wavenumber selection forvariousforcingratios with ( M 1 M 2 ) =(3,4)forthesystemofFigure 2-9 Dashedlinesrepresentsubharmonicandsolidlinesreprese ntharmonic thresholds.Circledpointsrepresentthetongueminimafro mFigure 2-9 0 10 20 30 40 50 60 70 80 90 0.025 0.03 0.035 0.04 0.045 0.05 0.055 frequencyratio, amplitude A ,cm subharmonicthresholds harmonicthresholds Figure2-11.Minimumthresholddependenceuponthefrequen cyratio foraninnite system.PredictionsareforthesystemofFigure 2-9 oscillatedatxed frequency f = != 2 =3Hz.Dashedlinesrepresentsubharmonicthresholds andsolidlinesrepresentharmonicthresholds.Circledpoi ntsinFigure 2-10 (b)representthewavenumberselectionofthesystem. 57

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0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0 1 2 3 4 5 6 basicfrequency f = /2 ,Hzamplitude A ,cm 1.6 1.8 2 2.2 2.4 2.6 0 0.1 0.2 0.3 0.4 0.5 (2,1) sh =0 (a) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 basicfrequency f = /2 ,Hzamplitude A ,cm 1.6 1.8 2 2.2 2.4 2.6 0 0.1 0.2 0.3 0.4 0.5 =10 (b) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0 0.5 1 1.5 2 2.5 basicfrequency f = /2 ,Hzamplitude A ,cm 1.6 1.8 2 2.2 2.4 2.6 0 0.05 0.1 0.15 0.2 0.25 =30 (c) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0 0.4 0.8 1.2 1.6 2 basicfrequency f = /2 ,Hzamplitude A ,cm 1.6 1.8 2 2.2 2.4 2.6 0 0.05 0.1 0.15 =60 (d) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0 0.4 0.8 1.2 1.6 2 basicfrequency f = /2 ,Hzamplitude A ,cm 1.6 1.8 2 2.2 2.4 2.6 0 0.05 0.1 0.15 =70 (e) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0 0.4 0.8 1.2 1.6 basicfrequency f = /2 ,Hzamplitude A ,cm 1.6 1.8 2 2.2 2.4 2.6 0 0.05 0.1 0.15 =90 (f) Figure2-12.Minimumthresholddependenceuponbasicfrequ encyforacylinder. Predictionsaremadewithvariousratios (a)-(f)forthesystemofFigure 2-9 ina R =2.55cmcylinder.ThemodeselectionofFigures(a)and(f) followsthatofFigure 2-6 beusedtodeducethatanadjustmentof caneitherincreaseordecreasetheoverall threshold A ,andthisisclearlypresentedinFigure 2-11 wheretheminimumthresholdis plottedasafunctionof foraxedfrequency. 58

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0 10 20 30 40 50 60 70 80 90 0 0.2 0.4 0.6 0.8 1 1.2 angle amplitude A ,cm 1.18 (a) (1,1) sh f =1.22Hz 1.20 (0,1) h 0 10 20 30 40 50 60 70 80 90 0.06 0.24 angle amplitude A ,cm (b) (1,1) sh modes (0,1) h modes 1.80 1.82 f =1.78Hz Figure2-13.Minimumthresholddependenceupontheratio foracylinder.Predictions areforbasicfrequencies of(a)1.2Hzand(b)1.8Hzinthesystemof 2-12 .Modeselectiontotheleftandrightoftheco-dimension2po intsin(a) are(0,1) h and(1,1) sh ,then(1,1) sh and(0,1) h in(b).Circledandsquared pointscorrespondtothethresholdsfromFigures 2-12 (a)-(f). Double-FrequencyPredictionsforCylindricalContainers Theadjustmentsmadetothepredictionsoncethemodesaredi scretizedby applicationofthestress-freeboundarycondition,asdone forthesingle-frequencycase, arepresentedinFigures 2-12 and 2-13 .ThepredictionsofFigures 2-12 (a)and(f)for =0and90 forcingrepresentsinglefrequencyforcingandcanbecompa reddirectly withthesinglefrequencypredictions 2-6 todeterminemodeselection.Startingwith =0 ,additionofthesecondforcingcomponenteffectivelysupe rimposesthesame setofresonanttongues,locatedatthe M 1 componentfrequenciesmultipliedbythe factor M 2 = M 1 .Slightinteractionisseen,butthethresholdamplitudesof thegrowing M 2 ordiminishing M 1 tonguesareproportionalto .Co-dimension2pointsaretherefore createdforthevalueof wherean M 2 thresholdsurpassesthe M 1 tonguethresholdfor agivenfrequency.Suchco-dimension2pointsarevisualized inFigure 2-13 (a)and(b), whichalsohelpfurthertovisualizethethree-dimensional A = A ( f ) stabilityspace. Itshouldbenotedthatthepurposeofthisanalysishasbeent omakepredictions whichpertaintotheavailableexperiment.Intheexperimen t,theamplitudeand frequencyareadjustedindependently,andduetothisthepr edictionshavebeen 59

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reportedasthresholdamplitudesincontrasttothresholda ccelerations,whichisoften doneinFaradayliterature.Forthisreasonitisdifcultto makestatementsregarding whetheradditionofasecondfrequencycomponent“stabiliz es”or“destabilizes”a system,becauseonestatementcouldbetruefortheamplitud ebutnottheacceleration. Intuitionsuggeststherewouldbecooperativeanddestruct iveeffectsofthetwo components.Anotheravenuethatwasnotpursedwastheeffect ofdifferentinteger sets ( M 1 M 2 ) .Theintegers(3,4)representaclosesetoffrequencieswhe ncomparedto otherpossibilitieswithagreaterdifference M 2 M 1 .Ofgreatexperimentalpotentialis thatbytailoringtheintegerset ( M 1 M 2 ) ,onecouldpositionanytwodesiredmodesnext toeachother,allowingstudyoftheirco-dimension2point. Lastly,theroleofthephase anglewasnotstudied.Whilethephaseanglehasbeendetermin edimportanttopattern formation[ 28 ],ithasbeenreportedtohaveverylittleinuenceonthelin earstability threshold[ 12 ].However,potentialinuenceofthephaselagontheintera ctionbetween theexperimentalnon-idealityandtheinstabilitywillbed iscussed. 60

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CHAPTER3 PREVIOUSEXPERIMENTSONSINGLE-MODEEXCITATION Oneofthecentralpurposesofthisworkistoreproducetheth resholdspredicted inFigure 2-6 .SeveralFaradayexperimentswereperformedinsmallcylind erswith theaimofmeasuringtheonsetofinstability,intheregimew heremodediscretization isimportant.Mostoftentheexperimentsinvestigatedthes tabilityboundaryforonly oneortwomodesinasystemwithlimitedviscouseffectsandt helargedissipation bycontrastarisingfromthesidewallsresultsinsystemsex hibitingnon-idealbehavior. Comparisonofthedatafromtheseexperimentstothepredict ionsofthemodelof Chapter 2 highlightsmanyofthediscrepanciesinthepreviousexperi mentsandraises manyquestionsregardingthemechanismsatplaycausingthi s,towhichthecurrent studywillprovidenewinsight.InFigures 3-1 (a)through(f)theresultsoftheseworks havebeencomparedtotheviscousmodelandwillbediscussed TheearlyexperimentsofBenjamin&Ursell(Figure 3-1 (a))utilizedadeeplayer ofwaterastheoperatinguidinatestcylinderofdiameter5 .4cm.Usingthenodal indexingofthiswork,theyinvestigatedthe(1,2) sh mode,appearingwithanatural frequencyat15.82Hz,incontrasttotheinviscidnaturalfr equencyof15.87Hz. Respectableagreementwiththeirinviscidmodelwasfounda ftershiftingthenatural frequencytotheobservedfrequency,buttheamplitudesare noticeablyhigheratthe tongueminimum,aresultofsystemdissipation.Upontting thedatawiththeviscous model,itisseenthatthestabilitytongueismodiedonlysl ightly,indicatingthatthe viscouscontributionoftheinteriorisnegligible.Onethe reforecanonlyconcludethatthe hiddendissipationcomesfromeithersidewallorfrominter facialdissipativeeffects.A ndingoftheexperimentsofChapter4wasthattheoreticala greementoftheobserved thresholdsimprovedwiththeincreaseoftheBondnumber, Bo = g r k 2 61

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12 13 14 15 16 17 18 0 0.02 0.04 0.06 0.08 frequency f ,Hzimposedamplitude A ,cm 15.7 15.9 16.1 0 2 4 x 10 -3 (3,1) sh (0,1) sh (1,2) sh (a) (4,1) sh 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 0 0.1 0.2 0.3 0.4 0.5 frequency f ,Hzimposedamplitude A ,cm h =2 R predictions h =2 R thresholds h = R= 2predictions h = R= 2thresholds (b) 15.4 15.6 15.8 16 16.2 16.4 0 0.005 0.01 0.015 frequency f ,Hzimposedamplitude A ,cm (c) (2,4) sh (0,4) sh (7,2) sh (4,3) sh 9.2 9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 11 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 frequency f ,Hzimposedamplitude A ,cm viscoustheory lineardampingmodel (3,1) sh (0,1) sh (0,3) h (d) 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5 6.6 0 0.05 0.1 0.15 0.2 0.25 frequency f ,Hzimposedamplitude A ,cm =19.3dyn/cm =21.5dyn/cm =23dyn/cm (e) (3,1) sh (0,1) sh 8 8.2 8.4 8.6 8.8 9 9.2 9.4 0 0.02 0.04 0.06 0.08 frequency f ,hzimposedamplitude A ,cm viscoustheory lineardampingmodel (f) (2,4) h (4,3) h (0,4) h (2,1) sh (0,1) sh (3,1) sh Figure3-1.Pastexperimentalthresholdsincylinders.Syst emdataarefrom(a) Benjamin&Ursell[ 11 ]:waterandair, =1000kgm 3 =1cSt, r =72.5dyncm 1 R =2.70cm, h =25.4cm;(b)Dodge etal. [ 24 ]:waterand air, R =14.5, h =29and7.25;(c)Ciliberto&Gollub[ 20 ]:waterandair, R =6.35, h =1;(d)Henderson&Miles[ 36 ]:water/surfactantandair, r =42.3, R =3.725, h =2.04;(e)Tipton&Mullin[ 77 ]:waterandsiliconeoil, 1 =997.5 and 2 =766, 1 =1.033and 2 =0.670, r unreportedandsetto23, R =1.76cm, h 1 =5.31and h 2 =0.739;(e)Das&Hopnger[ 23 ]:FC72andair, =1690, =0.406, r =11, R =5, h =6.Alltheoreticalcomparisonsaremadetothemodel of[ 52 ],assumingstress-freesidewallsandnointerfacialdissi pativeeffects. ThedashedlinesinFigures(d)and(f),however,aretheline ardamping predictionsoftheoriginalworksandarereproducedherefo rcomparisonto thecaseofstress-freesidewalls.Allunitsarethesameasli stedfor(a). 62

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ameasureoftherelativeimportanceofthesidewallcapilla ryeffects.Fortheexperiments ofBenjamin&Ursell,Bo=3.47.Carefulinspectionofallthepr edictionsshowsthe(4,1) sh modeoccupiesnearlythesamestabilityspaceasthatofthe( 1,2) sh ,whichisquickly understoodbynotingthatthedimensionlesswavenumbers k 41 R and k 12 R areidentical toeachotherwithin1%(seeFigure 2-4 ).Thereforetheselectionofthe(1,2) sh modein theexperimentisanimportantobservationtomakeandwewil lpresentargumentslater thisisaresultofsidewalleffects.Also,thersttwodatapo intsfallwellinsidetheregion ofinstabilityforthe(3,1) sh and(0,1) sh modes,butthesemodeshaveapparentlybeen dampedentirelyandinsteadthe(1,2) sh moderemains. Dodge etal. [ 24 ]ranexperiments,alsousingwater,inacylinderwitharath er largediameterof14.5centimeters,measuringcriticalthr esholdsandwaveamplitudes. Figure 3-1 (b)showstheironsetmeasurementsforthe(1,1) sh modefortwodifferent layerheights,alongwiththepredictedviscousthresholds .Theirthresholdsobserved betteragreementatthetongueminimathanthatofBenjamin&U rsell,likelydueto thelargetankdimensionsandtherelativelysmallersidewa llcontributiontotheoverall dissipation.AquantitativeindicatoristheBondnumberof8 39,whichisthelargestofall thepreviousexperiments.However,acknowledgedbyDodge etal ,ofgreatestsurprise istheshiftingoftheexperimentalnaturalfrequenciestoh igherfrequencies,incontrast totheshifttowardlowerfrequenciesobservedbyBenjamin&U rsell.Notablythe frequencybandwidthsof0.5and0.9Hzwererathersmall,inp artduetothelargetank dimensions.Ashifttowardhigherfrequencies(andamplitu des)thanpredictedisalso encounteredwiththethresholdsofCiliberto&Gollub[ 20 ](Figure 3-1 (c),Bo=3.39),who exploredchaosappearingneartheco-dimension2pointofa( 4,3) sh anda(7,2) sh mode. Theirfrequencybandwidthsof0.4Hzand0.3Hzwerealsonota blysmall.Comparison ofthepredictionsforthesystemsofDodge etal andCiliberto&Gollubalsoillustrates howtheviscousdampingofthethresholdscandifferforsyst emsofthesameworking uidwhentheheightsandmodearedifferent. 63

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OfgreatsignicancearetheexperimentsofHenderson&Mile s[ 36 ],whomade therstattemptofmatchingsingle-modeexperimentstoath eoryincorporatingviscous effects.Henderson&MilesperformedexperimentsinaR=3.7 25cmcylindricalcell withwaterastheoperatinguid,usingasurfactanttominim izethepinningofthe interfacetothesidewalls.Viscousdampingwasmodeledusin gthetheoryofMiles [ 58 ],assumingthattheeffectswereconstrainedtolaminarbou ndarylayersalong theinterfaceandthesidewalls.However,inthedampingmod el,thewaterviscosity hadtobetakentobe3cSttoproducetheexperimentallyobserv eddampingrates inthecylinder.Nonetheless,incorporationofthisdampin grateintothemodelforthe thresholdsproducedrespectableagreementwiththeexperi mentforthe(0,1) sh mode inthecylinder.Thenegativeshiftintheobservednaturalf requencywasaccountedfor, aswasthedampingofthethresholds.Thepredictionsofthis lineardampingmodel andtheirdataarereproducedalongsidethepredictionsoft heviscousmodelinFigure 3-1 (d).Incomparison,itisseenthattheregionofinstability isonlyslightlydamped fortheviscoustheory,muchlikethesystemsofBenjamin&Urs ellandDodge etal implyingthatagainbulkviscouseffectswerenottheprimar ysourceofdampinginthese experiments.Thelineardampingcoefcientaccountingfor viscouseffectsinthebulk phases,givenbyKumar&Tuckerman[ 52 ]andderivedbyLandau[ 54 ]uponignoring interfacialeffectsis r visc =2 k 2 1 coth kh 1 + 2 coth kh 2 1 coth kh 1 + 2 coth kh 2 (3–1) Itfollowsthatthelinearcontributiontothebulkviscouse ffectintheseexperimentswas r visc =0.0216s 1 ,onlyabout5%ofthemeasureddampingrateof0.44s 1 ,implying thatthesystem'sdissipationwasdominatedbythewallandi nterfacialdissipative effects.Thefrequencyshiftandslightdeviationofthe(0, 1) sh thresholdsinthework ofHenderson&Miles(Bo=21.9)isalsoseenintheexperiments ofDas&Hopnger [ 23 ],whoalsomeasuredthethresholdofthe(0,1) sh mode,butinalarge R =5cmcell 64

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(Bo=257).Figure 3-1 (f)revealsthedeviationofthedatafromtheviscousmodelt o bequalitativelythesamealthougharguablylessthanthato fHenderson&Milesfor boththe(0,1) sh modeanditsneighboring(3,1)hmode.Negativefrequencysh iftswere observedalongwiththresholddamping.Thedampingofthetu ned(3,1) sh thresholds appearstobegreaterthanthetunedthresholdsofthe(0,1) sh mode,aresultconsistent withthendingsofourexperiments. OtherexperimentsofnotearethoseofIto etal. [ 44 ]andTipton&Mullin[ 77 ], because,liketheexperimentsinthiswork,bothwereexperi mentsrunforliquid bilayersasopposedtoaliquidwithapassiveairlayer.Thee xperimentsofIto et al arequalitativelydifferentfromtraditionalsingle-mode experiments,astheywere interestedinmodelingtheeffectofsidewallowperturbat ionsontheinstability.Intheir experimentacolumnofwaterwithkerosenelyingontopofitw assinusoidallypumped usingapiston,producingamovinginterfaceliketheFarada yproblem,butbaseow perturbationsarosefromshearowsatthesidewalls.Theyo bservedthegrowingcell modesatdifferentfrequenciesmuchliketheFaradayexperi ment,butingeneralthe datadoesnotagreewiththeviscousFaradaymodelandisnots hownhere.Ofnote, however,aretheobservationsbyIto etal ofthedevelopmentofalmproducedby keroseneonthesidewalls.Qualitatively,thislmwasthes ameasobservedinthiswork, andIto etal provideexcellentdiscussionofitsdynamics.Additionally ,Ito&Kukita[ 43 ] furtherstudytheeffectofthelmonthenonlineardynamics oftheinstability. Tipton&Mullinprobedthebifurcationstructureofthe(0,1 ) sh modeinaclosed stainlesssteelcylindercontainingwaterandsiliconeoil foravarietyofinterfacial heights,andobservedanon-dimensionalizedcollapsetoth epredictionsofalinearly dampedMathieuequation.Theynotedthattheirdampingpara meterwas8timesthe valuepredictedbyEquation 3–1 ,suggestingwalleffectswereimportantintheircell. Theinterfacialtensionbetweentheiroilandwaterwasalso notmeasured,requiring guessestobemadetottheviscousmodeltotheirdata.Dueto thesmalldensity 65

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differencebetweenwaterandoil,thissystemstandsoutfor beingtheonlysetof previousexperimentswherethesurfacetensionmadeasigni cantcontributiontothe modedispersion.Takingtheinterfacialtensiontobe21.5d yncm 1 ,amatchcanbe madebetweentheobservedandpredictednaturalfrequencie s,butisquicklylostwhen thevalueisadjustedslightly.Ataninterfacialtensionofa round19dyncm 1 (avalue suggestedinprivatecommunicationwithTipton&Mullin),t hediscrepancyissignicant, cf.Figure 3-1 .Itappearsthethresholdslayabovetheviscousprediction regardless oftheexactvaluechosen,andthecelldiameterof35.22mm,t hesmallestofthe reviewedexperiments,suggeststhatthedissipationdueto walleffectswascontrolling. Additionally,theBondnumberhereof2.4wasthelowestofallr eviewedexperiments. Applicationoftheviscousmodeltothepastresultsclearlys howsthattherehave beenmanyexperimentalunknownswhicharenotaccountedfor inthelinearstability model.Thereforeresultsdifferentformsofexperimentalm ismatch,includingmodeshifts towardbothhigherandlowerfrequencies,increasesinthet hresholdamplitudesandthe completeabsenceofpredictedmodes.Broaderinterpretatio nofthesesystemscannot begivenduetothelimitedscopeofthesestudies,oftentime slimitedonlytooneortwo modes.Thusinadditiontoabetterreplicationoftheassump tionsofthelineartheory, anotherimportantgoalofthisworkistostudyanentirerang eofmodes,aspresented inFigure 2-6 ,wheretheinteractionbetweenthesidewallandtheinstabi litycanbemore deeplyunderstood. 66

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CHAPTER4 EXPERIMENTALMETHOD Whileunderstandingthetheoreticalpredictionsiscrucial tobuildinganunderstanding oftheinstability,themainndingsofthisdissertationar e,infact,experimental.Inthis chapter,thedifferentaspectsoftheexperimentaldesigno fasystemthatattemptsto replicatethemodelassumptionsandproducerepeatableand measurablephenomena willbediscussed.Theeffectoftemperatureuctuations,i mportanttothereplication ofthemainassumptionsofincompressibleNewtonianuidsw ithconstantdensityand viscositywereconsideredthroughoutdesignofthecelland theexperimentalmethod. Themostdifcultassumptiontoreplicatewasthethatofstr ess-freesidewalls,andthe behavioroftheinterfacenearthesidewallsthatitimplies .Thethree-phaseproblemof thetwoliquidscontactingasolidsidewallinevitablyprod ucesastaticmeniscus,which emitswavesduringvibration,andviolatestheassumptions ofano-owbasestateanda atinterface.Fluidshearingatthesidewallsandcapillar yhysteresisareotherexamples ofcomplexstresseswhichariseinrealsystemsandmustbeco nsidered.Allofthese phenomenahinderconnectionwiththeequationsofmotionan dtheexperimental methodmustrespectthem.Inadditiontothechoicesmadereg ardingthesystemuids, dimensions,andcelldesign,themethodsofmeasurementand analysisoftheimposed vibrationalwaveformwillbepresented. ChoiceofLiquids Onemethodtomitigatethenon-idealityandapproachidealb ehavioristoselect liquidsforminganinterfacethat“effortlessly”glidesac rossthecontainerwallwhen tilted,representingaminimizationoftheassociatedside wallstresses.Quickandsimple experimentswithdifferentsingleliquidlayersandimmisc ibleliquidbilayersrevealeda widevarietyofinterfacialcontactlinebehavior,strongl ydependentuponthecontainer material.Itwasbenecialtomoveawayfromsinglelayersof waterasaworkingsystem, consideringitwasshownthatthesidewallstressescancont roltheoverallsystem 67

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dissipation(seeChapter 3 ),evenwhenthecapillaryeffectswerelessenedbyaddition ofsurfactant,asintheexperimentsofHenderson&Miles[ 36 ].Thiswork,although somewhatnaively,shiftedtowardsimmiscibleliquidbilay ersystems,asthemeniscus waveamplitudesappearedtodecreasewhencomparedtothose observedwithsilicone oilsinplexiglascontainers,resultingfromalargeconcav emeniscus. Earlyexperimentswereconductedusingdifferentsetsofimm iscibleliquidsin rectangularplexiglascells.Whileimmisciblesiliconeoil andwatersystemswerenatural candidatesbasedonavailabilityandtheabilitytotunethe oilviscosity,waterdidnot wetplexiglasandformedagiganticbubblesurroundedbywal l-wettingsiliconeoil. ApplicationofScotchGuard TM dirtrepellanttothecellsidewallsattractedthewater andcreatedahorizontalinterface,buttheinterfacestill sufferedfromalarge,pinned meniscuswhichwasresistanttomotion.Notably,Tipton&Mu llin[ 77 ]usedsiliconeoil andwaterinastainlesssteelcell,butthebehavioroftheco ntactlineisunknownand non-idealbehaviorissuggestedbytheirdampingratesando bservedlinearthresholds. Octanolandwaterwasaninterestingandcomplexsystemcons idered,duetoits abilitytoabsorbisopropanolinbothphasesandtherebyred uceitsinterfacialtension. Bydoingso,itwaspossibletovirtuallyeliminatethesidewa llmeniscusandalso producefreemotionoftheinterfaceatthesidewalls.Howev er,thesolubilityofIPA inbothphaseswasdifculttocontrol,andproductionofasy stemwithaconstant concentrationofIPAinbothphaseswasverydifcultifnoti mpossible.Excitationofthe instabilityinawater-octanol-IPAsystemtypicallyshowe dvisual“streaming”ofmaterial nearthedeectinginterface,presumablyduetothediffusi onofIPAacrosstheinterface. Haltingoftheinstabilityandallowingthesystemtosettle subsequentlyresultedin stratiedlayersofdifferentrefractiveindex,alsolikel yduetothegraduationofIPA concentration.Whileextensivelinearthresholddatawasno ttakenforwater-octanol-IPA systems,thereexistmanyinterestingphenomenawhichcoul dbestudiedinacontrolled experimentandoughttobeasubjectoffutureinvestigation 68

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FC70,adenseandinertuorinatedhydrocarbon,and1.5cent iStokesilicone oilwerefoundtoproduceaninterfacewithonlyaslight( 1mm)meniscus.Tiltingof theexperimentalcellresultedinasmoothandveryquickret urnoftheinterfacetoa atposition,indicatinglowsidewallstresses.Itwasbeli evedthelowsurfacetension (reportedlessthan7dyncm 1 [ 75 ])wasinstrumentalinminimizingthesidewallstresses. Closeobservationofthesidewallinatiltedcellshowedtha tatinylmofthesiliconeoil hadformed,overwhichtheFC70glided,meaningtheinterfac eremainedpinnedtothe sidewallwellbelowwheretheapparentcontactlinehadrise nto.Lowinterfacialtension clearlyaidstheprocesswhere,toformalm,theinterfacem ustbendabruptlyfromits contactposition,andthenbestretchedalongthelengthoft heoil-coatedglass.Thelm thicknesswasfoundtodecreasewiththeoilviscosity,andw asthereforeminimizedby using1.5cStoil.Siliconeoilwith0.65cStviscositywasnotus edsinceitshowedslight miscibilitywithFC70. Liquiddensitiesweremeasuredusingapycnometerwithcali bratedvolumeof 51.490mL.ThedensityofFC70,listedas1940kgm 3 ,wasfoundtovaryfrombottle tobottleandthereforethemeasureddensitiesarenotedwit hthegures.Siliconeoils of10cSt( =940dyncm 1 )and1.5cSt( =846dyncm 1 )wereused.Thedensity differencebetweenthetwophasesplaysagreatroleinthepo sitioningofthefrequency bandsandthresholdsinthediscretizedregime,considerin gtheexcitedmodesare primarilygravitywaveswithlittlecapillarycontributio n.Asystemwithalargedensity differenceresultsinalargerfrequencyrangeforwhichmod ediscretizationisimportant duetotheselectionoflowerwavenumbers.Atlowfrequencies theinstabilitythreshold isalsoloweredforlargedensitydifferences,meaningharm onicandsuperharmonic modesaremoreeasilyaccessed.Thedensitydifferenceinth eseexperimentsof nearly1000kgm 3 isquitesimilartoawaterandairsystem,butthethresholdb ehavior changesduetotheincreasedviscousdampingofhighwavenum berharmonicand superharmonictongues. 69

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Itwasfoundthattheassumedvaluesoftheviscosityandthei nterfacialtension weresatisfactoryformakingexperimentally-veriablepr edictionswiththelinearmodel, andthereforeneitherwasmeasured.TheFC70viscositycont rolsthedampingof thethresholdtonguesintheseexperiments,andreasonable adjustmentofthevalue reportedbythemanufacturerhasverylittleeffectonthepr edictedthresholdcurves. Additionally,interfacialtensionshowsvirtuallynoeffec tonthetheoreticalpredictions, asthelargedensitydifferencebetweenFC70andsiliconeoi lsdominatesthecapillary contributionforthemodesexcitedbytheconsideredfreque ncies.Future,predictions mademodelsofgreatercomplexitywouldlikelybenetfromm oreaccurateknowledge ofthephysicalproperties. CellDesign Rectangularcellswereusedinitially,butlaterweresubst itutedforcylindrical cells,asitwassuspectedthecornersproducedanadverseef fectonthesidewall stresses.ThiswasessentiallyreportedbyHenderson&Mile s[ 36 ],whoalsoconducted experimentswithasquarecross-sectionandobservedgreat erdeviationfrompredicted thresholdsforsquaresthancylinders. Whileincreaseofthelateraldimensionsisappealingduetot hedwarngofthe sidewallcontributionbythebulkdampingeffects,adetrim entaleffectisthenarrowing ofmodefrequencybands,forwhichtighterexperimentalcon trolofimposedfrequency andamplitudewouldberequired.Althoughnotpursuedinthes eexperiments,testing ofmultiplecellradiiwouldoffertheopportunitytominimi zethesidewalldampingwith respecttotheexperimentalcontrol.Thecylindricalheigh tofthecellwasnotdeemed crucialtodesigningalinearlyidealsystem,butthesatura tionofthe tanh kH expression fromtheinviscidtheory(seeAppendix A )washelpfultodetermineatwhichlayer heightsawavenumber k doesnot“see”thetoporbottom. Schematicdiagramsofboththenalcelldesignandtheelectr omechanicalshaker arepresentedinFigure 4-1 .Amodularapproachwastakentowardsthecelldesign 70

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PC 4.1 or 6.4 cm total height = table 1 high speed camera counterweight cell Linear unit table 2 immiscible interface motor cell diameter = 5.1 cm pinhole reservoir channel interface Figure4-1.Schematicdiagramofthecellandtheelectromech anicalshaker.Forfurther informationseeZoueshtiagh etal. [ 83 ]. inordersimplifycleaningandaccountfortestcylindersof differentheight.Withthe availabilityof5.1cmIDglasstubes(ofsmallwallthicknes storeducetheopticaleffects duetoglasscurvature),twocylindricaltestcellsofappro ximately6cmand4cmtotal heightwerecut.Thecellswerebuiltbyclosingtheglasstub eendswithsquareacrylic plates,intowhichcirculartrencheswerecutando-ringswe replaced.Acompression sealwasformedtighteningofwingnutsonfourboltspositio nedneareachplatecorner. Finally,fourthinacrylicplatesandwaterresistantpuddy wereusedtobuildasquare cagearoundthecylinder.Thecagewaslledwithwater,whic hdramaticallyreducedthe opticaldistortionthatarisesfromcurvedgeometries. Theinuenceoftemperatureuctuationspervadedthrougho utallexperiments, andisbelievedtohaveasignicantimpactondevelopingare peatablenon-linear experiment.Anearlyproblemencounteredwasbeingabletoma intainafully-lled, non-leakingcellduetotheconstantexpansionandcontract ionoftheworkinguidsin ambientconditions.Thisissuewasexacerbatedbytheuseof ahalogenlampwhich generatedlargeamountsofheat.Useofthelampwastherefor elimited,butitremained normaltoperformexperimentsforanextendedperiodoftime ,allduringwhichtheuids wereperiodicallyheatedandleakedduetotheslightexpans ionoftheliquids.Duringa 71

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break,theliquidswouldagaincontract,typicallyintrodu cingairbubblestothecell.To managethesedensityuctuations,thecellwasdesignedwit hasmallchannelinthe topplateconnectingthemaintestcylindertoasmallerstai nlesssteelreservoir.The reservoirwasclosedbyascrew-capwithatinypinholethata llowedairow.Fillingof theentiresystemtoanitelevelinthereservoirallowedth emaintestcylindertobe completelylledindenitelywithoutneedforrelling. Tominimizecontaminationduringassemblyofthecell,allp ieceswererinsedwith isopropanol,followedbywater,andthendriedwithanti-st aticclothsandcompressedair. Neverthelesssmallamountsofdusttypicallyappearedinth etestcylinder.Oncethecell wascompletelyassembled,testliquidswereinjectedthrou ghtheexpansionreservoir intothemaintestcylinderusingsyringesandneedles.Toim provethecontrastand visualizationoftheinterfacialmodes,anelyspacedgrid wasprintedandafxedtothe backpaneofthewatercage. ElectromechanicalShaker,ImageCaptureandProcessing Experimentswereperformedbymountingthecelltoanelectro mechanical shaker(Figure 4-1 )capableofmovingtheplatformtoprogrammeddisplacement sat accelerationsofupto3 g .Thecellwasshakenusingbothsingleanddouble-frequency sinusoidalmotions.Allcellandinterfacialmotionwereexa minedusingtime-spacedata oftheimages,obtainedfromhighspeeddigitalimagingwith frameratesofupto2000 fps.Theshakeroutputfrequencywastakenastheactualexpe rimentalfrequency,as thenumberofimagesinacellperiodalwayscorrespondedtow ithinoneortwoimages basedonacalculationusingtheprogrammedvalue,suggesti ngpercenterrorsofless than0.5%. Theimposedexperimentalamplitude,ontheotherhand,wasq uicklyobserved todeviatefromtheprogrammedamplitude,especiallyforam plitudeslessthan3mm. Forthisreasontherealoutputamplitudesofthecellmotion weredeterminedbyimage analysiswithImageJandMATLAB R r .Nearlyalloftherestofthischapterwillbedevoted 72

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Figure4-2.Samplecameraimagedepictingexcitationofa(0, 1)mode.Thewhiteline depictsthesingle-pixelwidestripofdatausedingure 4-3 andthe graduationsonthefarrightisastripofpaperonthefrontof thecellusedto recordcellmotion.Cellradiusis2.55cm. tothedeterminationoftheoutputshakersingleanddoublefrequencyamplitudes,and analysisofthewaveformquality.Tobegintoillustratethe meansusedtodetermine theseamplitudes,picturedinFigure 4-2 isasamplecellimage.Themodebeingexcited isa(0,1) sh mode,behindwhichliesthebackdropgrid.Thedisparityint hegridspacing betweentheupperandlowerphaseisduetodifferentrefract iveindices,andcareful observationshowsdistortionofthegrid,owingtobothresi dualcurvaturedistortionand slighttemperatureanddensitygradients. TheactualcellamplitudeoftheexperimentinFigure 4-2 wascalculatedby measurementofthedisplacementofthecelloverthecourseo fseveraloscillations. Takingasingle-pixelwidesliceoftheimageinFigure 4-2 (denotedbythewhitearrow), andstackingitnexttothesamesetofdatafromeachsequenti alimageproducesaset oftime-spacedata,asinFigure 4-3 .ThewhitearrowinFigure 4-3 indicatesthelineof datatakenfrom 4-2 wherethewavedeectionisatamaximum.Thekeyinformation in thetimespacedataisthesinusoidalmotionofthebackgrid, representingtheoutputcell 73

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Figure4-3.Time-spacerepresentationoftheexcitationof asaturated(0,1) sh mode. VerticallengthscaleoftheimageisthesameasFigure 4-2 ,andthe horizontallengthoftheimagedepictsatotaltimeof0.374s econds.The whitearrowdepictsthedatatakenfromFigure 4-2 motion.Theinterfacialdeectionamplitudeofthe(0,1)mo de,subjecttoslightoptical distortions,canalsobeextractedfromFigure 4-3 .Thismethodhoweverbecomesmore difcultformodesotherthanthe(0,1) sh modeduetoazimuthalnon-uniformities.Dueto thenon-uniformbendingoflightthroughthecellandthento thecameraobjective,the backgridwasactuallynotuseddrawthecellmotiontimespac e,butratheranotched slipofpaperafxedtothefrontofthecell,asseeninthefar rightoftheimageinFigure 4-2 Collectingtimespacedatainthismanner,carefulcontrolo ftheimagebylighting andcamerapositioningallowedforquickandefcienttrans ferofthecellmotionfrom theexperiment,tothecamerasoftware,toImageJandnally toMATLAB R r .Large amplitudetestsrequiredthecameratobepositionedfurthe rawayfromtheexperiment tocapturetheentirecellmotion.Frameratesof1000fpswer edeemedsuitable,and capturedimagesizewasoftensettoa128pixelwidthsliveri nordertominimizethe amountofspaceconsumedonharddrivesandtimespentfortra nsfers.Typicallyaslip 74

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0 200 400 600 800 0 100 200 300 400 500 time t ,millisecondscelldisplacement,pixels (a) 0 1 2 3 4 5 0 50 100 150 200 basicfrequencymultipliers, MFFTamplitude115.34 A =1.16cm =0 (b) 0 200 400 600 800 0 100 200 300 400 500 600 time t ,millisecondscelldisplacement,pixels (c) 0 1 2 3 4 5 0 50 100 150 200 basicfrequencymultipliers, MFFTamplitude 113.96 30.73 A =1.19cm =15.1 (d) 0 200 400 600 800 0 100 200 300 400 500 time t ,millisecondscelldisplacement,pixels (f) 0 1 2 3 4 5 0 50 100 150 200 basicfrequencymultipliers, MFFTamplitude 40.87 67.00 A =0.79cm =31.4 (f) Figure4-4.ExperimentalcellmotionsignalsandFFTanalysi s.Thesignals(a),(c)and (e)andcorrespondingFFTspectra(b),(d)and(e)wereobtai nedfor programmedconditionsof f =1.14Hzwith(a) =0 A =1.20cm,(c) =15 A =1.23cm,and(f) =31 A =0.83cm. ofpaperlikeonthefarrightofFigure 4-2 withonlyonenotchwasusedtocreateaset ofimagesthatproducedatimespacewhichcouldquicklybeco nvertedtoasingle-pixel widthlineofbinarydataviaImageJ's“Threshold”and“Skele tonize”commands. 75

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SingleandDouble-FrequencyAmplitudeDetermination Theoutputamplitudeforsingle-frequencyexperimentscou ldbeobtainedsimply usingthetimespacedataobtainedinImageJ,withoutanyfur therprocessing.The single-frequencycellamplitude,inpixels,correspondst ohalfthepeaktotrough distanceofonecelloscillation,andconversiontoarealdi stancethenonlyrequiresa distanceofknownlengthontheimage.Theoutputamplitudef orthesignalpresented ingure 4-4 (a),obtainedbythismethodwas1.16cm,lessthantheprogra mmed amplitudeof1.2cm.Amajorobstaclethathadtoberespected wasthatthepercent errorbetweentheoutputandtheprogrammedamplitudewasac omplicatedfunction ofboththeprogrammedamplitudeandfrequency,anditwasth ereforenecessaryto analyzetheoutputcellmotionforeverydatapoint. TheoutputsignalspresentedinFigure 4-4 (a),(c),and(e)resultfromaprogrammed motionofEquation 2–26 where( M 1 M 2 )=(3,4)and f = != 2 =1.14Hz.Whilecrest-to-trough analysisworksforthesingle-frequencymotions,itisappa rentthismethodwillnot workforthedouble-frequencysignalsofFigures 4-4 (c)and(e).ThusMATLAB R r 's discreteFastFourierTransformfunctionwasusedtoanalyz edouble-frequencysignals. TakingthediscreteFouriertransformofadigitalsignalde composestherealdatainto thecomplexamplitudesofthevariousfrequencycomponents presentinthesignal. Tocombatdigitalnoiseandthelowresolutionthatarisesin shortsignals,FFTwas performedonasignalbuiltbystackingthesameperiodofcel lmotiondataontoitself manytimes.Analysisofthelong-timecellmotionshowedsign aldeterioration,butat timeswellpastthedurationofexperimentsinthiswork.Ina nalyzingsingleperiodsof cellmotion,thetwoamplitudesfor(3,4)forcingappearede xactlyatbasicfrequency multiplesof3and4,asseeninFigures 4-4 (b),(d),and(e),furthersupportingthe frequencycontroloftheshaker. Analysisofseveralsetsofcellmotionwith =0 andcomparisontothepeak-to-trough methodatdifferentfrequenciesandamplitudesshowedtheF FTpeakamplitudesto 76

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0 2 4 6 8 10 0 30 60 90 120 150 basicfrequencymultiplier, MFFTamplitude 139.6 11.7 (a) A =1.41cm =4.8 noisecontent=5.5% 0 2 4 6 8 10 0 3 6 9 12 15 basicfrequencymultiplier, MFFTamplitude 13.9 9.8 (b) A =0.17cm =35.3 noisecontent=28.0% Figure4-5.SampleFourierspectradepictingtheappearance ofundesiredfrequencies. Programmedparameterswere(a) f =1.12Hz, =5 A =1.45cmand(b)1.29 Hz,35 ,0.25cm. scalelinearlywithoutputcellmotionamplitudeandalsoto beindependentoffrequency. Thusthepeak-to-troughmethodwasusedtocalibratetheFFT peakamplitudesfor asingle-frequencyexperiment,andafterdoingsoitwaspos sibletocalculatethe outputamplitudes A M 1 and A M 2 forthe M 1 and M 2 componentsindouble-frequency experiments. Animportantimplicationofthevaryingpercenterrorofoutp uttoprogrammed amplitudemeantthatfordouble-frequencyexperimentsthe outputvalueof suffered fromthesamelackofcorrespondencetotheprogrammedvalue .Forexample,when aprogrammedvalueof speciesseparatecomponentamplitudesof,say,10and 5mm,buttheoutputamplitudesdisproportionatelyareadju stedto8and3mm, theoutputvalueof changes.However,giventherealoutputamplitudesofthe frequencycomponentsbytheFFTanalysis,therealvalueof wasdeterminedby taking tan 1 ( A M 2 = A M 1 ) viathedenition 2–26 .Finally,giventherealvaluesof and A M 1 and A M 2 ,theoverallamplitude A couldbedeterminedbyeither A M 1 = cos or A M 2 = sin 77

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0 0.5 1 1.5 0 5 10 15 20 25 30 35 amplitude A ,cmsignalnoisecontent,% (a) 0 10 20 30 40 50 60 70 80 90 0 5 10 15 20 25 30 35 angle signalnoisecontent,% (b) 1.1 1.15 1.2 1.25 1.3 1.35 1.4 0 5 10 15 20 25 30 35 frequency f = != 2 ,Hzsignalnoisecontent,% (c) Figure4-6.Outputcellmotionsignalqualitydependenceon parametricconditions. PercentquantitiesplottedaretheratiosoftheundesiredF FTnoisetothe amplitudesoftheprogrammedfrequencies. AnalysisoftheOutputCellMotionQuality Fourieranalysispresentedapowerfultechniquetoanalyze thequalityofthe imposedshakerwaveform.Inadditiontotheprogrammedfreq uencies,itwaspossible detectthepresenceofotherharmonicsinthecellmotion,ow ingtoeithermotor andstructuralvibrationsordigitalprocessingnoise.The presenceandeffectof theseharmonicsarehighlightedbythetwoFourierspectrai n 4-5 ,corresponding toexperimentswithdifferentamountsofnoiserelativetot heprinciplefrequency components.Figure 4-5 (a)presentsthespectrumforahigh-amplitudeexperiment withonlytheslightadditionofthe4 component(low).Heretheprinciple3 and 4 peakamplitudesdominateanyothernoiseorsystemresonanc es,whosetotal summedamplitudeaccountsfor5.5%oftheprogrammedfreque ncyamplitudes. 78

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Figure 4-5 (b)correspondstomotionwithfargreaterratioof4 component( =35.3 ), ataconsiderablyloweramplitude.Heretheamplitudesowin gtoundesiredsystem resonancesandnoisesummedtonearly30%ofthecombined3 and4 amplitude, raisinggreatercallforconcern. Thisnon-idealityshowedtrendswithrespecttotheparamet ers A ,and f which couldbegeneralized.PresentedinFigure 4-6 isthisrelativepercentofundesired frequenciesforalldouble-frequencydatapresentedinthi swork.Loweramplitudes, higherfrequencies,anddeparturefromthesingle-frequen cyvaluesof =0or90 alltendtointroduceagreaterproportionoftheseaddition alresonances.Thistrend dependsmostconsistentlyontheimposedamplitude,becaus eexperimentsathigh frequencyorfrequencymixingcanstillproducegoodsignal siftheoverallamplitude ishigh.Whilethisbehaviorcallsintoquestionthewaveform integrityforlowoutput amplitudes,itisbelievedtheamplitudesoftheundesiredf requenciessuchasthosein Figure 4-5 (b)arestilltoosmalltoaffectthelinearthresholdobserv edbytheprinciple 3 and4 frequencies.Onceagainthismaybeanavenueforimprovemen tnecessary toperformanon-linearexperiment. Figure 4-6 (b)illustratesanotherconcern,wherethedatapointsshow slight deviationsfromtheprogrammedintegervaluesof =[5,10,22.5,35,45].Again,this arisesduetotheinabilitytocontrolthedeviationofbotho utputfrequencyamplitudes simultaneously.Itwouldbepossibletomakeslightadjustm entstotheprogrammed untilthedesiredoutputwasreached,butinthemidstofsear chingforalinearthreshold thiswouldbetedious.Fortunately,thetypicaldeviationo bservedbetweenoutputand theprogrammedvalueof wasneverconsideredsignicantenoughtoaffectlinear stabilitypredictionsortheconclusionsdrawnfromtheexp eriments. ExperimentalRepeatability Themostcommonlyperformedexperimentwasthesearchforal inearthreshold amplitude,donebysuccessiveexperimentsatconstantfreq uency,whileinterpolating 79

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betweenamplitudeswhichdidanddidnotseetheinstability .Thismethodrequiresa tolerancefortheinstability,where,iftheamplitudeislo weredbyatolerableamount andtheinstabilityisnotseen,thentheoriginalpointisma rkedasthethreshold.The normaltoleranceintheseexperimentswasabout0.1mm.Atro ublingbehaviorthat wasencounteredwastheincreaseinthemarkedthresholdfro monedayuntilthe nextmorning,where,forexample,ifattheendofonedaythei nstabilityatacertain frequencywasrstfoundtogrowatanamplitudeof10mm,itwa scommontonot observetheinstabilityuntil,say,12mmamplitudethenext morning. Thereasonfortheincreaseinamplitudestemsfromfromthew ettingbehaviorof theFC70-siliconeoillmandthestick-slipbehaviorofthe contactline.Itwasfoundthat theexcitationoftheinstabilitydevelopedasiliconeoil lmbelowtherestinginterface position,andthegeneralsenseisthatthepresenceofthis lmgreatlyreducedthe stressesrelatingtomotionoftheapparentcontactline.Th elmisvisibletothenaked eyeandgraduallyrecedesafterthecellvibrationisstoppe dandtheinstabilitydiesout. “Sidewallwetting”ontheotherhandwasbelievedtobeamolec ularlevelphenomenon, inferredbecauseexcitationofalong-stagnantinterfacea ndre-formingofthesidewall lmdidnotinstantlyre-adjustthethresholdtothevalueof thepreviousday.Instead, manyexperimentsformingthelmhadtobecarriedout,throu ghoutwhichtheinstability couldbeexcitedatlowerandloweramplitudesforaspecicf requency. Thisprocessof“wetting”thesidewalltoitssaturatedstat ecouldtakenearlytwo hours,andthereforeadatasetofthresholdswasbestdeterm inedbyensuringthe instabilitywascontinuallybeingexcitedandthewallwett ingconditionswerekept constant.Maintenanceofthewettingwas,however,complic atedbytheexistenceof modeinductiontimes:apparentdeadperiodsbeforewhichth einstabilityappearedand grew.Thesetimesincreasedwithproximitytothethreshold ,approachingveminutes, meaningthemaintenanceofconstantwettingconditionsbec amemoredifcultasone nearedtheactualthreshold.Oneapproachtakentogettingd atanearthethreshold 80

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0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 imposedamplitude A ,cminterfacedeection A 1 ,cm (a) f =3.675Hz 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0 0.1 0.2 0.3 0.4 0.5 imposedamplitude A ,cminterfacedeection A 1 ,cm (b) f =3.725Hz Figure4-7.Experimentalrepeatabilityissues.Saturatedwa veamplitudesof(0,1) h modesforaseriesofexperiments(a)withoutacoffeebreaka nd(b)witha coffeebreak. wastouseintermittentexperimentswellabovethethreshol dwherethemodecouldbe quicklyexcitedandthewallconditionsrestored. Whilethediscussedmethodformaintainingconstantwetting issatisfactoryfor reproducingthereportedthresholdsofthiswork,thismeth odwouldlikelyneed tobecomemorespecictoensurerepeatabilityofnonlinear phenomena.Thisis illustratedsomewhatcomicallyinFigures 4-7 (a)and(b),twosetsofexperiments measuringthesaturatedwaveamplitudeasafunctionofthef orcingamplitude,a traditionalbifurcationcurve.Inordertopreservethesid ewallwettingfromexperiment toexperiment,measurementsweremadequicklyandeffortwa smadetobeconstantly excitingamode.Thegreatsensitivityofthenonlinearsyst emtothesidewallwettingis seeninFigure 4-7 (b),whereinsteadofrunningthesetofexperimentswithout pause, thecellsatstillforafteenminutecoffeebreak.Uponretu rn,thechangeofthesidewall wettingwassignicantenoughtoproducedatainconsistent withthatfrombeforethe break,around A =0.8cm.Inadditiontothesaturatedwaveamplitudes,there existsan abundanceofmorecomplexnonlinearbehaviorwhichwouldal soshowgreatsensitivity totheinitialcondition. 81

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CHAPTER5 SINGLE-FREQUENCYEXPERIMENTSANDDISCUSSION SidewallMeniscusandFilmBehavior Thekeyresultsareconnectedtothebehavioroftheinterfac eclosetothesidewalls, thesourceofthenon-ideality.FillingofthecellwithFC70 andsiliconeoilproduceda convexdownmeniscusintheglasscylinders,aresultofthep referentialwettingofthe glassbythesiliconeoil.Uponvibrationofthecelleitherb eloworabovetheFaraday threshold,theharmonicmodulationofthegravityeldcaus esadjustmenttothedesired meniscusprole,resultinginharmonicemissionofanaxisy mmetricwavefromthe sidewall.Forasystemthatcommencesoscillation,thereex istsatransientperiodduring whichtheemissionofthemeniscuswaveandthereectionoft hewavethroughthe cylinder's z -axisequilibratestoasteadystatewavethatappearsascon centricripples. Thequiescentstate,initialemissionofawave,andasteady statemeniscusproleare depictedinFigure 5-1 .Themagnitudesandcharacteristicwavelengthsoftheseri pples areverymuchafunctionoftheparametricamplitudeandfreq uency.Finally,although thetemporalbehavioroftheinteriorproleiscomplexduet othepersistentemission andreectionofmeniscuswaves,thealterationofthemenis cusremainsaxisymmetric andharmonicallyperiodicwiththecellmotion.Thisperiod icityissuspectedtointeract withtheharmonicmodesofinstability,whichwillbepresen tedanddiscussedfurtheron. PicturedinFigure 5-2 istheexcitationofa(0,1) sh modeafterithassaturatedto asteadyamplitude.Whilethenonlineargrowthisnotthemain focusofthiswork,the transitiontothisstateisessentialtobeabletoreproduce thelinearthresholds.The emissionofmeniscuswavesdoesnotalterthecontactpositi onoftheinterfaceatthe sidewalls,butexcitationofaFaradaywaveresultsinamoti onthatbeginstoseparate theapparentcontactpositionfromtheactualposition.Asth eFaradaywavebeginsto grow,therstdownwardmotionoftheinterfaceatthesidewa llcausesboththeapparent 82

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Figure5-1.Experimentalvisualizationofthemeniscusdyna mics.Imagesdepictthe staticFC70and1.5cStsiliconeoilinterfaceandmeniscus(t op),initial meniscuswaveemission(middle)andsteadystatemeniscusw aves (bottom).Thewaveprolegeneratedbythemeniscusdynamic sremains axisymmetricthroughouttheperiodofinductionpriortogr owthofthe instability. 83

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Figure5-2.Experimentalvisualizationoftheexcitationof a(0,1) sh mode.Excitationat f =7.5Hzisshownforthemaximum(above)andminimumcycle(be low)ina FC70and1.5cStsiliconeoilsystem.Thedouble-headedwhite arrow denotesthe z -directiondifferencebetweentheapparentandactualcont act linesoftheinterface.Cellradius R =2.55cm. andtheactualcontactlinetobepusheddownward.Fortherev erseupwardmotion oftheinterface,theapparentcontactlinemovesupward,wh iletheactualcontactline remainsxedtothelowestpositiontheinterfacehadreache d,visualizedinFigure 5-2 .Whiletheactualandapparentcontactlinescoincideforthe upwardcycle,the actualcontactlinelies0.56cmbelowtheapparentcontactl ineinthedownwardcycle, evidencedbytheslightopticaldeformationsofthebackgri dinthefarleftandrightsides oftheimage.AftersaturationoftheFaradaywavetoaniteam plitude,theinterface remainstetheredtothislowposition,stretchingandcontr actingverticallyasthebulk 84

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FC70glidesupanddownoveratinylmofsiliconeoil.Thisis clearlyvisualizedin Figures 5-3 and 5-4 ,withsetsofimagesshowingthelmdynamicsduringaperiod of asaturated(0,1) sh mode,likeFigure 5-2 .Asystemwithsiliconeoilviscosity50cStis showninFigure 5-3 ,andthelmformationisevidencedbythedeformationofthe grid astheapparentcontactlineadvancesabovethepinnedposit ion.Temporalasymmetry isseenwhentheapparentcontactlinebeginstorecede,andt heproleoftheinterface morecloselyrepresentsa90 anglewiththewall.Boththegriddeformation(lm thickness)andasymmetryaremoresubtleinFigure 5-4 ,whereasetofimageshas beenshownfora(0,1) sh modewhentheupperviscosityis1.5cSt.Theformationof thislmisbelievedtobecriticaltotherealizationofthes tress-freeboundarycondition. Obviouslytherearestressesintroducedthroughthismecha nism,butweshallargue thatitislesssignicantthantheviscousstressesarising frombulkuidmotion.The dependenceontheupperphaseviscositywillbeestablished byrstpresentingthe experimentalthresholdsfora(2,1) sh modewhen10cStoilisusedastheupperphase, followedbytheresultsusing1.5cStoil. ExperimentalThresholdDependenceUpontheUpperPhaseViscos ity Figure 5-5 presentstheexperimentalonsetsofthe(2,1) sh modeandthecorresponding theoreticalpredictionsfortwoFC70andsiliconeoilsyste msofidenticallayerheights inwhichtheupperphaseviscosityis(a)10cStand(b)1.5cSt.T hedatapoints correspondtosetsofconditionsatwhichtheinstabilitywa sobserved.Aseparate experimentperformedatanamplitudeslightlylowerthanth atofthedatapointsresulted inasystematwhichtheinstabilitydoesnotappearandonlyt hemeniscuswaveswere present.Hysteresiseffectshavebeenwellreported(Benjam in&Ursell[ 11 ],Henderson &Miles[ 36 ],Tipton&Mullin[ 77 ],Das&Hopnger[ 23 ])onthedetunedbranchofthe curve,wherethe“offset”ofwavemotionoccursatafrequenc yoramplitudelowerthan theonset,butthisphenomenonwasnotstudiedinthiswork.T heendpointsofthedata setcorrespondtoco-dimension2points,inthiscaseforbot hsetsofresultstheleft 85

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Figure5-3.FilmdynamicsinaFC70and50cStsiliconeoilsyst em.Imagesaretakenof thesidewallduringoneperiodofthesaturatedexcitationa t f =6.7Hzofa (0,1) sh mode.Movingfromlefttorightandthentoptobottom,thetem poral spacingbetweeneachimageis0.299seconds.Imagesarepres entedsuch thattheinterfacecontactlineremainsconstantineachima ge. co-dimension2pointisasuperpositionofthe(2,1) sh witha(1,1) sh mode,anda(2,1) sh witha(0,1) sh fortherightpoint.Thelatterco-dimension2pointinthe10 cStsystemis visualizedinFigure 5-6 ,exhibitingthesamecontactlinebehaviorasinFigure 5-2 .In thecaseofthisinstabilityitshouldbenotedthatthelmis notazimuthallyuniformdue tothepresenceofthe(2,1) sh mode. Ofgreatinterestisthedeviationbetweenthethresholdsan dthepredictionsinboth datasets;theobserved1.5cStthresholdsclearlyprovidebe tteragreementwiththe predictionscomparedtothatofthe10cStoil.Infacttheside walllmwasmuchsmaller anddifculttonoticeinthe1.5cStoilsystems,andtherefor e,inadditiontolower stressesarisingfromsidewallboundarylayers,weconclud edthatthetotalsidewall 86

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Figure5-4.FilmdynamicsinaFC70and1.5cStsiliconeoilsys tem.Imagesaretaken ofthesidewallduringoneperiodofthesaturatedexcitatio nat f =7.5Hzofa (0,1) sh mode.Movingfromlefttorightandthentoptobottom,thetem poral spacingbetweeneachimageis0.267seconds,andtheimagele ngthscale isthesameas 5-3 .Imagesarepresentedsuchthattheinterfacecontactline remainsconstantineachimage. contributiontotheoverallsystemdissipationwasmuchsma llerwhenusing1.5cSt oil.Attheendofthesection,thiswillbequantitativelyver iedbycomparisonofthe predictedinteriorviscouscontributionwiththeexperime ntaldampingrates.Thusinthe interestofidealitytheremainderofexperimentstobepres enteduse1.5cStoil,which allowsustomorecarefullypinpointtheinteractionbetwee nthenon-idealityandthe instability.Beforepresentingmorethresholddata,thenon linearbehaviorofthewave growthwillbediscussedusingFigures 5-5 (a)and(b)asitischaracteristicofallthe observedmodalbehaviorandhasatendencytoobscurethelin earthreshold. 87

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5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 0 0.1 0.2 0.3 0.4 0.5 frequency f ,Hzimposedamplitude A ,cm (a) 5.5 6 6.5 7 0 0.1 0.2 0.3 0.4 0.5 frequency f ,Hzimposedamplitude A ,cm (b) Figure5-5.Dependenceoftheupperphaseviscosityontheor eticalagreement. Experimentalthresholdsandpredictionsareforexcitation ofthe(2,1) sh modeinFC70(1916kgm 3 )andsiliconeoilbilayers( h 1 =2.1cm, h 2 =2.0cm) ina R =2.55cmcylinderfor(a)10cStoil(944kgm 3 )and(b)1.5cStoil (846kgm 3 ).Opencirclesrepresentsingle-modethresholds,blackdo ts representco-dimension2points.Neighboringmodesarethe (1,1) sh modeat lowerfrequenciesandthe(0,1) sh athigher. NonlinearGrowthandSaturation Thenonlinearbehaviorchangesconsiderablyasonemovesfr omthedetuned branchofthe(2,1) sh modetothetunedbranchinFigure 5-5 .Ithasalsobeenreported thatthebifurcationissubcriticalforfrequenciesbelowt henaturalfrequency,i.e., detunedmodes,andsupercriticalforfrequenciesabovethe naturalfrequency,i.e.,tuned modes[ 25 59 ].Inthiswork,theexperimentalobservationsareconsiste ntwiththis, giventhatthereexistsa“jump”inthesaturatedamplitudeo ftheexcitedmodeforthe datapointsatfrequencieslessthanthenaturalfrequency, whileatfrequenciesabove thenaturalfrequencythemodesaturationisnearlyzeroatt hecriticalthreshold.Infact, formanyofthepointsbelowthenaturalfrequency,theexcit edmodegrowsuntilthe interfacerupturesforimposedamplitudesslightlyabovet hethreshold.Becausesuch amarkedchangecanbeobservedbysuchslightchangesinthei mposedamplitude, detectionoftheinstabilityisquiteeasyforthesubcritic albranch.Theinstabilityismore difculttodetectonthesupercriticalbranch,becauseasl ightincreaseintheimposed amplitudespastthethresholdresultsinonlyaslightincre aseinthesaturatedwave 88

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Figure5-6.Experimentalvisualizationofaco-dimension2p oint.Presentedisthe(a) maximumcycleand(b)minimumcycleduringthegrowthofacodimension 2pointconsistingofa(2,1) sh modewitha(0,1) sh modefromthedatasetin Figure 5-5 (a). amplitude.Thisdifcultyiscompoundedfortunedharmonic modes,asitbecomesmore difculttodifferentiatetheinstabilityfromthemeniscu swaves,whoseemissionisalso harmonic. Finally,thetemporaldynamicsofthegrowthofdetunedmode sisveryinteresting. Intheexcitationofdetunedmodes,therewereperiodsoftim eduringwhichtheinterface showednounstablemotionlongafterinitializationofthec ellmotion.Onlythemeniscus wavespersistedduringtheseinductionperiods,andsudden lytheinterfacewouldbegin todeectandgrowinarapidmanner.Theseinductiontimesin creasewithproximityto 89

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2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 3.5 4 frequency f ,Hzimposedamplitude A ,cm 5.5 6 6.5 7 7.5 8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 (1 ; 1) h (1 ; 2) su (2 ; 1) h (0 ; 1) h (3 ; 1) h (1 ; 2) h (1 ; 1) sh (2 ; 1) sh (2 ; 1) sh (0 ; 1) sh (0 ; 1) sh Figure5-7.Theoreticalcomparisonforthelargeheightexp erimentalsystem. Experimentalthresholdsandthecorrespondingtheoretical curveisshown forabilayerofFC70( h 1 =3.1cm, 1 =1888kgm 3 )and1.5cStsiliconeoil ( h 2 =3.3cm, 2 =846kgm 3 )inthe R =2.55cmcell.Datapointsenclosedbya boxindicatetheexcitedmode.Blackdotsrepresentco-dimen sion2points. thecriticalthreshold,sometimesapproachingfourorvem inutes,asnotedbyDas& Hopnger[ 23 ]. FC70and1.5cStSiliconeOilInstabilityThresholds PresentedinFigures 5-7 and 5-8 arethesetsofthresholddataforcellslled withlargeandsmallheightsofFC70and1.5cStsiliconeoil.F requencybandsand thresholdamplitudeswerefoundformodesrangingfrom2toa bout8Hz.Above8Hz thewaveeldwasexcitedatconsiderablysmalleramplitude s,andthesystembehaved morelikealaterallyinnitesystemastheeffectsofmodedi scretizationbegantovanish. Inbothsystems,superharmonicandharmonicmodeswereobse rvedinadditiontothe traditionalsubharmonicresponse.Co-dimension2points, predictedtheoreticallybythe 90

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modelasthecuspsinthestabilitycurve,werefoundthrough outthefrequencyrange bycarefulactuationofbothfrequencyandamplitude.Theex perimentalthresholdsfor bothlayerheightsarepredictedquitewellbythemodel,esp eciallyfromtheperspective ofmodeselection.Foronlyahandfulofpointsistheobserve dmodedifferentfromthe theoreticalprediction.Howevertherearecertainlysomen oticeabletrendsofthedata notmatchingthepredictions,andforthisreasontheresult sofeachcylinderwillbe discussedseparately. Theexperimentaldataaregroupedtogetheraccordingtomod e.Thesystem presentedinFigure 5-7 isthelargeheightsystemwithaFC70layerof3.1cmanda 1.5cStsiliconeoillayerof3.3cm.Themodesobservedinthis system,inincreasing frequency,are(1,1) h ,(1,2) su ,(2,1) h ,(0,1) h ,(3,1) h ,(1,2) h ,(1,1) sh ,(2,1) sh ,and(0,1) sh .The observedmodes,notedbytheenclosingboxesinthegure,ma tchthepredictedmode exceptwhennoted.Thethresholdamplitudesshowgoodagree ment,saveforthepoints nearthetongueminimaforthesubharmonicandsuperharmoni cmodes,duetoresidual walldissipation.Ontheotherhand,forthe(0,1) h and(1,2) h tongues,itcanbeclearly seenthatthethresholdsoftheexperimentliebelowthepred ictedthresholds.Thisisa noteworthyresult,andthebestexplanationseemstobethei nteractionoftheharmonic, axisymmetricmeniscuswaveswiththeinstability,lending itselftothresholdsbelowthe prediction. Asimilarsetofmodeswereobservedforexperimentswithred ucedlayerheights, whereaFC70layerof h 1 =2.1cmanda1.5cStlayerof h 2 =2.0cm,wereused(cf.Figure 5-8 ).Theeffectoflayerheightsfromtheinviscidtheoryfortw oliquidswasshownby Kumar&Tuckerman[ 52 ]whereheightdependenceinthedispersionrelationarises via theexpression 1 + 2 1 coth kh 1 + 2 coth kh 2 91

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2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 3.5 4 frequency f ,Hzimposedamplitude A ,cm 5.5 6 6.5 7 7.5 8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 (1 ; 1) h (1 ; 2) su (0 ; 1) h (2 ; 1) sh (2 ; 1) sh (0 ; 1) sh (4,1) h (3,1) h (1 ; 1) sh (2 ; 1) h (0 ; 1) sh (1 ; 2) h Figure5-8.Theoreticalcomparisonforthesmallheightexp erimentalsystem. Experimentalthresholdsandthecorrespondingtheoretical curveisshown forabilayerofFC70( h 1 =2.1cm, rho 1 =1916kgm 3 )and1.5cStsiliconeoil ( h 2 =2.0cm, 2 =846kgm 3 )inthe R =2.55cmcell.Thedashedline representsthepredicted(1,2) sh threshold. whichshowssaturationtowardunitywhen kh 1 and kh 2 aregreaterthanabout3.Inthis system,theexpressionleadstovaluesof0.89forthe(1,1)m odes,0.98forthe(2,1) modes,and0.99forthe(0,1)modesindicatingonlymoderate effectofheight.Still,the resultingshiftoftheharmonicandsubharmonic(1,1)modes areobservedaspredicted bytheviscoustheory.Howeveritshouldbenotedthatthe(1, 2) h modeobserved between4and4.4Hzjustabovethresholdisnotthepredicted mode.Thepredicted thresholdforthe(1,2) h modeisshowninFigure 5-8 andispredictedtobehigherthan the(1,1) sh mode.Theearlypresenceofthe(1,2) h modeobservedinthesmallandlarge heightsystemsgivefurtherreasontobelievetheexistence ofaninteractionwiththe meniscuswaves.TheorderingofthemodesinFigure 5-8 islargelythesameasseen 92

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0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 imposedamplitude A ,cmwaveamplitude A w ,cm instabilitythreshold 0.3 0.5 0.7 0.9 1.1 1.3 0 0.1 0.2 0.3 0.4 0.5 imposedamplitude A ,cmwaveamplitude A 1 ,cm Figure5-9.Saturatedinterfacedeectionowingtomeniscus andparametricexcitation. Waveamplitudesarefromthelargeheightsystem(seeFigure 5-7 )for imposedfrequenciesof(a)2.52Hzand(b)3.68Hz. forthelargeheightsystem,savefortheregionfrom3.8to4H z,whereinsteadofonly (3,1) h modes,a(4,1) h modewasalsoobserved.Thepredicted(3,1) h modewasdamped considerably.Alsoabsentfromtheexperimentalsystemaret he(4,1) h modespredicted around2.6Hz.Thetrendhereappearstobethatwalldampingi ncreasessubstantially withthenumberofazimuthalnodes.Mostclearlynoticeable inthelowheightsystem isthatthe(0,1) h resonanceisagainmuchlowerthanpredicted,forthesamere asonas givenearlierforthelargeheightsystem. SaturatedInterfaceAmplitudes Figures 5-9 (a)and(b)presentthesaturatedwaveamplitudewithrespec ttothe imposedvibrationalamplitudeinthelargeheightsystem(c f.Figure 5-7 )fortwodifferent frequencies,roughlycorrespondingtothenaturalfrequen ciesofthe(1,1) h and(0,1) h modes.Herethewaveamplitudehasbeentakenashalfthediff erencebetweenthe maximumandminimuminterfacialheights.Thegeneralshape softheresponseof Figures 5-9 (a)and(b)resemblethoseofHenderson&Miles[ 36 ]andDas&Hopnger [ 23 ],butwhatisintendedtobeshownissmoothingoftherespons enearthecritical thresholdasopposedtoasharpbifurcation.Thissmoothing oftheresponsenearthe thresholdwasdescribedasa“tailing”byVirnig etal. [ 79 ],whoreportedsimilarbehavior 93

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forthewaveresponseof(1,1) sh modesinarectangularcellusingwater.Additionof surfactantminimizedthetailingintheirexperiments.Inc ontrasttotheirobservation ofthisbehaviorforasubharmonicmode,thetailinginthecu rrentworkseemsbe characteristicoftheinteractionwiththemeniscuswaves, asitwasadditionallynoticed forthe(2,1) h mode,butwasnearlyabsentforthe(1,1) sh ,(2,1) sh ,and(0,1) sh modes. HigherOrderNonlinearPhenomenaandthePathtoTurbulence Inadditiontopracticalreasons,thenonlinearbehavioral soservestodene thesingle-modeFaradayexcitationasauidmechanicalsys tem,separatingitfrom themoregeneraldescriptionasaresonantnonlinearoscill ator.Amajorbenetof theexperimentisthereforetheopportunitytoobservereal nonlinearphenomena, whichmorebroadlyareaccountedforasnonlineardampingin dynamicalsystems analysis.Thesephenomenarepresenttheinitialdeparture sfromthe“regular”behavior predictedbyanevolutionequationobtainedfromaweaklyno nlinearanalysis,where alinearmodegrowsandsaturatestoaniteamplitude.Knowin gtheamplicationsat whichthesedeparturesappear,onecanreasonablysetexpec tationsforthepossible agreementbetweentheexperimentandaweaklynonlinearthe ory.Thesephenomena arereportedanddiscussedqualitatively,astherepeatabi lityofnonlinearexperiments wasnotrigorouslyinvestigated. Saturationtoaregularstandingwavewascommonforexcitati onnearthethreshold forfrequenciesaboveorslightlybelowthenaturalfrequen cy,standingincontrasttothe breakingbehaviorobservedforexcitationwellbelowthena turalfrequency.Theimages inFigure 5-10 depicttheinitialgrowthofa(0,1) sh modeabovethenaturalfrequency, atanamplitudesufcientlyhightoproducecomplexnonline arbehavior.Itisseenthat asthemodegrowsandapproachesamaximumdeectionamplitu dethereappear concentricripplesonthesurfacewhichmodulatethespatia lformofthemode,decrease modeamplitude,andthendissipate.Originatinginalllike lihoodfromsheareffects resultingfromhighervelocitiesassociatedwithhigheram plication,thisbehavioradds 94

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Figure5-10.Shearinstabilitiesona(0,1) sh mode.Transientsurfaceripplesappearduring thegrowthofa(0,1) sh mode,excitedat7.65HzinthesystemofFigure 5-7 Thesequencedepictedspanselevenwaveoscillations,andt hepresented imagescorrespondtothemaximumwavedeectionsofthe1st, 3rd,5th, 7th,9th,and11thoscillations. additionaltemporalperiodicitiestothemodedynamics.Fu rtherincreaseoftheforcing amplitudeatxedfrequencycancausethemechanismtoeithe rcontinuetoreappear, oratevenhigheramplitudes,sustainitself.Thisbehavior maybesimilartotheperiodic modulationobservedbyDas&Hopnger[ 23 ],butcomparisontotheirimagesforthe (0,1) sh moderevealshighlynonlinearwaveforms,suggestinggreat ereffectsofsurface tension. Anothermechanismbywhichnewfrequenciesenteredintothes ystemispresented in 5-11 ,whichshowsthesaturatedexcitationofa(0,1) sh mode,exceptatalower frequencythan 5-10 .Hereitisseenbycarefulinspectionoftheimagesthatthep olar positionofthemaximumdeectionamplitudeisdifferentea chsubsequentcycle.Inthis casethemodeactuallyorbitsthecentral z -axisofthecell,inaclockwisemotionwhen observedfromabove.Theappearanceofthismotion,referre dtoasa“precessional state,”isdescribedbyTipton&Mullin[ 77 ],andwithevengreaterdetail,including 95

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Figure5-11.Precessionofa(0,1) sh mode.Imagesdepictingtheorbitingofthe(0,1) sh modeaboutthecentral z -axis,excitedat7.2HzinthesystemofFigure 5-7 .Eachimagedepictsthecellateveryothertroughoftheimpos edcell motion,i.e.attimes t =0,0.28,0.56,0.83,1.11,1.39seconds. measurementsofthefrequencyoftheprecession,inthePh.D. dissertationofTipton [ 76 ].Theapparentcontactlineisclearlynothorizontalinthe 2ndand5thimages, suggestingthepossibilityofthesuperpositionofanother linearmode,albeitunpredicted bythetheory.Quiteinterestingworkontheinteractionoft woFaradaymodesand theresultingphasespaceattractorshasbeendonebyCilibe rto&Gollub[ 20 ]and Henderson&Miles[ 36 ]. Secondaryshearinstabilitieswerecommonlyobservedon(1, 1)and(2,1)modes, asseeninFigures 5-12 and 5-13 .Theseripplesappearedduringthesaturationof themodeandpersistedthroughoutexcitation,incontrastt otheripplesofFigure 5-10 whichwouldappearanddisappear.Theseripplesappearedon cethesaturatedmode amplicationsurpassedathresholdvalue,belowwhichthem oderetaineditslinear form.Furtherincreaseinthemodeamplicationcausedbubb lestotearfromtheripples, seenslightlyin 5-12 andmoresoin 5-13 ,andlargeramplicationnaturallyresultedin 96

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Figure5-12.Shearinstabilitiesona(1,1) sh mode.Visualizationofastanding(1,1) sh modewithsurfaceripples,excitedat5.6HzinthesystemofF igure 5-7 Figure5-13.Shearinstabilitiesona(2,1) sh mode.Visualizationofastanding(2,1) sh modewithsurfaceripples,excitedat6.8HzinthesystemofF igure 5-7 97

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Figure5-14.Breakupofa(2,1) sh mode.Visualizationoftheunboundedgrowthofa (2,1) sh mode,excitedat5.75HzinthesystemofFigure 5-7 theunrestrictedgrowthandbreakupoftheinterface,seeFi gure 5-14 .Theseripples appeartohavebeensuppressedbyexperimentswithlargervi scosityoils,rangingfrom 10to50cSt. TheimagesequenceofFigure 5-15 showshighlynonlinearbehaviortypicalofthe breakingmodesonthesubcriticalbranchesofthe(0,1) sh .Herethebreakuppatternis guidedbythecontinuedgrowthofthesecondaryinstabiliti espresentinFigure 5-10 whichturnover,collapse,andcompressasthewaveproceeds throughitsreversecycle andbreaksup.Thesepatternsremainazimuthallyuniformth roughoutmostofthis process,althoughnon-uniformpatternswerealsoobserved .Althoughsuchpatternsare wellknownintherealmsoftheRayleigh-TaylorandKelvin-H elmholtzinstabilities,none havebeenreportedinthereviewedexperimentsofsingle-mo deexcitation.Itisbelieved thecharacteristiclengthscalesofthebreakuppatternsar everydependentuponthe 98

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Figure5-15.Orderedbreakupofa(0,1) sh mode.Theimageseriesdepictsasequence ofinstabilitiesinthecatastrophicbreakupofa(0,1) sh ,excitedat7.4Hzin thesystemofFigure 5-8 99

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dynamicviscositiesofthetwophases,asthisseemstocontr olthegrowthratesofthe secondaryinstabilitieswhichproducethebehavior. SystemDampingStudy Avaluableandcommonmeasurementinthesingle-modeexperi mentsistherate atwhichanexcitedmodedecaysoncetheforcinghasbeenstop ped.Thesedamping ratescanbeusedtotalinearlydampedMathieuequationtot heobservedthresholds, asinHenderson&Miles[ 36 ]andTipton&Mullin[ 77 ],butinthecaseoftheexperiments inthisstudytherateswillprimarilybeusedtogaugetheamo untofdissipationinthe systemduetobulkviscouseffectsnotincludedinEquation 3–1 .Theexponentialrate ofdecayisdeterminedbytheslopeofthelogarithmoftherat ioofthewaveamplitude scaledbytheinitialvalueplottedagainsttime.Thiswasdo neforbothharmonicand subharmonic(0,1)modes,forwhichthemaximumwaveheighto ccursatthe z -axis. Imagingofthemodeandselectionofthetimespaceofthiscen tralpositionproduces adecay.Thewaveamplitudewasthenmeasuredashalfofthedi fferencebetween themaximumwaveamplitudeandtheapparentcontactline.Ext ensivedetailsonthis methodcanbefoundinKeulegan[ 48 ],Henderson&Miles[ 36 ]andDas&Hopnger [ 23 ]. Toquantifythedampinginthesystem,the(0,1)modewasboth harmonicallyand subharmonicallyexcitedintheFC70-10cStsiliconeoilsyst emandalsointhelarge andsmallheightFC70-1.5cStsiliconeoilsystems.Atotalof eightmeasurements ofthedecayofanexcited(0,1) h modeforthe10cStsystemweretakenatexcitation frequenciesof3.3and3.4Hzwithimposedamplitudesof15an d16mm.Analysis ofthetimeseriesofthewaveamplitudesatthecenterofthec ellyieldedanaverage dampingrateof1.15s 1 .Measurementsforthe(0,1) sh modeweretakenatfrequencies of7and7.1Hzwithimposedamplitudesof2.2and2.3mm,yield inganaverage rateof1.12s 1 .Calculationofthelinearviscouscontributionfromthein teriorwith Equation( 3–1 )predictsarateof0.51s 1 forboththeharmonicandsubharmonic 100

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modes,suggestingthattheremaindermightbeattributedto walleffects.Damping measurementsforsimilarforcingparametersinthe1.5cStsi liconeoilsystems yieldedharmonicandsubharmonicdampingratesof0.65s 1 and0.67s 1 inthe largeheightsystemand0.58s 1 and0.57s 1 inthesmallheightsystems.Herethe interiorcontributionsfromEquation 3–1 are0.397s 1 and0.395s 1 forthesmalland largeheightsystems,respectively.Comparisonoftheinte riorviscousdampingtothe overallmeasureddampingrevealsthesidewalleffectsofro ughly0.6s 1 inthe10cSt systemand0.18to0.25s 1 inthe1.5cStsystems.Aconjecturethatcanthereforebe madeisthatanupperuidviscosityhasagreateffectonthed issipationowingtothe lmformation,asthethicknessofthislmwasnoticeablysm allerfortheexperiments with1.5cStsiliconeoilversusthe10cStoil.Henderson&Mile sandDas&Hopnger reportmeasured(0,1) h decayratesofabout0.38s 1 and0.45s 1 whereastheinterior contributionsare0.022s 1 and0.0049s 1 101

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CHAPTER6 DOUBLE-FREQUENCYEXPERIMENTS Thischapterpresentstheresultsofalimitedstudyonexcit ationoftheinstability withvibrationscomposedoftwofrequencies,thelinearthe oryforwhichhasbeen presentedanddiscussedinChapter 2 .Theseexperimentsrevealregimesinwhich thethresholdbehaviormimicsthatofthesinglefrequencyc aseandalsowhereit showsnew,unexpectedbehavior.PresentedinFigure 6-1 (a)-(g)aredatasetsand thecorrespondinglinearpredictionsfor( M 1 M 2 )=(3,4)excitationoftheFC70and1.5 cStsiliconeoilsystemforvariousvaluesofthefrequencyra tio .Forlowvaluesof ,Figures(a)-(c),theonsetthresholdsforthe(0,1) h modeowingresonancewiththe M 1 componentareobservedinaccordancewiththelinearpredic tions,followedby theemergenceofa(1,1) sh modeathigher inFigures 6-1 (d)-(g),owingtoresonance withthe M 2 component.Figure 6-1 (h)bestdisplaysthetransitionfrom M 1 to M 2 resonance,wherethethresholdismeasuredforincreasingv aluesof whenthebasic frequency f = /2 isheldxed. LinearBehavior Excitationoftheinstabilityusingtwofrequencycomponent swasqualitativelythe sameaswithasinglefrequency,wheremeniscuswavespersis tedforamplitudesbelow threshold,andinductionperiodsfollowedbygrowthandsat urationorbreakupprevailed atamplitudesabovethethreshold.Thelineartheorywasals oshowntoagainvery accuratelypredictthethresholdbehavior,especiallyfor the(1,1) sh modeatratiosof 22.5 ,35 ,and45 (Figures 6-1 (d)-(f)).Thefrequencybandforeachofthesedata setswasalittlelargerthanpredicted,duetothetonguedam pingobservedinthe neighboring(3,1) h and(2,2) h modes,inaccordancewiththedeviationseeninthe singlefrequencyexperimentsformodeswithhighernumbers ofazimuthalnodes. Theremarkablendingfromthedouble-frequencyexperimen tsstemmedfrom themodicationsseentothe(0,1) h thresholdduepresumablytotheinteractionwith 102

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1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 0.4 0.6 0.8 1 1.2 1.4 frequency f = != 2 ,Hzamplitude A ,cm =0 (0,1) h (2,1) h (3,1) h (a) 1.1 1.13 1.16 1.19 1.22 1.25 1.28 0.5 0.7 0.9 1.1 1.3 1.5 frequency f = != 2 ,Hzamplitude A ,cm =5 (0,1) h (2,1) h (3,1) h (b) 1.1 1.14 1.18 1.22 1.26 1.3 0.3 0.6 0.9 1.2 1.5 frequency f = != 2 ,Hzamplitude A ,cm =10 (2,1) h (0,1) h (1,1) sh (c) 1.07 1.13 1.19 1.25 1.31 1.37 0 0.2 0.4 0.6 0.8 1 1.2 frequency f = != 2 ,Hzamplitude A ,cm =22.5 (2,1) h (1,1) sh (d) 1.05 1.1 1.15 1.2 1.25 1.3 1.35 0 0.2 0.4 0.6 0.8 1 1.2 frequency f = != 2 ,Hzamplitude A ,cm =35 (3,1) h (1,1) sh (2,2) h (e) 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 0 0.2 0.4 0.6 0.8 1 frequency f = != 2 ,Hzamplitude A ,cm =45 (3,1) h (1,1) sh (2,2) h (f) 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 0 0.2 0.4 0.6 0.8 frequency f = != 2 ,Hzamplitude A ,cm =90 (1,1) sh (2,1) sh (g) 0 5 10 15 20 25 30 35 40 0.3 0.5 0.7 0.9 1.1 1.3 frequencyratioangle, amplitude A ,cm 1.20 1.14 1.18 (h) (1,1) sh (0,1) h Figure6-1.Thresholddataandpredictionswith( M 1 M 2 )=(3,4)excitation.Systemis FC70(1905kgm 3 ,3.1cm)and1.5cStsiliconeoil(846kgm 3 ,3.3cm)in the R =2.55cmcylinder. 103

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themeniscuswaves.Relativetothesingle-frequencythres holdsinFigure 6-1 (a),the datasetobtainedat =5 (Figure(b)),indicatingslightadditionofthe4 component, showsadecreaseinthedeviationbetweentheobservedandpr edictedthresholdsas onemovesfromthenaturalfrequency(about1.22Hz)totheco -dimension2point. Excitationat =10 withgreater M 2 componentinFigure(c)showsthisdeviationto continuetodecrease,wheretheexperimentalthresholdsac tuallycrossthepredictions, withhigher-thanpredictedthresholdsnearthe(2,1) h -(0,1) h co-dimension2point.This behavioroftheobservedcrossingthepredictedthresholdi shighlightedinFigure 6-1 (h),byxedfrequencydatasets,whereitisseenthattheobs ervedthresholdsremain nearlyconstantas isincreased,incontrasttothepredicteddecrease. NonlinearBehaviorandInteraction Double-frequencyexcitationwillbefurtherdescribedusi ngthetimespacedata forsetsofharmonicallyandsubharmonicallyexcited(0,1) modes.Thesewaveshave saturatedtoasteady-statestandingwaveandarehelpfulfo rinferringbothlinearand characteristicnonlinearbehavior.Theparametriccondit ionswerealsosettoinorder toproducehighwaveamplication,butnotsogreatthatthel inearformofthemode wasalteredbysecondaryinstabilitiesorhigherorderdamp ingmechanisms(see Chapter 5 ).Relativetothresholdexperimentswhereonlytheminimum thresholdwas required,theseexperimentswereperformedinanespeciall ycontrolledmanner,where thetemperatureuctuationsduetolampusewereminimizeda ndthewallconditions weremaintainedatmaximumwetting(seeexperimentalrepea tability,Chapter 4 ). Eachexcitedwaveowesitsresonancetothe M 1 =3 component,meaningthewave executesthreeperiodsinthesinglecellperiodpresentedi nthegures. Forreference,the(0,1) h wavewasexcitedatthebasicfrequency f =1.2Hzwithout additionofthe4 component( =0 )atanamplitudeof A =0.84cm,andtheresulting standingwavetimespaceandcelldisplacementisshowninFi gure 6-2 .Evidentfrom thewavemotionisthatthemaximumwaveamplitudesremainco nstant,denoted 104

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by A 1 A 3 and A 5 inTable 6-1 .Theconsistencyofthereectionsonthewavesalso indicatestheabsenceofadditionalfrequenciesintheresp onseandlendscredenceto theexperimentalcontrol.Theminimumwaveamplitudesalso remainconstant,butare notablylessthanthemaximumwaveamplitude,indicatingan on-symmetricalmotion ofthewaveaboutthemidplaneandthereforealsothenonline arrestoringforce.This resultsfromnotonlythedensityandviscositycontrastoft hetwophases,butalsofrom thebiaseddirectionofgravity,theconcavityofthemenisc usandthelmdynamics.The apparentcontactlinelagsbehindthewavecrests,possibly duetothehighmeniscus waveamplitudes,whicharegreatlyreducedinsubharmonice xcitationwherethephase shiftisnearlyabsent. The(0,1) h wavesresultingfromexcitationwith4 componentaddedareshown inFigures 6-3 (a)and(b).Themainsimilaritiesbetweenthesewavesandth e single-frequencycaseofFigure 6-2 arethattheminimumwaveamplitudesareagain lessthanthemaximumamplitudes,andthatthecontactlinel agsthewavedeection. ExaminationofTable 6-1 revealsthatnotonlyhavetheaveragewavemaxima( A 1 A 3 A 5 )andminima( A 2 A 4 A 6 )decreasedwiththeincreasedratioofthe4 component, buttherealsoentersinternalvariancefromoneofthesemax imaorminimatothenext. Theimagesvisuallyconrmthisbytheslightdistortionoft hereectionfromonewave tothenext,whichwasnotpresentinthesingle-frequencyca se.Thismaybeexpected theoretically,astheFloquetexpansioniswrittenfortheb asicfrequency ,andthis periodicityintheexperimenthasremainedintact. Acharacteristicdifferencebetweensingleanddouble-fre quencyexperiments, importanttobothlinearandnonlinearbehavior,resultsfr omthemeniscuswaves adoptingtheperiodicityoftheparametricvibrations.Thu swhenasecondfrequency isaddedtotheforcing,thematchingofthemeniscuswaveper iodicitywiththatof theinstabilityislost,becausethemeniscuswaveperiodic itynowalsocontainstwo frequencies,whiletheresponseoftheinstability,torst approximation,containsonly 105

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onefrequency.Theresultingmismatchofthemeniscuswavep eriodicitywiththatofthat oftheinstabilityisthecentralexplanationofferedforth eobservedthresholdscrossing thepredictionsseeninFigure 6-1 (h). Giventhelimitedquantityofdata,theintentionofthisana lysisistoprovideinitial insightintothethecharacteristicsanddifferencesinthe nonlinearbehaviorofmodes excitedbyoneandtwofrequencies.However,onequantitati vecomparisonthatwillbe madeusingthisdataiswithregardstothedifferencesbetwe entheaveragemaximum andminimumwaveheights.Saturatedwavedatasimilartothe( 0,1) h excitationof Figures 6-2 and 6-3 ispresentedforthesubharmonicanalogueinFigure 6-4 for f =2.5 Hz,andA=0.098cmexcitationwithfrequencyratiosof =0 ,13.9 ,and26.9 .Muchlike theharmonicexperiments,deviationfromtheconstantpeak amplitudesandperiodicity isseenasthesecondfrequencyisadded.Notablydifferenti sthereductionofthe phaselagofthecontactline.Comparingtheaveragepeakamp litudestothetrough amplitudes,showninTable 6-1 ,thisratioisconsistentlysmallerforthesubharmonic excitation(about1.33)tothatoftheharmonicexperiments (about1.47),indicatinga biasedincreaseofthepeakmaximarelativetotheminimaint heharmonicexperiments. Thisbiasisaccreditedtotheinteractionwiththemeniscus waves,whoseamplitudes werenotedtobeconsiderablygreaterinharmonicexperimen tsthanthesubharmonic analoguewheretheywerenearlyabsent(seeChapter 5 ). ExperimentalConclusion Thisconcludesthepresentationandanalysisoftheobserve dexcitationof single-modeFaradaywaveswithoneandtwofrequencycompon ents.Thestudy revealsthelineartheorytobesufcientforpredictingthe criticalthresholdsforsome regimes,whilelesssoforthoseinwhichthesidewallnon-id ealityisabletointeract withtheinstability.Mostnotableistheinteractionbetwe entheharmonicfundamental axisymmetricmodeandthemeniscuswavesofthesameperiodi city,andsimilarspatial form.Thenatureofthisinteractionisobservedtochangewh entheinstabilityisexcited 106

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Figure6-2.Saturatednonlineardataforaharmonicmodeexci tedwithasingle frequency.Shownisthetimespace,intheoscillatingrefere nceframe,ofthe singlefrequency(3 )excitationofasaturated(0,1) h modeat f = != 2 =1.2 Hzand A =0.84cmusingthesystemofFigure 6-1 .Timespacedatawas collectedforthemaximumwaveamplitude,asinFigure 4-2 .Verticallength oftheimagerepresentsanexperimentallengthof3.902cman dthetotal time,representedbythehorizontallengthoftheimage,is1 .666seconds. Belowtheimageforcomparisonistheoutputcelldisplacemen t.Allimages inthischapterareofthesameresolution.Waveamplitudeme asurements arereportedinTable 6-1 withtwofrequencies,duetotheresultingmismatchbetween thewaveperiodicities. Analysisofthenonlinearsaturationofexperimentsdisplay sdifferencesbetween thesingleanddouble-frequencycases,andalsorevealsmor econsequencesofthe interactionbetweentheinstabilityandthemeniscuswaves .Aconsequenceofthe additionofasecondfrequencycomponentistheintroductio nofthreemoredegrees offreedom,therebygreatlyincreasingtherequiredscopet ofullycaptureitseffecton thephenomena.Thedouble-frequencyexperimentsofthisst udywererestrictedto thecasesof( M 1 M 2 )=(3,4)andthephaselag =0 .Whilethephaselagisknownto havelittleeffectonthelinearpredictions,intuitionsug gestsanabilityofittoadjustthe interactionbetweenthemeniscuswavesandtheinstability andoughttobethesubject offutureinvestigation. 107

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Figure6-3.Saturatednonlineardataforharmonicmodesexci tedwithtwofrequencies. Shownisthetimespacedatafromsystem 6-1 forsaturated(0,1) h modes excitedatafrequency f = != 2 =1.2Hzwith( M 1 M 2 )=(3,4)where =5.2 and A =0.84cm(above)and =10.6 and A =0.84cm(below).Waveamplitude measurementsreportedinTable 6-1 108

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Figure6-4.Saturatednonlineardataforsubharmonicmodese xcitedwithtwo frequencies.Shownisthetimespacedatafromsystem 6-1 forsaturated (0,1) sh modesexcitedatafrequency f = != 2 =2.5Hzwith( M 1 M 2 )=(3,4) where =0 and A =0.097cm(top), =13.9 and A =0.098cm(middle),and =26.9 and A =0.098cm(bottom).Waveamplitudemeasurementsreported inTable 6-1 109

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Table6-1.Parametricconditionsandwaveheightmeasureme nts. f ,Hz AA c A 1 A 2 A 3 A 4 A 5 A 6 A odd A even 1.2000.841.001.430.981.430.981.430.981.471.205.20.840.871.430.951.441.001.380.921.471.2010.60.840.781.350.881.390.981.240.841.482.5000.0970.0441.521.141.521.141.521.141.332.5013.90.0980.0331.511.131.521.141.491.151.322.5026.90.0980.0301.471.081.471.101.361.101.32 DatatabulatedfromFigures 6-2 6-3 ,and 6-4 .Theamplitude A c representsthe xed-frequencythresholdamplitudepredictedbythelinea rtheory.An A subscripted withanumberdenoteshalfthewaveheightsmeasuredaccordi ngtotheblackarrowsin Figure 6-2 .Imposedandwaveamplitudesmeasuredincm. 110

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CHAPTER7 FINALDISCUSSIONANDCONCLUDINGREMARKS Despiteoccurringinauidmechanicalsystem,singlemodeF aradaywave excitationisuniqueinitsabilitytobegeneralizedundert heumbrellaofparametric excitationofanonlinearsystemwithasingledegreeoffree dom.Animportant realizationmadeoverthecourseofthisworkthereforewast hatallpreviousexperiments measuringthecriticalthresholdshadmoreorlessadoptedt hisgeneralizationinregards tothetheoreticalexplanationoftheirresults.Therstex perimentsofBenjamin&Ursell andDodge etal simplynotedthedeviationbetweentheirthresholdsandthe inviscid model,astheydidnotincludeanyformofsystemdamping.The laterexperimentsof Henderson&Miles,Tipton&Mullin,andDas&Hopngereachsh owedtheassumption oflineardampingtobevalidastheyeachsawsufcientagree mentwiththelinearly dampedoscillator,andmovedontostudyhigherorderphenom ena.Whathasset thisworkapartfromthepreviousexperimentswasthemotiva tiontodesignasystem whichrespectedtheassumptionsnecessarytomakeconnecti onwiththeequationsof uidmechanics,andmostsignicantly,thestress-freebou ndarycondition.Thismeant considerableeffortwasspentsearchingforaliquidsystem thatcouldapproximatethe idealbehaviordictatedbythetheory.Whilenotinitofitsel fanoblerwayofinvestigating thephenomenon,theresultwasanincreasedefciencyofint erpretingthenuanced phenomenawhichleadtotheoreticalmismatch,openingrobu stavenuesforfuture investigation.Thesephenomena,alongwiththekeyaspects ofthesystemdesignand methodaretheprincipleresultsofthisscienticinvestig ationandwillbediscussed. ExperimentalMethod RealizationoftheStress-FreeBoundaryCondition Primarily,anextensivestudyoftheFaradaythresholdwasco nductedwithsingle anddouble-frequencyparametricexcitationintheregimeo fmodediscretization,andfor thersttimecomparisonhasbeenmadetotheviscouslinears tabilitytheoryofKumar 111

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&Tuckerman[ 52 ].Thistheory,whichiswrittenforhorizontallyinnitesy stems,was adaptedtonitesizecylindersbyapplicationofasidewall stress-freecondition,allowing separationofvariablesandthestudyofindividualmodesof instability.Whilefromone perspectivethisisaninviscidtheorycondition,becausea llrealsystemsexperience sidewallstressesowingtoviscousboundarylayersandcapi llaryeffects,inaviscous systemcloseapproximationhasbeenshownpossiblewhenthe sesidewallstresses paleincomparisontothebulkstresses.Thismethodofappro ximationrequirescareful selectionoftheparticipatinguids,whichcouldonlybede terminedbytrialanderror,as thebehaviorofaninterfaceatthesidewallisextremelydif culttopredict.Thechosen liquidsinthisexperiment,theuorinertFC70andsilicone oil,producedaninterface whichhappenedtomoveveryeffortlesslyoverglass,overa lmofthesiliconeoil(see Figures 5-3 and 5-4 ).Whilethislmisnotaccountedforbythemodel,itsassocia ted stressesalongwiththoseofviscousboundarylayerswerefo undtobemuchlessin comparisontothebulkdomainlinearviscouscontribution( seeChapter 5 ).Furthermore thesidewallstresseswereminimizedbyoptimizingtheuppe rphaseviscosity,which bothdecreasestheboundarylayerstressesandthelmthick ness. ExperimentalRepeatability Thepathofinvestigationwasstronglyaffectedanddrivent owardsstudyofthe linearthresholdbyboththeexperimentalrepeatabilityan dthedataanalysistechniques. Repeatabilitywasrstinvestigatedinthecontextofbeing abletoreproducethe observedlinearthresholdfromoneexperimenttoanother.T roublingbehaviorwas observedwhenexcitationoftheinstabilitycouldrequirea considerablyhigheramplitude fromtheafternoonofonedaytothemorningofthenext.Additi onallyitwasseenthe nonlinearbehaviorcouldchangesignicantlyduringaseri esofexperimentswhenthe systemwasleftsedentaryforafairlyshortperiodoftime,a sseenintheinconsistency ofsaturatedwaveamplitudesinFigures 4-7 (a)and(b).Alongwiththepresenceofthe sidewalllm,thisbehaviorisbelievedtobedependentupon molecular-scalebehavior 112

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ofsiliconeoiladsorptionontotheglass,whichsetsahighs tandardforsettingan experimentalinitialcondition.ExperimentalInitialCondition Forthereasonofperformingcontinuedexperimentsoverthe courseoftheday,and alsothefactthatsidewallwettingwasobservedtoproducel owerthresholds,theinitial conditionofacompletelyde-wettedsidewallcorrespondin gtothestagnantsystemwas deemedimpractical.Theotheroption,acompletely“wetted ”sidewallinitialcondition waschosen,andwasemployedbycontinuedexcitationofthei nstability,withbreaks takenwhileastandingmodewasleftinexcitation.Inregard stosearchingforalinear threshold,thismethodwassufcientasthethresholdthere foredecreasedoverthe courseofarunofexperiments,andtheactualthresholds,re portedsuchasinFigures 5-7 and 5-8 ,arethelowestobservedthresholds,believedtorepresent thecasewhen themaximumwettingconditionwasmet.Whilethetime-spaces ofFigures 6-2 6-3 and 6-4 representnonlinearexperimentswherethegreatestcarewa stakentoensurethe maximumwettinginitialcondition,futurenonlinearexper imentswiththisliquidsystem wouldrequireamorewell-denedmethodofpreparingtheini tialconditionandseveral experimentsberuntoverifyrepeatability.Thedependence oftheobservedphenomena onthesidewallshearkenstotheoriginalresultsofKeulega n[ 48 ]wherewavedamping dependedstronglyonthecontainermaterial.Apromisingmo dern-daysolutionwhich couldoffertightercontrolofthewettingandthustheiniti alconditionisnanoscale surfacemodicationsuchassilanization,summarizedbyBlo ssey etal. [ 14 ]. DataAnalysisTechniques Theelectromechanicalshakerwasfoundtoproduceamismatc hbetweenthe programmedandtheoutputvibrationalamplitudes,whichre quiredtheoutputamplitude tobemeasuredindividuallyforeachexperimentviaimagean alysis.Thismethodwas tedious,butatthesametimeallowedforasmoothtransition totheFFTanalysis usedforthedouble-frequencyexperiments.Imageanalysis forwaveamplitude 113

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measurements,ontheotherhand,wasmostinefcient,asitw asrstrequiredto removethevibrationalreferenceframefromtimespacedata ,followedbymanual tracingoftheinterface(example,Figure 6-2 ).Thiswasanoverlytediousmethod whichseriouslyimpedednonlinearanalysis,andtherefore relegatedfocustothelinear threshold.Thepotentialscopeofexperimentswouldincrea sedramaticallyfromthe abilitytoaccuratelymeasureatimeseriesoftheinterfaci aldeection,aslongtime dynamicsandphaseportraitbehaviorcouldbeinvestigated .Henderson&Miles[ 36 ] employeda1.15mmdiameterinsitucapacitancetypeprobeto measuredeections. Tipton[ 76 ]employedanon-intrusivemethodwhichmaybebettersuited forimmiscible systems,wherethemeasurementofthedeectionofalasersh ownverticallythrough thesystemwasusedtoinferboththesurfacegradientandde ection.Withouttheability toefcientlymeasurewavedeections,muchfocusremained onthelinearthreshold, whereonlyvisualconrmationoftheinstabilitywasrequir edtomarkadatapoint. ExperimentalResults ThemeasuredthresholdsofFigures 5-7 and 5-8 formthecentralresultsofthis work,andthecomparisontothelineartheoryadaptedtostre ss-freecaseenables veryclearseparationofthedifferentnon-idealitiespres entinthesystem.Firstand foremost,however,theagreementbetweentheobservations andthepredictionsis quitegood,fromseveralperspectives.First,theobserved thresholdsacrosstheentire frequencyrangefollowthepredictionsverycloselygivent hethresholdamplitudes rangefromnearly4centimetersatlowfrequencytolessthan amillimeterathigh frequency.Lookingatindividualmodes,theexperimentalm odeselectionisalso excellent,wheretheco-dimension2pointsformingtheboun dsofthefrequencybands arepositionedverynearboththepredictedfrequenciesand amplitudes.Careful examinationoftheobservedthresholdsrevealsthepossibi lityoffrequencybandshifts tobothlowerandhigherfrequencies,whereitappearsthe(1 ,1) sh modeexperiences apositiveshiftinFigure 5-7 andanegativeshiftinFigure 5-8 .However,makingthese 114

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claimswithcertaintyisdifcultbecausetheshiftsarever ysmallandlikelywithin theexperimentalcontrol,giventheaccuracytowhichtheph ysicalpropertieswere known,andchangesduetoambienttemperatureuctuations. However,bothpositive andnegativeshiftswereinfactobservedinthepreviousexp erimentsnegativeshifts wereobservedbyBenjamin&Ursell,Miles&Henderson,andDas &Hopngerwhile Dodge etal andCiliberto&Gollubobservedpositivefrequencyshifts. Notablythe experimentsofDodge etal andCiliberto&Gollubwereperformedinlargediameter cells,resultinginremarkablysmallfrequencybands,sugg estingapossibledependence ofthefrequencyshiftonthecellsize.Predictionofdamping ratesandfrequencyshifts ofwavesdependingonfactorssuchascontactconditionsand capillaryeffectshave beenperformedbythelikesofMiles[ 58 ],Mei&Liu[ 56 ]andHocking[ 40 ]. Beyondtheambiguityofmodefrequencyshifts,denitetrend scanbegleaned fromthethediscrepanciesseenbetweentheexperimentsand thepredictionsbased onthemodalstructureandtheresponseperiodicity.Deviat ionisobservedfornearly allthemodesnearthenaturalfrequencies,wheretheobserv edthresholdslieabove thetheoreticalpredictions,muchlikeallofthepreviouse xperiments.Thisdeviationcan beseentogrowformodeswithgreaternumbersofazimuthalno des,asseenwhen comparingthe(0,1) sh thresholdstothe(2,1) sh thresholdsinbothFigures 5-7 and 5-8 Thiskindoftrendisalsoseeninthe(0,1) sh and(3,1) sh thresholdsobservedbyDas& Hopnger,Figure 3-1 (f). TheFC70andsiliconeoilsystemusedintheseexperimentsof fersaverypromising startingpointofconnectingtheinstabilitytothesidewal lbehavior,becauseitispossible toestablishawell-denedinitialcondition,ifexcitatio nisstartedforasystemwhere thelmhasalreadybeencreatedandthewall“wetted”(seeab ove,andChapter 4 ).Inpreparingaconsistentinitialcondition,considerat ionswouldhavetobemade regardingtheratesatwhichthelmdrainedandtheactualco ntactlinere-advanced inthesedentarysystembeforeexcitationre-commenced.Th ereafter,theinterface 115

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wouldideallyremaintetheredtothesamecontactpositionf orthedurationofthe experiment,andtheapparentcontactlinemotionwouldbeco nnectedtothestretching andcontractionoftheinterfacealongthelm.Ito etal. [ 44 ]presentedadetailed comparativestudybetweenamodeltheyproposedandthedyna micsofthelm formedintheirexperimentswithwaterandkerosene,whichc ouldpossiblybeadapted totheinitialconditionproposedbythiswork.Thislmisal soqualitativelysimilar tothebehaviorobservedintheclassicallm-formingbehav iorofLandau&Levich [ 53 ].Anotherimportantresultthatdeservesexplanationwasth eimprovementofthe theoreticalagreementwhentheoilviscositywasreducedfr om10cStto1.5cSt,as inFigure 5-5 .Ifbothsetsofdatawereobtainedforthesameinitialcondi tion,witha lmpresentandmaximumsidewallwetting,theonlydifferen cenoticeabletothelinear theoryoughtbetheincreaseddampingduetotheupperphaseb oundarylayers. Intheseexperimentsthestabilitythresholdsarealsorepo rtedforseveralharmonic andsuperharmonicmodes,whichhavenotreceivedmuchatten tionfromprevious reports.Thereexistsanabundanceofworkstudyingtheforc edexcitationofvibrations inasystemviaexcitationatornearthesystemnaturalfrequ ency(seeclassical worksbyRayleigh[ 68 ]andNayfeh&Mook[ 63 ]),butthisremainsseparatefromthe harmonicexcitationoftheFaradayinstabilityindiscreti zedsystems,asitresultsfrom time-periodic,homogeneoustermsincontrasttotheinhomo geneoustermsthatgive risetoforcedoscillations.Harmonicexcitationofthesin gle-modeinstability,nearthe naturalfrequencyofthemodeinsteadoftwiceasinsubharmo nicexcitation,naturally occursathigheramplitudes.Thisaresultofthegreaterdam pingexperiencedbythe harmonicmode,asitexecutestwiceasmanywaveoscillation srelativetothecellmotion asthesubharmoniccase.MeniscusWaveInhomogeneity Themostinterestingndinguncoveredbytheseexperiments istheinteraction betweenthemeniscuswavesandtheharmonicFaradaymodes.T hemeniscus 116

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effectivelyrepresentstheloose,inhomogeneousconnecti onrequiredfortheaforementioned forcedoscillations,andthelower-than-predictedthresh oldsforthe(0,1) h modesnear3.6 HzinFigures 5-7 and 5-8 provideevidenceofinteractionbetweentheforcedandthe parametricoscillations.Inclusionofaperiodicinhomoge neityintheMathieuequation meansthatforparametricamplitudesbelowthethresholdth erenolongerexistsa quiescentbasestate,butratheranite,butnon-growingam plitudethatoscillateswith theinhomogeneity.Onepossibleavenuefortheoreticalcom parisonismodicationof thecurrentlineartheorywiththeinhomogeneousFloquetth eoryofSlane&Tragesser [ 73 ],whoshowadditionofaninhomogeneitycantransitionanas ymptoticallystable systemtounboundedgrowth.Tipton[ 76 ]notedthepossibilityofinteractionbetween meniscuswavesandthesubharmonicexcitationofthe(0,1)m ode,buttheeffectwould inalllikelihoodbeconsiderablylessdetectablethantheh armoniccasewherethe meniscuswaveamplitudesarefargreaterandthefrequencyi sequaltothatofthe instability.Theharmonicexcitationofthe(0,1)modether eforeservesasanexcellent experimentforstudyingthemixedparametricandforcedres onances.Theworkof HaQuang etal. [ 35 ]providesanexcellentfoundationforthenonlinearbehavi orofsuch asystem,wheretheytreatthegeneralizedcaseofaparametr icallyandforcedoscillator subjecttobothquadraticandcubicnonlinearities.Theoth erharmonicmodesinthis Faradaysystemappeartoalsoexperienceinteractionwitht hemeniscuswaves,butdue totheazimuthalmismatchofthesemodeswiththemeniscuswa vestheinteractionis diminished.Double-FrequencyPhenomena Theexperimentspresentedforthedouble-frequencyexcita tionofsingle-mode FaradaywavesinChapter 6 arebelievedtobetherstoftheirkind.Double-frequency excitationopensthedoortoalargeincreaseinthenumberof parametricstudies whichcanbeconducted,includingthedependenceonthefreq uencyspacingdened by M 1 M 2 ,theratioofthecomponentspresentintheparametricsigna l ,andthe 117

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phaselagbetweenthetwocomponents .Intheseexperimentsitwasshownthere existregionsofthestabilityspacefordifferentvaluesof thatexhibitagreement withthelineartheorythatisqualitativelythesameasthes ingle-frequencycase, forthestudiedcaseof( M 1 M 2 )=(3,4)and =0 .Aninterestingco-dimension2point appearsinthedouble-frequencysystem,whereincreasein causesthetransition from M 1 to M 2 resonance.Theco-dimension2pointcontainedinthistrans ition therebybreakstheharmonicorsubharmonictimesymmetryof theresponse,amethod usedinhigh-frequencyexperimentstoaccessnewstatesofp atternformation(see Edwards&Fauve[ 28 ],Kudrolli etal. [ 50 ],andArbell&Fineberg[ 3 ]).Insingle-mode experimentsthisisinterestingforbeingabletochoosewhi chcellmodesparticipate intheco-dimension2point,byadjustmentoftheparameters M 1 and M 2 ,offering aversatileextensiontothespatiotemporalchaosCilibert o&Gollub[ 20 ]observed betweenthe(4,3) sh and(7,2) sh modes.The(0,1) h modecontinuestoshowinteresting thresholdbehavior,wheretheadditionofthesecondfreque ncycomponentdoesnot changethethresholdamplitudeaccordingtotheprediction s.Againthisresultsfrom theinteractionbetweenthemeniscuswavesandtheinstabil ity,whichbecomesfurther complicatedbyadditionofthesecondfrequencycomponenta sthematchingofthe periodicityisbetweenthetwowavesislost.NonlinearBehavior ThetimespacedataofFigures 6-2 6-3 and 6-4 displaysbehaviorofsaturated(0,1) modesexcitedbothharmonicallyandsubharmonically,andi smeanttoqualitatively outlinebehaviorwhichcouldbematchedbyaweaklynonlinea rtheory.Henderson &Miles[ 36 ]notablycomparedtheir(0,1) sh modeamplitudestothepredictionsofthe nonlineartheoryofMiles[ 59 ],againrequiringthemeasuredexperimentaldamping ates.Whileaweaklynonlinearsingle-modeFaradaytheoryin cludingtherigorous calculationofviscosityhasnotbeenwritten,thetheoryof Skeldon&Guidoboni[ 72 ] isacompletetheorywhichactsasthenonlinearextensionto theFloquetanalysisof 118

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Kumar&Tuckerman,exhaustivelyanalyzingtheLyupanovfun ctionsforallpossible waveformsinordertodeterminepatternselection.Extensio nofaweaklynonlinear Floquetanalysistoacylindrical,discretizedsystemforp redictionofmodeamplitudes wouldbemuchsimplerasthenumberofparticipatingmodesis considerablysmaller.A complicationinmakingcomparisontotheamplitudesofacar efulnonlinearexperiment thatwouldhavetobenegotiatedisthemismatchbetweentheo bservedandtheoretical thresholds.Beyondthisthenonlineartheorywouldbevalidf orexcitationthatresultedin thegrowthandsaturationofthelinearwaveform,belowthet hresholdsatwhichhighly nonlinearbehaviorbeginstoappear,asdetailedinChapter 6 .Akeyobservationofthe timespaceimagesisthatboththewavepeakamplituderatios andcontactlinephase lagsaremuchmorepronouncedintheharmonicexperimentsth anthesubharmonic experiments,indicativeofstronginteractionwiththemen iscuswaves,meaninga nonlineartheorywouldrequireincorporationoftheinhomo geneoustermsuggested above.Theinductiontimesrepresentaphenomenononthebou ndaryofunderstanding, asitmayeitherberelatedtostick-slipbehaviorofthewall ornitetimephenomena. ThephenomenonisreportedbyDas&Hopngerbutnotablynots obyMullin&Tipton. 119

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APPENDIXA INVISCIDRESULTANDCOMMENTS WhileRayleigh[ 68 ]presentedmathematicaljusticationfortheexistenceof a parametricthresholdforsubharmonicexcitation,thesemi nalworkofBenjamin &Ursell[ 11 ]offeredtherstconnectionbetweentheinstabilityandth eequations ofmotion.TheinviscidtheoryforFaradaywavesishelpfulf orunderstandingthe fundamentalphysicalbehavior,includingmodedispersion .Theirresultequatesthe instabilitytotheparametricallyexcitedlinearpendulum ,andwillbepresentedhereas aspecialcaseoftheviscousmodelpresentedinChapter 2 .Takingtheupperlayerto bepassiveandre-settingtheinterfaceto z = h andthelowersurfaceto z =0,Benjamin &Ursell'sinviscidresultcanbeadaptedbysetting =0intheperturbednormalstress balance,Equation 2–16 @ t @ z w + g + A 2 cos t r 2H = r r 4H at z = h (A–1) Thetimederivativeofthevelocityprole @ t w iscalculateddirectlybysetting =0 in Equation 2–18 r 2 @ t v =0 (A–2) whosecharacteristicsolutionis @ t w = A sinh kz + B cosh kz (A–3) when r 2H = k 2 .Notingtheno-slipconditiondoesnotapplytoinviscidthe ory,the pertinentboundaryconditionsarethetimederivativesoft heno-owconditionatthe bottomsurfaceandthekinematicconditionattheinterface @ t w =0 at z =0 (A–4) and @ t w = @ tt at z = h (A–5) 120

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Applyingtheseconditions,theproleevaluatesto @ t w = @ tt sinh kz sinh kh (A–6) andsubstitutionintoEquation A–1 yields @ tt +tanh kh gk + r k 3 + A 2 k cos t (A–7) ThenalresultofBenjamin&Ursellisobtainedbyre-casting withthetimetransformation T = 1 2 t andtheparameters p = 4tanh kh h gk + r k 3 i 2 and q = 2 kA 2 tanh kh 2 (A–8) revealingtheclassicalform(seeMcLachlan[ 55 ])oftheMathieuequation @ tt +[ p +2 q cos T ] =0. (A–9) Especiallyusefulareboththedenitionofamode'snaturalf requencyvia ( p 2 ) 1 2 andtheappearanceofthe tanh kh term,whichcanbeusedtogaugethewhether ornottheeffectoflayerheightissaturated.Inthecontext oflinearstability,the interpretationofthestabilitychartofBenjamin&Ursell(F igure A-1 )isidenticaltothat oftheviscoustheorypredictions,i.e.Figure 2-1 ,wheretheregioninsideandoutside thenscorrespondtogrowingandboundedsolutions,respec tively.PresentinFigure A-1 arethepointsofperfectresonanceoccurringforinnitesi malforcingamplitudes, whichvanishuponadditionofadampingsource.TheMathieue quationlendsitself tonumericalsolutionbytimeintegrationandspecication ofaninitialcondition,and representativestableandunstablesolutionsarepresente dinFigure A-2 .Kumar& Tuckerman[ 52 ]presenttheinviscidtheoryforsystemswithtwoactivelay ers,towhich theyaddlineardampingforthepurposeofcomparisontothep redictionsoftheirviscous theory. 121

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FigureA-1.ThestabilitydiagramofBenjamin&Ursell. 0 50 100 150 200 250 300 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 time,Tfreesurfacevariable, (a) 0 50 100 150 200 250 300 -50 -30 -10 0 10 30 50 time,Tfreesurfacevariable, (b) FigureA-2.SampleMathieuequationsolutions.Timedevelopm entofthefreesurface variable for(a)astablesolution ( p q ) = (2,1.15) and(b)anunstablesolution ( p q ) = (2,1.213) 122

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APPENDIXB LINEARSTABILITYMATLAB R r CALCULATION ThefollowingMATLAB R r codeisasimpleexampleofthetypesofcalculationsused tomakethepredictionsthroughoutthiswork.Thephysicalp ropertiesofthesystem inputintothecalculationarethelayerheights,layerdens itiesandviscosities,interfacial tensionandgravitationalacceleration.Specicationofth eimposedfrequencyand thewavenumberofinterestalongwiththeFourierseriescut offcompletestheinput, andexecutionofthecodeperformsthelinearstabilitycalc ulationoutlinedinChapter 2 .Theparameterinputas-isoutputsharmonicandsubharmoni cthresholdsforthe wavenumber k =259.5m 1 inFigure 2-1 ,ofwhichthesubharmonicminimumis A =0.091 cm.Thiscodeservesasthefoundationforallthecalculatio nsperformedinthiswork. 1 clearall 2 %%%%%%%%%%%%RequiredParameterINPUT%%%%%%%%%%% 3 rho=[1880,846]; %densities,kg/mˆ3 4 nu=[12,1.5]; %kinematicviscosities,centiStokes 5 gamma =.007; %interfacialtension,kg/sˆ2 6 h=[.005,.005]; %layerheights,m 7 g=9.81; %gravitationalacceleration,m/sˆs 8 f=9; %imposedfrequency,Hz 9 k=259.5; %wavenumber,mˆ 1 10 sigma=0; %modegrowthrate,sˆ 1,zeroforneutralstability 11 N=10; %Fourierseriescutoff 1213 %%%%%%%%CalculationproceedsfromHERE%%%%%%%%% 14 thresholds= zeros (4,2); %calculationoutputvariable,andcontains theharmonicandsubharmonicneutralstabilityamplitudes 15 nu=nu/1e6; %kinematicviscosityconversion,cSttomˆ2/s 123

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16 mu=rho. nu; %dynamicviscositycalculation,kg /m s 17 omega=2 pi f; %frequencyconversion,Hztoradians/s 18 INDEX1=0; 19 for alpha=[0,1/2 omega]; %harmonicandthensubharmonicsolutions 20 INDEX1=INDEX1+1; 21 D= zeros (2 (N+1)); %eigenvalueproblemD matrixinitialize 22 for n=0:N; %looptocalculatetheD matrixcoefficients foreachFouriermoden 23 fexp=(mu+1i (alpha+n omega)); %Fourier Floquet exponent 24 q1= sqrt (kˆ2+1/nu(1) fexp); %characteristicsolution exponent1 25 q2= sqrt (kˆ2+1/nu(2) fexp); %characteristicsolution exponent2 26 b= zeros (8,1); %initializationofinhomogeneityin boundaryconditionsystemofequations 27 b(8)=fexp; %insertionofFloquetexponentarising fromkinematiccondition 28 %Herethematrixdefinedbythesystemofequations arisingfromtheboundaryconditionsis constructed.Thecharacteristicvelocity solutionsincludez exp(k z)andz exp( kz) insteadofexp(q1 z)andexp(q2 z)whenq1=q2=0 andthecoefficientsforthesesolutionsare calculatedfirst. 29 if fexp==0 124

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30 fhs=[ exp ( k h(1)), exp (k h(1)),h(1) exp ( k h(1)) ,h(1) exp (k h(1)),0,0,0,0; %noflow condition,bottomsurface 31 0,0,0,0, exp (k h(2)), exp ( k h(2)),h(2) exp (k h(2)),h(2) exp ( k h(2)); %noflow, topsurface 32 k exp ( k h(1)), k exp (k h(1)),h(1) k exp ( k h(1))+ exp ( k h(1)), h(1) k exp (k h(1))+ exp (k h(1)),0,0,0,0; %noslip condition,bottomsurface 33 0,0,0,0,k exp (k h(2)), k exp ( k h(2)),h (2) k exp (k h(2))+ exp (k h(2)), h(2) k exp ( k h(2))+ exp ( k h(2)); %noslip,top surface 34 1, 1, 1, 1,1,1,1,1,; %continuityof velocityattheinterface 35 k,k, 1, 1,k, k,1,1; %continuityof velocityz derivativeattheinterface 36 mu(1) 2 kˆ2, mu(1) 2 kˆ2, mu(1) (kˆ2+2 k), mu(1) (kˆ2 2 k),mu(2) 2 kˆ2,mu(2) 2 k ˆ2,mu(2) (kˆ2+2 k),mu(2) (kˆ2 2 k); % continuityoftangentialstressesatthe interface 37 1,1,1,1,0,0,0,0]; %kinematiccondition .equallycanbewrittenas0,0,0,0, 1,1,1,1] 125

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38 coeff=fhs n b; %calculationofthevelocityprofile coefficients 39 dz1=coeff(1) k coeff(2) k+coeff(3) coeff(4); % evaluationoffirstz derivativeofthelower layervelocityprofileatz=0 40 dzzz1=coeff(1) kˆ3 coeff(2) kˆ3+coeff(3) 3 kˆ2+ coeff(4) 3 kˆ2; %thirdz derivativeofthe lowerlayer 41 dz2=coeff(5) k coeff(6) k+coeff(7) coeff(8); % firstz derivativeoftheupperlayer 42 dzzz2=coeff(5) kˆ3 coeff(6) kˆ3+coeff(7) 3 kˆ2+ coeff(8) 3 kˆ2; %thirdz derivativeofthe upperlayer 43 Dn=(rho(2) fexp+3 mu(2) kˆ2) dz2 mu(2) dzzz2 (( rho(1) fexp+3 mu(1) kˆ2) dz1 mu(1) dzzz1)+(( rho(2) rho(1)) g gamma kˆ2) kˆ2; %evaluation oftheD matrixcoefficientsforeachFourier moden 44 else %calculationforwhenq1=q2arenotzero.Same formatasabove. 45 fhs=[ exp ( k h(1)), exp (k h(1)), exp ( q1 h(1)), exp (q1 h(1)),0,0,0,0; 46 0,0,0,0, exp (k h(2)), exp ( k h(2)), exp (q2 h(2)), exp ( q2 h(2)); 47 k exp ( k h(1)), k exp (k h(1)),q1 exp ( q1 h (1)), q1 exp (q1 h(1)),0,0,0,0; 126

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48 0,0,0,0,k exp (k h(2)), k exp ( k h(2)), q2 exp (q2 h(2)), q2 exp ( q2 h(2)); 49 1, 1, 1, 1,1,1,1,1,; 50 k, k,q1, q1, k,k, q2,q2; 51 mu(1) 2 kˆ2,mu(1) 2 kˆ2,mu(1) (kˆ2+q1ˆ2), mu(1) (kˆ2+q1ˆ2), mu(2) 2 kˆ2, mu(2) 2 k ˆ2, mu(2) (kˆ2+q2ˆ2), mu(2) (kˆ2+q2ˆ2); 52 1,1,1,1,0,0,0,0]; 53 coeff=fhs n b; 54 dz1=coeff(1) k coeff(2) k+coeff(3) q1 coeff(4) q1 ; 55 dzzz1=coeff(1) kˆ3 coeff(2) kˆ3+coeff(3) q1ˆ3 coeff(4) q1ˆ3; 56 dz2=coeff(5) k coeff(6) k+coeff(7) q2 coeff(8) q2 ; 57 dzzz2=coeff(5) kˆ3 coeff(6) kˆ3+coeff(7) q2ˆ3 coeff(8) q2ˆ3; 58 Dn=(rho(2) fexp+3 mu(2) kˆ2) dz2 mu(2) dzzz2 (( rho(1) fexp+3 mu(1) kˆ2) dz1 mu(1) dzzz1)+(( rho(2) rho(1)) g gamma kˆ2) kˆ2; 59 end 60 D(2 n+1,2 n+1)= real (Dn); %placementoftherealand imaginarycomponentsDnintheD matrix 61 D(2 n+1,2 n+2)= imag (Dn); 62 D(2 n+2,2 n+1)= imag (Dn); 63 D(2 n+2,2 n+2)= real (Dn); 64 end 127

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6566 B= zeros (2 (N+1)); %constructionoftheBmatrix 67 if alpha==0; %harmonic 68 B(1,3)=2; 69 B(3:2 (N+1),1:2 N)= eye (2 N); 70 for i=1:2 (N 1) 71 B(2+i,4+i)=1; 72 end 73 else %subharmonic 74 B(1,1)=1; 75 B(2,2)= 1; 76 B(3:2 (N+1),1:2 N)= eye (2 N); 77 for i=1:2 N 78 B(i,2+i)=1; 79 end 80 end 8182 [eigenvecs,eigenvals]= eig (D,1/2 (rho(2) rho(1)) kˆ2 B); %solutionoftheeigenvalueproblem 83 eigenvals= diag (eigenvals); 8485 index1=0; %routineusedtosortthroughtheobtained eigenvalues 86 for index2=1: length (eigenvals) % 87 ifisnan (eigenvals(index2)) %NaNeigenvaluesignored 88 elseifisinf (eigenvals(index2)) %infiniteeigenvalues ignored 128

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89 elseifabs ( imag (eigenvals(index2))) > 1e 6 %supertiny eigenvaluesignored 90 elseif eigenvals(index2) < 0 %negativeeigenvalues ignored 91 else 92 index1=index1+1; %indexsetforpositive eigenvalues 93 evals(index1)=eigenvals(index2); %eigenvalue storedinevalsvector 94 ifabs ( imag (evals(index1))) < 1e 6 %disposalof supertinycomplexparts 95 evals(index1)= real (evals(index1)); 96 end 97 end 98 end 99 evals= sort (evals); %sortingoftheobtainedeigenvalues 100 evals=100/omegaˆ2 evals; %conversionfromacceleration Aomegaˆ2toamplitudeA,centimeters 101 thresholds(:,INDEX1)=evals(1:4)'; %outputvariable 102 end 129

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APPENDIXC B M 1 AND B M 2 For ( M 1 M 2 )=(2,3) and N =9,wehave B M 1 = 26666666666666666666666666664 0000200000000000000010000010000 100000100 100000001001000000010010000000000100000000001000000000010000 37777777777777777777777777775 B M 2 = 26666666666666666666666666664 0000002e i 000 000000000000e i 00000e i 0 000 e i 00000e i e i 000000000 0 e i 00000000 e i 000000000 0e i 00000000 00e i 0000000 000e i 000000 37777777777777777777777777775 130

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forharmonic( =0)responses,and B M 1 = 26666666666666666666666666664 0010100000000 1010000 10000010000 100000100 100000001001000000010010000000000100000000001000000000010000 37777777777777777777777777775 B M 2 = 26666666666666666666666666664 0000e i 0e i 000 00000 e i 0e i 00 00e i 00000e i 0 000 e i 00000e i e i 000000000 0 e i 00000000 e i 000000000 0e i 00000000 00e i 0000000 000e i 000000 37777777777777777777777777775 forsubharmonic( = 2 )responses. 131

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APPENDIXD OUTREACHSUMMARYOFWORK Theworkofthisdissertationistheresultofadetailedtheo reticalandexperimental studyofaphenomenonoriginallydiscoveredbyFaradayin18 31.Faradaysawthat whenheheldavibratingviolinbowtothebottomofaplatehol dingaliquid,tinywaves wouldformandvibrateonthesurface.Hisexperimentsshowe dthatthepatternsformed bythewaves,whichhecalled“beautifulcrispations,”chan geddependingontheliquids heusedandthenotehestruckwithhisviolinbow.Over150yea rslater,wehavea detailedunderstandingofthemathematicsthatexplainthe sepatterns,andhavealso performedmanyexperimentswhichshownewbehaviornotseen byFaraday.What willbedescribedinthisappendixisthatthisphenomenonof Faraday'swaves,canbe deeplyunderstoodfromtheperspectiveofsomesimpleexper iments,analogies,and physicalprinciplesofeverydaylifesuchasresonanceandf riction. WhatFaraday'sviolinboweffectivelydidtotheplateofliqu idwastocauseit tomoveupanddowninaregularmotionatacertainfrequency. Beforediscussing themeaningofthisupanddownmotion,itwillbenotedthatFa radaymadeavery importantobservation,whichwasthatthetinyripplesonth esurfacewouldmoveupand downeachtimetheplatemovedupanddown.Describedhereisa resonancethatthe frequencyoftheupanddownripplemotionwasequaltotheupa nddownmotionof theplate.Healsoobservedthatthenotestruckonthebowhad tobeloudenoughfor anythingtohappenonthesurface.Thecombinationoftheset woobservationsarethe signatureofparametricresonance,aphenomenonobservedi nmanyothersystems, andmostnotablythependulum.Tofurtherexplainthenature ofthisworkandFaraday's experiment,ananalogybetweenaliquidinterfaceandapend ulumwillbemade. Apendulum,likethatofagrandfatherclock,consistsofama ssattachedtoarodor stringwhichisheldataxedposition.Atrest,thependulumb obpointsinthedirection ofgravity,andwhenpulledawayandreleased,itswingsback andforthabouttherest 132

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positionataregularfrequency.Thiswastakenadvantageof byChristianHuygens [ 41 ]whoinventedtherstpendulumclock.Thefrequencyofosci llationabouttherest positiondependsuponthelengthandthemassofthependulum bob.Itcanbeseen thatshortclockpendulumsoscillatequiterapidlyrelativ etothe67meterlongFoucault pendulumwhichhangedatthePantheoninParis,France,andr equiredover16seconds tocompleteoneperiodofoscillation. Anotherimportantideainbeingabletocompletetheanalogyb etweenapendulum andthesurfaceripplesobservedbyFaradayisthatofenergy dissipation.Whenthethe pendulumispulledawayfromtherestposition,energyisput intothesystem,which, whenthebobisreleased,tendstokeepthesysteminmotion.H owever,inanyreal system,maintenanceofthisenergyinthesystemisdifcult andisgraduallylost,or dissipated.Inthecaseofthependulum,energyislostdueto frictionaleffectssuchas thestringconnectiontothexedpointorevenairresistanc eduetothebobmotion. Eventuallythelossofenergyresultsinthedecayofthebobmo tionuntilitstopsatthe restposition.Thisenergylossiscompensatedinpendulumc locksbywindingofthe clock,thatis,astorageofenergyinaspring. InthecontextofexplainingFaraday'sphenomenonofwavesa ppearingon averticallyvibratedliquidsurface,thesewavesexhibitt hesamebehaviorasjust describedforpendulumswherethereexistsanaturalfreque ncyandthereexistsources ofenergyloss.Imaginingkidsplayinginapool,thesurface getsdisturbedandbecomes verywavyintime.Afterthekidsleavethepool,thewavesonth esurfacecontinue tooscillatebackandforthatasomewhatregularfrequency, justlikethependulum bobmovingbackandforthaboutitsrestposition.Whilethepe ndulumfrequencyis determinedbythestringlengthandthependulummass,thewa terwavefrequency isprimarilydeterminedbythedistancebetweentwowavecre sts.Largedistances betweenwavecrests,orwavelengths,oscillateveryslowly ,whereassmallripples,like thoseofFaraday'sexperiment,oscillatequickly.Returni ngtothepool,eventuallythe 133

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wavesdieoutjustlikethependulumbobreturnedtorest.One oftheexplanationsfor thiswhichismorewellknownistheuidviscosity,whichact sasasourceofenergy dissipationinuids. BeforeexplaininghowthewavesofFaraday'sexperimentmani festthemselves,a realexperimentdemonstratingparametricresonancewillb edemonstrated.Walking homefromthegrocerystorewithshoppingbags,onemightnot icethatafterpicking upacertainpacetheswayingofone'sarmsbeginstonaturall yfallintorhythmwiththe stepstheyaretaking.Thisisverymuchparametricresonanc e!Lettingone'sarmhang loose,theshoppingbagactsasapendulumbob,andwhenwalki ngiscommenced, theupanddownmotionofone'sshoulderactsintheexactsame wayasFaraday's vibrationofhisplate.Asanexperiment,yououghttryusings hoppingbagsofdifferent lengthsandtheamountof“apples”inthebagtosensethediff erentresonanceswhich canbeachievedbetweenthemotionofthebagandyourstride. Notablyonewillsee thatnormallythebagexecutesoneperiodforevertwosteps. Thisisactuallywhat Faradayalsoobservedthefrequencyoftheripplesontheliq uidsurfacewasexactlyhalf thatoftheviolinbowfrequency.Trysittingonaswingandse eingifyouobserveany similarities! Justliketheexperimenthere,parametricresonanceisobse rvedinapendulum whenthepointtowhichitisconnectedismovedupanddown.Sim ilartotheinitial inclineordeclineinanelevator,movingofthependulumcon nectioncausesthe magnitudeofgravitytoincreaseanddecreasewiththeupand downmotion.Inthe casethebobismovedslightlyawayfromrest,ifitsmotionba cktowardstherest positioncoincideswiththedownwardmotionofthependulum connection,energycan betransferredtothebob,becauseitsmotionisinphasewith theconnectionmotion. Likewise,coincidenceofthemotionofthebobpasttherestp ositionwiththeupward motionoftheconnectionalsotransfersenergytothebob.Th isisthemethodbywhich parametricresonanceisachievedinthependulum,andiside nticaltothatofFaraday's 134

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experimentinthesensethattheeffectivegravityisbeingc hangedbytheupand downmotionofthesystem,andthatthebackandforthmotiono fthependulumbobis substitutedbytheupanddownmotionofthestandingwaves. Thisexplanationformsthefoundationfortheunderstandin gofthephenomenon exploredinthisdissertation.However,asmentionedbefor e,Faradayhadtostrike hisbowwithanoteloudenoughtoexcitethewaves.Equally,in theshoppingbag experiment,onemustwalkatapacebriskenoughforanynotic eablebehaviortooccur. Thisconditionrepresentsathresholdenergyinputwhichmu stbesurpassedinorder forthephenomenontobeobserved,andisdirectlyrelatedto thedissipationpresentin anexperiment.Measurementofthesethresholdsandcompari sontoamathematical predictionisacentralfocusofthiswork. PreviousFaradaywaveexperimentshavebeenperformedintwo different regimes,whichlendthemselvestotwodifferentmeansofpre dictingthisthreshold. Athighfrequencies,liketheoriginalexperimentsofFarada y,theexcitedwavelength isverysmall,andthesourcesofenergydissipationduetoth esidewallsareless importantthanthedissipationduetoviscosity.Inthiscas ethecriticalthresholdcanbe predicteddirectlybyanalysisofthewellknownequationsd escribinguidmotion,the Navier-Stokesequations.Replicationofthesidewallbehav iorrequiredbythisanalysis isdifcultinexperiment,andthereforeinthelow-frequen cyregime,wherethesidewall dissipationisbycontrastmoresignicant,experimentsha vebeendesignedtouselow viscosityuidsandcircumventthenecessityforsuchanaly sis.Thethresholdsarethen matched,notpredicted,byamodelwhichincludesthedissip ationmeasuredfromthe experiment. Whiletherearecertainbenetstothistypeofapproachinthe lowfrequency regime,inthisdissertationanexperimentalsystemwhiche xcitestheparametric resonanceatlowfrequencieshasbeendesignedwhichminimi zesthedissipationdueto sidewalleffects,producingagoodagreementwiththepredi ctionsoftheNavier-Stokes 135

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equations.IndoingsoanovelFaradaywavesystemhasbeenst udiedandmany interestingphenomenahavebeenrevealedthatwerenotobse rvedintheprevious experiments.Oneofthesebehaviorsistheuseofaliquidlm toallowidealsidewall behavior,muchlikeonemightapplyalubricanttoapendulum connectioninorder tominimizetheassociateddissipation.Theothernovelasp ectisanobservanceof aninteractionbetweentheforcedandtheparametricresona nces.Forcedresonance representsthemotiononecouldinduceinapendulumiftheyw eretooscillatethe connectionfromsidetosideratherthanupanddown.IntheFa radayexperiment thisoccursbecauseofaripplethatappearsinthesystemdue totheliquidsidewall meniscus.Inthecourseofpresentingtheseresults,themat hematicalmodelusedto predictthethresholdsisdescribedindetail,aswellasthe experimentalmethod. 136

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BIOGRAPHICALSKETCH WilliamRichardBatsonIII,affectionatelyknownasTrey,was bornin1985toSusan andBillyBatsoninLexington,SouthCarolina.Hewasinspiredb yscienceandmathat anearlyage.Whenhewasn'tenjoyinglearninghismultiplica tiontableshewasenvying hisfriendswhohadalreadyguredoutwhata“squareroot”wa s.Inelementaryschool hereadbooksonDNAandelectricityforfun,andoncetoldhis grandmotherhewould growuptobeabrainsurgeon.Beforemiddleschoolheplayedsu mmerbaseball,which hebelievestohavebeen“decent”at,yetstillmostvividlyr emembersthetimehewas hitintheheadbyawarm-upyball,andthencriedaboutstill havingtogotobatagainst theleague'sacepitcher.Intheendbaseballwasfun,butthe mosthumblingwasgolf whichhelovedevenmore,yetcouldneverseemtondtheright mindsettoputitall together.AtIrmoHighSchoolhebuiltonhismother'sloveofmu sicbydiscoveringhis own,andembracedtheintellectofhisfatherashesurrounde dhimselfwithanincredibly brightcrowd,manyofwhomhavealreadybecomedoctors,lawy ers,andsiliconevalley entrepreneurs.HefollowedhisfatherbyattendingClemson UniversityinupstateSouth Carolina,wherehechosechemicalengineeringbasedonhisa ppreciationofchemistry andknackforcalculus.Heconductedundergradresearchwit hDavidBruceatClemson fortwosummers,cementinghisdecisiontopursuegradschoo lbecauseheenjoyed the“slowprocess”ofresearch.HenishedwithhisB.S.inch emicalengineeringand aminorinmathin2008,thenwoundupinRangaNarayanan'sof cebecausetheman waswearingNewBalanceshoes.Subsequentlyhespenthistimel earningtolove researchattheUniversityofFlorida.Girlfriendsuttere dabouthereandtherebutat thispointheiscontenttowaitforthespecialsomeone,andi sgratefulforthefriends hehasmadeandallofthetimehehasbeenaffordedtopursuehi sdreams,whichhe willcontinuetodo,ashehopesgivebacksomeofwhathehasre ceivedonedaywitha careerinacademia.HereceivedhisPh.D.fromtheUniversity ofFloridainthespringof 2013. 143