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The Effects of Wavy Boundaries on Flow Structures and Fluid Instabilities

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Title:
The Effects of Wavy Boundaries on Flow Structures and Fluid Instabilities
Creator:
Messmer, Brad M
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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english
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1 online resource (128 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Chemical Engineering
Committee Chair:
NARAYANAN,RANGANATHA
Committee Co-Chair:
KOPELEVICH,DMITRY I
Committee Members:
CURTIS,JENNIFER S
FAN,ZHONGHUI HUGH
UENO,ICHIRO
Graduation Date:
8/9/2014

Subjects

Subjects / Keywords:
Amplitude ( jstor )
Curvature ( jstor )
Flow structures ( jstor )
Fluids ( jstor )
Heat transfer ( jstor )
Liquids ( jstor )
Mathematical variables ( jstor )
Temperature gradients ( jstor )
Velocity ( jstor )
Wavelengths ( jstor )
Chemical Engineering -- Dissertations, Academic -- UF
hydrothermal -- off-center -- recirculation -- thermocapillary -- wavy
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Chemical Engineering thesis, Ph.D.

Notes

Abstract:
The work presented here is concerned with flow patterns and how they arise from, or interact with wavy surfaces. Three case studies are analyzed with the aim of understanding the physics behind the flow phenomena involved. These include shear induced flow over a wavy boundary, thermocapillary flow in a rectangular film with two free surfaces, and interface stability in a two-phase liquid jet contained within an annulus. In the first case study, the flow driven between two plates, one wavy and the other flat is considered. Flow is induced by inclining the arrangement with respect to gravity and also by moving the flat wall. It is found that when the wave amplitude is around a third of the gap, recirculation is obtained if the inclination is zero. If flow is allowed only by gravity then no recirculation is possible. This peculiar result is due to the flow adjusting itself to meet the back pressure caused by the wall undulation. A discussion on perturbation methods in deformed domains follows. The best way to obtain solutions at interior points using these methods is presented. The second case study considers thermocapillary flow. Experiments were performed for a liquid film suspended between four solid side walls, leaving its top and bottom surface free. A temperature gradient is then applied between two opposite walls by holding one at a constant high temperature and the other at a constant low temperature. When the temperature gradient is small there will be one of two stable flow structures present. An explanation, based on heat transfer and simple scaling arguments, shows why different flow structures exist and reveals the underlying physics in the flow structure selection. It also shows that only one of two of these flow structures can lead to secondary states which are called hydrothermal waves. Finally, linear stability theory is applied to an inviscid two-phase liquid jet which is contained within an annulus having solid inner and outer boundaries. The dynamics show and explain the variation of the maximum growth rate relative to the position of the inner boundary. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: NARAYANAN,RANGANATHA.
Local:
Co-adviser: KOPELEVICH,DMITRY I.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31
Statement of Responsibility:
by Brad M Messmer.

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UFRGP
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Applicable rights reserved.
Embargo Date:
8/31/2015
Classification:
LD1780 2014 ( lcc )

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THEEFFECTSOFWAVYBOUNDARIESONFLOWSTRUCTURESANDFLUIDINSTABILITIESByBRADM.MESSMERADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014BradM.Messmer 2

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Idedicatethistomyfamilyandfriends. 3

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ACKNOWLEDGMENTS Thankstomyfamilyandfriendsforallthewonderfulexperiencestheyhavesharedwithmeovertheyears.Ioweallofyou.Iamtrulygratefultomyadvisors:RangaNarayanan,IchiroUeno,andSatoshiMatsumotoforallofthegreatopportunitiesthattheyhavegivenme.Thankstomyfundingsourcesformakingmygraduateeducationpossible;NSF0968313,NSF1402151,NASANNX11AC16G. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTIONANDROADMAP ......................... 13 2CIRCULATIONCELLSINFLOWPASTAPERIODICWALL ........... 17 2.1FlowNearWavyBoundaries ......................... 17 2.2TheWavyWallProblem ............................ 19 2.3LongWavelengthModel ............................ 20 2.4StokesandNavier-StokesModels ...................... 25 2.5PredictionsofRecirculation .......................... 39 2.6FinalRemarks ................................. 46 3PERTURBATIONEXPANSIONMETHODSANDTHEIRACCURACY ..... 48 3.1MappingofVariablesNearaPerturbationSource .............. 48 3.2ExpansionMethods .............................. 51 3.2.1Notation ................................. 51 3.2.2TraditionalPerturbationExpansion .................. 52 3.2.3BoundaryExpansionMethod ..................... 52 3.3ComparisonofExpansionMethods ...................... 54 3.3.1RecirculatingHalos-AnExamplebyWayoftheLid-DrivenWavyBoundarySystem ............................ 56 3.3.2ExpansionsonanEllipticalDomain .................. 58 3.3.2.1Exactsolution ........................ 58 3.3.2.2Perturbationsolution:traditionalandboundaryexpansiontechniques .......................... 59 3.3.2.3Comparisonofexpansionmethodstoanexactsolution . 61 3.4FinalRemarksonExpansionMethods .................... 62 4THERMOCAPILLARYFLOWINADOUBLEFREESURFACEFILMATLOWIMPOSEDTEMPERATUREGRADIENT ...................... 64 4.1IntroductiontoThermocapillaryFlow ..................... 64 4.2LowImposedTemperatureGradientFlowStructures ............ 66 4.3TheExperiment ................................ 68 4.3.1ExperimentalTechnique ........................ 68 5

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4.3.2ObservationsFromtheExperiment .................. 71 4.4PhysicalExplanationofFlowStructureSelection .............. 71 4.4.1TheScalingLaw ............................ 71 4.4.2FlowStructureSelection ........................ 72 4.5ConrmationofResultsbyaRestrictedModel ............... 76 5THERMOCAPILLARYINSTABILITYINADOUBLEFREESURFACEFILMATHIGHTEMPERATUREGRADIENT ....................... 84 5.1ExperimentalTechnique ............................ 84 5.2PhysicsoftheInstability ............................ 85 5.3TheCriticalPointandFilmGeometry ..................... 88 5.4InstabilityCulmination ............................. 92 6STABILITYOFALIQUIDTHREADINANANNULUS .............. 93 6.1On-CenterInterfaceStability ......................... 96 6.2TwoPhaseAnnularLiquidJetModel ..................... 100 6.3Off-CenterInterfaceStability ......................... 106 7CONCLUDINGREMARKS ............................. 109 7.1TheLid-DrivenWavyBoundary ........................ 109 7.2ThermocapillaryFlowinaDoubleFreeSurfaceFilm ............ 110 7.3StabilityofaLiquidThreadinanOff-CenterAnnulus ............ 111 7.4PerturbationExpansionsonanInconvenientDomain ............ 112 APPENDIX ANAVIER-STOKESEQUATIONSFORLID-DRIVENWAVYBOUNDARYSYSTEM 114 BDERIVATIVESONAPERTURBEDDOMAIN ................... 119 CCOMPARINGCURVEDSURFACEHEATTRANSFERTOHEATTRANSFERFROMAFLATSURFACE .............................. 121 REFERENCES ....................................... 126 BIOGRAPHICALSKETCH ................................ 128 6

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LISTOFTABLES Table page 2-1Comparisonofvaluesfoundbyanalyticconstantcoefcientsolutionversusspectralconstantcoefcientsolution.Erroristhedifferencebetweenthetwomodelsdividedbytheaverageofthetwomodels.ValuesshownontheleftareforU=1,G=0,R=10,k=2,A=0.01.OntherightU=1,G=0,R=10,k=2,A=0.33. .................................. 37 2-2DependencyofcriticalamplitudeonkwhenR=0andG=0. ........... 42 2-3DependencyofcriticalamplitudeonRforlargervaluesofk,G=0. ....... 43 2-4Asymmetry()andnon-dimensionallength(Lcell)forvaryingRwithG=0,U=1,k=2,=0.25 ............................... 45 2-5Asymmetry()andnon-dimensionallength(Lcell)forvaryingRwithG 2U=2,k=2,=0.25 ................................... 45 3-1Comparisonofanexactsolutionandadomainvariablefoundbytraditionalperturbationexpansionandbytheboundaryexpansiontechnique ....... 63 4-1PhysicalPropertiesof6cstSiliconeoil ....................... 69 5-1Valuesforcriticaltemperaturedifferencewherehydrothermalwaveswereobservedinvariouslms .................................... 91 7

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LISTOFFIGURES Figure page 2-1Diagramofthesystem. ............................... 19 2-2Asketchcastingdoubtonthescaling. ....................... 24 2-3Predictionofrecirculationcells,asafunctionoflidspeedandgravity,bylongwavelengthmodelandStokescalculation. .................... 41 2-4vxasafunctionofzforlargegravitationalterms. ................. 44 2-5FlowrateversusamplitudeforvariousvaluesofG 2U. ................ 47 2-6Denitionofparametersusedtoquantifytheasymmetryofacirculationcell. . 47 3-1Variouspointsinthecurrentdomainandtheircorrespondinglocationwhensuperimposedontheancestordomain.Themultiplelocationsmustbetreateddifferentlytoobtainaccurateresults. ........................ 49 3-2Systemshowinganarbitraryperturbation. .................... 51 3-3Falsecirculationcellsshowinghalosaroundapeakoftheperturbedboundary. ............................................. 57 3-4Threedifferentlocationsforcomparingvariablesfoundbytraditionalexpansionandboundaryexpansionmethods ......................... 58 3-5Ellipsefromaperturbedcircle. ........................... 59 3-6Locationneartheboundary,withpointsonboththecurrentandreferencedomains,whichwillbeusedtocomparesolutionmethods ........... 61 4-1Flowattwointerfacesduetoatemperaturegradientcausingachangeinsurfacetensionalongthesurface. .............................. 64 4-2Exampleofopen-boatcrystalgrowthobtainedfromBagdasarovCrystalsGroupwebsite[ 1 ]. ...................................... 65 4-3Caricatureofthetwoowstatesseenatlowtemperaturegradients. ...... 67 4-4Idealsystemgeometry ............................... 68 4-5Photoofatestsection ................................ 68 4-6Sheet(left)andCellular(right)owstructuresforlowtemperaturethermocapillaryeffect ......................................... 71 4-7Interfacedeformationalongcenterofthelm. .................. 74 8

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4-8Experimentalimagesforadepthof0.6mm.IntegratedCCDimageontheleftandIRimageontheright.Theimposedtemperaturedifferencewas4.1Kwiththeleftwallheated. ................................. 75 4-9Experimentalimagesforadepthof0.2mm.IntegratedCCDimageontheleftandIRimageontheright.Theimposedtemperaturedifferencewas4.1Kwiththeleftwallheated. ................................. 75 4-10Experimentalimagesforadepthof0.6mm,Ly=18mmandLx=4mm,leftimage,andLx=2mmrightimage.Theimposedtemperaturedifferencewasabout4Kwiththeleftwallheated. ......................... 76 4-11TheregularizationfunctionforvariousintegersP. ................ 80 4-12ComputationalimagesofpathlinesforLxof2mm,Lyof4mmanddepthsof0.6mmand0.2mmassumingatemperaturedifferenceof4.1Kusinghighdegreeregularization.Thegray-scaleindicatesthetransitioninisothermswiththeleftwallheated.Thetemperaturescaleisnormalizedwithrespecttothetotaltemperaturedifference.RegularizationpolynomialofP=9isused. ...... 81 4-13Computationalimagesforadepthof0.6mm,Ly=18mmandLx=4mm,topimage,andLx=2mm,bottomimage.Theimposedtemperaturedifferencewasabout4Kwiththeleftwallheated.Thetemperaturescaleisnormalizedwithrespecttothetotaltemperaturedifference.RegularizationpolynomialofP=9isused. .................................... 82 4-14ComputationalimagesforLxof2mm,Lyof4mmanddepth0.2mmandLxof4mm,Lyof18mmanddepth0.6mm.Thegray-scaleindicatesthetransitioninisothermswiththeleftwallheated.RegularizationpolynomialofP=18isused. ......................................... 83 5-1Heatowthroughrealandidealtestsections ................... 85 5-2Instabilitymechanismforthecaseofsheetow ................. 86 5-3Instabilitymechanismforthecaseofcellularow ................ 87 5-4Depictionoftheowalongthecoldwallasthelmdepthischanged.Theuidbecomescolderfordeeperlms. ....................... 89 5-5Depictionofvariouswavelengthdisturbancesasthespanwiselength,Ly,ischanged.LongerwavelengthscanenterintothelargerLylms. ........ 90 5-6DepictionofthemagnitudeoftheinterfacialowwhenLxischangedwhileholdingthetemperaturedifferenceconstant. ................... 91 6-1Pictureofacylindricalthreadofhoneyatthreedifferenttimesonitswaytobreakup.Theearliestpictureisontheleftandthelatestontheright.PictureisfromJohnsandNarayanan[ 2 ]. .......................... 94 9

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6-2Depictionoftransversecurvatureinaperturbedjet.Theboldarrowsindicatetheshapeofthecurvaturebeingconsidered.Thethinarrowsrepresentthedirectionofow.Flowwillgofromtroughtopeak. ................ 95 6-3Depictionoflongitudinalcurvatureinaperturbedjet.Theboldarrowindicatestheshapeofthecurvaturebeingconsidered.Thethinarrowsrepresentthedirectionofow.Flowwillgofrompeaktotrough. ................ 95 6-4Thesetupinvestigatedtodeterminestabilitycharacteristics.Thedashedlineontherightgurerepresentstheinitial(oncenter)positionoftheboundary.Thecenterrodandouterboundaryarebothsolidsurfaces. ........... 96 6-5Diagramofthereferencedomainofatwouidjetwithinanannulus. ..... 96 6-6On-centergrowthratesforinviscidandvaryingviscosities.=0.25m,Rinterface=1m,Rout=1.5m,=1500kg m3,=1000kg m3,=0.1N m .............. 97 6-7Changeinthemaximumgrowthrateforinvisciduidsastheradiusratioincreases.=1500kg m3,=1000kg m3,=0.1N m ........................ 98 6-8Changeinthemaximumgrowthrateforviscousuidsastheradiusratioincreases.=1Pas,=1Pas,=1500kg m3,=1000kg m3,=0.1N m .......... 99 6-9Thelowestordercorrectiontothegrowthratewhenthesystemisshiftedoff-center.Parametersusedare=0.25m,Rinterface=1m,Rout=1.5m,=1500kg m3,=1000kg m3,=0.1N m. ............................... 107 7-1Variouspointsinthecurrentdomainandtheircorrespondinglocationwhensuperimposedontheancestordomain.Themultiplelocationsmustbetreateddifferentlytoobtainaccurateresults. ........................ 113 B-1Systemshowinganarbitraryperturbationwhichgivesageneralcurrentdomain. ............................................. 119 C-1Diagramofthesystem.Asurfacecanhaveconcaveorconvexcurvaturedependingonthechoiceof. .................................. 121 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTHEEFFECTSOFWAVYBOUNDARIESONFLOWSTRUCTURESANDFLUIDINSTABILITIESByBradM.MessmerAugust2014Chair:RangaNarayananMajor:ChemicalEngineeringTheworkpresentedhereisconcernedwithowpatternsandhowtheyarisefrom,orinteractwithwavysurfaces.Threecasestudiesareanalyzedwiththeaimofunderstandingthephysicsbehindtheowphenomenainvolved.Theseincludeshearinducedowoverawavyboundary,thermocapillaryowinarectangularlmwithtwofreesurfaces,andinterfacestabilityinatwo-phaseliquidjetcontainedwithinanannulus.Intherstcasestudy,theowdrivenbetweentwoplates,onewavyandtheotheratisconsidered.Flowisinducedbyincliningthearrangementwithrespecttogravityandalsobymovingtheatwall.Itisfoundthatwhenthewaveamplitudeisaroundathirdofthegap,recirculationisobtainediftheinclinationiszero.Ifowisallowedonlybygravitythennorecirculationispossible.Thispeculiarresultisduetotheowadjustingitselftomeetthebackpressurecausedbythewallundulation.Adiscussiononperturbationmethodsindeformeddomainsfollows.Thebestwaytoobtainsolutionsatinteriorpointsusingthesemethodsispresented.Thesecondcasestudyconsidersthermocapillaryow.Experimentswereperformedforaliquidlmsuspendedbetweenfoursolidsidewalls,leavingitstopandbottomsurfacefree.Atemperaturegradientisthenappliedbetweentwooppositewallsbyholdingoneataconstanthightemperatureandtheotherataconstantlowtemperature.Whenthetemperaturegradientissmalltherewillbeoneoftwostable 11

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owstructurespresent.Anexplanation,basedonheattransferandsimplescalingarguments,showswhydifferentowstructuresexistandrevealstheunderlyingphysicsintheowstructureselection.Italsoshowsthatonlyoneoftwooftheseowstructurescanleadtosecondarystateswhicharecalledhydrothermalwaves.Finally,linearstabilitytheoryisappliedtoaninviscidtwo-phaseliquidjetwhichiscontainedwithinanannulushavingsolidinnerandouterboundaries.Thedynamicsshowandexplainthevariationofthemaximumgrowthraterelativetothepositionoftheinnerboundary. 12

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CHAPTER1INTRODUCTIONANDROADMAPThisthesisdescribestheinvestigationofuidowinthepresenceofwavyboundaries.Ofspecicinterestistheinteractionofwavyboundarieswiththeoweldscausedbyuidmotionadjacenttothoseboundaries.Nowwavyboundariesincontactwithmovinguidsareseeninmanycommonaswellasindustrialsituations.Commonexamplesofwavysurfacesareuidinteractionintheuidowbelowseawaves,estuariesandthelike.Needlesstosayalargebodyofliteratureisavailableonthissubject,mostlyintheeldofwaterwaves[ 3 ][ 4 ].Asaspecicexampleofindustrialapplications,aregroovedwallheatexchangerswherethewavinessorgroovesinthewallshelpenhanceheattransfer.Anotherexampleisthecleaningofasubstrateviachemicaletching.Herealiquid(etchant),achemicalsolution,owsoverasurfaceand,atthesametime,dissolvesthatsurface.Someregionswilldissolvefasterorslowerthanothersandabumpysurfacecanresult.Whenthishappens,theetchingcanbecontrolledbymanipulatingtheamountofcirculationinthechemicalsolution.Thiswillallowforbumpsonthesurfacetogroworcansmooththesurfaceasthechemicalsolutionbecomessaturatedinregionsofcirculation.Asecondindustrialexampleisthatofoatzonecrystalgrowthwhereapolymorphiccrystalistranslatedbetweenradialheatersgeneratingameltzone.Thecrystalre-solidiesaftertheheaterpassesbyandeventuallymostoftheimpuritiesareattheendofthecrystalwhichisthelastparttore-solidify.Howeverinthepresenceoftheheaters,atemperaturegradientissetupalongthesurfaceofthemeltandtheambient,generatingsurfacetensiongradientsandhencemotionaswellasadeformedorwavyinterface.Thewavinesscontrolsthenatureoftheowandthestabilityoftheowstructureforlargetemperaturegradientsbeyondacriticalvalue.Inotherwordswhentheinterfaceiswavy,itwillinuencetheamountofheattransferalongthesurface 13

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ofthelm.Understandinghowthesetemperatureuctuationscanbecontrolledallowsformoreconsistentcrystalgrowth.YetathirdapplicationinwhichwavyboundariesarespontaneouslyformedisintheJouleheatingofsheathedwires.Hereawireundergoingelectricalheatingonaccountofitsresistance,mayheatandmeltaninsulation.Firstwavesandthenbeadeddropsofinsulantmeltbegintoform.Thewaveformationisaresultofcompetitionbetweenthetransverseandlongitudinalcurvaturesofthemeltsurroundingacylindricalsolidwire.Thecompetitioninitiallyleadstounequalpressuregradientsthat,inturn,causeinertialow.Suchowcanbeslowedbyviscosity.Ourinterestintheproblemofwavyboundary-uidinteractionisconnedtounderstandingtheroleofasecondaryboundaryontheowwhentherstboundaryiswavy,thewavinessoccurringeitherbydesignorbyvirtueoftheboundarysveryinteractionwiththeuid.Tounderstandtheroleofasecondaryboundary,threecasestudiesareinvestigatedandineachcasesurprisingresultsareobtaineddemandingexplanationsthataredelivered.Therstproblemthatisinvestigatedistheowstructureinthepresenceofawavywallwhenasecondatwallissetintomotionwithandwithoutagravityeldbeingpresent.Ofinterestaretheconditions,ifany,underwhichrecirculationmayoccur.Acommontechniquewhichisusedtounderstandsuchproblemsisvia“perturbationtheory”.Therearemultiplewaystohandleperturbationtheoryanditiscommonlythoughtthateachofthesemultiplewaysareequal,oratleastthattherearenomajordifferencesinthemethods.Eachmethodsolvesforvariablesintermsofaninniteseriesanditisfoundthatwhentheseseriesaretruncated,thencertainmethodswillgivebetterresults.Inthepresentwork,thesevariousmethodsarecomparedandanunderstandingoftheirdifferencesispresented. 14

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Thesecondproblemisalsoconnectedtoadrivenowbutnowthedrivingmechanismisheat.Atemperaturegradientatthesurfacegeneratesasurfacetensiongradientandthisinturncausesow.Nowbecausetemperaturegradientsmustbeniteandhotandcoldendboundariesarenecessarytogeneratethegradientitfollowsthatthefreesurfacemustinstantlybecomewavyonceowensues.Thedepthofthewavesdependsonthesurfacetensionandthegeometricaldimensionsaswellasthestrengthofthedrivingforce.Themagnitudeofthewaveheightsaffecttheverytemperaturegradientsthatcausedthemandtherebyaffecttheowstructures.Ourprimaryinteresthereisindeterminingwhenandwhysuchstructuresareobtained.Thereisalsoasecondaryinterestandithastodowiththeexperimentalstabilityofthesestructuresforlargeimposedtemperaturedifferences.Thereforethisworklooksatcriticalpointswhenowstructureschangewithdrivingforces.Ifevermoretemperatureforcingisaddedthenatsomecriticaltemperaturedifferencethelm'ssurfacetemperaturewillbegintouctuate.Belowthiscriticaltemperaturethelmtemperatureisconstantateachlocation.Thethirdandnalproblemwhichisinvestigatedisthestabilityofaliquidthreadsurroundingawall.Theimportantdimensionhereisthelengthofthethreadand,forlongthreads,theuidinterfacewillnaturallybreakdown.Thisarrangementisunstablebecausetheliquidthreadinterfacehassmallperturbationscreatingripples,andthiswavysurfacecausespressuredifferencestoarise,duetoacompetitionbetweentransverseandlongitudinalcurvatures.Thepressuredifferencecausesthesurfacewavestogrowandthethreadwillbreakapart.Asstudiedhere,thisproblemhasusesinowthroughoilpipelinesandinJouleheatingofwires.Eachoftheseproblemscontainssomeunexpectedresults.Inthelid-drivenwavyboundaryproblem,circulationcellswillnotformwhendrivenbygravityalone,butrecirculationwilloccuriftheuidisdrivenbyshear. 15

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Forthermocapillaryowinarectangularlm,theuidistypicallyfoundtomoveuni-directionallyacrosstheinterfaceandthendipintothelmandreturnalongthemid-plane.Bychangingthesizeofthesystemadifferentowstructureisobservedandanexplanationispresentedforthemechanismofowselection.Liquidjetsarenaturallyunstable.Byshiftingathreadofuidoff-center,withrespecttoasolidinnercoreandasolidouterboundary,thejetinterfacecanbemademorestableandthebreakupwavelengthcanbeadjusted.Thesetopicsarediscussedinthefollowingchapters.Theoutlineisasfollows.Thersttwochaptersdealwiththedrivenowpastawavywall,followedbyachapterondomainperturbations.Thisisthenfollowedbytwochaptersonowdrivenbytemperaturegradientsinwhichawavysurfaceemerges.Followingtheseisachapteronthedynamicsofathreadofliquid.Ineachcasethepasthistoryisdiscussedonlyinthecontextoftheresearchandthroughthechapterratherthanatthebeginningofthechapter.Analchapteronconclusionstobedrawnispresented. 16

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CHAPTER2CIRCULATIONCELLSINFLOWPASTAPERIODICWALL 2.1FlowNearWavyBoundariesTounderstandhowandwhyrecirculationcellsaregeneratedinthepresenceofwavywalls,werstmakeapredictionofrecirculationcellsinasheardrivenwavyboundarysystemundersomeapproximations.Alongwavelengthmodelisusedwhichimposescertainconditionssimilartothoseofalubricationapproximation.Atwo-dimensionallubricationapproximationusesascalingargumentthatleadstoseveralcharacteristicsoftheowviz.,(1)Pressureisonlyafunctionofthedirectionofbulkuidmovement;(2)Momentumdiffusionismuchlargerinthedirectionofthegapheightascomparedtomomentumdiffusioninthedirectionofbulkmotionand(3)theReynoldsnumbermustbesmall.Thesesameconditionsholdinalongwavelengthmodelaswell.ThelongwavelengthmodelmaybecontrastedwithaStokesmodelwhichisvalidforallwavelengths,albeitsmallperturbationamplitudes.Weusethistoverifythelongwavelengthmodelinthephasespacewherebotharevalid.Finally,aNavier-Stokesmodelisderivedinordertoinvestigatetheeffectsofinertiaontherecirculationcells.Tosetourprobleminthecontextofearlierwork,wereviewthereleventliterature.Flowoverawavyboundaryhasabroadclassofapplicationsrangingfrombloodowthroughdiseasedarteries[ 5 ]tomachineryinvolvinglubrication.Understandingtheformationofrecirculationcellsisimportantinandofitsselfascirculatinguideddiesarefoundtoenhancemassandheattransferandalsooccurinindustrialrollcoating[ 6 ].Decadesago,Wang[ 7 ]providedsomeoftherstworkonthissystemwhenhepresentsanamplitudeperturbationoftheStokesequationsallowinghimtoconcludethatthewavyboundaryincreasesdrag,ascomparedtotheatboundarycase.SomeoftherstexperimentalworkwasperformedbyMunsonetal.in1985whentheyinvestigatedtheformationandsizeofrecirculationcellsincylindricalCouetteow[ 8 ].PozrikidisfoundgoodagreementbetweentheexperimentsofMunsonetal.anda 17

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numericalsolutionforcreepingowinatwo-dimensionalchannel[ 9 ].TheadvantageofthisnumericalsolutionofStokesequationsisthattheresultsarevalidforarbitraryamplitudesanddisturbancesarenotrestrictedbythesmallamplitudeassumptionofperturbationtheories.ThisinuentialworkbyPozrikidisdescribestheonsetconditionsforrecirculationwhenvariousvaluesofamplitudeorwallwavinessareconsidered.Hisworkalsoanalyzescirculationcellsizeforspecicvaluesofamplitudeandwavinessandtheeddyheightispredictedtobezerowhenthedisturbanceamplitudeislessthan30.1%oftheaveragechannelheight,i.e.thereisnorecirculation.AtlargeramplitudesPozrikidisndsthatthecirculationcellscanactuallygrowtoprotrudeabovethewavywallamplitudeandnallytheeddysizeapproaches100%ofthechannelheightwhenthedisturbanceamplitudebecomesnearlyequaltotheaveragechannelheight.Inmorerecentwork,someparametersarederivedtorelatetheowratesordragfoundforthewavywallsystemtocomparableheightsintheplane-boundary,undisturbedsystem[ 10 ].Morecomplicatedsystemshavealsobeenanalyzed,suchasthecaseofhavingtwoimmiscibleuids,oneovertheother,inbetweentheplanemovingwallandthewavyboundary[ 11 ].Theredoesnotappeartobeanyreportsintheliteratureforexperimentalobservationsofcreepingowoverawavyboundarydrivenbylidmovementastherearesomedifcultiesinmanufacturingthissetup.Numerousexperimentsfortheboundedliddrivencavityarereported,aswellasexperimentswhenbothsurfacesarewavy[ 12 ].Liddrivenowinamicrochannelisalsoofinteresttothescienticcommunitywheremixingcanbedifculttoachieveandtherecirculationcellspredictedinthisworkcanbeofassistance.ItwasfoundbyXiaetal.thatelectro-osmoticowalonecannotcauserecirculationtooccurinasinusoidallyvaryingmicro-channel[ 13 ].Onlybyapplyingapressuregradientoppositethedirectionoftheelectro-osmoticpotentialcanrecirculationoccur.Experimentsinvestigatingrecirculationinasinusoidallyvaryingchannel,withbothboundarieswavy,wereperformedbybothLeneweitetalandStephanoffetal[ 14 ][ 15 ]. 18

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Stephanoffobservedthat,foragivenchannelgeometry,circulationcellswouldappearandgrowwhentheuidacceleratedordeceleratedaslongasthepeakReynoldsNumberwasbeyondacriticalvalue,wherethecriticalvalueistheReynoldsNumberneededtocausecirculationforsteadyow.Leneweitlookedintohowthelocationoftheincipientrecirculationcellchangesforvariousamplitudes,wavelengthsandReynoldsNumbers.TheeffectsofinertiaonthissystemhavealsobeenstudiedextensivelyviatheNavier-StokesequationswhentheReynoldsNumberistakentobenon-zero.Zhouetal.studiedPouseuilleowforchannelsofsinusoidal,arced,andtriangulargeometries,withthesecondwallbeingat[ 16 ]. 2.2TheWavyWallProblemFigure 2-1 presentsasketchofthecurrentproblemlabeledwithdimensionlessvalues.Theowofaviscousuidtotherightiscausedbythemotionofthewallatz=0,movingatspeedU.Thewallatz=1+Acos(2x)isnotmoving.Theowissteadyandtwo-dimensional. Figure2-1. Diagramofthesystem. TheaimistondthevalueofAatwhichtheowdirectionisjustatthepointofreversing.AcirculationcellmightbeareasonableexpectationasAincreases,holdingUconstant,duetotheanticipatedpressureincreaseatthepointwheretheowis 19

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squeezed.Agravitationalforceisalsoincluded,actingtotheright,andanunexpectedresultisobtained.Theproblemissolvedusingseveralmodels.First,alongwavelengthmodelisusedinordertodeterminethecriticalvalueofAatwhichowreversaloccursasafunctionoftheinputsGandU.Next,aStokesmodelissolvedusingperturbationtechniquesaroundsmallvaluesoftheamplitude.Thismodelisusedfordualpurposes.TheStokesmodelisusedasacheck,when2H !0,forthelongwavelengthmodelandalsotondasimplepredictionoftheonsetofcirculationcellsatsmallwavelengths,i.e.,as2H !1.Lastly,aNavier-Stokesmodelisusedtoinvestigatetheeffectsofinertiaonthesystemforanyvalueofthewavelength. 2.3LongWavelengthModelThemainassumptionofthismodelisthatthedisturbancewavelengthmustbeverylong,oratleastmuchlongerthantheheightofthesystem.AreviewandexplanationofthisapproximationispresentedbyOronetal[ 17 ].Forsteadystateowofaviscousuidthedomainequationsare x)]TJ /F3 11.955 Tf 11.96 0 Td[(component:vx@vx @x+vz@vx @z=)]TJ /F6 11.955 Tf 10.49 8.08 Td[(@P @x+g+r2vx(2) z)]TJ /F3 11.955 Tf 11.95 0 Td[(component:vx@vz @x+vz@vz @z=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(@P @z+r2vz(2)andcontinuity,r~v=0.Herethesuperscriptrepresentsthatthesearedimensionalvariables.Now,thehorizontalvelocityisscaledbyV,acharacteristicvelocity,verticalheightsarescaledbytheaveragegapheight,H,andhorizontaldistancesscaledbythedisturbancewavelength,.Finally,afterobservingthatforverylongwavelengthdisturbances,vxshouldbemuchlargerthanvzduetothesourcesofmomentumactingonlyinthex-direction,thenvzisscaledbyH V.Scalingtheequationsgives x)]TJ /F3 11.955 Tf 11.52 0 Td[(component:V2 vx@vx @x+HV2 Hvz@vx @z=)]TJ 10.49 8.09 Td[(P @P @x+g+V H2(@2 @z2+H2 2@2 @x2)vx(2) z)]TJ /F3 11.955 Tf 11.95 0 Td[(component:HV2 2vx@vz @x+H2V2 2Hvz@vz @z=)]TJ 10.54 8.09 Td[(P H@P @x+HV H2(@2 @z2+H2 2@2 @x2)vz(2) 20

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and,r~v=0.Nowmultiplyingthex-componentequationbyH2 Vandthez-componentequationbyH V: x)]TJ /F3 11.955 Tf 9.36 0 Td[(component:H2V vx@vx @x+H2V vz@vx @z=)]TJ /F3 11.955 Tf 10.61 8.09 Td[(H2P V@P @x+gH2 V+(@2 @z2+H2 2@2 @x2)vx(2) z)]TJ /F3 11.955 Tf 11.96 0 Td[(component:H2V vx@vz @x+H2V vz@vz @z=)]TJ /F6 11.955 Tf 10.96 8.09 Td[(P V@P @x+(@2 @z2+H2 2@2 @x2)vz(2)Finally,sinceH issmallinthelongwavelengthmodel,theequationsaregroupedinordersofH ,givingazeroorderequationoftheform @P @x=@2vx @z2+G(2)whichistheequationtobeusedforndingvx.@P @xisretainedbecausethehorizontalpressuregradientisexpectedtobelargewherethegapclearanceissmall.Also,@P @xisindependentofzbecausethewavelengthsarelargeandthereforetheslopeoftheboundaryvariesgraduallyinthex-direction,beingnearlyat.Now,thelidismovingwithaconstantspeedu,orindimensionlessterms,U.Theconditionsatboundariesare,vxequaltozeroatz=1+Acos(2x)andvxequaltoUatz=0.SolvingEquation 2 gives vx=1 2(@P @x)]TJ /F3 11.955 Tf 11.95 0 Td[(G)z2+1z+2(2)where1and2areconstantsofintegration.Applyingno-slipboundaryconditionsresultsin vx=1 2(@P @x)]TJ /F3 11.955 Tf 11.96 0 Td[(G)(z2)]TJ /F3 11.955 Tf 11.96 0 Td[(zh)+U(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(z h)(2)Theverticlecomponentofvelocity,vzisobtainedfromcontinuityby,vz=)]TJ /F12 11.955 Tf 11.29 9.63 Td[(R@vx @xdzwhichgives vz=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2@2P @x2(z3 3)]TJ /F3 11.955 Tf 13.15 8.09 Td[(z2h 2)+1 4(@P @x)]TJ /F3 11.955 Tf 11.96 0 Td[(G)z2@h @x)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2Uz2 h2@h @x(2)Now,allofthetermsontheright-handsideofEquation 2 and 2 arenotyetknown.Extraconstraintsofthesystemareneededinordertondthenecessaryterms 21

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andsothevolumetricowrateiswrittenas Q=h(x)Z0vxdz=1 12()]TJ /F6 11.955 Tf 10.49 8.09 Td[(@P @x+G)h3+1 2hU(2)Sincetheowratemustbeaconstantatanypositioninx,theconditionof@Q @x=0mustholdandEquation 2 gives @Q @x=@ @x()]TJ /F6 11.955 Tf 10.5 8.09 Td[(@P @x+G)+3 h@h @x()]TJ /F6 11.955 Tf 10.5 8.09 Td[(@P @x+G)+6 h3@h @xU=0(2)whereEquation 2 canbesolvedfor@P @x.Todothis,Equation 2 isrstmultipliedbyanintegratingfactor,f(x),and()]TJ /F10 7.97 Tf 10.5 4.71 Td[(@P @x+G)isreplacedby(x). f(x)d dx+3 hdh dxf(x)(x)+6 h3dh dxUf(x)=0(2)Sincewehavetherelation,d dx[f(x)(x)]=fd dx+df dx,thenitisseenthatthesecondterminEquation 2 mustbeequaltodf dxwhichresultsin df dx=3 hdh dxf(2)Rearrangingthisgives, df f=3 hdh(2)anduponintegratingthisequationanexpressionforfisobtained:ln(f)=3ln(h)=ln(h3).Herethearbitraryintegrationconstanthasbeensettozeroforthesimplestvalidfunction.Theequationforfcanbesimpliedto f(x)=h3(2)NowrewritingEquation 2 d dx[f(x)(x)]=)]TJ /F4 11.955 Tf 9.3 0 Td[(6Udh dx(2) 22

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andafterintegratingfromx=0toanarbitrarydownstreamposition,x f(x)(x)=f(x=0)(x=0))]TJ /F4 11.955 Tf 11.95 0 Td[(6U[h(x))]TJ /F3 11.955 Tf 11.95 0 Td[(h(x=0)](2)Finally,afterdividingbyfandrewritingas()]TJ /F5 7.97 Tf 10.5 4.71 Td[(dP dx+G),Equation 2 gives )]TJ /F6 11.955 Tf 13.15 8.08 Td[(@P @x+G=[)]TJ /F6 11.955 Tf 10.5 8.08 Td[(@P @x(x=0)+G]h3(x=0) h3)]TJ /F4 11.955 Tf 13.15 8.08 Td[(6U h3[h)]TJ /F3 11.955 Tf 11.96 0 Td[(h(x=0)](2)NowPisperiodicinxhavingaperiod1,hencewehaveR10dx=0andintegratingEquation 2 overaperiod,weobtain [)]TJ /F6 11.955 Tf 10.49 8.09 Td[(@P @x(x=0)+G]h3(x=0)=G+6UR10h)]TJ /F5 7.97 Tf 6.59 0 Td[(h(x=0) h3dx R101 h3dx(2)AftersolvingEquation 2 for)]TJ /F10 7.97 Tf 10.49 4.7 Td[(@P @x(x=0)+GandsubstitutingtheresultingexpressionintoEquation 2 ,then)]TJ /F5 7.97 Tf 10.49 4.71 Td[(dP dx+Gisfoundtobe )]TJ /F6 11.955 Tf 13.15 8.09 Td[(@P @x+G=1 h3G+6UR101 h2dx R101 h3dx)]TJ /F4 11.955 Tf 13.15 8.09 Td[(6U h2(2)meaningthatvxandQarenowknownvia vx=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2()]TJ /F6 11.955 Tf 10.49 8.09 Td[(@P @x+G)(z2)]TJ /F3 11.955 Tf 11.95 0 Td[(zh)+U(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(z h)(2)and Q=1 12()]TJ /F6 11.955 Tf 10.5 8.09 Td[(@P @x+G)h3+1 2hU(2)TondthevalueofAcorrespondingtotheonsetofcirculationcells,weassumethattherstsignofrecirculationoughttoappearatx=0andoughttocorrespondtodvx dzvanishingthere.Thereforewehave, @vx @z(x,z=h)=)]TJ /F4 11.955 Tf 10.5 8.08 Td[(1 2((x)+G)h)]TJ /F3 11.955 Tf 13.15 8.08 Td[(U h(2)andbysettingthistozeroweobtain, G 2U=2hZ101 h3dx)]TJ /F4 11.955 Tf 11.95 0 Td[(3Z101 h2dx(2) 23

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whichallowsthepredictionofthenecessaryamplitudeforrecirculationtooccuratanygivenvaluesofGandU.Thelongwavelengthmodelispeculiarinthesensethatitseemstoparadoxicallypredictphenomenalikecirculationcells.Oneoftheconditionsimposedonthelongwavelengthsystemisthatvzmustbemuchsmallerthanvx.ThisistakenintoaccountbyscalingvzwithH V.Evenundertheseconditions,whichmustnotbetakingregion2ofFigure 2-2 intoaccount,themodelstillcorrectlypredictscriticalamplitudesofrecirculation. Figure2-2. Asketchcastingdoubtonthescaling. Addingtothispeculiarityisthefactthat-eventhoughrecirculationiscorrectlypredicted-themodelcannot,atthesameorder,predicttheeffectsofthisrecirculationonthetransportphenomena.Togivethisstatementcredibility,considerthewavyboundarysystemwithconstanttemperaturesurfacesatz=0andz=h(x).Energyconservation,atsteadystate,givesthedomainequation, vx@T @x+vz@T @z=r2T+Hv Cp(2)Now,scalingTbythecharacteristictemperatureTcandothervariablesbythesamescalesaswereusedearlier,Equation 2 isobtained. VTc vx@T @x+HVTc Hvz@T @z=Tc H2(@2 @z2+H2 2@2 @x2)T(2) 24

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AftermultiplyingbyH2 Tcthisbecomes H2V (vx@T @x+vz@T @z)=(@2 @z2+H2 2@2 @x2)T(2)andtozerothorderinH ,Equation 2 reducesto @2T @z2=0(2)whichissolvedwiththedimensionlessconstraints T=Thotatz=h(x)(2)and T=Tcoldatz=0(2)togiveT=(Thot)]TJ /F5 7.97 Tf 6.59 0 Td[(Tcold)z h(x)+Tcold.Now,theheatuxduetothistemperaturegradientwillbe~q=)]TJ /F3 11.955 Tf 9.3 0 Td[(krT,resultingin~q=)]TJ /F3 11.955 Tf 9.3 0 Td[(k()]TJ /F7 7.97 Tf 10.5 6.83 Td[((Thot)]TJ /F5 7.97 Tf 6.59 0 Td[(Tcold)@h @xz h2^ix+(Thot)]TJ /F5 7.97 Tf 6.59 0 Td[(Tcold) h^iz).Integratingtheheatuxtimesunitnormalvector(givenby~n=^iztozeroorderinH )overaperiodinxandfromzerotoh(x)inzresultsinaheattransfergivenby,Q=)]TJ /F3 11.955 Tf 9.29 0 Td[(k(Thot)]TJ /F3 11.955 Tf 12.19 0 Td[(Tcold),whichisthesameexpressionasfoundforheattransferinthecaseofaatboundary.Therefore,theeffectsofthewavywallonheattransfercannotbeobtainedatthelowestorderofthelongwavelengthmodeleventhoughthemodelcancorrectlypredicttheonsetofcirculation. 2.4StokesandNavier-StokesModelsStokesandNavier-Stokesmodelsmayalsobesolvedforthewavywallproblem.Thewavyboundarywastreatedbyexpandingtheequationsintermsofthedisturbanceamplitude,A,aboutthebasestateofaatsurface.Herethetopsurfaceisdenedas,Z=1+1 2A(^Z1eikx+~Z1e)]TJ /F5 7.97 Tf 6.59 0 Td[(ikx).^Z1and~Z1aresettoavalueof1sothatthetopboundarycanbecomparedwiththecosinedependencyusedinthelongwavelengthmodel.TheStokesmodelcanbesolvedanalyticallyat0th,1st,and2ndordersintermsofafourthorderdifferentialequation.TheNavier-Stokesmodelbecomesmorecomplicated, 25

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generatingvariablecoefcientfourthorderequationsandisthereforesolvedusingaspectralmethod.Obtaininganumericalsolutiongivestheadvantagethatthevelocityeldcanbeconstructedandfromthisthecriticalamplitudecanbedeterminedbylookingdirectlyatthevelocityvectors,ratherthananalyzingviscousstressesasinthelongwavelengthmodel.Thevariablesvxandzhavethesamescalesasthelongwavelengthmodel,butnowvzisscaledbyVandxbytheaveragegapheight,H.FortheStokesmodel,withonlygravityandshearowonthebasedomainandthenexpandinginpowersofA,thedimensionlessdomainandboundaryconditionsare,atzerothorder 0=r2~v0+G^ix(2) r~v0=0(2)subjectto ~v0=U^ixatz=0(2) ~v0=0atz=1(2)Atrstorderwehave 0=rP1+r2~v1(2) r~v1=0(2)subjectto vx1+Z1@vx0 @z=0atz=1(2) 26

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vz1=0atz=1(2) vx1=0atz=0(2) vz1=0atz=0(2)Atsecondorder,wehave 0=rP2+r2~v2(2) r~v2=0(2)subjectto vx2+2Z1@vx1 @z+Z21@2vx0 @z2=0atz=1(2) vz2+2Z1@vz1 @z=0atz=1(2) vx2=0atz=0(2) vz2=0atz=0(2)Thezeroordervelocityisfound,bysimpleintegration,tobe vx0=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(Gz2 2+[1 2G)]TJ /F3 11.955 Tf 11.96 0 Td[(U]z+U(2)Atrstandsecondorderdomainequations,thesystemofequationscanbesolvedintermsofasinglevariable,vz.Thevariable,vz,mustbechosen,ratherthanvx,because 27

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continuityhastobeusedinordertoconvertfromonevelocitycomponenttotheother.Attheboundary,wecannotaskwhatthederivativeiswithrespecttozbecausethiswouldessentiallybeaskingwhatishappeningoutsideofthedomainandthisquestionisirrelevant.Therefore,x-derivativesaretakenattheboundaryinordertoputvxintermsofvz.Takinganx-derivativeofthez-componentofEquation 2 and 2 andaz-derivativeofthex-componentequationsandthensubtractingthetwoateachorderofA,resultsin r2(@vx1 @z)]TJ /F6 11.955 Tf 13.15 8.09 Td[(@vz1 @x)=0(2)and r2(@vx2 @z)]TJ /F6 11.955 Tf 13.15 8.09 Td[(@vz2 @x)=0(2)Now,takingonemorex-derivativeoftheseequationsandusingcontinuity,i.e.,@vx @x=)]TJ /F10 7.97 Tf 10.5 4.7 Td[(@vz @z,theusableformofthedomainequationsareobtainedas 1st:r4vz1=0(2) 2nd:r4vz2=0(2)Next,theboundaryconditionsmustbeputintermsofvzandtodothiscontinuityisonceagainusedaftertakingx-derivativesofEquations 2 , 2 , 2 ,and 2 toget @vz1 @z=Z1x@vx0 @zatz=1(2) @vz1 @z=0atz=0(2) @vz2 @z=2Z1x@vx1 @z+2Z1@2vx1 @x@z+2Z1Z1x@2vx0 @z2atz=1(2) 28

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@vz2 @z=0atz=0(2)Togetthenalformoftheequationsthex-dependencyisrstdeterminedbylookingattheboundaryconditions.Thevariablesarethenpluggedintodomainandboundaryconditionsleavingasetofequationsdependingonlyonz.Thezeroorderproblemisx-independentsincethisisthecaseoftwoplanewalls.Atrstorder,thex-dependencycanbeseenfromEquations 2 and 2 ,as@vz1 @z=[1 2(^Zikeikx)]TJ /F4 11.955 Tf -424.25 -21.25 Td[(~Zike)]TJ /F5 7.97 Tf 6.59 0 Td[(ikx)]@vx0 @z,whichmeansvz1takestheform,vz1=^vz1eikx+~vz1e)]TJ /F5 7.97 Tf 6.59 0 Td[(ikx.Herethe“hat”and“tilde”notationisusedtorepresentcomplexconjugates.Then,bycontinuity,thex-dependencyofvx1is,vx1=^vx1eikx+~vx1e)]TJ /F5 7.97 Tf 6.58 0 Td[(ikx.Bysubstitutingtheseexpressionsforvx1andvz1the1storderequations,intermsofhattedvariables,become (@2 @z2)]TJ /F3 11.955 Tf 11.95 0 Td[(k2)2^vz1=0(2) ik^vx1=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(@^vz1 @z(2) B.C.@^vz1 @z=1 2^Zik@vx0 @zatz=1(2) ^vz1=0atz=1(2) ^vz1=0atz=0(2) @^vz1 @z=0atz=0(2)Likewise,at2ndorder,Equations 2 and 2 showthatthe2ndordervelocitycomponentshaveane2ikx,e)]TJ /F7 7.97 Tf 6.59 0 Td[(2ikx,andx-independentterms.However,ifcontinuityis 29

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considered,itisseenthatthex-independentterminvz2isnotbalancedduetothex-independenttermfromvx2vanishingupondifferentiation.Therefore,thisshouldbecomeapparentafterinspectingtheformoftheequations.Notingthatvx0isentirelyreal,Equation 2 showsthat^vz1mustbeentirelyimaginary.Also,fromEquation 2 itisseenthateverytermontheRHSisentirelyimaginaryandtherefore^vz2isimaginary.Thismeansthatthex-independenttermscomingfromEquation 2 give)]TJ /F4 11.955 Tf 10.7 2.65 Td[(^Z1@~vz1 @z)]TJ /F4 11.955 Tf 14.51 2.65 Td[(~Z1@^vz1 @zandsincethe“hat”and“tilde”termsareconjugatesandentirelyimaginarytheysimplysumtozero.Thisleavesasetofhomogenousequationsforthex-independentportionofvz2meaningthesolutionisthetrivialcase.Theresultofthisisvz2intheformvz2=^vz2e2ikx+~vz2e)]TJ /F7 7.97 Tf 6.59 0 Td[(2ikxandthe2ndorderequationsbecome (@2 @z2)]TJ /F3 11.955 Tf 11.95 0 Td[(k2)2^vz2=0(2) 2ik^vx2=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(@^vz2 @z(2)subjectto @^vz2 @z=ik^Z1@^vx1 @z)]TJ /F4 11.955 Tf 13.36 2.65 Td[(^Z1@2^vz1 @z2+^Z21ik1 2@2vx0 @z2atz=1(2) ^vz2=)]TJ /F4 11.955 Tf 10.7 2.65 Td[(^Z1@^vz1 @zatz=1(2) @^vz2 @z=0atz=0(2) ^vz2=0atz=0(2)Equations 2 2 willprovideasolutionfor^vz2andthex-dependentportionof^vx2=^vx2e2ikx+~vx2e)]TJ /F7 7.97 Tf 6.58 0 Td[(2ikx+vx2ind,butthex-independentsolutionstillremains.Toobtain 30

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this,thex-componentoftheStokes'equationisusedalongwithno-slipconditionsatbothsurfaces 0=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(dP2 dx+r2vx2(2) vx2+2Z1@vx1 @z+Z21@2vx0 @z2=0atz=1(2) vx2=0atz=0(2)andtakingthex-independenttermsfromtheseequationsgives @2vx2ind @z2=0(2) vx2ind=)]TJ /F4 11.955 Tf 9.3 0 Td[(2real(^Z1@~vx1 @z))]TJ /F4 11.955 Tf 13.16 8.09 Td[(1 2^Z1~Z1@2vx0 @z2atz=1(2) vx2ind=0atz=0(2)leavinguswiththenecessaryequationstondvz1,vx1,vz2,andvx2.ThedifferentialEquations 2 and 2 canbesolvedbyobtainingthecharacteristicequationofbothandthendeterminingtherootsoftheseequations.Thisresultsin^vz1and^vz2intheform: ^vz1=1cosh(kz)+1sinh(kz)+1zcosh(kz)+1zsinh(kz)(2)and ^vz2=2sinh(2kz)+2cosh(2kz)+2zsinh(2kz)+2zcosh(2kz)(2)wheretheconstants,1)]TJ /F6 11.955 Tf 12.15 0 Td[(1and2)]TJ /F6 11.955 Tf 12.14 0 Td[(2canbefoundfromEquations 2 2 and 2 2 respectively.Fromhere^vxcanbeobtainedfromcontinuitybytherelation,^vx1=i k@^vz1 @zand^vx2=i 2k@^vz2 @ztogive: ^vx1=i kf1kcosh(kz)+1[cosh(kz)+kzsinh(kz)]+1[sinh(kz)+kzcosh(kz)]g(2) 31

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and ^vx2=i 2kf2k2cosh(2kz)+2k2sinh(2kz)+2[2kzcosh(2kz)+sinh(2kz)]+2[2kzsinh(2kz)+cosh(2kz)]g (2) Now,theonlyvelocitythatremainstobedeterminedisthex-independentportionofvx2whichcanbeobtainedbyintegrationofEquation 2 togive vx2ind=2indz+2ind(2)wheretheconstantsareobtainedviaEquations 2 and 2 .Thevelocityis,therefore,knownuptosecondorderandresultsofthiscalculationwillbediscussedlater.ThefullNavier-StokesmodelismorecomplicatedandmustbesolvednumericallyforanythingpastzeroorderinA.Atorderzerothereisonlyadrivingforceinthex-directionandnoinhomogeneoustermstocreateowinthez-direction.Forthex-componentequationsthenonlinearterminNavier-StokescontainsR(vx0@vx0 @x+vz0@vx0 @z)whichmustbezerosincevz0hasnosourceandvx0isonlyafunctionofz.Additionally,atzeroorder@P @xiszerobecausetheowisdrivenbydiffusionofmomentumfromthemovinglidandgravity.Therefore,thebaseowwillbethesameasthatobtainedfromtheStokesmodelwherethesystemisdescribedbyEquations 2 , 2 ,and 2 .Again,vx0isthenequalto)]TJ /F5 7.97 Tf 10.49 4.71 Td[(Gz2 2+[1 2G)]TJ /F3 11.955 Tf 11.96 0 Td[(U]z+U.Summarizingthegeneraldomainandboundaryequations: R(~vr~v)=rP+r2~v+G~ix(2) ~v=Uatz=0(2) ~v=0atz=1(2)whereeachofthesewillbeexpandedinordersofthewallamplitudeA. 32

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UponexpansionoftheNavier-Stokesmodel,wehaveatrstorder, R(~v0r~v1+~v1r~v0)=rP1+r2~v1(2) r~v1=0(2) ~v1=0atz=0(2) ~v1+Z1@~v0 @z=0atz=1(2)andatsecondorder, R(~v0r~v2+~v2r~v0+2~v1r~v1)=rP2+r2~v2(2) r~v2=0(2) ~v2=0atz=0(2) ~v2+2Z1@~v1 @z+Z21@2~v0 @z2=0atz=1(2)Now,similartotheStokessolutionmethod,thex-andz-componentequationswillbewrittenandthentheseequationswillbecombinedinordertocancelthepressuretermleavinganequationintermsofonlyvelocities.Thereforeatrstorder,afternotingthatvz0iszeroandanyx-derivativesofvx0arezero,Equation 2 becomes x)]TJ /F3 11.955 Tf 11.95 0 Td[(componentR(vx0@vx1 @x+vz1@vx0 @z)=)]TJ /F6 11.955 Tf 10.49 8.08 Td[(@P1 @x+r2vx1(2) z)]TJ /F3 11.955 Tf 11.95 0 Td[(componentR(vx0@vz1 @x)=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(@P1 @z+r2vz1(2)Togetridofthepressureterm,aderivativewithrespecttozwillbetakenofthex-componentequationandfromthisthederivativewithrespecttoxofthez-componentequationwillbesubtractedwhichgives, R(@vx0 @z@vx1 @x+vx0@2vx1 @x@z+@vz1 @z@vx0 @z+vz1@2vx0 @z2)]TJ /F3 11.955 Tf 11.95 0 Td[(vx0@2vz1 @x2)=r2(@vx1 @z)]TJ /F6 11.955 Tf 13.15 8.09 Td[(@vz1 @x)(2) 33

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Finally,togettheequationintermsofonlyvz1anx-derivativeistakenofEquation 2 R(@vx0 @z@2vx1 @x2+vx0@3vx1 @x2@z+@2vz1 @x@z@vx0 @z+@vz1 @x@2vx0 @z2)]TJ /F3 11.955 Tf 9.62 0 Td[(vx0@3vz1 @x3)=r2(@2vx1 @x@z)]TJ /F6 11.955 Tf 10.82 8.09 Td[(@2vz1 @x2)(2)andapplyingcontinuityintheform,@vx1 @x=)]TJ /F10 7.97 Tf 10.49 4.88 Td[(@vz1 @z,simpliesEquation 2 into R(@vz1 @x@2vx0 @z2)]TJ /F3 11.955 Tf 11.96 0 Td[(vx0@3vz1 @x@z2)]TJ /F3 11.955 Tf 11.95 0 Td[(vx0@3vz1 @x3)=r2()]TJ /F6 11.955 Tf 10.49 8.08 Td[(@2vz1 @z2)]TJ /F6 11.955 Tf 13.15 8.08 Td[(@2vz1 @x2)(2)Rewritingr2r2asr4,wehave r4vz1+R(@2vx0 @z2@ @x)]TJ /F3 11.955 Tf 11.96 0 Td[(vx0@ @x(@2 @z2+@2 @x2))vz1=0(2)Thesystemisnowintermsofasinglevariable,buttheequationscanstillbesimpliedsincethex-dependencyisknown.Thex-componentofEquation 2 isthesameasEquation 2 ,fromtheStokesmodel,andsothex-dependencewillbethesameaswasfoundforStokes,i.e.,vz1=^vz1eikx+~vz1e)]TJ /F5 7.97 Tf 6.59 0 Td[(ikx.ThisallowsEquation 2 tobewrittenintermsof^vz1as r4^vz1+R(ik@2vx0 @z2)]TJ /F3 11.955 Tf 11.96 0 Td[(ikvx0(@2 @z2)]TJ /F3 11.955 Tf 11.96 0 Td[(k2))^vz1=0(2)wherer4isnow(@2 @z2)]TJ /F3 11.955 Tf 11.95 0 Td[(k2)2andvx0isafunctionofz.Now,theonlythingthatremainsatrstorderistoconverttheboundaryconditionstobeintermsof^vz1.AfterwritingEquations 2 and 2 inx-andz-componentformsitisseenthattheyarethesameastheboundaryconditionsfortheStokesmodel.Thismeansthatthenalformofthe1storderNavier-Stokesmodelfor^vz1isdescribedbyEquations 2 2 and 2 .Thesecondorderproblemwillbetreatedbythesameprocessasrstorderuntiltheequationsareintermsofonly^vz2.Therefore,writingEquation 2 intermsofit'sx-andz-componentsgives, R(vx0@vx2 @x+vz2@vx0 @z+2vx1@vx1 @x+2vz1@vx1 @z)=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(@P2 @x+r2vx2(2) 34

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and R(vx0@vz2 @x+2vx1@vz1 @x+2vz1@vz1 @z)=)]TJ /F6 11.955 Tf 10.49 8.08 Td[(@P2 @z+r2vz2(2)Takingd dz(x)]TJ /F3 11.955 Tf 11.95 0 Td[(component))]TJ /F5 7.97 Tf 15.1 4.71 Td[(d dx(z)]TJ /F3 11.955 Tf 11.96 0 Td[(component)leaves R(vx0@2vx2 @x@z+vz2@2vx0 @z2+2vx1@2vx1 @x@z+2vz1@2vx1 @z2)]TJ /F3 11.955 Tf 11.95 0 Td[(vx0@2vz2 @x2)]TJ /F4 11.955 Tf 11.96 0 Td[(2vx1@2vz1 @x2)]TJ /F4 11.955 Tf 9.3 0 Td[(2vz1@2vz1 @x@z)=r2(@vx2 @z)]TJ /F6 11.955 Tf 13.16 8.09 Td[(@vz2 @x) (2) andupontakinganotherderivativewithrespecttoxandmakinguseofcontinuityintheform,@vx2 @x=)]TJ /F10 7.97 Tf 10.5 4.88 Td[(@vz2 @z,gives R(@vz2 @x@2vx0 @z2)]TJ /F3 11.955 Tf 11.95 0 Td[(vx0r2@vz2 @x)]TJ /F4 11.955 Tf 11.96 0 Td[(2@vx1 @xr2vz1)]TJ /F4 11.955 Tf 11.95 0 Td[(2vx1r2@vz1 @x+2@vz1 @xr2vx1+2vz1r2@vx1 @x)=r4vz2 (2) Uponrearrangingterms, r4vz2+R(@2vx0 @z2@ @x)]TJ /F3 11.955 Tf 11.96 0 Td[(vx0r2@ @x)vz2=R(2@vx1 @xr2vz1+2vx1r2@vz1 @x)]TJ /F4 11.955 Tf 11.95 0 Td[(2@vz1 @xr2vx1)]TJ /F4 11.955 Tf 11.95 0 Td[(2vz1r2@vx1 @x) (2) Onceagainthedomainequationisobtainedintermsofasinglevariable.Therelationvz2=^vz2e2ikx+~vz2e)]TJ /F7 7.97 Tf 6.58 0 Td[(2ikx+vz2indalongwithvz1=^vz1eikx+~vz1e)]TJ /F5 7.97 Tf 6.59 0 Td[(ikxandtherstordercontinuityequationwillbeusedtoputthedomainequationintermsof^vz2.DoingsoreducesEquation 2 to (@2 @z2)]TJ /F4 11.955 Tf 11.95 0 Td[(4k2)2^vz2)]TJ /F4 11.955 Tf 11.96 0 Td[(2ikR@2vx0 @z2^vz2)]TJ /F4 11.955 Tf 11.95 0 Td[(2ikRvx0(@2 @z2)]TJ /F4 11.955 Tf 11.96 0 Td[(4k2)^vz2=4R[^vz1@3^vz1 @z3)]TJ /F6 11.955 Tf 13.15 8.08 Td[(@^vz1 @z@2^vz1 @z2] (2) Justlikeatrstorder,theboundaryconditionsareagainthesameasthosefromtheStokesmodelandsotheonlythingthatremainsistondthex-independentsecondorderterm.Thex-independentportionofvz2iszero,butthex-independenceofvx2is 35

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givenby @2vx2ind @z2=[~vr~v]x)]TJ /F5 7.97 Tf 6.58 0 Td[(ind(2) vx2ind+[2Z1@vx1 @z+Z21@2vx0 @z2]x)]TJ /F5 7.97 Tf 6.59 0 Td[(ind=0atz=1(2) vx2ind=0atz=0(2)Afterdeterminingthex-independentquantitiestheseequationsfurtherreducetogive @2vx2ind @z2=4Rreal(^vz1@~vx1 @z)(2) vx2ind=)]TJ /F4 11.955 Tf 9.3 0 Td[(2real(^Z1@~vx1 @z))]TJ /F4 11.955 Tf 13.16 8.09 Td[(1 2^Z1~Z1@2vx0 @z2atz=1(2) vx2ind=0atz=0(2)whichresultsinthenecessaryequationstondvz2andvx2.Equations 2 and 2 2 aresolvednumericallytogivevz2andthecontinuityequationisusedtondpartofvx2via,^vx2=i 2k@^vz2 @z.ThenEquations 2 2 areusedtoobtainvx2ind,whichallowsthesystemtobeknownuptosecondorderinA.InthecaseofNavier-Stokessolution,aspectralmethodwithChebyshevbasisfunctionswasemployed.Severalcheckswereusedtovalidatethespectralsolution.ThesimplestvalidationwastonoticethatwhenReynoldsnumberissettozerointhenumericalcodeitmatchesalmostexactlywiththeanalyticStokesresults.AsecondcheckwastosetthevariablecoefcientintheN-Sdomainequationstoaconstantvalueandthensolvetheequationsanalytically.Then,thecoefcientsweresettothesameconstantvalueinthenumericalsolutionandthesetwomodiedN-Smodels,analyticandnumericalsolution,werefoundtomatchasshowninTable 2-1 .Somediscrepancyisseenbetweenthevariousmodelsfornodesnearthewavyboundary,particularlyatlargeramplitudes.ThisisbecausethevelocityvaluesobtainedfromtheN-Smodelwerecomparedwithvaluesobtaineddirectlyfromthesolutionofthedomainequationswithappropriateboundaryconditions.AnexplanationforcorrectionofvelocitynearthewavyboundaryingiveninChapter 3 . 36

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Table2-1. Comparisonofvaluesfoundbyanalyticconstantcoefcientsolutionversusspectralconstantcoefcientsolution.Erroristhedifferencebetweenthetwomodelsdividedbytheaverageofthetwomodels.ValuesshownontheleftareforU=1,G=0,R=10,k=2,A=0.01.OntherightU=1,G=0,R=10,k=2,A=0.33. z-coordinateError 0.9900.986.31E-040.956.84E-050.905.26E-050.833.96E-050.742.61E-050.651.47E-050.557.01E-060.442.90E-060.341.11E-060.254.49E-070.162.10E-070.091.04E-070.044.41E-080.011.08E-080.000 z-coordinateError 0.6700.652.99E-020.559.50E-030.453.43E-030.351.24E-030.254.89E-040.172.26E-040.101.12E-040.044.77E-050.011.17E-050.000 Asanalstepofvalidation,afakepolynomialsolutionwasassumed.Thisistosaythat,ateachorder,thevelocitywassetequaltoaknownpolynomialfunction.Thispolynomialfunctionisthensubstitutedintothedomainandboundaryconditionstoobtainan“errorvector”(_err),wheretheerrorisrelativetotheoriginalsetofequations.Thiserrorvectoristhenaddedintothenumericalequationssothatthegeneralsystemgoesfrom,A_v=_RHStoA_v=_RHS+_err.Whenthemodiedsystemissolvedusingtheoriginaldomainandboundaryoperators,thepolynomialsolutionisrecoveredexactly,validatingtheoperatorsusedforthisproblem.TheowrateisdeterminedthesamewayforbothStokesandNavier-Stokesmodels.Startingwiththebasicequationforowthroughachannelgives Q=ZZ(x,)0vxdz(2) 37

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butQitselfisequaltoQ0+AQ1+(1=2)A2Q2+...andsoQmustbedeterminedateachorderofAuptothedesiredaccuracy.Todothis,dQ dAandd2Q dA2mustbefound.Upontakingasinglederivative dQ dA=ZZ(x,A)0dvx dAdz+vx(Z(x,A))dZ dA(2)andafterthesecondderivativeinA d2Q dA2=ZZ(x,)0d2vx dA2dz+2dZ dAdvx dAjZ+(dZ dA)2dvx dzjZ+d2Z dA2vxjZ(2)sinceQ0,Q1,Q2,etc.areevaluatedatA=0,theequationsforQconvertto Q0=Z10vx0dz(2) Q1=dQ dAjA=0=Z10vx1dz+vx0Z1j1(2) Q2=d2Q dA2jA=0=Z10vx2dz+2Z1vx1j1+Z21dvx0 dzj1(2)Now,theowrateforthissystemmustbeindependentofxduetoconservationofmass.Thiscanbeproveninastraightforwardmannerbyconsideringtheequationforvolumeow. Q=ZZ(x,)0vxdz(2)Thex-derivativeofthisequationmustbezeroifQisindependentofx.Takingd dxofEquation 2 , dQ dx=ZZ(x,)0dvx dxdz+vx(Z)dZ dx(2)Sincevxisalwayszero,duetono-slip,atthewavyboundarythesecondterminEquation 2 iszero.Continuitygivestherelationthat,dvx dx=)]TJ /F5 7.97 Tf 10.5 4.71 Td[(dvz dz.Equation 2 thenbecomes, dQ dx=ZZ(x,)0)]TJ /F3 11.955 Tf 10.5 8.08 Td[(dvz dzdz(2) 38

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afterintegrationthisbecomes dQ dx=)]TJ /F3 11.955 Tf 9.3 0 Td[(vz(Z)+vz(0)(2)andsincevziszeroatboththetopandbottomsurfaceEquation 2 reducesto,dQ dx=0.Infact,thismustbetrueforeveryorderofQsothat,dQ0 dx=0,dQ1 dx=0,dQ2 dx=0,etc.InadditiontoQbeingindependentofx,itcanalsobesaidthatQ1,andinfactalloddordersofQ,mustbezero.Thisisanimplicationofthesymmetryofthedisturbanceintroducedintothesystem.Thatistosay,thesameresultsshouldbeexpectedwhetheraninitiallyatwallisgivenadisturbanceamplitudeofAor)]TJ /F3 11.955 Tf 9.3 0 Td[(A.SinceQisaglobalvariableitwillhavethesamevaluesofQ0,Q1,etcforeitherA.TheonlywaythatthiscanbetruewithoutchangingthevalueofQisforalloddordersofQtobeexactlyzero.Evenordersmaytakeonanyvaluebecausetheamplitudewillmultiplyitselfanevennumberoftimesattheseordersandthevalueresultingfromtheamplitudetermwillalwaysbepositive.Applyingtheseprinciplesresultsinanequationforowrateoftheform Q=Z10vx0dz+1 2A2[Z10vx2inddz+2real(^Z1~vx1j0))]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2^Z1~Z1dvx0 dzj0](2) 2.5PredictionsofRecirculationThecriticalamplitudeforrecirculationwasdeterminedintwodifferentways.Forthelongwavelengthmodelthestressattheboundary,wherex=0,wassettozero.Circulationmeansthattheowmustchangedirections,withthevelocityandhenceviscousstressgoingtozerosomewhereduringthistransition,andresultsinEquation 2 .FortheStokesandNavier-Stokesmodelsthevelocityiscalculatedand,foragivenvalueofA,thesystemisanalyzedtoseeifthereareanynegativevaluesofvx. 39

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WhenU=0,thecriticalamplitudeisA=1.Thisistosaythat,whentheowisdrivenbygravityalone,vx!0asA!1andtherewillnotbeanyrecirculation.AtanyvalueofUtherecirculationcanbeeliminatedbymakingGdominant.TheeffectofUistoincreasethepressureinthegapbetweenabumpandthemovingwall.Thus,uidapproachingabumpwillseealocationofrelativelyhighpressureandthispressurewillincreasewithincreasingU,thuscausingowreversalatsomecriticalpoint.Gactstoincreasevxeverywhere;inparticularclosetothewavywall.Therefore,thesetwocompetingeffectswillcreateabalanceatsomepointwheretheowbeginstomovebackwards.Theconditionswheretheseeffectsarejustinbalanceisthecriticalpointforrecirculation.AscanbeseeninFigure 2-3 ,whengravityiszerointhelongwavelengthapproximation,thecriticalpointforowreversalinabumpisatA=0.31.ThisagreesverycloselywiththeresultsobtainedfromaStokesowmodelatlongwavelength(smallk)whichshowthatthecriticalpointoccursatA=0.32,whichalsocloselymatchestheexpectationofPozrikidis[ 9 ].Infact,forthesmallwavenumbercaseaddinginReynoldsnumberdoesnothaveasignicanteffectontheow.ForReynoldsvaluesrangingfrom,R=1toR=1000,thecriticalamplitudeisconsistentlyfoundtobe32%ofthechannelheightwhenthedisturbancewavelengthislongandgravityissettozero.So,inthelongwavelengthlimitthereisnotastrongdependenceonR,butthereisasignicanteffectduetogravityasisseenfromFigure 2-3 .WhenthedisturbancewavesbecomechoppierthesystemshowsmoresensitivitytochangesinRandk.Askisincreasedaboveavalueof0.3,thecriticalamplitudebeginstodrop,whereasforanyvaluesofksmallerthan0.3theresultsagreewiththelongwavelengthmodel.ForthecaseofnogravityandzeroReynolds,atableshowingthedependencyofthecriticalamplitudeonlargerkcanbeseeninTable 2-2 .Itmakessensethatthecriticalamplitudelowersforthechoppierwaves.Asthewavelengthsshortenthedisturbedboundarybecomessteeperandtheowhaslesstimetodevelop 40

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Figure2-3. Predictionofrecirculationcells,asafunctionoflidspeedandgravity,bylongwavelengthmodelandStokescalculation. intheoriginaldirectionofmotionbeforeitispushedbackbylocalpressuresinthenarrowinggapheightregion.Itisworthmentioningherethat,althoughmuchofScholle'sworkresultsinpertinentconclusions,onestatementseemstobemistaken.Whenitisclaimedthattherangeofvalidityforthelubricationapproximationisrestrictedtocaseswithsmallwaviness('perturbationamplitudes'intermsofthepresentwork)[ 18 ],thereisaawinthereasoningusedtodeterminethisstatement.ThesmallamplituderestrictionisimpliedaftercomparisonofthesolutionfromStokesequationswithresultsfromalubricationapproximation.Theresultsarecomparedforthecaseofadimensionlesschannel 41

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height(h)of,h=3 2,wherehisdenedas,h=2H ;andwhenthepredictionsdonotagreethenitisassumedthatthelubricationapproximationisgivingerroneousresults.However,thelubricationapproximationisonlyvalidforcertaingeometricalscales,andanyresultsfromlubricationmustmaintainthesescales,otherwisetheresultsshouldnotbecorrect.Thepresentworkndsthatlubricationconstraintsarevalidforthissystemwhenthewavenumberislessthan0.3.ThisgivesaconditionalstatementforH as,k=2H <0.3,sothat,H /0.048.ThelubricationresultsusedforcomparisonwiththeStokesmodelfromSchollewerepresentedforH =0.75meaningthattheconclusionoflubricationbeingrestrictedtosmallamplitudesisinvalid.ThisispointedoutinordertoalleviateanydoubtsabouttheresultspresentedinFigure 2-3 whichpredictslargecriticalamplitudeswhengravitybecomesstrong. Table2-2. DependencyofcriticalamplitudeonkwhenR=0andG=0. kAcritical 0.40.310.70.310.2920.2250.1 ThecriticalamplitudeatveryshortwavelengthscanbefoundbywritingtheequationfortangentialstressatthewavywallfortheStokesmodel,whichisgivenby2Advx0 dz[ksinh(k)cosh(k))]TJ /F5 7.97 Tf 6.59 0 Td[(k2 k2)]TJ /F5 7.97 Tf 6.59 0 Td[(sinh2(k)]+Ad2vx0 dz2+dvx0 dz=0[ 19 ].Thestresswillbezeroattheveryonsetofcirculation,sothatinthelimitofsmallwavelength,thecriticalamplitudeisgivenbylimk!1A!1 2k.Meaningthatforverysmallwavelengths,therewillalwaysberecirculation.ThismayhelpexplainthereasoningofScholleetal.whentheysaythatforlargeR,butsmallwavelength,theamplitudehasverylittleinuenceonthecirculationcellpositionandsymmetry[ 10 ].Table 2-3 displaysthesensitivityofthecriticalamplitudeonR,againonlyforlargerk.Sincethereisnoviscouseffectforcingtheuidtowardsrecirculation,butthereareinertialeffects,duetolocalpressures,thenitisunderstoodthattheRequaltozerocase 42

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shouldrequirethelargestamplitudeforcirculation.Observationsfromthemodelagreewiththisreasoning.ThecriticalamplitudedecreaseswithincreasingR,asshouldbeexpected. Table2-3. DependencyofcriticalamplitudeonRforlargervaluesofk,G=0. RkAcritical 10.40.31100.40.311000.40.27120.221020.2110020.14 AcomparisonofthecriticalamplitudefoundforthelongwavelengthmodelandthatobtainedbytheNavier-StokessolutionisshowninFigure 2-3 .Thelongwavelengthcalculationisvalidforanyamplitude,albeitonlyatsmallwavenumbers.Ontheotherhand,theNavier-Stokesmodelisonlyvalidforsmallenoughamplitudes,butanywavenumber.Figure 2-3 isusedtogetanestimateofwhatrangeofamplitudestheNavier-Stokesmodelisvalidforanditisseenthatatdisturbanceamplitudesgreaterthanabout0.4theN-Smodelgivespoorresults.ThismeansthattheNavier-Stokesmodelisstillinformative,butwhenevergravityhasastrongenougheffectonthesystem,thecriticalamplitudetendstowardslargervaluesandtheselargeramplitudeswillnotbecorrectlypredictedbytheNavier-Stokesmodel.AninterestingthinghappenswhenGismadenegative.AscanbeseenfromFigure 2-3 ,foravalueofU=1,whenGisvariedbetween)]TJ /F4 11.955 Tf 9.3 0 Td[(2
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reversalwellbeforetheoweverencountersabump.Infact,forverylargeG,owreversalwilloccurrightnexttothemovingwall.Figure 2-4 displayhowthex-componentofvelocitychangeswiththez-coordinateforlargegravitationaleldsactingoppositethedirectionoflidmovement. Figure2-4. vxasafunctionofzforlargegravitationalterms. Initially,itisnotclearhowrecirculationwillaffecttheowrateofthesystem.Thetwoopposingfactorsconcerningtheowratearethattheowfeelslessfrictionasitnowslipsacrossthecirculatingcells,butatthesametimetheowseesareducedcrosssectionforow.Itisseenthatthelongwavelengthmodel,atsmallamplitudes,agreeswiththeStokesmodelatlongwavelengthsandthatthewavywalldecreasesvolumetricowrate.TheseresultsarepresentedinFigure 2-5 .Inertiainuencesnotonlythecriticalamplitudeforrecirculation,butalsothesymmetryofthecirculationcells.ForStokesow,thecirculationcellswillalwaysbecircular,i.e.symmetric.AsRisincreasedabovezerothecirculationcellswillbecomeskewedinamannerthatelongatesthecellsintheupstreamdirection.Theasymmetryisquantiedbyaparameter,=xL xR,wherexLandxRarethedistancesfromthewavywallmaximum(x=0asseeninFigure 2-1 )totheleftandrightedgesofthecirculationcellrespectively.ThisisdepictedinFigure 2-6 Ifisgreaterthanone,thecelliselongatedintheupstreamdirectionandvice-versawhenlessthanone.When 44

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isequaltoone,thecellissymmetric,whichisonlypreciselytruefortheStokesowsystem.AtabledisplayingtheeffectofRonasymmetry,foramoderatewavelengthdisturbance,isgiveninTable 2-4 anditisseenthattheonsetofrecirculationoccursupstreamofthedisturbancepeakfornon-zeroR.ThisisinagreementwiththeasymmetricdevelopmentofrecirculationforbothPouseuille[ 14 ]andCouetteow[ 20 ].Thetotalcirculationcellnon-dimensionallength,Lcell=xL+xR,isalsolistedtoshowthatthecellsarebothskewedandstretchedduetoinertia.Theseresultsareforthecasewhengravityiszero. Table2-4. Asymmetry()andnon-dimensionallength(Lcell)forvaryingRwithG=0,U=1,k=2,=0.25 RLcell 11.251.29101.381.36501.631.501001.561.64 Whengravityisincluded,againactinginthedirectionoflidmotionasshowninFigure 2-1 ,thecirculationcellsareskewedevenfartherupstream.AsinStokesow,therecirculationisdampedbygravity,butwheninertiaisincluded,theoriginofthecirculationcellsisalsoshiftedintheupstreamdirection.Thisisapparentforvaluesofequaltoinnity,whichcorrespondstorecirculationtakingplaceentirelyupstreamofthedisturbancemaximum(x=0).TheseresultsaregiveninTable 2-5 foramoderatesizewavelength. Table2-5. Asymmetry()andnon-dimensionallength(Lcell)forvaryingRwithG 2U=2,k=2,=0.25 RLcell 110.071010.50501.861.431001.781.79 45

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2.6FinalRemarksDeterminingcriticalamplitudesatwhichowreversaloccursistheaimofthisstudy.Itisfoundthatwhengravityisabsent,thereisalwayssomepointatwhichowreversaloccursforliddrivenow.TakinginertiaintoaccountwithReynoldsnumberactstodecreasethecriticalamplitudeandalsoskewthesymmetryoftherecirculationcells.Additionally,havingasteeperboundaryorshorterwavelengthsalsolowersthecriticalamplitude.Whengravityisincluded,itcanhavetwodifferenteffects.Ifitisactinginthesamedirectionasthemovinglid,gravitycanactuallydampouttheowreversaltosuchanextentthatforverylargegravityforcesanamplitudenearlythesizeofthechannelisneeded.Whengravityactsoppositethedirectionoflidmovement,thereisalwaysapointatwhichreversalwilloccur.Inthiscase,ifgravityislargeenough,thenitdoesnotmatterhowsmallthedisturbanceamplitudeis,therewillalwaysbeowreversal.Malevichetaldevelopedatheoremforthissystemstatingthat,“foragivenvalueofR,inany3Dchannelthecriticalamplitude(3D)wheneddiesstartisalwayslessthanthecriticalamplitudeforthecorresponding2Dchannel”[ 20 ].Thisistosaythatthe2Dmodel,aspresentedhere,canbesafelyusedasaconservativeestimateofthecriticalvaluesinregardstodesignparametersinanappropriatesystem. 46

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Figure2-5. FlowrateversusamplitudeforvariousvaluesofG 2U. Figure2-6. Denitionofparametersusedtoquantifytheasymmetryofacirculationcell. 47

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CHAPTER3PERTURBATIONEXPANSIONMETHODSANDTHEIRACCURACY 3.1MappingofVariablesNearaPerturbationSourceThefocusofthischapteristoprovideabetterunderstandingofperturbationmethods,particularlyinrelationtodomainvariablessuchastemperature,velocity,concentration,etc.Manyphenomenaoccurinaninconvenientgeometryandwewouldliketondasolutionfortheseproblemsintermsofasimple,orregular,domain.Now,intheprocessofconvertingaproblemfromacomplicatedtoasimplegeometry,thevariablesinvolvedwillbeintermsofaninniteseriesofquantities.Thereareafewdifferentperturbationmethodsbywhichtoobtaintheseinniteseriesandeachofthemwillprovideanexpressionfortheexactvalueofthevariable.However,whentheinniteseriesistruncated,asitmustbeforanysolution,thendifferencesarisewhencomparingtheexpansionmethods.Ourdiscussioncontinuesalongthisthought.Intraditionalmethodsofperturbationexpansions,thedependentvariable,onlyatinteriorpointswhichdonothaveacorrespondingpointontheancestordomain,areexpandedintermsofboundaryvariables.ThesepointsaredepictedbylineAinFigure 3-1 .Hence,theonlyoptionforndingvariablesatAistoputthemintermsoftheknownboundaryvalues.Ontheotherhand,pointsBandCbothhavereferencepointsontheancestordomain,asshowninFigure 3-1 .Therefore,thesetwopointsarenormallyfoundintermsofperturbationvariablesbeingevaluatedat(x,z)wherexandzarepointsonthecurrentdomainwhicharealsopointsontheancestordomain.Now,thevariablesatAmustbeacceptablebecausetheyaretheonlyexpressionsavailable.PointAmustbeclosetotheboundarysinceissmallandsotheexpressionsareexpectedtobereasonable.Likewise,thevariablesatCareexpectedtobeaccuratebecausewesimplyhaveanexpressionforcorrectingthebasevariableandthepointCisalreadyfarawayfromtheperturbation,meaningthatitshouldnotchangesignicantlyfromthebasevalue.However,questionsariseatpointBwhereatraditionalexpression 48

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forcorrectingthebasevaluemightbeinaccurateunlessalargenumberoftermsintheseriesareused.So,itwouldappearthatpointBcanbetreatedinthesamewayaseitherpointAorC.Initially,itisnotclearwhichofthismethodswouldgiveabetterapproximationofB.Theremainderofthischapterdiscussesthisissue. Figure3-1. Variouspointsinthecurrentdomainandtheircorrespondinglocationwhensuperimposedontheancestordomain.Themultiplelocationsmustbetreateddifferentlytoobtainaccurateresults. Duetothenatureofperturbationexpansionmethods,theresultsobtainedarenecessarilyanapproximation.Alsoduetothenatureofthemodel,theapproximationwillbecomemoreandmoreaccurateasmoretermsareincludedintheirexpansions,withhigherordertermscontributinglesstotheaccuracyofthevariables.Sincethesehigherordertermshavelesssignicanceandtheybecomeincreasinglymorecomplicatedtoobtain,thenthemodelisusuallyworkedouttorstorsecondorderatwhichpointtheeffectsoftheperturbationonthesystemcanbeunderstood.Ifhigherordertermsareneededtoobtainaccuratevariablesthenanapproximation,asdescribedinthefollowingparagraphs,canbeusedasaquickerandsimplerwaytogetaccuratevariables.Whenusingperturbationtechniques,whatishappeningonthe“currentdomain”,isofinterest,i.e.Figure 3-1 .Theissuehereisthatwewanttoknowthevalueofa 49

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domainvariableatlocation(x,z)(asshowninFigure 3-1 )butthevalueofourvariablesateachorderisknownonlyintermsofpoints(x0,z0).Now,wehavenowayofknowingwhich(x0,z0)valuecorrespondstoeachpoint(x,z)becausethe(x,z)pointhasshiftedbysomearbitraryamountrelativetoit'sancestorpoint(x0,z0).Thisiswhyweobtainaninniteseriesforavariableonthecurrentdomainintermsofvariablesontheancestordomain,suchas:u(x,z)=u0(x0,z0)+[u1(x0,z0)+z1du0 dz0(x0,z0)]+1 22[u2(x0,z0)+2z1@u1 @z0(x0,z0)+z21@2u0 @z20(x0,z0)+z2@u0 @z0(x0,z0)]+...,whereuisadomainvariableandisthemagnitudeoftheperturbation.Uponre-summingthisseries,weget,u(x,z)=u0(x,z)+u1(x,z)+1 22u2(x,z)+....Ontheotherhand,apointontheboundaryofthecurrentdomain,(xbndry,zbndry),hasaknownancestorpoint.Therefore,aboundaryvaluewillbemoreaccuratethananinteriorpointwhichisneartheperturbationsourcebecausetheinteriorpointisbeingevaluatedatan(x0,z0)whichmayormaynotbetheactualancestorlocationcorrespondingtothecurrentlocation.Toremedythisinaccuracyoftheinteriorpoints,thedomainvariablescanbeexpandedintermsoftheboundaryvariable.Atsomepoint,fartherawayfromtheperturbationsource,theboundaryexpansionwillnolongerbeneededbecausetheinteriorpointsinthisregionarenotsignicantlyaffectedbytheperturbation.Hence,theancestorpointshavenotshiftedmuchonthecurrentdomainandthedomainvariableswillbeaccurateasgivenbythenormalperturbationexpansion.ThisregionofpointsisdepictedbylineCinFigure 3-1 .Theboundaryvariableexpansioncanbedoneatanyorder.Afterthedomainvariablesareobtainedintermsoftheboundaryvariables,thentheboundaryvariableexpansioncanbecomparedwiththetraditionalperturbationexpansionateachpointasyoumoveawayfromtheperturbedboundary.Thepointwherethetwoexpansionsmatchisthelocationatwhichtheboundaryexpansionisnolongerneeded.Theboundaryexpansionwillonlybeneededatincreasinglysmallerintervalsawayfromtheboundaryashigherordertermsareaddedtotheperturbationexpansion.Atsomehigh 50

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orderoftheperturbationexpansion,theboundaryexpansionwillnolongerbeneededasthetwowillmatchateverylocation.Reasonsforusingeitherexpansionareconsideredinthefollowingsections. 3.2ExpansionMethodsAnoverviewofperturbationexpansionsonthedomainispresentedintheAppendix.Itisshownthatinternalmappingsnaturallycanceloutofdomainequations.Thiscalculationoughttobeexpectedduetothechoiceofaninternalmappingbeingarbitrary,orevenunknown.Toseethedifferencesbetweenthedifferentperturbationexpansionmethods,anexpressionwillrstbeobtainedforatraditionalperturbationexpansionnearaperturbedsourceafterwhichanalternativeexpressionwillbeobtainbytheboundaryexpansionmethod.Onemethodisexpectedtobemoreaccuratethantheotheraswillbehighlighted.TheexpansionswillbewrittenforanarbitrarycurrentdomainasshowninFigure 3-2 . Figure3-2. Systemshowinganarbitraryperturbation. 3.2.1NotationThenotationusedhereisthesameasisfoundintheappendix.Expansionsarewrittenforageneraldomainvariable,u,intermsofthecoordinatesxandy.isused 51

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fortheperturbationamplitude.Thebasesystemislabeledasthe“ancestordomain”inFigure 3-2 .Itisalsoreferredtoasthe“referencedomain”andthesetwotermswillbeusedinterchangeably.Thesubscriptofavariablerepresentshowmanyderivativesinarebeingtakenofthatvariable,afterwhichthevariableisevaluatedatequaltozero.Forexample,un,isequivalentto,@nu @nj=0.Naturally,itfollowsthatasubscriptofzeromeansthatthevariableisevaluatedonthereferencedomain. 3.2.2TraditionalPerturbationExpansionAsderivedintheappendix,adomainvariableobtainedthroughperturbationexpansionistraditionallyexpressedas u(x,y,)=u0(x,y)+u1(x,y)+1 22u2(x,y)+...(3)Itisnowofinteresttoknowhowthedomainvariableactsneartheperturbationsource.Sincetherighthandsideofthisequationisknownentirelyintermsofthereferencedomain,eachtermofEquation 3 willbeexpandedintermsofthereferenceboundaryby,u0(x,y)=u0(x,Y0,)+(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)@u0 @y0+1 2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)2@2u0 @y20.Then,Equation 3 becomes u(x,y,)=[u0+(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)@u0 @y0+1 2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)2@2u0 @y20]jY0 (3) +[u1+(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)@u1 @y0+1 2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)2@2u1 @y20]jY0+1 22[u2+(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)@u2 @y0+1 2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)2@2u2 @y20]jY0Equation 3 willbeleftinthepresentformforcomparisonwiththedomainvariableafteritisexpandedaboutthecurrentdomainboundary. 3.2.3BoundaryExpansionMethodSinceaboundarypointisknownexactlywhentransformingbetweentheancestorandthecurrentdomain,thenthedomainvariablesnearaperturbedboundaryshouldbemoreaccurateiftheyareknownintermsoftheboundary,ratherthananinteriorpoint.It 52

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canbestatedthat,neartheboundary,variablesinthecurrentdomainaregivenby, u(x,y,)=u(x,Y,)+(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y)@u @y(x,Y,)+1 2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y)2@2u @y2(x,Y,)(3)ExpandingeachofthetermsontherighthandsideofEquation 3 resultsin, u(x,Y,)=u0(x0,Y0,0)+[u1(x0,Y0,0)+Y1@u0 @y0(x0,Y0,0)]+1 22[u2+2Y1@u1 @y0+Y21@2u0 @y20+Y2@u0 @y0]j(x0,Y0,0)+1 3!3[u3+3Y1@u2 @y0+3Y2@u1 @y0+3Y1Y2@2u0 @y20+3Y21@2u1 @y20+Y31@3u0 @y30+Y3@u0 @y0]j(x0,Y0,0)(3) @u @y(x,Y,)=@u0 @y0(x0,Y0,0)+[@u1 @y0(x0,Y0,0)+Y1@2u0 @y20(x0,Y0,0)]+1 22[@u2 @y0+2Y1@2u1 @y20+Y21@3u0 @y30+Y2@2u0 @y20]j(x0,Y0,0)+1 3!3[@u3 @y0+3Y1@2u2 @y20+3Y2@2u1 @y20+3Y1Y2@3u0 @y30+3Y21@3u1 @y30+Y31@4u0 @y40+Y3@2u0 @y20]j(x0,Y0,0)(3)and @2u @y2(x,Y,)=@2u0 @y20(x0,Y0,0)+[@2u1 @y20(x0,Y0,0)+Y1@3u0 @y30(x0,Y0,0)]+1 22[@2u2 @y20+2Y1@3u1 @y30+Y21@4u0 @y40+Y2@3u0 @y30]j(x0,Y0,0)+1 3!3[@2u3 @y20+3Y1@3u2 @y30+3Y2@3u1 @y30+3Y1Y2@4u0 @y40+3Y21@4u1 @y40+Y31@5u0 @y50+Y3@3u0 @y30]j(x0,Y0,0)(3)Now,theperturbedcoordinateattheboundaryis,Y=Y0+Y1+1 22Y2.Fromthis,itisknownthat, (y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y)=(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0))]TJ /F6 11.955 Tf 11.96 0 Td[(Y1)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 22Y2(3)and (y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y)2=(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)2)]TJ /F4 11.955 Tf 11.96 0 Td[(2Y1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0))]TJ /F6 11.955 Tf 11.95 0 Td[(2Y2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)+2Y1(3) 53

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FurtherexpansionoftherighthandsideofEquation 3 leadsto, (y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y)@u @y(x,Y,)=(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)@u0 @y0+[(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)(@u1 @y0+Y1@2u0 @y20))]TJ /F3 11.955 Tf 11.96 0 Td[(Y1@u0 @y0] (3) +1 22[(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)(@u2 @y0+Y21@3u0 @y30+2Y1@2u1 @y20+Y2@2u0 @y20))]TJ /F3 11.955 Tf 9.3 0 Td[(Y2@u0 @y0)]TJ /F4 11.955 Tf 11.95 0 Td[(2Y1(@u1 @y0+Y1@2u0 @y20)]and 1 2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y)2@2u @y2(x,Y,)=1 2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)2@2u0 @y20 (3) +[(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)2(@2u1 @y20+Y1@3u0 @y30))]TJ /F3 11.955 Tf 11.95 0 Td[(Y1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)@2u0 @y20]+1 22[1 2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)2(@2u2 @y20+2Y1@3u1 @y30+Y21@4u0 @y40+Y2@3u0 @y30))]TJ /F3 11.955 Tf 9.3 0 Td[(Y2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)@2u0 @y20+Y21@2u0 @y20)]TJ /F4 11.955 Tf 11.96 0 Td[(2Y1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)(@2u1 @y20+Y1@3u0 @y30)]GroupingtermsfromEquations 3 , 3 ,and 3 inordersof,uptosecondorder,resultsin, u(x,y,)=[u0(Y0)+(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)@u0 @y0jY0+1 2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)2@2u0 @y20jY0] (3) +f[u1(Y0)+(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)@u1 @y0jY0+1 2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)2@2u1 @y20jY0]+1 2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)2Y1@3u0 @y30g+1 22f[u2(Y0)+(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)@u2 @y0jY0+1 2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)2@2u2 @y20jY0])]TJ /F4 11.955 Tf 9.3 0 Td[((y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)Y21@3u0 @y30+(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)2Y1@3u1 @y30+1 2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y0)2Y21@4u0 @y40+1 2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Y0)2Y2@3u0 @y30gandEquation 3 canbeusedforcomparisonwiththetraditionalperturbationexpansionaboutthebasedomain. 3.3ComparisonofExpansionMethodsTherstthingtonoticeaboutbothexpansionsisthatthereisaseriesateachorderofwhichisequalbetweenthetraditionalandtheboundaryexpansion,i.e.[un(Y0)+(y)]TJ /F3 11.955 Tf 11.85 0 Td[(Y0)@un @y0jY0+1 2(y)]TJ /F3 11.955 Tf 11.85 0 Td[(Y0)2@2un @y20jY0],wherethesubscriptndenotesanarbitraryorderofthedomainvariableu.Hence,ifEquation 3 issubtractedfromEquation 3 54

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thenitisseenthatextratermsoccurintheboundaryexpansionmethodwhichdonotariseinatraditionalexpansion.Inspectionoftheextratermsshowsthattheywillonlybenettheexpansion'saccuracyneartheperturbedboundary.Atdomainpointsfartherawayfromtheboundary,thequantity(y)]TJ /F3 11.955 Tf 12.52 0 Td[(Y0)becomeslargeandtheseriesshouldnotnecessarilyconvergequickly.However,thesamewillbetrueforeachofthe(y)]TJ /F3 11.955 Tf 10.02 0 Td[(Y0)quantitieswhichoccurinthetraditionalexpansionneartheboundaryandsoneithersolutionshouldbegivenmuchcredibilityfartherawayfromthecurrentwavyboundary.Neartheperturbedboundarythequantitywillberelativelysmallandsincetheextratermsfoundviatheboundaryexpansionmethodareatleastoforder,(y)]TJ /F3 11.955 Tf 12.5 0 Td[(Y0)2,thenthesetermsshouldconvergeandthereforewillprovideamoreaccuratedomainvariable.Thiswillbeshownbyexampleinthefollowingsection.Theearlierderivationswereperformedsothateachexpansioncouldbecomparednearthedisturbedboundary,whichwillbetheregionmostsignicantlyinuencedbytheperturbation.Inapplicationsthough,Equations 3 and 3 areusedtoobtainvaluesofthedomainvariables.Now,inadditiontomorequicklyconvergingtermsintheboundaryexpansiontechnique,lookatwhatEquation 3 tellsincomparisontothetraditionalmethodwhereitiscommontouseEquation 3 everywhereontheinteriorofthedomainwherethereareancestorpoints.ThelargestcontributiontoEquation 3 ,thersttermintheseries,isknownexactlyonthecurrentdomain.ThisisincontrastwiththelargestcontributiontoEquation 3 whichisknownontheancestordomainateachsetofreferencepoints.Moreover,thequantities(y)]TJ /F3 11.955 Tf 12.84 0 Td[(Y)appearingintheboundaryexpansionmethodwillbesmallverynearthecurrentboundaryandasaresult,willquicklyconvergeinthisboundary-region.Infact,(y)]TJ /F3 11.955 Tf 12.32 0 Td[(Y)willbesmallerthantheperturbationamplitude,,foraportionofthedomain.Itisthereforeclearthatboundaryexpansiontechniqueshouldgivemoreaccurateresultsinregionsclosetotheperturbationsource. 55

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Considerationshouldbegivenastohowpointsfartherawayfromtheperturbedboundaryshouldbetreated.Pointsfartherawayfromaperturbationsourcewillnotbestronglyaffectedbythedisturbance.Thisistosaythat,notonlywillthedisturbance,,besmall,butthecorrectionsthemselves,u1,u2,etc.,willalsobesmall.However,thequantity,(y)]TJ /F3 11.955 Tf 12.13 0 Td[(Y),willbecomeverylarge.Therefore,thetraditionalboundaryexpansiontechniqueshouldquicklyconvergetoavalidsolution,whereastheboundaryexpansionshouldgiveinaccurateresultsatlocationsfarfromtheperturbationsource.Forthesereasons,bothexpansionmethodsneedtobecalculatedateachorder.Thisisstillanadvantageasitisasimplertaskthansolvingforhigherordersoftheperturbedvariables. 3.3.1RecirculatingHalos-AnExamplebyWayoftheLid-DrivenWavyBoundarySystemAsjustdiscussed,therearemultiplewaystotreatperturbationvariablesandsomemethodswillbemoreaccuratethanothersdependingonthelocationinthedomain.Infact,forthewavyboundarylid-drivenowsystemdiscussedinChapter 2 ,whenthetraditionalperturbationmethodisusedthereappearstobefalsecirculationtakingplacewheretheboundaryjutsintothedomain.Thesefalsecirculationcellsarehenceforthreferredtoas“halos”andaredepictedinFigure 3-3 .Theerroneoushalosarenotdiscussedquantitativelybecauseitisactitiousresultandtherewouldbenopointincharacterizingthehalosforvariouswavelengthsandamplitudes.Thehalosareseentooccuratbothlargeandsmallwavelengthdisturbances.HalosarepredictedtooccuratlowerperturbationamplitudesthanthosepredictedinChapter 2 .Similartothetruecirculationcells,thehaloswilldisappeariftheamplitudeismadesufcientlysmall.Thismakessenseas,forverysmallperturbations,eventhetraditionalexpansionmethodwillconvergewhenincludingasmallnumberoftermsintheseries. 56

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Figure3-3. Falsecirculationcellsshowinghalosaroundapeakoftheperturbedboundary. Ontheotherhand,iftheboundaryexpansionmethodisused,asoutlinedpreviously,thehalosareseentodisappearatthesameorderofwheretheyareobservedforthetraditionalexpansion.Again,thisisduetoknowingexactvaluesforthevelocityontheboundaryandhence,alowerorderoftermswillgiveconvergenceoftheseriesandcorrectresults.Thelong-wavelengththeorydiscussedinChapter 2 providedadditionalcredibilitytotheobservationthattheserecirculatinghalosarefalse.Sincetheboundaryexpansionmethodisneededclosetotheperturbationsource,butbecomeslessaccuratefartherawayfromthesource,acombinationofthetwoexpansions,traditionalandboundary,mustbeused.Ateachpointinthedomain,therewillbetwodifferentvaluesforadomainvariable.Onevaluefromeachexpansion.Ifitisseenthatthetwovaluesmatchateachpoint,thenthetraditionalmethodmustbeasaccurateatthatorderastheboundaryexpansionandsothetraditionalvaluesshouldbeused.Ifthevaluesaredifferentateachlocation,thentheboundaryexpansionshouldbeusedneartheperturbedwall.Asthelocationofinterestmovesawayfromtheboundary,therewillbesomepointwherethetwoexpansionsmatchveryclosely.ReferringtoFigure 3-4 ,foragivenx-value,iftheboundaryexpansionisbeingusedatlocation1,thenthenextpointofconsiderationislocation2.Here,boththe 57

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boundaryandtraditionalexpansionvalueswillbecomparedwithlocation1.Whicheverisclosertothevalueatlocation1willbethevaluewhichisdeterminedtobecorrect.Iftheboundaryexpansionisstillcorrect,thenthesameprocesswillbeperformedatlocation3.Eventually,thetraditionalexpansionwillgivebetterresultsandthetraditionalexpansionwillbeusedfromthispointonwarduntiltheedgeofthedomain(themovinglidinthiscase). Figure3-4. Threedifferentlocationsforcomparingvariablesfoundbytraditionalexpansionandboundaryexpansionmethods Assoonasthelocationisfoundwherethetraditionalvariablesbecomemoreaccuratethen,fromthereon,theboundaryexpansionwillcontinuetodivergefromthecorrectvaluesbecauseofthe,(y)]TJ /F3 11.955 Tf 11.95 0 Td[(Y),quantityseeninEquation 3 .Incasethereaderwouldliketobelievethatrecirculatinghalosdoindeedoccurinthelid-drivenowsystem,thenconsiderthefollowingsectiondiscussingthetwoexpansionmethodsforanellipticaldomain. 3.3.2ExpansionsonanEllipticalDomain 3.3.2.1ExactsolutionHere,anellipticaldomainistakentobethesystemofinterest.Thedomainequationtosolveis r2u=)]TJ /F4 11.955 Tf 9.3 0 Td[(1(3) 58

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Thiswillrstbesolvedanalyticallytoobtainanexactsolutionandthensolvedviaaperturbationexpansionwiththegoalofcomparingtheperturbedsolutions(i.e.traditionalandboundaryexpansionvariables)totheexactsolution.Theboundaryconditionisgivenbysettinguequaltozeroontheboundary,givenby x2 a2+z2 b2=1(3)Thesolutiontothissystemofequationsisseentobe u=)]TJ /F4 11.955 Tf 9.3 0 Td[((x2 a2+y2 b2)+1 2 a2+2 b2(3) 3.3.2.2Perturbationsolution:traditionalandboundaryexpansiontechniquesNow,fortheperturbedsystemshowninFigure 3-5 ,theperturbedradiimustbefound.Todothis,theconstraintthatcrosssectionalareaisconservedwhentransformingfromacircletoanellipseisimposed.Thisgivestherelation ab=R20(3) Figure3-5. Ellipsefromaperturbedcircle. 59

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IfaischosentobeequaltoR0+,thenEquation 3 gives,b=R20 R0+andexpandingthisinpowersofgives b=R0(1)]TJ /F6 11.955 Tf 17.27 8.09 Td[( R0+2 R20+...)(3)Toobtaintheperturbedradii,thepolarcoordinatesaresubbedintotheequationforanellipsecircumference,meaningthat,b2x2+a2y2=a2b2=R40,becomes R20(1)]TJ /F6 11.955 Tf 17.27 8.09 Td[( R0+2 R20+...)2R2cos2()+R20(1+ R0)2R2sin2()=R40(3)whereRisjusttheperturbedradiiexpandedinas,R=R0+R1+1 22R2.AfterrearrangingthetermsinEquation 3 ,itisfoundthat [2R0R1)]TJ /F4 11.955 Tf 11.95 0 Td[(2R0]cos2()+[2R0R1+2R0]sin2()=0(3)andupongroupingsimilartermsEquation 3 gives,2R0R1)]TJ /F4 11.955 Tf 10.32 0 Td[(2R0[cos2())]TJ /F3 11.955 Tf 10.31 0 Td[(sin2()]=0.Usingtheidentity,cos2())]TJ /F3 11.955 Tf 11.95 0 Td[(sin2()=cos(2),R1isfoundtobe R1=cos(2)(3)Now,turningtothedomainequationgivenbyEquation 3 ,tozerothorderin r2u0=)]TJ /F4 11.955 Tf 9.3 0 Td[(1(3)with u0=0atr=R0(3)Incyllindricalcoordinateswethenhave 1 r0@ @r0(r0@u0 @r0)=)]TJ /F4 11.955 Tf 9.3 0 Td[(1(3)whichleadstoasolutionforu0oftheform u0=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 4(r20)]TJ /F3 11.955 Tf 11.95 0 Td[(R20)(3) 60

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Toobtaintherstordersolution,thedomainequationis r2u1=0(3)with u1+R1@u0 @r0=0atr0=R0(3)AtrstorderthevariablesareafunctionofasisseenfromEquation 3 .Inthiscase,thesquareofnablais,r2=1 r0@ @r0(r0@ @r0)+1 r20@2 @r20.Aftersubbinginthecos(2)dependency,therstordersolutionisfoundtobe u1=1 2R0r20cos(2)(3)Now,withtheboundaryexpansionforageneraldomainvariablegivenbyEquation 3 ,theresultsfromatraditionalexpansionandaboundaryexpansioncanbecomparedwithanexactknownsolution. 3.3.2.3ComparisonofexpansionmethodstoanexactsolutionApointneartheboundarywherethecurrentdomainhasanancestordomainlocationwillbeanalyzedforseveralparametersets. Figure3-6. Locationneartheboundary,withpointsonboththecurrentandreferencedomains,whichwillbeusedtocomparesolutionmethods Table 3-1 showshowtheexpansionsandexactsolutioncomparewitheachother. 61

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Itisevidentthattheboundaryexpansiongivesaccurateresultsatalowerorderthanthetraditionalmethod.Foreachcasepresented,theresultsfromtheboundaryexpansionareclosesttotheexactsolution.Thereisatrendforbothexpansionmethodstodeviatefartherawayfromtheexactsolutionastheperturbationamplitude()isincreased.Thismakessenseastheexpectationforanyperturbationvariableisthatitshouldgivelessaccurateresultsastheamplitudeofthedisturbancebecomeslarger.Therearenoapparenttrendsinexpansionaccuracyfordifferentsizedomains. 3.4FinalRemarksonExpansionMethodsFormanyproblemsofscienticinterest,anexactsolutionforthephenomenainvolvedcannotbefound.Someoftheseproblemscanbeaddressedviaaperturbationmethodtoobtainresultswhichdescribethephysics.Ifaperturbationmethodisusedandthereisinterestinknowinginformationneartheboundary/interface,andalsotheentiredomain,thenconsiderationmustbegivenastohowaccuratethesolutionisatanygivenorderoftheperturbation.Togetthebestresults,twoexpansionmethods,boundaryexpansionandtraditionalexpansion,shouldbeusedsimultaneously.Thetwoexpansionsareequivalentforaninniteseries,butwhenusingonlyanitenumberoftermsintheseriesthenonewillconvergefasterthantheother.Infact,onewillconvergefasternearadisturbedboundaryandtheotherwillconvergefasteratsomedistanceawayfromadisturbance.Knowingthis,thetwomethodsshouldbecomparedatconsecutiveintervalsmovingawayfromtheboundary,startingwiththeboundaryexpansionbeingcorrect,andoncethetwomethodsmatch,switchovertousingthetraditionalmethodfortheremainderofthedomain.Ofcourse,itisalwaysadvisedtohaveanideaofwhatisexpectedoutofasystembeforeanalyzingit.Afterknowingwhatisexpected,perturbationtechniquescanbeapowerfultoolinpredicting,understanding,anddescribingscienticphenomena. 62

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Table3-1. Comparisonofanexactsolutionandadomainvariablefoundbytraditionalperturbationexpansionandbytheboundaryexpansiontechnique SolutionMethoduforR0=10,=0.1uforR0=10,=1uforR0=10,=2.5 TraditionalExpansion0.50495.36113.66BoundaryExpansion0.50245.20013.28Exact0.50245.10511.97 SolutionMethoduforR0=0.1,=0.01uforR0=1,=0.1uforR0=7,=0.7 TraditionalExpansion5.361e-040.05362.627BoundaryExpansion5.200e-040.05202.548Exact5.105e-040.05102.501 63

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CHAPTER4THERMOCAPILLARYFLOWINADOUBLEFREESURFACEFILMATLOWIMPOSEDTEMPERATUREGRADIENT 4.1IntroductiontoThermocapillaryFlowAuidinterfaceexperiencesmotionwhenatemperaturegradientisappliedacrossthesurface.Thevaryingtemperaturecausesachangeinsurfacetensionofthelmwithhotregionscorrespondingtoalowersurfacetension.Duetothischangeinsurfacetension,apressuregradientexistsalongthesurfaceandtheuidattheinterfacewillbepulledfromlowsurfacetensionregionstowardshighsurfacetensionregions,i.e.fromhottocoldareasasseeninFigure 4-1 .Theterm“thermocapillary”owisusedtodescribethisphenomenabecauseoftheinteractionbetweentemperatureandsurfacetension. Figure4-1. Flowattwointerfacesduetoatemperaturegradientcausingachangeinsurfacetensionalongthesurface. Inexperiments,almisconstrainedinallthreedimensionswithauidbeingsuspendedbetweenopposingwallsandopentotheatmosphereonthetop,bottom,orbothsurfaces.Theuiddepthisofgreatimportanceasthinnerlmswillbemoresignicantlyeffectedbythenaturalsurfacedeformationswhichareboundtobepresentattheinterface.Thepresentworkfocusesonobservingandexplainingtheowstructuresseeninatwofree-surfacelmunderlowimposedtemperaturegradients.Twodistincttypesof 64

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owareseenandaheuristicnumericalmodelisusedtosupportthereasoningwhichdeterminesowstructureselection.Thermocapillaryowsarecommonincrystalgrowthprocesses[ 21 , 22 ]duetothenon-uniformheatingneededtomelttheprecursormaterial.OneoftherstexperimentsdonetoinvestigatethesignicanceofthermocapillaryeffectsincrystalgrowthwasperformedbyEyeretal.[ 23 ].Intheirexperiment,theyloadedaphosphorus-dopedsiliconcrystalaboardasoundingrocketandprocessedthematerialinthesamemannerasonewouldtreatthecrystalinaterrestrialprocess.Bydoingsoinamicro-gravityenvironment,theywereabletoobservethesamestructureofstriationsinthenishedproductandconcludethatbuoyancywasnotresponsibleforthepatternsinvolved,butthermocapillaryowsarethemainculpritofdopantinhomogeneitiesinthenalcrystal. Figure4-2. Exampleofopen-boatcrystalgrowthobtainedfromBagdasarovCrystalsGroupwebsite[ 1 ]. Crystalgrowthisperformedinvariousmanners.Somecommonlyusedproceduresaretheopen-boatandtheoating-zonetechniques.Adiagramoftheopen-boatprocessusedbytheBagdasarovCrystalsGroupisshowninFigure 4-2 [ 1 ],whichisthetechniquemostcloselyrelatedtothepresentgeometry.Intheopen-boatprocess 65

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anamorphouscrystallinematerial,orlowpuritysinglecrystallinematerial,ispulledthroughtheheatingzonecausingthesolidtomelt.Thiscreatesaliquidzoneinwhichimpuritiesaremoresoluble,ascomparedtothesolidmaterial,thuspushingtheimpuritiesdownstreamoftheheaters.Sincetheheatersmaintainanonuniformtemperaturedistributionalongthecrystalmelt,therewillnecessarilybeatemperaturegradientandthermocapillaryowpresentwithintheliquidsection.Whenthematerial,aidedbythesinglecrystalseed,resolidies,itwillformasinglecrystallinedomain.Theresultisapuried,highlyorderedcrystal.Understandingthephenomenainvolvedinthisprocesscanhelpinimprovingqualityofthenalproduct. 4.2LowImposedTemperatureGradientFlowStructuresThetypesofowsthathavebeenobservedinliquidlmsaretypicallysheetowswheretheuidleavesthehotboundary,movesasasheettowardthecoolwallandthendipsintotheuidreturningtothehotsource[ 24 ].However,therehavealsobeenexperimentswheretheowstructureisnotsheet-likebutcellular-likewhenviewedfromaboveandwheretheuidnowleavesthehotsourcetowardthecoldwallreturningalongthesidesinacellular-likestructure.AdepictionoftheseowsisseeninFigure 4-3 .Examplesofbothoftheseowtypesindoublefree-surfaceexperimentshavebeenseenbyUenoandTorii[ 25 ]andPettit[ 26 ].Theformerfocusedonthetransitionfromlowtemperaturegradientbase-owstohighertemperaturegradientchaotic-ows.UenoandToriiwerealsothersttoobservecellularowinarectangulargeometryasinvestigatedinthepresentwork.ThelatterworkbyPetit,performedqualitativeexperimentsinmicro-gravityonlargespanlmsoftheorder10centimeters.Theaimofthecurrentworkistore-examinethesemultiplesteadyowstatesandexplaintheconditionsfortheirappearanceassuchowstatesmustbearuponthenatureoftheinstabilitiesthatmightariseathigherimposedtemperaturegradients. 66

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Figure4-3. Caricatureofthetwoowstatesseenatlowtemperaturegradients. ThisresearchfocusesonthegeometrydepictedinFigure 4-4 .ThelengthbetweentheheatedandcooledwallsislabeledasLx,thedistanceinthespan-wisedirectionisdenotedbyLyandthedepthisgivenbyd.TheuidissurroundedbysolidboundariesintheLxandLydirectionsbutisopentotheatmosphereatthetopandbottomsurfaces.Atemperaturegradient,b=Thot)]TJ /F5 7.97 Tf 6.59 0 Td[(Tcold Lx,ismaintainedalongLxwhereThotandTcoldarethehotandcoldwalltemperatures.Thisgradientcausestheowwhichisdrivenbythermocapillarystressesactingontheupperandlowerfreesurfaces.Thestrengthofthethermo-capillaryowintheliquidischaracterizedbytheMarangoniNumber,denedbyMa=Tbd2=whereTistheabsolutevalueofthesurfacetensioncoefcientwithrespecttotemperature,thedensity,thekinematicviscosity,andthethermaldiffusivityoftheliquid.Thepropertiesarereasonablyassumedtobeconstantforthetemperaturegradientsused. 67

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Figure4-4. Idealsystemgeometry 4.3TheExperiment 4.3.1ExperimentalTechniqueTheexperimentfollowedthemethodofUenoandTorii[ 25 ]andconsistedofimagingtheowstructureswhenadoublefree-surfacelmwassubjecttoaunidirectionaltemperaturegradient.Torealizethis,auidwassuspendedwithinasmallrectangularholethatwasmachinedinsideofametalplate.Eachdimensionwasmachinedtowithin0.05mmprecision.Thesensitivityofthepredictedresultstoanyerrorindepthsduetomachiningwillbecommentedonlater.AphotographofthisgeometryisgiveninFigure 4-5 . Figure4-5. Photoofatestsection 68

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Theuidvolumewassmallenoughthattheeffectofbuoyancyinthesystemwasnegligiblecomparedtotheeffectofsurfacetensiongradients.ThismeansthatthedynamicBondNumber,denedasBo=Ra Ma=gd2 T,wassmall.Here,isthethermalexpansioncoefcient,gtheaccelerationduetogravityandRatheRayleighNumber[ 27 ].Now,thedynamicBondNumbervarieswiththetestsectiondepth,butevenforthedeepestuidthevaluewasBo=0.056,indicatingthatowduetosurfacetensiongradienteffectswasdominantovergravitationaldrivenconvection,i.e.buoyancy.Theuidofchoicewas6cstSiliconeoilratherthanalowerviscosityoilinordertominimizeevaporationduringthetimethatimageswererecorded.Toensurethatastablelmcouldbeformed,theRayleigh-Taylor(R-T)limitforarectangularcrosssectionwascalculated.Thecriticallengthatwhicharectangularplanarlmwithlengthratioof1:2becomesR-Tunstableisgivenby,Lc=q 5 4q gwhereisthedensitydifferencebetweentheuidandthesurroundingair[ 27 ].For6cstSiliconeoil,producedbyShin-etsuandwhosepropertiesaregivenintheTable 4-1 ,thevalueofLciscalculatedtobeLc=5.1mm,implyingthattheheavieroilcouldbestablysuspendedwithinthe4x2mmtestsectionsabovethelighterair. Table4-1. PhysicalPropertiesof6cstSiliconeoil QuantityUnitsValue KinematicViscositym2 s6x10)]TJ /F7 7.97 Tf 6.59 0 Td[(6SpecicGravity-0.925SurfaceTensionN m1.98x10)]TJ /F7 7.97 Tf 6.59 0 Td[(2SurfaceTensionChangewithTemperatureN mK6.37x10)]TJ /F7 7.97 Tf 6.59 0 Td[(5ThermalDiffusivitym2 s7.21x10)]TJ /F7 7.97 Tf 6.59 0 Td[(8ThermalExpansionCoefcientK)]TJ /F7 7.97 Tf 6.59 0 Td[(11.09x10)]TJ /F7 7.97 Tf 6.59 0 Td[(3 Toallowoilinjectiondirectlyintotherectangularareaandattainanaccuratevolume,themetalsectionwaspretreated.First,themetalwascleanedbydippingacottonswabin99%pureAcetoneandwipingthesurface.Then,anothercottonswabwasdippedinRyokoRFH-10Fluorinatedoilwhichcoatedthemetalsurfacearoundthetestsection.ThiswasdonebecauseFluorinatedoilisoleophobictowardSilicone 69

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oil,thusallowingtheSiliconeoiltobeeasilyinjectedintothetestsection.Finally,themetalwasannealedinanovensetat130Cfor4minutes.ThisallowedtheAcetonetoevaporateleavingbehindathinlayeroftheFluorinatedoil.AfterthisprocessthemetalplatewasplacedintotheexperimentalsetupandlledwithanamountofSiliconeoilequaltothevolumeofthetestsection.Thiswasveryimportantasvolumeratiosdifferentthanunitywereknowntogiveanomalousresults.Oncealmwasformed,onesideofthemetalplatewasheatedwhiletheoppositesidewascooledtocreateatemperaturegradientalongthelm.ThetemperatureprolewasmeasuredusingaNECAvioInfraredCamera-R300Model.Thesensitivityofthecamerawas0.05Kwithanaccuracyof1K.Thecamerawasplacedabovethetestsectionobtaininganimageoftheuppersurface.Theowwasvisualizedbysuspending15mdiametergold-coatedAcrylicparticlesusingaCCDcamerawithaframerateof56frames/sandashutterspeedof1=60s.Afteratemperaturegradientwasappliedandthesystemhadreachedsteadystate,asmallamountofparticleswereintroducedintothelm.Thiswasdonebycoveringthetipofathinmetalwirewithparticlesandthensubmergingthetipofthewireintotheoilforabout1swhiletheparticlesenteredtheuid.Asthesolidparticleswereintroducedfromasourcelocationwithinthelm,theparticleswouldfollowthestreamlinesandavoidpassingintoothersectionsoftheuid.TheCCDdataprovidedaseriesofimagesthatcouldbeusedtocreateavideooftheuidowortocreateanimagedisplayingtheparticletrajectorieswithintheuid.TheopensourcesoftwareImageJwasusedtoobtainuidtrajectoryimages.TheoriginaldataacquiredbytheCCDcamerawasrstlteredtosubtractoutlightreectionswhichwerenecessarytovisualizethelm.Then,theresultingimagesweresummedtogethertogiveasinglepictureshowingthepathofparticlesowingthroughthelm. 70

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4.3.2ObservationsFromtheExperimentSeveralobservationsweremade,someforthicklmsandothersforthinlms.Forthicklms,atypeofdoublelayersheetowwasobservedasdepictedinFigure 4-6 .Asdescribedearlierthesheetowisonewheretheuidleavesthehotwallmovingtowardthecoldwallandthendipsintothedepthofthelayerreturningalongthemid-planeofthetestsection.Forthinnerlms,thebaseowtakesadifferentform(cf.Figure 4-6 ).Heretheuidmovesfromthecenterofthelmatthehotwall,againtowardthecoldwall,butnowturnsspan-wardtowardthethesidewallsatwhichpointitturnsaroundandowsbacktothehotwallcreatingacellularlikestructure. Figure4-6. Sheet(left)andCellular(right)owstructuresforlowtemperaturethermocapillaryeffect 4.4PhysicalExplanationofFlowStructureSelection 4.4.1TheScalingLawWheneverauidinaconnedgeometryisselectingaowpath,itmustalwaysmaintainconservationofmass.Therefore,takingvx,vy,andvztobethespatialcomponentsofvelocity,thenthesevelocitycomponentsmustobeythecontinuityequation,@vx @x+@vy @y+@vz @z=0.Uponscalingthisequation,thereobtains,vx Lx@vx @x+vy Ly@vy @y+vz d@vz @z=0.Thisgivesascalinglawbalanceintheformof: vx Lx+vy Ly+vz d/0(4) 71

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whichisusedinthefollowingsectiontounderstandtheuidmovementinaregionwheretheowmustchangedirections.WhenusingthescalinglawtopredictowstructuresinthisthinlmsystemitisunderstoodthatonlytwoofthethreetermsinEquation 4 areinbalancewitheachother.Thisisbecausetheowstructuresthemselvesaretwo-dimensional.Initially,thedrivingforcepushestheuidinthex-direction,alongtheinterface.Whentheowapproachesthecoldwallitmustdecidewhethertoturnsideward,twistdownward,orpushthroughthesolidboundary.Assoonastheowchoosesapath,thentherestoftheuidwillfollowandastableowstructureresults.Ifallthreevelocitycomponentswereinbalance,thentheresultingowstructurewouldlikelyappearmixedandhavenoeasilyrecognizablepatterns. 4.4.2FlowStructureSelectionThetypesofowsthathavebeenobservedinliquidlmsaretypicallysheetowswheretheuidleavesthehotboundary,movesasasheettowardthecoolwallandthendipsintotheuidreturningtothehotsource.However,therehavealsobeenexperimentswheretheowstructureisnotsheet-likebutcellular-likewhenviewedfromaboveandwheretheuidnowleavesthehotsourcetowardthecoldwallandcurvesoutwardbeforereturningalongthesidesinacellular-likestructure,cf.Figure 4-3 .Toseewhytheuidcongurationisdifferentforthetwodepthsanexplanationisadvancedthatinvolvessurfaceelevationsanddepressions,andalsotheslowingoftheowduetoviscosityastheuidreachesthecoldwall.Now,whiletheowstructurewillbedeterminedbysurfacedeformationsandviscouseffects,itcanalsobesaidthatthelocationwheretheuid“decides”onit'sstructurewillbethatlocationwheretheuidrsthastochangedirection.Thiswilloccurastheuidmovesfromhottocoldandasitnearsthecoldboundaryitmustturntostaywithinit'ssolidcontainer.Thelocationwheretheuidrstreachesthecoldwallwillbewhereitisfarthestawayfromtheside 72

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walls,i.e.inthecenterofthex-yplane(asshowninFigure 4-4 )andtheargumentspresentedbelowaregivenfortheuidinthisregion.ThesurfacedeformationisproportionaltotheCapillaryNumber,denedhereas,Ca=L2x Vch d2where Vch=Tbd isacharacteristicvelocityrepresentativeoftheMarangonieffect.ThisimpliesthatCa=Lx dTT withTbeingthetemperaturedropacrossthesystem.Itisseenimmediatelythatshortdepthuidsaremorestronglyinuencedbysurfacedeformationthandeepuids.Forreference,typicalCaforcellularowlms,asseeninFigure 4-6 ,areabout0.13andCavaluesforsheetowsare0.04.Inamannerthatisanalogoustowinddrivenowsoveraconnedchannelwherethesurfaceelevationoccursdownstream,thethermocapillarymotioninthecurrentproblemalsogeneratesanelevationdownstream,nearerthecoldwall.Theincreasedpressureunderneathsuchanelevationwillretardtheowmovingtowardthecoldwall.Butequallyimportant,thesurfaceelevationenhancestheheattransfertotheambienttherebyalsorenderingareductionintheaxialtemperaturegradient,i.e.@T @x,nearthecoldwall.Seeappendix C foracalculationonheattransferfromacurvedboundary.Likewisetheaxialtemperaturegradientisalsolowerednearthehotwallasthesurfacedepressioninitsvicinitydecreasestheheattransferthere.Thegradientisthereforegreatestneartheinectionpoint,seeFigure 4-7 .Theweakgradientsintheupanddownstreamsectionsinthemid-planeofthesurfaceowi.e.nearthehotandcoldboundariesrenderweakowinthexdirection,becauseoftheresultingweakMarangonistresses.Asevidenceoftheabovereasoning,afteratemperaturegradientwasapplied,itwasobservedthattheuidmovedfastestinthecenterofthelmwheretheowisfarthestfromsidewalls.Thiswasmostdramaticforthethinlmsasseenintheattachedmoviewithparticleseeding.Thevaryingspeedofowwasmoredifculttoobserveinthethicklmwheresurfaceeffectsplayamoreminimalroleandthetracerparticlesappearedtomoveataconstantspeed. 73

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Figure4-7. Interfacedeformationalongcenterofthelm. Additionalslowingoftheuidoccursduetoviscosityasitapproachesthecoldwall.ForlmswithhighCatherewillbeverylittledrivingforcenearthecoldwallforowinthex-directionwhich,coupledwiththeviscousdamping,means,@vx @xissmallinthisregion.Consequentlyfrommassconservationorthecontinuityequationi.e.,@vx @x+@vy @y+@vz @z=0,thereisabalanceprincipallybetween@vy @yand@vz @z.UsinganideafromlongwavelengththeorywecanscaletheycoordinatewithLyandthezcoordinatewithdtoseethatvy Lyisproportionaltovz d,whichcanalsobeseenwhenthex-componentofthescalinglaw,Equation 4 ,isnegligible.Thus,forthinnerlmswheresurfacedeformationsplayastrongerrole,itisthenseenthatifLyismuchlargerthand,thenvymustbemuchlargerthanvzandtheowwillturnsidewardsratherthanintothedepthofthelm,formingacellular-likestructure.Ontheotherhand,iftheuidslowsdownonlyverynearthecoldwallduetoviscosity,then@vx @xislargerinthisregion.ForthiscasetheowstructurewillbedeterminedbasedpurelyonthemagnitudesofLyandd.Sincethex-componentofthescalinglawisretained,thereisabalancebetweeneithervxandvyorvxandvz.IfLyismuchlargerthand,thenthebalance,asseenfromEquation 4 ,willbevx Lx+vz d/0andtheuidwillspeedforwardandtwistdownwardbeforecollidingwiththe 74

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wall,creatingsheet-likeow.IfdweremuchlargerthanLy,thenthebalancegoesasvx Lx+vy Ly/0andcellularowisexpected.Figures 4-8 and 4-9 clearlyshowthethermalimagesandthestreamlinesfromparticletracerswhichsupportthisreasoning. Figure4-8. Experimentalimagesforadepthof0.6mm.IntegratedCCDimageontheleftandIRimageontheright.Theimposedtemperaturedifferencewas4.1Kwiththeleftwallheated. Figure4-9. Experimentalimagesforadepthof0.2mm.IntegratedCCDimageontheleftandIRimageontheright.Theimposedtemperaturedifferencewas4.1Kwiththeleftwallheated. Tolendvaliditytothescalingargument,considerathickergeometrywherecellularowshouldbeobserved.Basedonthescaling,itisunderstoodthatcellularowwillonlybeseenatthickerdepthsifthespan-wiselengthisalsoincreased.Yet,foranygeometrywherecellularowisobserved,thescalinglawalsoimpliesthat,ifLxisgraduallyreducedthenthereissomeLxvalueatwhichthevxcomponentcannolongerbeignoredinthebalanceofthecontinuityequation.AtthisLxvalue,andanyshorterLx,sheet-likeowisexpected. 75

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Twomoreexperimentswereperformedandarepresentedhere.InbothcasesalmwithLyof18mmanddepthof0.6mmwereused.Inonecase,Lxof2mmwasusedwhileintheotherLxwas4mm.Asexpected,theshortlmleadstoasmallerCapillaryNumber,weaksurfaceelevation,strongertemperaturegradientsandsharperaxialgradientsinthevelocityi.e.,@vx @x,andthussheet-likeowwasobtained,whilethelonglmledtoare-circulatoryowdespitethedepthofthelmbeinglargerthantheoneshowninFigure 4-9 .Thislendscredibilitytothescalingargumentjustgiven.TheexperimentalimagesforbothofthesegeometriesareshowninFigure 4-10 . Figure4-10. Experimentalimagesforadepthof0.6mm,Ly=18mmandLx=4mm,leftimage,andLx=2mmrightimage.Theimposedtemperaturedifferencewasabout4Kwiththeleftwallheated. Thecompetitionbetweenlengthscalesthatpromotesurfacedeectionwithnonuniformsurfaceheattransferandlengthscaleswheretheinterfaceisapproximatelyatcausesatransitionatsomedepthwheretheowstructurechanges.This,therefore,isthemainargumentandthemainnding. 4.5ConrmationofResultsbyaRestrictedModelAmodelisproposedtosimulatethesystemasdepictedinFigure 4-4 .Inthismodelweassumethatthesurfacesremainatwithoutanydeformation.Nowthisisindeed 76

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asevereapproximationasitisatoddswithourreasoningthatweakvelocitygradientsarisefromtheweakenedtemperaturegradientsassociatedwithsurfacedeections.Hence,onlysheetowoughttobeexpectedinthesimulations.Howeverdeparturesfromthisexpectationwillbeobservedandthusexplanationswillbeoffered.Wenowmoveontothemodel.Thedimensionlessversionofthemodelingequationsusesthescales,d, Tb,Tbd ,Tbforthelength,thetime,velocity,andpressureelds.Thetemperatureeldisexpressedasadifferencefromthecoldend-walltemperature,TCandthenscaledbythedifferenceTH)]TJ /F3 11.955 Tf 11.95 0 Td[(TC,whereTHstandsforthetemperatureatthehotend-wall.Thedimensionlessgoverningequationsthustaketheform Ma Pr"@~v @t+~v.r~v#=rp+r2~v(4) r.~v=0(4)and Ma@T @t+~v.rT=r2T(4)wherePr= isthePrandtlnumber.Weconsidertheendwallstobeatconstanttemperatures,whereasthesidewallsaretakentobeconductingwithalineargradient.Newton'slawofcoolingappliestothefreesurfaceswithasurfaceBiotNumbertakingvariousvaluesincludingzero.Theseassumptionsleadtothethermalboundaryconditions T(x=0)=1,(4) T(x=Lx)=0.(4) 77

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and T(y=0)=T(y=Ly)=)]TJ /F3 11.955 Tf 13.14 8.09 Td[(x Lx+1,(4)Theno-slipboundaryconditionsareappliedtothevelocityeldonallofthewallsoftherectangularcavity.Ontheatfreesurfaces,thermo-capillarystressesmustbebalancedbyviscousstresses.Thisleadsto ~v(x=0,x=Lx,y=0,y=Ly)=~0,(4) vz(x,z=1;x,z=0)=0,(4) @vx @z(z=0,z=1)=@T @x(4)and @vy @z(z=0,z=1)=@T @y(4)TheresultingspatialproblemisdiscretizedbyusingChebyshevcollocationbasedonNx,Nz,andNyGauss-Lobattopointsalongthex,z,andydirections.TheEquations 4 and 4 aretimeintegratedbyusingasecond-ordernite-differencescheme.Thediffusiontermisimplicitlytreatedintime,whereastheconvectivetermisexplicitlyevaluated.Thenumericalproceduretosolvetheproblemistorstcalculatethepressure,thendeterminethevelocityeld,andnallysolvethetemperaturedistribution.TheuncouplingbetweenthevelocityandpressureeldsisperformedbyusingtheProjection-DiffusionmethodandSuccessiveDiagonalization.Allofthenumericalproceduresareexplainedindetailelsewhere[ 28 , 29 ].Thecodewasextensivelyvalidatedforthepresentthermo-capillaryowbycomparingtheresultsobtainedinafreeliquidlayerwiththoseavailablefromtheliterature[ 30 – 34 ].Convergenceinthe 78

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timemarchingstepaswellaswiththecut-offGauss-Lobattopointswasassuredusingstandardrelativeerrorcriteria[ 28 , 29 ].Acommentontheboundaryconditionsisinorder.Tomaintaincompatibilitywiththeedgeconditionsandtoremovethevorticitysingularityatthecorners[ 35 ],twopolynomialregularizingfunctions,viz.,fP(x)andgP(y),areusedintheboundaryconditions,Equations 4 and 4 .Thusatz=0,wehave @vx @z=@T @xfP(x)gP(y)(4)and @vy @z=@T @ygP(y)(4)whileatz=1wehave, @vx @z=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(@T @xfP(x)gP(y)(4)and @vy @z=)]TJ /F6 11.955 Tf 10.5 8.09 Td[(@T @ygP(y)(4)wherethepolynomialfunctionsareexpressedas,fP(x)=(1)]TJ /F4 11.955 Tf 13.03 0 Td[((2x Lx)]TJ /F4 11.955 Tf 13.03 0 Td[(1)2P)2andgP(y)=(1)]TJ /F4 11.955 Tf 11.96 0 Td[((2y Ly)]TJ /F4 11.955 Tf 11.96 0 Td[(1)2P)2.TheregularizationfunctionsaredepictedatvariousordersofPinFigure 4-11 .Itisseenthat,forhigherordersoftheregularizationfunction,thedropfromunitytozerooccursverysharplyneartheboundaries.Forlowerorderfunctionsthechangefromunitytozeroismoregradual.Now,whiletheregularizationfunctionsmaintainedgeconditioncompatibility,theyalsohavetheeffectofsimulatingaweakeningtemperaturegradientneartheboundaries.ThisfeaturecanbeseenfromEquations 4 and 4 .Thelowertheorderoftheregularizationthemorepronouncedtheeffectofweakenedtemperaturegradients.Thus,forexampleinEquation 4 ,whilemaintainingedgecondition 79

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Figure4-11. TheregularizationfunctionforvariousintegersP. compatibility,thenumericalalgorithmgeneratesaweaktemperaturegradientandthisinturnweakensthevelocitygradientwhichwouldencouragecellularow.Thusweconjecturethatthelowerregularizationwouldbeexpectedtohaveaneffectsimilartotheweakeninggradientgeneratedbyadeectingsurface.Ifourconjecturewerecorrectwemightalsoexpectareductioninthecellularowforthethinlayercaseathigherorderregularization.Tothisendtwocomputationsareproduced,oneforlowregularizationandtheotherforhigherorderregularization.IneachcasethesurfaceBiotnumberofzeroistakenthusemphasizingtheregularizationeffectonthemodulationofthe'x'and'y'gradientsofthesurfacetemperature.FirstobservetheeffectontheowstructureswiththeregularizingpolynomialofP=9asdepictedinFigure 4-12 .ThecomputationsaredoneforthesamelmdimensionsthatwerepresentedintheexperimentalworkshowninFigures 4-8 and 4-9 .Thesesimulatedowsaredistinctlysheetowforthethickerlmandcellularowforthe 80

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Figure4-12. ComputationalimagesofpathlinesforLxof2mm,Lyof4mmanddepthsof0.6mmand0.2mmassumingatemperaturedifferenceof4.1Kusinghighdegreeregularization.Thegray-scaleindicatesthetransitioninisothermswiththeleftwallheated.Thetemperaturescaleisnormalizedwithrespecttothetotaltemperaturedifference.RegularizationpolynomialofP=9isused. thinnerlm.Thismakessenseasthetemperaturegradientnowgraduallydecreasesneartheboundariesduetothedegreeofregularizationthatwasused.Theowisthenprincipallydeterminedbythesamescalinglawthatwaspresentedearlierinreferencetotheexperiment.CalculationscorrespondingtotheconditionsofFigure 4-10 alsoshowqualitativeagreementandaredepictedinFigure 4-13 assuringusthatevenarestrictedmodelcancapturethemainfeaturesoftheowstructures.NownotetheeffectontheowstructurewhentheregularizationofP=18isused.Twointerestingobservationscanbemade.First,fortheexperimentalconditionsofLx=2,Ly=4andd=0.2mmthecellular-likenatureoftheowisseverelyreducedandreplacedwithasheet-likeowprole.ThisisseenclearlywhenFigure 4-14 iscomparedwithFigure 4-9 .Ontheotherhand,forthecaseofLx=4,Ly=18andd=0.6mmevenavalueofP=18doesnotsufcetogetridofthecellularow,asseen 81

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inFigure 4-14 .ThereasonforthisistheweakvelocitygradientthatpersistsforlargeLx,indicatingthatevenhighervaluesofPareneededwhenthedimensioninthexdirectionisincreased.InnocasewerethecomputationscarriedoutforpolynomialslargerthanP=18asverylargeChebychevgridsand,consequentlylargecomputationtimes,areneededwithincreasingpolynomialdegrees. Figure4-13. Computationalimagesforadepthof0.6mm,Ly=18mmandLx=4mm,topimage,andLx=2mm,bottomimage.Theimposedtemperaturedifferencewasabout4Kwiththeleftwallheated.Thetemperaturescaleisnormalizedwithrespecttothetotaltemperaturedifference.RegularizationpolynomialofP=9isused. Inlightofthedifferencesbetweenthehigherorderandlowerorderpolynomialsimulations,thenumericalmodellendscredencetothehypothesisthatthemainfactorcausingmultipleowstatesisthemagnitudeofthetemperaturegradientsneartheboundaries.Whenthesetemperaturegradientsarerelativelyweak,ascomparedtoaperfectlyatinterfacetemperaturedistribution,thenthescalingargumentpresentedinthesectiononexperimentalresultswilldeterminetheow. 82

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Figure4-14. ComputationalimagesforLxof2mm,Lyof4mmanddepth0.2mmandLxof4mm,Lyof18mmanddepth0.6mm.Thegray-scaleindicatesthetransitioninisothermswiththeleftwallheated.RegularizationpolynomialofP=18isused. Thepresentsimulationisnotmeanttobeareplacementforafullnumericalmodelincludingsurfacedeformations,buttheresultsarestillpresentedhereastheydoshowgoodqualitativeagreementwiththeexperiment,aswellasprovidingadditionalevidenceinsupportofthehypothesisforexplainingowstructures.ItshouldbenotedthatchangingthesurfaceheattransfercoefcientorBiotNumberfromanadiabaticcaseinthecomputationsdoesnotchangeanyoftheprolesqualitatively.Alsochangesindepthsbyasmuchas0.05mmforthecasesunderstudydidnotaffecttheresultsqualitatively.Thisisimportantasmachiningaccuracyforthesmalldepthswasonlywithin0.05mm.Finally,computationsalsoshowsimilarqualitativebehaviorforsinglefreesurfacelms;inotherwordsourreasoningholdsforsingleortwo-freesurfacelms. 83

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CHAPTER5THERMOCAPILLARYINSTABILITYINADOUBLEFREESURFACEFILMATHIGHTEMPERATUREGRADIENT 5.1ExperimentalTechniqueTheexperimentalprocedurefollowsthemethodoutlinedinSection 4.3.1 .However,forexperimentsintheinstabilityregimeamuchlargertemperaturegradientisneeded.ThecoldtemperaturewallposednosignicantissuesbecauseauidmixtureofwaterandethyleneglycolcouldbeheldataconstantlowtemperatureandusedtomaintainTcatasetvalue.Ontheotherhand,thehotwallneededtobeelevatedabove100oCandasuitableworkinguidcouldnotbefoundwhichwouldeliminateanypotentialsafetyhazardsinthecaseofevaporation.Therefore,aresistiveheatingelementwasusedwhichcouldbecontrolledbyadjustingavariacsettingtodeterminehowmuchelectriccurrentwouldpassthroughtheresistor,andhencewhattemperaturetheresistanceheaterwouldproduce.Typicaltemperaturedifferencesneededtoobservehydrothermalwaveswereintherangeof10)]TJ /F4 11.955 Tf 12.65 0 Td[(15oCandthecoldtemperaturelimitwasabout)]TJ /F4 11.955 Tf 9.3 0 Td[(30oC.Figure 5-1 illustrateswhythehottemperaturestillneededtogoabove100oCtorealizea10oCtemperaturedifferenceacrosstheliquidlm.Intheidealsetup,nearlyalloftheheatowingthroughthemetaltestsectionmustpassthroughtheoillmintheprocessofreachingthecoldsource.Forthecaseoftheactualsetup,theheattakesthepathofleastresistanceandsimplyowsthroughthemetalinordertoreachthecoldside.Ofcoursetherewillbeasmalleramountwhichtravelsthroughtheoillmbutsincetheconductivityofoilismuchlessthanthatofmetal,theamountofheatpassingthroughwillbecorrespondinglylowerintheoil.Asimpleremedyforthisheattransferscenariowastousealargeoveralltemperaturedifferencewhichwouldresultinadifferenttemperaturegradientattheoillm.Oneissuewithusingsuchalowcoldtemperature()]TJ /F4 11.955 Tf 23.29 0 Td[(30oC)isthepossibilityofcondensationformingonthemetalsection.Ifthecondensatemixedwiththeoillm 84

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Figure5-1. Heatowthroughrealandidealtestsections theexperimentwouldbecompromised.Toavoidthis,thecoldtemperatureandhottemperatureweredecreasedandincreasedgraduallysothattheheatwouldpreventcondensationformationonthemetal.Eventually,thecoldsidewouldreachit'slimitandthehotsidewouldbeslowlyincreasedbyitself.Duringthisprocess,thevariacwouldbeincreasedatsmallincrementsandthesystemlefttosettletoasteadystatefor10)]TJ /F4 11.955 Tf 12.22 0 Td[(15minutesbeforeadjustingthetemperatureagain.Atsomepointduringthisprocedure,hydrothermalwaveswouldarisewithinthesystemandavideowasinstantlyrecordedofthetemperatureuctuations.Thevideoframesofthehydrothermalwavescouldbeusedtoreadthetemperaturedifferenceatwhichthewavesoccurred.Theframescouldalsobetreatedbytakinganaverageofalltheimages,subtractingthisaveragefromeachindividualframe,andthenusingthetemperaturedifferenceimagestoviewavideoofthehydrothermalwavespropagatingthroughthelm. 5.2PhysicsoftheInstabilityThedoublefree-surfacesystemdiscussedhererelatestotheinnitelayer,recirculatingowsetuppresentedbySmithandDavis[ 36 ].Forrecirculatingow,theonlypossibleinstabilityisthatofhydrothermalwaves,whichwillappearandpropogatethroughthelmafteracriticalMarangonivalueisreached.Themechanismofthisinstabilityisdescribedasfollows. 85

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Imaginetravelingalongauidinterfacewiththermocapillaryforcing.Now,sinceanynaturalsystemissubjecttosmallperturbations,eventuallyapointonthesurfacewillbereachedwheretheinterfaceisdeformedbydisturbancesbetweenthetwosidewalls.Atthislocation,therewillbepeaksandtroughswithlocalheatingaffectstakingplace.Dependingonthestructureofthebaseow,differentinteractionswilloccur.Theinstabilitymechanismforsheetowandcellularowarepresented.ThesysteminvolvingsheetowispresentedinFigure 5-2 .Forsheetow,theuidstartsatthehotsource,movestowardsthecoldwall,losingheatalongtheway,andthendipsintotheuid,losingmoreheatastheuidtravelsverticallyalongthecoldwall,beforetravelingbacktowardsthehotwallalongthemid-plane.Thus,itisunderstoodthatthesurfaceoftheuidisatahighertemperaturethanthemid-planeoftheuid. Figure5-2. Instabilitymechanismforthecaseofsheetow Wheneveraperturbationdistortstheinterface,thetroughsofthesurfacewhicharenowclosestthethemidplanewillbecomealocallycoldregion.Now,thiscoldregionwillhaveacorrespondinglyhighersurfacetensionanditwillpulluidfromthepeaks. 86

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Theuidfromthepeakswillbegintoheatthecoldertroughs,butatthesametime,thisinterfacialowwillcausethepeaktosuckincolduidfromthemid-planetoconservemass.Sincethepeakisalsocooled,itwillnotbeabletoheatthetroughbacktoit'soriginalbasetemperature.Hence,theperturbedowwillpersist.Thebasethermo-capillaryowwillbesuperimposedonthisperturbedowsetup.Asaresult,acontinuousstreamofrelativelyhotuidisbeingpumpedintoeverypointalongtheinterface.Thecompetitionbetweencolduidbeingpumpedfromthemid-planetotheinterfaceandthehotuidfeedingintotheinterface(i.e.competitionbetweenperturbedowandbaseow)willcausetemperatureuctuationsandthesetemperatureuctuationsareknownashydrothermalwaves.Forthecaseofcellularow,Figure 5-3 providesaschematicofthearrangement.Forcellularow,theuidwillloseheatfromthefreesurfacescausingthemid-planeoftheliquidtobehotter. Figure5-3. Instabilitymechanismforthecaseofcellularow Now,whenaperturbationdistortstheinterface,thetroughsofthesurfacewillexperiencealocalheatingcausingahotpointattroughs.Thishotregionwillhavea 87

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relativelylowersurfacetensionandtheuidwillbepulledfromthepeaksoutofthetroughs.Thismovementcausesuidtoowfromthemid-planetothetroughs,causingfurtherheatingandtherebyreinforcingtheperturbedowintheformofcirculationcells.Again,thebaseowissuperimposedonthisandhotuidispumpedintoeverypointalongthissurface.Thisadditionofhotuidonlyreinforces,oratleastdoesnotoppose,theperturbedow.Hence,itwouldseemthatakindofMarangonicirculationcellisencouragedduetothecellularbaseowandhydrothermalwavesarenotexpected. 5.3TheCriticalPointandFilmGeometryThepredictionofinstabilitycriticalpointscanbeacomplicatedmatter.Thisisnodifferentforthethermo-capillarylmsystem.Sometimesdimensionlessnumberscanbeofassistanceinpredictingthesevaluesbutinathermo-capillarysystemtherearemultiplequantitiestoworryabout.Whenaninstabilityexists,therewillbenonlinearphenomenainvolved.Here,boththemomentumandenergybalancesdescribingthesystemcontaindifferentnonlinearquantitiesasshowninEquations 5 and 5 . Ma Pr"@~v @t+~vr~v#=rp+r2~v(5)and Ma@T @t+~vrT=r2T(5)BothequationscontainanonlineartermscalebytheMarangoninumber.ThisisinformativewhenworkingatlowMabecauseinthatscenariobothnonlineartermswillbeverysmallandhencenegligable.However,asthetemperaturegradientisincreasedandanunstableregimeisreachedthereisnowaytoknowwhichnonlineartermiscontributingorbyhowmucheachnonlineartermiscontributingtotheinstability.Thismakesitverydifcult,maybeevenimpossible,topredictcriticalvalueswithoutempericaldataornumericalsimulations. 88

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Theworkpresentedherefocusedonvaryingfamiliesoftestsectionssothatasinglelength-dimensioncouldbeanalyzedtodetermineit'seffectonthecriticaltemperaturedifferenceatwhichhydrothermalwavessetin.Thehypothesesforlmsdisplayingsheet-owareasfollows.Holdingotherlengthsconstant,whenthedepthofthetestsectionsisincreased,themid-planeoftheuidwillbecolderbecausetheliquidwillspendalongertimeincontactwiththecoldwallasithastotravelalongerdistancethroughthedepth.ThisisdepictedinFigure 5-4 .Asaresult,thetemperatureatatroughofaperturbationwillbelowerandtherewillbealargertemperaturedifferencebetweenthisperturbedtemperaturesetupandthehotuidenteringthesepointsduetothebaseow.Sincethedifferencebetweentheperturbedandthebasestateismoreextreme,itwillmakeiteasiertotipoverintotheunstableregion,thusloweringthecriticaltemperaturedifferenceforincreasingdepths. Figure5-4. Depictionoftheowalongthecoldwallasthelmdepthischanged.Theuidbecomescolderfordeeperlms. Now,holdingthedepthanddistancebetweentemperaturecontrolledwallsconstant,considerwhenLyisvaried.AsshowninFigure 5-5 ,ifLyisincreased,thiswillintroducelongerwavelengthdisturbancesintothesystem.Thesewavelengthswillonlyhaveaneffectiftheyaremoredisturbingthatthesmallwavelengthsthatalreadyexistwithinthesystem.Smallwavelengthsareexpectedtobemoredisturbingbecausethere 89

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isasmallerdistancebetweenpeaksandtroughsofaperturbationmakingiteasierfortheperturbedowtosetin.So,thetwopossibilitesherearethatthelongerwavelengthdisturbanceswillhavealmostnoeffectonthecriticalpointortheywillnotmakeiteasierforperturbedowtooccurandthereforewillincreasethecriticaltemperaturedifferencewhenLyisincreased. Figure5-5. Depictionofvariouswavelengthdisturbancesasthespanwiselength,Ly,ischanged.LongerwavelengthscanenterintothelargerLylms. Finally,holdingdandLyxedandvaryingLx.Aswaspreviouslymentioned,theMamustbecomesufcientlylargebeforenonlineareffectswillimpactthesystem.SinceMaisproportionaltothetemperaturegradient,thenifLxisincreasedtheMawilldecrease.TomakeMalargeenoughfornonlinearities,thetemperaturedifferenceshouldbeincreased.Inotherwords,forasettemperaturedifference,ashorterLxlmwillhavestrongerinterfacialow,aspresentedinFigure 5-6 .Thehydrothermalwaveinstabilityessentiallyarisesoncetheowreachesahighenoughspeed.Therefore,alargercriticaltemperaturedifferenceisexpectedforincreasingLx.Table 5-1 givestheexperimentallydeterminedcriticalpointsforseverallmgeometries.Thereisatleastapairoflmsvaryingeachlengthdimensionwhileholdingtheothersconstant.SeveralobservationsweremadefromthedatapresentedinTable 5-1 .Atrstglance,itwouldappearthatallthreehypothesesturnouttobetrue.Whenthedepthisincreasedfrom0.6to1.2mmforeitherthe4or18mmspanwiselms,thecriticaltemperaturedropssignicantly.WhenLyisincreasedfrom4to18mmatadepthof0.6 90

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Figure5-6. DepictionofthemagnitudeoftheinterfacialowwhenLxischangedwhileholdingthetemperaturedifferenceconstant. Table5-1. Valuesforcriticaltemperaturedifferencewherehydrothermalwaveswereobservedinvariouslms FilmGeometry(Ly,Lx,d)mmAverageCriticalToCStandardDeviation 18,1,0.612.42.218,2,0.614.91.318,2,1.26.30.84,2,0.611.30.84,2,1.26.30.84,2,1.57.21 thecriticaltemperaturedoesincreaseslightly,asexpected.WhenLxisincreasedfrom1to2mmforthe18mmspanwiselm,thetemperaturedoesappeartoincrease.Alloftheseagreewiththeexpectationsforchangeincriticalpoint,however,thereisnotyetenoughdatatobesureofthesestatements.Forthe1and2mm,18mmspanwise,lmsthestandarddeviationsoverlap.Sincethesamplesizesarestillrelativelysmall,about5datapointspertestsection,theconclusionsinferredfromthisdatamustbeputthroughfurthertesting.ThesameistrueforthedepthandLyvaryinglmsasthe4x2x1.2to4x2x1.5mmsectionsareunclearalongwithcomparisonofthe4x2x1.2with18x2x1.2mmlms.Asitstandsnow,thehypotheseslaidoutabovearesupportedbytheexperimentalresults,andthereasoningissound,butmoredataneedstobeobtainedbeforeconrmingthehypothesesastrue. 91

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Forlmsdisplayingcellularow,thehypothesisisasfollows.Sincethebaseowandtheperturbedowbothfeedrelativelyhotuidalongtheinterface,itwouldappearthatthecirculationcellsgeneratedintheperturbedstatewillcontinuetoexistandfeedoffofthebasestate.Upontestingthecellularowhypothesis,itwasfoundthatwhenthetemperaturegradientisincreased,thelmwillrupture.ThiscanbeexplainedintermsofthesurfacedeformationshowninFigure 4-7 .Athinregionexistsnearthehotwallandasthetemperatureincreasestheinterfacialowspeedsupandmakesthedeformationsevenlarger.Atsomepointthetopandbottomsurfacesofthelmwillcontacteachothernearthehotwallandthelmbreaks.Hence,aninstabilityhasnotbeenobservedforanygeometrywhichhasacellularbaseow. 5.4InstabilityCulminationForthermocapillaryow,ahydrothermalwaveinstabilityarisesatlargeenoughimposedtemperaturedifferences.Themechanismforhydrothermalwavescomesfromaninteractionbetweentheperturbedowandbaseowwheretheperturbedowcanbedescribedintermsofay-directiondisturbance.Resultsobtainedsofarappeartosupportthetrendshypothesized.Thecriticaltemperatureis:loweredwithincreasingdepths,increasedforincreasingLy,andalsoincreasedforincreasingLx.Therehasnotbeenanyinstabilityobservedtooccurwithinthecellularbaseowregime.Thisisbecauseofthebasetemperaturedistributionbetweenthelminterfacesandthemid-plane. 92

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CHAPTER6STABILITYOFALIQUIDTHREADINANANNULUSInearlierchaptersthewavysurfacewaseitherimposedorresultedfromforcingtheow.Flowstructuresthatappeareddependedongeometricaldimensionsandthepresenceofanon-wavysurfaceintheproblem.Inthischapterawavysurfacewillspontaneouslyappearandyettheowstructurewillagaindependonthegeometricaldimensionsandthepresenceofasecondnon-wavywall.Tothisendweconsideracylindricalliquidthread.Thecongurationofacylindricalliquidvolumeof”innite”extentintheaxialdirectionisknownasaliquidjet,alsocalledaliquidthread.Liquidjetshavebeenusedintechnologiesrangingfrominkjetprinterstoowthroughanoilpipeline.JosephandRenardyhavedoneseveralinvestigationsoncore-annularowwhichcanbeusedtotransportoilviaalubricatedpipeline[ 37 ].Thetopicisalsoofinterestinsafetyaboardtheinternationalspacestation(ISS),orothermicrogravityenvironments.OneofthemostlikelycausesofreaboardtheISSiswhenashortcircuitoccursandawirebecomesoverloadedwithcurrent.Ifthishappens,theinsulationaroundthewirecancombustandanunderstandingofhowthisinsulationmeltwillbreakupcanbehelpfulinoptimizingsafety[ 38 ].Perhapsthemostimportantcharacteristicofaliquidjetisit'sstability.Acylindricalvolumeofliquidbecomesunstablewhenit'slengthbecomesgreaterthanit'scircumference.Sincealiquidthreadisverylong,itwillalwaysbeunstable.However,ifthegrowthrateofthejetinstabilityisverylow,thenthejetmightappeartoretainit'scylindricalstateovertheperiodwhichitisobserved.Ifthegrowthrateislargeenough,asisusuallythecase,thenapatternofdropletsisformedasthejetbreaksup,showninFigure 6-1 .Inthecontextoftheapplicationsmentioned,liquidjetinstabilityisneededforinkjetprinterswherethedropletsformedfromathreadofinkareguidedontoasheetofpaper 93

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Figure6-1. Pictureofacylindricalthreadofhoneyatthreedifferenttimesonitswaytobreakup.Theearliestpictureisontheleftandthelatestontheright.PictureisfromJohnsandNarayanan[ 2 ]. tocreatepictures.Ontheotherhand,incore-annularowthroughapipeline,jetstabilityissoughtsothatthelubricatinguidandtheoilcanbeseparatedmoreeasily.Inthecaseofaliquidjet,therearetwocompetingforceswhichallowforpatternformation.Thedestabilizingeffectarisesfromthetransversecurvatureofthesystem.AsshowninFigure 6-2 ,thereisapressuregeneratedfromtheperturbedjetsurface,similartothepressurecreatedinsideofabubble.Thepressureinsideofabubbleisgreaterthanthepressureoutsideandintheliquidjet,thepressureinsideoftheinterfacewillalsobehigherthanambient,duetotransversecurvature.Thispressureincreasewillbegreaterinthinregionsandlessinthickregions.Thiscreatesowgoingfromthetroughtowardsthepeakregionsandactstoripthejetapart.Ontheotherhand,thereisalsoapressurecreatedfromthelongitudinalcurvatureofthesurface.AsseeninFigure 6-3 ,sincethepressure(P)frommeancurvature(2H)goesas,P/)]TJ /F4 11.955 Tf 24.07 0 Td[(2H,thenthepressurewillbehigherwherethenormalvectorgivesnegativecurvatureatacrest.Likewise,thelongitudinalpressurewillbelowerwherethe 94

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Figure6-2. Depictionoftransversecurvatureinaperturbedjet.Theboldarrowsindicatetheshapeofthecurvaturebeingconsidered.Thethinarrowsrepresentthedirectionofow.Flowwillgofromtroughtopeak. normalgivespositivecurvatureatatrough.Thisresultsinowfromthepeakstowardsthetroughsandwillacttoreformthecylindricalsurface,i.e.tostabilizethesystem. Figure6-3. Depictionoflongitudinalcurvatureinaperturbedjet.Theboldarrowindicatestheshapeofthecurvaturebeingconsidered.Thethinarrowsrepresentthedirectionofow.Flowwillgofrompeaktotrough. Theaimofthisworkwastodeterminewhetherornotshiftinganencapsulatedliquidjetoff-centercouldenhancethestabilityofthejet.Thisuidarrangementissimilartothatwhichisfoundincore-annularow.Ajetwithasolidinnercoreiscoveredbyanimmiscibleencapsulatinguidandcontainedwithinasolidboundary.ThediagramoftheproblemisshowninFigure 6-4 foranr)]TJ /F6 11.955 Tf 12.47 0 Td[(crosssectionofthejet.AsseeninFigure 6-4 ,thejetextendsinnitelylongoutoftheFigure 6-4 .Wepresenttwosetsofcalculations.Therstoneisthestabilityofaviscousandinviscidthreadintheannularconguration.Weobtainmaximumgrowthratesandthenplotthevariationofthemaximumgrowthratewiththeratioofthecoreradiustothebaseradiusoftheliquid-liquidinterface.Somephysicallyinterestingresultsareobtained. 95

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Figure6-4. Thesetupinvestigatedtodeterminestabilitycharacteristics.Thedashedlineontherightgurerepresentstheinitial(oncenter)positionoftheboundary.Thecenterrodandouterboundaryarebothsolidsurfaces. Wethentakethecaseoftheinviscidjetaloneascalculationsaremuchsimplerhereandshowtheeffectoftheoff-centeringofthecoreonthegrowthratesoftheinstability. 6.1On-CenterInterfaceStabilityThereferencedomainofthesystem,withinner,interface,andouterradiilabeled,isshowninFigure 6-5 . Figure6-5. Diagramofthereferencedomainofatwouidjetwithinanannulus. Now,whenthesystemisoncenter,theproblemisverysimilartoaninviscidliquidjetstabilitywhichhasbeencoveredelsewhere[ 2 ][ 27 ].Theinstabilitywillarisesimplyfromnaturalperturbations. 96

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SeveralstabilitycurvesarepresentedinFigure 6-6 .Theabscissaislabeledasthewavenumber,k2,whichisproportionaltoadisturbancewavelengthas,k=2 ,whereisthewavelengthofaperturbationwavealongtheuidinterface.Thepointwherethestabilitycurvesintersecttheabscissaisknownastheneutralpoint.Here,thejetwillbeneitherstableorunstable,butallofthevelocitiesandpressureswillbesteadyatzero.Wheneverthejetislongerthanthewavelengthcorrespondingtotheneutralpoint,thenkwillbesmallerthantheneutralkvalueandtheinterfacewillbeunstable.Sinceajetisinnitelylongbydenition,thenthistypeofsystemisalwaysunstable.Infact,sincethejetisalwaysunstableandislongenoughtoassumeallwavelengths,thenthewavelengthwhichweexpecttoobserveisonlythatonecorrespondingtothelargestgrowthrate.Theon-centerstabilitywasdeterminedforboththeviscidandinviscidcases.Itmakessensethatastheuidsbecomemoreviscous,thenthiswilldamptheinstabilitybecausemoreenergyisbeingdissipatedduetoshear.ThisiswhatweobserveasisshowninFigure 6-6 . Figure6-6. On-centergrowthratesforinviscidandvaryingviscosities.=0.25m,Rinterface=1m,Rout=1.5m,=1500kg m3,=1000kg m3,=0.1N m Thestabilityoftheinterfacecanalsobeadjustediftheviscositiesoftheinnerandouteruidsaredifferent.Whentheouteruidismoreviscousthantheinneruid,thetheinterfacewillbeslightlymorestablethanwhenviscositiesarematched.Ontheother 97

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hand,whentheinneruidismoreviscousthantheouteruid,thentheinterfacewillbemoreunstable.Aninterestingobservationarisesforthetwouidjet.Iftheinnersolidradiusisclosetotheuidinterface,thenitwouldseemtoattentheinterfacedeections,asiftheareabetweenthesolidandinterfacewerealmostasheet,andhenceweakenthetransversecurvature.ThisaccountsforthedecreasingslopeseeninFigure 6-7 . Figure6-7. Changeinthemaximumgrowthrateforinvisciduidsastheradiusratioincreases.=1500kg m3,=1000kg m3,=0.1N m Forviscousuids,whentheradiusratioofinnerradiustotheinterfaceradius, R,isvariedthentherewillbeanincreaseinthemaximumgrowthrateuntilsomeratiowherethemaximumgrowthratedecreases.ThisisseeninFigure 6-8 .Again,thedecreasingslopecanbeattributedtoaweakeningoftransversecurvaturewhichmakesthesystemmorestable.Theincreasingpartofthecurveisaresultofviscosity.Thepressurearisingfromlongitudinalcurvaturewillbeproportionaltok2.Thisistosaythat,forshorterwavelengths,theslopeinthelongitudinaldirectionbecomessteeper.Thissteepercurvewillalsoincreasethelongitudinalpressureaswellasmakek2larger.Additionally,theeffectofviscositywillalsobeproportionaltok2asisseenfromther2terminthedomainequation. 98

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Figure6-8. Changeinthemaximumgrowthrateforviscousuidsastheradiusratioincreases.=1Pas,=1Pas,=1500kg m3,=1000kg m3,=0.1N m Whenviscosityisincludedintheequations,thenitwillacttodampoutboththeowfrominterfacialpeakstotroughs,aswellastroughstopeaks.Sinceboththelongitudinalpressureandtheviscosityscaleask2,theywillbecompetingwitheachother.Whilethelongitudinalpressuretriestostabilizethejet,theviscositywilldampoutthisre-stabilizingow.Hence,thedestabilizingtransversecurvaturewilltakeoverandmakethesystemmoreunstableatsmallradiusratios.Astheradiusratiocontinuestoincrease,thedestabilizingtransversepressurewillbecomeweakeraswasmentionedearlier.Goingfromlowtohighradiusratio,therewilloccuraradiusratiowherethedestabilizingtransversepressureisweakenedenoughsothatthestabilizinglongitudinalcurvaturecanbegintodecreasethemaximumgrowthrate.Thejetwillultimatelybeunstableforeachoftheseradiusratios,butthemaximumgrowthrateforeachcongurationwillvarybasedonthecompetitionbetweenviscosityandradiusratio.Viscositydampensthelongitudinallyinducedowandtheradiusratioweakenstheeffectofthetransversecurvature. 99

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6.2TwoPhaseAnnularLiquidJetModelThissectionwillpresentsomeofthemainequationsneededfordeterminingthestabilityoftheuidinterfacebetweenaninnerliquidandanouterliquid(-uid)whentheyarecontainedwithinanannulushavingsolidinnerandouterboundaries.Whenthesystemisshiftedoff-center,theproblembecomesmoreinteresting.Abase,on-center,stabilitywillbeobtainedforthecasewhennothingisshifted,butnaturalperturbationsexist.Thevariables,duetothenaturalperturbation,willhavetheform:u= u+[u0+R0@ u @r].Here,uisageneraldomainvariableandthe”bar”variablesrepresenttheunperturbedinterfacevariableswhilethe”prime”notationisusedforvariablesaftertheinterfacehasbeenperturbed.Theseperturbedvariableswillstopatrstordersincetheperturbationisinnitesimallysmall.Afterthevariablesareexpandedintermsoftheperturbation,theneachofthe”bar”and”prime”variableswillalsohavetobeexpandedintermsofadisturbance,,whichcharacterizestheoff-centershiftingofthesystem.Thisshiftingcanbecarriedouttohigherorderssincetheoff-centeringrequiresanitedisturbance.Thesedisturbedvariableswillhavetheform:u0=u00+[u01+R01@u00 @r]+....Here,weusethesamenotationasinpreviousperturbationexpansionswherethesubscriptrepresentsderivativesintheexpansionparameter,,andthevariablesareevaluatedatequaltozero.Now,understandinghowbothlevelsoftheexpansionswilltakeplace,wewritetheoriginalnon-linearequationswhichdescribethesystem.Fortheinneruid,ourdomainequationis, @~v @t+~vr~v=rP(6)and r~v=0(6) 100

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whereisthedensity.Likewisefortheouteruid,whereanasterisksrepresentsvariablesintheouteruid, @~v @t+~vr~v=rP(6)and r~v=0(6)Foraninvisciduid,theboundaryconditionsattheinnerandoutersolidboundaryaresimplynoowacrossthoseboundaries.Attheinnersurface,r=, ~v=0(6)andattheoutersurface,r=Rout, ~v=0(6)Additionally,attheinterfacewehavetwomoreconditionsfromourassumptionofimmiscibleuids.Theuidateachsideoftheinterfacemustmoveatthespeedoftheinterfacesothat,atr=R, ~v~n=u(6)and ~v~n=u(6)whereuisthespeedoftheinterfaceand~nisthenormalvectorattheinterface.Additionally,toaccountforthecurvatureoftheinterface P)]TJ /F3 11.955 Tf 11.96 0 Td[(P=)]TJ /F6 11.955 Tf 9.3 0 Td[(2H(6)where2Histhemeancurvatureoftheinterfaceandisthesurfacetensionbetweentheouterandinnerliquids.Now,todeterminewhathappenstotheinterfacestability,wemustrstexpandEquations 6 6 intermsoftheinterfaceperturbation.Doingsowillresultintwo 101

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setsofequations:onefortheunperturbedinterfaceandanotherfortheperturbedstate.Inthecaseofanunperturbedinterface,evenshiftingthesystemoff-centerwillnotgenerateow.Therewillbenodeformationoftheinterfacebywhichtocreatepressuregradients.Thiswillbetakenintoaccountfromhereonout,andallofthe”bar”variableswillbeconsideredzerowhilecontinuingontotheproblemoftheperturbedinterfacestate.However,notethattermssuchastheunperturbedinterfacenormalvector, ~nwillstillbenon-zero.Forthecaseofaperturbedinterface,thedomainEquations 6 6 become @~v0 @t+( ~vr~v0)+(~v0r ~v)=rP0(6)andcontinuityintheinneruid r~v0=0(6)andintheouteruid @~v0 @t+( ~vr~v0)+(~v0r ~v)=rP0(6)withcontinuityintheouteruid r~v0=0(6)Likewise,theperturbedinterfaceboundaryequationsatr=become ~v0+R0@ ~v @r=0(6)andthenoowconditionatr=Routis ~v0+R0@ ~v @r=0(6)Theimmiscibleuidconditionsnowgive ~v~n0+[~v0+R0@ ~v @r] ~n)]TJ /F3 11.955 Tf 11.96 0 Td[(u0=0(6) 102

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andfortheouteruidattheinterface ~v~n0+[~v0+R0@ ~v @r] ~n)]TJ /F3 11.955 Tf 11.95 0 Td[(u0=0(6)Meancurvaturefortheperturbedinterfacewillbe P0)]TJ /F3 11.955 Tf 11.96 0 Td[(P0=)]TJ /F6 11.955 Tf 9.3 0 Td[(2H0(6)Thisgivesoursetofequationsfortheperturbedinterface.Now,sincethe”bar”variablesarezero,theequationssimplifyandwecanbeginexpandingintermsofthedomaindisturbance,.simplycharacterizestheoff-centershiftingasshowninFigure 6-4 .Wecanseehowentersourequationsbyconsideringthetransformationfromthebasetothedisturbedproblem.Lettingtheouterboundarybeshiftedbyanamount,thenwecanrelatethereferenceandcurrentdomainby [X)]TJ /F6 11.955 Tf 11.95 0 Td[(]2+[Y]2=R2out(6)whereXandYarethexandy-coordinatesattheboundary.ChangingfromCartesiantopolarcoordinatesgives,X=Routcos()andY=Routsin().SubbingtheseintoEquation 6 resultsincorrectionstothedisturbedradiusateachorderof.Tozerothorder, Rout0=Rout0(6)atrstorder Rout1=cos()(6)andtosecondorder Rout2=cos2() Rout0)]TJ /F4 11.955 Tf 22.79 8.08 Td[(1 Rout0(6) 103

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Thecorrectionstotheouterradiusiswheretheproblemwillpickupit'sthetadependancyandsowearenowinapositiontoobtaintheremainingequationsfortheoff-centerperturbedjet.Droppingthe”bar”terms,Equations 6 and 6 areexpandedinasfollows.Thezeroorderdisturbeddomainequationsare @~v00 @t=rP00(6)andcontinuity r~v00=0(6)andwewillhavethesamedomainequationsintheouteruidexceptwithvariables.Thenoowconditionattheinnersurfaceis ~v00=0(6)andsimilarly,attheouterboundary ~v00=0(6)Attheinterface,theimmiscibleuidsconditiongives ~v00 ~n0)]TJ /F3 11.955 Tf 11.95 0 Td[(u00=0(6)andtobalancetheouteruid ~v00 ~n0)]TJ /F3 11.955 Tf 11.96 0 Td[(u00=0(6)Meancurvaturetozeroordergives 2H0=[1 R20+1 R20@2 @2+@2 @z2]R0(6)Therstorderdomainequationis @~v01 @t=rP01(6) 104

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withcontinuityas r~v01=0(6)Theboundaryconditionsaresimilartothoseatzerothorder,withnoowattheinnerboundarybeing ~v01=0(6)Noowattheouterboundarybecomes ~v01+R0out1@~v00 @r=0(6)Theimmiscibleinterfaceresultsin ~v00 ~n1+~v01 ~n0=u01(6)and ~v00 ~n1+~v01 ~n0=u01(6)andtollouttherstordersetofequations,meancurvatureis P01)]TJ /F3 11.955 Tf 11.95 0 Td[(P01=)]TJ /F6 11.955 Tf 9.29 0 Td[(2H01(6)Atsecondorder,thedomainequationsagainhavethesamestructuresincetherearenoinertialterms.Onthedomainwehave @~v02 @t=rP02(6)andcontinuityas r~v02=0(6)asusual,thesameequationsexistfordescribingtheouteruid.Noowattheinnerboundarygives ~v02=0(6) 105

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andnoowattheouterboundary, ~v02+2Rout1@~v01 @r+R2out1@2~v00 @r2+Rout2@~v00 @r=0(6)Theimmiscibleconditionattheinterfaceproduces ~v02 ~n0+2~v01 ~n1+~v00 ~n2)]TJ /F3 11.955 Tf 11.96 0 Td[(u02=0(6)andfortheouteruidattheinterface ~v02 ~n0+2~v01 ~n1+~v00 ~n2)]TJ /F3 11.955 Tf 11.96 0 Td[(u02=0(6)andthenalcondition,meancurvature,attheinterfaceis P02)]TJ /F3 11.955 Tf 11.95 0 Td[(P02=)]TJ /F6 11.955 Tf 9.29 0 Td[(2H02(6)Wherevertheyappearinthepreviousequations,thenormalvectorandmeancurvaturearegivenby, ~n=^ir)]TJ /F4 11.955 Tf 10.77 2.66 Td[(^i1 R@ R @)]TJ /F4 11.955 Tf 10.76 2.66 Td[(^iz1 R@ R @z [1+(1 R@ R @)2+(1 R@ R @z)2]1=2(6)and 2H0=[1 R2+1 R2@2 @2+@2 @z2]R0(6)Now,ateachorder,thedependencecanbeseenfromtheconditionsattheouterboundary,Equations 6 , 6 ,and 6 .Thezdependencewilltakenormalmodesoftheformeikz.Timewillalsobedescribedbyanexponentialdependence,etwherethegrowthrate,willdeterminetherateatwhichavariablegrowsordies.Thesystemofequationsissolvedunderthesedependenciesandthegrowthratesoftheinstabilityareobtained. 6.3Off-CenterInterfaceStabilityThelowestordercorrection,duetooff-centeringthesystem,comesinatsecondorderin.Thisshouldbeexpectedbecauseourproblemactsthesamewhetheritis 106

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shiftedby+=)]TJ /F6 11.955 Tf 12.3 0 Td[(.Thisistosaythat,regardlessofwhetherthesystemisshiftedbyanamount,,totheleftortotherightthentheinterfaceshouldstillhavethesamegrowthrate.Hence,torstorderwendthatthegrowthrateissimplyzero,i.e.21=0.Infact,alloddordersofthegrowthratewillbezeroforthissamereason.Onlytheevenordertermswillprovideacorrection.Theoff-centershiftingwasonlysolvedfortheinviscidjet.AnanalyticalsolutionwasobtainedandtheresultsareshowninFigure 6-9 . Figure6-9. Thelowestordercorrectiontothegrowthratewhenthesystemisshiftedoff-center.Parametersusedare=0.25m,Rinterface=1m,Rout=1.5m,=1500kg m3,=1000kg m3,=0.1N m. Althoughitmaynothavebeenobviousinitially,itisseenthattheoff-centershiftingopensupextrapathwaysbywhichtheuidpressurecandissipate.Theseextrapathwaysrelievesomeofthepressurebuiltupinsidetheuidandthisactstostabilizetheinterface.Noticethattheneutralpointdoesnotchangeuponoff-centering.Thisisbecauseattheneutralpointallofthepressuresandvelocitiesareexactlyzero.Thereisnothingtobeaffectedbyshiftingtheboundariesandsotheneutralpointwillnotvarybetweentheon-andoff-centerdomains. 107

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Thejetisstabletosmallwavelength(largewavenumber)perturbations.Inthisstableregion,theoff-centeringappearstomaketheperturbationslessstable.However,thetimedependenceactuallygoesasetandwemusttakethesquarerootofthesenegative2terms.Thisresultsintwovaluesofinthestableregion,andthesevaluesareimaginarycomplexconjugatesofeachother.Thismeansthatthesystemwouldcontinuetooscillateaboutthesesmallinterfacedistortionsbecausethereisnoviscositytodampoutthestabilizingperturbations.However,sincethelongerwavelengthsexistintandemwiththeshorterones,thenthejetwillultimatelybreakupataspeeddeterminedbythemaximumgrowthrate.Anotherobservationisthatthewavenumbercorrespondingtothemaximumgrowthratecanalsobeadjustedbytheoff-centershifting.Noticethattheminimumofthesecondordercorrection,inFigure 6-9 ,correspondstoadifferentwavenumberthanthemaximumvalueofthezeroordergrowthrate.Thenewmaximumgrowthrate,resultingfrom2=20+1 2222,canhaveamaximumcorrespondingtoawavelengthwhichislargerthanthezeroorderwavenumber. 108

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CHAPTER7CONCLUDINGREMARKSThethreecasestudiespresentedinthisdocumenthavemanysimilarfeaturesandyetdifferintheirndingsandtheirresults.Thephysicalproblemsareconnectedtowavysurfaces,thesurfacessometimesbeingimposedbyconstruction,byconsequenceofthedrivingforceitself,orarisingfromaninequlibriumofahydrostaticstate.Ineachoftheproblems,pressuregradientsplayamajorrole.Pressuregradientsareinducedasaresultofthewavysurfaceandacttocauserecirculationinsystemsthatcommencedowwithoutrecirculation.Finally,ineachcasetheproximityofthesecondaryboundaryalsochangeseithertheowstructureorthedynamicsinsuchawayastochangetheowstructureswithineachcaseunderconsideration.Theresultsforeachsystemwillbesummarized. 7.1TheLid-DrivenWavyBoundaryForlid-drivenowoverawavyboundary,circulationcellsariseduetopressurebuildupintheregionswherethewavyboundaryisclosesttothemovinglid.Thepressureincreasesinthisregionsincetheowissqueezedandwhentheboundaryamplitudecreatesahighenoughpressure,thentheuidapproachingtheconstrictedregionwillreverseitsdirectionandcirculationcellsmayresult.Thisispredictedevenforveryslowlymovinguidinapurelylaminarregimebutonlyifthewaveamplitudeislargeenough.Thepredictionthatisbasedonlongwaveanalysisissurprisingbecauseitiscounterintuitivetoexpectlongwavetheorytopredictanyformofrecirculation.Takingtheinertiaoftheuidintoaccountactstoskewthecirculationcellssothattheyarenotsymmetricabouttheapexofawave.Inertiaisalsopredictedtohaveasmallinuenceinthecriticalamplitudeatwhichrecirculationsetsin.Theamplitudeneededtocreatecirculationcellswillbeslightlylowerwheninertiaistakenintoaccount.Pressureisneededtoinducecirculationcellsinthissystembutatverysmallwavelengthsgravityalonecancauserecirculationtotakeplace. 109

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Theworkdonehereistheoreticalandsoexperimentalvericationisneeded.Itwouldbeinterestingtouseamicro-uidicdevicetotestthelongwavelengththeory.Inamicro-uidicsystem,itshouldbeeasiertocreateawavyboundarywithwavelengthmuchlongerthanthechannelheight.Iftheuidisdrivenbyapressuregradientthennorecirculationisexpected,butiftheuidissheared,thencirculationcellsshouldbepresentforamplitudeslargerthanabout30%ofthechannelheight. 7.2ThermocapillaryFlowinaDoubleFreeSurfaceFilmThemaingoalofstudyingthermocapillaryowwastoprovideanunderstandingoftwodifferentowstructureswhichwererstobservedaboutadecadeago.Thisresearchprovidesanexplanationofthephysicsbehindtheowstructureselection.Theargumentsarebasedaroundheattransferfromthelminterfaceandsimplescalingarisingfromcontinuity.Essentially,whenthetemperaturegradientbecomesweakastheowapproachesthetemperaturecontrolledwalls,thentheowwillobtainacellular-likestructure,butiftheuidonlyslowsveryclosetothetemperaturecontrolledwalls,thentheowwillobtainasheet-likestructure.Sincethedifferenceintemperaturegradientsforthetwoowstructuresiscausedbyinterfacecurvatureeffectsontheheattransfer,thennaturallytherewillalsobeadifferenceinpressureduetosurfacecurvatureforbothcases.Whentheinterfacecurvatureismostextreme,thetemperaturegradientsbecomeweakandcellularowresults.Thissameinterfacecurvaturewillalsocreaterelativelyhigherpressureunderacrestnearthecoldwallandlowerpressurebetweentroughsnearthehotwall.Thiswillworkintandemwiththeweaktemperaturegradientandcauseadditionalslowingoftheuidasitmovesfromthehotwalltowardsthecoldwall.Ontheotherhand,whenthesurfacecurvatureisweaker,thenthetemperaturegradientswillbeclosertothatofaatinterfaceandthehighandlowpressurenearthecoldandhotboundarieswillalsobeminimal.Hencetheowwillspeedtowardsthe 110

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coldwalluntilitmusttwistintothedepthandreturnalongthemid-planeintheformofsheet-likeow.Whenhighertemperaturegradientsareapplied,thisworkproposeshypothesisforhowthecriticaltemperaturegradient,wherehydrothermalwavesappear,willbeaffectedbythesystemgeometry.Experimentaldatainsupportofthesehypotheseswasprovided.Itwouldbebenecialtoseemeasurementsoftheinterfaceshapeforsomeoftheselms.Althoughthesurfaceshapemightnotbeasexactaswasdepictedinthisheuristicmodel,thesurfaceshapeshouldstillbesimilar.Itisexpectedthatmeasurementswouldshowabulgeclosetothecoldwallandathinnerinterfaceshapenearthehotwall.Cellularowisexpectedtogeneratemultiplecellsasthespan-wisedimensionisincreased.Itwouldbeinterestingtoverifyandunderstandifthereissomewaytopredictthiscellsplitting. 7.3StabilityofaLiquidThreadinanOff-CenterAnnulusThisworkcomparedthestabilityofatwo-phaseliquidjetinsideofasymmetricannulustothatofatwo-phaseliquidjetinanoff-centerannulus.Aliquidjetbecomesunstablebecausetherearetwocompetingpressures.Onetriestoripthethreadapartwhiletheotheractstostabalizetheinterface.Thiscompetitionbetweentransversepressureandlongitudinalpressureresultsinthebead-likepatternwhichisobservedforliquidjetbreakup.Resultsfromthetheoryonaninviscidtwo-uidjetinsideofanannulusshowthatthestabilityoftheinterfacecanbeenhancedfromshiftingthesystemoff-center.Theinterfacewillstillbeunstableandtheresultingpatternsafterbreakupwillundoubtedlybesomewhatdifferent,butthejetwillbreakupmoreslowlyintheoff-centercase.Additionally,themostdestabilizingwavelengthcanbechangedduetotheoff-centering.Foranevenbetterunderstandingoftheeffectsofoff-centeringonaliquidjet,aviscousmodelneedstobeobtainedandisrecommendedforfuturework. 111

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Inmoregeneralregardstoaliquidjet,acuriousinteractionseemstotakeplacebetweenthevelocityofajetanditsstability.Thiscanbeseeneasilyinakitchensinkbyrunningalowowratethroughthefaucet.Itiscleartoseethatthethinjetofwaterbecomesunstable.Iftheowrateisincreased,andhencethevelocityofthewaterincreased,thenthesurfaceofthismovingthreadseemstoremainstable.Abetterunderstandingofthisphenomenaisneededanditseemsthatatheoreticaltreatmentofthisproblemcouldbeveryinsightful. 7.4PerturbationExpansionsonanInconvenientDomainInadditiontothephysicalinterpretationoftheresultsforthevariouscasestudieswehavemadedenitediscoveriesonthePerturbationExpansionsonanInconvenientDomain.ThemainpointsofthediscussiononperturbationexpansionsaresummarizedinreferencetoFigure 7-1 ,redrawnhereforconvenience.ForavariableintheregionofpointA,theexpressionofthisvariableisunderstoodintermsofanexpansionabouttheboundary.ForavariableintheregionofpointC,thevariableisunderstoodintermsofanexpansionaboutanearbyreferencedomainwherethesolutionsmaybemoreconvenientlyobtained.ForvariablesintheregionofpointB,itisnotimmediatelyclearonhowtoexpressthesolutions.Bothways,theboundaryexpansionandthetraditionalperturbationexpansion,appeartobeusable.Theissuewiththeperturbationexpansionisthatthevariableisclosetotheperturbationsourceandso,atlowordersofthesolution,thecorrectionsmaynotbeenoughtogetaccurateresults.Thebenetofusingtheboundaryexpansiontechniqueisthatthevaluesattheboundaryareknownexactly,andnearbytheperturbationsourcetheleadingcorrectionstothisexactvaluewillbeaccurate. 112

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Figure7-1. Variouspointsinthecurrentdomainandtheircorrespondinglocationwhensuperimposedontheancestordomain.Themultiplelocationsmustbetreateddifferentlytoobtainaccurateresults. ThisworkconcludesthattheboundaryexpansionmethodoughttogivebetterresultsforlocationBandthisissupportedbyexampleswiththewavyboundarylid-drivenowandheattransferinslightlyeccentricellipticaldomains.BothexpansionmethodsneedtobeobtainedforanyproblemtotreatpointsAandC.IntheregionofpointB,theboundaryexpansionwillbemoreaccurateneartheperturbedboundaryandatsomelocationfartheraway,theperturbationexpansionwilltakeoverasthemoreaccuratevariable. 113

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APPENDIXANAVIER-STOKESEQUATIONSFORLID-DRIVENWAVYBOUNDARYSYSTEMDetailsabouttherstandsecondorderequationsfromtheNavier-StokesmodelpresentedinChapter 2 aregivenhere.Thegeneraldomainandboundaryequationsare: R(~vr~v)=rP+r2~v+G~ix(A) ~v=Uatz=0(A) ~v=0atz=1(A)UponexpansionoftheNavier-Stokesmodel,intermsofthedisturbanceamplitude,A,wehaveatrstorder, R(~v0r~v1+~v1r~v0)=rP1+r2~v1(A) r~v1=0(A) ~v1=0atz=0(A) ~v1+Z1d~v0 dz=0atz=1(A)andatsecondorder, R(~v0r~v2+~v2r~v0+2~v1r~v1)=rP2+r2~v2(A) r~v2=0(A) ~v2=0atz=0(A) ~v2+2Z1d~v1 dz+Z21d2~v0 dz2=0atz=1(A)Now,similartotheStokessolutionmethod,thex-andz-componentequationswillbewrittenandthentheseequationswillbecombinedinordertocancelthepressuretermleavinganequationintermsofonlyvelocities.Thereforeatrstorder,afternotingthat 114

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vz0iszeroandanyx-derivativesofvx0arezero,Equation A becomes x)]TJ /F3 11.955 Tf 11.95 0 Td[(componentR(vx0dvx1 dx+vz1dvx0 dz)=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(dP1 dx+r2vx1(A) z)]TJ /F3 11.955 Tf 11.95 0 Td[(componentR(vx0dvz1 dx)=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(dP1 dz+r2vz1(A)Togetridofthepressureterm,aderivativewithrespecttozwillbetakenofthex-componentequationandfromthisthederivativewithrespecttoxofthez-componentequationwillbesubtractedwhichgives, R(dvx0 dzdvx1 dx+vx0d2vx1 dxdz+dvz1 dzdvx0 dz+vz1d2vx0 dz2)]TJ /F3 11.955 Tf 11.95 0 Td[(vx0d2vz1 dx2)=r2(dvx1 dz)]TJ /F3 11.955 Tf 13.15 8.09 Td[(dvz1 dx)(A)Finally,togettheequationintermsofonlyvz1anx-derivativeistakenofEquation A R(dvx0 dzd2vx1 dx2+vx0d3vx1 dx2dz+d2vz1 dxdzdvx0 dz+dvz1 dxd2vx0 dz2)]TJ /F3 11.955 Tf 9.53 0 Td[(vx0d3vz1 dx3)=r2(d2vx1 dxdz)]TJ /F3 11.955 Tf 10.73 8.09 Td[(d2vz1 dx2)(A)andapplyingcontinuityintheform,dvx1 dx=)]TJ /F5 7.97 Tf 10.5 4.88 Td[(dvz1 dz,simpliesEquation A into R(dvz1 dxd2vx0 dz2)]TJ /F3 11.955 Tf 11.96 0 Td[(vx0d3vz1 dxdz2)]TJ /F3 11.955 Tf 11.95 0 Td[(vx0d3vz1 dx3)=r2()]TJ /F3 11.955 Tf 10.49 8.08 Td[(d2vz1 dz2)]TJ /F3 11.955 Tf 13.15 8.08 Td[(d2vz1 dx2)(A)Rewritingr2r2asr4,wehave r4vz1+R(d2vx0 dz2d dx)]TJ /F3 11.955 Tf 11.96 0 Td[(vx0d dx(d2 dz2+d2 dx2))vz1=0(A)Thesystemisnowintermsofasinglevariable,buttheequationscanstillbesimpliedsincethex-dependencyisknown.Thex-componentofEquation A isthesameaswhatwasfoundfortheStokesmodel,andsothex-dependencewillbethesameasforStokes,i.e.,vz1=^vz1eikx+~vz1e)]TJ /F5 7.97 Tf 6.59 0 Td[(ikx.ThisallowsEquation A tobewrittenintermsof^vz1as r4^vz1+R(ikd2vx0 dz2)]TJ /F3 11.955 Tf 11.95 0 Td[(ikvx0(d2 dz2)]TJ /F3 11.955 Tf 11.96 0 Td[(k2))^vz1=0(A)wherer4isnow(d2 dz2)]TJ /F3 11.955 Tf 11.95 0 Td[(k2)2andvx0isafunctionofz. 115

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Now,theonlythingthatremainsatrstorderistoconverttheboundaryconditionstobeintermsof^vz1.AfterwritingEquations A and A inx-andz-componentformsitisseenthattheyarethesameastheboundaryconditionsfortheStokesmodel: B.C.d^vz1 dz=1 2^Zikdvx0 dzatz=1(A) ^vz1=0atz=1(A) ^vz1=0atz=0(A) d^vz1 dz=0atz=0(A)Thismeansthatthenalformofthe1storderNavier-Stokesmodelfor^vz1isdescribedbyEquations A A and A .Thesecondorderproblemwillbetreatedbythesameprocessasrstorderuntiltheequationsareintermsofonly^vz2.Therefore,writingEquation A intermsofit'sx-andz-componentsgives, R(vx0dvx2 dx+vz2dvx0 dz+2vx1dvx1 dx+2vz1dvx1 dz)=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(dP2 dx+r2vx2(A)and R(vx0dvz2 dx+2vx1dvz1 dx+2vz1dvz1 dz)=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(dP2 dz+r2vz2(A)Takingd dz(x)]TJ /F3 11.955 Tf 11.95 0 Td[(component))]TJ /F5 7.97 Tf 15.1 4.71 Td[(d dx(z)]TJ /F3 11.955 Tf 11.96 0 Td[(component)leaves R(vx0@2vx2 @x@z+vz2@2vx0 @z2+2vx1@2vx1 @x@z+2vz1@2vx1 @z2)]TJ /F3 11.955 Tf 11.95 0 Td[(vx0@2vz2 @x2)]TJ /F4 11.955 Tf 11.96 0 Td[(2vx1@2vz1 @x2)]TJ /F4 11.955 Tf 9.3 0 Td[(2vz1@2vz1 @x@z)=r2(@vx2 @z)]TJ /F6 11.955 Tf 13.16 8.08 Td[(@vz2 @x) (A) 116

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andupontakinganotherderivativewithrespecttoxandmakinguseofcontinuityintheform,dvx2 dx=)]TJ /F5 7.97 Tf 10.49 4.88 Td[(dvz2 dz,gives R(@vz2 @x@2vx0 @z2)]TJ /F3 11.955 Tf 11.95 0 Td[(vx0r2@vz2 @x)]TJ /F4 11.955 Tf 11.96 0 Td[(2@vx1 @xr2vz1)]TJ /F4 11.955 Tf 11.95 0 Td[(2vx1r2@vz1 @x+2@vz1 @xr2vx1+2vz1r2@vx1 @x)=r4vz2 (A) Uponrearrangingterms, r4vz2+R(@2vx0 @z2@ @x)]TJ /F3 11.955 Tf 11.96 0 Td[(vx0r2@ @x)vz2=R(2@vx1 @xr2vz1+2vx1r2@vz1 @x)]TJ /F4 11.955 Tf 11.95 0 Td[(2@vz1 @xr2vx1)]TJ /F4 11.955 Tf 11.95 0 Td[(2vz1r2@vx1 @x) (A) Onceagainthedomainequationisobtainedintermsofasinglevariable.Therelationvz2=^vz2e2ikx+~vz2e)]TJ /F7 7.97 Tf 6.59 0 Td[(2ikx+vz2indalongwithvz1=^vz1eikx+~vz1e)]TJ /F5 7.97 Tf 6.58 0 Td[(ikxandtherstordercontinuityequationwillbeusedtoputthedomainequationintermsof^vz2.DoingsoreducesEquation A to (@2 @z2)]TJ /F4 11.955 Tf 11.95 0 Td[(4k2)2^vz2)]TJ /F4 11.955 Tf 11.96 0 Td[(2ikR@2vx0 @z2^vz2)]TJ /F4 11.955 Tf 11.95 0 Td[(2ikRvx0(@2 @z2)]TJ /F4 11.955 Tf 11.96 0 Td[(4k2)^vz2=4R[^vz1@3^vz1 @z3)]TJ /F6 11.955 Tf 13.15 8.09 Td[(@^vz1 @z@2^vz1 @z2 (A) Justlikeatrstorder,theboundaryconditionsareagainthesameasthosefromtheStokesmodel: B.C.d^vz2 dz=ik^Z1d^vx1 dz)]TJ /F4 11.955 Tf 13.36 2.66 Td[(^Z1d2^vz1 dz2+^Z21ik1 2d2vx0 dz2atz=1(A) ^vz2=)]TJ /F4 11.955 Tf 10.7 2.66 Td[(^Z1d^vz1 dzatz=1(A) d^vz2 dz=0atz=0(A) ^vz2=0atz=0(A) 117

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andsotheonlythingthatremainsistondthex-independentsecondorderterm.Thex-independentportionofvz2iszero,butthex-independenceofvx2isgivenby d2vx2ind dz2=[~vr~v]x)]TJ /F5 7.97 Tf 6.58 0 Td[(ind(A) vx2ind+[2Z1dvx1 dz+Z21d2vx0 dz2]x)]TJ /F5 7.97 Tf 6.58 0 Td[(ind=0atz=1(A) vx2ind=0atz=0(A)Afterdeterminingthex-independentquantitiestheseequationsfurtherreducetogive d2vx2ind dz2=4Rreal(^vz1d~vx1 dz)(A) vx2ind=)]TJ /F4 11.955 Tf 9.3 0 Td[(2real(^Z1d~vx1 dz))]TJ /F4 11.955 Tf 13.16 8.08 Td[(1 2^Z1~Z1d2vx0 dz2atz=1(A) vx2ind=0atz=0(A)whichresultsinthenecessaryequationstondvz2andvx2. 118

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APPENDIXBDERIVATIVESONAPERTURBEDDOMAINThissectionwillgiveanexampleforageneralperturbedvariabletoillustratethatderivativestakenonacurrentdomainareequivalenttoevaluatingderivativesonareferencedomain.ThesystemofinterestisshowninFigure B-1 . FigureB-1. Systemshowinganarbitraryperturbationwhichgivesageneralcurrentdomain. Takingutobeanarbitrarydomainvariable,thenexpandingaboutequaltozerotogetthevariableonacurrentdomaingives u(x,y)=u(=0)+du d(=0)+1 22d2u d2+...(B)and,lookingattherstordersinceallhigherordertermswillgobythesametechnique,evaluatingthederivativesofuwithrespecttoshowsthat du d=@u @+@u @y@y @(B)Sincethisistobeevaluatedatequalto0,Equation B becomes du d(=0)=@u @(=0)+@u @y(=0)@y @(=0)(B)Usingthenotationthat,foranyknownvariableevaluatedatequaltozero,thosevariableswillbewrittenwithasubscripttodenotehowmanyderivativeshavebeen 119

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takenwithrespectto,thenEquation B gives du d(=0)=u1+y1@u0 @y0(B)Notethatthederivativewithrespecttoyisnowaderivativewithrespecttoy0sincethederivativeisbeingtakenatequaltozero,i.e.onthereferencedomain.Thatistosay,whenspatialderivativesarebeingtakenforaperturbationvariable,thenitcanbeconsideredthat@ @y=@ @y0. 120

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APPENDIXCCOMPARINGCURVEDSURFACEHEATTRANSFERTOHEATTRANSFERFROMAFLATSURFACEThissectionisinreferencetothethermocapillaryowchapters.Oneofthecrucialargumentsusedinunderstandinghowdifferentowstructuresariseistheideathatheattransferisenhancedordecreaseddependingonthecurvatureofasurface.Here,anexampleispresentedtoshowthatheattransferdoesindeedincreaseforaconvexsurfaceanddecreasesforaconcavesurface.Thesechangesinheattransferarerelativetotheheattransferfromaatsurface.TheshapeofthecurvedboundaryshowninFigure C-1 canbecharacterizedbytheequation y=yB+Acos( 2x)(C) FigureC-1. Diagramofthesystem.Asurfacecanhaveconcaveorconvexcurvaturedependingonthechoiceof. 121

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Bygeometry 1+=yB+A(C)Tomakecomparisonsonthebasisofconstantarea,wehave Z1)]TJ /F7 7.97 Tf 6.59 0 Td[(1ydx=2(C)NowtoobtainA,Equation C isevaluatedwithy=1+)]TJ /F3 11.955 Tf 12.24 0 Td[(A+Acos( 2x),whichleadsto A= 1)]TJ /F7 7.97 Tf 13.51 4.71 Td[(2 (C)andhence,fromEquation C yB=1+)]TJ /F6 11.955 Tf 25 8.08 Td[( 1)]TJ /F7 7.97 Tf 13.51 4.71 Td[(2 (C)yieldsyas y=1+)]TJ /F6 11.955 Tf 25 8.09 Td[( 1)]TJ /F7 7.97 Tf 13.51 4.71 Td[(2 + 1)]TJ /F7 7.97 Tf 13.51 4.71 Td[(2 cos( 2x)(C)Now,theonlyfreeparameteris.Consideratransientconductionstategivenby@T @t=r2T,orindimensionlessformas @ @=r2(C)whereisscaledtemperatureandisscaledtime.Now,makingthethreesidewallsadiabaticandallowingheattransferonlythroughthecurvedboundary,theconditionthereis =0aty0=1(C)ThesolutiontoEquation C isgivenby =^e)]TJ /F10 7.97 Tf 6.59 0 Td[(2t(C) 122

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Then,Equation C becomesaneigenvalueproblemforthegrowth/decayconstant,i.e. )]TJ /F6 11.955 Tf 11.95 0 Td[(2=r2(C)Atzeroandrstorderin,theeigenvalueproblemgives )]TJ /F6 11.955 Tf 11.96 0 Td[(200=r20(C)and )]TJ /F6 11.955 Tf 11.95 0 Td[(210)]TJ /F6 11.955 Tf 11.95 0 Td[(201=r21(C)Exactsolutionsoftheseequationsarenotneeded.Theinteresthereisindeterminingifheattransferisincreasedordecreasedbycurvedsurfaces.Inlightofthis,onlythesignof21isneededasthiswilldescribehowthereference,atsurface,isaffectedbythecurvature.TondtheconditionunderwhichEquation C canbesolved,multiplyEquation C by1andEquation C by0,thensubtracttheresultingexpressionsandintegrate.Doingsogives )]TJ /F6 11.955 Tf 11.96 0 Td[(21Z1)]TJ /F7 7.97 Tf 6.59 0 Td[(1Z1020dydx=Z1)]TJ /F7 7.97 Tf 6.59 0 Td[(1Z10[0r21)]TJ /F6 11.955 Tf 11.95 0 Td[(1r20]dydx(C)Uponintegratingtheright-handsidebyparts,weget )]TJ /F6 11.955 Tf 11.95 0 Td[(21Z1)]TJ /F7 7.97 Tf 6.58 0 Td[(1Z1020dydx=)]TJ /F12 11.955 Tf 11.29 16.27 Td[(Z1)]TJ /F7 7.97 Tf 6.58 0 Td[(1[1@0 @y]jy=1dx(C)andusingtheboundarycondition,fromEquation C ,1=)]TJ /F3 11.955 Tf 9.29 0 Td[(Y1@0 @y,andthefollowingresultattains )]TJ /F6 11.955 Tf 11.96 0 Td[(21Z1)]TJ /F7 7.97 Tf 6.59 0 Td[(1Z1020dydx=Z1)]TJ /F7 7.97 Tf 6.59 0 Td[(1Y1[@0 @y]2jy=1dx(C)Sincethebasetemperaturemustbeindependentofx, )]TJ /F6 11.955 Tf 11.95 0 Td[(21Z1)]TJ /F7 7.97 Tf 6.59 0 Td[(1Z1020dydx=[@0 @y]2jy=1Z1)]TJ /F7 7.97 Tf 6.58 0 Td[(1Y1dx(C) 123

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whereY1isknownfromEquation C tobe,Y1=1)]TJ /F7 7.97 Tf 19.92 4.71 Td[(1 1)]TJ /F8 5.978 Tf 8.4 3.25 Td[(2 +1 1)]TJ /F8 5.978 Tf 8.4 3.25 Td[(2 cos( 2x).EvaluatingtheintegralforY1providesuswiththesignof21whichisallthatisneededtodrawconclusions.TheintegralofY1showsthat Z1)]TJ /F7 7.97 Tf 6.59 0 Td[(1Y1dx=2(1)]TJ /F4 11.955 Tf 24.23 8.08 Td[(1 1)]TJ /F7 7.97 Tf 13.51 4.71 Td[(2 )+2 2<0(C)Hence,21mustbepositiveasseenfromEquation C .Notethatthisanalysishasbeencarriedoutregardlessofthesignof,andsotheresultsof21willbethesameforeithersignof.Whenispositive,then,asgivenby2=20+21,willbelargerthantheatboundarysystem.Sincethetemperaturegoesas,=^e)]TJ /F10 7.97 Tf 6.59 0 Td[(2t,thenthetemperaturewilldecaymorequicklyandtheheattransferofthesystemobviouslyincreasesinordertocausethistemperaturechange.However,whenisnegative,becomessmallerthantheatboundarycaseandhencethetemperaturewillfallataslowerrate.Thismeansthattheheattransferhasdecreasedduetotheconcavecurvature.Recallthatthesearethesameheattransferargumentsusedtounderstandowstructureselectioninthermocapillaryow.Now,asadditionalproofthattheheattransferisaffectedmainlybysurfacecurvature,thelengthacrosswhichheatistransferrediscalculatedfortheatandcurvedsystems.Fortheatcase,thislengthissimplythedistanceofxgoingfrom)]TJ /F4 11.955 Tf 9.3 0 Td[(1to1atthetopsurface.So,theatsurfaceheattransferlengthisequalto2.Forthecurvedsurfaces,itisknownthatforadifferentialtrianglealongthetopboundary ds2=dx2+dy2(C)wheresrepresentsthearclength.Equation C readilysimpliesto ds=dxr 1+(dy dx)2(C) 124

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IntegratingEquation C alongthecurvedarcgives s=Z1)]TJ /F7 7.97 Tf 6.59 0 Td[(1dxr 1+(dy dx)2(C)andthederivativeofyisknownfromEquation C tobedy dx=)]TJ /F10 7.97 Tf 6.59 0 Td[( 1)]TJ /F8 5.978 Tf 8.4 3.26 Td[(2 [ 2sin( 2x)].ThesquarerootquantityfromEquation C isthen f1+2 (1)]TJ /F7 7.97 Tf 13.5 4.7 Td[(2 )2[2 4sin2( 2x)]g1=2(C)Whenissmall,Equation C canbewrittenas r 1+(dy dx)2t1+1 22 (1)]TJ /F7 7.97 Tf 13.51 4.71 Td[(2 )2[2 4sin2( 2x)](C)Finally,integratingEquation C showsthat s=2+1 22 (1)]TJ /F7 7.97 Tf 13.51 4.71 Td[(2 )2[1 22)]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 2sin(x)1)]TJ /F7 7.97 Tf 6.58 0 Td[(1](C)Thearclengthisthensimply,s=2+1 22 (1)]TJ /F8 5.978 Tf 8.4 3.26 Td[(2 )2.Infact,itisseenthatatrstorderinthereisnodifferencebetweentheatboundaryheattransferlengthandthecurvedsurfacescenarios.Thismeansthattheheattransferlengthsarethesameinallcasesforthelargestgrowthrates,yettheheattransferisaffectedsimplyduetothecurvatureofthesystem.Additionally,whenthecorrectiontotheheattransferlengthisincludedatsecondorder,thereisanincreaseinnon-insulatedlength,buttheheattransferincreasesforonecurvatureyetdecreasesfortheother.Insummary,theactiveheattransferlengthisthesameforatorcurvedsystemsuptorstorderinwhenissmall.However,duetothedifferentcurvatures,theheattransferincreaseswhenispositiveanddecreaseswhenisnegative. 125

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BIOGRAPHICALSKETCH BradwasbornandraisedinIndianabeforemovingtheFloridaattheageof22.Heearnedabachelor'sdegreeinchemicalengineeringfromPurdueUniversityinMayof2010andaPhDinchemicalengineeringfromtheUniversityofFloridainAugustof2014.WhileattendingschoolinFlorida,hehasalsoworkedwithresearchersfromtheTokyoUniversityofScienceandtheJapaneseAerospaceExplorationAgency(JAXA). 128