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Evaporative Instability in Binary Mixtures

Permanent Link: http://ufdc.ufl.edu/UFE0044573/00001

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Title: Evaporative Instability in Binary Mixtures
Physical Description: 1 online resource (124 p.)
Language: english
Creator: Uguz, Kamuran E
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: binary -- convection -- evaporation -- instability -- interfacial -- marangoni -- rayleigh
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This study focuses on the understanding of the physics of convective flow resulting from evaporative instability in binary mixtures. The system of interest consists of a liquid mixture underlying its own vapor sandwiched between two conducting plates with insulated sidewalls in a closed container. In this system it is important to understand how the fluid dynamics and the heat and mass transfer interact competitively to form patterns. The main goal of this work is to identify the conditions for the system going from the conductive no-flow state to a convection state when the applied vertical temperature gradient exceeds a certain value called the critical value. Convection arises due to; evaporation, density gradients, and interfacial-tension gradients. These convective forces are opposed by the diffusion effects (vorticity diffusion, heat diffusion, and mass diffusion) that try to keep the system in the conductive state. The problem is investigated both in the presence and in the absence of gravitational effects. Four major results arise from this work. First, in a multi-component system in the absence of gravity, an instability arises only when the system is heated from the vapor side as opposed to evaporation in a single-component. The implication is that evaporative processes in thin layers or in micro-gravity are best conducted with heat from the liquid side if instabilities are to be avoided. Second, in the presence of gravity, a multi-component system may become unstable no matter the direction of heating. This means that the applied temperature difference must be kept below a threshold in order to avoid flow instabilities for either heating direction. Third, whenever instability occurs in the absence of gravity, patterns will not result in the case of a pure component but may result in the case of multi-components. Likewise, patterns will result when gravity is taken into account provided the aspect ratio of the container lies in a suitable range. As a result, aspect ratios can be chosen to avoid multi-cellular patterns even if convective flow instabilities arise during evaporation. Lastly, oscillations are not ordinarily predicted despite opposing effects of solutal and thermal convection in the evaporation problem.
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.
Statement of Responsibility: by Kamuran E Uguz.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Narayanan, Ranganathan.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044573:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044573/00001

Material Information

Title: Evaporative Instability in Binary Mixtures
Physical Description: 1 online resource (124 p.)
Language: english
Creator: Uguz, Kamuran E
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: binary -- convection -- evaporation -- instability -- interfacial -- marangoni -- rayleigh
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This study focuses on the understanding of the physics of convective flow resulting from evaporative instability in binary mixtures. The system of interest consists of a liquid mixture underlying its own vapor sandwiched between two conducting plates with insulated sidewalls in a closed container. In this system it is important to understand how the fluid dynamics and the heat and mass transfer interact competitively to form patterns. The main goal of this work is to identify the conditions for the system going from the conductive no-flow state to a convection state when the applied vertical temperature gradient exceeds a certain value called the critical value. Convection arises due to; evaporation, density gradients, and interfacial-tension gradients. These convective forces are opposed by the diffusion effects (vorticity diffusion, heat diffusion, and mass diffusion) that try to keep the system in the conductive state. The problem is investigated both in the presence and in the absence of gravitational effects. Four major results arise from this work. First, in a multi-component system in the absence of gravity, an instability arises only when the system is heated from the vapor side as opposed to evaporation in a single-component. The implication is that evaporative processes in thin layers or in micro-gravity are best conducted with heat from the liquid side if instabilities are to be avoided. Second, in the presence of gravity, a multi-component system may become unstable no matter the direction of heating. This means that the applied temperature difference must be kept below a threshold in order to avoid flow instabilities for either heating direction. Third, whenever instability occurs in the absence of gravity, patterns will not result in the case of a pure component but may result in the case of multi-components. Likewise, patterns will result when gravity is taken into account provided the aspect ratio of the container lies in a suitable range. As a result, aspect ratios can be chosen to avoid multi-cellular patterns even if convective flow instabilities arise during evaporation. Lastly, oscillations are not ordinarily predicted despite opposing effects of solutal and thermal convection in the evaporation problem.
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.
Statement of Responsibility: by Kamuran E Uguz.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Narayanan, Ranganathan.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044573:00001


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EVAPORATIVEINSTABILITYINBINARYMIXTURESByKAMURANERDEMUGUZADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012KamuranErdemUguz 2

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Topeoplewhoareproudofme 3

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ACKNOWLEDGMENTS Firstofall,IwouldliketothankProfessorRangaNarayananforhissupportandadvice.Hehasbeenbothamentorandafriend.Healwaysemphasizestheimportanceofenjoyingyourwork.Prof.Narayananisenthusiasticabouthisworkandthisisthebestmotivationforastudent.Hisdedicationtoteachingandhisphilosophyhaveinspiredme.ThemembersofmyPhDcommittee,Prof.OscarD.Crisalle,Prof.CorinSegal,andProf.KirkZiegleralsodeservemygratitude.Also,IwouldliketothankProf.GerardLabrosseforsharinghisknowledgeandexperiencesandacceptingtobeinmydefense.Iwouldliketothankmybrother,KeremUguz,whohasalwaysbeenwithme,andhasmotivatedmeformywork.Iwouldliketoexpressmyhighestappreciationformyparentsandmybrotherfortheirloveandsupportthroughoutmyeducationalcareer.Ithasbeendifcultforthemandformebecauseofthelargedistance.Thankyouforyourpatience,encouragementandyourmoralsupport.IwouldliketothankthePartnerUniversityFundandtheEmbassyofFranceforaChateaubriandFellowshipduringSpring2011andtheUniversityofFloridaforthepart-alumnifellowshipforamajorpartofmyDoctoralwork.TheNationalScienceFoundationprovidedfundingviagrantNSFOISE0968313isgratefullyacknowledged.Iamgratefultoalloftheminnosmallmeasure. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 12 CHAPTER 1INTRODUCTIONTOTHEPHYSICS ........................ 14 1.1PureEvaporativeInstability .......................... 14 1.2RayleighorBuoyancy-DrivenConvection .................. 17 1.3PureMarangoniorInterfacialTensionGradient-DrivenConvection .... 20 2LITERATUREREVIEW ............................... 25 3MATHEMATICALMODEL .............................. 28 3.1ScaledNonlinearEquations .......................... 29 3.2TheScaledLinearEquations ......................... 33 3.3NumericalMethod ............................... 37 4RESULTSANDDISCUSSIONS .......................... 42 4.1EffectoftheSidewallConditionsintheSingleComponentSystem .... 42 4.2Non-ConstantViscosityModelforSingleComponentSystem ....... 45 4.3TheEffectofaSecondVolatileComponentontheOnsetPoint,BinarySystem ..................................... 46 4.3.1PureEvaporativeConvectioninBinaryMixtures ........... 47 4.3.2SurfaceTensionDrivenInstabilitywithPhaseChange ....... 51 4.3.3BuoyancyDrivenInstabilitywithPhaseChange ........... 60 4.3.41-DModelto3-DModel ........................ 69 5EXPERIMENTALSETUPANDRESULTS ..................... 73 6CONCLUDINGREMARKSANDFUTURESCOPE ................ 81 APPENDIX ATHERMODYNAMICEQUILIBRIUM ........................ 84 BBOUSSINESQAPPROXIMATION ......................... 87 CRAOULT'SLAW ................................... 88 DDERIVATIONOFTHEUNITNORMALANDTHEINTERFACESPEED .... 90 5

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EENERGYBALANCEFORA1-DBINARYSYSTEM ............... 92 FCOMPONENTBALANCEFOR1-DSYSTEM ................... 95 GSORETEFFECT ................................... 96 H3-DSCALEDLINEAREQUATIONSFORSINGLECOMPONENTSYSTEM .. 98 IMODIFICATIONSFORNON-CONSTANTVISCOSITYMODEL ......... 103 JHOWTOCOMPARE3-DPERIODICMODELWITH1-DMODEL ....... 107 KDOMAINVARIABLEEXPANSIONANDMAPPINGS ............... 110 LBINARYMIXTUREPROPERTIES ......................... 114 MSIMPLEEVAPORATIVEINSTABILITYEXPERIMENTFORMIDDLESCHOOLS ............................................. 118 REFERENCES ....................................... 121 BIOGRAPHICALSKETCH ................................ 124 6

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LISTOFTABLES Table page 1-1Summarytable.(Thermalproblem:Ther.,Solutalproblem:Sol.,Unstable:Unst.,Stable:Stab.) ................................. 24 4-1Physicalpropertiesofliquidethanolandsec-butanoland50=50liquidweightpercentbinaryliquidmixtureat30Cand1atm. .................. 43 4-2Physicalpropertiesofethanolandsec-butanolvaporsand50=50liquidweightpercentbinaryvapormixtureat30Cand1atm. ................. 44 J-1RootsoftheJ0m,Jrm,j. ................................ 109 7

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LISTOFFIGURES Figure page 1-1Heatinputfromtheliquidside,unstabletoallwavenumbers. .......... 15 1-2Heatinputfromthevaporside,stabletoallwavenumbers. ........... 16 1-3Physicswithuiddynamics. ............................. 17 1-4ThermalRayleighconvection. ............................ 18 1-5SolutalRayleighconvection. ............................ 19 1-6ThermalMarangoniconvection. ........................... 21 1-7SolutalMarangoniconvection. ........................... 22 3-1GridpointsforGauss-Labottorepresentation. ................... 40 3-2GridpointsforGauss-Radaurepresentation. ................... 40 4-1Effectofsidewallboundaryconditions,aspectratio,theonsetofinstabilityforliquidheight=5mmandvaporheight=5mm.(Barswithoutlabel:3-Dperiodicsystem) ........................................ 42 4-2Non-constantviscosityvs.constantviscosityforliquidheight=5mmandvaporheight=9mm.Columnsontheleftaretheconstantviscositysystem. ..... 45 4-3Onsetofpureevaporativeconvection.ScaledTCriticalvs.wavenumber,liquidheight=2mmandvaporheight=1mm. ....................... 47 4-4Effectofviscositychangeontheonsetofpureevaporativeconvectionforethanol,liquidheight=2mmandvaporheight=1mm. ................... 48 4-5Binarymixturepureevaporativeconvection.ScaledTCriticalvs.massfraction,liquidheight=2mmandvaporheight=1mm(a)Smallandlargewavenumbers(b)Maximumstabilitydepictedforamiddlerangewavenumber. ........ 49 4-6Onsetofbinarymixturepureevaporativeconvection.ScaledTCriticalvs.k2wavenumber,(a)Effectofliquidheight,vaporheight=1mmand!A=0.5(b)Effectofvaporheight,liquidheight=1mmand!A=0.5. ............ 50 4-7Surfacetensionofthemixture. ........................... 52 4-8OnsetofsolutalMarangonidrivenconvectioninthepresenceofphasechange.ScaledTCriticalvs.massfraction,liquidheight=2mmandvaporheight=2mm.Tref=)]TJ /F3 11.955 Tf 9.29 0 Td[(7C. ................................. 52 4-9SolutalMarangoninumbervsmassfractionforliquidheight=2mm. ....... 53 8

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4-10NumericalexperimentforonsetofconvectionSolutalMarangoninumbervsmassfractionforliquidheight=2mmandvaporheight=2mm.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(7C. 54 4-11OnsetofsolutalMarangonidrivenconvectioninthepresenceofphasechange.ScaledTCriticalvs.wavenumber,liquidheight=2mm,andvaporheight=2mm.Tref=)]TJ /F3 11.955 Tf 9.29 0 Td[(7C. ................................. 55 4-12ComparisonbetweenthemodelwithsurfacedeectionandwithoutforsolutalMarangonidrivenconvectioninthepresenceofphasechange.ScaledTCriticalvs.wavenumber,liquidheight=2mm,andvaporheight=2mm. ......... 56 4-13AppearanceofthelocalmaximumforsolutalMarangonifor!A=0.5.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(7C. ......................................... 57 4-14Growthconstantvs.wavenumber,liquidheight=2mm,andvaporheight=2mmfor!A=0.5.InputT=)]TJ /F3 11.955 Tf 9.3 0 Td[(7.3C. ....................... 57 4-15OnsetofsolutalMarangonidrivenconvectioninthepresenceofphasechangefor!A=0.5(a)Effectofthevaporheightontheonset,liquidheight=2mm(b)Effectoftheliquidheightontheonset,vaporheight=2mm. ........... 58 4-16EffectofvaporphaseSoretdiffusionontheonsetpoint.Liquidheight=1mm,vaporheight=1mmfor!A=0.5. .......................... 59 4-17Onsetofconvectionforheatingfromabove.ScaledTCriticalvsmassfraction,liquidheight=5mmandvaporheight=4mm.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(2.6C. ......... 61 4-18Stabilizationeffectofgravityforliquidheight=3mmandvaporheight=3mmandwavenumberk2=1. .............................. 62 4-19Onsetofconvectionforheatingfromabove.ScaledTCriticalvs.wavenumber,liquidheight=5mmandvaporheight=4mm.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(2.6C. ......... 64 4-20Onsetofconvectionforheatingfromabove,liquidheight=2mmandvaporheight=4mm:(a)ScaledTCriticalvswavenumber:localmaximum,(b)Growthconstantvswavenumber. .............................. 65 4-21Onsetofconvectionforheatingfrombelow.ScaledTCriticalvs.massfraction,liquidheight=5mmandvaporheight=4mm.Tref=13.7C. .......... 66 4-22Onsetofconvectionforheatingfrombelow,theeffectofsolutalconvection(Ra!,Ma!)withincreasingwavenumber.ScaledTCriticalvs.massfraction,liquidheight=5mmandvaporheight=4mm. .................... 67 4-23Onsetofconvectionforheatingfrombelow.ScaledTCriticalvs.wavenumber,liquidheight=5mmandvaporheight=4mm.Tref=1.8C. .......... 68 9

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4-24Onsetofconvectionforheatingfrombelow,theeffectofthevaporheight.ScaledTCriticalvs.scaledvaporheight,liquidheight=4mm,!A=0.5andk2=6.25.Tref=3.7C. .................................... 69 4-25Differentazimuthalmodes. ............................. 70 4-26Onsetofconvectionversuswavenumberforliquidheight=2mmandvaporheight=3mmand!A=0.5. ............................. 70 4-27Onsetofconvectionversuswavenumberforliquidheight=2mmandvaporheight=3mmand!A=0.5.Availablewavenumberswithrespecttoazimuthalmodes. ........................................ 71 4-28Growthrateversuswavenumber. ......................... 72 5-1Experimentalsetup. ................................. 73 5-2Schematicoftheexperimentalsetup. ....................... 75 5-3ThinlmcondensationontheZnSewindow.HeatingfrombelowforT=1.2C. ......................................... 76 5-4Condensationontheglasswindow.Heatingfrombelowforabinarymixtureof50/50.InputT=5.Liquidheight=10mm,vaporheight=15mm,anddiameter=38.1mm. .................................. 77 5-5HeatingfromaboveconvectiveowforT=)]TJ /F3 11.955 Tf 9.3 0 Td[(7.5.Longwavelengthinstabilityfor50/50binarymixture.Liquidheight=6mm,vaporheight=14mm,anddiameter=18mm. .......................................... 78 5-6HeatingfromaboveconvectiveowforT=)]TJ /F3 11.955 Tf 9.3 0 Td[(15for50=50binarymixture.Liquidheight=12mm,vaporheight=13mm,anddiameter=38.1mm. ...... 79 H-13-Dmodelillustration. ................................ 98 I-1ViscosityofLiquidEthanolversusTemperature(Forlargerange). ........ 103 I-2ViscosityofLiquidEthanolversusTemperature(Forsmallrange). ....... 103 J-1JkindBesselfunctions. ............................... 108 J-2DerivativeofJkindBesselfunctions. ........................ 109 L-1Densityoftheliquidmixturewithrespecttomassfraction. ............ 114 L-2Viscosityoftheliquidmixturewithrespecttomassfraction. ........... 115 L-3Viscosityofthevapormixturewithrespecttomassfraction. ........... 116 L-4Thermalconductivityofthevapormixturewithrespecttomassfraction. .... 117 10

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L-5Thermalconductivityoftheliquidmixturewithrespecttomassfraction. .... 117 M-1Evaporatingethanollayer.Patternformation. ................... 119 M-2Waterlayer.No-Flow. ................................ 119 M-3Waterlayerheatingfrombelow.Patternformation.(Foodcoloringisadded) 120 11

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyEVAPORATIVEINSTABILITYINBINARYMIXTURESByKamuranErdemUguzAugust2012Chair:RangaNarayananMajor:ChemicalEngineeringThisstudyfocusesontheunderstandingofthephysicsofconvectiveowresultingfromevaporativeinstabilityinbinarymixtures.Thesystemofinterestconsistsofaliquidmixtureunderlyingitsownvaporsandwichedbetweentwoconductingplateswithinsulatedsidewallsinaclosedcontainer.Inthissystemitisimportanttounderstandhowtheuiddynamicsandtheheatandmasstransferinteractcompetitivelytoformpatterns.Themaingoalofthisworkistoidentifytheconditionsforthesystemgoingfromtheconductiveno-owstatetoaconvectionstatewhentheappliedverticaltemperaturegradientexceedsacertainvaluecalledthecriticalvalue.Convectionarisesdueto;evaporation,densitygradients,andinterfacial-tensiongradients.Theseconvectiveforcesareopposedbythediffusioneffects(vorticitydiffusion,heatdiffusion,andmassdiffusion)thattrytokeepthesystemintheconductivestate.Theproblemisinvestigatedbothinthepresenceandintheabsenceofgravitationaleffects.Fourmajorresultsarisefromthiswork.First,inamulti-componentsystemintheabsenceofgravity,aninstabilityarisesonlywhenthesystemisheatedfromthevaporsideasopposedtoevaporationinasingle-component.Theimplicationisthatevaporativeprocessesinthinlayersorinmicro-gravityarebestconductedwithheatfromtheliquidsideifinstabilitiesaretobeavoided. 12

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Second,inthepresenceofgravity,amulti-componentsystemmaybecomeunstablenomatterthedirectionofheating.Thismeansthattheappliedtemperaturedifferencemustbekeptbelowathresholdinordertoavoidowinstabilitiesforeitherheatingdirection.Third,wheneverinstabilityoccursintheabsenceofgravity,patternswillnotresultinthecaseofapurecomponentbutmayresultinthecaseofmulti-components.Likewise,patternswillresultwhengravityistakenintoaccountprovidedtheaspectratioofthecontainerliesinasuitablerange.Asaresult,aspectratioscanbechosentoavoidmulti-cellularpatternsevenifconvectiveowinstabilitiesariseduringevaporation.Lastly,oscillationsarenotordinarilypredicteddespiteopposingeffectsofsolutalandthermalconvectionintheevaporationproblem. 13

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CHAPTER1INTRODUCTIONTOTHEPHYSICSEvaporationisanimportantprocessinmanyindustrialapplicationssuchasheatpipes,spincoatingofmaterials,paintdrying,andglassfabricationwhereatleasttwophases,likeliquidandvapor,areincontactwithoneanotherviaaninterface.Inmanyoftheseapplicationsevaporationisgenerallyaccompaniedbyeitheratemperaturegradientoraconcentrationgradient,ifnotboth.Duetotheseappliedorinducedgradients,anevaporationprocesscanleadtoaninstabilitywhichcanaffectthenalqualityoftheproduct.Theinstabilitymanifestsitselfasanundulationoftheinterfacebetweenthephasesandanonsetofconvectiveow.Therefore,abetterunderstandingoftheevaporationprocess,especiallyinmulti-componentmixtures,isnotonlyofscienticinterestbutalsoofindustrialimportance.Theuidowthatarisesonaccountofinstabilityinanevaporativeprocessisnotonlycausedbytheevaporationitselfbutalsoduetodensitygradientsandforcesarisingfromsurfacetensiongradients.Whenyoucombinealltheseeffectsforamulti-componentsystem,thephysicsoftheproblembecomesverycomplicated.Therefore,inthefollowingthreesub-sectionsallthesephenomena;owduetoevaporation,owduetobuoyancyforcesandowduetosurfacetensiongradientswillbediscussedoneatatimeemployingexplanatorypicturearguments. 1.1PureEvaporativeInstabilityAnevaporatingliquidcanconvectevenintheabsenceofgravityorsurfacetensiongradienteffects.Thisistermedpureevaporativeconvection.Tounderstandwhypureevaporativeconvectioncanoccurinamulti-componentsystem,considerFigures 1-1 and 1-2 whichdepictaliquidmixtureinequilibriumwithitsvaporinaclosedvesselsubjecttoaverticaltemperaturegradient.Figure 1-1 showstheheatingarrangementwheretheheatneededforevaporationissuppliedfromtheliquidsidewhileFigure 1-2 showsheatinputfromthevaporside.Forthesakeofunderstandingthephysics 14

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imaginethattherearenouiddynamicspresentinthesystem.Asthesystemisclosed,thenetevaporationrateiszero.However,inthepresenceofmechanicalperturbationlocalevaporationandlocalcondensationwilltakeplaceattheliquid-vaporinterfaceforbothheatingarrangements.Intheguresthedottedwaverepresentsaperturbedinterfacewhilethedottedlinesrepresentthetemperatureprolesintheperturbedstateattheinterface. Figure1-1. Heatinputfromtheliquidside,unstabletoallwavenumbers. WhentheheatissuppliedfromtheliquidsideasinFigure 1-1 ,thetemperaturegradientatatroughbecomesmorepronouncedonaccountofitsproximitytothehotboundary.Therateofevaporationfromthetroughisthusincreasedandthetroughbecomesdeeperwhichenhancestheinstability.Ontheotherhand,whenthesystemisheatedfromthevaporsideasinFigure 1-2 ,thecrestgetsclosertotheheatsourceandevaporationisenhancedtherewhichstabilizestheinterface.Observethatwhentheheatissuppliedfromtheliquidsidethesystemisunstabletoalldisturbances.Inotherwords,thedisturbancewillgrowwithtime,independentofthewavelengthofthedisturbance.However,whentheheatissuppliedfromabove,i.e.fromthevaporside,thesystemisalwaysstable.Figure 1-3 depictspureevaporativeconvectionwithuiddynamicspresentinbothphases;but,otherconvectiveeffectslike 15

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Figure1-2. Heatinputfromthevaporside,stabletoallwavenumbers. buoyancydrivenowsandtheinterfacialtensiongradientdrivenowsarestillignored.Asbefore,ontheperturbedinterfacethetroughisthelocalevaporationandthecrestisthelocalcondensationpoint.Duetotheevaporation,thereisanupwardowfromthebulktowardthetrough.Theupwardowfromthebulkbringswarmliquidtothetroughenhancingtheinstability.Ontheotherhand,theowinthevaporphasetakesplacefromtroughtocrestonaccountofahigherpressureatthetrough.Thisowbringswarmvaportothecoldcrestandhelpstore-establishthetemperatureuniformity.Thisinstabilityoccursatanywavenumberwhentheappliedtemperaturedifferenceislargerthanthecriticaltemperature.Interfacialtensionactstostabilizethesurfaceandthereforeselectsthecriticalwavelength.Duetothelargedensitydifferencebetweenthevaporandtheliquid,theowinthevaporismuchstrongerthantheowintheliquid.Observefromthepictureargumentthatpureevaporativeinstabilitywouldnotoccurifthebilayerarrangementweretobeheatedfromthevaporsidebecause,inthatcase,likevaporow,thebulkowintheliquidwouldalsobestabilizing.Notethatwehavenotmentionedanythingaboutthesystembeingmulti-componentuntilthispointbecausethephysicsofmulti-componentandsinglecomponentsystemsforpureevaporativeprocessareidentical.However,inthepresenceofgravityand/or 16

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Figure1-3. Physicswithuiddynamics. interfacialtensiongradienteffects,havingabinarymixturechangesthephysicsdrastically. 1.2RayleighorBuoyancy-DrivenConvectionBuoyancy-drivenconvection,oftentermednaturalconvectionorRayleighconvection,occurswhenauidissubjecttoatemperaturegradientoraconcentrationgradientinagravitationaleldandwhenthereisavariationofdensitywithrespecttotemperatureand/orconcentration.TounderstandthephysicsoftheRayleighconvectionletusseparatetheproblemintotwoparts:rst,thethermalproblemwherethedensitydependenceonthemassfractionisignoredandthen,thesolutalproblemwherethedensityofthemixturedoesnotdependontemperature.Figure 1-4 isdrawntoshowthephysicsofthethermalproblemwherethebilayerisheatedfromtheliquidside.Knowingthatdensityordinarilydecreaseswithincreasingtemperature,inthissetupthedensityoftheuidneartheupperplateishigherthanthedensityoftheuidnearthebottomplate.Inotherwords,thetemperaturegradientcreatesagravitationallyunstablestratication.Forsufcientlysmalltemperaturedifferencesthisstraticationisstableandthesystemisabletoconductheatfromthelowerplatetotheupperonebycreatingalineartemperaturegradient. 17

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Figure1-4. ThermalRayleighconvection. Nowimaginegivingamechanicalperturbationtotheconductivesystembypushinganelementofliquiddownwards.Thedensityofthisliquidelementishigherthantheonesthatunderlieit,soitwillcontinuemovingdownward.Meanwhileduetothemassconversation,theliquidfrombelowoughttomoveupward.Thedownward-movingcolduidelementsareheatedastheyreachthebottomofthesystemwhiletheupward-movinguidelementsarecooledastheyreachthetopofthesystem.Thisowwillcontinueunabatedunlesstheliquid'skinematicviscosityandthermaldiffusivityarelargeenoughtosettledowntheperturbation.Thermaldiffusivityhelpsthedownwardmovinguidelementtoreachthesametemperatureasitsenvironment,anddecreasethedensity.Kinematicviscosityalsoworksagainstconvectionbyslowingdownthemechanicalperturbationsintheow.However,thereisacriticalappliedtemperaturegradientwheretheviscosityandthethermaldiffusivitycannotoutweighthedestabilizingeffectsanddisturbanceswillgrow.ThebalancebetweentheseforcesisrepresentedbyadimensionlessnumbercalledthermalRayleighnumber,whicharisesfromscalingthenonlinearequationsandisgivenby,RaT=TgTd3 (1) 18

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Here;T:Negativethermalexpansioncoefcient(apositivenumber).g:Magnitudeofgravity.T:Temperaturedifference.d:Heightoftheliquid.:Kinematicviscosity.:Thermaldiffusivity.Convectioncaninitiateineitherphasesimultaneouslyorpropagatefromonetotheother,dependingonthedomaindimensions,thermal,andmechanicalpropertiesofthephases.Observethatinstabilitywouldnotoccurifthebilayerweretobeheatedfromthevaporsidebecauseitcreateslightoverheavyconguration,whichisastablecongurationwithrespecttogravity. Figure1-5. SolutalRayleighconvection. NowletuslookatthesolutalRayleighconvectionignoringthetemperaturedependenceofthedensity.Concentrationgradientscanbeimposedsimilartothetemperaturegradientbykeepingthetopandbottomplatesatconstantconcentrationwhichisexperimentallyhardtoachieve.Alternativelytheconcentrationgradientcanbeinducedviaevaporation.Inthelattercase,theconcentrationgradientiscreatedby 19

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thedifferentevaporationratesofthecomponentswherethelocaltemperaturegradientisadjustedbythelocalevaporationandthelocalcondensation.Figure 1-5 depictsthesystemheatsuppliedfromthevaporsidewheretheconcentrationgradientisaresultofdifferentevaporationratesofthecomponents.Tounderstandthephysicsweconsideracaseofamulti-componentmixturemadeupoftwosimplealcoholssuchasethanolandsecondarybutanol(sec-butanol),theformerbeingthemorevolatileandthelessdenseone.Inorderwords,thedensityofthemixturedecreaseswithincreasingethanolconcentration.Uponperturbation,ethanolevaporatesmorethansec-butanolatthecrestcreatingasec-butanolrichheavypointwhilethetroughisethanolrichandlight.Therefore,theuidpocketatthecrestsinksandpushesthecolduidfromthebottomofthedomaintothecoldtroughenhancingtheinstability,whilethemassdiffusionandtheviscosityenhancethestability.Herethebalancebetweendestabilizingdensitygradienteffectandstabilizingmassdiffusionandviscosityisgivenby,Ra!=!gd3 DAB (1)Here:!:Negativesolutalexpansioncoefcient.DAB:Massdiffusivity.ObservethatthesolutalRayleighnumberdoesnotcontainatermsuchasCwhichwouldbeanalogoustoTinthethermalRayleighnumberbecausethereisnoappliedconcentrationgradientinthesystem.Theotherpointthatrequiresattentionisthatwhentheheatissuppliedfromtheliquidside,solutalRayleighcannotinitiateconvectioninthesystemsimplybecausethecrestwouldberichinethanolandthereforeitwouldbelight. 1.3PureMarangoniorInterfacialTensionGradient-DrivenConvectionInterfacialtensiongradient-drivenconvectionorMarangoniconvectionisunlikebuoyancy-drivenconvectionandcanoccurinauidevenintheabsenceofagravitational 20

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eld.Likethedensity,interfacialtensionalsodependsonbothtemperatureandconcentration;therefore,itisbesttoinvestigatethephysicsoftheMarangoniconvectionintwopartsasdonefortheRayleighconvection:sorst,thethermalproblemwherethemassfractiondependenceofsurfacetensionisignored,andsecondthesolutalproblemwherethetemperaturedependenceisignored. Figure1-6. ThermalMarangoniconvection. Similartodensity,interfacialtensiondecreaseswithincreasingtemperature.Imaginethevapor-liquidbilayersubjecttoverticaltemperaturegradientwheretheheatissuppliedfromtheliquidsideasdepictedinFigure 1-6 .Ontheperturbedinterfacethereisaninterfacialtensiongradientcausedbythetemperaturegradientwheretheinterfacialtensionishigheratthecoldcrest.Duetothisgradientaowoccursfromtroughtocrestdragginghotterliquidfromthebulktothecrest,enhancingtheMarangoniow.Inthegasphasetheowbringsthecolduidfromthecoldplatetothecrestworkingforthestability;however,theeffectofthisowinthevaporphaseislessthantheliquidphasebecauseofgasphase0shighkinematicviscosity.Inadditiontovaporow,thedestabilizingeffectoftheliquidowcanbeoutpoweredbythethermaldiffusivityandbydynamicviscosityiftheappliedtemperaturegradientissufcientlysmall.Inotherwords,muchlikethecaseofbuoyancy-drivenconvection,thereisa 21

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criticaltemperaturedifferencewheretheMarangoniowcannotbedampenedbythethermaldiffusivityortheviscosity,andatthatpointconvectionstarts.Againlikethebuoyancy-drivenconvectioncase,adimensionlessgrouparisesfromthescaledmodelingequationsthatexhibitthebalancebetweenstabilizingdiffusiveeffectsandthedestabilizinginterfacialtensiongradienteffects.ThisdimensionlessgroupiscalledthethermalMarangoninumberandisgivenby,MaT=TTd (1)Here;T:Changeintheinterfacialtensionwithrespecttothetemperature.:Dynamicviscosity.Notethat,asinthepureevaporativeandthethermalRayleighconvectionproblems,herealsoheatingfromthevaporsideisastableconguration. Figure1-7. SolutalMarangoniconvection. TounderstandthesolutalMarangoniconvection,againconsidertheethanolandsec-butanolbinarymixture.Inthismixtureethanolisthemorevolatilecomponentwithasmallerinterfacialtension.Therefore,theinterfacialtensionofthemixturedecreaseswithincreasingethanolconcentration.Figure 1-7 depictsthebilayerbinarymixture 22

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heatedfromthevaporside.Ontheperturbedinterface,i.e.,thedottedlines,thecrestwouldbeethanolrichcomparedtothetrough;asaresult,theinterfacialtensionisloweratthecrest.Duetothisinterfacialtensiongradientalongtheinterface,owtakesplacefromcresttotrough,draggingcoolerliquidfromthebulktothecrestenhancingthecondensationandenhancingtheinterfacialtensiongradient.Thegradientalsocausesaowinthevaporphasewhichbringsthehotowtothecoldcresttryingtore-establishtemperatureuniformityandenhancingthestability.However,theeffectofvaporphaseowislessthantheliquidphaseowbecauseofgasphase0shighkinematicviscosity.Again,thereisacriticalappliedtemperaturedifference,whichistheonlycauseofconcentrationgradientinthesystem,wherethedynamicviscosityandmassdiffusionofthesystemisnotstrongenoughtosustainthestabilityandconvectionsetsin.ThistimethebalancebetweenthedestabilizingsolutalinterfacialtensiongradientandstabilizingdynamicviscosityandmassdiffusionisrepresentedbyasolutalMarangoninumbergivenby,Ma!=!d DAB (1)Here;!:Changeintheinterfacialtensionwithrespecttotheconcentration.AlsoinsolutalMarangoninumber,likeinsolutalRayleighnumberthereisnoCterm.Iftheheatingarrangementweretobereversed,i.e.heatingfromtheliquidside,thesolutaleffectsontheinterfacialtensioncreatesastablesystembecausethistimetheowwouldtakeplacefromcresttothetrough,dragginghotliquidtothecoldcrestandre-establishingthetemperatureuniformity.Itisimportanttopointoutagainthatthephysicalobservationofsolutalconvectiondependsonthecomponentsthatareused.Ifthemorevolatilecomponent,inourcaseethanol,weretobedenserandhadahigherinterfacialtensionthanthesecondcomponent,heatingfromthevaporsidewouldbesolutallystableasinthethermalproblems. 23

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Table1-1. Summarytable.(Thermalproblem:Ther.,Solutalproblem:Sol.,Unstable:Unst.,Stable:Stab.) PureEvaporationMarangoniRayleighTher.Sol.Ther.Sol.Ther.Sol. HeatingfromtheLiquidsideUnst.N/AUnst.Stab.Unst.Stab.HeatingfromtheVaporsideStab.N/AStab.Unst.Stab.Unst. Althoughwehavediscusseddifferenttypesofconvectionseparately,inanexperimentorapplicationallthreemodesofconvectionwouldoccursimultaneously,bothsolutallyandthermally.However,byarrangingtheliquidheightonecanmakethesystemRayleighorMarangonidominant.ThesystemswithdeepliquidswouldbeRayleighdominantbecauseRayleighconvectionisproportionaltothecubeoftheliquiddepth,seeequations( 1 )and( 1 ).Ontheotherhand,forshallowliquiddepthsthesystemwouldbeMarangonidominantsimplybecauseMarangoniconvectionisdirectlyproportionaltotheliquiddepthgiveninequations( 1 )and( 1 ).Forintermediateliquiddepths,RayleighandMarangoniconvectionhavesimilareffectsonthesystem. 24

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CHAPTER2LITERATUREREVIEWEvaporationcoupledwithconvection,whichhasmanyindustrialapplications,hasbeenextensivelyinvestigated.Evaporativeinstabilityisarichprobleminmanyaspectsandopentoadetailedparametricstudyofthephysicalconditionsandproperties.Therefore,theremightbevariouswaystoanalyzetheliterature.Inthissectionwewillpresenttheliteraturedealingwithaphysicalsetupclosetoours.Earlyworksonthissubjectfocusedonsinglecomponentsystemswithdifferentaspectsoftheproblemsuchasevaporationaccompaniedbyonlysurfacetensioneffects,passivevaporassumption,openorclosedsystemsorexperimentsineitheropenorclosedsystems.Pearson[ 1 ]gavethersttheoreticalanalysisoftheMarangoniconvectionusingamethodologysimilartothatusedbyRayleigh[ 2 ]forthecaseofbuoyancydrivenconvection.Pearson'sworkwasmeanttodescribeBenard's[ 3 ]experiment.Heusedthetechniquesoflinearizedstabilitycf.Chandrasekhar[ 4 ].AlthoughhisworkisnotdoneforevaporativemediathephysicsofMarangoniconvectioninnon-evaporativesystemsissimilartoMarangoniconvectionasitoccursinevaporativesystems.InevaporativesystemsthepassivevapormodelwasusedintheearlyworksofBurelbachetal.[ 5 ]andrecentworksbyMargeritetal.[ 6 ]andOron[ 7 ].Inthistypeofmodel,aBiotnumberisdenedtomodelthethermalresistanceinthevaporbehavior.TheworksthatincludeuidmechanicsintheuppervaporphasearetheworksofHuangandJoseph[ 8 ],OzenandNarayanan[ 9 10 ],HautandColinet[ 11 ],McFaddenetal.[ 12 ],andGuoandNarayanan[ 13 ].Theotherpointofinterestinevaporativeinstabilityproblemsaretheboundaryconditions,specicallyhowtheinterfacetemperatureconditionsareintroduced.Ordinarily,thecontinuityoftemperatureisusedalongwithanotherthermodynamicequilibriumcondition[ 8 ]orarelationderivedfromthekinetictheoryisusedinstead[ 14 ].WardandStanga[ 15 ]havequestionedtheassumptionofcontinuityoftemperatureand 25

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haveveriedtheexistenceofatemperaturediscontinuityattheinterface.Margeritetal.[ 16 ]studiedtheroleofsuchinterfacialnon-equilibriumeffectsonBenard-Marangoniinstability.However,WardandStangareportedthatthemagnitudeofthetemperaturediscontinuityincreaseswiththeevaporationuxandnotedthatwhenequilibriumexistsbetweentheliquidandvaporphases,thetemperatureissameineachphase.ThisisalsoreportedbyShankarandDeshpande[ 17 ].Consequently,whentheevaporationrateisverysmallwemayassumethatthereisthermodynamicequilibriumatthephase-changeboundaryandthetemperatureofbothuidsareequaltoeachotherattheinterface.Theworksmentionedabovefocusedonsinglecomponentsystems.Studiesthatincludemulti-componentsystemsarelimitedcomparedtosinglecomponentsystems.deGennes[ 18 ]workedonascalinganalysisforasolventevaporationinnon-glassypolymer.Heshowedthatintheearlytimestheconvectiveinstabilityrisesduetoconcentrationgradientsandalsoconcludedthatconcentrationeffectsdominatethethermaleffects.Trouetteetal.[ 19 ]numericallyinvestigateddryingofapolymer/solventsolutionatthethermaltransientregime.Theypointedoutthatthethresholdsinthe2-Dmodeland3-Dmodel0sarenotverydifferentfromoneanotherandconvectioncellsareintheformofnon-regularpentagonsorhexagonswithincreasingsizewithtimewhileconvectionfadesaway.Machraetal.[ 20 21 ]numericallyinvestigatedtheonsetoftheconvectioninabinarysystemconsistingofethanolandwater.TheyshowedthattheirsystemisdominatedbythesolutalMarangonimechanism.Inadditiontothetheoreticalworksthereareseveralexperimentalworksdonetoshowtheinstabilitiesinevaporatingliquidlayers.ManciniandMasa[ 22 ]workedonthepatternformationinanevaporatinghexametildisiloxaneliquidlayer.Theyshowedthatiftheevaporationrateislargeenoughthereisnoexternalheatingneededtoinitiatetheconvection.Thecoolingofevaporationontheinterfaceissufcientenoughtostartconvectionforcertainliquidheights.Alsotheyshowedthedynamicbehaviorofthe 26

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convectioncellswithrespecttotheliquidheight.Zhang[ 23 ]studiedconvectioninevaporatingliquidssuchasethanolandR-113whereR-113is10timesmorevolatilethanethanol.Heshowedthatinthinnerliquidlayers,1mmthickorlessforethanoland0.5mmthickorlessforR-113,Marangoni-BenardconvectionisalwaysdominantcomparedtoRayleigh-Benard.Therearealsosomeexperimentalworksdoneinbinarysystems,e.g.byZhangetal.[ 24 25 ]TheyrstinvestigatedthetransientMarangoniconvectioninthinliquidlmsusingsolutionsofNaCl-waterandobservedthepatterns,consistingofrollsandpolygons,evolvingwithtimetilldryingandsaltislandoccurrence.Inthesecondstudytheychoseethanolandwatersystemandobservedcircularringtypeconvectioncells.AnotherexperimentalworkisduetobyDehaecketal.wheretheyinvestigatedthebinarymixtureevaporationandreportedRayleigh-Taylortypeofinstabilityduetodensitystratication[ 26 ].Toussaintetal.didanexperimentalstudyofthedryingofapolymersolution[ 27 ].Theyconcludedthatforsmallinitialthickness,convectionoccursduetothesurfacetensioneffectsandthelifetimeofthisconvectionislessthanthethermaltransientregime.Ontheotherhandforlargerthicknesses,convectionlastslonger.Also,thereisstronginteractionbetweentheconvectionandtheskinformationontheinterface. 27

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CHAPTER3MATHEMATICALMODELTocapturethephysicsoftheevaporativeconvectionphenomenathatisdescribedinChapter 1 acompletemathematicalmodelmustbedevelopedandthatistheobjectiveofthischapter.Inthemodelwetakeintoaccountofphases0momentumdynamics,heattransfer,andmasstransferassumingallalongthattheinterfacedeects.Theequationsarerepresentedintheirscaledformswherewerstintroducethedomainequationsineachphasethenthetopandbottomboundaryconditionsandnally,theinterfaceconditions.Theinstabilityoccursasconvectioncellsandinmathematicaltermsarisesfromthenonlinearityofthesystem.Thisnonlinearitymanifestsitselfviaacyclicaldependenceofthepressure,temperature,velocity,andmassfractioneldsattheinterface.Forexample,attheinterface,thelocaltemperaturegradientcreatesapressuregradientwhichinturncausesowandalsochangestheconcentrationeld.Thisowwillofcoursemodifythetemperatureproleandwillalsomodifythecompositionproleviaconvectivetransport.Thecyclecontinuesonwiththetemperatureandthecompositionmodifyingthepressureeld.Anothersourceofthenonlinearityistheinterfacepositionwhichdependsupontheowdynamicsandviceversa.Infactthecurvatureoftheinterfaceisanonlinearfunctionalofitsposition.Thisnonlinearityis,however,nottheessentialonethatgovernstheinstability.Interfacedeectionisonlyonemanifestationoftheinstability,theessentialmanifestationbeingtheonsetofowfromaquiescentstate.Thenonlinearequationsthatmodelthephysicsmaybescaled.Wechoosetheliquiddepth,d,asthelengthscale,=dasthevelocityscale,d2=asthetimescaleand=d2asthepressurescalewhilethetemperatureiswrittenasadeviationfromTcoldandscaledbyThot)]TJ /F6 11.955 Tf 12.5 0 Td[(Tcold.Intheequationsthatwillfollow,asterisksdenotethevaporphaseand,,,andDABarethedensity,thekinematicviscosity,thethermal 28

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diffusivity,thedynamicviscosityandthemassdiffusion.Inthemodel,theuidmixturesaretakentobeidealthisisreasonableforthepresentexamplewherethemixturemadeupoftwolowweightalcohols. 3.1ScaledNonlinearEquationsTheuidowinbothphasesismodeledusingscaledNavier-Stokesequationsormomentumbalances.Inthefollowingmomentumbalanceequationsthedensitiesaretakentobeconstantinallofthetermsotherthanthetermsmultipliedbygravitationalacceleration.ThisapproachiscalledBoussinesqapproximationanddetailsaregivenintheAppendix B .Thisassumptionisfairprovidedthetemperaturedropinthesystemisnotverylargeanddensityisnotastrongfunctionoftemperature.ThemomentumbalancesforbothphasesusingtheBoussinesqapproximationaregivenby,1 Pr@~v @t+~vr~v=rP+r2~v+RaTT~k+Ra! Le)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(!A)]TJ /F4 11.955 Tf 11.96 0 Td[(!Aref~k (3)and 1 Pr@~v @t+~vr~v=rP+r2~v+T T RaTT~k+! Ra! Le(!A)]TJ /F4 11.955 Tf 11.96 0 Td[(!Aref)~k (3)Intheabove,~kistheunitvectorinthepositivezdirectionwith!Arefistheinitialreferencemassfraction.Thedimensionlessvariablesare;Prandtlnumber,Pr= ,thermalRayleighnumber,RaT=TgTd3 ,solutalRayleighnumber,Ra!=!gd3 DABandtheLewisnumber,Le= DAB.TheowsinbothphasesareconnectedtotemperatureviatheRaTterm.Inordertosolveforthetemperatureprolesweneedtheenergybalancesineachphases.Theyaregivenby,@T @t+~vrT=r2T (3) 29

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and@T @t+~vrT= r2T (3)Likethetemperaturedependencetheowalsodependsontheconcentrationeldineachphase.Inordertosolvefortheoweldweneedtosolvefortheconcentrationeldaswellandforthatweneedtheconcentrationbalancesineachphase,i.e. @!A @t+~vr!A=1 Ler2!A (3) and @!A @t+~vr!A=DAB DAB1 Ler2!A (3) Finallyweneedtosatisfymassconversationineachphasebyusingthecontinuityequations,i.e.,r~v=0andr~v=0.Intheequationsabove,Pisthescaledmodiedpressurewherethemodiedpressureisequaltothescalarpressure,p,addedtothedensitymultipliedbythegravitationalpotential.Thevariables,~vandTarethescaledmodiedpressure,velocityandtemperatureelds.ThemassfractionofcomponentA,!A,isalreadydimensionless.NotethatinabinarymixturetherearetwocomponentsAandBwhere!A+!B=1.Inordertosolvethesystemofdomainequationsweneed16boundaryconditionsandoneadditionalconditiontodenethesurfacedeection.Thetopandbottomwallsaresolidandimpermeablewithuniformtemperature,i.e.,atthebottomwall,z=)]TJ /F6 11.955 Tf 9.3 0 Td[(d,wehavevz=0,@!A @z=0andT=TBottomandatthetopwall,z=d,wehavevz=0,@!A @z=0andT=TTop 30

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No-slipconditionsareassumedforthetopandbottomwalls.Therefore,atthebottomwallvx=0andatthetopwallvx=0.Thetotalmassbalanceattheinterface,locatedatz=Z(x,t),is (~v)]TJ /F4 11.955 Tf 11.49 .49 Td[(~u)~n= (~v)]TJ /F4 11.955 Tf 11.49 .49 Td[(~u)~n (3) andthecomponentbalanceattheinterfaceis )-222(r!A~n+Le!A(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n=)]TJ /F4 11.955 Tf 10.49 8.09 Td[( )]TJ /F6 11.955 Tf 10.49 8.09 Td[(DAB DABr!A~n+Le!A(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n (3) Notethatthemassux,JA,isgivenbyJA=)]TJ /F4 11.955 Tf 9.3 0 Td[(DABr!Aisthemassux.Theunitnormalvectoris, ~n=)]TJ /F9 7.97 Tf 10.49 4.71 Td[(@Z @x~i+~k h1+)]TJ /F9 7.97 Tf 6.68 -4.97 Td[(@Z @x2i1/2Theinterfacespeedis, ~u~n=@Z @t h1+)]TJ /F9 7.97 Tf 6.67 -4.98 Td[(@Z @x2i1/2Attheinterfacethetangentialcomponentsofthevelocitiesofbothuidsareequaltoeachother,i.e.,~v~t=~v~tholdwhere, ~t=~i+@Z @x~k h1+)]TJ /F9 7.97 Tf 6.67 -4.98 Td[(@Z @x2i1/2Thederivationsoftheunitnormalvector,interfacespeed,andtheunittangentisgivenintheAppendix D .Theinterfacialforcebalanceequationinscaledfromis, CaT1 Pr~v(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u))]TJ /F3 11.955 Tf 16.1 8.09 Td[(1 Pr ~v(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n)]TJ /F6 11.955 Tf 11.96 0 Td[(CaTMaTrsTt)]TJ /F6 11.955 Tf 11.96 0 Td[(Ca!Ma!rs!At+2H~n=CaT~~T)]TJ /F4 11.955 Tf 13.24 6.53 Td[(~~T~n (3) 31

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where2H=@2Z @x2 h1+)]TJ /F9 7.97 Tf 6.68 -4.98 Td[(@Z @x2i3/2istwicethemeancurvature,~~T=)]TJ /F6 11.955 Tf 9.3 0 Td[(P~~I+~~Sisthestresstensor,CaT= disathermalcapillarynumber,MaT=TdT isthethermalMarangoninumber,Ca!=DAB disasolutalcapillarynumberandMa!=!d DABisthesolutalMarangoninumber.Observethattheforcebalancehastwocomponents,thenormalandthetangentialcomponents.Theenergybalanceattheinterfaceis 1+KPC1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)2)]TJ /F3 11.955 Tf 13.16 8.09 Td[(1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n+E2r!A)]TJ /F6 11.955 Tf 9.3 0 Td[(E1rT~n)]TJ /F4 11.955 Tf 13.15 8.09 Td[( rT~n)]TJ /F6 11.955 Tf 11.95 0 Td[(VPC~~S(~v)]TJ /F4 11.955 Tf 11.5 .5 Td[(~u))]TJ /F4 11.955 Tf 13.15 8.09 Td[( ~~S(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n=0 (3) Herethedimensionlessnumbersare;E2=DAB(~A)]TJ /F14 11.955 Tf 11.95 0 Td[(~B)/()]TJ /F14 11.955 Tf 9.3 0 Td[(~A!A)]TJ /F14 11.955 Tf 11.96 0 Td[(~B!B),KPC=2d2()]TJ /F14 11.955 Tf 9.3 0 Td[(~A!A)]TJ /F14 11.955 Tf 11.95 0 Td[(~B!B),E1=T/()]TJ /F14 11.955 Tf 9.3 0 Td[(~A!A)]TJ /F14 11.955 Tf 11.95 0 Td[(~B!B),VPC=d2()]TJ /F14 11.955 Tf 9.3 0 Td[(~A!A)]TJ /F14 11.955 Tf 11.96 0 Td[(~B!B),isthethermalconductivityand~isthelatentheatperunitmass.AttheinterfacethecompositionsofbothphasesarecoupledwithRaoult'slaw,i.e., yAp=xAPvapA (3) whereyAandxAarevapormolarandliquidmolarcompositions.Attheinterface,thetemperaturesofbothuidsareequaltoeachother,i.e.,T=TbesideslocalthermostaticequilibriumwhichleadstotheClausius-Clapeyronequation,i.e., YKE(p)]TJ /F6 11.955 Tf 11.96 0 Td[(pbase)=lnT Tbase (3) whereYKE= ~d2,pbaseandwhereTbasearethescaledinterfacialpressureandtemperatureofthevaporatareferencestate. 32

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3.2TheScaledLinearEquationsTodeterminethestabilityofthesystemandtheonsetpointofconvection,requireslinearizationofthedomainequationsandboundaryconditionsaboutabasestate[ 28 ].Thebasestateforoursystemistheno-owconductivestate.Inordertostudythestabilityofareferencestate,arbitraryinnitesimaldisturbancesareappliedtothedomainstatevariablesandtheboundaryvariables.Ifthestateisunstabletoinnitesimaldisturbancesitmustbeunstabletoalldisturbancesandsoitissufcienttoconsiderinnitesimaldisturbances.Thus,stabilitytosmalldisturbancedoesnotgiveanyinformationofthesystembutinstabilitydoes.Usingsmalldisturbancesalsoallowsustodroptermswithquadraticandhigherinteractions.Hereon,thebasestatevariablesaredenotedbysubscript0andthevariablesoftheperturbedstatebythesubscript1.Anyperturbeddomainvariableevaluatedonthedeectingsurfaceisgiveby,U=U0+"U1+dT0 dzz1+O)]TJ /F4 11.955 Tf 5.48 -9.69 Td[("2where"isthearbitraryinnitesimaldisturbancethatrepresentsthedeviationfromthebasestateandz1isthemappingoftheperturbedstatetothebasestate.ThedetailedexplanationofthemappingisgivenintheAppendix K .Attheinterfacethemappingissimplytheperturbationofthesurfacedeectionandatitsrstorderitistermed,Z1,avariablethatalsoneedstobedetermined.Theperturbedvariablesareassumedtoeithergrowordecayaset,whereistheinversegrowthordecayconstanthereaftertermedasjustthegrowthconstant.Theperturbedvariablesarefurtherexpandedintonormalmodes,thusU1=bU1(z)eteikxwherekisthewavenumberofagivendisturbancethatarisesbecausethesystemisinniteinlateralextent.Thewavenumbercanberelatedtotheradiusofthesystemifthesidewallsareassumedtohaveperiodicboundaryconditions. 33

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Aswementionedaboveinthebasestatethereisnoow,i.e.~v0=0=~v,andithasasimpleconductionsolution.ThetemperatureprolesinbothphasesintheirscaledformaregivenbyT0=)]TJ /F4 11.955 Tf 24.04 8.09 Td[( +z+ +andT0=)]TJ /F4 11.955 Tf 26.4 8.09 Td[( +z+ +where=d/distheratioofvaportoliquiddepth.Thelinearizeddomainequationsintheliquidphasebecome, Prvx1=)]TJ /F6 11.955 Tf 9.3 0 Td[(ikp1+d2 dz2)]TJ /F6 11.955 Tf 11.96 0 Td[(k2vx1 (3) Prvz1=)]TJ /F6 11.955 Tf 10.5 8.09 Td[(dp1 dz+d2 dz2)]TJ /F6 11.955 Tf 11.96 0 Td[(k2vz1 (3) T1=d2 dz2)]TJ /F6 11.955 Tf 11.95 0 Td[(k2T1)]TJ /F6 11.955 Tf 13.15 8.09 Td[(dT0 dzvz1 (3) !A1=1 Led2 dz2)]TJ /F6 11.955 Tf 11.96 0 Td[(k2!A1 (3) and 0=ikvx1+dvz1 dz (3) Likewiseforthevaporphasewehave Pr vx1=)]TJ /F6 11.955 Tf 9.3 0 Td[(ik p1+d2 dz2)]TJ /F6 11.955 Tf 11.95 0 Td[(k2vx1 (3) Pr vz1=)]TJ /F4 11.955 Tf 12.86 8.09 Td[( dp1 dz+d2 dz2)]TJ /F6 11.955 Tf 11.96 0 Td[(k2vz1 (3) !A1=1 LeDAB DABd2 dz2)]TJ /F6 11.955 Tf 11.95 0 Td[(k2!A1 (3) 34

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and 0=ikvx1+dvz1 dz (3) Atthelowerplate,z=)]TJ /F3 11.955 Tf 9.29 0 Td[(1,wehavevx1=vz1=T1=d!A1 dz=0andatthetopplate,z=,wehavevx1=vz1=T1=d!A1 dz=0Attheinterface,themassbalanceandcomponentbalanceturninto, vz1)]TJ /F4 11.955 Tf 13.15 8.09 Td[( vz1=1)]TJ /F4 11.955 Tf 13.15 8.09 Td[( Z1 (3) and )]TJ /F6 11.955 Tf 13.15 8.09 Td[(d!A1 dz+ DAB DABd!A1 dz+Le!A0vz1)]TJ /F4 11.955 Tf 13.15 8.09 Td[( !A0vz1=Le !A0)]TJ /F4 11.955 Tf 11.96 0 Td[(!A0Z1 (3) whereZ1istheperturbedsurfacedeectionand!A0istheinitialmassfractionatthebasestate.Thecontinuityoftemperatureandtheno-slipconditionsbecome T1+dT0 dzZ1=T1+dT0 dzZ1 (3) and vx1=vx1 (3) Thelinearizedequationsforthetangentialandnormalstressaregivenby, dvx1 dz+ikvz1)]TJ /F11 11.955 Tf 11.96 16.86 Td[(dvx1 dz+ikvz1)]TJ /F6 11.955 Tf 11.95 0 Td[(ikMaTT1+dT0 dzZ1)]TJ /F6 11.955 Tf 11.95 0 Td[(ikMa!!A1=0 (3) 35

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and Ca(p1)]TJ /F6 11.955 Tf 11.96 0 Td[(p1))]TJ /F3 11.955 Tf 11.96 0 Td[(2Cadvz1 dz)]TJ /F4 11.955 Tf 13.16 8.09 Td[( dvz1 dz)]TJ /F11 11.955 Tf 11.96 9.69 Td[()]TJ /F6 11.955 Tf 5.48 -9.69 Td[(Bo+k2Z1=0 (3) HereBo=()]TJ /F4 11.955 Tf 11.96 0 Td[()gd2istheBondnumber.Theinterfacialenergybalanceinitslinearformgivenby, vz1+E2d!A1 dz)]TJ /F6 11.955 Tf 11.96 0 Td[(E1dT1 dz)]TJ /F4 11.955 Tf 11.95 0 Td[(dT1 dz=Z1 (3) UponperturbationRaoult'slawbecomes, dPVapA dT!T=T0(xA0yA0)]TJ /F6 11.955 Tf 11.96 0 Td[(xA0)+ dPVapB dT!T=T0(yA0)]TJ /F6 11.955 Tf 11.96 0 Td[(xA0yA0)#T1+dT0 dzZ1+hPVapA0(yA0)]TJ /F3 11.955 Tf 11.95 0 Td[(1))]TJ /F6 11.955 Tf 11.95 0 Td[(PVapB0yA0ixA1+hPVapA0xA0+PVapB0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(xA0)iyA1=0 (3) AndthecoefcientsyA1andxA1aregivenby, xA1=MB !A0(MB)]TJ /F6 11.955 Tf 11.96 0 Td[(MA)+MA)]TJ /F4 11.955 Tf 27.44 8.09 Td[(!A0MB(MB)]TJ /F6 11.955 Tf 11.96 0 Td[(MA) (!A0(MB)]TJ /F6 11.955 Tf 11.96 0 Td[(MA)+MA)2!A1and yA1="MB !A0(MB)]TJ /F6 11.955 Tf 11.95 0 Td[(MA)+MA)]TJ /F4 11.955 Tf 28.04 8.95 Td[(!A0MB(MB)]TJ /F6 11.955 Tf 11.96 0 Td[(MA) )]TJ /F4 11.955 Tf 5.48 -9.69 Td[(!A0(MB)]TJ /F6 11.955 Tf 11.96 0 Td[(MA)+MA2#!A1Theremainingcondition,i.e.,theClausius-Clapeyroninitslinearizedfromis YKEp1)]TJ /F11 11.955 Tf 11.95 11.36 Td[(YPEZ1=T1+dT0 dzZ1 T0 (3) Inunscaledterms,theinputstothecalculationbesidestheuidthermo-physicalproperties,depthsanddisturbancewavenumberk,arethecompositionoftheliquidphaseofthebinarymixtureinthereststateandT,thetemperaturedropacrossthebilayer.Theoutputvariableisthegrowthconstant,.ThecalculationsaredoneusingtheSpectralChebyshevcollocationmethodwhichisexplainedinthenextsection. 36

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3.3NumericalMethodThespectralmethodapproximatesthesolutionofadifferentialequationascontinuousfunctions,resultinginaglobalsolution.Inourwork,weusethespectralmethodwithChebyshevgridpointstosolvecoupledpartialdifferentialequationsconsistingofmomentum,energy,andconcentrationeldsintwophases[ 29 ].Considerafunctionu(x),whichisasolutiontoadifferentialequation.ThegoaloftheSpectralChebyshevmethodistoprovideapolynomialu(N)(x)whichapproximatesthesolutionu(x)dependingonthenumberoftermsNtaken.TherststepoftheSpectralChebyshevmethodistodenethegridpoints,whichwillbeusedtocreatedifferentiationmatrices.Thebestwayofdescribingthedifferentiationmatrixisthroughanexample.Letsdiscretized dxwithxdenedover[)]TJ /F3 11.955 Tf 9.3 0 Td[(1,1].WerstneedtodecidethenumberofintervalsNcorrespondingtoN+1numberofgridspoints.LetusillustratethiswithN=2,i.e.,wedenethreegridpointslocatedatx0=)]TJ /F3 11.955 Tf 9.3 0 Td[(1,x1=0,x2=1whereu(x)isevaluated.Then,u(x)isapproximatedwithaquadraticpolynomial. u(x)u(2)(x)=a(x)]TJ /F6 11.955 Tf 11.96 0 Td[(x0)(x)]TJ /F6 11.955 Tf 11.96 0 Td[(x1)+b(x)]TJ /F6 11.955 Tf 11.95 0 Td[(x1)(x)]TJ /F6 11.955 Tf 11.95 0 Td[(x2)+c(x)]TJ /F6 11.955 Tf 11.96 0 Td[(x2)(x)]TJ /F6 11.955 Tf 11.96 0 Td[(x0) (3) u0,u1,andu2canbewrittenas u0=u(x0)=b(x0)]TJ /F6 11.955 Tf 11.96 0 Td[(x1)(x0)]TJ /F6 11.955 Tf 11.95 0 Td[(x2) (3) u1=u(x1)=c(x1)]TJ /F6 11.955 Tf 11.96 0 Td[(x2)(x1)]TJ /F6 11.955 Tf 11.95 0 Td[(x0) (3) u2=u(x2)=a(x2)]TJ /F6 11.955 Tf 11.96 0 Td[(x0)(x2)]TJ /F6 11.955 Tf 11.95 0 Td[(x1) (3) Fromthissetofequationsa,b,andcareobtainedintermsofu0,u1,andu2asa=u2 2,b=u0 2andc=)]TJ /F6 11.955 Tf 9.3 0 Td[(u1.Thederivativeofu(2)(x), du(2) dx=2x(a+b+c))]TJ /F3 11.955 Tf 11.96 0 Td[([a(x0+x1)+b(x1+x2)+c(x2+x0)] (3) 37

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du dxx=x0=)]TJ /F3 11.955 Tf 9.29 0 Td[(1.5u0+2u1)]TJ /F3 11.955 Tf 11.95 0 Td[(0.5u2 du dxx=x1=)]TJ /F3 11.955 Tf 9.3 0 Td[(0.5u0+0.5u2 (3) du dxx=x2=0.5u0)]TJ /F3 11.955 Tf 11.96 0 Td[(2u1+1.5u2ThisequationsetcanberepresentedinmatrixformasA u =b ,whereA isthedifferentiationmatrix. )]TJ /F3 11.955 Tf 9.3 0 Td[(1.52)]TJ /F3 11.955 Tf 9.3 0 Td[(0.5)]TJ /F3 11.955 Tf 9.3 0 Td[(0.500.50.5)]TJ /F3 11.955 Tf 9.3 0 Td[(21.5u0u1u2=du0/dxdu1/dxdu2/dx (3) Toobtainthe2ndand3rdorderderivativematrices,wecansimplytakethe2ndand3rdpowerofthedifferentiationmatrix. d2u(2) dx2=2(a+b+c)=u0)]TJ /F3 11.955 Tf 11.95 0 Td[(2u1+u (3) Observethatthesecondderivativedoesnotdependontheposition,sothesecondderivativeissameateverypositionandthesecondorderdifferentiationmatrixisgivenby, 1)]TJ /F3 11.955 Tf 9.29 0 Td[(211)]TJ /F3 11.955 Tf 9.29 0 Td[(211)]TJ /F3 11.955 Tf 9.29 0 Td[(21 (3) Onecangetthesamematrixbysimplytaking2ndpowerof1storderdifferentiationmatrix.NotethatyoucannotuseN=1forsolvinga2ndorderdifferentialequationbecausethesecondpowerofdifferentiationmatrixturnsouttobezero. 38

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Thismethodworksovertheinterval[)]TJ /F3 11.955 Tf 9.3 0 Td[(1,1]becausethenodesaredenedbyusingcosinesfunction.Ifthedomainisdenedinanotherinterval,thedifferentiationmatrixshouldbemultipliedbyalineartransformationcoefcient.Asanexample,imaginethatthedomainliesbetween0andL.Thelineartransformationcoefcientcalculationgoesasfollows;x0=Ax+Bisthelinearfunctionwhichconnectsyourdomaintospectralinterval,wherex0isthespectralintervalandxisyourdomain.Whenx=0!x0=)]TJ /F3 11.955 Tf 9.3 0 Td[(1andx=L!x0=1)]TJ /F3 11.955 Tf 9.29 0 Td[(1=B1=AL)]TJ /F3 11.955 Tf 11.96 0 Td[(1!A=2 LItisclearthatAd dx0=d dx,soAisthelineartransfercoefcientwhichis2 Linthesamplecalculation.ChebyshevGauss-LobattonodesaredenedforNnumberofsubintervalsoverrange[)]TJ /F3 11.955 Tf 9.3 0 Td[(1,1]; xj=)]TJ /F3 11.955 Tf 11.29 0 Td[(cosj N,j=0,....,N (3) Notethatthegridpointsarenotequallyspaced.Thenodesaredenserneartotheboundaries.Thishasadvantageoncapturingboundaryeffectsonthedomain,whichisimportantfortransportphenomenaproblems.TwoexamplesofhowgridslooklikefordifferentpointsaregiveninFigure 3-1 ;Adifferenttypeofgridpointsstructureneedstodeneforacylindricalcoordinatesystembecauseofthesingularitythatoccursintheoperatoratthecenterlineposition,r=0.InordertoovercomethisproblemGauss-RadaugridpointshavetobeintroducedseeninFigure 3-2 .IntheGauss-Labottostructurebothboundariesofthesystemareincludedtothemodel.OntheotherhandintheGauss-Radaunodesr=0pointisexcludedfromthesystem.InGauss-Radaustructurethereisanodeclosetocenterlineposition(r=0)whichwithincreasingnodenumbersgetsclosertor=0pointbutnever 39

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Figure3-1. GridpointsforGauss-Labottorepresentation. Figure3-2. GridpointsforGauss-Radaurepresentation. reaches.Thegridpointsaredenedas, xp=)]TJ /F3 11.955 Tf 11.29 0 Td[(cos(2p+1) 2N+1,p=0,...,N (3) Therightboundarynodeforp=Nisxp=1,buttheleftonedependsonNforexample;forN=10,xp=)]TJ /F3 11.955 Tf 9.3 0 Td[(0.9888andforN=20,xp=)]TJ /F3 11.955 Tf 9.3 0 Td[(0.9970.ThisisshowninFigure3-2. d2u(2) dx2=2(a+b+c)=u0)]TJ /F3 11.955 Tf 11.95 0 Td[(2u1+u (3) Inordertosolvethatthesetoftheequations,thedifferentiationmatrixhastobeinverted.However,thedeterminantoftheAmatrixiszero,sotheAmatrixcannotbeinverted.Indeed,iftheAmatrixwasinvertibleitwouldmeanthatwecansolvea 40

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differentialequationwithoutboundaryconditions.Forsolvingthedifferentialequation,thematrixAshouldbemodiedbyinsertingtheproperboundaryconditions.Thediscretizedsystemisrepresentedaseigenvalueproblemintheformof A (T)u=B u (3) A andB arefoundbythediscretizationofthegoverningequationsandtheboundaryconditions. uistheeigenvectoroftheproblemwhichincludesthevelocity,temperature,andconcentrationelds. istheeigenvalueoftheprobleminourcaseisthegrowthconstant.Inputstothecomputationbesidestheuidthermo-physicalproperties,depthsandthedisturbancewavenumberk,arethecompositionoftheliquidphaseofthebinarymixtureinthereststateandT,thetemperaturedropacrossthebilayer.Theoutputvariableisthegrowthconstant,.Calculationsrevealthatthevalueoftheleadingarealwaysreal,nomatteriftheTispositiveornegative.AsourinterestisindeterminingtheonsetconditionfortheinstabilitywesettozeroanddeterminethecriticalTinitsscaledform.Alsoseveralcalculationsaregivenintheresultsanddiscussionsection. 41

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CHAPTER4RESULTSANDDISCUSSIONSInthischapterwediscussthesolutionsofthelinearizedequationsthatweregiveninchapter 3 ,titledMathematicalModel,aswellasintheAppendix.Thecalculationsarepresentedrstforthethreedimensional(3-D)modelforasinglecomponentsysteminordertoshowtheeffectofthesidewallsonthesystem.The3-DmodelingequationsaregivenintheAppendix H .Wewillthenbrieyintroducetheeffectofthenon-constantviscosityforthesinglecomponentsystem,againforthe3-Dmodel.Themodiedmodelingequationsforthenon-constantviscositycasearegivenintheAppendix I .Inthelastpartofthissectionwewilldiscuss,ingreatdetail,thebinarysystemresults. Figure4-1. Effectofsidewallboundaryconditions,aspectratio,theonsetofinstabilityforliquidheight=5mmandvaporheight=5mm.(Barswithoutlabel:3-Dperiodicsystem) 4.1EffectoftheSidewallConditionsintheSingleComponentSystemInvestigatinganinstabilityprobleminthepresenceofphasechangeinvolvesheattransferanduiddynamicsinadditiontothecomplicationsthatarisefromthegeometry,aswellasmasstransferformulti-componentsystems.Inordertohavea 42

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betterunderstandingofthephysicsastepbystepapproachistakeninthisstudywhereweinvestigatethesidewalleffectontheonsetforsinglecomponentsystems.Thus,threedifferentcodesweredeveloped,aonedimensional(1-D)code,a3-Dcodewithperiodicsidewallconditions,anda3-Dcodewithno-slipsidewallconditions. Table4-1. Physicalpropertiesofliquidethanolandsec-butanoland50=50liquidweightpercentbinaryliquidmixtureat30Cand1atm. Ethanolsec-Butanol50=50mix. (kg m3)782800791(kg msec)9.5010)]TJ /F8 7.97 Tf 6.59 0 Td[(42.7410)]TJ /F8 7.97 Tf 6.58 0 Td[(31.5210)]TJ /F8 7.97 Tf 6.58 0 Td[(3(J msecC)1.6910)]TJ /F8 7.97 Tf 6.59 0 Td[(11.3410)]TJ /F8 7.97 Tf 6.58 0 Td[(11.4610)]TJ /F8 7.97 Tf 6.58 0 Td[(1(m2 sec)9.2010)]TJ /F8 7.97 Tf 6.59 0 Td[(86.2010)]TJ /F8 7.97 Tf 6.58 0 Td[(87.4510)]TJ /F8 7.97 Tf 6.58 0 Td[(8~(J kg)8.801056.631057.92105(N m)2.2710)]TJ /F8 7.97 Tf 6.58 0 Td[(22.3710)]TJ /F8 7.97 Tf 6.58 0 Td[(22.3110)]TJ /F8 7.97 Tf 6.58 0 Td[(2T(N mC)9.6010)]TJ /F8 7.97 Tf 6.58 0 Td[(59.6010)]TJ /F8 7.97 Tf 6.58 0 Td[(59.6010)]TJ /F8 7.97 Tf 6.58 0 Td[(5T(1 C)1.2010)]TJ /F8 7.97 Tf 6.58 0 Td[(31.2010)]TJ /F8 7.97 Tf 6.58 0 Td[(31.2010)]TJ /F8 7.97 Tf 6.58 0 Td[(3!(N m!)N=AN=A8.5010)]TJ /F8 7.97 Tf 6.58 0 Td[(4!(1 !)N=AN=A2.1810)]TJ /F8 7.97 Tf 6.58 0 Td[(2 Theperiodic3-Dcodeisanintermediatestepbetweenthe1-Dcodeandthe3-Dcodewithano-slipsidewallcondition.Theperiodiccodeisexpectedtogivethesameresultsasinthe1-Dcodeonaccountofaseparationofvariablesthatarisesthanks 43

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tostress-freesidewallconditions.Thismakesiteasiertocheckthe3-Dcodeandalsogivesuscondenceinourcodes.Asimplederivationonhowwemaycomparethe3-Dperiodiccodewiththe1-DcodeisgiveninAppendix J .Inthissectiononlytheresultsofthosecomparisonsaregiven.EthanolisusedasanexampleofthemediaandthepropertiesofethanolforliquidandvaporphasearegivenintheTables 4-1 and 4-2 [ 30 ]. Table4-2. Physicalpropertiesofethanolandsec-butanolvaporsand50=50liquidweightpercentbinaryvapormixtureat30Cand1atm. Ethanolsec-Butanol50=50mix. (kg m3)1.883.032.15(kg msec)0.910)]TJ /F8 7.97 Tf 6.58 0 Td[(50.7510)]TJ /F8 7.97 Tf 6.59 0 Td[(50.8710)]TJ /F8 7.97 Tf 6.59 0 Td[(5(J msecC)1.6710)]TJ /F8 7.97 Tf 6.59 0 Td[(21.4410)]TJ /F8 7.97 Tf 6.58 0 Td[(21.6210)]TJ /F8 7.97 Tf 6.58 0 Td[(2(m2 sec)6.1310)]TJ /F8 7.97 Tf 6.58 0 Td[(63.0810)]TJ /F8 7.97 Tf 6.59 0 Td[(65.1410)]TJ /F8 7.97 Tf 6.59 0 Td[(6T(1 C)3.5610)]TJ /F8 7.97 Tf 6.59 0 Td[(33.4010)]TJ /F8 7.97 Tf 6.59 0 Td[(33.5310)]TJ /F8 7.97 Tf 6.59 0 Td[(3!(1 !)N=AN=A0.53 Thecomparisonisgivenforaliquidheightof5mmandavaporheightof5mmforvariousaspectratiosinFigure 4-1 .Observethatbykeepingtheliquidheightxedat5mmtheonlywaytochangetheaspectratioisthroughincreasingtheradiusofthesystem.Therstthingtonoticeisthatthe3-Dperiodiccalculationsand1-Dcalculationsgivethesameresultsasexpected.Ontheotherhand,the3-Dcalculationswithno-slipconditionsgiveahighercriticaltemperaturedifferencethanwhatisobtainedfromtheothertwocalculations.Thisisalsoanexpectedresultbecauseno-slipsidewallsstabilizethesystemagainstconvection.However,withincreasingaspectratio,i.e. 44

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largerradii,theeffectofthesidewallgetsweakerandallthreecasesconvergetothesameresult.Anotherpointthatrequiresattentionisthatwithincreasingaspectratiothedifferencebetweentheonsetpointsofanytwosuccessiveaspectratiosgetssmaller.Thissupportsthefactthatforlargeenoughcontainersthesystemisdominatedbytheheightsandnotbytheradius. Figure4-2. Non-constantviscosityvs.constantviscosityforliquidheight=5mmandvaporheight=9mm.Columnsontheleftaretheconstantviscositysystem. 4.2Non-ConstantViscosityModelforSingleComponentSystemPreviouscalculationsweredoneusingconstantthermo-physicalproperties,whichareevaluatedatthereferencetemperatureforbothliquidandvaporphasesexceptinthecaseofthedensitywherereferencetemperatureisthecoldplatetemperature.Inthissectiontheresultsofnon-constantviscosity,i.e.(T),modelarecomparedwiththeconstantviscosityresults.Togofromaconstantviscositymodeltoanon-constantviscositymodelwehavetomodifythemomentumbalancesinbothphasesaswellastheforcebalancesattheinterface(seeAppendix I ).Inthecaseofethanol,viscosityisanon-linearfunctionoftemperature.However,forsmallenoughtemperaturerangessuchasbetween30and40Ctheviscositycanberepresentedasalinearfunctionoftemperature.Usinga 45

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lineardependenceinsteadofanonlinearoneconsiderablysimpliesthecodinganddecreasesthecomputationtimebyavoidinganiterativecomputation.Inthemodiedmodeltheinputviscosityissmallerthantheconstantviscositymodel,resultinginamoreunstablesystemforallaspectratiosasseeninFigure 4-2 4.3TheEffectofaSecondVolatileComponentontheOnsetPoint,BinarySystemAddingasecondcomponenttothesystemconsiderablymodiesthemodelingequationsandalsothephysicsoftheproblem.Inordertohaveabetterunderstandingoftheconvectionphysicsinabinarysystemwecontinuewiththestepbystepapproach.Werstinvestigatethepureevaporativeconvection,thenthephasechangeproblemaccompaniedbyMarangoniowandnally,asalaststep,weaddtheeffectofgravity.Thebinarymixtureonsetpointdependsonseveralparameterssuchasthevaporandliquidheights,thedisturbancewavenumber,andthemassfractionofthemixture.Inthefollowingsectionstheeffectsofalltheseparameterswillbediscussed.TheresultswillmostlybegivenasscaledTCriticalversusoneofthoseparameterswherethereferencepointofscalingisgivenineachplot.Thisrepresentationisnotusualforthistypeofwork;however,inourcasethethermo-physicalpropertiesoftheliquidandvapormixtureschangewithrespecttoinitialthemassfraction.Therefore,presentingtheresultsbyadimensionlessnumbersuchasRayleighnumberorMarangoninumberthatcontainsthermo-physicalpropertieswouldbemisleading.Toinvestigatethephysicsofthebinarysystemweconsidertwosimilarlow-weightalcoholssuchasethanolandsec-butanol,whichallowsustoassumeanidealmixture.Anidealmixtureisamixturewhoseenthalpyofmixingisequaltozero.Inotherwords,uponmixingtheintermolecularattractionsdonotmodifythepropertiescomparedtopureliquids.Also,anidealmixture'svaporphaseobeysRaoult'slaw.Themixturepropertiesaretypicallycalculatedusingsimplemixingruleswithrespecttoeither 46

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massormolefraction.Wherepossible,empiricalformulasdocumentedintheliteraturewereusedsuchasforviscositiesandthermalconductivities.Forreferencepurposes,densities,viscosities,thermalconductivities,andsurfacetensionofthemixtureareplottedwithrespecttomassfractionintheAppendix L Figure4-3. Onsetofpureevaporativeconvection.ScaledTCriticalvs.wavenumber,liquidheight=2mmandvaporheight=1mm. 4.3.1PureEvaporativeConvectioninBinaryMixturesInthissectionweinvestigatetheonsetofconvectionintheabsenceofgravityandsurfacetensiongradienteffects.ThistypeofconvectionistermedasPureEvaporativeConvectionanditsphysicsisdescribedinChapter 1 ,theIntroductionofthePhysics.Recallthatpureevaporativeconvectionoccursonlyiftheheatissuppliedfromtheliquidside,i.e.,positivetemperaturedifferences.Inordertounderstandtheeffectofmassfractiononthesystem,werstneedtoinvestigatehowthesinglecomponentsystemsbehave.Westartbygivingacomparisonbetweentheonsetpointsoftwopurecomponents,i.e.,pureethanolandpuresec-butanol.Figure 4-3 depictstheonsetpoint,givenasscaledTCritical,versuswavenumberofthedisturbanceforaliquidheightof2mmandforavaporheight 47

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Figure4-4. Effectofviscositychangeontheonsetofpureevaporativeconvectionforethanol,liquidheight=2mmandvaporheight=1mm. of1mm.Bothsystemsbecomemorestablewithincreasingwavenumberasaresultofincreasedthermaldiffusioninsmallerdomains,thatis,thermaldiffusivityactsmoststronglyinsmallerwidthcontainersorlargerwavenumbersduetotheproximityofhotandcoldregions.Onecanarguethatviscositymayintroducesometypeofstability,too.Toshowthatthestabilizingeffectprincipallycomesfromthermaldiffusionforlargewavenumberslet'stakealittledetourfromFigure 4-3 andshowtheresultsofanumericalexperiment.Inthisnumericalexperimentwedeliberatelydecreasetheviscosityofethanolby50%(callitmodiedethanol).InFigure 4-4 theresultsofmodiedethanolarecomparedwiththeethanolofrealproperties.Itisclearthatchangingtheviscositychangestheonsetpointonlyinthesmallwavenumberregionandthischangeisminimal.Nowlet'sgobacktoFigure 4-3 ,thesecondpointthathastobemadeisthecross-overbetweenethanol'sandsec-butanol'sonsetpoints.Althoughbothpurecomponentsbecomemorestablewithincreasingwavenumber,therateofstabilization 48

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isnotequaltoeachother.Thisisduetothedifferentthermaldiffusivitiesofethanolandsec-butanol.Ethanol,whichhasthelargerthermaldiffusivity,getsmorestablecomparedtosec-butanolinthelargewavenumberregion.Thepresenceofacross-overhasnon-trivialconsequencesinthebinarysystem. Figure4-5. Binarymixturepureevaporativeconvection.ScaledTCriticalvs.massfraction,liquidheight=2mmandvaporheight=1mm(a)Smallandlargewavenumbers(b)Maximumstabilitydepictedforamiddlerangewavenumber. Duetothecross-overseeninFigure 4-3 ,thepureevaporativeinstabilityofabinarymixtureofethanolandsec-butanolexhibitsthreedifferentbehaviors.Theyareshownin 49

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Figure4-6. Onsetofbinarymixturepureevaporativeconvection.ScaledTCriticalvs.k2wavenumber,(a)Effectofliquidheight,vaporheight=1mmand!A=0.5(b)Effectofvaporheight,liquidheight=1mmand!A=0.5. Figures 4-5 (a)and 4-5 (b)andwillbediscussedmomentarily.OnceagaintheordinateisexpressedbydividingtheTCriticalforeachinputmassfractionbytheTCriticalforpureethanolatagivenwavenumber.InFigure 4-5 (a)thebehavioroftheonsetpoint,TCritical,forabinaryalcoholmixtureforpureevaporativeconvectionisdrawnagainstthemassfraction,withwavenumberasafreeparameter.Itisclearthatforeithersmallorlargewavenumbersthecurvesaremonotonicandunderstandablytheyeitherincreaseordecreaseinslopedependingonthewavenumber,smallorlarge.ObservethatthepointsontheextremeleftofFigure 4-5 (a),puresec-butanolpoints,canbereadofffromFigure 4-3 .WhatisnotapparentfromFigure 4-5 (a)istheeffectofthe 50

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cross-overpointseeninFigure 4-3 .ThisismademoretransparentinFigure 4-5 (b).Here,aweakmaximumisseentooccurinthemiddlerangeofwavenumbersclosetothecrossoverpoint.Inordertoinvestigatetheeffectofvaporandliquidheightsontheonsetpoint,severalcalculationsaredoneandshowninFigure 4-6 forafty-ftymassfraction.Itisconcludedthatbyincreasingthevaporheightthesystembecomesmorestableduetoincreaseinvaporowfromtroughtocrest.Thisowbringsinwarmvaportothecoldcrestandhelpstoreestablishthetemperatureuniformity.Thisbehavioroccursforallmassfractions.Conversely,increasedliquidheightscreateamoreunstablesystemduetoincreasedhotupwardowtowardthetroughsanddownwardowfromthecrestsintheliquidphase.Bothoftheseresultsarequalitativelysimilartoallmassfractionsincludingsinglecomponentevaporation. 4.3.2SurfaceTensionDrivenInstabilitywithPhaseChangeInthissectiontheadditiveeffectofsurfacetensiongradientsontheonsetofconvectionisinvestigatedwhilestillignoringthegravity.Thisisplausibleforsmallliquidandvapordepthsorforsystemsinamicro-gravityenvironment.Forthebinarysystemsurfacetensiondependsonbothconcentrationandtemperature.Thecalculationsshowthatthedependenceofsurfacetensionupontemperatureisinconsequentialinevaporationproblems.Recallthat,inourphysicalsetup,thereisnoexternalmassfractiongradientapplied.Themassfractiongradientisinducedbythedifferentevaporationratesofthecomponentswherethelocaltemperaturegradientisadjustedbythelocalevaporationandthelocalcondensation.Inoursystemethanolisthemorevolatilecomponentwithlowersurfacetension;therefore,withincreasingethanolconcentrationsurfacetensiondecreasesshowninFigure 4-7 .Theonsetofconvectiondependsonseveralfactorssuchasmassfractionofthemixture,wavenumberofthedisturbance,andphaseheights.Duetotheselectedbinarysystem0sthermo-physicalpropertiesethanolandsec-butanol,thesystembecomes 51

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Figure4-7. Surfacetensionofthemixture. unstableonlywhenitisheatedfromthevaporside,i.e.,negativecriticalTinthepresenceofsurfacetensiongradients.ThisisdepictedinFigure 4-8 inarangeofmassfractionswithwavenumberasafreeparameterforaliquidheightof2mmandavaporheightof2mm.Notethattheregionaboveeachcurveisthestableregionforthat Figure4-8. OnsetofsolutalMarangonidrivenconvectioninthepresenceofphasechange.ScaledTCriticalvs.massfraction,liquidheight=2mmandvaporheight=2mm.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(7C. 52

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curvebecausetheregionrepresentslessnegativetemperaturedifferencesthanthecriticalone.Itisclearthatforanywavenumberthemiddlerangeofmassfractionsismoreunstablethanthedilutemixtures.Inotherwords,middlerangemassfractionsrequirelessheatingfromabovecomparedwiththerestofthemassfractionrange.Thisisduetostrongerperturbationsonthemassfractioninthemiddlerangethantherest.Tosupportthishypothesis,anumericalexperimentisdonewherethesolutalMarangoninumberisxedtothatofmassfraction0.3andcomparedtotheresultswithnon-xedsolutalMarangoniresults.AMassfractionof0.3ischosenbecauseitisthemaximumvalueofthesolutalMarangoninumberasshowninFigure 4-9 .InFigure 4-10 thenumericalexperimentisdepicted.ItisclearthatxingthesolutalMarangoninumberdoesnotaffectthebehavioroftheonsetpoint;themiddlerangemassfractionsarestillmoreunstablethanthediluteregion.ThismeansthatthechangeinthesolutalMaranoninumberwithrespecttomassfractionisnotthereasonforthisminimumstabilityregion.Therefore,theonlymechanismresponsiblefortheminimumstabilityisthelargerperturbationsinthemassfractioninthemiddleregioncomparedwiththediluteregion. Figure4-9. SolutalMarangoninumbervsmassfractionforliquidheight=2mm. 53

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Figure4-10. NumericalexperimentforonsetofconvectionSolutalMarangoninumbervsmassfractionforliquidheight=2mmandvaporheight=2mm.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(7C. AfteridentifyingwhytheminimumstabilityoccursatthemiddlerangemassfractionletusgobacktoFigure 4-8 again.Althoughthegurerevealsthenon-monotonicbehaviorofthecriticalpointwithrespecttothewavenumber,Figure 4-11 isdrawntohaveaclearerview.Thisgurecanbedividedinto3sectionswithrespecttowavenumbertoidentifythecompetitionbetweenvariouseffects.Inthelargewavenumberregionthedisturbancesbecomechoppierandthesystembecomesmorestablebecausethetransversediffusioneffectincreasesquadraticallywiththewavenumberwhereasthesystembecomesmoreunstablebecausetransversevariationofthesurfacetensionisstrong.Theconverseistrueatthelowerwavenumbers.ThiscompetitionbetweensurfacetensiongradientsanddiffusiveeffectsleadstoalocalmaximuminthecriticaltemperaturedifferenceasseeninFigure 4-11 .Atverylowwavenumbersthecriticaltemperaturedifferencegoestozero(itisnotseeninthescaleofFigure 4-11 ).Thislowwavenumbertailisaresultoftheweakinterfacialdeectionthatshowsasignatureontheinstabilityatthelowestwavenumbersandistypicalofmanyinterfacialinstabilityproblems[ 31 ].ThetailisplottedinFigure 4-12 sidebysidewitharesultof 54

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Figure4-11. OnsetofsolutalMarangonidrivenconvectioninthepresenceofphasechange.ScaledTCriticalvs.wavenumber,liquidheight=2mm,andvaporheight=2mm.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(7C. amodelwithoutthesurfacedeection.ThistimethegureisplottedTCriticalversuswavenumberforA=0.5.Themodelwithoutsurfacedeectiondoesnotexhibitthetail.However,otherthanthelowwavenumberregionthetwomodelsgivethesameresults.Thismeansthatsurfacedeectionisnotthereasonfortheinstabilityitisjustonemanifestationandattheonsetpointitseffectisminimalotherthanthelowwavenumberregion.LetusgobacktoFigure 4-11 andobservethattheregionabovethecriticalcurveisinthestableregionandtheregionbelowisunstablebecausethephysicalsituationdepictsheatingfromthevaporside,leadingtoanegativeTCritical.Thelocalmaximumwhichdepictstheaforementionedcompetitioncanbemadetodisappearbyincreasingtheregionofinstability.Thisisdonebystrengtheningtheeffectofthesurfacetensiongradientsandmaybeachievedbyincreasingtheliquiddepth,decreasingthevapordepthorboth.Figure 4-13 showsdisappearanceofthedipbyincreasingtheliquidheightfrom2mmto3mmbykeepingthevaporheightconstantat2mmformass 55

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Figure4-12. ComparisonbetweenthemodelwithsurfacedeectionandwithoutforsolutalMarangonidrivenconvectioninthepresenceofphasechange.ScaledTCriticalvs.wavenumber,liquidheight=2mm,andvaporheight=2mm. fraction0.5.Presenceofalocalmaximum,inFigure 4-13 foraliquidheightof2mmandavaporheightof2mmisalsoanindicationofpatternformationattheonsetofinstability.Inotherwords,attheonsetpointthereisawavelengththatexhibitsamaximumgrowthrate.IntheFigure 4-14 thegrowthrateisplottedversusthewavenumberforamassfractionof0.5andappliedtemperaturedifferenceof)]TJ /F3 11.955 Tf 9.3 0 Td[(7.3Ctoillustratethe3neutralpointsthatappearonaccountofthelocalmaximumseeninFigure 4-13 .Theexistenceofmultipleneutralpointswhichconrmstheexistenceofalocalmaximumassuresusofhorizontalpatternsthatmayappearattheonsetofinstabilityinthisproblem.IfthelocalmaximuminFigure 4-11 weretodisappearthenonlyoneneutralpointwouldbeobtainedinthegrowthratecurveandtheonlypatternthatwouldprevailattheonsetwouldbethelongwavelengthow.Anothersetofcalculationsisdonetoinvestigatetheeffectofdomainheightsontheonsetpoint.Asindicatedinthepictureargumentgivenintheintroductionsectionthevaporowhasastabilizingeffectwhiletheliquid 56

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Figure4-13. AppearanceofthelocalmaximumforsolutalMarangonifor!A=0.5.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(7C. Figure4-14. Growthconstantvs.wavenumber,liquidheight=2mm,andvaporheight=2mmfor!A=0.5.InputT=)]TJ /F3 11.955 Tf 9.3 0 Td[(7.3C. owisdestabilizing.ThatpictureargumentissupportedbythecalculationsgiveninFigure 4-15 (a)and 4-15 (b);increasingvaporheightbringsstabilityshowninpart(a)andincreasingliquidheightmakesthesystemmoreunstable.Thisisshowninpart(b).Both 57

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Figure4-15. OnsetofsolutalMarangonidrivenconvectioninthepresenceofphasechangefor!A=0.5(a)Effectofthevaporheightontheonset,liquidheight=2mm.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(7C.(b)Effectoftheliquidheightontheonset,vaporheight=2mm.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(14C. thesegurearealsoagoodrepresentationofhowthedestabilizingeffectscanbemadestrongenoughtoremovethelocalmaximumsimilartoFigure 4-10 .Inthemulti-componentsystemsthatareunderatemperaturegradientthereisasecondmassdiffusionmechanismcalledtheSoreteffect.ThismechanismisquantiedbytheSoretcoefcientwhichcanbenegativeorpositivedependingonthemass 58

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Figure4-16. EffectofvaporphaseSoretdiffusionontheonsetpoint.Liquidheight=1mm,vaporheight=1mmfor!A=0.5. fractionand/orthetemperatureofthemixture.ForpositiveSoretcoefcients,theheavyuidelementsmovefromhottocoldregionsandviceversafornegativecoefcients.ToinvestigatethiseffectanewmodelisdevelopedthatincludesSoretdiffusioninbothphases.ThemodiedmodelingequationsaregiveninAppendix G .AsetofcalculationsisdonetoinvestigateallthepossibilitiessuchasnegativeorpositiveSoretcoefcients,theSoreteffectinbothphases,oronlyinonephase.TheabsolutevalueoftheSoretcoefcientissettobeequaltothethermalexpansioncoefcient.CalculationsrevealthattheSoreteffectintheliquidphasehasnoeffectontheonsetpointwhereasinthevaporphaseithasaminimaleffect.ToshowthisminimaleffectFigure 4-16 isdrawnfor!A=0.5,liquidheight1mm,andvaporheight1mm.InthegurebothnegativeandpositiveSoretcoefcientresultsarescaledusingthebasemodel'sonsetpointatcorrespondingwavenumber.Itisclearthatforlargewavenumbers,duetothermaldiffusiondominance,thereisalmostnodifferencebetweenthemodels.TheminimaldifferencemanifestsitselfforsmallwavenumberswherethepositiveSoreteffectintroducesasmallstabilityandnegativeSoreteffectintroducesasmallinstability. 59

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4.3.3BuoyancyDrivenInstabilitywithPhaseChangeInthissectionweintroducetheadditiveeffectofgravityontheonsetpoint.Gravityhasastronginuenceontheowcharacteristicsunlessthedepthsareveryshalloworunlesstheenvironmentischaracterizedbymicro-gravity.Inourproblemgravitymainlyactsonthedensitygradients.Likesurfacetension,densityisalsoafunctionoftemperatureandconcentration.Asbeforethebinarymixtureconsistsofethanolandsec-butanol.Inthismixtureethanolisthelighterandmorevolatilecomponent.Inotherwords,astheethanolconcentrationincreasesthedensitydecreases;likewiseasthetemperatureincreasesdensitydecreases.ThedensityvariationwithrespecttoconcentrationandtemperatureismodeledusingtheBoussinesqapproximationandisexplainedintheAppendix B .Severalcalculationsareperformedtorevealthephysicsandtodosoweseetheeffectofmassfractions,disturbancewavenumbersanddomainheights.Thesecalculationsnotonlyrevealthephysicsoftheonsetoftheinstabilitybutalsorevealthecompetingeffectsthatcomeintoplaywhendifferentheatingdirectionsareimposed.AgaintheonsetpointisrepresentedasscaledTCriticalwherethereferencescalingpointisgivenineachgure.Theintroductionofgravitymakesthephysicsofthisproblemconsiderablymoreinvolvedthantheproblemthatwasaddressedinthepreviousparts.Wepresenttwosituationsthatarephysicallyrealizable.Therstisheatingfromthevaporside,i.e.,negativecriticalTandthesecondisheatingfromtheliquidside,i.e.,positivecriticalT;bothsituationscorrespondtoneutralconditions.Therefore,uponsettingequaltozero,weareinterestedonlyinthelowestabsolutecriticalT,whetherthebilayerisheatedfrombeloworfromabove.IntheFigure 4-17 theonsetpointisdrawnversusmassfractionwhentheheatissuppliedfromthevaporside,i.e.negativecriticalTwithwavenumberasthefreeparameter. 60

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Figure4-17. Onsetofconvectionforheatingfromabove.ScaledTCriticalvsmassfraction,liquidheight=5mmandvaporheight=4mm.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(2.6C. Thiscaseiscertainlyastablecongurationforasinglecomponentsystemand,infact,weobservethatfordilutebinarysystemsthiscaseisalsoastableconguration.Figure 4-17 ,therefore,isdrawnwithintherangeofmassfractionsthatdonotincludediluteconditions.Now,thenegativecriticalTsaretheresultofthemassfractiondependenceofthesurfacetensionandthedensityor,inotherwords,duetosolutalMarangoniconvectionandsolutalRayleighconvection,bothintheliquidandinthevapor.ObservethatthethermalRayleighandthethermalMarangonieffectsplayonlyastabilizingrole,thethermalMarangonieffectbeingtheweakone.Thus,inthepresenceofgravityagreaternegativeTisneededtogeneratetheinstabilitythanintheabsenceofgravity.Againobservethatthesolutalgradientsalongtheinterfacearecausedbythedifferentevaporationratesuponperturbationandthedifferingsurfacetensionsofthetwocomponents.AsshowninFigure 4-8 forsurfacetensiongradientdrivenconvectionherealsotheonsetpointhasastrongdependenceonthe 61

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massfraction.Infact,thecriticalTincreaseswiththemassfraction,goesthroughamaximumandthendecreases.Thismaximumoccursinthemiddlerangeofmassfractions,principallyonaccountofstrongperturbationsontheconcentrationeldattheinterfaceinthatlocation.Thiswasanobservationmadeearlierevenwhengravitywasignored.ToshowthestabilizationeffectofthermalRayleighwhentheheatissuppliedfromthevaporsideFigure 4-18 isdrawnforTCriticalversusmassfractionforaliquidheightof3mmandforavaporheightof3mmwithawavenumber,k2=1.IntherangeofFigure 4-17 thestabilizationratioisaround1.25,i.e.,thesysteminthepresenceofgravityis1.25timesmorestablethanintheabsenceofgravity.Thisratioincreasesforthedilutecase.Thisratiomaybemadesmallerbydecreasingtheliquidheight,forexampleforaliquidheightof2mmthisratiodecreasesto1.05inthemiddlerangeofstartingcomponentmassfractions.Thissupportsthefactthatforsmallliquidheights,onecanignoretheRayleighconvectionasassumedearlierwhenMarangoniconvectionwasaloneconsidered. Figure4-18. Stabilizationeffectofgravityforliquidheight=3mmandvaporheight=3mmandwavenumberk2=1. 62

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AscanbeobservedfromFigure 4-17 theonsetpointalsodependsonthewavenumbertoo.ThisdependenceisdrawninFigure 4-19 wheretheonsetpointisplottedversuswavenumberwithmassfractionnowasthefreeparameter.Asintheprevioustwosections,i.e.pureevaporativeandMarangonidrivenconvection,appearstobeanincreaseinstabilitywithincreasingwavenumberforanygivenmassfraction.Inthegurethisincreaseismonotonic;however,thisisnotalwaysthecaseandwewillsee,momentarily,whythisisso.Forsmallwavenumbers,nearzero,wealwayshaveatail,i.e.,criticalTdecreasesfromzero,exhibitingthesignatureofadeectingsurfaceorinterfacialmode.WhentheliquidheightsarelargethestabilizationofferedbythelowerhorizontalboundaryisreducedandinasmuchasthetransversevariationoftheconcentrationthatisrequiredtoprecipitatethesolutalMarangoniconvectionisreducedatlowwavenumbers,evenaweakvariationisenoughtogeneratesomeow.FurtherincreasesinthewavenumberenhancethestabilityduetodiffusioncausingthedownwardtrendinthecurvesofFigure 4-19 .Tounderstandwhytheincreaseinstabilitywithwavenumbermightnotalwaysbemonotonic,considerthecaseofshorterliquidheightsandlargervaporheightsinwhichgreaterconvectivestabilizationobtains.Thiscausesthesmallwavenumberregionsofthecurvetobepulleddownandnowtransversevariationoftheconcentrationenhancedbylargerwavenumberscangeneratetherequiredconvectivedestabilizationleadingtoalocalmaximuminthecurve.Thislocalmaximumisasignatureofacompetitionandweseethiscompetitioninoursystembetweenthesurfacetensiongradientandthedensitygradientduetoconcentrationversusthediffusiveeffects.Thisismadeapparentmerelybyincreasingthestabilityofthesystemviadecreasingtheliquidheight.Figure 4-20 (a)isdrawntoshowtheappearanceofamaximumwiththedecreaseoftheliquidheight,from5mmto2mm,whileFigure 4-20 (b)depictsthebehaviorofthegrowthconstant,,forthatsystemforagivenappliedtemperaturedifference.Bothguresaredrawnformassfraction0.5.In 63

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Figure4-19. Onsetofconvectionforheatingfromabove.ScaledTCriticalvs.wavenumber,liquidheight=5mmandvaporheight=4mm.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(2.6C. Figure 4-20 (b)weseethreeneutralpointswherethegrowthrateisequaltozero.Theexistenceofmultipleneutralpointsalsoconrmstheexistenceofalocalmaximuminthecriticaltemperaturedifferenceandassuresusthathorizontalpatternsmayappearattheonsetofinstabilityinthisproblem.Unlikethesituationwherethegravityisabsent,thesystemcanalsoconvectwhenheatissuppliedfromtheliquidside,generatingapositivecriticalT,asdepictedinFigure 4-21 .Intheguretheonsetpointisplottedversusmassfractionwithwavenumberasafreeparameterandforthecaseofaliquidheightof5mmandavaporheightof4mm.Whentheheatissuppliedfromtheliquidsidethebinarymixtureexhibitsamaximumstabilityinthemiddlerangeofmassfraction,analogoustoFigure 4-17 .Thisderivesprincipallyfromthestabilizingeffectofsolutalconvectionwhenthesystemisheatedfromtheliquidside.Themaximumappearsnearthemiddlemuchasintheheatingfromtheabovecase.Theappearanceofthismaximumnearthemid-rangemassfractionsareduetotheconcentrationperturbationswhicharethestrongestthereandwhichonlyservetostabilizethesystembecauseoftheheatingarrangement.Anadditionalobservationthatonecanmakefromthegureisthatthe 64

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Figure4-20. Onsetofconvectionforheatingfromabove,liquidheight=2mmandvaporheight=4mm.(a)ScaledTCriticalvswavenumber:localmaximum.Tref=)]TJ /F3 11.955 Tf 9.3 0 Td[(17.8C.(b)Growthconstantvswavenumber:threeneutralpoints.InputT=)]TJ /F3 11.955 Tf 11.96 0 Td[(20C. maximumofthecurvesgetprogressivelyweakerwiththeincreasingwavenumberultimatelyleadingtothemonotonicdecaywiththemassfraction.Thisrequiresanexplanation.ToseewhythemaximumgetsweakerandthendisappearsconsiderFigures 4-22 (a)to(f)wheretwocurvesaredepictedineachsub-gure.TheuppercurveistheusualonedrawninthemannerofFigure 4-21 ;thelowercurveontheother 65

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Figure4-21. Onsetofconvectionforheatingfrombelow.ScaledTCriticalvs.massfraction,liquidheight=5mmandvaporheight=4mm.Tref=13.7C. handreferstoanarticialcase,onewherethesolutalexpansioncoefcientsinbothphasesaswellasthesolutalsurfacetensiongradientaresettozero.Thiscasewhichiscalledthepurethermalproblemmayalsobeviewedasasinglecomponentsystemthathasthethermo-physicalpropertiesofabinarymixtureforeachcorrespondingmassfraction.Acomparisonbetweenthetwomodels,asthewavenumberincreases,ismadeprogressivelyfromFigure 4-22 (a)to(f).Andprogressively,weseetheincreasingeffectofthethermaldiffusioni.e.,theincreasingeffectofthethermaldominance.Whatstandsoutisthatthebinaryproblem,givenbytheuppercurve,approachesthethermalproblem,losingitsmaximumandultimatelymergingwithitforverylargek2.ThebehaviorofthismaximumgettingweakerwithincreasingwavenumbercanalsobeseeninFigure 4-23 fromadifferentpointofviewwhilesimultaneouslyshowingtheeffectofwavenumberforheatingfromtheliquidsideforaxedmassfraction.Asseeninthegure,a50=50weightpercentmixtureismorestablethaneitherpureethanolorpuresec-butanolinthesmallwavenumberregion.Andsincethemaximumdisappearsinthelargewavenumberregion,thecriticalTforthemixtureliesbetween 66

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Figure4-22. Onsetofconvectionforheatingfrombelow,theeffectofsolutalconvection(Ra!,Ma!)withincreasingwavenumber.ScaledTCriticalvs.massfraction,liquidheight=5mmandvaporheight=4mm.(a)k2=1Tref=13.7C(b)k2=4Tref=1.8C(c)k2=8Tref=1.2C(d)k2=12Tref=1.2C(e)k2=16Tref=1.4C(f)k2=81Tref=8.1C. 67

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Figure4-23. Onsetofconvectionforheatingfrombelow.ScaledTCriticalvs.wavenumber,liquidheight=5mmandvaporheight=4mm.Tref=1.8C. thetwopurecomponents.Now,nomatterwhattheconcentrationofthemixtureis,aminimumoradipispresentinFigure 4-23 .Thisdipisaresultofacompetitionbetweenthedestabilizinghorizontaldensitygradientscausedbythethermalexpansionwhichgetsstrongerforsmallwavelengthsandthestabilizingthermaldiffusionwhichalsogetsstrongerforsmallwavelengths.Thesolutalgradientsplayonlyastabilizingroleandtheireffectisnotacauseforthedip.Onceagainthepresenceofadipisindicativeofcellularpatternsthatcanobtaininthisproblem.AndonceagainonemustexpectthecurvestoexhibitataildescendingtowardtheorigineventhoughitisnotdepictedwithinthescaleofFigure 4-23 .Finally,weconsidertheresultofonemorecalculationthatrevealsthephysicsoftheproblem.Thishasgottodowiththeeffectofthevaporheightandtobeconcretewerestrictourselvestoheatingfromtheliquidside.Figure 4-24 showsthreedifferentphysicalsituationsthatobtainuponincreasingthevaporheightforaxedliquidheight.TheinitialincreasedinstabilitydepictedbyanincreaseinthecriticalTariseson 68

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Figure4-24. Onsetofconvectionforheatingfrombelow,theeffectofthevaporheight.ScaledTCriticalvs.scaledvaporheight,liquidheight=4mm,!A=0.5andk2=6.25.Tref=3.7C. accountofstabilizationofferedbytheincreasedthermalresistance.ThesubsequentdecreaseinthecriticalTisaresultofincreasingthermallybuoyantowinthevaporandthenalincreaseinthestabilityisaresultofvaporowintheevaporationdominantregime.Thiscompetitionbetweenthermalresistanceandthermalbuoyancyhasbeenobservedbothexperimentallyandtheoreticallyinnon-evaporationsystemsbyOzenetal.[ 9 ] 4.3.41-DModelto3-DModelAsdescribedintherstpartofthissectionthankstostressfreeconditionona3-Dmodelwecandirectlycomparea1-Dmodeltothe3-Dmodelthatusesazimuthalexpansionmodes.ThedetailsofthiscomparisonaregivenintheAppendix J .Usinga1-Dmodelismuchsimplerthanusinga3-Dmodelinmanywayssuchascomputationtimeandthecomputationpowerneeded.Howeverinanexperiment,becauseofthenitedimensionsthesystemdoesnothaveaccesstoallwavenumbers.Theallowedwavenumbersaretheoriginofthepatternsthatwouldbeobservedinanexperimentat 69

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theonsetpoint.Theazimuthalmodearisesfromthe-directionexpansiongivenby,U1=bU1(z)eteiminthe3-Dmodel.Inthefollowingguresomepossiblepatternsaredepictedforvariousazimuthalmodes. Figure4-25. Differentazimuthalmodes. Figure4-26. Onsetofconvectionversuswavenumberforliquidheight=2mmandvaporheight=3mmand!A=0.5. TheFigure 4-27 isdrawntodepicttheonsetpoint,asinTCritical,versuswavenumberforliquidheight2mmandforvaporheight3mmandmassfraction0.5.ThereisacleardipatTCritical=12.8forwavenumber1.44.Allconvectivemechanismsareassumed. 70

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Theonsetcurveisacontinuouscurve;however,anitesystem,i.e.asystemwithsidewalls,doesnothaveaccesstoallthosewavenumbers.Tondouttheaccessibleonesweneedtousethewavenumbertoazimuthalconversionformulawhichisgivenbelow.Forthe1-Dmodel,theinputstoacalculationarethedomainheights,wavenumberofthedisturbance,andtheinitialmassfraction.Ontheotherhand,inthe3-Dmodel,insteadofwavenumberofthedisturbance,theradiusandtheazimuthalmodearetheotherinputsintoacalculation.Byinputtingtheradiustheaspectratioisset,whichisradiusovertheliquidheight,andbyinputtingtheazimuthalmodetheoutputplanformfunctionisisset(thisistheoreticallygivenbyaBesselsfunction).AgainfordetailsseetheAppendix J .WiththisinformationthedimensionlesswavenumberisgivenintermsoftheaspectratioandtherootofthederivativeoftheBesselfunctioni.e., km,j=Jrm,j (4) Where,istheaspectratioandJrm,jistherootofthederivativeoftheBesselfunctionkindm. Figure4-27. Onsetofconvectionversuswavenumberforliquidheight=2mmandvaporheight=3mmand!A=0.5.Availablewavenumberswithrespecttoazimuthalmodes. 71

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Figure4-28. Growthrateversuswavenumber. Tomakethecomparison,letussettheradiusto6mm.Thissetstheaspectratioto=3.WithBessel0sfunction,Jrm,j,wegetthecorrespondingwavenumbersforeachoftheazimuthalmodes.ThesepointsaregivenintheFigure 4-28 .Inthegurethem=0modehasthelowesttemperaturedifferencemeaningthatontheonsetpointtheobservedpatternwouldbeadonutshapem=0.ThisisalsosupportedbythegrowthratecalculationthatisgivenintheFigure 4-28 wherethegrowthrateofthesystemiscalculatedusingagivenTof12.9degrees.Onemajorresultthatstandsoutasaconsequenceofourstudyisthis:Givenathinuidmixture,wecandetermine,withoutanycalculation,whethertheheatingshouldbefromthevaporsideortheliquidsideinordertoavoidaninstability.Iftheuidmixtureisthickindimensionsuchthatgravityplaysarolethencalculationswilldeterminethedirectionofheatingandtheamountofheatingtoavoidtheinstability. 72

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CHAPTER5EXPERIMENTALSETUPANDRESULTSAnexperimentalset-upwasbuilttoverifymanyofthequalitativepredictions.Forexampleinpurecomponentstheheatingfromabovearrangementisastableconguration,whileforbinarysystemsinstabilitycanbetriggeredwhenheatedfromthevaporside.ThepictureoftheexperimentalsetupisgiveninFigure 5-1 wherethewholesetupwasplacedinsideatransparentbox.ThesetupconsistedofanIRcameraoraCCDcamera(notshowninthepicture)andtemperaturecontrolsystemsbothfortheboxandtheliquidsideofthetestsection.AschematicofthesetupisgiveninFigure 5-2 Figure5-1. Experimentalsetup. 73

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Asdescribedpreviouslytheonlycontrolparameteristheverticaltemperaturegradientandinanexperiment.Thisgradientisappliedtothetestsectionbetweenthezincselenidewindow(orglasswindow)andthecopperplateofthetestsection.AsshowninFigure 5-2 ,theliquidsidetemperaturecontrolsystemconsistedofseveralsub-parts.Therstpartwastheheatingelement,whichwasplacedinsideinsulatingcement.Thisheatingelementwasplacedoveramagneticstirrer.Ontopoftheheatingelementtherewasawaterbathwithamagneticstirbartoensuretheuniformtemperatureproleinit.Thetopandthebottomofthewaterbathwerecappedwithcopperplates.Toincreasetheefciencyofthetemperaturecontrolinsidethewaterbath,coldwaterwasfedintoandoutofthebathusingamini-pump.Tomeasurethetemperature,4RTDs(OMEGAR)wereplacedundertheuppercopperplate.ThetemperatureswerecontrolledbyaPI(OMEGAR)controllerwhichonlycontroledtheheatingelementnotthecoldwaterinlet.Thisinletcouldbecontrolledbytheuserdependingonthesetpointofthetemperature.Theoverallcontrolwasbetterthan)]TJ /F4 11.955 Tf 9.3 0 Td[(=+0.1C.Thetestsectionwasplacedoverthewaterbath.Itwasconsistsoflucite-air-tightcontainers.Theaccuracyofeachpiecewasmachinedwithin0.1mm.Thissectionhousedtheliquidandvapormixture.Theinsideofthetestsectionwasmadeaccessiblefromtheoutsideusinginletandoutletholes,piping,andvalves.Twodifferenttestsectionswereusedduringtheexperiments.Thersttestsectionwasmadeupoftwopieces:liquidinsertandvaporinsert.Air-tightnessofthistestsectionwasensuredbyusingO-ringsandscrews.Theliquidside'sbottomboundarywasacopperplateplacedoverthewaterbath.ThevaporinsertwasscrewedovertheliquidinsertbetweenthosetwoO-rings,whichwereplacedtoensuretheair-tightness.Thetopboundaryofthevaporinsertwasthezincselenidewindowthatissituatedinsidealuciteholder.Theholderwasclampedtothevaporinsert.Here,also,O-ringswereemployed.Thesecondinsertwasaone-pieceinsertdesignedtobesee-throughto 74

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Figure5-2. Schematicoftheexperimentalsetup. obtainasideviewoftheconvectiveow.Ithadacopperplateatbottomboundaryandaglasswindowatthetopboundary.Forbothinsertszincselenide(glasswindows)wereusedtoapplytheuppertemperaturesetpointtothetestsection.ThetemperaturewasmeasuredusinganRTDthatwasplacedoverthewindows.Thesetpointwasthetemperatureinsidetheboxthathousedthewholeset-up.Aheater,fan,andcarradiatorwereusedtocontrolthistemperature.HerealsothePIcontrolleronlycontrolledtheheatingelementwithanoverallcontrolwhichwasbetterthan)]TJ /F4 11.955 Tf 9.29 0 Td[(=+0.1C. 75

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AnInframetrics,model760,infraredcamerawasusedtovisualizetheowpatternswhenthezincselenidewindowwasused.Thisparticularmodelofinfraredcameraiscapableofmeasuringinthe3to5mrangeorthe8to12mrange;however,onlythe8to12mrangewasusedduetothetemperaturerangeoftheexperiments,i.e.20to40C,andalsointhatrangethezincselenidewindowis60%transparenttoinfraredradiation.Theinfraredcamerawasplaceddirectlyabovethetestsectionandmeasuredtheinfraredradiationbeingemittedbytheliquid.Aneffectiveemissivitycouldbecalibratedandprogrammedintotheinfraredcameratondthetemperatureoftheinterface.TheIRcamerawasneverusedtomeasurethetemperature,ratheritwasusedtodetectthevariationininterfacialtemperaturefromwhichtheowstructurecouldbededuced.Additionally,ananti-reectiveinfraredpolymerwascoatedonthezincselenidewindowbyII-VIInc.Thiscoatingwasusefultodecreasethefalseimagesgeneratedbyreected,ambientinfraredradiation. Figure5-3. ThinlmcondensationontheZnSewindow.HeatingfrombelowforT=1.2C. Fortheglasswindowsetup,KalliroscopetraceroraluminumakesandCCDcameracouplewereusedtovisualizetheow.BoththeCCDandIRcameraweredirectlyconnectedtoacomputerwhereatimelapsevideowascaptured.Inaseparateair-tight-metalcontaineralcoholvaporswereproducedtoushtheairinsidethetestsection.Thismetalcontainerhadoneoutletandoneinletholethatwerecontrolledbyvalves.Toproducethealcoholvapor,liquidswerefedintothecontainer 76

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Figure5-4. Condensationontheglasswindow.Heatingfrombelowforabinarymixtureof50/50.InputT=5.Liquidheight=10mm,vaporheight=15mm,anddiameter=38.1mm. andplacedoveraheater.First,bothvalveswerekeptopentoallowairtoescape.Afterclosingthevalves,butkeepingtheheateron,thevaporwassuckedbysyringes.However,thevaporequalto4timesofthemetalcontainervolumewasnotusedtoushtheairinsidethetestsectiontoensureverylowairconcentration.Furthervaporthatwasproducedfedintothetestsectionusingtheinletholesandvalvesystem.Againtoensureaverylowconcentrationofairthetestsectionwasushedbyalcoholvapor(eitherpureormixture)atleast20timesofitsvolume.Sigma-Aldrichethanol99.5+%purityandSigma-Aldrichsec-butanol99+%puritywereusedduringtheexperiments.IntheheatingfrombelowexperimentsduetoZnSeorglasswindowbeingcolderthantheinterface,acondensationoccurredonthewindowsevenforthetemperaturedifferenceslessthanadegree.ThiscondensationblockstheIRradiationgoingthroughtheZnSewindow;therefore,nosignalwasreceivedfromtheinterface.Thisisshownin 77

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Figure5-5. HeatingfromaboveconvectiveowforT=)]TJ /F3 11.955 Tf 9.3 0 Td[(7.5.Longwavelengthinstabilityfor50/50binarymixture.Liquidheight=6mm,vaporheight=14mm,anddiameter=18mm. 78

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Figure 5-3 forethanol.IntheleftpicturethebigcircleistheZnSewindow,whichhasauniformtemperatureprole.Thetestsectionisthelittleinnercircle;itismadeclearerintherightpicture.Condensationwasunavoidablewhentheheatwassuppliedfromtheliquidside.Foralargertemperaturedifferencethecondensationonthewindowalsomanifestedaninstabilitywhichmightbeworthresearchingabout.ThisisgiveninFigure 5-4 .Althoughcondensationonthetopplatewasnotpartofourmodeltoobservethephenomena,severalsideviewsoftheconvectiveowwereobtainedusingKalliroscopeparticles;However,theseparticlesdidnotcreateenoughcontrasttohaveclear,stillimages.Additionally,aluminumakeswereusedforowvisualizationwithbothethanolandsec-butanol;thus,theseparticlessettleddownreallyfastandhadatendencytocollidetogether. Figure5-6. HeatingfromaboveconvectiveowforT=)]TJ /F3 11.955 Tf 9.3 0 Td[(15for50=50binarymixture.Liquidheight=12mm,vaporheight=13mm,anddiameter=38.1mm. Intheheatingfromaboveexperimentsduetoconstantevaporationandcondensationthetemperatureattheinterfaceuctuates;therefore,itwasnotpossibletoobserveaclearno-owstate.TheconvectiveowthatisshowninFigures 5-5 and 5-6 forabinarymixturewereprincipallydonetoverifythephenomena.IntheFigure 5-5 along-wavelengthinstabilityisobserved.Theliquidcomingfromthebottomofthedomainrisesfromtheupperpartoftheinsertthatgoesalongtheinterfaceandfallsdownfromthedomainpartoftheinsert.InFigure 5-6 theazimuthalmode2isshown.Asis 79

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expectedinasimilarexperimentforasinglecomponentsystem,nopatternformationwasobserved.Meanwhile,toovercomethethermalreadingproblemsforbothheatingfromaboveandbelow,theZnSewindowwasreplacedbyaglasswindow.Aglasswindowallowedvisualizingtheowbyusingsometracerparticles;however,aspreviouslymentioned,itwasnoteasytoavoidthesettlingdownproblemand,thecontrastissueaswell.Thus,nogureisdisplayedwiththoseparticlespresented.Toovercometheproblemsthatarefacedduringtheseexperimentsusingadifferentbinarymixturecanbeagoodproposalforthefuture.Keepinmindthatourmodelisapplicableforidealmixingandsimplemixturepropertycalculations. 80

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CHAPTER6CONCLUDINGREMARKSANDFUTURESCOPEInthischapterasummaryoftheresultswithsomeideasforpossiblefutureworksisgiven.Evaporativeinstabilityinbinarymixturewasinvestigatedusingethanolandsec-butanolasthetwocomponentsofthebinarymixture.Itisshownthatduetotheconcentrationdependenceofsurfacetensionanddensitytheevaporativeinstabilityprobleminabinarymixtureismorecomplicatedthanasinglecomponentsystems.Theresearchproblemisdividedintomainlythreeparts;pureevaporativeconvection,surfacetensiondrivenconvection,andtheaddedeffectofthegravity.Inalloftheseconvectivemodeshavingasecondcomponentinthesystemshowsitssignatureontheonsetpoint.Inthepureevaporativeconvectiontheonsetpointhasmaximumstabilitywithrespecttomassfractionforacertainrangeofwavenumber.Thismaximumistheresultofachangeinthethermo-physicalpropertieswithmassfractioneventhoughthemixturepropertiesarecalculatedbyusingsimplemixingrules.Withtheadditionofthesurfacetensiongradienteffectsintothemodelthebinarymixturephysicsdepartgreatlyfromasinglecomponentsystem.Itisimportanttopointoutthatthesolutaleffectsstronglydependonthechoiceofthecomponents.Intheethanolandsec-butanolmixturethemorevolatilecomponentethanolhasalowersurfacetension;therefore,withincreasingethanolconcentrationthesurfacetensionofthemixtureislowered.Duetothisproperty,unlikeasinglecomponentsystemthebinarysystemcanbemadeunstablebyheatingfromthevaporside,i.e.,negativecriticaltemperaturedifference.Theonsetpointexhibitsastrongdependenceonthemassfractionandwavenumber.Themassfractiondependenceoftheonsetpointresultsinalessstableregioninthemiddlerange.Itisfoundthatthisisduetostrongermassfractionperturbationsinthatregionthantheremainingmassfractionranges.The 81

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liquidandvaporheightseffectontheonsetpointshowthatwithincreasingvaporheight,thesystemgetsmorestablewhilethecontraryholdswithincreasingliquidheighti.e.,thesystemgetmoreunstable.Inthelastpartgravityisintroducedintotheproblemandwithintroductionofgravitythedensitygradientsstarttoplayaroleonthephysics.Thedensityofthemixturedecreasesbothbyincreasingtemperatureandbyincreasingethanolconcentration.Inthissectionwestarttohavetwophysicallyrealizablepossibilitiestogetaninstabilityinoursystem;oneisbyheatingfromabove,i.e.,anegativetemperaturedifference,andthesecondisbyheatingfrombelow,i.e.,apositivecriticaltemperaturedifference.Theinstabilityobtainedbyheatingfromaboveistheconsequenceofsolutaldependenceofsurfacetensionanddensity.Asintheabsenceofgravitytheonsetpointshowsaminimumstabilityregioninthemiddlerangeofmassfraction.Thesecondpossibility,thatisheatingfromtheliquidside,occursonaccountofthetemperaturedependenceofdensitymuchlikeinasinglecomponentsystem.Inthisheatingarrangementthemiddlerangeofmassfractionofthebinarymixtureismorestablethantherest,i.e.largerpositivecriticaltemperaturedifferencesobtainuntilthewavenumberofthedisturbanceexceedsacertainlevel.Inthepresenceofgravityanincreaseinthevaporheightleadstoanincreaseinstabilityreachedamaximumandwithfurtherincreaseofvaporheightleadstoadecreaseinthestabilitybeforeattainingasecondregionofstabilityincreaseforfurtherheightincreases.Therstriseinstabilityisduetoincreaseinthethermalresistanceofthevaporphase.Afurtherincreaseinheightcausesthevaporphasetobecomegravitationallyunstable;nallyanincreaseintheheightevenmorebringsaboutstabilizationoftheevaporationmode.Theexistenceofinstabilitywhenheatedfromaboveresultsaresupportedbyexperiments.Thephysicsoftheproblemshowsthatthereisacompetitionbetweenthermalconvectionmodesandsolutalconvectionmodes.Nowheredoesoscillatorybehavior 82

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showitspresenceattheonset.Theoscillatorybehaviorcanbeignitedbydecreasingtheeffectofthesolutaldependenceofthesurfacetension.Inotherwords,ifthetwocomponentshaveaclosersurfacetensionvalue,wecouldseeoscillationsattheonset.Preliminarycalculationsshowthat1-PentanolandHexanolbinarypairisagoodcandidateforfutureinvestigationoftheoscillations.Theotherpointthatwouldbeworthinvestigatingistheeffectofinertsinbothphasessuchasairinthevaporphaseandnon-evaporatingsiliconeoilintheliquidphase.Airinthevaporphasewillmodifythepartialpressuresandthereforemodifythecompositionattheinterface.Thepresenceofoilwillgreatlymodifytheviscosityoftheliquidbringingthestabilitytotheproblemalsomodifythedynamicsofevaporation.Keepinmindthatthesemodicationsarenottrivialandmightrequirenon-idealmixingtobeaccountedforinthemodelasmorecomplicatedmixturepropertiescalculations. 83

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APPENDIXATHERMODYNAMICEQUILIBRIUMConsideranevaporatingliquidunderlyingitsownvaporatequilibrium.InthermodynamictermsthisequilibriumisrepresentedbytheequalityinspecicGibbsfreeenergieswhichisgivenby,^GL=^GVwhere^GLand^GVaretheliquidandvaporspecicGibbsfreeenergies,respectively.Thisrelationisvalidforanyinterfaceshape;therefore,wecanwrite^GLcurved=^GVcurvedand^GLat=^GGatSumandrearrangetheabovetwoequations,^GLcurved)]TJ /F3 11.955 Tf 13 2.65 Td[(^GLat)]TJ /F11 11.955 Tf 11.96 13.27 Td[(^GGcurved)]TJ /F3 11.955 Tf 13 2.65 Td[(^GGat=0TheGibbsfreeenergyisgivenbyd^G=)]TJ /F3 11.955 Tf 9.65 2.66 Td[(^SdT+^VdPwhere^Sand^V=1 arethespecicentropyandvolume,respectively.AssumingthatthespecicentropyandvolumedonotchangemuchonaccountofsmallchangesinTandP,weget)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(SG)]TJ /F3 11.955 Tf 11.95 0 Td[(SLdT+VLdPL)]TJ /F3 11.955 Tf 11.95 0 Td[(VGdPG=0Usingtheidealgaslaw,wecanwrite)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(SG)]TJ /F3 11.955 Tf 11.95 0 Td[(SLdT+VLdPL)]TJ /F3 11.955 Tf 13.15 8.09 Td[(RT PGdPG=0 84

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Andinserting(GG)]TJ /F6 11.955 Tf 11.96 0 Td[(GL)=(HG)]TJ /F6 11.955 Tf 11.96 0 Td[(HL))]TJ /F3 11.955 Tf 11.95 0 Td[((SG)]TJ /F6 11.955 Tf 11.96 0 Td[(SL)TintoaboveequationHG)]TJ /F3 11.955 Tf 11.96 0 Td[(HL TdT+VLdPL)]TJ /F3 11.955 Tf 13.15 8.09 Td[(RT PGdPG=0ObservethatHG)]TJ /F6 11.955 Tf 11.95 0 Td[(HLisequaltothelatentheatofevaporation,~.Therefore,~ T2dT+VL TdPL)]TJ /F3 11.955 Tf 16.28 8.08 Td[(R PGdPG=0Wecanintegratebothsidestogetarelationbetweenthetemperatureandpressureattheinterfacefromtheattocurvedstates.HenceZTCTF~ T2dT+ZPLCPLFVL TdPL)]TJ /F11 11.955 Tf 11.96 16.27 Td[(ZPGCPGFR PGdPG=0Integrationgives,~1 TTCTF+VL TPLCPLF)]TJ /F6 11.955 Tf 11.95 0 Td[(RlnPjPGCPGF=0and~1 TC)]TJ /F3 11.955 Tf 17.13 8.09 Td[(1 TF+VL T1 PLC)]TJ /F3 11.955 Tf 17.26 8.09 Td[(1 PLF)]TJ /F6 11.955 Tf 11.96 0 Td[(R)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(lnPGC)]TJ /F3 11.955 Tf 11.96 0 Td[(lnPGF=0Intheequationabovethesecondtermshowsthepressurechangeintheliquidphasewhichisnegligiblecomparetotherest.Therefore,wecanwritethethermodynamicequilibriumas,~1 TC)]TJ /F3 11.955 Tf 17.13 8.09 Td[(1 TF)]TJ /F6 11.955 Tf 11.96 0 Td[(R)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(lnPGC)]TJ /F3 11.955 Tf 11.95 0 Td[(lnPGF=0TheaboveequationiscalledtheClapeyronequationanditgivesthevaporpressureofacomponentasafunctionoftemperature.Thevaporpressureoftheidealbinarymixtureisgivenbythemolarweightedsumofthepurecomponentvaporpressures.Thiscanbedoneintwoways;rstusingaClapeyronequationforeachcomponentsandtakethemolarweightedsumorbyusingamixtureformoftheClapeyronequation.Let'squantifythisapproachwithanexample. 85

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Asanexampleassumethattheatstateisstate1andthecurvedstateisstate2withT1=303.15K,T2=304.15KPVapeth(2)=exp)]TJ /F11 11.955 Tf 11.29 16.86 Td[(~ R1 T2)]TJ /F3 11.955 Tf 16.82 8.09 Td[(1 T1)]TJ /F3 11.955 Tf 11.95 0 Td[(lnPVapeth(1)PVapbut(2)=exp)]TJ /F11 11.955 Tf 11.29 16.86 Td[(~ R1 T2)]TJ /F3 11.955 Tf 16.82 8.09 Td[(1 T1)]TJ /F3 11.955 Tf 11.95 0 Td[(lnPVapbut(1)andthevaporpressureofthemixture,ortotalpressure,isgivenby,P2=PVapeth(2)xeth+PVapbut(2)(1)]TJ /F6 11.955 Tf 11.95 0 Td[(xeth)Theotherwayofcalculatingthetotalpressureisusingthelatentheatofthemixtureandthevaporpressureofthemixtureatstate1.PVapmix(2)=exp)]TJ /F11 11.955 Tf 11.29 16.86 Td[(~mix R1 T2)]TJ /F3 11.955 Tf 16.81 8.09 Td[(1 T1)]TJ /F3 11.955 Tf 11.95 0 Td[(lnPVapmix(1)Thedifferencebetweenthesetwoapproachesislessthan0.3%.TheperturbedversionoftheClayperonequationisgivenby,YKEP1)]TJ /F11 11.955 Tf 11.96 11.35 Td[(YPEZ1=T1+dT0 dzZ1 TbasewhereYKEandYPEarethedimensionlessparametersfromthelinearizedClapeyronequationdenedasYKE= ~d2andYPE=gd ~. 86

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APPENDIXBBOUSSINESQAPPROXIMATIONIntheRayleighproblem,itisclearthatuidscannotbestrictlyincompressiblebecausedensityvariationscreatetheconvection.Thefactthatuiddensityisafunctionoftemperatureandconcentrationmakestheproblemdependentontheheattransfer.Forthatreason,theBoussinesqapproximationisemployed.Clearly,densitybeingafunctionoftemperatureandalsoconcentrationmakestheRayleighproblemverycomplicated.Thisapproximationallowsauidtobetreatedasincompressibleinalltermsofthemomentumequationexceptthosemultipliedbygravity.Inotherwords,intheBoussinesqapproximationthedensityofauidisafunctionoftemperatureandconcentrationonlywhenitismultipliedbygravity,intherestofthetermsdensityistakenasconstant.Boussinesqapproximationisgivenby,=R 1+1 R@ @TTR(T)]TJ /F6 11.955 Tf 11.96 0 Td[(TR)+1 R@ @!A!AR(!A)]TJ /F4 11.955 Tf 11.95 0 Td[(!AR)!here1 R@ @TT=TRand1 R@ @!!=!RarethedensityvariationoftemperatureandconcentrationwithsubscriptRrepresentsareferencestate.LetusapplyBossinesqapproximationtoamomentumequationgivenbelow@!v @t+!vr!v=rP+!g+r~~SUponemployingtheapproximationwehaveR@!v @t+R!vr!v=rP+R 1+1 R@ @TTR(T)]TJ /F6 11.955 Tf 11.95 .01 Td[(TR)+1 R@ @!A!AR(!A)]TJ /F4 11.955 Tf 11.95 0 Td[(!AR)!!g+r~~ST=)]TJ /F3 11.955 Tf 13.68 8.09 Td[(1 @ @TT=TRand!=)]TJ /F3 11.955 Tf 13.68 8.09 Td[(1 @ @!!=!RwithT&!arepositivewhenthederivativeofdensitywithrespecttotemperatureandconcentrationarenegative. 87

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APPENDIXCRAOULT'SLAWToderivetheRaoult'slawforvapor-liquidequilibriuminamulti-componentsystemweassumethatthevaporphasebehavesasanidealgasandtheliquidphaseasanidealsolution.Theequilibriumdenedastheequalityofthechemicalpotentials,isgivenby'LA='VAForanidealliquidmixturewecanwrite'LA='0,LA(T,PA)+RTlnxA'LA='0,LA(T,PAvap)+PAZPAvap@'0,LA(T,P0A) @PdP0+RTlnxAwhere@'0,LA(T,P0A) @P=V0,LA,isthemolarvolumeofpureliquidAandifweassumetheliquidvolumeisnotchangingwithpressuretheaboveequationbecomes'LA='0,LA(T,PvapA)+V0,LA(P)]TJ /F6 11.955 Tf 11.96 0 Td[(PvapA)+RTlnxAThesamederivationisvalidforthevaporphasebutforthevaporphaseweneedtoemploytheidealgaslawtocalculatethevolumechangewithrespecttopressurechange.Thechemicalpotentialofanidealvaporphaseisgivenby,'VA='0,VA(T,PvapA)+RTlnP PvapA+RTlnyARecallthat'LA='VAandalso'0,LA(T,PvapA)='0,VA(T,PvapA)leadstoV0,LA(P)]TJ /F6 11.955 Tf 11.95 0 Td[(PvapA)+RTlnxA=RTlnP PvapA+RTlnyA 88

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afterrearrangingweget,yAP=xAPvapAexpV0,LA(P)]TJ /F6 11.955 Tf 11.96 0 Td[(PvapA) RTTheexponentialtermiscalledthePoyntingfactorwhichiscloseto1formostofthesystemsatlowpressures.AndtheRaoult'slawinitsnalfromforidealvaporandidealliquidmixtureisgivenby,yAP=xAPvapAForthesystemswheretheliquidandthevaporphasescannotbeassumedidealamodiedRaoult'slawshouldbeused.ThemodiedRaoult'slawandthedenitionofthetermsaregivingbelowtogiveanideatothereader;although,non-idealsystemsarenotdiscussedinthisstudy.bAyAP=AvapAxAPvapAexpV0,LA(P)]TJ /F6 11.955 Tf 11.96 0 Td[(PvapA) RTWhere;bAisthefugacitycoefcientofAinmixtureatgivenTandP,accountsfornon-idealityinthemixture.AistheactivitycoefcientforAinmixtureatT,PcanbeafunctionofallxA,T,andPaccountsfornon-idealityinliquidmixture.vapAisthefugacitycoefcientforpureAatT,Pvap. 89

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APPENDIXDDERIVATIONOFTHEUNITNORMALANDTHEINTERFACESPEEDLetasurfacebedenotedbyf=z)]TJ /F6 11.955 Tf 11.95 0 Td[(Z(x,t)=0Then,f,ispositiveononesideoff=0,negativeontheother,andthenormalpointingintotheregionwherefispositivegivenby,~n=rf jrfjHere,rf=@f @x~i+@f @x~kThen,theequationforthenormalstressbalanceisgivenby ~n=)]TJ /F9 7.97 Tf 10.49 4.71 Td[(@Z @x~i+~k h1+)]TJ /F9 7.97 Tf 6.68 -4.98 Td[(@Z @x2i1/2Thederivationoftheunittangentvectorisstraightforwardusingthedenition~n~t=0fromwhichweget ~t=~i+@Z @x~k h1+)]TJ /F9 7.97 Tf 6.67 -4.97 Td[(@Z @x2i1/2Togettheinterfacespeed,letasurfacebedenotedby,f(~r,t)=0~r=~r(x,y,z).Recallunitnormalofthesurfaceisgivenby~n=rf jrfjLetthesurfacemoveasmalldistancesalongitsnormalintimet.Then,f(*rs*n,t+t)isgivenby,f(~rs~n,t+t)=f(~r,t)s~nrf(~r,t)+t@f(~r,t) @t+... 90

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whencef(~rs~n,t+t)=0=f(~r,t)requiress~nrf(~r,t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(t@f(~r,t) @tThenormalspeedofthesurface,u,isthengivenby,u=s t=)]TJ /F9 7.97 Tf 26.15 14.26 Td[(@f(~r,t) @t ~nrf(~r,t)Now,usingthedenitionoftheunitnormalgivenearlierwegetu=)]TJ /F9 7.97 Tf 17.31 13.49 Td[(@f @t jrfjInourproblems,thedenitionofubecomesu=@Z @t )]TJ /F9 7.97 Tf 6.68 -4.98 Td[(@Z @x2+11/2 91

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APPENDIXEENERGYBALANCEFORA1-DBINARYSYSTEMTheenergybalanceinitsrawformisgivenby,(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)bU+1 2~v2+~q)]TJ /F4 11.955 Tf 13.24 6.53 Td[(~~T~v~n+2H~n~u=(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)bU+1 2~v2+~q)]TJ /F4 11.955 Tf 13.24 6.53 Td[(~~T~vandthemomentumbalanceby,(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~v)]TJ /F4 11.955 Tf 13.24 6.53 Td[(~~T~n+2H~n=(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~v)]TJ /F4 11.955 Tf 13.24 6.53 Td[(~~T~nInordertoconverttheenergybalanceintoanobserverinvariantformthedotproductofthemomentumbalancewith~uissubtractedfromtheenergybalance.Workingonlywiththerighthandsideoftheequationsweget(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)+(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)1 2~v2)]TJ /F4 11.955 Tf 11.54 .5 Td[(~v~u+~q)]TJ /F4 11.955 Tf 13.24 6.53 Td[(~~T(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~nUsingthedenitionof~~Tthestresstensor,~~T=)]TJ /F6 11.955 Tf 9.3 0 Td[(P~~I+~~S,weget(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)+(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)1 2~v2)]TJ /F4 11.955 Tf 11.53 .5 Td[(~v~u+~q)]TJ /F11 11.955 Tf 11.95 16.86 Td[()]TJ /F6 11.955 Tf 9.3 0 Td[(P~~I+~~S(~v)]TJ /F4 11.955 Tf 11.5 .5 Td[(~u)~nandusingtheinternalenergydenition,bH=bU+P weget(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)+(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2~u2+~q)]TJ /F4 11.955 Tf 12.13 6.53 Td[(~~S(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~nThe(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)isanobserverinvarianttermbut~u2termisnotobserverinvariant,however,employing1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2~u2=1 2~v2)]TJ /F4 11.955 Tf 11.54 .5 Td[(~v~u!1 2)]TJ /F4 11.955 Tf 5.06 -9.18 Td[(~v2)]TJ /F3 11.955 Tf 11.96 0 Td[(2~v~u+~u2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2~u2=1 2~v2)]TJ /F4 11.955 Tf 11.53 .5 Td[(~v~uandfromthetotalmassbalance1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~u2=1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~u2.Wegettheobserverinvariantenergybalance,(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)bH+1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)2+~q)]TJ /F4 11.955 Tf 12.13 6.53 Td[(~~S(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n... 92

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=(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)bH+1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)2+~q)]TJ /F4 11.955 Tf 12.13 6.53 Td[(~~S(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~nForabinarymixture~q=)]TJ /F4 11.955 Tf 9.3 0 Td[(rT+NPiJibHiandthetotalmixtureenthalpyis=NPi!ibHi.Observethatforindividualcomponentsthereferencestatesenthalpiesforliquidandvaporarethesame;therefore,referencestatescanceloutfromlefthandsidetorighthandside.Fornowletusjustworkwithh(~v)]TJ /F4 11.955 Tf 11.49 .49 Td[(~u)bHi~n,~nNPiJibHifromthelefthandside(LHS)andwithh(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)bHii~n,~nNPiJibHifromtherighthandside(RHS).ForabinarymixtureusingtheaboverelationshbHA!A(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u) +JA +bHB(!B(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)+JB)iLHSandhbHA!A(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u) +JA +bHB(!B(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)+JB)iRHSbyusingcomponentbalance,!A(~v)]TJ /F4 11.955 Tf 11.49 .49 Td[(~u) +JA =!A(~v)]TJ /F4 11.955 Tf 11.49 .49 Td[(~u) +JA forcombiningtheLHSandRHS(ThisisalsovalidforcomponentB).Thenonlinearenergybalanceattheinterfaceforanidealbinarymixturebecomes,hbHA)]TJ /F11 11.955 Tf 13.34 3.15 Td[(bHA(!A(~v)]TJ /F4 11.955 Tf 11.49 .49 Td[(~u)+JA)+bHB)]TJ /F11 11.955 Tf 13.34 3.15 Td[(bHB(!B(~v)]TJ /F4 11.955 Tf 11.49 .49 Td[(~u)+JB)i~n+...(v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)2+()]TJ /F4 11.955 Tf 9.3 0 Td[(rT))]TJ /F4 11.955 Tf 12.13 6.53 Td[(~~S(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n...=(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)2+()]TJ /F4 11.955 Tf 9.3 0 Td[(rT))]TJ /F4 11.955 Tf 12.13 6.53 Td[(~~S(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~nHere)]TJ /F11 11.955 Tf 11.29 13.27 Td[(bHA)]TJ /F11 11.955 Tf 13.33 3.15 Td[(bHA=~AisthelatentheatofthecomponentAand)]TJ /F11 11.955 Tf 11.29 13.27 Td[(bHB)]TJ /F11 11.955 Tf 13.34 3.15 Td[(bHB=~BisthelatentheatofthecomponentB.[)]TJ /F14 11.955 Tf 9.3 0 Td[(~A(!A(~v)]TJ /F4 11.955 Tf 11.49 .49 Td[(~u)+JA))]TJ /F14 11.955 Tf 11.96 0 Td[(~B(!B(~v)]TJ /F4 11.955 Tf 11.49 .49 Td[(~u)+JB)]~n+...(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)2+()]TJ /F4 11.955 Tf 9.3 0 Td[(rT))]TJ /F4 11.955 Tf 12.13 6.53 Td[(~~S(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n 93

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=(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)2+()]TJ /F4 11.955 Tf 9.3 0 Td[(rT))]TJ /F4 11.955 Tf 12.13 6.53 Td[(~~S(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~nScalingtheenergybalancetermbytermgives1+KPC1 2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2(~v)]TJ /F4 11.955 Tf 11.5 .5 Td[(~u)2(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n+E2r!A~n+...)]TJ /F6 11.955 Tf 9.3 0 Td[(VPC~~S(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u))]TJ /F4 11.955 Tf 13.15 8.09 Td[( ~~S(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n)]TJ /F6 11.955 Tf 11.96 0 Td[(E1rT)]TJ /F4 11.955 Tf 13.15 8.09 Td[( rT~n=0whereKPC=2 d2()]TJ /F14 11.955 Tf 9.3 0 Td[(~A!A)]TJ /F14 11.955 Tf 11.95 0 Td[(~B!B)VPC= d2()]TJ /F14 11.955 Tf 9.29 0 Td[(~A!A)]TJ /F14 11.955 Tf 11.96 0 Td[(~B!B)E1=T ()]TJ /F14 11.955 Tf 9.29 0 Td[(~A!A)]TJ /F14 11.955 Tf 11.96 0 Td[(~B!B)E2=DAB(~A)]TJ /F14 11.955 Tf 11.95 0 Td[(~B) ()]TJ /F14 11.955 Tf 9.3 0 Td[(~A!A)]TJ /F14 11.955 Tf 11.95 0 Td[(~B!B) 94

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APPENDIXFCOMPONENTBALANCEFOR1-DSYSTEMThecomponentbalanceinrawfromisgivenby,A(~vA)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n=A(~vA)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~nwhereAisthedensityofcomponentAand~vAisthecomponentvelocityattheinterface.Becauseofthecomponentvelocitytermthecomponentbalanceisnotuserfriendlyinthisrepresentation;therefore,toreachmoreusableformofthecomponentbalanceweaddandsubtract~vfromrighthandsideandaddandsubtract~vfromlefthandside.A(~vA)]TJ /F4 11.955 Tf 11.53 .49 Td[(~v+~v)]TJ /F4 11.955 Tf 11.49 .49 Td[(~u)~n=A(~vA)]TJ /F4 11.955 Tf 11.54 .49 Td[(~v+~v)]TJ /F4 11.955 Tf 11.5 .49 Td[(~u)~nHerewecandenethemassuxasJA=A(~vA)]TJ /F4 11.955 Tf 11.54 .5 Td[(~v)=!A(~vA)]TJ /F4 11.955 Tf 11.53 .5 Td[(~v)andputthecomponentbalanceinitsnalnonlinearform.JA~n+!A(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n=JA~n+!A(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~nUponscalingcomponentbalancebecomes,[r!A+Le!A(~v)]TJ /F4 11.955 Tf 11.49 .49 Td[(~u)]~n= )]TJ /F6 11.955 Tf 10.49 8.08 Td[(DAB DABr!A+Le!A(~v)]TJ /F4 11.955 Tf 11.49 .49 Td[(~u)~nNotethatJA=)]TJ /F4 11.955 Tf 9.3 0 Td[(DABr!A 95

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APPENDIXGSORETEFFECTTheSoreteffectalsocalledthermodiffusion,isthediffusionofthespeciesinamulti-componentmixturewhenthesystemissubjecttoatemperaturegradient.ForpositiveSoretcoefcient,theheavyparticlesmovefromhottocoldregionandviceversa.TheSoretcoefcientisafunctionofbothtemperatureandconcentrationwherethesignofthecoefcientcanalsochangedependingonbothtemperatureandconcentration.IntroducingtheSoreteffecttoourbasemodelwouldonlymodiestheequationsthathasthemassuxterm,JA,JA=)]TJ /F4 11.955 Tf 9.3 0 Td[((DABr!A+DT!A0(1)]TJ /F4 11.955 Tf 11.96 0 Td[(!A0)rT)whereDTisthethermaldiffusionterm.TheSoreteffectmodiestotal6equationsinoursystem;theconcentrationbalanceequationsinbothphases,bottomandtopboundaryconditionsonconcentration,andthetwointerfaceconditions.Theliquidphasenon-linearconcentrationbalancewiththeSoreteffectis@!A @t+~vr!A=)]TJ /F3 11.955 Tf 10.49 8.08 Td[(1 rJA@!A @t+~vr!A=DABr2!A+DT!A0(1)]TJ /F4 11.955 Tf 11.95 0 Td[(!A0)r2TUponscalingtheliquidphasenon-linearconcentrationbalanceweget,@!A @t+~vr!A=1 Le)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(r2!A+(STT)r2TwhereST=DT DAB!A0(1)]TJ /F4 11.955 Tf 11.95 0 Td[(!A0)iscalledtheSoretcoefcientandthelinearizedconcentrationbalancegivenby@!A1 @t+~vr!A1=DAB DAB1 Le)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(r2!A1+(STT)r2T1 96

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Likewise,thevaporphasescalednon-linearconcentrationbalanceisgivenby,@!A @t+~vr!A=DAB DAB1 Le)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(r2!A+(STT)r2TInit0slinearfrom@!A1 @t+~vr!A1=DAB DAB1 Le)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(r2!A1+(STT)r2T1ThetopandbottomboundaryconditionsonmassfractionareJA~n=@!A @z+DT DAB!A0)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(!A0@T @z=0!@!A1 @z+(STT)@T1 @z=0andJA~n=@!A @z+DT DAB!A0)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(!A0@T @z=0!@!A1 @z+(STT)@T1 @z=0Therearetwointerfaceconditionsleftthatneedsmodicationduetothemassuxtermandoneofthemisthecomponentbalanceandinitsnon-linearformitisgivenby,JA~n+!A(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~n=JA~n+!A(~v)]TJ /F4 11.955 Tf 11.49 .5 Td[(~u)~nandthelinearformisgivenby,)]TJ /F4 11.955 Tf 10.5 8.09 Td[(@!A1 @z)]TJ /F3 11.955 Tf 11.96 -.17 Td[((STT)@T1 @z+Le!A0vz1)]TJ /F4 11.955 Tf 13.15 8.09 Td[(@Z1 @t=...)]TJ /F4 11.955 Tf 10.49 8.09 Td[( DAB DAB@!A1 @z+(STT)@T1 @z+ Le!A0vz1)]TJ /F4 11.955 Tf 13.15 8.09 Td[(@Z1 @tThelastequationthatrequiresmodicationwhentheSoreteffectispresentistheenergybalanceattheinterfacei.e.,@!A1 @z+Le)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F14 11.955 Tf 9.3 0 Td[(~A!A0+~B)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(!A0)]TJ /F3 11.955 Tf 11.96 0 Td[(1 [~A)]TJ /F14 11.955 Tf 11.96 0 Td[(~B]vz1)]TJ /F4 11.955 Tf 13.15 8.09 Td[(@Z1 @t+(STT)@T1 @z=...T (DAB[~A)]TJ /F14 11.955 Tf 11.96 0 Td[(~B])@T1 @z)]TJ /F4 11.955 Tf 11.95 0 Td[(@T1 @z 97

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APPENDIXH3-DSCALEDLINEAREQUATIONSFORSINGLECOMPONENTSYSTEMInthisAppendixwewillgiveonlythescaledlinearformofthe3-DmodelingequationsofthesystemthatisdepictedinFigure H-1 .Thegurerepresentstheclosedimpermeablecontainerwithaconductivetopplate,atz=,aconductivebottomplate,atz=)]TJ /F3 11.955 Tf 9.3 0 Td[(1,andaninsulatedsidewallatr=ASP.Theliquid-vaporinterfaceislocatedatz=0. FigureH-1. 3-Dmodelillustration. Beforegivingthelinearmodelingequationsletusrstintroducetheunitnormal,unittangents,andthesurfacespeedfor3-Dsystem.Thederivationsofthesevectorsaresimilarto1-Dsystem.~n=~iz)]TJ /F9 7.97 Tf 13.15 4.7 Td[(@Z @r~ir)]TJ /F9 7.97 Tf 13.15 4.7 Td[(@Z @~i 1+)]TJ /F9 7.97 Tf 6.67 -4.98 Td[(@Z @r2+1 r)]TJ /F9 7.97 Tf 6.67 -4.98 Td[(@Z @21=2~t1=@Z @1 r~iz+~i )]TJ /F9 7.97 Tf 6.67 -4.98 Td[(@Z @2+11=2&~t2=@Z @r~iz+~ir 1+)]TJ /F9 7.97 Tf 6.68 -4.98 Td[(@Z @r21=2 98

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andu=@Z @t 1+)]TJ /F9 7.97 Tf 6.67 -4.97 Td[(@Z @r2+1 r)]TJ /F9 7.97 Tf 6.68 -4.97 Td[(@Z @21=2Linearizationandscalingaredoneinthesamemannerwith1-Dmodel;however,inthenormalmodeexpansionhereweuseU1=^U1(z)eteimwheremrepresentazimuthalsymmetryofthesolutionandisstillthegrowthconstant.Duetothisazimuthalexpansiontherearenoderivativesinthefollowingequations.Liquidphasemomentumbalance,energybalance,andtotalcontinuityintheirlinearizedandscaledformsaregivenby, Pr~vr1=)]TJ /F3 11.955 Tf 11.02 8.09 Td[(1 @P1 @r+1 2@2~vr1 @r2+1 r@~vr1 @r)]TJ /F6 11.955 Tf 13.15 8.09 Td[(m2 r2~vr1)]TJ /F4 11.955 Tf 12.73 8.58 Td[(~vr1 r2)]TJ /F3 11.955 Tf 11.95 0 Td[(2im~v1+@2~vr1 @z2 Pr~v1=)]TJ /F3 11.955 Tf 11.02 8.09 Td[(1 1 rimP1+1 2@2~v1 @r2+1 r@~v1 @r)]TJ /F6 11.955 Tf 13.15 8.09 Td[(m2 r2~v1)]TJ /F4 11.955 Tf 12.73 8.59 Td[(~v1 r2+2im r2~vr1+@2~v1 @z2 Pr~vz1=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(@P1 @z+1 2@2~vz1 @r2+1 r@~vz1 @r)]TJ /F6 11.955 Tf 13.16 8.09 Td[(m2 r2~vz1+@2~vz1 @z2+RaT PrT1+~vz1dT0 dz0=1 2@2T1 @r2+1 r@T1 @r)]TJ /F6 11.955 Tf 13.15 8.09 Td[(m2 r2T1+@2T1 @z2and1 r@(r~vr1) @r+1 r@~v1 @+@~vz1 @z=0Inaboveequationsistheaspectratio.Likewiseinthevaporphasethemomentumandenergybalanceswithtotalcontinuityaregivenby, Pr~vr1=)]TJ /F4 11.955 Tf 12.86 8.09 Td[( 1 @P1 @r+1 2@2~vr1 @r2+1 r@~vr1 @r)]TJ /F6 11.955 Tf 13.16 8.09 Td[(m2 r2Vr1)]TJ /F4 11.955 Tf 12.73 8.58 Td[(~vr1 r2)]TJ /F3 11.955 Tf 11.95 0 Td[(2im~v1+@2~vr1 @z2 99

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Pr~v1=)]TJ /F4 11.955 Tf 12.86 8.09 Td[( 1 1 rimP1+1 2@2)777(!v1 @r2+1 r@~v1 @r)]TJ /F6 11.955 Tf 13.15 8.09 Td[(m2 r2~v1)]TJ /F4 11.955 Tf 12.73 8.59 Td[(~v1 r2+2im r2~vr1+@2~v1 @z2 Pr~vz1=)]TJ /F4 11.955 Tf 12.86 8.08 Td[( @P1 @z+1 2@2~vz1 @r2+1 r@~vz1 @r)]TJ /F6 11.955 Tf 13.15 8.08 Td[(m2 r2~vz1+@2~vz1 @z2+ RaT1 PrT1+~vz1dT0 dz0= 1 2@2T1 @r2+1 r@T1 @r)]TJ /F6 11.955 Tf 13.15 8.08 Td[(m2 r2T1+ @2T1 @z2and1 r@(r~vr1) @r+1 r@~v1 @+@~vz1 @z=0Weneedatotalof16boundaryconditionsandalsoaconditiontoidentifythesurfacedeection,Z1.Thesolidbottomwallwithuniformtemperaturegiveriseto,atz=)]TJ /F3 11.955 Tf 9.3 0 Td[(1,~vr1=0=~v1=~vz1=T1andlikewisethetopwall,atz=~vr1=0=~v1=~vz1=T1Theperiodicsidewallconditionsfortheliquidandvaporphasesare,atr=~vr1=0=@(r~v1) @r=@~vz1 @r=@Tz1 @rand~vr1=0=@(r~v1) @r=@~vz1 @r=@Tz1 @rFornon-periodicsidewalltheboundaryconditionsforbothphasesaregivenby,~vr1=0=~v1=~vz1=@Tz1 @r 100

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and~vr1=0=~v1=~vz1=@Tz1 @rThetotalmassbalanceandthecontinuityofthetemperatureattheinterfacegivenby~vz1)]TJ /F4 11.955 Tf 11.96 0 Td[(~vz1=()]TJ /F4 11.955 Tf 11.95 0 Td[()Z1andT1+dT0 dzZ1=T1+dT0 dzZ1Attheinterfacethetangentialcomponentsofthevelocitiesofbothuidsareequaltoeachother,i.e.,no-slipconditions~vr1=~vr1~v1=~v1Thetangentialstressbalancesassumethefollowinglineardimensionlessform)]TJ /F4 11.955 Tf 12.86 8.09 Td[( @~vr1 @z+1 @~vz1 @r+@~vr1 @z+1 @~vz1 @r+1 Ma@T1 @r+dT0 dz@Z1 @r=0)]TJ /F4 11.955 Tf 12.86 8.09 Td[( @~v1 @z+im ~vz1+@~v1 @z+im ~vz1+1 im rMaT1+dT0 dzZ1=0whereMa=TTd isthethermalMarangoninumber.Theenergybalanceattheinterfaceis~vz1)]TJ /F6 11.955 Tf 11.95 0 Td[(E dT1 dz)]TJ /F6 11.955 Tf 13.15 8.09 Td[(dT1 dz=Z1HereEstandsfortheEvaporationnumberandisgivenbyE=T ~.ThelocalthermodynamicequilibriumconditionattheinterfaceleadstoClausius-Clapeyron,i.e.,YKEP1)]TJ /F11 11.955 Tf 11.29 11.36 Td[(YPEZ1=T1+dT0 dzZ1 T0 101

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whereYKEandYPEarethedimensionlessparametersfromthelinearizedClapeyronequationdenedasYKE= ~d2andYPE=gd ~.Thelastequationtoclosethesystemisthenormalstressbalanceis.Ca(P1)]TJ /F6 11.955 Tf 11.96 0 Td[(P1)+BoZ1)]TJ /F3 11.955 Tf 11.95 0 Td[(2Cad~vz1 dz)]TJ /F4 11.955 Tf 13.15 8.09 Td[( d~vz1 dz+1 2d2Z1 dr2+1 rdZ1 dr)]TJ /F6 11.955 Tf 13.15 8.09 Td[(m2 r2Z1=0whereCa= dandBo=()]TJ /F4 11.955 Tf 11.95 0 Td[()gd3 aretheCapillarynumberandtheBondnumber. 102

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APPENDIXIMODIFICATIONSFORNON-CONSTANTVISCOSITYMODELInthispartoftheappendixwepresentthemodicationsfornon-constantviscosityforasinglecomponentsystemin3-D.Although,viscosityisanon-linearfunctionoftemperatureforalargerangeoftemperaturechange,showninFigure I-1 ,itcanberepresentedasalinearfunctionoftemperatureforasmallerrangeoftemperature,showninFigure I-2 .Thisapproachwillsimplythederivationandthecodingconsiderably. FigureI-1. ViscosityofLiquidEthanolversusTemperature(Forlargerange). FigureI-2. ViscosityofLiquidEthanolversusTemperature(Forsmallrange). 103

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TorepresentviscositydependenceofthetemperatureaBoussinesqlikeapproachisused.Recallthatdensitydependenceoftemperatureisgivenby=ref[1+(T)]TJ /F6 11.955 Tf 11.96 0 Td[(Tref)]&=ref[1+(T)]TJ /F6 11.955 Tf 11.95 0 Td[(Tref)]andviscositydependenceis=ref+ (T)]TJ /F6 11.955 Tf 11.95 0 Td[(Tref)&=ref+ (T)]TJ /F6 11.955 Tf 11.96 0 Td[(Tref)where,ref&refaretheviscositiesatareferencetemperature. & aretheviscositycorrectioncoefcients.Fortheliquidphase isnegativeandforthevaporphase ispositive.(Intheliquidphase=ref+)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F3 11.955 Tf 9.3 0 Td[(1.510)]TJ /F8 7.97 Tf 6.58 0 Td[(5(T)]TJ /F6 11.955 Tf 11.96 0 Td[(Tref),recallthatthereferencetemperatureisthecoldplatetemperature)Momentumequationsinbothphasesrequiresmodicationfornon-constantviscositymodelandthefollowingequationgivesmomentumequationinitsgeneralform,@~v @t+~vr~v=rP+~g+r~~Swhere~~S=2(T)r~v+r~vt 2!r~~S=r2~v+r(r~v+r~vt)andforcylindricalcoordinatesr~v=~ir~ir@~vr @r~ir~i@~v @r~ir~ir@~vr @r~i~ir1 r)]TJ /F9 7.97 Tf 6.68 -4.97 Td[(@~vr @)]TJ /F4 11.955 Tf 11.53 .5 Td[(~v~i~i1 r@~v @+~vr~i~iz1 r@~vz @~iz~ir@~vr @z~iz~i@~v @z~iz~iz@~vz @zandr~v=~ir~ir@~vr @r~ir~i1 r)]TJ /F9 7.97 Tf 6.68 -4.98 Td[(@~vr @)]TJ /F4 11.955 Tf 11.53 .5 Td[(~v~ir~ir@~vr @z~i~ir@~v @r~i~i1 r@~v @+~vr~i~iz@~v @z~iz~ir@~vr @r~iz~i1 r@~vz @~iz~iz@~vz @zBysubstitutingthedenitionofr~~Sintothemomentumequationweget@~v @t+~vr~v=rP+~g+r2~v+r)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(r~v+r~vt 104

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andusingthedensityandviscosityexpansionswehaveref@~v @t+~vr~v=rP+ref(T)]TJ /F6 11.955 Tf 11.95 0 Td[(Tref)g~iz+...[ref+ (T)]TJ /F6 11.955 Tf 11.95 0 Td[(Tref)]r2~v+r[ref+ (T)]TJ /F6 11.955 Tf 11.96 0 Td[(Tref)])]TJ /F2 11.955 Tf 5.48 -9.68 Td[(r~v+r~vtThescalednon-linearmomentumequationsforliquidandvaporphasesbecome,1 Pr@~v @t+~vr~v=rP+RaT~iz+r2~v+ETr2~v+ErT)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(r~v+r~vtand 1 Pr@~v @t+~vr~v=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(ref refrP+ RaT~iz+r2~v...+ETr2~v+ErT)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(r~v+r~vtUponperturbationthelinearizedmomentumequationintheliquidphase1 Pr@~v1 @t=rP1+RaT1~iz+r2~v1+ET0r2~v1+ErT0)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(r~v1+r~vt1andforthevaporphase 1 Pr@~v1 @t=)]TJ /F4 11.955 Tf 10.5 8.08 Td[(ref refrP1+ RaT1~iz+r2~v1+ET0r2~v1+ErT0)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(r~v1+r~v1twhereE= T refandE= T refInadditiontothedomainmomentumequations,theforcebalanceattheinterfaceneedsmodicationfornon-constantviscositymodel.Recallthattheforcebalanceisgivenby)]TJ /F4 11.955 Tf 10.58 6.53 Td[(~~T~n+2H~n+~trs=)]TJ /F4 11.955 Tf 10.58 6.53 Td[(~~T~nwhere~~T=)]TJ /F6 11.955 Tf 9.3 0 Td[(P~~I+~~Sand~~S=2(T)r~v+r~vt 2=ref)]TJ /F2 11.955 Tf 5.47 -9.68 Td[(r~v+r~vt+ (T)]TJ /F6 11.955 Tf 11.95 0 Td[(Tref))]TJ /F2 11.955 Tf 5.48 -9.68 Td[(r~v+r~vtDerivationsforthenormalstressbalanceandtangentialstressbalancesforconstantviscositywerealreadygivenearlier.Alsotheweshowedthemomentumequationforthenon-constantviscositycaseindetail;therefore,herewewillonlyshow 105

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thenalperturbedversionofstressbalancesforthenon-constantviscositymodel. @~vr1 @z+1 @~vz1 @r)]TJ /F11 11.955 Tf 11.95 16.86 Td[(@~vr1 @z+1 @~vz1 @r=Ma @T1 @r+dT0 dz@Z1 @r...)]TJ /F6 11.955 Tf 9.3 0 Td[(ET0 @~vr1 @z+1 @~vz1 @r+ET0@~vr1 @z+1 @~vz1 @rand @~v1 @z+im r~vz1)]TJ /F11 11.955 Tf 11.95 16.85 Td[(@~v1 @z+im r~vz1=im rMaT1+dT0 dzZ1)]TJ /F3 11.955 Tf 11.96 0 Td[(...ET0 @~v1 @z+im r~vz1+ET0@~v1 @z+im r~vz1Thenormalstressbalancebecomes,Ca(P1)]TJ /F6 11.955 Tf 11.95 0 Td[(P1)+2Ca @~vz1 @z)]TJ /F4 11.955 Tf 13.15 8.08 Td[(@~vz1 @z+1 2@2 @r2+1 r@ @r)]TJ /F6 11.955 Tf 13.16 8.08 Td[(m2 r2)]TJ /F6 11.955 Tf 11.95 0 Td[(BoZ1=...2QT0@~vz1 @z)]TJ /F6 11.955 Tf 11.95 0 Td[(T0 @~vz1 @zwhereQ= T h. 106

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APPENDIXJHOWTOCOMPARE3-DPERIODICMODELWITH1-DMODELTovalidateournumericalresultsof3-Dperiodiccalculationswith1-Dcalculations,weapplythecurloperatortwiceonthelinearizedmotionequationsforboththeupperuidandloweruid.Hence,thepressure,azimuthalvelocityandradialvelocityareeliminatedinfavoroftheperturbedverticalvelocityandtemperature.Weobtainfor=0.r4~vz+RaTr2T=0r4~vz+ RaTr2T=0r2T)]TJ /F4 11.955 Tf 11.54 .5 Td[(~vzdT0 dz=0r2T)]TJ /F4 11.955 Tf 15.52 8.09 Td[( ~vzdT0 dz=0Wecanexpandthevariablesintheliquidphaseasvz=cos(m)Jm(km,jr)w(z)T=cos(m)Jm(km,jr)(z)vr=1 km,jcos(m)J0m(km,jr)dw(z) dzv=)]TJ /F6 11.955 Tf 17.64 8.08 Td[(m k2m,jrsin(m)Jm(km,jr)dw(z) dzExpandedvariablesinthevaporphase;vz=cos(m)Jm(km,jr)w(z)T=cos(m)Jm(km,jr)(z)vr=1 km,jcos(m)J0m(km,jr)dw(z) dzv=)]TJ /F6 11.955 Tf 17.63 8.08 Td[(m k2m,jrsin(m)Jm(km,jr)dw(z) dzWhereJm&J0m:BesselfunctionJanditsderivative. 107

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FigureJ-1. JkindBesselfunctions. Theaboveexpansionsassumeseparabilityintherdirectionandthisofcourseputsarestrictionontheboundaryconditions.Thusthecomparisonbetween3-Dand1-Dcanbedonewhensuchexpansionsarevalidandcompatiblewiththeboundaryconditions,noslipbeingexcludedfromthecompatibleconditions.SubstitutingtheaboveexpansionsintothedomainequationsgiveninAppendixH,resultsina1-Dproblem,whichcanbedirectlycomparedwiththeresultsof3-Dcalculations.Thewaytodoitistondarelationbetweenwavenumber,kmj,andazimuthalmodemforeachaspectratioandthatisdonebysubstituting;vz=cos(m)Jm(km,jr)w(z)into@vZ @rr==0.Thedimensionlesswavenumberisthereforegivenaskm,j=Jrm,j 108

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whereJrm,jaretherootsoftherstderivativesofBesselfunctionforanyxedazimuthalmodenumbermandistheaspectratioofthesystem.ForreferencepurposesintheFigures J-1 and J-2 theBesselfunctionsandtheirderivativesareplotted. FigureJ-2. DerivativeofJkindBesselfunctions. TableJ-1. RootsoftheJ0m,Jrm,j. J00J01J02J03J04 1st3.831.843.054.205.312nd7.015.336.708.019.283rd10.178.539.9611.3412.684th16.4711.7013.1714.5815.96 Anexampletocomparethe3-Dcalculationwithaspectratio1.5andazimuthalmode1;km,j=Jrm,j =1.84 1.5=1.22 109

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APPENDIXKDOMAINVARIABLEEXPANSIONANDMAPPINGSInthisappendix,theperturbationequationsusedinthetheoreticalworkareexplained.Thisisdoneforaone-dimensionaldomainandthereaderisreferredtoJohnsandNarayanan[ 28 ]fordetailsLetudenotethesolutionofaprobleminanirregulardomainDwhereDisnotspeciedandmustbedeterminedaspartofthesolution.ImaginethatDliesinthevicinityofareferencedomainD0andcanbeexpressedintermsofthereferencedomainviaaparameter".Letubeafunctionofaspatialcoordinatez.Thenumustbeafunctionof"directlybecauseitliesonDandalsobecauseitisafunctionofz.Byanirregulardomainitismeantadomainthatisnotconvenient.NowtosolveforuandobtainDsimultaneouslywecansolveaseriesofcompanionproblemsdenedonthenearbyregularorconvenientdomain,D0,whichwecalledthereferencedomain.Whatneedstobedoneistodiscoverhowtodetermineuintermsofthesolutionstoproblemsdenedon.ThepointsofD0willbedenotedbythecoordinatez0andthoseofDbythecoordinatez.ImagineafamilyofdomainsD"growingoutofthereferencedomainD0.umustbedeterminedoneachofthese.ThepointzofthedomainD"isthendeterminedintermsofthepointz0ofthereferencedomainD0bythemapping.z=f(z0,")Theequationaboveisalittlemoregeneralthannecessarytoexplainthemappingforourpurposes.Intheproblemsofthisstudy,onlyonepartofthedomainisirregular,namelytheinterface.Now,togetgoingletusexpandthefunctionfinpowersof"asf(z0,")=f(z0,"=0)+"@f(z0,"=0) @"+1 2"2@2f(z0,"=0) @"2+... 110

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wheref(z0,"=0)=z0andthederivativesoffareevaluatedholdingz0xed.Thenintermsofthenotationz1(z0,")=@f(z0,"=0) @"z2(z0,")=@2f(z0,"=0) @"2...themappingcanbewrittenasz=z0+"z1(z0,"=0)+1 2"2z2(z0,"=0)+...Also,theboundaryofthereferencedomainmustbecarriedintotheboundaryofthepresentdomainbythesamemapping.Thefunction,Z,whichdescribestheboundaryofthenewdomain,inheritsitsexpansioninpowersoffromthemappinggiveninandcanbewrittenasZ=Z0+"Z1(Z0,"=0)+1 2"2Z2(Z0,"=0)+...ItistheZi'sthatneedtobedeterminedtospecifythedomainD"intermsofthedomainD0.Accordingly,u(z,")canbeexpandedinpowersof"alongthemappingasu(z,")=u(z=z0,"=0)+"du(z=z0,"=0) d"+1 2"2d2u(z=z0,"=0) d"2+...wheredu d"denotesthederivativeofthefunctionudependingonzand"takenalongthemapping.Toobtainaformulafordu(z=z0,"=0) d",differentiateualongthemappingtakingztodependon",holdingz0xed.Usingthechainrule,thisgivesdu(z,") d"=@u(z,") @"+@u(z,") @z@f(z0,") @"Now,set"tozerointheaboveequationtogetdu(z=z0,"=0) d"=@u(z0,"=0) @"+@u(z0,"=0) @zz1(z0,") 111

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Then,introducethedenitionofu1viau1(z0)=@u(z0,"=0) @"andobservethat@u(z0,"=0) @z=@u0(z0) @z0togetdu(z=z0,"=0) d"=u1(z0)+@u0(z0) @z0z1(z0,")Alltheotherordersofthederivativesofucanbedeterminedthesameway.Finally,ifadomainvariableneedstobespeciedattheboundaryitiswrittenasdu(z=Z0,"=0) d"=u1(Z0)+@u0(Z0) @z0Z1(Z0,")Whenadditionalderivativesareobtainedandpluggedintotheexpansionofu,itbecomesu(z,")=u0+"u1+@u0 @z0z1+1 2"2u2+2@u1 @z0z1+@2u0 @z20z21+@u0 @z0z2+...Thecarefulreaderwillnoticethatthemappingz1doesnotappearinthedomainequationsgiveninthework.Toshowwhy,letusworkoutanexampleandsoletusatisfyanequation@u @z=0inthenewdomain.Ourreferencedomain,however,isinthecoordinatesystem,z0.Thus,@u @z=@u @z0@z0 @zHence,wemustdifferentiatetherighthandsideoftheexpansionofuwithrespecttoz0,holding"xedandthenmultiplyitby@z0 @z=1)]TJ /F4 11.955 Tf 11.96 0 Td[("@z1 @z)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2"2@z2 @z)]TJ /F3 11.955 Tf 11.96 0 Td[(... 112

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Thisgivesustotherstorder@u @z=@u0 @z0+"@u1 @z0+@2u0 @z20z1+...Seehowthederivativesofz1andz2arelostduringthestepwherewemultipliedthederivativeofuwithrespecttoz0by@z0 @z.Nowgoingbacktoourexampleandpluggingthisin,weget@u @z=@u0 @z0+"@u1 @z0+@2u0 @z20z1+...=0Inthedomain@u0 @z0=0andaccordingly,@2u0 @z20=0,whichgives@u1 @z0=0Themappingislostfromdomainequations.Ifsurfacevariableswereconsideredthemappingwouldnotbelost. 113

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APPENDIXLBINARYMIXTUREPROPERTIESThemixturepropertiesaretypicallycalculatedusingsimplemixingruleswithrespecttoeithermassormolefractionsuchasfordensity,latentheat,andsurfacetension. FigureL-1. Densityoftheliquidmixturewithrespecttomassfraction. Forexampleforbothliquidandvapordensitiesarecalculatedusing;mix=!AA+(1)]TJ /F4 11.955 Tf 11.95 0 Td[(!A)Bhere!Aisthemassfractionofethanol.Latentheatandsurfacetensionofthemixturearecalculatedusingthemolefractioninsteadofthemassfractionsuchassurfacetensionofthemixtureisgivenby,mix=xAA+(1)]TJ /F6 11.955 Tf 11.95 0 Td[(xA)BherexAisthemolefractionofethanol.Refutasequationisusedtocalculatetheliquidmixtureviscosityanditisdoneinthreesteps.TherststepistocalculatetheViscosityBlendNumber(VBN)foreach 114

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component,VBNA=14.534log(log(A+0.8))+10.975VBNB=14.534log(log(B+0.8))+10.975 FigureL-2. Viscosityoftheliquidmixturewithrespecttomassfraction. NextstepisthecalculatetheVBNofthemixture,VBNmix=!AVBNA+(1)]TJ /F4 11.955 Tf 11.95 0 Td[(!A)VBNBandthelaststepistogettheviscosityofthemixture,mix=exp(exp(VBNmix)]TJ /F3 11.955 Tf 11.96 0 Td[(10.975)=14.534))]TJ /F3 11.955 Tf 11.95 0 Td[(0.8NotethattousetheRefutasequationweneedeachcomponentviscosityincentistokesunit.ThegasmixtureviscosityiscalculatedusingmethodofWilkewhichisanempiricalformulaforalcoholmixtures.Themethodrequirescalculatingtheviscositycoefcients 115

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ofeachcomponentusingthefollowingequation,AB= 1+A B0.5MA MB0.25!2,81+MA MB0.5AB=BABMA AMBandtheviscosityofthemixturemix=(yAA)((1)]TJ /F6 11.955 Tf 11.96 0 Td[(yA)+yABA)/(yA+(1)]TJ /F6 11.955 Tf 11.95 0 Td[(yA)AB) FigureL-3. Viscosityofthevapormixturewithrespecttomassfraction. Samemethodisusedforthermalconductivityofthevapormixturebyjustreplacingtheviscositytermswiththermalconductivities.AtlastfortheliquidthermalconductivityFilippovequationisusedwithmassfractionsoftheeachcomponent.kmix=!AkA+(1)]TJ /F4 11.955 Tf 11.95 0 Td[(!A)kB+abs(kA)]TJ /F6 11.955 Tf 11.95 0 Td[(kB)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(!A)0.5!A 116

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FigureL-4. Thermalconductivityofthevapormixturewithrespecttomassfraction. FigureL-5. Thermalconductivityoftheliquidmixturewithrespecttomassfraction. 117

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APPENDIXMSIMPLEEVAPORATIVEINSTABILITYEXPERIMENTFORMIDDLESCHOOLSInthisappendixasimplesetofexperimentsareproposed(alsoresultsareshown)formiddleschoolscienceclasses.Thefollowingsetofexperimentsissimpletoconductbutshowstheimportantphysicsofanevaporativeconvectivesystem.Themainideaoftheexperimentsistoshowtheeffectofevaporation,densitystraticationandviscosity.Wewillneed; 3easytondliquids;ethanol(insteadofethanolotheralcoholscanbeused),water,andglycerin, Petridishes,sampleexperimentsaredonewith6inchdiameterpetridishes, Containerswithalid(75ml,preferablyglassorplasticonesthroughwhichyoucansee), Foodcoloring, Kalliroscopetracer.(Thiscanbeeasilyorderedonline)Intherstexperimentwewillshowtheeffectofevaporationbycomparingtheevaporationofethanolwithwater.Notethatethanolismorevolatilethanwater.Tovisualizetheowwerstneedtomixthetracerwithourliquidsusingthecontainers.For50mlofethanolorwaterlessthanagramoftracerwouldbesufcient.Youcanalwaystryandseeifitissufcientbyaddingalittleatatimeandcheckifyoucanvisualizeaowwhenyoushakethecontainer.Addingfoodcoloringwillcreateacontrastthatmighthelptoseetheow.Aftermixingthetracerwiththeliquids,simplypourtheliquidsintopetridishes.Youwillobservesomepatternformationduetoowintheethanol,showninFigure M-1 ,butno-owinthewater,showninFigure M-2 .Letusrstexplainwhyweseeowinethanolbutnotinwater.Thisismainlybecauseethanolevaporatesmorethanwaterandduetothisevaporationthesurfaceoftheethanolgetscoolercomparedtothebottomofthepetridish.Inotherwordsthesurfaceoftheethanolbecomesheavierandwantstomovedownpushingtheliquidfromthebottomtothesurface.However,thisdensitystraticationisnottheonlyreasonwhy 118

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FigureM-1. Evaporatingethanollayer.Patternformation. FigureM-2. Waterlayer.No-Flow. youseeowinethanolbutnotinwater.Evaporationitselfcreatesowinethanoltoowhichisweakinwater.Ifyouwanttoseethepatternsandowinwateryouwillneedtoputthepetridishontopofaheater,showninFigure M-3 .(Youshouldkeeptheheaterinitslowestpositionbecauseyoudonotwanttoboilthewater).Heatingfrombelowcreatesthe 119

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densitystraticationlikeevaporationdidinethanol.Itmakesthebottomwaterlighterthanthetopandinitiateow. FigureM-3. Waterlayerheatingfrombelow.Patternformation.(Foodcoloringisadded) Toseetheeffectofviscositywewilladdaround3mlglycerinto50mlethanol.Thiswillmainlymaketheviscosityofethanolhigher.Afteraddingglycerinagainpourtheliquidintothepetridish.Youwillnotableseetheanyowinthepetridish.Thisismainlybecausebyaddingglycerinyouincreasedtheviscosityofethanolandviscosityworksagainstow. 120

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REFERENCES [1] J.R.Pearson,Onconvectioncellsinducedbysurfacetension,J.FluidMech.4(1958)489. [2] L.Rayleigh,Onconvectioncurrentsinahorizontallayerofuid,whenthehighertemperatureisontheunderside,Phil.Mag.32(1916)529. [3] H.Benard,Lestourbillonscellulairesdansunenappeliquide,Rev.Gen.SciencesPureAppl.11(1900)1261. [4] S.Chandrasekhar,HydrodynamicandHydromagneticStability,OxfordUniversityPress,Oxford,1961. [5] J.P.Burelbach,S.G.Bankoff,S.H.Davis,Nonlinearstabilityofevaporatingandcondensingliquidlms,J.FluidMech.195(1988)692. [6] J.Margerit,M.Dondlinger,P.Dauby,Improved1.5-sidedmodelfortheweaklynonlinearstudyofbenardmarangoniinstabilitiesinanevaporatingliquidlayer,J.ColloidInterfaceSci.290(2005)220. [7] A.Oron,Nonlineardynamicsofthinevaporatingliquidlmssubjecttointernalheatgeneration,FluidDynamicsatInterfaces(1999)3. [8] A.Huang,D.D.Joseph,Instabilityoftheequilibriumofaliquidbelowitsvapourbetweenhorizontalheatedplates,J.FluidMech.242(1992)235. [9] O.Ozen,R.Narayanan,Thephysicsofevaporativeandconvectiveinstabilitiesinbilayersystems:Lineartheory,Phys.Fluids16(2004)4644. [10] O.Ozen,R.Narayanan,Thephysicsofevaporativeandconvectiveinstabilitiesinbilayersystems:weaklynonlineartheory,Phys.Fluids16(2004)4653. [11] B.Haut,P.Colinet,Surface-tension-driveninstabilitiesofapureliquidlayerevaporatingintoaninertgas,J.ColloidInterfaceSci.285(2005)296. [12] G.B.McFadden,S.R.Coriell,K.F.Gurski,D.L.Cotrellc,Onsetofconvectionintwoliquidlayerswithphasechange,Phys.Fluids19(2007)104109. [13] W.Guo,R.Narayanan,Interfacialinstabilityduetoevaporationandconvection:linearandnonlinearanalyses,J.FluidMech.650(2010)363. [14] A.Oron,S.H.Davis,S.G.Bankoff,Long-scaleevolutionofthinliquidlms,Rev.Mod.Phys.69(1997)931. [15] C.Ward,D.Stanga,Interfacialconditionsduringevaporationorcondensationofwater,Phys.Rev.E64(2001)051509. 121

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[16] J.Margerit,P.Colinet,G.Lebon,C.S.Iorio,J.C.Legros,Interfacialnonequilibriumandbenard-marangoniinstabilityofaliquid-vaporsystem,Phys.Rev.E68(2003)041601. [17] P.Shankar,M.Deshpande,Onthetemperaturedistributioninliquidvaporphasechangebetweenplaneliquidsurfaces,Phys.FluidsA2(1990)1030. [18] P.G.deGennes,Instabilitiesduringtheevaporationofalm:Non-glassypolymer+volatilesolvent,Eur.Phys.16(2001)421. [19] B.Trouette,E.Chenier,C.Delcarte,B.Guerrier,Numericalstudyofconvectioninducedbyevaporationincylindricalgeometry,Eur.Phys.SpecialTopics192(2011)83. [20] H.Machra,A.Rednikov,P.Colinet,P.Dauby,Benardinstabilitiesinabinary-liquidlayerevaporatingintoaninertgas,J.ColloidInterfaceSci.349(2010)331. [21] H.Machra,A.Rednikov,P.Colinet,P.Dauby,Benardinstabilitiesinabinary-liquidlayerevaporatingintoaninertgas:Stabilityofquasi-stationaryandtime-dependentreferenceproles,Eur.Phys.SpecialTopics192(2011)71. [22] H.Mancini,D.Maza,Patternformationwithoutheatinginanevaporativeconvectionexperiment,Europhys.Lett.66(2004)812. [23] J.Zhang,SurfaceTension-DrivenFlowsandApplications,ResearchSignpost,India,2006. [24] J.Zhang,R.Behringer,A.Oron,Marangoniconvectioninbinarymixtures,Phys.Rev.E76(2007)016306. [25] J.Zhang,A.Oron,R.Behringer,Novelpatternformingstatesformarangoniconvectioninvolatilebinaryliquids,Phys.Fluids23(2011)0721102. [26] S.Dehaeck,C.Wylock,P.Colinet,Evaporatingcocktails,Phys.Fluids21(2009)091108. [27] G.Toussaint,H.Bodiguel,F.Doumenc,B.Guerrier,C.Allain,Experimentalcharacterizationofbuoyancy-andsurfacetension-drivenconvectionduringthedryingofapolymersolution,Int.J.HeatMassTransfer51(2008)4228. [28] L.E.Johns,R.Narayanan,InterfacialInstability,Springer-Verlag,NewYork,2002. [29] W.Guo,G.Labrosse,R.Narayanan,TheChebyshevSpectralMethodinTransportPhenomena,Springer-Verlag,ForthComing,2012. [30] C.L.Yaws,Yaws0HandbookofThermodynamicandPhysicalPropertiesofChemicalCompounds,Knovel,2003. 122

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[31] L.E.Scriven,C.V.Sternling,Oncellularconvectiondrivenbysurfacetensiongradients:effectsofmeansurfacetensionandsurfaceviscosity,J.FluidMech.19(1964)321. 123

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BIOGRAPHICALSKETCH KamuranErdemUGUZwasborninTurkey.HegraduatedfromYildizTechnicalUniversityinIstanbul,Turkey,receivingaB.S.degreeinchemicalengineeringin2004.HethenattendedBogaziciUniversityinIstanbul,Turkey,receivingaM.S.degreeinchemicalengineeringin2007.HethenattendedUniversityofFloridafordoctoralstudiesunderthesupervisionofProf.RangaNarayanan. 124