Hydrothermal Surface-Wave Instability and the Kuramoto-Sivashinsky Equation

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma

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aHYDROTHERMALSURFACE-WAVEINSTABILITYANDTHEKURAMOTO-SIVASHINSKYEQUATIONR.A.Kraenkel,J.G.PereiraInstitutodeF´ sicaTe´oricaUniversidadeEstadualPaulistaRuaPamplona14501405-900S aoPauloSP–BrazilM.A.MannaLaboratoiredePhysiqueMath´ematiqueUniversit´edeMontpellierII34095MontpellierCedex05–FranceAbstractWeconsiderasystemformedbyanin niteviscousliquidlayerwithacon-stanthorizontaltemperaturegradient,andabasicnonlinearbulkvelocitypro le.Inthelimitoflong-wavelengthandlargenondimensionalsurfaceten-sion,weshowthathydrothermalsurface-waveinstabilitiesmaygiveriseto

disturbancesgovernedbytheKuramoto-Sivashinskyequation.Apossible

connectiontohot-wireexperimentsisalsodiscussed.

TypesetusingREVTEX

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma

I.INTRODUCTION

Thermocapillarydynamicsinthintwo-dimensionalliquidlayers,andinparticulartheconvectiveinstabilitiesofsuch owshavebeenasubjectofmuchinterest[1,2].Whentheupperfreesurfaceofaplanarliquid,boundedbelowbyarigidplateandabovebyaninterfacewithapassivegas,issubmittedtoatemperaturegradient,acorrespondinggradientinthesurfacetensionwillappearwhichwillproducemotioninthebulk uid.Ifaverticaltemperaturegradientisappliedtoabasicstaticstate,convectivemotionsetsin,aphenomenoncalledMarangoniconvection.However,ahorizontaltemperaturegradientmayalsogiverisetoinstabilities,providedthebasicstateisnotstatic[1].Thesearethesocalledhydrothermalinstabilities,whichareacouplede ectproducedbybothtemperatureandvelocitygradients.SmithandDavis[1,2]identi edtwosuchinstabilities,whichmanifestthemselvesintheformofconvectionandsurface-wavemotion.

Ourconcerninthisletterwillbethestudyofthehydrothermalsurface-waveinstability.IthasbeenshowninRef.[1]thatitispossibletohavesuchinstabilitiescharacterizedbyazerowave-number,providedthebasicunderlying owisanonlinearreturn ow.Thelongwavelenghtnatureoftheinstabilityallowsustobroachtheproblembyalong-waveperturbativeanalisys,wherenonlinearitymaybetakenintoaccount.ByusingthereductiveperturbationmethodofTaniuti[3],wewillshowthat,inthelimitoflargenondimensionalsurfacetension,thewavesoriginatedbythisinstabilityturnouttobegovernedbytheKuramoto-Sivashinskyequation[4,5].Thisequationhasalreadybeenderivedindi erentphysicalcontexts[6].Inparticularithasbeenobtainedastheequationgoverningpertur-bationsfromareferencePoiseuille owofa lmlayeronaninclinedplane[7],alsointhespeci climitoflargenondimensionalsurfacetension[8].However,inthiscase,theinsta-bilityunderconsiderationwashydrodynamicalinnature,andnothydrothermal,asisthecaseconsideredinthepresentwork.

Thebasicinterestoftheresultspresentedinthisworkisconcernedwithrecentex-perimentalevidencesfortheexistenceofsurfacehydrothermalwaves.Forexample,ithas

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma

beenreportedintheliteraturetheobservationofsuchwavesinacylindricalcontainer[9].Moreover,hot-wireexperimentsperformedrecently[10,11]haveindicatedthepresenceofpropagativepatterns.Inthe nalsection,wewillspeculateonapossibleconnectionbetweenourresultsandtheseexperiments.

II.MATHEMATICALFORMULATION

Weconsidera uidlayerofheightd,boundedbelow,atz=0,byarigidperfectlyinsulatingplate,andabove,aty=d+η(x,t),byafreedeformablesurfaceincontactwithapassivegasofnegligibledensityandviscosity.Theliquidischaracterizedbyadensityρ,thermalconductivityk,thermaldi usivityκ,unitthermalsurfaceconductanceh,anddynamicviscosityµ.TothefreesurfaceweassociateasurfacetensionT,whichwillbeassumedtodependonthelocaltemperatureθaccordingtothelinearlaw

T=T0 γ(θ θ0),(1)

whereγisapositiveconstant,andT0,θ0arereferencevaluesforsurfacetensionandtem-perature,respectively.

Wewillbeconcernedwithe ectscomingfromthermocapillarityonly.Thereforewewillneglectgravity.Thisisagoodapproximationforathinenoughlayer,oralayerinamicrogravityenvironment.Theequationsgoverningthe uidmotionarewrittenas:

ux+wz=0,(2)

ρ(ut+uux+wuz)= px+µ(uxx+uzz),(3)

ρ(wt+uwx+wwz)= pz+µ(wxx+wzz),(4)

θt+uθx+wθz=κ(θxx+θzz),(5)

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma

wherethesubscriptsdenotepartialdi erentiation,uandware,respectively,thexandzcomponentsofthevelocity,andpisthepressure.Ontheupperfreesurface,theseequationsaresubjecttothefollowingboundaryconditions

ηt+uηx=w,

p

(6)ηxx,(7)

(8)2µN32µ1 (ηx)[uz+wx]+2µηx(wz+ux)=N[Tx+ηxTz],

ηxθx+θz=h

2.

Equation(6)isthekinematicboundarycondition,Eqs.(7)and(8)representthenormalandtangentialstressbalanceattheupperfreesurface,andEq.(9)istheequalityoftheheat uxattheinterface,whereθ∞isthetemperatureatz→∞.Onthelowerplate,theboundaryconditionsare

u=w=θz=0,

whichcharacterizesthenoslipandzeroheat uxconditions.

Perturbationtheorywillbeperformedonabasicstateofthesystem,de nedbyimposingtothein niteliquidlayeraconstanthorizontaltemperaturegradient

dθb(10)

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma

temperature,γβastheunitofpressure,andT0astheunitofsurfacetension.Inthisprocessthefollowingdimensionlessnumbersappear:

R=ργβd2

ρκ,B=hd

µ2,

whereRistheReynoldsnumber,σisthePrandtlnumber,BistheBiotnumber,andSisthesurfacetensionnumber.Inaddition,wehavealsotheMarangoninumberM,whichisde nedbyM=Rσ.

Thebasicstate,fromwhichwewillconsidersmalldisturbances,isthereturn owsolutiontotheEqs.(2-10)which,inthelimitS→∞,isgivenby:

ub=3

2

pb=

z3 (12a)wb=0;

161 z4 1

4Sx3+R

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma

III.PERTURBATIONTHEORY

Letusconsiderthefollowingperturbationstothebasicstate:

u=ub+ (u0+ u1+....),(13a)

w= 2(w0+ w1+...),(13b)

p=pb+ (p0+ p1+...),(13c)

θ=θb+ (θ0+ θ1+...),(13d)

η=ηb+ (η0+ η1+...),

where isasmallparameter.Next,weintroducetheslowvariablesaccordingto

ξ= (x ct),(13e)(14)

τ= 2t,(15)

andwesupposethattheperturbationsdependon(x,t)through(ξ,τ)only.Wenowmakemoreprecisethemeaningoflargealbeit niteforS,byassumingthat

S= 2

S～O( 0).

Weareable,atthispoint,toobtainanorderbyordersolutiontotheproblembyintegrationsinthezvariable.Ateachorder,thebulkequationsandtheboundaryconditionsatthebottomyieldu,w,θ,pintermsofthreearbitraryfunctions,whichwillbedeterminedintermsofη0bytheboundaryconditionsattheupperfreesurface.However,aswehavefourequations,acompatibilityconditionwillariseateachorder.Explicitly,atorder 0,weget:

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma

u0= 3

4

θ0=Rσ z2η0ξ;1(17b)z3

2B

S

2.(18)

Atthenextorder,therelevantsolutionsare

u1=RS

160 z 5z54 η0ξξ+6Rzη0ξξξξ 3z2

4η1ξ+1

4η0η0ξ+RS

4η1ξ+1

4 σ2 η0ξξ+2Rη0ξξξξ.(21)

SubstitutingintoEq.(20),weobtainanevolutionequationforη0:

η0τ 5

4B+1S

S= 2Sandη0= 1η,

Eq.(22)reads

ηt+c 5

4B+

13Rηxxxx=0.(23)

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma

De ninganew eldby

ζ=c 5

1

4B+3Rζxxxx=0.(24)

Byasimplerescalingofvariables,thisequationcanbewrittenas

ζt+ζζx+ζxx+ζxxxx=0,(25)

whichistheKuramoto-Sivashinsky(KS)equationinitsstandardform[4,5].

ThelinearanalysisperformedbySmithandDavis[1]canberecoveredfromEq.(24)ifweneglectthenonlinearterm,andtake

ζ～exp[ (αx ωt)].

Thisleadstothedispersionrelation

ω= α2R σ

10 α4S

S,weobtainthecriticalReynoldsnumber

Rσ 1

c= 3 10 1

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma

bythecoexistenceofcoherentspatialstructureswithtemporalchaos[14].Inparticular,thesearchforabetterunderstandingofthelongwavelengthpropertiesoftheKSequationhasbeenintensivelypursuedlately[15].Anditispreciselyinthelong-wavelimitthatwehaveobtainedtheKSequationasgoverninghydrothermalsurface-waveinstabilitiesinathinviscouslayerof uidwithhorizontaltemperatureandvelocitygradients,andinthelimitoflargenondimensionalsurface-tension.Thisresultcouldpossiblybeconnectedtoarecentexperiment[9]whichhasbeenperformedtostudythefeaturesofhydrothermalwavesappearingwhenacylindricalliquidlayerislaterallyheated.Theexistenceofhydrothermalwaveinstabilitieshasbeenobservedexperimentally,andtheresultswerefoundtopartiallyagreewiththeprevioustheoreticalpredictionsofSmithandDavis[1].Ifwebelievethat,despitethedi erenceinthegeometrytheresultspresentedinRef.[9]cansomehowbeconsideredasanapproximationtothetheoreticalpredictions,thenthelong-wavedynamicsofthefreesurfacedescribedbytheKSequationshouldplayaroleintheexplanationofthepropagativepatternsobservedinthiskindofexperiment.

Atlast,wewouldliketospeculateonapossibleconnectionbetweentheresultsobtainedinthisletterandhot-wireexperiments[10,11].Theseexperimentsconsistofahotwireplacedhorizontallyjustbelowafreesurfaceofa uid.Aboveacertaincriticalvalueoftheelectricalpowersuppliedtothehotwire,propagatingpatternsareobserved.Theexistenceofbothhorizontaltemperaturegradientsandabasicvelocity ow,asthe uidisconvecting,makescontactwiththetheoreticalsettinginwhichhydrothermalwavesshowup.Ofcourse,thisisnotenoughtoensurethattheaboveresultscanhaveanimmediateapplicationtothehot-wireexperiments.However,sincethesystemunderconsiderationherepresentssimilaritieswiththeabovementionedexperiments,theycouldhelptoelucidatethephysicalmechanismresponsiblefortheobservedpropagativepatterns.Theideatomimicfeaturesofaconvecting uidbyatwo-dimensionalsystemwithavelocitygradienthasbeenexaminedbyRoz`e[16],basedondataofanexperimentwithamuchshorterwire[17].However,thisapproach,whichusedalinear owpro leandlinearizedequations,turnedouttoberatherdisappointing.Infact,asshowninRef.[1],thereisnolong-waveinstabilityinthiscase.

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma

Onlywhenanonlinearreturn owsolutionissupposed,whichisindeedmoreappropriateforaclosedsystem,long-wavehydrothermalphenomenaappear.Therefore,wethinkitwouldbeinterestingtoreexaminetheideasproposedinRef.[16]forthehotwireprobleminthelightofthepresentresults.

Insummary,wehavemadeinthisletteranonlinearanalysisofthelongwavelengthhy-drothermalinstabilities,whichcanbeconsideredasanextensionofaproblem rstdiscussedbySmithandDavis[1]inalinearapproximation.Wehaveshownthatthefree-surfacedy-namicsisgovernedbytheKuramoto-Sivashinskyequation,andthatthepreviousresultscanberecoveredfromoursasaparticularcase.

ACKNOWLEDGMENTS

Twooftheauthors(R.A.K.andJ.G.P.)wouldliketothankConselhoNacionaldeDesenvolvimentoCient´ coeTecnol´ogico(CNPq),Brazil,forpartialsupport.

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma

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