Hydrothermal Surface-Wave Instability and the Kuramoto-Sivas(2)

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

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