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K.Takahashietal./ProcedingsoftheCombustionInstitute30(205)271-27 273
his自amespreadtheoryusedaliquidpoldepthas anaproximationforacharacteristiclengthand pointedoutthatinrelationtothechoiceofchar- acteristiclengths:“.• • itisbothpos sibleanddesir- able to pursue more careful analysis to 0btain improvedestimate(ofspreadrateusingapropri- atecharacteristiclengths)"[1].Acordingly，we experimentaly measured both characteristic lengths，whichcanhelpimproveWil1iams'theory， andalsousedtheminouranalyses.
2.Theoreticalanalysisandresults 2.1.Bαsicequation
Ourobjectivewastostudytheinstabilityof laminar sub問 surface layer flow ahead of the spreadingflame.Temperaturecoeficientofsur- facetensionforceandotherphysicalproperties arebothtreatedasconstants.Ourinstabilityanal- ysisisbasedonourpreviousstudy[12，13J.Asche- maticofsub-surfacelayerflowisshowninFig.2. Althesymbolsusedaredefinedinthenomencla- ture.Liquidtemperaturejustaheadoftheflame leadingedgeishigherthanthebulkliquidtemper- ature，becausethehightemperatureliquidgener- ated by the flame moves to the upstream directionbysurfacetensionforce.Thecontinuity and momentum equations for the sub-surface layercanbe，respectively，expresedas:
ん+(hu)x= 0， (1) 附([内十t+(学)x
(2)
nomenclature.InEq.(2)，τiisthesurfaceshear stresduetoMarangoniefectandisexpresedas:
ThethirdtermontheLHSinEq.(2)isduetosur- face tension， where d = A exp ik(x - ct) is the expresionforthesurfacewave，k(三 2n/A)isthe wavenumber，A isthewavelength，cisthewave velocity，andalothersymbolsaredefinedinthe
土terminthedynamicwaveexpresion.
If the surface perturbation is expres sed as d=Aexpi(kx-ωt)，we obtain the folowing
Surfacewave
λ'
Wal Fig.2.Sub-surfacelayerflowmodelandsymbols.
1
3μ
4
.3μl
6σ oT dσ
τi=-=σT一 σT三 一 (3)
ß(三 J~
内
dT
l/uh) andthecontinuity叩 tion，
ox -.ox -
If the momentum displacement thicknes:
2
<
Eq.(1)， aresubstitutedintoEq.(2)，wecanobtain
thefolowingequation: 九
十九
hUt+(2s-l)hux+(s-l)l
re._ +22;(Sxh+Shx)=-(τ i - τb)・
P
ghh x
(4)
Conducting perturbation analysis (detailed in [14])forEqs.(1)and(2)，thefolowingwaveequa- tionscanbe0btained.
合(+C+ where
o
ま)
(:t+C_~)ð十長(~十C
，
pAz' (6)
一 l一 一ーゥ "1，12h，__qi
C:I:=su:l:~I(s-1)ßu2 十gh+σe~+ασT一 Iρ
Co=u.
ThefirsttermontheLHSofEq.(5)representsthe dynamicwavewhereC+andC，respectively，rep- resenttheforwardandreversepropagationveloc- ities. The second term on the LHS of Eq. (5) representsthekinematicwave，where Coisthe propagationvelocityinthedirectionofflow.Eq. (5)representsthesurfacewaveasalinearcombi- nationofthedynamicwaveandthekinematic wave. The 1¥在arangoni efect apears in the
equationfromEq.(5):
.3μ
Hence，ωmaybeaproximatedas
ph-
c
(ω -kC_)(ω-kC+)+iっ (ω -kCo)=O. (7) ph-
r1•
ω空く kC+-iっ (C+-Co)/(C+-C_)
+kC一 1寸 (Co-C_)/(C+-C_)>. (8) ph-. .J
IfIm(ω)>0，thedynamicwaveisunstable，result- inginCo~ C+.Then，itbecomesaneutralysta回 blecondition，Co=C+Thefolowingequation describesthegenerationofasurfacewave:
n¥-? "1， 12h， qi
(1-s)グ =gh+σkー十 ασTよ . (9) p ， - - - < pA[
x+sxl
か
=0