This article was downloaded by: On: 24 January 2010 Access details: Access Details: Free Access Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37- 41 Mortimer Street, London W1T 3JH, UK Combustion Theory and Modelling Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713665226 Collective effects and dynamics of non-adiabatic flame balls Yves D'Angelo a; Guy Joulin a a Laboratoire de Combustion et de Détonique, Futuroscope, Poitiers, France Online publication date: 21 December 2000 To cite this Article D'Angelo, Yves and Joulin, Guy(2001) 'Collective effects and dynamics of non-adiabatic flame balls', Combustion Theory and Modelling, 5: 1, 1 — 20 To link to this Article: DOI: 10.1088/1364-7830/5/1/301 URL: http://dx.doi.org/10.1088/1364-7830/5/1/301 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. 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INSTITUTEOFPHYSICSPUBLISHING COMBUSTIONTHEORYANDMODELLING Combust.TheoryModelling5(2001)1–20 www.iop.org/Journals/ct PII:S1364-7830(01)14932-6 Collective effects and dynamics of non-adiabatic flame balls YvesD’Angelo1andGuyJoulin LaboratoiredeCombustionetdeDe´tonique,UPR9028CNRS,ENSMA,BP109,86960 Futuroscope,Poitiers,France E-mail:dangelo@lcd.ensma.fr Received23June2000,infinalform1November2000 Abstract The dynamics of a homogeneous, polydisperse collection of non-adiabatic flame balls (FBs) is investigated by analytical/numerical means. A strongly temperature-dependentArrheniusreactionrateisassumed,alongwithalight enough reactant characterized by a markedly less than unity Lewis number (Le). Combining activation-energy asymptotics with a mean-field type of 0 1 0 treatment,theanalysisyieldsanonlinearintegro-differentialevolutionequation 2 ry (EE) for the FB population. The EE accounts for heat losses inside each FB a nu and unsteadiness around it, as well as for its interactions with the entire FB a J population, namely mutual heating and faster (Le < 1) consumption of the 4 2 3 reactant pool. The initial FB number density and size distribution enter the 4 5: EE explicitly. The latter is studied analytically at early times, then for small 0 : totalFBnumberdensities;itissubsequentlysolvednumerically,yieldingthe t A d wholepopulationevolutionanditslifetime. Generalizationsandopenquestions e ad relatingto‘spotty’turbulentcombustionarefinallyevoked. o l n w o D Nomenclature A Heat-lossintensity,equations(1)and(4) c Specificheatatconstantpressure,equation(1) D Reactantdiffusivity,equation(2) D Thermaldiffusivity,equation(5) th D1/2(·) Fractionalderivativeoforder 21,equation(23) hi(ρ) ithterminearlyexpansionofR,equation(39) L Laplacetransform,equation(A2) Le Lewisnumber,equation(5) n(r ) InitialdistributionofFBsizes,equations(11)and(12) 0 N(ρ) NormalizedinitialdistributionofFBsizes,equation(21) 1 Authortowhomcorrespondenceshouldbeaddressed. 1364-7830/01/010001+20$30.00 ©2001IOPPublishingLtd PrintedintheUK 1 2 YD’AngeloandGJoulin n InitialFBnumberdensity 0 p Variableconjugatetoτ inLaplacetransforms,equation(27) Q Heatofreaction,equation(1) r DistancetocentreofasampleFB,equation(1) rF Flameballcurrentradius,equation(8) r Initialflameballradius,equations(11)and(12) 0 rZ RadiusofZel’dovich’sFB,equation(6) r R = F Scaledflameballradius,equation(9) r 0 R∗ RatthemaximumoflogR−αR2,equation(9) s Variableconjugatetot inLaplacetransforms,equation(A2) ST Collectiveheat-source,equation(1) Sy Collectivereactant-sink,equation(2) t Time,equation(1) T Temperature,equation(1) T Activationtemperature,equation(3) ac T∗ Referencereactiontemperature,equation(5) Tu Initialtemperatureinthebulkofmixture,equation(5) t Collisiontime,equation(3) coll 0 w Chemicalsourceterm,equation(1) 1 20 y Reactantmassfraction,equation(2) y uar yu Initialreactantmassfraction,equation(5) n Ja Ze(cid:5)1 Zel’dovichnumber,seeaboveequation(6) 4 2 3 4 : Greekandothersymbols 5 0 t: α Scaledheat-losscoefficient,equation(9) A ed θ(ρ) Time-scaleofFBextinctionforρ (cid:6)1,equation(B3) d a lo λ Heatconductivity,equation(1) n w Do ν Measureofcollectiveeffects,equation(22) ρ =r0/rZ ScaledinitialFBsize,equation(20) ρu =constant Mixturedensity,equation(1) σ Dummytimevariable,equation(23) τ Scaledtime,equation(20) τ τ atendofquasi-steadyphase,equation(36) end τ IntegralFBpopulationlifetime,equation(37) life 1. Introduction Adiabatic flame balls (FBs, for short), first invented theoretically by Zel’dovich (1944), constitute exact convection-free solutions to the reaction/conduction/diffusion balances. However, an adiabatic FB is unstable (Zel’dovich et al 1985, Deshaies and Joulin 1984), apropertypreviouslyexploitedinignitionmodels(Joulin1985). The experimental discovery (Ronney et al 1998 and references therein) that stable FB-looking combustion objects exist when lean mixtures of light fuels (e.g. H ) burn in 2 Collectiveeffectsanddynamicsofnon-adiabaticflameballs 3 microgravity conditions, has motivated the search for stabilizing mechanisms. The matter is not quite settled quantitatively, yet theoretical investigations have identified plausible candidates: radiantheatlossesfrominsidetheFB(Buckmasteretal1990)and/orfromaround it(Buckmasteretal1991);conductivelossestowalls(JoulinandBuckmaster1993);overall convectivelossescausedbysteadyvelocitygradientsintheneighbouringfluid(Buckmaster andJoulin1991). A more recent analysis (Joulin et al 2000) gives converging arguments that fluctuating velocitygradientswithzerotimeaverageactsimilarlytoglobalheatlossesoftime-dependent intensityandnon-zerotimeaverage;also,thattheycanmaintainthethrobbingFBradiusrF(t) aroundafinitevaluethatexceedsthevaluerZ foundbyZel’dovich. Astheanalysisdoesnot restricttherangeofchemicaltimesinvolved,itpointstothepossibilityofa‘spotty’regime ofturbulentcombustionoflean(andpossiblypreheated)mixturesofmobilefuels: manyFBs would then be competing for the common fuel and mutually heating one another, while the localvelocitygradientswouldextractheatfromeachindividualFBatafluctuatingratewhile thespotisadvectedbyturbulence. Analysingthiskindofconfigurationisaverydifficultmatter; yetoneknowsforsurea step that ought to be understood theoretically: how to tackle the collective effects among a populationofevolvingnon-adiabaticFBsofvarioussizes? Thisiswhatthepresentpaperaims todealwith. Forsimplicity,thenon-adiabaticityofeachFBismodelledherebyaconstant, commonrate,asifthelosseswerelocalones,e.g.radiant. The paper is organized as follows. Section 2 recalls the governing equations for a single quasi-steady FB embedded in a slowly varying atmosphere. Far-field unsteadiness 0 1 and a mean-field account of collective effects, resulting in an evolution equation for the 0 2 y whole FB population are also analysed. In section 3, the system dynamics is studied r ua analytically for some limiting cases. Sections 4 and 5 deal with the numerical procedure n a J and solutions of the integro-differential evolution equation for different levels of number 4 2 density of flame balls. We end up with concluding remarks and remaining open problems 3 :4 (section6). 5 0 : t A d e d 2. Mean-fieldanalysis a o l n w o D 2.1. Outlineofthemethod We consider a reactive gaseous premixture of initial ambient temperature Tu, density ρu and light-fuel mass fraction yu, through which combustion spots idealized as Zel’dovich- type FBs of various sizes rF(0) are randomly distributed at time t = 0. Our goal is to determinethebehaviourofthesystemfort > 0. BecauseFBsareknowntospontaneously evolveonamuchlongertimethantZ ≡ rZ2/Dth (whereDth isthethermaldiffusivity)when the reactant is mobile enough and the Zel’dovich number (Ze) is large, each of them will vary quasi-steadily if the concentration and temperature fields around it also change only slowly. In turn, because the latter drift under the collective effects, the FB number density has tobesmall,forconsistency. Specifically,theanalysisassumesaninitialnumberdensitythat correspondsroughlytooneFBofradiuscomparabletorZ inanyvolumeofO(Ze3rZ3)size, up to numerical factors; the typical spacing between neighbouring FBs is then O(ZerZ), i.e. comparable to the range of unsteady effects around each of them. At such a scale, each FB looks point-like. A rigorous treatment of the problem would require one to cope with all the direct interactions among the FB population to determine each individual 4 YD’AngeloandGJoulin Figure1. SampleflameballofradiusrF(t),surroundedbymanyothersthatmayhavedifferent sizes,e.g.initially(two-dimensionalsketch). behaviour. Foranalyticalconvenience,weshallpresentlyinvokeanapproximatemean-field treatment, like in studies of spray combustion (Williams 1985) or of charge screening in plasmas (Diu et al 1989). Focusing on a sample FB (cf figure 1), the influence of the others is modelled as that of a homogeneous continuous medium characterized by mean 0 quantitiesonly: numberdensity,‘granulometry’;consistently,theconsideredtestFBmaybe 1 20 consideredspherical. Suchacontinuous-mediumapproximationisbelievednottointroduce y r importantbiases: eachFBbehavesinitiallyasifalone,whereasnon-trivialcollectiveeffects a u an can be felt only late enough that information about the precise FB locations is already J 4 lost. 2 3 Thus, the chosen procedure consists in three steps: (a) the local analysis of a spherical 4 : 05 sample FB embedded in a provisionally unknown, slowly evolving environment; (b) a t: determination of the latter’s evolution; (c) acknowledging that the selected test FB is A ed totally anonymous, to obtain a two-variable (t and rF(0)) evolution equation for the whole d oa population. To further simplify the matter, all fluid-mechanical effects caused by density l wn changesaretobeomitted(seeJoulin(1985)forajustification),aswillbethedensitychanges o D themselves. 2.2. LocalstructureofasampleFB Let r denote the distance from the centre of a sample FB (figure 1), whose current radius (tobebetterdefinedlateron)isrF(t)attimet (cid:1) 0. Asexplainedabove,thepresentmodel considersthemixturedensityρuasaconstant,neglectsconvection,andassumeslocalspherical symmetry. ThelocalprofilesoftemperatureT andreactantmassfractiony willfollowthecoupled balances: (cid:1) (cid:2) ∂T 1 ∂ ∂T ρuc ∂t =λr2∂r r2 ∂r +Qw−A(r,t)+ST(r,t) (1) (cid:1) (cid:2) ∂y 1 ∂ ∂y ρu∂t =ρuDr2∂r r2∂r −w+Sy(r,t) (2) Collectiveeffectsanddynamicsofnon-adiabaticflameballs 5 where,uponthefurtherassumptionofaone-stepArrheniusburningprocess,therateofreactant consumptionreadsas e−Tac/T w =ρuy t . (3) coll The constant positive parameters c, λ, Q, D, T , t represent the mixture specific heat ac coll andthermalconductivity,thespecificheatofreaction,thereactantmoleculardiffusivity,the activationtemperatureandatypicalcollisiontime,respectively. ST(r,t)denotesmixtureheatingby‘theother’FBsinthepresentmean-fieldmodelling, whereasSy(r,t)<0accountsfortheirdepletingthereactantpool. A(r,t)isalocalheat-loss term,chosenheretohavetheform(Buckmasteretal1990): (cid:3) A>0 constantfor r (cid:2)rF(t) A(r,t)= (4) 0 otherwise. Forasteady(∂/∂t ≡0)adiabatic(A=0)isolated(ST =0=Sy)FB,equations(1)–(3)admit thefirstintegralλT +QρuDy ≡constant;areferencereactiontemperatureT∗,corresponding toy =0,maybedefinedas Qy T∗ =Tu+ cLeu (5) where Tu and yu are the values of T and y far from the FB (r (cid:5) rF) and Le = Dth/D 010 (Dth =λ/ρuc)istheLewisnumberofthereactant. ry 2 InthelimitZe≡Tac(T∗−Tu)/T∗2 −→∞ofalargeZel’dovichnumberthatweconsider ua from now on, the chemical activity is known (Joulin 1985) to be confined in a thin layer of n Ja O(rF/Ze) width about a special value rF(t), called the flame ball radius, of r, and may be 24 neglected outside. If A = 0, ST = 0 = Sy and Ze −→ ∞, rF (then noted rZ) is given by 3 :4 Zel’dovich’s(1944)analysis,namely: 5 0 (cid:1) (cid:2) ed At: 2rZ2 =DtcolleTac/TR Tac(LTe∗T−2Tu) 2 TR =T∗. (6) ad R o wnl Still for steady isolated FBs, accounting for A (cid:10)= 0 and allowing the values of (T,y) o D at r (cid:5) rF (yet (cid:6) ZerF) to be shifted from (Tu,yu) to (Tu +δT∞,yu +δy∞), changes the reaction temperature TR (≡ T(rF,t)) from T∗ to T∗ + δTR, with (Buckmaster and Joulin 1991) δT δT T −T δy Ar2 R (cid:11) ∞ + ∗ u ∞ − F (7) T∗ T∗ T∗ yu 3λT∗ whentheright-handsideof(7)isO(Ze−1)atmost. Theflameballradiuschangesaccordingly, fromrZ torF (cid:10)=rZ,with (cid:1) (cid:2) (cid:1) (cid:2) r 2 T δT F =exp − ac R (8) r T T Z ∗ ∗ asaconsequenceof(6). Forexample,R ≡rF/rZ isgivenby T Ar2 logR−αR2 =0 α ≡ ac Z (9) T∗ 6λT∗ whenδT∞/T∗ =0=δy∞/yu,anequationtobecommentedlater. 6 YD’AngeloandGJoulin Interestinglyenough,equations(7)and(8)stillholdwheneverδT∞ andδy∞,andhence rF/rZ,evolveslowlyenough,andifST,Sy maybeneglectedin(1)and(2)forr =O(rF). In particular,thisisthecaseforspontaneousevolutionsofasingleFB(ST =0=Sy)whenthe reactantLewisnumberismarkedlylessthanone(0 < 1−Le = O(1)),forsuchevolutions takeplaceoverthet =O(Ze2r2/D )time-scale(Joulin1985);the∂/∂t termsin(1)and(2) F th maythenbesafetyneglectedwhenanalysingthelocalFBstructure(r =O(rF)). Accordingly, onemaystillemploy(7)and(8)todescribethelocalstructureofasampleFBwhencollective effectsexist(ST (cid:10)= 0,Sy (cid:10)= 0),providedthatST andSy areatmostcomparable(uniformly) to ρuc∂T/∂t and ρu∂y/∂t, respectively. This is the case if the FB number density is small enough,O(Ze−3r−3)orless. Z Forfuturereferenceonemaynotethat,forrF (cid:6) r (cid:6) ZerF,thetemperatureandmass- fractionprofilesassociatedwiththequasi-steadysampleFBweselectedaregivenby (cid:1) (cid:2) r T T −Tu =(T∗−Tu) rF +δT∞+o Ze∗ (cid:4) (cid:5) (10) r y y−yu =−yu rF +δy∞+o Zue asistypicalofthree-dimensionalLaplacianfieldsawayfroman‘object’offinitesize((cid:11)rF). 2.3. Far-fieldandcollectiveeffects As indicated by (10), neglecting the transient effects and (ST,Sy) cannot be a uniformly 0 01 valid procedure, for example because the presumed dominant contributions ∂2/∂r2 and 2 y r−1∂/∂r to the Laplacians of T and y decay like 1/r3, whereas the small transient terms r nua decay like 1/r; also, ST and SL will not vanish at r (cid:5) rF. Accordingly, a far-field a 4 J analysis, corresponding to the range r = O(ZerZ) of unsteadiness (t = O(Ze2rF2/Dth)) 3 2 is needed. Before handling the latter problem proper, one must specify ST(r,t) and 5:4 Sy(r,t). 0 t: Let rF(t,r0) denote the current radius of a FB whose initial value at t = 0 is r0. As d A impliedby(10),thisFBactsaspointheatsource(ofstrength4πλ(T∗−Tu)rF(t,r0))andas oade asinkofreactant(ofstrength−4πρuDyurF(t,r0))whenseenfromadistancer (cid:5)rF(t,r0). wnl Ifn(r0)dr0 denotesthenumberperunitvolumeofFBswithinitialradiiin[r0,r0+dr0],the Do presentmean-field(i.econtinuous-medium)approximationstoST andSy read (cid:6) ∞ ST =4πλ(T∗−Tu) rF(t,r0)n(r0)dr0 (11) 0 (cid:6) ∞ Sy =−4πρuDyu rF(t,r0)n(r0)dr0 (12) 0 andarefunctionsoftimeonly,becausetheinitialdistributionofFBsinthemediumisassumed spatiallyhomogeneous. Accordingly, the O(T∗/Ze) profile of T(r,t)−Tu corresponding to the far-field scale r =O(ZerF)isgovernedbythePDE: (cid:1) (cid:2) ∂T 1 ∂ ∂T ρuc ∂t =λr2∂r r2 ∂r +ST(t). (13) Equation(13)isendowedwiththeboundaryconditions r (t,r ) T −Tu =(T∗−Tu) F r 0 +δT∞(t)+O(r) when r −→0 (14) Collectiveeffectsanddynamicsofnon-adiabaticflameballs 7 asisimpliedby(11)viamatchingarguments,and ∂T =0 at r =+∞. (15) ∂r Asfortheinitialcondition,wechose r T(r,0)−Tu =(T∗−Tu)r0. (16) Itisimportanttonotethatknowingthelimitofr(T −Tu)asr −→0andtheinitialprofile(16) isenoughtospecifythefarfieldT −Tu. Therefore,δT∞isexpressibleasalinearfunctionalof rF(t,r0)andST(t). AsshowninappendixA,timewiseLaplacetransformationfinallyyields2 (cid:6) (cid:6) (cid:6) TδT∞−(Tt) =4πDth tdt(cid:12) ∞rF(t(cid:12),r0)n(r0)dr0− t √r˙πF(Dt(cid:12),r(t0)−dtt(cid:12)(cid:12)). (17) ∗ u 0 0 0 th The first term on the right-hand side of (17) corresponds to the collective mixture heating by the whole FB population around the selected sample object; as expected, it is positive. The second one, negative if r˙F (cid:1) 0, accounts for unsteadiness of the selected sample FB: outwardly displacing a nearly isothermal surface (TR (cid:11) T∗) into cold gases induces an additive cooling at the surface’s exterior side, as the displacement brings about cold material. Thefar-fieldprofileofy(r,t)−yucanbeprocessedsimilarlytoproduce 0 (cid:6) (cid:6) (cid:6) ary 201 δy∞yu(t) =−4πD 0tdt(cid:12) 0∞rF(t(cid:12),r0)n(r0)dr0+ 0t √r˙Fπ(Dt(cid:12),(rt0−)dtt(cid:12)(cid:12)). (18) u n Ja Apartfromachangeofsign, thisisthesameas(17)onceD issubstitutedforD ; because th 3 24 D >Dth(i.e.Le<1)thecombinedinfluencesofδT∞andδy∞willnotvanishin(7),however. 4 : 5 0 t: 2.4. Evolutionequation A d de Once(17)and(18)arepluggedinto(7)and(8),anintegro-differentialevolutionequationis a nlo obtainedforrF(t,r0),namely w o (cid:1) (cid:2) (cid:7) (cid:6) D logrF −α rF 2 = Ze (D−1/2−D−1/2) t √r˙F(t(cid:12),r0)dt(cid:12) rZ rZ 2 th 0 πDth(t −t(cid:12)) (cid:6) (cid:6) (cid:8) t ∞ +(D−Dth)4π dt(cid:12) rF(t(cid:12),r0)n(r0)dr0 . (19) 0 0 Atthisstage,itisconvenienttointroducethescaledvariablesanddata τ = 4Dtht√ ρ = r0 R(τ,ρ)= rF (20) Ze2rZ2(1− Le)2 rZ rZ r n(r ) N(ρ)= (cid:9) Z 0 (21) ∞n(r )dr 0 0 0 √ (cid:6) π(1−Le)(1− Le)2 ∞ ν =(ZerZ)3 n(r0)dr0 (22) 2Le 0 2 r˙F denotes∂rF/∂t. 8 YD’AngeloandGJoulin Figure2. RepresentationofthefunctionlogR−αR2 forthreevaluesofα (cfequation(9)), ρ−>1andρ+>ρ−denotethetworootsofequation(9);R∗isthemaximumofthisfunctionfor α(cid:1)1/2e.(Valuesofρ−,ρ+andR∗areindicatedhereforα= 310.) 0 1 0 2 y r a u whichtransform(19)intothemorecompactform n a J 24 D1/2(R(τ,ρ))+νF(τ)=logR−αR2 (23) 3 (cid:6) (cid:6) :4 τ ∞ 05 F(τ)= dσ N(ξ)R(σ,ξ)dξ (24) : At 0 0 ded (cid:6) τ R˙(σ,ρ)dσ loa D1/2(R(τ,ρ))≡ √π(τ −σ) (25) wn 0 o D whereα isdefinedin(9)andR˙ nowdenotes∂R/∂τ. Equation(23)istobesolvedwiththe initialcondition R(0,ρ)=ρ. (26) (cid:9) FOBnenummaybenrodteicnesitthya(cid:9)t ∞0∞n(Nr()ρd)rd.ρW=h1e,na3nνd(cid:2)thaOt(ν1)=thOer(e1p)recsoernretasptiovnedesvotoluatinonOt(iZmee−3ofrF−th3e) 0 0 0 system is τ (cid:1) O(1), i.e. t (cid:1) O(Ze2r2/D ), as anticipated. This justifies a posteriori the F th quasi-steadyanalysisleadingto(6)and(7). √ Forν =0(isolatedFB),equation(23)admitstwofixedpointsρ− >1andρ+ (cid:1) e(cid:1)ρ−, thataretherootsof(9). Forthosetobereal,thescaledheat-losscoefficientα (equa√tion(9)) hastobelessthan1/2e;α (cid:6)1leadstoρ− −→1+andρ+ −→+∞,whereasρ− = e=ρ+ atα =1/2e(cffigure2). When non-zero, the νF-term acts in the same direction as αR2 and tends to make the ‘fixed’pointsshift: thecollectiveeffects(mutualheatingandreactantdepletion)haveanoverall coolinginfluence,becausethereactantdiffusesfasterthanheatdoes(Le<1). ThisνF-term 3 Actually,ν=1isalreadyaratherlargevalue(seesection5andfigure4). Collectiveeffectsanddynamicsofnon-adiabaticflameballs 9 is expected to make all the FBs ultimately shrink to zero size, since it cannot decrease as timeelapses; theonlywaytostopitsgrowthistohaveR(τ,ρ) = 0 forallρ, i.e.complete extinction. 3. Limitingcases Two analytical studies of equation (23) are summarized below: the limit of small ν, giving overalltrends;theearly-timehistoriesofR(τ,ρ),neededinthenumericaltreatmentspresented insection4. 3.1. Small-ν limit When ν is small, i.e. for ‘rarefied’ FB populations, the second, νF-term of (23) (collective effect)isnegligibleinitially; itwillbecomeimportantonlylater(forτν = O(1)), whenthe firsttermontheleft-handsideinturnbecomesnegligible. • τ =O(1). Neglecting the νF-term in equation (23) decouples the FB evolutions for a while. We recallthatwedenotebyρ+andρ− <ρ+thetworootsoflogρ−αρ2,α <1/2e. Eventhough onecannotsolve(23)exactlyforν = 0,differenttypesofFBfatesareexpected,depending on whether ρ is greater or smaller than ρ−. The case ρ < ρ− will lead to R(τ,ρ) = 0 shortly(seeappendixBforthecasewhereρ (cid:6) 1),whereasρ > ρ− willpresumably4 yield 0 201 R(τ,ρ)=ρ+attheendofthetimewiseboundarylayercorrespondingtoτ =O(1),ν −→0+. y Noexactproofisavailablethatsuchtrajectoriesexist,yetthefollowingmaybeputforward. r ua IfR(τ −→+∞,ρ)=ρ forsomeinitialρs,then an + J 24 pL(R(τ,ρ)−ρ)−→ρ+−ρ (27) 3 4 5: asthevariable(p)conjugatetotime(τ)intheLaplacetransformationL(·)goestozero(Luikov 0 : 1968). Usingthen(23),stillwithν =0,showsthat t A d ρ −ρ oade L(logR−αR2)−→ +√p as p −→0 (28) l n w Do wherebyRapproachesρ fromabove(respectively,below)ifρ >ρ (respectively,ρ <ρ ), + + + inthemanner (cid:1) (cid:2) (cid:1) (cid:2) 1 ρ −ρ 1 −2αρ (R−ρ )−→ √+ +o √ as τ −→+∞ (29) ρ + + πτ τ + because 2αρ2 > 1. Accordingly, no trajectory-crossing will take place during their (slow) + approachtoρ . Weshalltakeforgrantedthatν =0in(23)yields + R(τ,ρ <ρ−)−→0 R(τ,ρ >ρ−)−→ρ+ (30) as τ −→ +∞. The exceptional cases ρ (cid:11) ρ− have a negligible measure, generically. The conjecture (30) was tested upon looking at the auxiliary problem (a linearized form of (23) whenν =0andR (cid:11)ρ±) (cid:6) τ F˙(σ)dσ D1/2(F)≡ √π(τ −σ) =−mF (31) 0 4 Themethodsofsuper-solutionandsub-solutiondevelopedbyAudounetandRoquejoffre(2000)arelikelytosettle thepoint.

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