Review articles on the study of combustion. Combustion Theory and Modelling. Volumes Part 5 1-13

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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
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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:[email protected]
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.