CHAPTER DYNAMICS OF PARTICLES IN A FLUID
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36 CHAPTER 1. DYNAMICS OF PARTICLES IN A FLUID Notes on Mathematical Problems on the Dynamics of Dispersed Particles interacting through a Fluid P.E. Jabin (?) and B. Perthame (?)(??) (?) Ecole Normale Superieure, DMI 45, rue d'Ulm 75230 Paris Cedex 05, France (??) INRIA-Rocquencourt, Projet M3N BP 105 78153 Le Chesnay Cedex, France 1 Introduction In this Chapter, we present some mathematical problems related to the dy- namics of particles interacting through a fluid. We are interested in the dilute cases. We mean the cases where a transport Partial Differential Equations in the phase space can be expected for the particles density. In order to derive these transport equations explicitely, some assumptions on the fluid dynam- ics are necessary. They limit the validity of the model but still represents many possible applications. Namely we assume that the fluid dynamics can be reduced to two simple situations. The first situation is the simple case of a potential flow (perfect incompressible and irrotational flow). This is relevant to describe for instance the motion of bubbles in water (see G.K. Batchelor [2]) and focuses mainly on the added mass effect which means that to acce- larate bubbles requires to accelerate some part of the water too. The second situation is the more standard case of particles in a Stokes flow, for which the domains of application are suspensions or sedimendation.
- mathematical formalism
- potential ?
- particle density
- stokes flow
- active field
- equa- tion
- particles
- concerning bubbly-potential
- fi
36
CHAPTER1.DYNAMICSOFPARTICLESINAFLUID
NotesonMathematicalProblemsonthe
Dynamicsof
DispersedParticlesinteractingthroughaFluid
P.E.Jabin(
)andB.Perthame(
)(
)
(
)EcoleNormaleSuperieure,DMI
45,rued’Ulm
75230ParisCedex05,France
(
)INRIA-Rocquencourt,ProjetM3N
501PB78153LeChesnayCedex,France
1Introduction
InthisChapter,wepresentsomemathematicalproblemsrelatedtothedy-
namicsofparticlesinteractingthroughauid.Weareinterestedinthe
dilute
cases.Wemeanthecaseswherea
transport
PartialDierentialEquationsin
thephasespacecanbeexpectedfortheparticlesdensity.Inordertoderive
thesetransportequationsexplicitely,someassumptionsontheuiddynam-
icsarenecessary.Theylimitthevalidityofthemodelbutstillrepresents
manypossibleapplications.Namelyweassumethattheuiddynamicscan
bereducedtotwosimplesituations.Therstsituationisthesimplecaseofa
potentialow
(perfectincompressibleandirrotationalow).Thisisrelevant
todescribeforinstancethemotionofbubblesinwater(seeG.K.Batchelor
[2])andfocusesmainlyonthe
addedmass
eectwhichmeansthattoacce-
laratebubblesrequirestoacceleratesomepartofthewatertoo.Thesecond
situationisthemorestandardcaseofparticlesina
Stokesow
,forwhich
thedomainsofapplicationaresuspensionsorsedimendation.
1.INTRODUCTION
37
Thecaseofa
potentialow
aroundtheparticles,leadstoadicultyin
establishingtheequationfortheparticledensity.Amathematicalformalism
wasdevelopedbyG.RussoandP.Smereka[26]whichwewillpresenthere,
intheimprovedversionofH.Herrero,B.LucquinandB.Perthame[19].
Wewillrecallherehowonecanderive,fromtheinteractingsystemofpar-
ticles,aVlasovtypeofequationfortheparticledensityinthephasespace
g
(
t,x,p
),here
t
0isthetime,
x
∈
IR
3
representsthespacepositionand
p
∈
IR
3
representsthetotalimpulsionofparticles(dualofthevelocityinthe
Lagrangian-Hamiltonianduality).Thisequationis
∂∂t
g
+grad
p
H
grad
x
g
grad
x
H
grad
p
g
=0
,
(1.1)
H
(
t,x,p
)=
1
|
p
+(
t,x
)
|
2
,
(1.2)
2(
t,x
)=
B
(
P
+
)(
t,.
)
.
Here
B
=
B
(
x
)isagiven3
3matrix,
isthekineticparameter(relating
theradiusoftheparticlestothedensitiesoftheparticlesandoftheuid)
andthemacroscopicdensityandimplusion
,
P
aredenedby
(
t,x
)=
3
g
(
t,x,p
)
dp,
(1.3)
ZRIP
(
t,x
)=
3
pg
(
t,x,p
)
dp.
(1.4)
ZRIThedicultytoestablishthisequation,comesfromtheLagrangianaspect
ofthenaturaldynamicsfortheparticles.ItturnsoutthattheHamiltonian
variablesarebetteradaptedtomathematicalmanipulationsandtomechan-
icalinterpretation(noticethattheHamiltonianvariableisjustthetotal
impulsionofparticles).Butthederivationofthe
meaneld
equation(1.1)-
(1.3)fortheparticlesdensityiseasierinLagrangianvariables.Then,one
issueistounderstandhowtodene,inthekineticP.D.E.,theLagrangian
andHamiltonianvariables(andtounderstandalsochangeofvariables).
Thesecondsituationconsistsinconsideringa
Stokesow
aroundthepar-
ticles.Itleadstoquitedierentmathematicalissues.Inordertoestablish
equationsfortheparticledensityonecanfollowthesamederivationasbe-
fore.Fromthefulldynamicsofparticles-Nbodyinteraction-arst(and
restrictive)assumptionistomakeadipoleapproximationfortheuidequa-
tion.Thisreducesthedynamicstotwo-bodyinteractionsandthusallows
38
CHAPTER1.DYNAMICSOFPARTICLESINAFLUID
tosettlethekineticequationfortheparticledensity
f
(
t,x,v
),here
v
isthe
velocityoftheparticle.OneobtainsaVlasovtypeequation.
∂∂t
f
+
v
grad
x
f
+
div
v
((
g
+
A?
x
j
v
)
f
)=0
,
(1.5)
j
(
t,x
)=
3
vf
(
t,x,v
)
dv.
(1.6)
RIRThematrix
A
(
x
)isnowrelatedtotheStokesEquation,aswellas
B
,in
thepotentialcase,isrelatedtotheLaplaceEquation.Also,
g
denotesthe
gravityvector,
thekineticparameterand
=
43
Na
,with
N
thenumber
ofparticles,
a
theirradius.Eventhoughthereisnomathematicaldiculty
inestablishingthissystem,severalmathematicalquestionsariseconcerning,
forinstance,variousasymptoticbehaviors(largetimebehaviorcf[21],
vanishing...etc)Theyarisebecausethefrictiontermplaysamajorrolein
theparticlesdynamicsforaStokesow.Aparticularlyinterestingsituation
isthelimit
→∞
.Itgivesanexampleofa
macroscopiclimit
whichisnot
obtainedbythecollisionalprocess,butbyastrongforceterm.Inthecase
athand,itisprovedinP.E.Jabin[22]thatthemacroscopiclimitgivesrise
totheequation
∂∂t
+div(
u
)=0
,
A?
x
(
u
)
u
=
g.
((11..78))
Thetopicofthesenotesrepresentparticularexamplesofaveryactive
eldofuidmechanicswherekineticphysicsplaysafundamentalrole.Usu-
allyitisusedinthederivationofmodelsforparticularsituations,butalsoof
eectiveequationsforthemotion.Innowaywecangiveacompleteaccount
oftheliteratureinthisdomainandweprefertorefertosomegeneralworks.
Concerningbubbly-potentialows,thepaperbyY.YurkovetskyandJ.F.
Brady[32]containsnumerousrecentreferencesaswellasconsiderationson
statisticalphysicsaspectsofthemodelandtheeectofcollisisons.Forthis
eect,seealsoG.RussoandP.Smereka[27],J.F.Bourgat
etal
[6].The
derivationofpdemodelsandtheuseofkineticdescriptionisaratherrecent
subject,conferH.F.Bulthuis,A.ProsperettiandA.S.Sangani[7],A.S.San-
ganiandA.K.Didwana[28],P.Smereka[30]andthereferencestherein.On
theotherhand,thedynamicsofparticlesinaStokesowhaveleadtovery
numerousworks.Letusquotesomeofthem:G.K.BatchelorandC.S.Wen
2.DYNAMICSOFBALLSINAPOTENTIALFLOW
39
[8],F.Feuillebois[12],E.J.Hinch[16],R.HerczynskiandI.Pienkowska[20]
andthebookbyJ.HappelandH.Brenner[17].Otherregimeshavealso
beenstudiedandleadtomathematicalmodelswhichhavebeenanalyzedfor
instancebyK.Hamdache[18]forthecaseofamoregeneralincompressible
ow(andsmallparticles),byD.Benedetto,E.CagliotiandM.Pulvirenti
[3]forgranularow.Complexnumericalsimulationshavebeenperformed
byB.MauryandR.Glowinski[25],R.Glowinski,T.W.PanandJ.Periaux
[15],forhighconcentrationsofparticles(seealsothereferencestherein).
TheoutlineofthisChapterisasfollows.Thenexttwosectionsare
devotedtothecaseofapotentialow;insection2wederivethemodel
dynamicalsystemandsection3isdevotedtothemean
