A kinetic description of particle fragmentation

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A kinetic description of particle fragmentation? Pierre-Emmanuel Jabin†& Juan Soler‡ June 2, 2006 Abstract This paper is concerned with the formulation and analysis of par- ticle fragmentation by a mathematical approach of the kinetic theory. We consider a fairly general model which may require a description of the internal configuration of each particle, like internal energy. The fragmentation process is supposed to occur due to the configuration of the corresponding particle; An easy modification would allow to con- sider the interaction with some external medium (typically a fluid) but we do not deal here with fragmentation processes induced by particles collision. The proposed model is therefore linear and may be analyzed with the use of correct entropies. Key words: Fragmentation kernels, kinetic equations, Boltzmann equation, coagulation kernels, sprays, polymers. 1 Introduction and setting of the problem The aim of this paper is to propose a framework to understand and analyze particle fragmentation in a kinetic context. The idea to describe fragmenta- tion in this setting is based on including the mass and the internal energy of ?This research was partially supported by the EU financed network IHP-HPRN-CT- 2002-00282 and by MEC (Spain), Proyecto BFM2002–00831. †Laboratoire J. A. Dieudionne CNRS-Universite Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice cedex 02, France.

• ticle fragmentation


• duce fragmentation


• fluid


• coagulation-fragmentation models


• another daughter


• kinetic description


• fragmentation kernel


• turbulent fluid


• surrounding fluid


• interaction between

Sujets

• Antipolis

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• Publié par: pefav
• Nombre de lectures: 10
• Langue: English

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Akineticdescriptionofparticlefragmentation ∗ Pierre-EmmanuelJabin † &JuanSoler ‡ June2,2006 Abstract Thispaperisconcernedwiththeformulationandanalysisofpar- ticlefragmentationbyamathematicalapproachofthekinetictheory. Weconsiderafairlygeneralmodelwhichmayrequireadescriptionof theinternalconfigurationofeachparticle,likeinternalenergy.The fragmentationprocessissupposedtooccurduetotheconfigurationof thecorrespondingparticle;Aneasymodificationwouldallowtocon- sidertheinteractionwithsomeexternalmedium(typicallyafluid)but wedonotdealherewithfragmentationprocessesinducedbyparticles collision.Theproposedmodelisthereforelinearandmaybeanalyzed withtheuseofcorrectentropies. Keywords:Fragmentationkernels,kineticequations,Boltzmannequation, coagulationkernels,sprays,polymers. 1Introductionandsettingoftheproblem Theaimofthispaperistoproposeaframeworktounderstandandanalyze particlefragmentationinakineticcontext.Theideatodescribefragmenta- tioninthissettingisbasedonincludingthemassandtheinternalenergyof ∗ ThisresearchwaspartiallysupportedbytheEUfinancednetworkIHP-HPRN-CT- 2002-00282andbyMEC(Spain),ProyectoBFM2002–00831. † LaboratoireJ.A.Dieudionne´CNRS-Universite´Nice-SophiaAntipolis,ParcValrose, F-06108Nicecedex02,France.E-mail: [e-mail] ‡ DepartamentodeMatema´ticaAplicada,FacultaddeCiencias,Universidadde Granada,18071Granada,SPAIN.E-mail: [e-mail] 1

theparticlesasadditionalvariablesinthedistributionfunctionwhichmakes themodelingofthefragmentationkernelclearerintheequation.Weanalyze thismodel(thepropertiesofthefragmentationkernel)intheframeworkof aLlogLtheory. Theevolutionofcomplexfluidsconsistingofagasinteractingwithdroplets orsolidparticlesisaninterestingproblemwithawiderangeofapplications inastrophysics,biology,chemistryormeteorologyfromwhichwemention heretheproductionofdieselmotors,spraysandpolymers,forexample.The particles(ordroplets)inmersedintothefluidcouldcollideandthismaypro- ducefragmentationorcoagulationbetweenthem,whichmodifiesthedensity orthevelocityofthesurroundingfluidandthusaffectsagaintheevolutionof theparticles.Thisproblemisusuallystudiedthroughatwo-phase(Eulerian– Eulerian)descriptionoffluidsorthroughafluid-kineticdescription.Inthis paperwefollowthesecondapproachandfocusoureffortintheanalysisand understandingofthekineticdescriptionofthefluid-kineticcoupling.Inthis sense,weconsiderherethekinetic(particle)partofthemodelasisolatedof thefluid,sothatthiscanbealsoviewedasaweakcouplinglimitmodel.Ina forthcomingpaperwewillfocusourattentiononintroducingandanalyzing somemodelsofinteractionwiththesurroundingfluid. Beforeproceedingtothedescriptionandanalysisofourmodel,letus brieflysummarizethedifferentapproachestothisproblemstudiedinthe literatureandwhicharerelevantinourcontext.Animportantdirection isthestudyofthetimeevolutionoftheaverageconcentrationofparticles ofagivensizeinsomespatiallyhomogeneousphysicalsystemdescribedby theSmoluchowski-typeequations,see[2,10,28]forastochasticpointof view.Ontheotherhand,adeterministicpointofviewfortheSmoluchowski diffusivemodelswithcoagulation-fragmentationkernelshasbeenrecently analyzedin[16]forsolutionsinL 1 .Theconnectionbetwenthedeterministic discreteandthecontinuouscoagulation-fragmentationmodels(connectedto thescaleofobservationandthedegreeofsaturationofthemixtures)hasbeen investigatedin[8,17,18].Anintroductiontothemathematicalinvestigation ofcoagulationandfragmentationwasfirstgivenin[9](seealso[4]or[21]). Inthefluid-kineticmodelthedensityfunctionassociatedwiththeparti- clessatisfiesaBoltzmann-typeequation f∂ ∂t +(v ∙r x )f+ r v ∙ (Ff)=Q(f),(1.1) whereFistheforceperunitmass,whichisessentiallyduetotheinteraction 2