Fluid Dynamic Limits of the Kinetic Theory of Gases

In this paper, we propose a formal derivation of the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic gases. We use a direct extension of the model devised in [8,16] for treating the internal energy with only one continuous parameter. This model is based on the Borgnakke-Larsen procedure [6]. We detail the dissipative terms related to the interaction between the gradients of temperature and the gradients of concentrations (Dufour and Soret effects), and present a complete explicit computation in one case when such a computation is possible, that is when all cross sections in the Boltzmann equation are constants.

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“…[11]). A survey on recent advances on fluid-dynamic limits of kinetic models, with both formal and rigorous proofs, may be found in [22].…”

mentioning

confidence: 99%

On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases

Baranger¹,

Bisi²,

Brull³

et al. 2018

Kinetic &Amp; Related Models

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“…From assumption (a), the initial conditions for (4.3) are assumed to be local Maxwellian functions centered at the average mean velocity of the species. From the first assertion of (4.2) and following [17,16], such local Maxwellians are assumed to be of the form…”

Section: The Scaled Equations and Our Assumptionsmentioning

confidence: 99%

From the simple reacting sphere kinetic model to the reaction-diffusion system of Maxwell–Stefan type

Anwasia

Gonçalves

Soares

2019

Communications in Mathematical Sciences

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In this paper we perform a formal asymptotic analysis on a kinetic model for reactive mixtures in order to derive a reaction-diffusion system of Maxwell-Stefan type. More specifically, we start from the kinetic model of simple reacting spheres for a quaternary mixture of monatomic ideal gases that undergoes a reversible chemical reaction of bimolecular type. Then, we consider a scaling describing a physical situation in which mechanical collisions play a dominant role in the evolution process, while chemical reactions are slow, and compute explicitly the production terms associated to the concentration and momentum balance equations for each species in the reactive mixture. Finally, we prove that, under isothermal assumptions, the limit equations for the scaled kinetic model is the reaction diffusion system of Maxwell-Stefan type. 1 2. The continuum reaction-diffusion system of Maxwell-Stefan type In this section, we introduce a mathematical model for a reactive multi-species gaseous mixture in the context of continuum mechanics. The mixture is influenced by two processes, namely the diffusion, which causes the species to spread in space, and a chemical reaction, which results in the transformation of the species into each other. The model equations consist of the concentration balance equations for the reactive species in the mixture coupled with the MS equations for the momentum of the species. These equations describe how both processes affect the evolution of the mixture and will be referred to as the reaction-diffusion system of MS type.Let Ω ⊂ R 3 be a bounded domain with boundary denoted by ∂Ω and outward normal vector at each point x of the boundary given by ν(x). We consider a mixture of four species, say A 1 ,A 2 ,A 3 and A 4 , that participate in a chemical reaction of type(2.1)This means that species A 1 ,A 2 react to produce species A 3 ,A 4 and conversely, species A 3 ,A 4 also react to produce species A 1 ,A 2 . We say that A 1 ,A 2 and A 3 ,A 4 are the reactive species (or the reactive pairs), more specifically A 1 ,A 2 are reactants and A 3 ,A 4 are products of the forward chemical reaction. For each species A i , with i = 1,2,3,4, let ̺ i (t,x) ≥ 0 be the mass density, u i (t,x) the mean velocity and r i (t,x) the production rate of mass density due to the chemical reaction, where x ∈ Ω and t > 0.

show abstract

“…[11]). A survey on recent advances on fluid-dynamic limits of kinetic models, with both formal and rigorous proofs, may be found in [22].…”

Section: Introductionmentioning

confidence: 99%

On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases

Baranger

Bisi

Brull

et al. 2019

31st International Symposium on Rarefied Gas Dynamics: Rgd31

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In this paper, we propose a formal derivation of the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic gases. We use a direct extension of the model devised in [8, 16] for treating the internal energy with only one continuous parameter. This model is based on the Borgnakke-Larsen procedure [6]. We detail the dissipative terms related to the interaction between the gradients of temperature and the gradients of concentrations (Dufour and Soret effects), and present a complete explicit computation in one case when such a computation is possible, that is when all cross sections in the Boltzmann equation are constants.

show abstract

Fluid Dynamic Limits of the Kinetic Theory of Gases

Cited by 19 publications

References 83 publications

On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases

On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases

From the simple reacting sphere kinetic model to the reaction-diffusion system of Maxwell–Stefan type

On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases

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