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

Abstract

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.

From the Boltzmann Description for Mixtures to the Maxwell–Stefan Diffusion Equations

“…However, when the mixture is composed of more than one species, the local Maxwellian state is still an equilibrium for the whole system only when the velocities and the kinetic temperatures of the species composing the mixture are exactly the same. 4 The Boltzmann system for polyatomic reactive or non-reactive gas mixtures…”

“…Starting from the SRS kinetic model [24] for a mixture of four ideal gases undergoing a reversible chemical reaction of bimolecular type, Anwasia, Gonçalves and Soares derive in [4] the corresponding Maxwell-Stefan asymptotics in the diffusive scaling, in the vanishing limit of the mean free path, when all the collisional integrals have the same order of magnitude. The authors first define the reactive cross sections for the direct and reverse chemical reactions in terms of their threshold relative velocities.…”

“…After describing these formal results, we summarize the result by Bondesan and Briant [8], which provides a rigorous description of the limiting procedure. The next chapter is devoted to the reactive case, and is essentially based on the recent articles [2], [3] and [4]. We conclude this review by quoting the most recent literature on some strictly related subjects.…”

On the Maxwell-Stefan diffusion limit for a reactive mixture of polyatomic gases in non-isothermal setting

“…By following the research line initiated by Bardos, Golse and Levermore in the Nineties -whose goal was the derivation of the equations of fluid mechanics starting from the Boltzmann equation [2,3] -several articles have carried out the formal derivation of isothermal multicomponent Maxwell-Stefan type diffusion equations starting from the Boltzmann system for monatomic non-reactive gaseous mixtures [12,13,15,26,27]. The diffusive limit in a reactive mixture described by the simple reacting sphere kinetic model (SRS), which retains the main features of the reaction mechanism without taking into account the internal degrees of freedom of the particles, has been investigated in [1].…”

On the relaxation of the Maxwell–Stefan system to linear diffusion

“…The Maxwell-Stefan equations have been written in the nineteenth century [16,18], but the interest in their rigorous mathematical study is very recent and not yet complete. After some works, mainly devoted to the matrix formulation of the gradient-flux relationships and described in [10], the study of existence and uniqueness has been carried out in [2,5,7,14,15,11], the formal derivation of multicomponent diffusion equations from the Boltzmann system has been investigated in [4,6,3,12,13,1] and some numerical discretizations of the Maxwell-Stefan system have been proposed in [5,17].…”