In this article we deduce a mathematical model of Maxwell-Stefan type for a reactive mixture of polyatomic gases with a continuous structure of internal energy. The equations of the model are derived in the diffusive limit of a kinetic system of Boltzmann equations for the considered mixture, in the general non-isothermal setting. The asymptotic analysis of the kinetic system is performed under a reactive-diffusive scaling for which mechanical collisions are dominant with respect to chemical reactions. The resulting system couples the Maxwell-Stefan equations for the diffusive fluxes with the evolution equations for the number densities of the chemical species and the evolution equation for the temperature of the mixture. The production terms due to the chemical reaction and the Maxwell-Stefan diffusion coefficients are moreover obtained in terms of the collisional kernels and parameters of the kinetic model.

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.

We study multicomponent diffusion in a gaseous mixture undergoing a reversible chemical reaction. Under isothermal and non-isothermal conditions, we extend the Maxwell-Stefan (MS) theory to the reactive system, by deriving the reactive MS equations containing additional diffusion terms resulting from the chemical reaction. We start from the simple reacting spheres (SRS) kinetic model and consider a diffusive regime, for which both mechanical collisions and chemical reactions have comparable relaxation times, which are much smaller than the characteristic times of the flow. In the corresponding hydrodynamic limit, we formally derive the reactive MS equations from the momentum equation of the species and deduce explicit expressions for the diffusion coefficients.

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