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

Abstract

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.

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

“…In this case, the velocities and the internal energies are modified by the collision and, moreover, the reactants are transformed into the products of the reaction. By following the notation of [2], the rearrangement of mass and the redistribution of chemical binding energy are formalized as follows.…”

“…The formal limit as ε → 0, when the angular collision kernels of both the elastic bi-species and the reactive operators are odd functions of cos θ , has been studied by Anwasia, Bisi, Salvarani and Soares [2]. The resulting target equations have the structure of a system of balance equations coupled with the corresponding temperature-dependent flux-gradient relationships and with an equation describing the energy balance: Φ i j (I, I j , R, r)…”

“…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.…”

Maximum entropy principle approach to a non-isothermal Maxwell-Stefan diffusion model

“…The Maxwell-Stefan diffusion equations for the mixture of Maxwell molecules were derived in [13], and in the non-isothermal setting in [22], while in [16] the case of reactive mixture of polyatomic gases was considered. Since our interest is on inert mixtures, we shall give the non-isothermal Maxwell-Stefan diffusion equations here for the completeness.…”

“…From the macroscopic point of view, Maxwell-Stefan diffusion model consists of mass balance equations for the constituents, and a kind of momentum balance relations adjoined with a closure relation due to their linear dependence. It can be derived either in the continuum framework [11,12], or starting from the Boltzmann equations (BEs) for mixtures [13,14,15,16]. In the latter approach, crucial step in derivation of Maxwell-Stefan equations is the asymptotic limit enabled through appropriate scaling of the governing equations.…”