On the Maxwell–Stefan diffusion limit for a mixture of monatomic gases

Abstract

Multi‐species Boltzmann equations for gaseous mixtures, with analytic cross sections and under Grad's angular cutoff assumption, are considered under diffusive scaling. In the limit, we formally obtain an explicit expression for the binary diffusion coefficients in the Maxwell–Stefan equations. Copyright © 2016 John Wiley & Sons, Ltd.

A numerical scheme for a kinetic model for mixtures in the diffusive limit using the moment method

“…of the system (33) with boundary conditions (20), which also solves (28). Moreover, thanks to Proposition 4, for > 0 small enough, it is also a solution of system (21) with boundary conditions (20).…”

“…It has also been shown in that Maxwell–Stefan's equations can be seen as the limit in the small Mach and Knudsen number regime of the Boltzmann equations for mixtures in the case of Maxwellian molecules. This result has been extended to some analytical cross sections in , and generalized to general cross sections in , as well as in a nonisothermal setting in .…”

“…4,p. 509] ensures that f has a fixed pointỹ n+1 1 , andỹ n+1 = (ỹ n+1 1 , y n+1 2 ) ⊺ is thus a solution of the new auxiliary system (33). By construction, since we added positive parts on every concentration in this new auxiliary system, all components ofỹ 1 are nonnegative, which means that it is also a solution to the auxiliary system (28).…”

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

“…Subsequently, the wellposedness and the long-time behavior of the solutions of the Maxwell-Stefan system (or some variants) have been studied in [11,14,16,25,28,29], the numerical simulation of the Maxwell-Stefan system has been the subject of [9,14,22,32], and the relationships between Fickian diffusion and the Maxwell-Stefan model have been analyzed in [12,37]. 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].…”

Compactness of Linearized Kinetic Operators

“…some specific physical phenomena (such as uphill diffusion in the purely diffusive case, see [26,29,16,3,5,24]). Consequently, it is not surprising that compactness properties in the mixture case cannot be deduced through a straightforward adaptation of the standard methods of proof from the mono-species case.…”