The Maxwell–Stefan diffusion limit for a kinetic model of mixtures with general cross sections

Abstract

In this article, we derive the Maxwell-Stefan formalism from the Boltzmann equation for mixtures for general cross-sections. The derivation uses the Hilbert asymptotic method for systems at low Knudsen and Mach numbers. We also formally prove that the Maxwell-Stefan coecients can be linked to the direct linearized Boltzmann operator for mixtures. That allows to compute the values of the Maxwell-Stefan diusion coecients with explicit and simple formulae with respect to the cross-sections. We also justify the specic ansatz we use thanks to the so-called moment method.

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

“…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 .…”

“…As formal theoretical asymptotic results are obtained by a moment method , assuming that the distribution function of each species i is at a local Maxwellian state with a small velocity for any ( t , x ) ∈ ℝ + × Ω, we apply the same approach in order to derive a numerical scheme which nicely behaves when ϵ tends to 0. More precisely, the results are obtained under the following ansatz, for any 1 ≤ i ≤ p , The same ansatz is also made on the initial condition .…”

“…The numerical scheme described in this paper has been especially designed in order to nicely degenerate for arbitrary small values of ϵ . Moreover, it has been proved, at least formally , that the Boltzmann equations for mixtures tend to the Maxwell–Stefan equations when ϵ vanishes. Therefore, for each species, we compute the L ∞ ‐norm in both x and t of the difference between its concentration and the concentration computed by Maxwell–Stefan equations.…”

“…Our approach is to mimic the analytical method used to obtain the Maxwell–Stefan equations in the low Mach and Knudsen numbers limit from the Boltzmann equations for mixtures. Whereas the Fick equations are naturally obtained from the kinetic equations by a perturbative method, the Maxwell–Stefan ones are obtained, as we already stated, by a moment method . Thus the scheme we propose relies on the computation of the moments of the distribution functions , under the ansatz that these distributions functions are at local Maxwellian states.…”

High-friction limits of Euler flows for multicomponent systems

“…, ρ ε n ) and ρ ε = n i=1 ρ ε i . When the barycentric velocity v ε vanishes, we recover the Maxwell-Stefan equations analyzed in, e.g., [2,5,14].…”