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

TL;DR

This paper derives a Maxwell-Stefan reaction-diffusion system from a kinetic model of reacting gases by asymptotic analysis, highlighting the transition from microscopic interactions to macroscopic behavior.

Contribution

It provides a formal derivation of Maxwell-Stefan type equations from a kinetic model for reactive mixtures, including explicit production terms under specific scaling.

Findings

Derivation of reaction-diffusion system from kinetic model

Explicit computation of production terms

Validation of Maxwell-Stefan equations as limit

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.