A space-dependent Boltzmann-BGK model for gas mixtures and its hydrodynamic limits

TL;DR

The paper develops a space-dependent hybrid Boltzmann-BGK model for inert gas mixtures, enabling local switching between detailed Boltzmann collisions and BGK relaxation while preserving mass, momentum, energy, and entropy dissipation; the approach yields Maxwellian equilibria with a common mean velocity ${\bf u}$ and temperature $T$ independent of the collision description. Through a Chapman-Enskog analysis, it derives three hydrodynamic limits: (i) a collision-dominated Euler regime, (ii) a dominant intra-species collision regime with multi-velocity/multi-temperature behavior, and (iii) a heavy-light regime producing a kinetic-fluid coupling where the heavy species obeys fluid dynamics and the light species remains kinetic. The results demonstrate that Euler-level dynamics are insensitive to the Boltzmann vs BGK choice, while more detailed constitutive relations emerge at higher orders; the framework offers a flexible tool for modeling regions with strong nonequilibrium features (e.g., shocks) alongside near-equilibrium areas. This hybrid model provides a rigorous bridge from kinetic to macroscopic descriptions and supports the derivation of kinetic-fluid systems used in multi-phase flow studies. All mathematical notation is consistently presented with $...$ delimiters for clarity and SEP-compatible embedding.

Abstract

We present a hybrid Boltzmann-BGK model for inert mixtures, where each kind of binary interaction may be described by a classical Boltzmann integral or by a suitable relaxation-type operator. We allow also the possibility of changing the option Boltzmann/BGK operator according to the space position. We prove that this model guarantees conservations of species masses, global momentum and energy, as well as the entropy dissipation, leading to the expected Maxwellian equilibria with all species sharing the same mean velocity and the same temperature. We investigate then such mixed kinetic equations in three different hydrodynamic limits: the classical collision dominated regime, a situation with dominant intra-species collisions, and a mixture with heavy and light particles leading to a kinetic-fluid description.