Multi-temperature Hydrodynamic Limit from Kinetic Theory in a Mixture of Rarefied Gases

Abstract

Starting from the Boltzmann kinetic equations for a mixture of gas molecules whose internal structure is described by a discrete set of internal energy levels, hydrodynamic equations at Euler level are deduced by a consistent hydrodynamic limit in the presence of a two-scale collision process. The fast process driving evolution is constituted by mechanical encounters between particles of the same species, whereas inter-species scattering proceeds at the macroscopic scale. The resulting multi-temperature and multi-velocity fluid-dynamic equations are briefly commented on, and some results in closed analytical form are given for special simplified situations like Maxwellian collision kernels, or monoatomic hard sphere gases.

Shock Wave Structure of Multi-Temperature Euler Equations from Kinetic Theory for a Binary Mixture

“…However, we are able to build up a weak solution with two discontinuities. Specifically, we allow a first discontinuity with the same features of the previous case (same type of jump for u 1 and T 1 , no jump for species 2) and consider the tail originating from it. On the other hand, we move backward from + ∞ starting tangent to its stable manifold.…”

“…They give rise typically to hyperbolic systems of balance laws which can be studied in the frame of extended thermodynamics [9]. An interesting point in this respect is a comparison with the derivation of fluid-dynamic equations of this type, with momentum and energy exchange rates, as suitable hydrodynamic limit, starting from a kinetic theory description [7,1]. Indeed this paper belongs to the latter research line, and is aimed at testing a simple fluid-dynamic model at Euler level, derived from the relevant Boltzmann equations, on a very classical problem like the one dimensional steady shock waves in a binary mixture.…”

On the Boltzmann gas mixture equation: Linking the kinetic and fluid regimes

“…The derivation of incompressible Navier-Stokes equations from Boltzmann equations that model the dynamics of gases whose particles may undergo nonelastic collisions has been proposed in [9]. In [11], starting from a kinetic model for a chemical reaction, the authors have derived multi-temperature reactive Euler equations. In [23,25,37] the authors have considered the Vlasov-Boltzmann system for a fluid of two species of interacting particles and have performed diffusive expansions, in terms of Knudsen numbers, for the solutions of the rescaled system around Maxwellians of equal weights for the two species.…”

Multi-velocity and multi-temperature model of the mixture of polyatomic gases issuing from kinetic theory

“…On the other side, the source terms N i and E i can be determined using the kinetic theory of gases, provided that the collisional cross-section is specified. In particular, in [3,4] these source terms are calculated for some specific choices of the cross-section, and polyatomic gases are modeled with discrete energy levels.…”