Multi-temperature hydrodynamic limit from kinetic theory in a mixture of rarefied gases

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 mono-atomic hard sphere gases.