Formal passage from kinetic theory to incompressible Navier-Stokes equations for a mixture of gases (Q2877719)

The formal passage from a kinetic model to the incompressible Navier-Stokes equations is presented for a mixture of \(N\) elastically scattering monatomic rarefied gases with different masses. The starting point of the work is the collection of coupled Boltzmann equations for the mixture of the gases. In comparison to a former work by \textit{C. Bardos} et al. [``Fluid dynamic limits of kinetic equations I'', J. Stat. Phys. 63, 323--344 (1991)] deriving the Navier-Stokes equation for a monatomic gas, \(N-1\) convection-diffusion equations for the densities of the species have to be taken into account. Also, the assumption of different particle masses complicates the formal derivation of the equations for velocity and temperature. In the work, first the perturbations of the particle distribution functions from the Maxwellian distribution are obtained. The hydrodynamic incompressibility and Boussinesq relations are derived and discussed. Then, the convection-diffusion equations for the densities, momenta and the temperature of the species are found. Finally, in an appendix, technical lemmas and evaluations of suitable collision contributions are reported, which are used to obtain explicit expressions for the diffusion coefficients in the equations for the densities, velocities, and temperature. In doing so, only Maxwellian molecules are taken into account. It is shown that the set of incompressible Navier-Stokes equations derived from kinetics is equivalent to incompressible Navier-Stokes equations obtained from compressible Navier-Stokes equations considering low Mach numbers.