Grad's moment method for relativistic gas mixtures of Maxwellian particles

Abstract

Mixtures of relativistic gases are analyzed within the framework of Boltzmann equation by using Grad's moment method. A relativistic mixture of r constituent is characterized by the moments of the distribution function: particle four-flows, energy-momentum tensors, and third-order moment tensors. By using Eckart's decomposition and introducing 13r + 1 scalar fields – related with the four-velocity, temperature of the mixture, particle number densities, diffusion fluxes, non-equilibrium pressures, heat fluxes, and pressure deviator tensors – Grad's distribution functions are obtained. Grad's distribution functions are used to determine the third-order tensors and their production terms for mixtures whose constituent's rest masses are not too disparate, so that it follows a system of 13r + 1 scalar field equations. By restricting to a binary mixture characterized by the six fields of partial particle number densities, four-velocity and temperature, the remainder 21 scalar equations are used to determine the constitutive equations for the non-equilibrium pressures, diffusion fluxes, pressure deviator tensors, and heat fluxes. Hence the Navier-Stokes and generalized Fourier and Fick laws are obtained and the transport coefficients of bulk and shear viscosities, thermal conductivity, diffusion, thermal-diffusion, and diffusion-thermal are determined. Analytic expressions for the transport coefficients in the non-relativistic and ultra-relativistic limiting cases are given. Furthermore, solutions of the relativistic field equations for the binary mixture are obtained in form of forced and free waves. In the low frequency limiting case the phase velocity and the attenuation coefficient are determined for forced waves. In the small wavenumber limiting case it is shown that there exist four longitudinal eigenmodes, two of them corresponding to propagating sound modes and two associated with non-propagating diffusive modes.