Macroscopic descriptions of rarefied gases from the elimination of fast variables

Abstract

The Boltzmann equation describing a dilute monatomic gas is equivalent to an infinite hierarchy of evolution equations for successive moments of the distribution function. The five moments giving the macroscopic mass, momentum, and energy densities are unaffected by collisions between atoms, while all other moments naturally evolve on a fast collisional time scale. We show that the macroscopic equations of Chen, Rao, and Spiegel [Phys. Lett. A 271, 87 (2000)], like the familiar Navier-Stokes-Fourier equations, emerge from using a systematic procedure to eliminate the higher moments, leaving closed evolution equations for the five moments unaffected by collisions. The two equation sets differ through their treatment of contributions from the temperature to the momentum and energy fluxes. Using moment equations offers a definitive treatment of the Prandtl number problem using model collision operators, greatly reduces the labor of deriving equations for different collision operators, and clarifies the role of solvability conditions applied to the distribution function. The original Chen-Rao-Spiegel approach offers greatly improved agreement with experiments for the phase speed of ultrasound, but when corrected to match the Navier-Stokes-Fourier equations at low frequencies, it then underestimates the phase speed at high frequencies. Our introduction of a translational temperature, as in the kinetic theory of polyatomic gases, motivates a distinction in the energy flux between advection of internal energy and the work done by the pressure. Exploiting this distinction yields macroscopic equations that offer further improvement in agreement with experimental data, and arise more naturally as an approximation to the infinite hierarchy of evolution equations for moments.