Internal Energy Relaxation Processes and Bulk Viscosities in Fluids

Abstract

Internal energy relaxation processes in fluid models derived from the kinetic theory are revisited, as are related bulk viscosity coefficients. The apparition of bulk viscosity coefficients in relaxation regimes and the links with equilibrium one-temperature bulk viscosity coefficients are discussed. First, a two-temperature model with a single internal energy mode is investigated, then a two-temperature model with two internal energy modes and finally a state-to-state model for mixtures of gases. All these models lead to a unique physical interpretation of the apparition of bulk viscosity effects when relaxation characteristic times are smaller than fluid times. Monte Carlo numerical simulations of internal energy relaxation processes in model gases are then performed, and power spectrums of density fluctuations are computed. When the energy relaxation time is smaller than the fluid time, both the two temperature and the single-temperature model including bulk viscosity yield a satisfactory description. When the energy relaxation time is larger than the fluid time, however, only the two-temperature model is in agreement with Boltzmann equation. The quantum population of a He-H2 mixture is also simulated with detailed He-H2 cross sections, and the resulting bulk viscosity evaluated from the Green–Kubo formula is in agreement with the theory. The impact of bulk viscosity in fluid mechanics is also addressed, as well as various mathematical aspects of internal energy relaxation and Chapman–Enskog asymptotic expansion for a two-temperature fluid model.