Fourteen moment theory for granular gases

Abstract

A fourteen moment theory for a granular gas is developed within the framework of the Boltzmann equation where the full contracted moment of fourth order is added to the thirteen moments of mass density, velocity, pressure tensor and heat flux vector. The spatially homogeneous solutions of the fourteen moment theory implied into a time decay of the temperature field which follows closely Haff's law, besides the more accentuated time decays of the pressure deviator, heat flux vector and fourth moment. The requirement that the fourth moment remains constant in time inferred into its identification with the coefficient $a_2$ in the Chapman-Enskog solution of the Boltzmann equation. The laws of Navier-Stokes and Fourier are obtained by restricting to a five field theory and using a method akin to the Maxwellian procedure. The dependence of the heat flux vector on the gradient of the particle number density was obtained thanks to the inclusion of the forth moment. The analysis of the dynamic behavior of small local disturbances from the spatially homogeneous solutions caused by spontaneous internal fluctuations is performed by considering a thirteen field theory and it is shown that for the longitudinal disturbances there exist one hydrodynamic and four kinetic modes, while for the transverse disturbances one hydrodynamic and two kinetic modes are present.