Continuum description of rarefied gas dynamics. I. Derivation from kinetic theory

We describe an asymptotic procedure for deriving continuum equations from the kinetic theory of a simple gas. As in the works of Hilbert, of Chapman, and of Enskog, we expand in the mean flight time of the constituent particles of the gas, but we do not adopt the Chapman-Enskog device of simplifying the formulas at each order by using results from previous orders. In this way, we are able to derive a new set of fluid dynamical equations from kinetic theory, as we illustrate here for the relaxation model for monatomic gases. We obtain a stress tensor that contains a dynamical pressure term (or bulk viscosity) that is process dependent and our heat current depends on the gradients of both temperature and density. On account of these features, the equations apply to a greater range of Knudsen number (the ratio of mean free path to macroscopic scale) than do the Navier-Stokes equations, as we see in the accompanying paper. In the limit of vanishing Knudsen number, our equations reduce to the usual Navier-Stokes equations with no bulk viscosity.