NONEQUILIBRIUM THERMODYNAMICS AND DISSIPATIVE FLUID THEORIES .2. VISCOUS FLOWS IN RAREFIED-GASES

The first paper in this series investigated, from the mathematical poi nt of view, several aspects of the thermodynamic fluid theories (of di vergence type). The object of this second paper is to present a simple example of such a theory. Here the Grad moment method is applied to a classical Boltzmann equation to obtain a determined system of the qua si-linear, first-order partial differential equations for the evaluati on of the usual hydrodynamic variables and the stress deviator. As dem onstrated already by Loose and Hess (Phys. Rev. A, 37, 2099 (1988)), t hese equations provide valuable information about the shear-rate depen dence of the viscosity coefficients and on other non-Newtonian propert ies of the pressure tensor. Accidentally, for the above-mentioned choi ce of independent gas-state variables, the truncation scheme of Grad n ot only leads to a symmetric hyperbolic system of differential equatio ns, but also is in complete agreement with the variational approach to Maxwell's equations of transfer. Thus the proposed method enables one to obtain the supplementary balance law, interpreted as the equation of balance of entropy, satisfied by a certain function h of the origin al variables. This function, which in the present case can be calculat ed explicitly, is referred to as the specific entropy (per unit mass). The resulting non-linear expression for h is investigated with a view to a deeper understanding of a status of the extended Gibbs relation. Due to the existence of this relation, one easily arrives at the natu ral definitions of temperature, pressure, and thermodynamic potentials for gaseous systems <<not infinitesimally near to equilibrium>>.