Statistical derivation of hydrodynamic equations of the grad type

V. A. Savchenko ; T. N. Khazanovich

Teoretičeskaâ i matematičeskaâ fizika, Tome 14 (1973) no. 3, pp. 388-399

Résumé

Zubarev's method is used to derive equations corresponding to Grad's ten-moment approximation for a fluid of structureless particles. The nonequilibrium state of the system is described by not only the ordinary hydrodynamic variables but also by the components of viscosity stress tensor. It is assumed that the correlation radius is small compared with the inhomogeneity scales of the macroscopic variables. Equations that are analogous to Grad's equations linearized with respect to the gradients are obtained under the assumption that the correlation time scales for the coordinate functions and the momenta are very different. Expressions are obtained for the times of shear and dilatational relaxation in terms of the time correlation functions. The shear relaxation time calculated by means of this expression for a rarefied gas coincides with Grad's result. If approximations are not introduced for the correlation functions, agreement is obtained with the result of the ordinary statistical approach. Thus, the introduction of fluxes to describe a macroscopic equilibrium state does not lead to new hydrodynamic equations, but it may be useful for various approximations.