Generalized Fluid Models of the Braginskii Type. Part 2. The Boltzmann Operator

Abstract

In our previous paper (Hunana et al. 2022) we have employed the Landau collisional operator together with the moment method of Grad and considered various generalizations of the Braginskii model, such as a multi-fluid formulation of the 21- and 22-moment models valid for general masses and temperatures, where all of the considered moments are described by their evolution equations (with fully non-linear left-hand-sides). Here we consider the same models, however, we employ the Boltzmann operator and calculate the collisional contributions via expressing them through the Chapman-Cowling collisional integrals. These ``integrals'' just represent a useful mathematical technique/notation introduced roughly 100 years ago, which (in the usual semi-linear approximation) allows one to postpone specifying the particular collisional process and finish all of the calculations with the Boltzmann operator. We thus consider multi-fluid 21- and 22-moment models which are valid for a large class of elastic collisional processes describable by the Boltzmann operator. Reduction into the 13-moment approximation recovers the models of Schunk and Burgers. We only focus on the particular cases of hard spheres, Coulomb collisions, purely repulsive inverse power force $|K|/r^\nu$ and attractive force $-|K|/r^\nu$ with repulsive rigid core (or potential $V(r)=\delta(r)- |c|/r^n$, so that the particles bounce from each other when they meet), but other cases can be found in the literature. In the Appendix, we introduce the Boltzmann operator in a way suitable for newcomers and we discuss a surprisingly simple recipe how to calculate the collisional contributions with analytic software.