Mathematical structure of transport equations for multispecies flows

Abstract

In this review we attempt to present a unified picture of transport in multispecies gas mixtures. To accomplish this task, it is necessary to outline the mathematical structure of the transport equations. Starting from Boltzmann's equation, we derive a general system of transport equations using an approach that is valid for flow situations in which there are large temperature and drift velocity differences between the interacting species. However, this system of equations, which is obtained by taking velocity moments of the Boltzmann equation, does not constitute a closed set, since the equation governing the velocity moment of order r contains the velocity moment of order r + 1. To close the system of transport equations, it is therefore necessary to adopt an approximate expression for the species velocity distribution function. For near‐equilibrium flows, various levels of approximation are considered, including the 5‐, 8‐, 10‐, 13‐, and 20‐moment approximations. The procedure for obtaining closed sets of transport equations for far‐from‐equilibrium flows is also discussed. When the transport equations are ordered with respect to the collisional mean free path, the result is the Euler, Navier‐Stokes, or extended Navier‐Stokes equations depending upon whether terms proportional to the zeroth, first, or second power of the mean free path are retained. For a collisionless plasma the analogous expansion using the Larmor radius yields the Chew‐Goldberger‐Low (CGL) equations to zeroth order and the extended CGL equations to first order.