A single-fluid, self-consistent formulation of particle transport and fluid dynamics
I present a formulation of fluid dynamics that is consistent with particle transport and acceleration. This formulation consists of two parts: a transport equation that describes the evolution of a particle distribution function in terms of a fluid velocity in which the distribution is embedded, and an equation for the fluid velocity that involves integrals of the distribution function. The motivation of this work is to provide a formalism for calculating the effect of particle acceleration on the flows of typical astrophysical plasmas.
It is shown that the equation to be solved simultaneously with the transport equation is just the momentum equation for the system, and that the number and energy equations are implicit in the transport equation. There is no restriction on the energies of particles constituting such systems. Connections are made to the cosmic-ray transport equation, two-fluid models of cosmic-ray - thermal gas interaction, and self-consistent Monte Carlo models of particle acceleration at parallel shocks.
The formalism is developed for non-relativistic flow speeds. It is assumed that particle distributions are nearly isotropic in the fluid frame, an assumption that is generally valid in space plasmas. It is assumed that particle scattering mean-free-paths are much less than the length scales associated with changes in the fluid velocity or particle distribution.