Radiation-Hydrodynamics

Principal Governing Equations

First, referencing §2 of J. C. Hayes et al. (2006, ApJS, 165, 188 - 228) — see also, D. C. Marcello & J. E. Tohline (2012, ApJS, 199, id. 35, 29 pp) — we see that the set of principal governing equations that is typically used in the astrophysics community to include the effects of radiation on self-gravitating fluid flows includes the,

Poisson Equation

<math>\nabla^2 \Phi = 4\pi G \rho</math>

the,

Continuity Equation

<math>\frac{d\rho}{dt} + \rho \nabla \cdot \vec{v} = 0</math>

and — ignoring magnetic fields — a modified version of the,

Lagrangian Representation
of the Euler Equation,

<math>~ - \frac{1}{\rho}\nabla P - \nabla \Phi + \frac{1}{\rho}\biggl(\frac{\chi}{c}\biggr) \vec{F} \, , </math>

plus the following pair of additional energy-conservation-based dynamical equations:

<math>~\frac{d}{dt} \biggl( \frac{e}{\rho}\biggr)</math>	<math>~=</math>	<math>~ - \frac{P}{\rho}\nabla \cdot \vec{v} + \frac{1}{\rho} \biggl[c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p\biggr] \, , </math>
<math>~\frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>	<math>~=</math>	<math>~ - \frac{1}{\rho}\nabla \cdot \vec{F} - \vec{\bold{P}}:\nabla{\vec{v}} - \frac{1}{\rho} \biggl[c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p\biggr] \, . </math>