six linearly independent equations for the six unknowns $U^{\mu}$, $e'$, and $\rho'$.

In addition to the energy density of the radiation, $e'$, one can also calculate the number density of photons by requiring that the number flux, $F^{\mu}$, be conserved. One can show [@cou14b] that the equation $\nabla_{\mu}F^{\mu} = 0$ becomes, in the viscous limit,

$$\begin{gathered}
\nabla_{\mu}\bigg{[}N'U^{\mu}\bigg{]} \\ 
= \nabla_{\mu}\bigg{[}\frac{1}{\rho'\kappa}\bigg{(}\frac{10}{9}N'U^{\mu}\nabla_{\sigma}U^{\sigma}+N'U^{\alpha}\nabla_{\alpha}U^{\mu}+\frac{1}{3}\Pi^{\mu\sigma}\nabla_{\sigma}N'\bigg{)}\bigg{]} \label{fluxeq},\end{gathered}$$

where $N'$ is the rest-frame number density of photons. Once we solve the equations of radiation hydrodynamics for the four-velocity of the fluid and the mass density of scatterers, we can solve equation for the number flux of photons.

The goal of the next two sections is to apply equations and – to the boundary layers established between fast-moving jets and their ambient media. For a more thorough discussion of the nature of the equations of radiation hydrodynamics in the viscous limit, we refer the reader to @cou14b.

Jetted boundary layer
=====================

In this section we consider the problem where a narrow stream of material is continuously injected into a plane-parallel, static, ambient medium, known as the free-streaming jet problem. If the Reynolds number of the outflow is high, the transition between the stream of material and the external 