their equation (49);]{} see also Paper I):

$$\begin{gathered}
\nabla_{\mu}\bigg{[}\bigg{\{}\rho'+\frac{4}{3}e'\bigg{(}1-\frac{10}{9}\frac{1}{\rho'\kappa}\nabla_{\alpha}U^{\alpha}\bigg{)}\bigg{\}}U^{\mu}U^{\nu}\bigg{]}+\frac{1}{3}g^{\mu\nu}\partial_{\mu}e' \\ 
-\frac{8}{27}\nabla_{\mu}\bigg{[}\frac{e'}{\rho'\kappa}\Pi^{\mu\sigma}\Pi^{\nu\beta}\bigg{(}\nabla_{\sigma}U_{\beta}+\nabla_{\beta}U_{\sigma}+g_{\beta\sigma}\nabla_{\alpha}U^{\alpha}\bigg{)}\bigg{]} \\ 
-\frac{1}{3}\nabla_{\mu}\bigg{[}\frac{e'}{\rho'\kappa}\bigg{(}\Pi^{\mu\sigma}U^{\nu}+\Pi^{\nu\sigma}U^{\mu}\bigg{)}\bigg{(}4U^{\beta}\nabla_{\beta}U_{\sigma}+\partial_{\sigma}\ln{}e'\bigg{)}\bigg{]} = 0 \label{radhydroco}.\end{gathered}$$

Here the speed of light has been set to one, Greek indices range from 0 – 3, $\rho'$ is the fluid rest frame mass density of scatterers, $e'$ is the fluid rest frame radiation energy density, $\kappa$ is the scattering opacity (in units of cm$^2$ g$^{-1}$), $g_{\mu\nu}$ is the metric of the spacetime, $\nabla_{\mu}$ is the covariant derivative, $U^{\mu}$ is the four-velocity of the flow, and $\Pi^{\mu\nu} = U^{\mu}U^{\nu}+g^{\mu\nu}$ is the projection tensor. The Einstein summation convention has been adopted here, meaning that repeated upper and lower indices imply summation. This equation also shows, in agreement with previous findings [@bla85; @loe92], that the coefficient of dynamic viscosity, $\eta$, for an optically-thick, radiation-dominated gas is

$$\eta = \frac{8}{27}\frac{e'}{\rho'\kappa} \label{etaeq}.$$

We will also write down the gas energy equation, obtained by contracting equation with the four-velocity $U_{\nu}$, which gives [(see equation (50) of @cou14b)]{}

$$\begin{gathered}
\nabla_{\mu}(e'U^{\mu})+\frac{1}{3}e'\nabla_{\mu}U^{\mu} = \frac{4}{3}\frac{10}{9}\nabla_{\mu}\bigg{[}\frac{e'}{\rho'\kappa}U^{\mu}\nabla_{\alpha}U^{\alpha}\bigg{]} \\ 
+\frac{8}{27}\frac{e'}{\rho'\kappa}\bigg{(}\nabla_{\sigma}U_{\beta}+\nabla_{\beta}U_{\sigma}+g_{\sigma\beta}\nabla_{\alpha}U^{\alpha}\bigg{)}\Pi^{\mu\sigma}\nabla_{\mu}U^{\beta} \\
+\frac{1}{3}\Pi^{\mu\sigma}\nabla_{\mu}\bigg{[}\frac{e'}{\rho'\kappa}\bigg{(}4U^{\beta}\nabla_{\beta}U_{\sigma}+\partial_{\sigma}\ln{e'}\bigg{)}\bigg{]} \\ 
+\frac{1}{3}\frac{e'}{\rho'\kappa}\bigg{(}4U^{\beta}\nabla_{\beta}U_{\sigma}+\partial_{\sigma}\ln{e'}\bigg{)}\bigg{(}2U^{\mu}\nabla_{\mu}U^{\sigma}+U^{\sigma}\nabla_{\mu}U^{\mu}\bigg{)}
\label{gasenergyco}.\end{gathered}$$

To close the system, we require that the normalization of the four-velocity be upheld and that particle flux be conserved:

$$U_{\mu}U^{\mu} = -1,$$

$$\nabla_{\mu}\bigg{[}\rho'U^{\mu}\bigg{]} = 0 \label{masscont}.$$

Equations and – constitute 