Before the derivation from a physical view, we'd like introduce the full compressible Naiver-Stokes(-Fourier) system in $\Omega \subset \R^d$ directly:
\begin{equation}
\label{NSF}
\begin{cases}
\partial_t \rho +\nabla \cdot (\rho u )=0, \\
\partial_t (\rho u)+\nabla \cdot (\rho u\otimes u ) = \rho g+ f+ \nabla \cdot \sigma,\\
\partial_t ( E)+\nabla \cdot ( Eu)+\nabla \cdot q=\rho g\cdot u+ f\cdot u+\rho h+\nabla\cdot (\sigma\cdot u )
\end{cases}
\end{equation} usually with rheological assumption:
\begin{equation}
\label{rheological}
\sigma =-pI+\tau,
\end{equation} and total energy
\begin{equation} \label{total energy}
E=\frac 12\rho |u|^2+\rho e.
\end{equation} The physical meanings of quantities in the system are listed below: