The Navier-Stokes equations written in the conservative form

Continuity: \begin{equation} \frac{ \partial \rho }{ \partial t } + \nabla \cdot \rho \boldsymbol{v} = 0 \end{equation}

Momentum: \begin{equation}
\frac{ \partial \rho \boldsymbol{v} }{ \partial t }
+ \nabla \cdot ( \rho \boldsymbol{v} \otimes \boldsymbol{v} )
=~-  \nabla p
+ \nabla \cdot \sigma
\end{equation}

Energy: \begin{equation}
\frac{ \partial \rho e_t }{ \partial t }
+ \nabla \cdot \rho h_t \boldsymbol{v}
= \nabla \cdot ( \sigma  \boldsymbol{v} ) ~- \nabla \cdot \boldsymbol{q}
\end{equation}

Viscous stress and heat flux: \begin{equation}
\sigma = \mu [\nabla \boldsymbol{v} +(\nabla \boldsymbol{v})^T] -\frac{2}{3} \mu (\nabla \cdot \boldsymbol{v}) I ~~,~~
\boldsymbol{q} = -\kappa \nabla T
\end{equation}

\begin{equation}
I =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\end{equation}