Fluid total-energy equation (${\partial }_{t}E+\nabla \cdot [(E+p)\mathbf{u}]=-\rho \mathbf{u}\cdot \nabla \Phi +Q$)

For an inviscid fluid with specific internal energy $e$, let $E=\rho (e+|\mathbf{u}{|}^2/2)$. In a prescribed potential $\Phi$, ${\partial }_{t}E+\nabla \cdot [(E+p)\mathbf{u}]=-\rho \mathbf{u}\cdot \nabla \Phi +Q$, where $Q$ is external heat supply. A static gravitational potential can instead be included in the conserved energy.