1.3.2 Hydrodynamics equations including gravity

The Euler equations with gravity are

$${ \!\!\! \renewedcommand{arraystretch}{2.0} \begin{ar... ...dmath$\scriptscriptstyle v$}}} \right) & = & 0 \end{array}}$$ (23)

with the gravitational potential $\Phi$. The total energy $e_{\rm ikg}$ now contains a contribution due to the potential,

$$\textstyle e_{\rm ikg}= e_{\rm i}+ \frac{1}{2} \ensuremat... ... {\mbox{\boldmath$\scriptscriptstyle v$}}} + \Phi \enspace .$$ (24)

Note: the momentum equation now contains a true source term.