 Solution

For the polytropic equation of state $p=K{\rho }^{1+1/n}$, take $K>0$ and $n>0$ for the physical model. The specific enthalpy relative to vacuum isIts gradient obeys $\nabla Q={\rho }^{-1}\nabla p$, so the Euler equations becomeFor the material derivative $D_t={\partial }_t+\mathbf{u}\cdot \nabla$, $D_{t}Q=(Q/n\rho )D_t\rho$. The continuity equation thus becomesThe choice $Q=0$ at zero mass density fixes the additive convention in this equation. Although the momentum equation alone is unchanged by adding a constant to $Q$, the polytropic relation and enthalpy evolution must use the same vacuum reference.