 Polytropic enthalpy

For $p=K{\rho }^{1+1/n}$ with $K,n>0$, the specific enthalpy relative to vacuum is $Q=(n+1)K{\rho }^{1/n}=(n+1)p/\rho$. It obeys $\nabla Q=\nabla p/\rho$ and $n{D}_{t}Q=-Q\nabla \cdot \mathbf{u}$. The vacuum reference is fixed when these equations are combined with the polytropic equation of state.