Specific enthalpy ($h$)

Specific enthalpy is enthalpy per unit mass. For a simple compressible fluid of fixed composition, the first law of thermodynamics giveswhere $T$ is temperature, $s$ is specific entropy, $p$ is pressure, and $\rho$ is mass density. Along an isentropic flow, $dh=dp/\rho$. For a perfect gas with constant specific-heat ratio $\gamma$, $h=\gamma p/[(\gamma -1)\rho ]$.

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.