Barotropic enthalpy function ($h(\rho )={\int }_{{\rho }_1}^{\rho }\frac{dp/d{\rho }^{\prime }}{{\rho }^{\prime }}d{\rho }^{\prime }$)

This barotropic pressure potential obeys $\nabla h=\nabla p/\rho$ for a barotropic fluid. It gives the pressure contribution to a steady Bernoulli equation. In isentropic thermodynamics it agrees with specific enthalpy up to a constant; for an externally maintained isothermal ideal gas it is $c_s^2\log \rho$ and is not the constant thermodynamic enthalpy at fixed temperature.