Centrifugal potential of cylindrical rotation ($\Psi =\Phi -{\int }_0^R{R}^{\prime }{\Omega }^2(R^{\prime })d{R}^{\prime }$)

For angular velocity depending only on cylindrical radius, the centrifugal acceleration is a gradient. Steady barotropic balance is $\nabla p/\rho =-\nabla \Psi$, hence specific enthalpy satisfies $h(\rho )+\Psi =\text{constant}$. On an invertible equation-of-state branch, mass density and pressure are functions of $\Psi$. Cylindrical rotation is essential to this scalar centrifugal-potential representation.