 Hydrodynamic transfield equation for an axisymmetric wind

Using an axisymmetric hydrodynamic mass flux function, set $s=s(\psi )$, $\ell =R{u}_{\varphi }=\ell (\psi )$ and $\epsilon =\epsilon (\psi )$. The azimuthal vorticity is ${\omega }_{\varphi }=-R\nabla \cdot (\nabla \psi /(\rho R^2))$. The meridional projection of the Crocco form of the unsteady Euler equation gives the displayed equation. The mass density also satisfies the Bernoulli relation $\epsilon =|\nabla \psi {|}^2/(2{\rho }^2{R}^2)+{\ell }^2/(2R^2)+w+\Phi$, so an equation of state is part of closing the system.