 Narrow-gap centrifugal stability equation

For axisymmetric inviscid modes between impermeable cylinders in a thin annulus, let $A=2\Omega +r{\Omega }^{\prime }$ and ${\kappa }_{\mathrm{ep}}^2=2\Omega A=r^{-3}(r^4{\Omega }^2{)}^{\prime }$. The linear azimuthal equation gives $u_{\varphi }=-iA{u}_r/\omega$ and the axial equation gives $u_z=k\pi /\omega$, where $\pi$ is the kinematic pressure perturbation. Thin-gap incompressibility implies $\pi =i\omega u_r^{\prime }/k^2$. Eliminating it from the radial equation gives the title equation with $u(R_{<})=u(R_{>})=0$. If a plus sign is used for the potential, its coefficient must be defined as $\Phi =-{\kappa }_{\mathrm{ep}}^2$, the negative of the usual Rayleigh discriminant.