= Narrow-gap centrifugal stability equation
{title2=$-u''-(k^2\kappa_{\rm ep}^2/\omega^2)u=-k^2u$}
For axisymmetric inviscid modes between impermeable cylinders in a thin annulus, let $A=2\Omega+r\Omega'$ and $\kappa_{\rm ep}^2=2\Omega A=r^{-3}(r^4\Omega^2)'$. The linear azimuthal equation gives $u_\phi=-iAu_r/\omega$ and the axial equation gives $u_z=k\pi/\omega$, where $\pi$ is the <kinematic pressure> perturbation. Thin-gap incompressibility implies $\pi=i\omega u_r'/k^2$. 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 $\Phi=-\kappa_{\rm ep}^2$, the negative of the usual <Rayleigh discriminant>.
Back to article page
