Away from the <cylindrical vortex sheet>, the disturbance is both incompressible and irrotational, so its potential is harmonic. Regularity at the axis and decay at infinity select
$$
\phi_1=A r^m e^{\sigma t+im\theta},
\qquad
\phi_2=B r^{-m}e^{\sigma t+im\theta}
$$
for a positive integer $m$. The kinematic conditions give their values on the sheet:
$$
\phi_1(R_0)=\frac{R_0\sigma}{m}\xi,
\qquad
\phi_2(R_0)=-\frac{R_0}{m}
(\sigma+im\Omega_0)\xi.
$$
Substitution into the dynamical condition gives the <dispersion relation>
$$
\sigma^2+im\Omega_0\sigma
+\frac{m(1-m)}2\Omega_0^2=0.
$$
Thus
$$
\boxed{
\sigma=\frac{\Omega_0}{2}
\left[-im\pm\sqrt{m^2-2m}\right]}.
$$
For every $m\geq3$, one root has positive real part. The cylindrical sheet is therefore subject to a <Kelvin-Helmholtz instability>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 331 / 1 / iii / c / Solution

A cylindrical vortex sheet is a vortex sheet supported on a cylindrical surface. Azimuthal Fourier disturbances have harmonic velocity potentials proportional to $r^m$ inside and $r^{-m}$ outside.


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 Incoming links: Cylindrical vortex sheet