Vortex rings with swirl

Vortex rings with azimuthal vorticity $\zeta \propto r$, where $r$ is the distance from the axis of symmetry, have an unsteady contour dynamical formulation on the one hand and a family of steady solutions due to Norbury on the other hand. We consider the effect of adding swirl to such rings. Taking the swirl $w \propto r^{-1}$ maintains the contour dynamics formulation, but it becomes necessary to add a vortex sheet at the boundary of the rings. Steady and unsteady solutions are presented, and the relation of these results to previous work is discussed.