Steady ellipsoidal vortex rings with finite cores

The present paper provides explicit numerical descriptions of vortex rings that are exact solutions to the Navier-Stokes equations inside of the chosen ellipsoidal boundaries. It is a particular feature of the resulting rings that they have finite cores within which the vorticity is nonzero whereas outside the cores no vorticity exists. The vorticity distribution is in agreement with the Navier-Stokes equations and yield the shape of the vortex ring core as a result of the described predictions. The theoretical approach implies that the shape of the resulting rings does not change as the rings propagate steadily through an inviscid and unbounded fluid. Hill's spherical vortex and O'Brien's ellipsoidal vortices are limiting cases of the vortex rings described in this paper. Analytical expressions for their properties are provided and results computed from these expressions are compared with values numerically obtained for the range of vortex rings described by the authors. Further comparisons are made with experimental results by Sullivan, Widnall and Ezekiel.