The point vortex has been used as a simple model for flows with circulation, and has been desingularized into vortex patches and vortex sheets. In this work, we investigate the steady states of a combination of these two formulations, the Sadovskii vortex. The Sadovskii vortex is a uniform patch of vorticity surrounded by a vortex sheet. Numerical continuation is used to follow families of solutions. In the limiting cases of the vortex patch and vortex sheet cases, we confirm previous research, and in the vortex patch case show new solutions.
In the regime where both sources of circulation exist, we show the relationship between the vortex patch and vortex sheet solution families. The more complicated vortex patch solution families lead to the simpler vortex sheet solution family due to a splitting of the vortex patch families at bifurcation points in the presence of the vortex sheet. The more circular elliptical family remains attached to the family with a single pinch off, and this family extends all the way to the pure vortex sheet solutions. More elongated families below this one also split at bifurcation points, and these families do not exist in the vortex sheet regime.
In the presence of surface tension, vortex patch shapes are deformed where the background straining flow is into the vortices. This leads to oscillations on the boundary, similar to those found by Tanveer (1986) for bubbles in Hele-Shaw cells. The pinch off cusp for the patch vortex becomes desingularized in the presence of surface tension, and no longer touches. The effects of surface tension on the vortex sheet and Sadovskii vortex shapes are shown to be similar but smaller.