In this talk, we discuss complex-variable methods for computing vortex equilibria. The first part of the talk focuses on conformal-mapping techniques for obtaining exact solutions describing rotating vortex sheet–point vortex equilibria. Results for both simply and doubly connected configurations will be presented and discussed.
The second part of the talk introduces a new numerical method that uses the AAA algorithm’s approximation of the Schwarz function to compute rotating vortex patch–point vortex equilibria and the time-dependent evolution of rotating vortex patches. Comparisons with results obtained using contour dynamics will also be presented and discussed.