White, Cyan; McDonald, Robb; (2025) Exact solutions for vortex equilibria by conformal mapping. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 10.1098/rspa.2025.0362. (In press).

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Abstract

Conformal mapping is used to find exact, closed-form solutions for three classes of vortex sheet rotating equilibria. The first involves multi-sheet equilibria of the Protas-Sakajo class: N-fold symmetric equilibria consisting of multiple sheets stemming from a common origin. Conformal mapping of the exterior of the vortex structure to the exterior of the unit disk enables the solution construction using Fourier series. The solutions describe both the stream function field and the circulation density along the sheets and are found for N = 2, 3, and 4. The approach is effective in reproducing equilibria of a second class due to O’Neil: a single, straight sheet in the presence of one or more point vortices. Finally, the method is used to construct new equilibrium families sharing features of both Protas-Sakajo and O’Neil classes. That is, a N = 4 Protas-Sakajo equilibria together with four point vortices located unit distance from the origin either (i) off each sheet tip, or (ii), on the bisector of the sheets. Members of each family are determined by a parameter γ measuring the total circulation of the sheets. For given γ, equilbria properties are determined numerical solution of a nonlinear algebraic equation. In case (ii), a non-rotating stationary equilibria is found.

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