Translating hollow vortex pairs

A new derivation, and representation, of the classical solution for a translating hollow vortex pair originally discovered by Pocklington in 1895 [H.C. Pocklington, The configuration of a pair of equal and opposite hollow straight vortices of finite cross-section, moving steadily through fluid, Proc. Cambridge Philos. Soc. 8 (1895) 178–187] is presented. The derivation makes use of a special function known as the Schottky–Klein prime function. The new representation of the hollow vortex pair facilitates an investigation of the linear stability properties of this configuration, something apparently not previously studied in the literature. We show here that the hollow vortex pair is linearly stable to infinitesimal irrotational perturbations provided the area of the two vortices is below a critical value.

► Pocklington’s solution for the travelling hollow vortex pair is rederived.
► A one-parameter family of hollow vortex pairs exists.
► The hollow vortex pair is unstable when its area is large enough.