×

Hollow vortices, capillary water waves and double quadrature domains. (English) Zbl 1307.76014

Summary: Two new classes of analytical solutions for hollow vortex equilibria are presented. One class involves a central hollow vortex, comprising a constant pressure region having non-zero circulation, surrounded by an \(n\)-polygonal array of point vortices with \(n\geqslant 2\). The solutions generalize the non-rotating polygonal point vortex configurations of G. K. Morikawa and E. V. Swenson [“Interacting motion of rectilinear geostrophic vortices”, Phys. Fluids 14, 1058–1073 (1971)] to the case where the point vortex at the centre of the polygon is replaced by a hollow vortex. The results of Morikawa and Swenson would suggest that all equilibria for \(n\neq 3\) will be linearly unstable to point vortex mode instabilities. However even the \(n = 3\) case turns out to be unstable to a recently discovered displacement instability deriving from a resonance between the natural modes of an isolated circular hollow vortex. A second class of analytical solutions for periodic water waves co-travelling with a submerged point vortex row is also described. The analysis gives rise to new theoretical connections with free surface Euler flows with surface tension and, in particular, with Crapper’s classical solutions for capillary water waves. It is pointed out that the equilibrium fluid regions found here have a mathematical interpretation as an abstract class of planar domains known as double quadrature domains.

MSC:

76B47 Vortex flows for incompressible inviscid fluids
76B45 Capillarity (surface tension) for incompressible inviscid fluids
PDFBibTeX XMLCite
Full Text: DOI