Rotating equilibria of vortex sheets
Abstract
We consider relative equilibrium solutions of the twodimensional Euler equations in which the vorticity is concentrated on a union of finitelength vortex sheets. Using methods of complex analysis, more specifically the theory of the RiemannHilbert problem, a general approach is proposed to find such equilibria which consists of two steps: first, one finds a geometric configuration of vortex sheets ensuring that the corresponding circulation density is realvalued and also vanishes at all sheet endpoints such that the induced velocity field is welldefined; then, the circulation density is determined by evaluating a certain integral formula. As an illustration of this approach, we construct a family of rotating equilibria involving different numbers of straight vortex sheets rotating about a common center of rotation and with endpoints at the vertices of a regular polygon. This equilibrium generalizes the wellknown solution involving single rotating vortex sheet. With the geometry of the configuration specified analytically, the corresponding circulation densities are obtained in terms of a integral expression which in some cases lends itself to an explicit evaluation. It is argued that as the number of sheets in the equilibrium configuration increases to infinity, the equilibrium converges in a certain distributional sense to a hollow vortex bounded by a constantintensity vortex sheet, which is also a known equilibrium solution of the twodimensional Euler equations.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 February 2020
 DOI:
 10.1016/j.physd.2019.132286
 arXiv:
 arXiv:1906.03803
 Bibcode:
 2020PhyD..40332286P
 Keywords:

 Vortex dynamics;
 Relative equilibria;
 RiemannHilbert problem;
 Physics  Fluid Dynamics;
 Mathematics  Complex Variables
 EPrint:
 22 pages, 5 figures