Quantisation of free-surface waves in a hollow vortex with surface tension

Equilibrium shapes of hollow vortices with surface tension in a corner geometry are obtained by solving a free-boundary problem. Using the integral hodograph method, we derive the complex velocity potential in an auxiliary parameter plane, which includes the velocity magnitude along the free surface. A singular integral equation for the velocity magnitude is obtained by applying the dynamic boundary condition. Numerical solutions to this equation reveal a wave quantisation phenomenon on the boundary of the hollow vortex due to the surface tension. The number of waves allocated on the free surface is arbitrary, starting from some minimal value depending on the strain-to-circulation ratio, the corner angle and the surface tension. In the limiting case of zero surface tension, the solution is obtained analytically and shown to agree with previous studies based on alternative mathematical formulations. These findings provide the first known equilibrium configurations of hollow vortices with surface tension in the presence of solid boundaries.