Dependence of Time-periodic Vortex Sheets with Surface Tension on Mean Vortex Sheet Strength

In recent work, the authors have computed time-periodic solutions of the vortex sheet with surface tension in the spatially periodic setting. In these prior results, the mean vortex sheet strength was taken to be zero, so that there is no net shear across the fluid interface. In the current work, we explore the effect of allowing such a net shear, finding time-periodic vortex sheets that overturn without rolling up. We also find that resonances between Fourier modes lead to disconnections in the bifurcation curves that describe a two-parameter family of time-periodic solutions.