Singularities in electrified vortex sheets

We investigate the effect of a tangential electric field on the curvature singularity of vortex sheets. Linearly the electric field provides a dispersive regularization of short waves - above a critical electric field strength we have neutral stability. We address the question of whether this physical regularization can prevent the classical curvature singularities in vortex sheets (cf. surface tension). We do this by studying weakly nonlinear models derived using asymptotic expansions based on the Dirichlet-Neumann operators, and direct computations based on boundary integral methods using angle-arclength coordinates and a Lagrangian framework. Both weakly nonlinear models and fully nonlinear simulations suggest that the curvature singularity persists regardless of the electric field strength. Comparisons between the models and simulations show that the former can perform exceptionally well in predicting the singularity times and structure. For subcritical electric fields and small initial amplitudes δ, the singular time scales with log(1/δ), while for electric fields above critical a different scale 1/δ is found. Fully nonlinear simulations confirm that the curvature singularity can be significantly delayed when the electric field strength takes some special values. As the electric field strength increases, and starting with initial conditions proportional to the traveling-wave linear eigenmodes, we find that the singular time decreases and scales like the inverse square root of the electric filed strength.