Singularity formation in electrified vortex sheets

This study investigates the effect of a tangential electric field on the curvature singularity encountered by evolving vortex sheets. Weakly nonlinear models are derived by using asymptotic expansions based on the Dirichlet–Neumann operators. A fully nonlinear boundary integral method based on Cauchy’s integral formula and an angle –arclength derivative formulation in a Lagrangian framework is proposed. Linear stability analysis reveals a transition from the unstable regime to a neutrally stable regime at a critical electric field strength. However, both weakly nonlinear models and fully nonlinear simulations suggest that the curvature singularity persists regardless of the value of the electric field strength. The analysis demonstrates that the singularity time obeys two different asymptotic rules. For subcritical values of the tangential electric field and small initial amplitudes, the singularity time scales with the logarithm of the inverse of the amplitude, and monotonically increases with the electric field strength. For supercritical values of the tangential electric field the singular time is asymptotically larger, follows an inverse power law with respect to the initial amplitude and reaches local maxima as the electric field strength varies. Fully nonlinear simulations of the underlying electrodynamic Euler equations confirm that the curvature singularity can be significantly delayed when the electric field strength takes some special values. In the limit of large electric field strengths the numerical simulations indicate that the time to singularity formation is proportional to the inverse square root of the electric field strength.