Surface-charge blowup in Melcher–Taylor electrohydrodynamics

We show by local analysis that the Melcher–Taylor (“leaky-dielectric”) electrohydrodynamic model supports a steady, planar singularity structure where surface-charge density blows up like |s|^-1/3sgn(s) as s→0, where s is an arc length coordinate. The scenario is that opposite charges are advected—by a self-induced electrohydrodynamic flow—to the singularity, where they “annihilate”. In the classical problem of a drop in an electric field, this blowup corresponds to an equatorial singularity that forms in the steady-symmetric solution, at sufficiently strong fields and for fluid pairs such that the drop polarizes anti-parallel to the external field. We use a numerical scheme (with singularity removal based on the local analysis) to study the onset of the singularity and variation of the blowup prefactor in parameter space, considering for simplicity the 2D problem of a quasi-circular drop. The numerical results are complemented by a “universal” asymptotic solution valid far past the singularity threshold. We shall briefly discuss the relation between this work and the surface-charge “shocks” previously observed by others in initial-value simulations; possible links to the various dynamical instabilities observed in strong-field experiments; and the probable generality of the singularity structure in electrohydrodynamics beyond the drop problem.