Electrified cone formation in perfectly conducting viscous liquids: Self-similar growth irrespective of Reynolds number

Abstract

Above a critical field strength, the free surface of an electrified, perfectly conducting viscous liquid, such as a liquid metal, is known to develop an accelerating protrusion resembling a cusp with a conic tip. Field self-enhancement from tip sharpening is reported to generate divergent power law growth in finite time of the forces acting in that region. Previous studies have established that tip sharpening proceeds via a self-similar process in two distinguished limits—the Stokes regime and the inviscid regime. Using finite element simulations to track the shape and forces acting at the tip of an electrified protrusion in a perfectly conducting Newtonian liquid, we demonstrate that the conic tip always undergoes self-similar growth irrespective of the Reynolds number. The blowup exponents at the conic apex for all terms in the Navier-Stokes equation and the normal stress boundary condition at the moving interface reveal the dominant forces at play as the Reynolds number increases. Rescaling of the tip shape by the power law representing the divergence in capillary stress at the apex yields an excellent collapse onto a universal cone shape with an interior half-angle dependent on the Maxwell stress. The rapid acceleration of the liquid interface also generates a thin interfacial boundary layer characterized by a significant rate of strain. Additional details of the modeled flow, applicable to cone growth in systems such as liquid metal ion sources, help dispel prevailing misconceptions that dynamic cones resemble conventional Taylor cones or that viscous stresses at a finite Reynolds number can be neglected.