Self-Similar Blowup of Cuspidal Protrusions in an Electrified Liquid Metal on a Conducting Needle

Above a critical electric field strength, a flat layer of perfecting conducting liquid will overcome capillary leveling to form a cusp-like protrusion. The cusp tip undergoes rapid sharpening with accelerated enhancement of the local field strength. The extreme field conditions, which ultimately generate ion emission, form the basis of focused ion beam systems using liquid metals. An insightful asymptotic analysis by Zubarev (2001) has demonstrated that leading up to emission, acceleration of an axisymmetric tip emerging from an inviscid (ideal) liquid is governed by a self-similar process with a distinct blowup exponent defining the tip Maxwell pressure, capillary pressure and kinetic energy. Here we present results of high fidelity ALE-FEM simulations with axisymmetric geometry for a viscous liquid metal coating the surface of a slender needle near a circular apertured extractor electrode. We quantify evolution of the liquid structures resembling axial, coronal and wave-like protrusions. For the parameter range explored, the blowup exponents for the Maxell and capillary pressure tend to differ and even exceed the ideal limit due to the influence of needle tip curvature, liquid protrusion asymmetry and liquid viscosity.