# Electrified Cone Formation in Perfectly Conducting Viscous Liquids: Self-Similar Growth Irrespective of Reynolds Number

## TL;DR

This study demonstrates that the self-similar growth of electrified conic tips in perfectly conducting viscous liquids occurs regardless of Reynolds number, revealing universal shape scaling and force dynamics through finite element simulations.

## Contribution

It shows that conic tip growth is always self-similar across all Reynolds numbers, challenging previous assumptions about the influence of viscous and inertial forces.

## Key findings

- Self-similar tip growth is universal across Reynolds numbers.
- Universal conic shape with a half-angle dependent on Maxwell stress.
- Viscous stresses at finite Reynolds number cannot be neglected.

## 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 distinct limits - the Stokes regime at $\textsf{Re}=0$ and the inviscid regime $\textsf{Re} \to \infty$. Using finite element simulations to track the acceleration of an electrified protrusion in a perfectly conducting Newtonian liquid in vacuum held at constant capillary number, we demonstrate that the conic tip \textit{always} undergoes self-similar growth irrespective of Reynolds number. The computed blow up exponents at the tip for the terms in the Navier-Stokes equation and interface normal stress condition reveal the different forces at play as $\textsf{Re}$ increases. Rescaling of the tip shape by the capillary stress exponent yields excellent collapse onto a universal conic tip shape with interior half-angle dependent on the magnitude of the Maxwell stress. The rapid acceleration of the liquid interface also generates a thin surface boundary layer with very high local strain rate. 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 finite $\textsf{Re}$ can be neglected.