Singular Shape of a Fluid Drop in an Electric or Magnetic Field

TL;DR

This paper investigates the conditions under which dielectric or ferromagnetic fluid drops develop conical tips in electric or magnetic fields, analyzing the shape transition through force balance and energy considerations.

Contribution

It identifies the critical dielectric constant for conical tip formation and calculates the critical field strength for stable conical shapes, providing a theoretical framework for shape transitions.

Findings

Conical tips form only when dielectric constant exceeds 17.59.

Two possible conical angles exist, with one stable and one unstable.

Stable conical shapes are energetically favored at high fields.

Abstract

Beyond a threshold, electric or magnetic fields cause a dielectric or ferromagnetic fluid drop respectively to develop conical tips. We analyze the appearance of the conical tips and the associated shape transition of the drop using a local force balance as well as a global energy argument. We find that a conical interface is possible only when the dielectric constant (or permeability) of the fluid exceeds a critical value $\epsilon_c=17.59$. For a fluid with $\epsilon>\epsilon_c$, a conical interface is possible at two angles, one stable and one unstable. We calculate the critical field required to sustain a drop with stable conical tips. Such a drop is energetically favored at sufficiently high field. Our results also apply to the formation of conical dimples when a pool of fluid is placed in a normal field.