An electrohydrostatic analysis of equilibrium shape and stability of stressed conducting fluids: Application to liquid metal ion sources

Abstract

An exact mathematical treatment of the problem of an electrically stressed fluid from zero field to the onset of instability gives rise to the nonlinear electrohydrodynamic equations which, in general, are not amenable to analytic solution. To make the problem more tractable, one considers two limiting regimes, the electrohydrostatic (EHS) and the electrohydrodynamic (EHD) limits. In the EHS case, the fields and the velocities are assumed to be small so that quasistatic equilibrium exists and the fluid surface is essentially at rest. In this paper we consider the electrohydrostatic analysis of the equilibrium shape and stability of the electrically stressed fluids. The current work reintroduces the EHS stability criterion due to Zeleny, as well as a new set of equations and numerical procedure for analyzing the stability of an axially symmetric fluid with an arbitrary shaped surface. These are contrasted with a stability criterion, introduced by Taylor, which it is argued, is only an equilibrium condition and not a proper criterion for analyzing the general stability of electrified fluids. The Taylor and Zeleny criteria are applied to fluid sources modeled as simple coordinate surfaces, such as the cone, the cusp, and the hyperboloid. These results lead to a new physical interpretation of the onset of fluid instability in the EHS limit. A set of partial differential equations is derived, whose solution describes the equilibrium shape of a conducting fluid as a function of the applied electric field. Numerical results are presented for the evolution of the equilibrium shapes of several liquid metals as a function of the applied voltage. Values of the critical or breakdown voltage are obtained from these results and found to be in good agreement with experiment. Finally, the EHS analysis indicates that a realistic and accurate treatment of the onset of instability requires fluid flow in a dynamical model.