Effects of viscosity on capillary wave instabilities of a planar liquid-metal surface in an electric field

Abstract

An electrohydrodynamic surface capillary wave theory has previously been developed for ion and droplet formation in electrically stressed conducting viscous fluids. In this paper the formalism has been used to derive the dispersion relation for the simplest model, a planar liquid-metal ion source, which includes both gravitational and viscous effects. The calculation required the simultaneous solution of the linearized Navier–Stokes equation, the Maxwell equation, and the time-dependent Laplace–Young stress condition. The critical field necessary for the onset of instability of the fluid surface is obtained, as well as its dependence on surface tension, viscosity, and gravity. In addition, the effects of viscosity and gravity on the growth and decay rates of particle emission are investigated. It is found that, as a function of applied field, the effects of viscosity and gravity on the dominant capillary wave mode and growth rate range from a few percent for liquid Li to about 20%–40% for liquid Ga and Au, respectively. However, for the liquid metals Li, Al, Ga, Sn, and Au viscosity dominates gravity: a factor of 3 larger for Li to more than an order of magnitude greater for the heavier liquid metals. Finally, it is found that the field above a model protrusion on the perturbed surface is significantly larger than the average field over the liquid surface.