Stability of a Conducting Droplet under the Influence of Surface Tension and Electrostatic Forces

Abstract

The Lagrange Equations of Motion are written in generalized coordinates which describe small departures from the spherical equilibrium configuration of a conducting liquid droplet. It is initially assumed that the actual shape differs only very slightly from the equilibrium sphere. The equation representing the surface is, then, written as a series of surface zonal harmonics in which the coefficients are shown to be the normal coordinates of the droplet. The frequency of oscillation of the normal coordinates is shown to depend on the total charge on the droplet in such a manner that for all values of charge below a certain limit, the frequency is real. For all values of charge above a certain limit, the frequency is imaginary; and, thus, the droplet is unstable. This paper presents a detailed derivation of a result communicated by Rayleigh in 1882. The results of this communication have been widely quoted but, until now, this particular derivation has not appeared in the literature.

Linear oscillations of a drop in uniform alternating electric fields

“…A first-order perturbation model (weighted in terms of deformation amplitude) is necessary to describe this charge redistribution (Rayleigh, 1882;Hendricks and Schneider, 1963). When an external alternating electric field exists, direct interaction between the charge and the field is the primary effect on drop motion and oscillations.…”

A measuring system for the time variation of size and charge of a single spherical particle and its applications

“…The critical charge for the Coulombic fission is called the Rayleigh limit. For inviscid conductive liquids, the critical charge was theoretically predicted to be (Rayleigh, 1882, Hendricks andSchneider, 1963):…”

Electrically charged conducting drops revisited

“…Hence, the final result of the leading-order deviation in the constant surface potential of a deformed drop (3.28) is completely expressed in terms of the first-order drop shape deformations aj^. This might be why without rigorously considering the second-order problem, Hendricks and Schneider [2] could still obtain the correct final result.…”