The role of three-dimensional shapes in the break-up of charged drops

Abstract

A nonlinear analysis of the non-axisymmetric shapes and oscillations of charged, conducting drops is carried out near the Rayleigh limit that gives the critical amount of charge for which the spherical equilibrium form loses stability. The Rayleigh limit is shown to correspond to a fivefold singular point with only axisymmetric spheroidal shapes bifurcating from the family of spheres. The oblate spheroids that exist for greater amounts of charge are unstable to non-axisymmetric disturbances, which control the evolution of drop break-up. The bifurcating prolate spheroids that exist for values of charge less than the Rayleigh limit are only unstable to axisymmetric perturbations that elongate the drop along its symmetry axis; hence, the initial stage of the droplet break-up is through a sequence of lengthening prolate shapes. An external uniform electric field or a rigid-body rotation of the drop breaks the symmetry of the spherical base shape and is an imperfection to the Rayleigh limit. Addition of an electric field leads to slightly prolate shapes that end at a limiting value of charge. Rigid rotation leads to slightly oblate forms that lose stability to triaxial shapes. For values of charge just less than the Rayleigh limit, the amplitude equations that are derived from a multiple timescale analysis are equivalent to the dynamical equations of the Hénon‒Heiles Hamiltonian. The remarkable and complicated properties of the bounded solutions to this set of equations are well known and reviewed briefly here.