Axisymmetric shapes and stability of isolated charged drops

Axisymmetric equilibrium shapes and stability of isolated charged drops are found by solving simultaneously the Young-Laplace equation for surface shape and the Laplace equation for the electric field. Families of two-, three-, and four-lobed shapes that branch from the trunk family of spheres are treated systematically by means of the Galerkin/finite element method and a tessellation that deforms with the free surface. The results show that at the limit found by Rayleigh in 1882 the spherical family exchanges stability with a family of two-lobed shapes, a transcritically bifurcating family, one arm of which proves to consist of stable shapes. The results are reinforced by those of approximating the stable drop shapes as oblate spheroids. Thus oblate drops carrying charge in excess of the Rayleigh limit ought to be seen in experiments, though none have yet been reported.