Fontelos, M. & Friedman, A. Arch. Rational Mech. Anal. (2004) 172: 267. doi:10.1007/s00205-003-0298-x
It has been observed experimentally that an electrically charged spherical drop of a conducting fluid becomes nonspherical (in fact, a spheroid) when a dimensionless number X inversely proportional to the surface tension coefficient γ is larger than some critical value (i.e., when γ<γc). In this paper we prove that bifurcation branches of nonspherical shapes originate from each of a sequence of surface-tension coefficients ), where γ2=γc. We further prove that the spherical drop is stable for any γ>γ2, that is, the solution to the system of fluid equations coupled with the equation for the electrostatic potential created by the charged drop converges to the spherical solution as tρ∞ provided the initial drop is nearly spherical. We finally show that the part of the bifurcation branch at γ=γ2 which gives rise to oblate spheroids is linearly stable, whereas the part of the branch corresponding to prolate spheroids is linearly unstable.