Symmetry-breaking bifurcations of charged drops

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 gamma is larger than some critical value (i.e., when gamma<gamma(c)). In this paper we prove that bifurcation branches of nonspherical shapes originate from each of a sequence of surface-tension coefficients ), where gamma(2)=gamma(c). We further prove that the spherical drop is stable for any gamma>gamma(2), 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 trhoinfinity provided the initial drop is nearly spherical. We finally show that the part of the bifurcation branch at gamma=gamma(2) which gives rise to oblate spheroids is linearly stable, whereas the part of the branch corresponding to prolate spheroids is linearly unstable.