Singularities on charged viscous droplets

We study the evolution of charged droplets of a conducting viscous liquid. The flow is driven by electrostatic repulsion and capillarity. These droplets are known to be linearly unstable when the electric charge is above the Rayleigh critical value. Here we investigate the nonlinear evolution that develops after the linear regime. Using a boundary elements method, we find that a perturbed sphere with critical charge evolves into a fusiform shape with conical tips at time $t_0$, and that the velocity at the tips blows up as $(t_0-t)^\alpha$, with $\alpha$ close to -1/2. In the neighborhood of the singularity, the shape of the surface is self-similar, and the asymptotic angle of the tips is smaller than the opening angle in Taylor cones.