Dynamic Point-Formation in Dielectric Fluids

We use boundary-integral methods to compute the time-dependent deformation of a drop of dielectric fluid immersed in another dielectric fluid in a uniform electric field E. Steady state theory predicts, when the permittivity ratio, \beta, is large enough, a conical interface can exist at two cone angles, with \theta_<(\beta) stable and \theta_>(\beta) unstable. Our numerical evidence instead shows a dynamical process which produces a cone-formation and a transient finite-time singularity, when E and \beta are above their critical values. Based on a scaling analysis of the electric stress and the fluid motion, we are able to apply approximate boundary conditions to compute the evolution of the tip region. We find in our non-equilibrium case where the electric stress is substantially larger than the surface tension, the ratio of the electric stress to the surface tension in the newly-grown cone region can converge to a \beta dependent value, \alpha_c(\beta)>1, while the cone angle converges to \theta_<(\beta). This new dynamical solution is self-similar.