Explosive instability of the surface of a liquid dielectric in a vertical electric field in confined geometry

Nonlinear dynamics of the free boundary of an ideal dielectric liquid in a vertical electric field is studied in a confined axisymmetric geometry. We derive an amplitude equation which describes the behavior of a fluid surface in the weakly nonlinear approximation. It is demonstrated that, due to the effect of quadratic nonlinearities, the regime of excitation of the surface electrohydrodynamic instability is always hard. It is also shown that cubic nonlinearities suppress the instability development for liquids with relatively small permittivity. On the contrary, the nonlinearity begins to play a destabilizing role for high values of dielectric constant. As a consequence, the instability development takes an explosive character: the amplitude of boundary perturbations grows unlimitedly over a finite time. Using an analogy with the motion of an effective Newtonian particle in a certain potential, we formulate conditions for explosive buildup of perturbations. Differences in the behavior between axisymmetric perturbations of the free surface and the cases of hexagonal, square, and planar symmetries are discussed.