We study the evolution of a viscous fluid drop rotating about a fixed axis at constant angular velocity Ω or constant angular momentum L surrounded by another viscous fluid. The problem is considered in the limit of large Ekman number and small Reynolds number. The analysis is carried out by combining asymptotic analysis and full numerical simulation by means of the boundary element method. We pay special attention to the stability/instability of equilibrium shapes and the possible formation of singularities representing a change in the topology of the fluid domain. When the evolution is at constant Ω, depending on its value, drops can take the form of a flat film whose thickness goes to zero in finite time or an elongated filament that extends indefinitely. When evolution takes place at constant L, and axial symmetry is imposed, thin films surrounded by a toroidal rim can develop, but the film thickness does not vanish in finite time. When axial symmetry is not imposed and L is sufficiently large, drops break axial symmetry and, depending on the value of L, reach an equilibrium configuration with a 2-fold symmetry or break up into several drops with a 2 or 3-fold symmetry. The mechanism of breakup is also described.

We study the evolution of drops of a very viscous and conducting fluid under the influence of an external electric field. The drops may be neutral or may be charged with some amount of electric charge. If both the external electric field and total drop charge are sufficiently small, then prolate spherical shapes develop according to Taylor's observations. For sufficiently large charge and/or external field a self-similar conelike singularity develops in a mechanism different from Taylor's prediction. The opening semiangle of the cones both for uncharged and charged drops in a constant electric field is typically around 30°with a very slight dependence on the viscosity ratio and independence from both total charge and external field. We also discuss the structure of electric and velocity fields near the tip.

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 t0, and that the velocity at the tips blows up as (t0 − t) α , with α 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.One of the leading problems in fluid dynamics is the formation of singularities on charged masses of fluid. These problems are relevant in a variety of 1

We present analytical and numerical studies of the initial stage of the branching process based on an interface dynamics streamer model in the fully three-dimensional case. This model follows from fundamental considerations on charge production by impact ionization and balance laws, and leads to an equation for the evolution of the interface between ionized and nonionized regions. We compare some experimental patterns with the numerically simulated ones, and give an explicit expression for the growth rate of harmonic modes associated with the perturbation of a symmetrically expanding discharge. By means of full numerical simulation, the splitting and formation of characteristic treelike patterns of electric discharges is observed and described.

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