Numerical simulation of the shape of charged drops over a solid surface.

In this work we study the static shape of charged drops of a conducting fluid placed over a solid substrate, surrounded by a gas, and in absence of gravitational forces. The problem can be posed, since Gauss, in a variational setting consisting of obtaining the configurations of a given mass of fluid that minimize (or in general make extremal) a certain energy involving the areas of the solid-liquid interface and of the liquid-gas interface, as well as the electric capacity of the drop. In [6] we have found, as a function of two parameters, Young's angle θY and the potential at the drop's surface V0, the axisymmetric minimizers of the energy. In the same article we have also described their shape and showed the existence of symmetry-breaking bifurcations such that, for given values of θY and V0, configurations for which the axial symmetry is lost are energetically more favorable than axially symmetric configurations. We have proved the existence of such bifurcations in the limits of very flat and almost spherical equilibrium shapes. In this work we study all other cases numerically. When dealing with radially perturbed equilibrium shapes we lose the axially symmetric properties and need to do a full three-dimensional approximation in order to compute area and capacity and hence the energy. We use a boundary element method that we have already implemented in [3] to compute the surface charge density. From the surface charge density we can obtain the capacity of the body. One conclusion of this study is that axisymmetric drops cannot spread indefinitely by introducing sufficient amount of electric charges, but can reach only a limiting (saturation) size, after which the axial symmetry would be lost and finger-like shapes energetically preferre

Thermal rupture of a free liquid sheet

We consider a free liquid sheet, taking into account the dependence of surface tension on temperature, or concentration of some pollutant. The sheet dynamics are described within a long-wavelength description. In the presence of viscosity, local thinning of the sheet is driven by a strong temperature gradient across the pinch region, resembling a shock. As a result, for long times the sheet thins exponentially, leading to breakup. We describe the quasi one-dimensional thickness, velocity, and temperature profiles in the pinch region in terms of similarity solutions, which posses a universal structure. Our analytical description agrees quantitatively with numerical simulations

Time decay of scaling invariant Schroedinger equations on the plane

We prove the sharp L^1-L^{\infty} time-decay estimate for the 2D-Schroedinger equation with a general family of scaling critical electromagnetic potentials.Comment: 26 page

Evolution of neutral and charged droplets in an electric field

En el artículo se estudia la evolución de gotas de fluido conductor y muy viscoso sometidas a la influencia de un campo eléctrico. Las gotas pueden estar inicialmente cargadas o descargadas. Si tanto el campo eléctrico como la carga inicial son suficientemente pequeños las gotas adquieren forma de esferoides prolatos siguiendo las observaciones de Taylor. Por el contrario, si el campo eléctrico y/o la carga inicial de la gota son suficientemente grandes aparecen singularidades de tipo cónico mediante un mecanismo distinto al predicho por Taylor. El semiángulo de apertura de las puntas cónicas para gotas (inicialmente cargadas o descargadas) sometidas a un campo eléctrico constante tiene un valor aproximado de 30º. Este ángulo es independiente de la carga total y del campo eléctrico y tiene una pequeña dependencia de la relación de viscosidades entre el fluido externo e interno de la gota. En el trabajo también se analiza la estructura del campo eléctrico y el campo de velocidades en las proximidades de las puntas