Numerical simulations of electrostatically driven jets from nonviscous droplets

Effect of the Quadrupolar Trap Potential on the Rayleigh Instability and Breakup of a Levitated Charged Droplet

The experimental demonstration of Rayleigh instability that results in the breakup of a charged droplet, levitated in a quadrupole trap, has been investigated in the literature, but only scarcely. We report here the asymmetric breakup of a charged drop, levitated in a loose trap, wherein the droplet is stabilized at an off-center location in the trap. This aspect of levitation leads to an asymmetric breakup of the charged drop, predominantly in a direction opposite to that of gravity. In the present work, we report the evidence of successive events of the deformation and breakup of a charged drop and its subsequent relaxation after jet ejection using high-speed imaging at a couple of hundred thousand frames per second. Several relevant aspects of this phenomenon such as the effect of the electrodynamic (ED) trap parameters in terms of the applied potential as well as physical parameters such as the size of the drop, gravity, and conductivity on the characteristics of droplet breakup are explored. A clear effect of the trap strength on the deformation (both symmetric and asymmetric) is observed. Moreover, the cone angle at the pole undergoing asymmetric breakup is almost independent of the applied field investigated in the experiments. All of the experimental observations are compared with numerical simulations carried out using the boundary element method (BEM) in the Stokes flow limit. The BEM simulations are also extended to other experimentally achievable parameters. It is observed that the breakup in our study is mostly field-influenced and not field-induced. A plausible theory for the observations is reported, and a sensitive role of the sign of the charge on the droplet and the sign of the end-cap potential, as well as the off-center location of the droplet in the trap, is elucidated.

Numerical study on electrohydrodynamic multiple droplet interactions

We present a numerical study of inviscid multiple droplet coalescence and break-up under the action of electric forces. Using an embedded potential flow model for the droplet hydrodynamics, coupled with an unbounded exterior electrostatic problem, we are able to perform computations through various singular events and analyze the effects of the electrical field intensity on droplet interactions. Laboratory experiments on the electrodynamics of droplet pairs show a much richer, and sometimes unexpected, behavior than that of isolated droplets. For example, it has been found that opposite charged droplets tend to repel each other when the electric field intensity is above a certain critical value. Although the mathematical model employed in this work incorporates very simple flow and electric assumptions, many of the droplet coalescence patterns seen in laboratory experiments can be reproduced. In this model, the interaction pattern of two droplets of radii R_{0} separated a distance D_{0}, depends on the ratio X_{0}=D_{0}/R_{0} and the applied uniform electric field intensity, E_{∞}. By performing a vast number of numerical simulations we are able to characterize the coalescence modes before and after drop merging as a function of these two parameters. The simulations predict that droplet repulsion occurs within a narrow interval of E_{∞} values, different for each X_{0}. Surprisingly, in this E_{∞} interval, a sharp transition between two power-law precoalescence flow regimes is seen. The evolution of several flow characteristics before and after coalescence, and the shape of the deformed droplets at coalescing time and the double cone angle, are also addressed and analyzed in detail. Cone angles below 35^{∘} lead to droplet coalescence for any X_{0} value, which is in accordance with previously reported studies. Finally, it is shown that the model and algorithm can handle multiple droplet interactions. The simulations qualitatively match results from water in oil experiments in microchannels, despite the fact that the exterior fluid is not considered in the mathematical model.

Electrohydrodynamic coalescence of droplets using an embedded potential flow model

The coalescence, and subsequent satellite formation, of two inviscid droplets is studied numerically. The initial drops are taken to be of equal and different sizes, and simulations have been carried out with and without the presence of an electrical field. The main computational challenge is the tracking of a free surface that changes topology. Coupling level set and boundary integral methods with an embedded potential flow model, we seamlessly compute through these singular events. As a consequence, the various coalescence modes that appear depending upon the relative ratio of the parent droplets can be studied. Computations of first stage pinch-off, second stage pinch-off, and complete engulfment are analyzed and compared to recent numerical studies and laboratory experiments. Specifically, we study the evolution of bridge radii and the related scaling laws, the minimum drop radii evolution from coalescence to satellite pinch-off, satellite sizes, and the upward stretching of the near cylindrical protrusion at the droplet top. Clear evidence of partial coalescence self-similarity is presented for parent droplet ratios between 1.66 and 4. This has been possible due to the fact that computational initial conditions only depend upon the mother droplet size, in contrast with laboratory experiments where the difficulty in establishing the same initial physical configuration is well known. The presence of electric forces changes the coalescence patterns, and it is possible to control the satellite droplet size by tuning the electrical field intensity. All of the numerical results are in very good agreement with recent laboratory experiments for water droplet coalescence.

On well-posedness of variational models of charged drops

Electrified liquids are well known to be prone to a variety of interfacial instabilities that result in the onset of apparent interfacial singularities and liquid fragmentation. In the case of electrically conducting liquids, one of the basic models describing the equilibrium interfacial configurations and the onset of instability assumes the liquid to be equipotential and interprets those configurations as local minimizers of the energy consisting of the sum of the surface energy and the electrostatic energy. Here we show that, surprisingly, this classical geometric variational model is mathematically ill-posed irrespective of the degree to which the liquid is electrified. Specifically, we demonstrate that an isolated spherical droplet is never a local minimizer, no matter how small is the total charge on the droplet, as the energy can always be lowered by a smooth, arbitrarily small distortion of the droplet's surface. This is in sharp contrast to the experimental observations that a critical amount of charge is needed in order to destabilize a spherical droplet. We discuss several possible regularization mechanisms for the considered free boundary problem and argue that well-posedness can be restored by the inclusion of the entropic effects resulting in finite screening of free charges.