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.