On well-posedness of variational models of charged drops

TL;DR

This paper demonstrates that classical variational models for charged liquid drops are mathematically ill-posed, as they allow energy-lowering distortions of spherical droplets regardless of charge, contrasting with experimental stability thresholds.

Contribution

It reveals the ill-posedness of classical models and proposes that including entropic effects and charge screening can restore well-posedness.

Findings

Classical models are mathematically ill-posed for charged drops.

Spherical droplets are never local minimizers in these models.

Regularization via entropic effects can restore well-posedness.

Abstract

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 irrespectively 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, since the energy can always be lowered by a smooth, arbitrarily small distortion of the droplet's surface. This is in sharp contrast with 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.