Analysis of a soap film catenoid driven by an electrostatic force

Abstract

The interplay between surface tension and electrostatics is the underlying mechanism of many processes taking place on small length-scales. There are various examples in nature and technology, including the whole field of micro-electro-mechanical systems.

This thesis is devoted to a free boundary problem modelling a prototypical set-up in which both surface tension and electrostatics have the ability to break the set-up. The set-up consists of a conductive soap film spanned between two parallel rings, which are placed inside an outer metal cylinder. On the one hand, if the gap between the rings is not too big, surface tension forces the soap film to take the shape of a catenoid. In particular, surface tension pushes the film inwards. On the other hand, applying a voltage between the catenoid and the outer cylinder results in an electrostatic force pulling the film outwards. While a previous mathematical investigation focused on a simplified small aspect ratio model of the set-up and did not yet include time, we drop the small aspect ratio assumption which yields a completely different type of model. We also include time into our considerations.

In the first part of this thesis, we derive the new model for the soap film catenoid subjected to an electrostatic force. The model consists of a quasilinear parabolic equation for the evolution of the film coupled with an elliptic equation for the electrostatic potential in the unknown domain between outer cylinder and soap film catenoid. Then, for the rotationally symmetric case, we show local well-posedness of this free boundary problem by recasting it as a single quasilinear parabolic equation with a non-local source term. As the source term turns out to have slightly weaker regularity than required, the proof of local well-posedness contains a refinement of a classical fixed point argument based on semigroup theory.

In the second part of this thesis, we discuss different kinds of behaviour that the soap film displays depending on the strength of the applied voltage. For small voltages as well as voltages for which surface tension and electrostatics are balanced, we show the existence of stationary solutions and study their stability. Moreover, we prove that stable stationary solutions behave physically in the sense that they always deflect outwards if the applied voltage is increased. Finally, for large applied voltages, we show that solutions to the evolution problem do not exist globally for a large class of initial values. The proofs in the second part of this thesis mostly, but not exclusively, rely on the implicit function theorem, the principle of linearized stability, (anti-)maximum principles as well as positivity of a certain Fourier series.