Dynamics of Topological Transitions in Soap Films Spanning Deformable Contours

The study of minimal surfaces dates back to the work of Euler and Lagrange. 'Plateau's problem,' that of proving the existence of a minimal surface spanning a given contour, was solved in the 1930s, and subsequent mathematical work has focused chiefly on statics, involving, for example, proofs of the existence of such surfaces of prescribed topology in higher dimensions and classification of periodic minimal surfaces. With few exceptions, little attention has been paid to transitions that take one surface to another. Elsewhere in the mathematical sciences topological transitions have been studied extensively; from splitting fluid drops to reconnecting solar magnetic field lines, such transitions abound in nature, and are often associated with singular structures that evolve rapidly to a new state. In the field of Fluid Mechanics there has been a longstanding emphasis on interface collapse in viscous flows, and on the more inviscid problems of fluid and soap film motion and networks of film junctions. Surprisingly, these techniques, so successful in other contexts, have yet to be applied to understand the dynamical processes that take one minimal surface to another.

In an elegant article in 1940, the mathematician Courant laid out a number of fundamental questions about minimal-area surfaces that could be visualized with soap films spanning wire frames of various shapes. He noted that when the frame is a double loop it can support a film in the form of a Mobius strip, a one-sided surface. Pulling apart and untwisting the loops leads to an instability where the film jumps to a two-sided solution. This constitutes the simplest known topological transition which converts a one-sided surface to a two-sided one. While Courant concentrated on the key static issues in these systems, the dynamical processes that accompany such transitions were not considered; this is not surprising because they constitute a very modern class of problems in the field of free-boundary dynamics, the tools for which have been developed only over the last two decades.

The research described in this proposal will build upon our recent theoretical and experimental studies of the transition from one- to two-sided soap films driven by boundary deformation, in which a rich and complex dynamics was discovered. We found that this transition is associated with a singularity at which the linking number between the Plateau border of the film and the boundary jumps from two to zero. Moreover, that singularity occurs always at the film boundary and is preceded by collapse dynamics that displays an apparent crossover from viscous-dominated motion to an inertial regime, and ultimately leads to reconnection of the Plateau Border. The static minimal surface left after the singularity exhibits a localized region of high twist of the rearranged Plateau border which is a singular perturbation phenomenon on the scale of the wire radius.

This research will use a combination of analytical, numerical, and experimental methods to understand more deeply this and other topological transitions involving interconversion of minimal surfaces, with the ultimate goal of classifying these transitions and their singularities. Using the Mobius strip problem as a paradigm, we will develop a suitable free-boundary theory that incorporates air inertia and Plateau border dissipation to study the collapse dynamics, with particular attention to the twist singularity experienced by the Plateau border at the transition point. Development of these theoretical descriptions will be done in close contact with experimental studies in which variations in material properties (viscosity, surface tension, boundary geometry) will be used to provide insight and verification. Mathematical models based on ruled surfaces will be further developed to elucidate various topological connections that have been conjectured.

This work involves dynamical transformations that take one minimal surface to another, addressed from a holistic
point of view that combines theory, numerics, and experiments.

Although the mathematical details of the work in this proposal may be technical and abstract, the problem that we shall investigate is so easily demonstrated with little equipment, and has such a natural beauty, that it is a perfect vehicle to introduce abstract mathematical concepts to a lay audience. In particular, the fact that a simplified version of the experiment can be done in one's kitchen means that children will naturally be drawn to the subject. There are many opportunities to present this research to the general public, for DAMTP and the University of Cambridge have in place programs specifically for this. These include the annual Cambridge Summer Science Festival, lectures for school-age children through the Millenium Mathematics Project, and annual Open Houses in DAMTP.

Moreover, the very visual aspects of this research, involving striking high-speed movies and fascinating surface shapes will no doubt be of great interest to the public. As evidenced by BBC programs like Richard Hammond's Invisible Worlds such phenomena can be used to reach a very wide audience. In this regard, we have established a precedent for work with the BBC, as the G.K. Batchelor laboratory in DAMTP has more recently been used by it for a program involving ice.

The Principal Investigator has a longstanding consultancy with Unilever Corporation, the world's largest manufacturer of soaps. We expect that problems involving wetting of fibers, where Plateau border dynamics are of great importance, would be of interest to such surfactant manufacturers.

Finally, we will involve undergraduate maths students in aspects of this research, providing them with an opportunity to develop experimental skills, programming expertise, and experience in communicating scientific results.