Boundary Singularities Produced by the Motion of Soap Films
Seminar Room 1, Newton Institute
AbstractCo-authors: Adriana I. Pesci (University of Cambridge), Keith Moffatt (University of Cambridge), James McTavish (University of Cambridge), Renzo Ricca (University of Milano-Bicocca)
Recent experiments have shown that when a soap film with the topology of a Mobius strip, is rendered unstable by slow deformation of its frame past a threshold, the film changes its topology to that of a disc by means of a ``neck-pinching'' singularity at its boundary. This behaviour is unlike the more familiar catenoid minimal surface supported on two parallel circular loops, a two-sided surface which, when rendered unstable, transitions to two disks through a neck-pinching singularity in the bulk. There is at present neither an understanding of whether the type of singularity is in general a consequence of the topology of the surface, nor of how this dependence could arise from a surface equation of motion. We investigate experimentally, computationally, and theoretically the neck-pinching route to singularities of soap films with several distinct topologies, including a family of non-orientable surfaces that are sections of Klein bottles, and provide evidence that the location of singularities (bulk or boundary) may depend on the path along which the boundary is deformed. Since in the unstable regime the driving force for soap film motion is the surface's mean curvature, the narrowest part of the neck, which can be associated with the shortest nontrivial closed geodesic of the surface at each instant of time, has the highest curvature and is thus the fastest-moving. Just before the onset of the instability there exists on the stable surface also a shortest closed geodesic, which serves as an initial condition for the evolution of the geodesics of the neck, all of which have the same topological relationship to the surface boundary. We find that if the initial geodesic is linked to the boundary then the singularity will occur at the boundary, whereas if the two are unlinked initially then the singularity will occur in the bulk. Numerical study of mean curvature flows and experiments show consistency with these conjectures.
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