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Geometry of catenoidal soap film collapse induced by boundary deformation

Raymond E. Goldstein, Adriana I. Pesci, Christophe Raufaste, and James D. Shemilt
Phys. Rev. E 104, 035105 – Published 8 September 2021
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Abstract

Experimental and theoretical work reported here on the collapse of catenoidal soap films of various viscosities reveal the existence of a robust geometric feature that appears not to have been analyzed previously; prior to the ultimate pinchoff event on the central axis, which is associated with the formation of a well-studied local double-cone structure folded back on itself, the film transiently consists of two acute-angle cones connected to the supporting rings, joined by a central quasicylindrical region. As the cylindrical region becomes unstable and pinches, the opening angle of those cones is found to be universal, independent of film viscosity. Moreover, that same opening angle at pinching is found when the transition occurs in a hemicatenoid bounded by a surface. The approach to the conical structure is found to obey classical Keller-Miksis scaling of the minimum radius as a function of time, down to very small but finite radii. While there is a large body of work on the detailed structure of the singularities associated with ultimate pinchoff events, these large-scale features have not been addressed. Here we study these geometrical aspects of film collapse by several distinct approaches, including a systematic analysis of the linear and weakly nonlinear dynamics in the neighborhood of the saddle node bifurcation leading to collapse, both within mean curvature flow and the physically realistic Euler flow associated with the incompressible dynamics of the surrounding air. These analyses are used to show how much of the geometry of collapsing catenoids is accurately captured by a few active modes triggered by boundary deformation. A separate analysis based on a mathematical sequence of shapes progressing from the critical catenoid towards the Goldschmidt solution is shown to predict accurately the cone angle at pinching. We suggest that the approach to the conical structures can be viewed as passage close to an unstable fixed point of conical similarity solutions. The overall analysis provides the basis for the systematic study of more complex problems of surface instabilities triggered by deformations of the supporting boundaries.

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  • Received 9 May 2021
  • Accepted 24 August 2021

DOI:https://doi.org/10.1103/PhysRevE.104.035105

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Nonlinear DynamicsFluid Dynamics

Authors & Affiliations

Raymond E. Goldstein1,*, Adriana I. Pesci1,†, Christophe Raufaste2,3,‡, and James D. Shemilt1,§

  • 1Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
  • 2Université Côte d'Azur, CNRS, Institut de Physique de Nice, 06100 Nice, France
  • 3Institut Universitaire de France (IUF), 75005 Paris, France

  • *r.e.goldstein@damtp.cam.ac.uk
  • a.i.pesci@damtp.cam.ac.uk
  • christophe.raufaste@univ-cotedazur.fr
  • §james.shemilt@manchester.ac.uk

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Vol. 104, Iss. 3 — September 2021

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