A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

Dallaston, Michael C. & McCue, Scott W. (2016) A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area. Proceedings of the Royal Society A, 472, Article Number:-20150629.

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Abstract

Motivated by a problem from fluid mechanics, we consider a generalization of the standard curve shortening flow problem for a closed embedded plane curve such that the area enclosed by the curve is forced to decrease at a prescribed rate. Using formal asymptotic and numerical techniques, we derive possible extinction shapes as the curve contracts to a point, dependent on the rate of decreasing area; we find there is a wider class of extinction shapes than for standard curve shortening, for which initially simple closed curves are always asymptotically circular. We also provide numerical evidence that self-intersection is possible for non-convex initial conditions, distinguishing between pinch-off and coalescence of the curve interior.

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ID Code: 93810
Item Type: Journal Article
Refereed: Yes
Keywords: curve shortening flow, geometric partial differential equation, extinction behaviour, pinch-off, coalescence, self-similar solutions, non-local geometric flows, curvature driven flow, Hele-Shaw flow
DOI: 10.1098/rspa.2015.0629
ISSN: 1471-2946
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > PURE MATHEMATICS (010100) > Algebraic and Differential Geometry (010102)
Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Approximation Theory and Asymptotic Methods (010201)
Divisions: Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Funding:
Copyright Owner: 2016 The Authors
Copyright Statement: Published by the Royal Society under the terms of the Creative Commons Attribution License
http://creativecommons.org/licenses/
by/4.0/, which permits unrestricted use, provided the original author and source are credited.
Deposited On: 16 Mar 2016 00:13
Last Modified: 17 Mar 2016 04:45

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