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

Michael C. Dallaston, Scott W. McCue

Research output: Contribution to journalArticle

6 Citations (Scopus)
22 Downloads (Pure)

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 nonconvex initial conditions, distinguishing between pinch-off and coalescence of the curve interior. 2016 The Authors.

Original languageEnglish
Article number20150629
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume472
Issue number2185
DOIs
Publication statusPublished - 1 Jan 2016
Externally publishedYes

Fingerprint

Rate of change
Plane Curve
Closed
Curve
Fluid mechanics
curves
Coalescence
Extinction
Simple Closed Curve
Self-intersection
extinction
Fluid Mechanics
Numerical Techniques
fluid mechanics
Interior
Initial conditions
intersections
coalescing
Decrease
Dependent

Bibliographical note

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.

Keywords

  • Coalescence
  • Curve shortening flow
  • Extinction behaviour
  • Geometric partial differential equation
  • Pinch-off
  • Self-similar solutions

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

Cite this

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.

In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 472, No. 2185, 20150629, 01.01.2016.

Research output: Contribution to journalArticle

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