### 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 language | English |
---|---|

Article number | 20150629 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 472 |

Issue number | 2185 |

DOIs | |

Publication status | Published - 1 Jan 2016 |

Externally published | Yes |

### Fingerprint

### 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

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*472*(2185), [20150629]. https://doi.org/10.1098/rspa.2015.0629

**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.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 472, no. 2185, 20150629. https://doi.org/10.1098/rspa.2015.0629

}

TY - JOUR

T1 - A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

AU - Dallaston, Michael C.

AU - McCue, Scott W.

N1 - 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.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

KW - Coalescence

KW - Curve shortening flow

KW - Extinction behaviour

KW - Geometric partial differential equation

KW - Pinch-off

KW - Self-similar solutions

UR - http://www.scopus.com/inward/record.url?scp=84956853178&partnerID=8YFLogxK

U2 - 10.1098/rspa.2015.0629

DO - 10.1098/rspa.2015.0629

M3 - Article

VL - 472

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

IS - 2185

M1 - 20150629

ER -