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 |
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)