title: A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area.
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.
authors.affiliation: Department of Chemical Engineering , Imperial College London , London SW7 2AZ, UK.
authors.forename: Michael C
authors.initials: MC
authors.lastname: Dallaston
authors.affiliation: School of Mathematical Sciences , Queensland University of Technology , Brisbane, Queensland 4000, Australia.
authors.forename: Scott W
authors.identifier: 0000-0001-5304-2384
authors.initials: SW
authors.lastname: McCue
keywords: coalescence; curve shortening flow; extinction behaviour; geometric partial differential equation; pinch-off; self-similar solutions