Stable and unstable singularities in the unforced Hele‐Shaw cell
作者:
Robert Almgren,
Andrea Bertozzi,
Michael P. Brenner,
期刊:
Physics of Fluids
(AIP Available online 1996)
卷期:
Volume 8,
issue 6
页码: 1356-1370
ISSN:1070-6631
年代: 1996
DOI:10.1063/1.868915
出版商: AIP
数据来源: AIP
摘要:
We study singularity formation in the lubrication model for the unforced Hele‐Shaw system, describing the breaking in two of a fluid droplet confined between two narrowly spaced glass plates. By varying the initial data, we exhibit four different scenarios: (1) the droplet breaks in finite time, with two pinch points moving toward each other and merging at the singular time; (2) the droplet breaks in finite time, with two asymmetric pinch points propagating away from each other; (3) the droplet breaks in finite time, with a single symmetric pinch point; or (4) the droplet relaxes to a stable equilibrium shape without a finite time breakup. Each of the three singular scenarios has a self‐similar structure with different scaling laws; the first scenario has not been observed before in other Hele‐Shaw studies. We demonstrate instabilities of the second and third scenarios, in which the solution changes its behavior at a thickness that can be arbitrarily small depending on the initial condition. These transitions can be identified by examining the structure of the solution in the intermediate scaling region. ©1996 American Institute of Physics.
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