Stable and unstable singularities in the unforced Hele-Shaw cell
- The University of Chicago, Department of Mathematics, Chicago, Illinois 60637 (United States)
- Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois 60439 (United States)
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (United States)
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. {copyright} {ital 1996 American Institute of Physics.}
- Research Organization:
- Argonne National Laboratory (ANL), Argonne, IL
- DOE Contract Number:
- W-31109-ENG-38
- OSTI ID:
- 286067
- Journal Information:
- Physics of Fluids (1994), Journal Name: Physics of Fluids (1994) Journal Issue: 6 Vol. 8; ISSN 1070-6631; ISSN PHFLE6
- Country of Publication:
- United States
- Language:
- English
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