Cusp Development in Hele–Shaw Flow with a Free Surface
- 1 February 1986
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Applied Mathematics
- Vol. 46 (1) , 20-26
- https://doi.org/10.1137/0146003
Abstract
Free surface flow in a Hele-Shaw cell is considered. It has been known for some time that when the fluid region is contracting, a finite time blow-up can occur, in which a cusp forms in the free surface, the solution does not exist beyond the time of blow-up. Examples are presented of this blow-up, and also of a new process in which a cusp of a different type forms in the free surface, but after which the solution continues to exist. After a transformation of the dependent variable (the pressure) the new cusps are related to permissible singularities in the free boundary of the 'obstacle' problemKeywords
This publication has 9 references indexed in Scilit:
- Bubble growth in porous media and Hele–Shaw cellsProceedings of the Royal Society of Edinburgh: Section A Mathematics, 1986
- SINGULARITY DEVELOPMENT IN MOVING-BOUNDARY PROBLEMSThe Quarterly Journal of Mechanics and Applied Mathematics, 1985
- Two-phase displacement in Hele Shaw cells: theoryJournal of Fluid Mechanics, 1984
- The ill-posed Hele-Shaw model and the Stefan problem for supercooled waterTransactions of the American Mathematical Society, 1984
- Moving boundary problems in the flow of liquid through porous mediaThe Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 1982
- A variational inequality approach to Hele-Shaw flow with a moving boundaryProceedings of the Royal Society of Edinburgh: Section A Mathematics, 1981
- Generic unfoldings and normal forms of some singularities arising in the obstacle problemDuke Mathematical Journal, 1979
- Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channelJournal of Fluid Mechanics, 1972
- The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquidProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1958