Ceniceros, Hector D. and Hou, Thomas Y. and Si, Helen (1999) Numerical study of Hele-Shaw flow with suction. Physics of Fluids, 11 (9). pp. 2471-2486. ISSN 1070-6631 http://resolver.caltech.edu/CaltechAUTHORS:CENpof99
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We investigate numerically the effects of surface tension on the evolution of an initially circular blob of viscous fluid in a Hele-Shaw cell. The blob is surrounded by less viscous fluid and is drawn into an eccentric point sink. In the absence of surface tension, these flows are known to form cusp singularities in finite time. Our study focuses on identifying how these cusped flows are regularized by the presence of small surface tension, and what the limiting form of the regularization is as surface tension tends to zero. The two-phase Hele-Shaw flow, known as the Muskat problem, is considered. We find that, for nonzero surface tension, the motion continues beyond the zero-surface-tension cusp time, and generically breaks down only when the interface touches the sink. When the viscosity of the surrounding fluid is small or negligible, the interface develops a finger that bulges and later evolves into a wedge as it approaches the sink. A neck is formed at the top of the finger. Our computations reveal an asymptotic shape of the wedge in the limit as surface tension tends to zero. Moreover, we find evidence that, for a fixed time past the zero-surface-tension cusp time, the vanishing surface tension solution is singular at the finger neck. The zero-surface-tension cusp splits into two corner singularities in the limiting solution. Larger viscosity in the exterior fluid prevents the formation of the neck and leads to the development of thinner fingers. It is observed that the asymptotic wedge angle of the fingers decreases as the viscosity ratio is reduced, apparently towards the zero angle (cusp) of the zero-viscosity-ratio solution.
|Additional Information:||©1999 American Institute of Physics. (Received 30 July 1998; accepted 27 May 1999) We would like to thank Professor Sam Howison and Professor John Ockendon for suggesting this problem to us and for their valuable comments. We also thank Dr. Mark Kunka and Professor Saleh Tanveer for a number of helpful discussions. Finally, we would like to thank Professor Stephen Cowley and Professor John Hinch for many constructive comments and suggestions. T.Y.H. acknowledges support from National Science Foundation Grant No. DMS-9704976 and Office of Naval Research Grant No. N00014-96-1-0438.|
|Subject Keywords:||two-phase flow; surface tension; viscosity|
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|Deposited On:||23 Feb 2006|
|Last Modified:||26 Dec 2012 08:46|
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