Singularities in Hele-Shaw flows

SIAM Journal on Applied Mathematics

Volumn 58, Issue 1, 1998, Pages 34-54

  Nie, Qing ^a   Tian, Fei Ran ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

Hele Shaw; Interface; Singularity

Indexed keywords

COMPUTER SIMULATION; INTERFACES (MATERIALS); SURFACE TENSION; THEOREM PROVING;

HELE SHAW FLOWS;

FLOW OF FLUIDS;

EID: 0032000710     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/s0036139996297924     Document Type: Article

References (19)

1. R. ALMGREN, Singularity formation in Hele-Shaw bubbles, Phys. Fluids, 8 (1996), pp. 344-352.
2. G. R. BAKER, D. I. MEIRON, AND S. A. ORSZAG, Generalized vortex methods for free-surface flow problems, J. Fluid Mech., 123 (1982), pp. 477-501.
3. D. BENSIMON, L. KADANOFF, S. LIANG, B. SHRAIMAN, AND C. TANG, Viscous flows in two dimensions, Rev. Mod. Phys., 58 (1986), pp. 977-999.
4. P. CONSTANTIN AND M. PUGH, Global solutions for the small data to the Hele-Shaw problem, Nonlinearity, 6 (1993), pp. 393-415.
5. R. COURANT AND D. HILBERT, Methods of Mathematical Physics, Vol. II, Interscience Publishers, New York, 1962.
6. E. DIBENEDETTO AND A. FRIEDMAN, The ill-posed Hele-Shaw model and the Stefan problem for supercooled water, Trans. Amer. Math. Soc., 282 (1984), pp. 183-204.
7. J. DUCHON AND R. ROBERT, Évolution d'une interface par capillarité et diffusion de volume I. Existence locale en temps, Ann Inst. H. Poincaré, Anal. Non Linéaire, 1 (1984), pp. 361-378.
8. D. GOTTLIEB AND S. A. ORSZAG, EDS., Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1977.
9. T. Y. HOU, J. S. LOWENGRUB, AND M. J. SHELLEY, Remove the stiffness from interfacial flows with surface tension, J. Comp. Phys., 114 (1994), pp. 312-338.
10. Y. E. HOHLOV AND S. D. HOWISON, On the classification of solutions to the zero surface tension model for Hele-Shaw free boundary flows, Quart. Appl. Math., 51 (1993), pp. 777-789.
11. E. D. KELLEY AND E. J. HINCH, Numerical Simulations of Sink Flow in the Hele-Shaw Cell with Small Surface Tension, preprint, 1996.
12. Q. NIE, Topics in the Motion of Bubbles in Incompressible Liquids, Ph.D. thesis, Ohio State University, Columbus, OH, 1995.
13. L. PATERSON, Radial fingering in a Hele-Shaw cell, J. Fluid Mech., 113 (1981), pp. 513-529.
14. S. RICHARDSON, Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, J. Fluid Mech., 56 (1972), pp. 609-618.
15. S. RICHARDSON, On the classification of solutions to the zero-surface-tension model For Hele-Shaw free boundary flows, Quart. Appl. Math., 55 (1997), pp. 313-319.
16. Y. SAAD AND M. H. SCHULTZ, GMRES: A generalized minimal residual algorithm for nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856-869.
17. A. SIDI AND M. ISRAELI, Quadrature methods for singular and weakly singular Fredholm integral equations, J. Sci. Comput., 3 (1988), pp. 201-231.
18. F. R. TIAN, On the breakdown of Hele-Shaw solutions with non-zero surface tension, J. Nonlinear Sci., 5 (1995), pp. 479-484.
19. F. R. TIAN, A Cauchy integral approach to Hele-Shaw problems with a free boundary: The case of zero surface tension, Arch. Rational Mech. Anal., 135 (1996), pp. 175-196.