On the existence of finger solutions of the Hele-Shaw equation

Nonlinearity

Volumn 14, Issue 1, 2001, Pages 153-166

  Su, Jianzhong ^a,b  

^a UNIVERSITY OF TEXAS AT ARLINGTON   (United States)

^b OHIO STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035218554     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/14/1/309     Document Type: Article

References (32)

1. Alikakos N D, Bates P W and Chen X 1994 Convergence of the Cahn-Hillard equation to the Hele-Shaw model Arch. Rat. Mech. Anal. 128 165-205
2. Almgren R F 1995 Crystalline Saffman-Taylor fingers SIAM J. Appl. Math. 55 1511-35
3. Bazalli B V 1997 Stefan problem for the Laplace equation with regard for the curvature of the free boundary Ukrain. Math. J. 49 1465-84
4. Caginalp G 1989 Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations Phys. Rev. A 39 5887-96
5. Chen X 1993 Hele-Shaw problem and area-preserving curve shorting motion Arch. Rat. Mech. Anal. 123 117-51
6. Chen X, Hong J X and Yi F H 1996 Existence, uniqueness and regularity of classical solutions of the Mullins-Sekerka problem Commun. Part. Diff. Eq. 21 1705-27
7. Chouke R L, van Meurs P and van der Poel C 1959 The instability of slow immiscible viscous liquid-liquid displacements in permeable media Trans AIME 216 188-94
8. Constantin P and Pugh M 1993 Global solutions for small data to the Hele-Shaw equation Nonlinearity 6 393-415
9. Duchon J and Robert R 1984 Evolution d'une interface par capilarite et diffusion de volume I. existence locale en temps Ann. Inst. Poincaré H, Analyse Non Lineaire 1 361-78
10. Elliott C M and Ockendon J R 1982 Weak and Variational Methods for Moving Boundary Problems (Pitman Advanced Publishing Program) (New York: Pitman)
11. Escher J and Simonett G 1996 On Hele-Shaw models with surface tension Math. Res. Lett. 3 467-74
12. Escher J and Simonett G 1997 Classical solutions to Hele-Shaw models with surface tension Adv. Diff. Eq. 2 439-59
13. Gilbarg D and Trudinger N S 1983 Elliptic Partial Differential Equations of Second Order (Berlin: Springer)
14. Hele-Shaw H J S 1898 On the motion of a viscous fluid between two parallel plates Nature 58 34-6
15. Hill S 1952 Channelling in packed columns Chem. Eng. Sci. 1 247-53
16. Homsy G M 1987 Viscous fingering in porous media Ann. Rev. Fluid Mech. 19 271-311
17. Hong D C and Langer J S 1986 Analytic theory of the selection mechanism in the Saffman-Taylor problem Phys. Rev. Lett. 56 2032-5
18. Howison S D 1986 Cusp development in Hele-Shaw flow with a free surface SIAM J. Appl. Math. 46 20-6
19. Kessler D A, Koplik J and Levine H 1988 Pattern selection in fingered growth phenomena Adv. Phys. 37 255-329
20. McLean J W and Saffman P G 1981 The effect of surface tension on the shape of fingers in a Hele-Shaw cell J. Fluid Mech. 102 455-69
21. Mikhlin S G and Prossdorf S 1986 Singular Integral Operators (translated from the German by A Bottcher and R Lehmann) (Berlin: Springer)
22. Mullins W W and Sekerka R F 1963 Morphological stability of a particle growing by diffusion of heat flow J. Appl. Phys. 34 323-8
23. Nie Q and Tian F R 1998 Singularities in Hele-Shaw flows SIAM J. Appl. Math. 58 34-54
24. Otto F and E W 1997 Thermodynamically driven incompressible fluid mixtures J. Chem. Phys. 107 10 177-84
25. Protter M H and Weinberger H F 1967 Maximum Principles in Differential Equations (Englewood Cliffs, NJ: Prentice Hall)
26. Saffman P G and Taylor G I 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid Proc. R. Soc. A 245 312-29
27. Tanveer S 1987 Analytic theory for the selection of symmetric Saffman-Taylor finger Phys. Fluids 30 1589-605
28. Tanveer S 1990 Analytic theory for the selection of Saffman-Taylor finger in the presence of thin film effects Proc. R. Soc. A 428 511-45
29. Tanveer S 1993 Evolution of Hele-Shaw interface for small surface tension Phil. Trans. R. Soc. A 343 155-204
30. Tian F R 1996 A Cauchy integral approach to Hele-Shaw problems with a free boundary: the case of zero surface tension Arch. Rat. Mech. Anal 135 175-96
31. Tryggvason G and Aref H 1983 Numerical experiments on Hele-Shaw flow with a sharp interface J. Fluid Mech. 136 1-30
32. Zeidler E 1985 Nonlinear Functional Analysis and its Applications (translated by P R Wadsack) (Berlin: Springer) p 1