On the geometry of Hele-Shaw flows with small surface tension

Interfaces and Free Boundaries

Volumn 5, Issue 2, 2003, Pages 183-192

  Vasil'Ev, Alexander ^a   Markina, Irina ^a  

^a DEPARTAMENTO DE FÍSICA   (Chile)

Author keywords

Conformal map; Free boundary problem; Hele Shaw flow; Surface tension

Indexed keywords

EID: 84982864356     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/77     Document Type: Article

References (31)

1. CENICEROS, H. G., HOU, T. Y., & SI, H. Numerical study of Hele-Shaw flow with suction. Phys. Fluids 11 (1999), 2471-2486. MR 2000d:76051
2. DI BENEDETTO, E. & FRIEDMANN, A. The ill-posed Hele-Shaw model and the Stefan problem for supercooled water. Trans. Amer. Math. Soc. 282 (1984), 183-204. Zbl 0621.35102 MR 85g:35121
3. DUCHON, J. & ROBERT, R. Evolution d'une interface par capillarité et diffusion de volume, 1: existence locale en temps. Ann. Inst. H. Poincaré 1 (1984), 361-378. Zbl 0572.35051 MR 86i:35133
4. ELLIOTT, C. M. & JANOVSKÝ, V., A variational inequality approach to Hele-Shaw flow with a moving boundary. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 93-107. Zbl 0455.76043 MR 82d:76031
5. ESCHER, J. & SIMONETT, G. On Hele-Shaw models with surface tension. Math. Res. Lett. 3 (1996), 467-474. Zbl 0860.35149 MR 97i:35145
6. ESCHER, J. & SIMONETT, G. Classical solutions for Hele-Shaw models with surface tension. Adv. Differential Equations 2 (1997), 619-642. Zbl 0888.35142 MR 98b:35204
7. ESCHER, J. & SIMONETT, G. A center manifold analysis for the Mullins-Sekerka model. J. Differential Equations 143 (1998), 267-292. Zbl 0896.35142 MR 98m:35228
8. FOKAS, A. S. & TANVEER, S. A Hele-Shaw problem and the second Painlevé transcendent. Math. Proc. Cambridge Philos. Soc. 124 (1998), 169-191. Zbl 0918.76020 MR 99b:7602
9. GALIN, L. A. Unsteady filtration with a free surface. Dokl. Akad. Nauk SSSR 47 (1945), 246-249 (in Russian). Zbl 0061.46202
10. GOODMAN, A. W. Univalent Functions. Vols. I, II. Mariner, 1983.
11. GUSTAFSSON, B. Applications of variational inequalities to a moving boundary problem for Hele-Shaw flows. SIAM J. Math. Anal. 16 (1985), 279-300. Zbl 0605.76043 MR 86m:35155
12. HOHLOV, Yu. E., PROKHOROV, D. V. & VASIL'EV, A. Yu. On geometrical properties of free boundaries in the Hele-Shaw flows moving boundary problem. Lobachevskii J. Math. 1 (1998), 3-12 (electronic). Zbl 0939.76025 MR 99m:76054
13. HOWISON, S. D. Complex variable methods in Hele-Shaw moving boundary problems. European J. Appl. Math. 3 (1992), 209-224. Zbl 0759.76022
14. HOWISON, S. D. & HOHLOV, Yu. E. On the classification of solutions to the zero-surface-Tension model for Hele-Shaw free boundary flows. Quart. Appl. Math. 54 (1994), 777-789. Zbl 0793.76093
15. KIMURA, M. Time local existence of a moving boundary of the Hele-Shaw flow with suction. European J. Appl. Math. 10 (1999), 581-605. Zbl 0955.76020 MR 2001e:76038
16. MCLEAN, J. W. & SAFFMAN, P. G. The effect of surface tension on the shape of fingers in a Hele Shaw cell. J. Fluid Mech. 102 (1981), 455-469. Zbl 0472.76111
17. MUSKHELISHVILI, N. I. Singular Integral Equations. Noordhoff, Groningen, 1953. Zbl 0051.33203
18. OCKENDON, H. & OCKENDON, J. R. Viscous Flow. Cambridge Univ. Press, 1995. Zbl 0837.76001 MR 95k:76025
19. POLUBARINOVA-KOCHINA, P. Ya. On a problem of the motion of the contour of a petroleum shell. Dokl. Akad. Nauk SSSR 47 (1945), 254-257 (in Russian).
20. POLUBARINOVA-KOCHINA, P. Ya. Concerning unsteady motions in the theory of filtration. Prikl. Mat. Mekh. 9 (1945), 79-90 (in Russian).
21. POMMERENKE, C. Univalent Functions (with a chapter on quadratic differentials by G. Jensen). Vandenhoeck & Ruprecht, Göttingen, 1975. Zbl 0298.30014
22. POMMERENKE, C. Boundary Behaviour of Conformal Maps. Springer, Berlin, 1992. Zbl 0762.30001
23. PROKERT, G. Existence results for Hele-Shaw flow driven by surface tension. European J. Appl. Math. 9 (1998), 195-221. Zbl 0919.35005 MR 2000d:76052
24. QING, N. & TIAN, F. R. Singularities in Hele-Shaw flows. SIAM J. Appl. Math. 58 (1998), 34-54. Zbl 0917.76020 MR 2000c:76023
25. REISSIG, M. & VON WOLFERSDORF, L. A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane. Ark. Mat. 31 (1993), 101-116. Zbl 0802.35168
26. PRIMICERIO, M. & RODRIGUES, J.-F. The Hele-Shaw problem with nonlocal injection condition. Nonlinear Mathematical Problems in Industry, II (Iwaki, 1992), GAKUTO Internat. Ser. Math. Sci. Appl. 2, Gakkotosho, Tokyo, 1993, 375-390. Zbl 0875.35157 MR 96j:3527
27. SAFFMAN, P. G. & TAYLOR, G. I. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. Roy. Soc. London Ser. A 245 (1958), 312-329. Zbl 0086.41603
28. SIEGEL, M., TANVEER, S., & DAI,W. S. Singular effects of surface tension in evolving Hele-Shaw flows. J. Fluid Mech. 323 (1996), 201-236. Zbl0885.76022 MR 97h:76041
29. TANVEER, S. Evolution of Hele-Shaw interface for small surface tension. Philos. Trans. Roy. Soc. London Ser. A 343 (1993), 155-204. Zbl 0778.76029
30. TIAN, F.-R. Hele-Shaw problems in multidimensional spaces. J. Nonlinear Sci. 10 (2000), 275-290. Zbl 0955.76024 MR 2000m:76041
31. VINOGRADOV, Yu. P. & KUFAREV, P. P. On a problem of filtration. Prikl. Mat. Mekh. 12 (1948), 181-198 (in Russian). Zbl 0032.27901 MR 9,540j