Singular effects of surface tension in evolving Hele-Shaw flows

Journal of Fluid Mechanics

Volumn 323, Issue , 1996, Pages 201-236

  Siegel, Michael ^a   Tanveer, Saleh ^b   Dai, Wei Shen ^c  

^a NEW JERSEY INSTITUTE OF TECHNOLOGY   (United States)

^b OHIO STATE UNIVERSITY   (United States)

^c ELECTROTECHNICAL LABORATORY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

CAPILLARITY; CONTACTS (FLUID MECHANICS); GAS DYNAMICS; PERTURBATION TECHNIQUES; SURFACE TENSION; VISCOUS FLOW;

DAUGHTER SINGULARITY EFFECT; HELE-SHAW FLOWS;

CHANNEL FLOW;

FLOW; VISCOUS FLUIDS;

EID: 0030239360     PISSN: 00221120     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0022112096000894     Document Type: Article

References (50)

1. ARNEODO, A., COUDER, Y., GRASSEAU, G., HAKIM, V. & RABAUD, M. 1989 Uncovering the analytical Saffman-Taylor finger in unstable viscous fingering and diffusion limited aggregation. Phys. Rev. Lett. 63, 984.
2. BAKER, G., SIEGEL, M. & TANVEER, S. 1995 A well-posed numerical method to track isolated conformal map singularites in Hele-Shaw flow. J. Comput. Phys. 120, 348.
3. BENSIMON, D. 1986 Stability of viscous fingering. Phys. Rev. A 33, 1302.
4. BENSIMON, D., KADANOFF, L.P., LIANG, S., SHRAIMAN, B.I., & TANG, C. 1986 Viscous flows in two dimensions. Rev. Mod. Phys., 58, 977.
5. COMBESCOT, R., DOMBRE, T., HAKIM, V., POMEAU, Y. & PUMIR, A. 1986 Shape selection for Saffman-Taylor fingers, Phys. Rev. Lett. 56, 2036.
6. COMBESCOT, R., DOMBRE, T., HAKIM, V., POMEAU, Y. & PUMIR, A. 1987 Analytic theory of the Saffman-Taylor fingers, Phys. Rev. A 37, 1270.
7. CONSTANTIN, P. & PUGH, M. 1993 Global solutions for small data to the Hele-Shaw problem. Nonlinearity 6, 393.
8. DAI, W., KADANOFF, L.P & ZHOU, S. 1991 Interface dynamics and the motion of complex singularities. Phys. Rev. A 43, 6672.
9. DAI, W. & SHELLEY, M. 1994 A numerical study of the effect of surface tension and noise on an expanding Hele-Shaw bubble. Phys. Fluids A 5, 2131.
10. DEGREGORIA, A. J. & SCHWARTZ, L. W. 1985 Finger breakup in Hele-Shaw cells. Phys. Fluids 28, 2313.
11. DEGREGORIA, A. J. & SCHWARTZ, L. W. 1986 A boundary-integral method for two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 164, 383.
12. DORSEY, A. & MARTIN, O. 1987 Saffman-Taylor fingers with anisotropic surface tension. Phys. Rev. A 35, 3989.
13. DUCHON, J. & ROBERT, R. 1984 Evolution d'une interface par capillarité et diffusion de volume. Ann. Inst. H. Poincaré 1, 361.
14. GALIN, L. A. 1945 Dokl. Akad. Nauk. SSSR 47, 246 (in Russian).
15. HOMSY, G. 1987 Viscous fingering in porous media. Ann. Rev. Fluid Mech. 19, 271.
16. HONG, D. & LANGER, J. 1986 Analytic theory for the selection of Saffman-Taylor fingers. Phys. Rev. Lett. 56, 2032.
17. Hou, T., LOWENGRUB, J. & SHELLEY, M. 1994 Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114, 312.
18. HOWISON, S. D. 1985 Bubble growth in porous media and Hele-Shaw cells. Proc. R. Soc. Edin. A 102, 141.
19. HOWISON, S. D. 1986 Fingering in Hele-Shaw cells. J. Fluid Mech. 167, 439.
20. HOWISON, S. D. 1986 Cusp development in Hele-Shaw flow with a free surface, SIAM J. Appl. Math 46, 20.
21. HOWISON, S. D. 1991 Complex variable methods in Hele-Shaw moving boundary problems. Eur. J. Appl. Maths 3, 209.
22. KESSLER, D., KOPLIK, J. & LEVINE, H. 1988 Pattern selection in fingered growth phenomena. Adv. Phys. 37, 255.
23. KRASNY, R. 1986 A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167, 65.
24. KUNKA M., FOSTER, M. R., & TANVEER, S. 1996 In preparation.
25. LACEY, A. A. 1982 Moving boundary problems in the flow of liquid through porous media. J. Austral. Math. Soc. B24, 171.
26. MAXWORTHY, T. 1987 The nonlinear growth of a gravitationally unstable interface in a Hele-Shaw cell. J. Fluid Mech. 177, 207.
27. MCLEAN, J. W. & SAFFMAN, P. G. 1981 The effect of surface tension on the shape of fingers in a Hele-Shaw cell. J. Fluid Mech. 102, 455.
28. MEIBURG, E., & HOMSY, G. M. 1988 Nonlinear unstable viscous fingers in Hele-Shaw flows. II. Numerical simulation. Phys. Fluids 31, 429.
29. PARK, C. W. & HOMSY, G. M. 1985 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291.
30. PELCE, P. 1988 Dynamics of Curved Fronts. Academic.
31. POLUBARINOVA-KOCHINA, P. 1945 Dokl. Akad. Nauk. SSSR 47, 254 (in Russian).
32. REINELT, D. A. 1987 Interface conditions for two-phase displacement in Hele-Shaw cells. J. Fluid Mech 183, 219.
33. REINELT, D. A. 1987 The effect of thin film variations and transverse curvature on the shape of fingers in a Hele-Shaw cell. Phys. Fluids 30, 2617.
34. RICHARDSON, S. 1972 Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56, 609.
35. SAFFMAN, P. G. 1959 Exact solution for the growth of fingers from a flat interface between two fluids. Q. J. Mech. Appl. Maths 12, 146.
36. SAFFMAN, P. G. 1986 Viscous fingering in a Hele-Shaw cell. J. Fluid Mech. 173, 73.
37. SAFFMAN, P. G. & TAYLOR, G. I. 1958 The penetration of a fluid into a porous medium of Hele-Shaw cell containing a more viscous fluid. Proc. R. Soc. Lond. A 245, 312.
38. SARKAR, S. & JASNOW, D. 1987 Quantitative test of solvability theory for the Saffman-Taylor problem. Phys. Rev. A 35, 4900.
39. SCHWARTZ, L. & DEGREGORIA, A. 1987 Simulation of Hele-Shaw fingering with finite capillary number effects included. Phys. Rev. A 35, 276.
40. SHRAIMAN, B. I. 1986 On velocity selection and the Saffman-Taylor problem. Phys. Rev. Lett. 56, 2032.
41. SHRAIMAN, B. I. & BENSIMON, D. 1985 Singularities in nonlocal dynamics. Phys. Rev. A 30, 2840.
42. SIEGEL, M. & TANVEER, S. 1996 Singular perturbation of smoothly evolving Hele-Shaw solutions. Phys. Rev. Lett. 56, 419.
43. TANVEER, S. 1987 Analytic theory for the linear stability of Saffman-Taylor finger. Phys. Fluids 30, 2318.
44. TANVEER, S. 1990 Analytic theory for the selection of Saffman-Taylor finger in the presence of thin-film effects. Proc. R. Soc. Lond. A 428, 511.
45. TANVEER, S. 1991 Viscous displacement in a Hele-Shaw cell, In Asymptotics Beyond All Orders (ed. H. Segur, S. Tanveer & H. Levine) p. 131. Plenum.
46. TANVEER, S. 1993 Evolution of a Hele-Shaw interface for small surface tension. Phil. Trans. R. Soc. Lond. A 343, 1 (see also erratum 1994).
47. TANVEER, S. 1995 Asymptotic calculation of thin-film effects on Hele-Shaw fingering. J. Fluid Mech. (submitted).
48. TRYGGVASSON, G. & AREF, H. 1983 Numerical experiments on Hele-Shaw flow with a sharp interface. J. Fluid Mech. 136, 1.
49. TRYGGVASON, G. & AREF, H. 1985 Finger-interaction mechanisms in stratified Hele-Shaw flow. J. Fluid Mech. 154, 287.
50. ZHURAVLEV, P. 1956 Zap Leningrad Com. Inst. 133, 54 (in Russian).