Singularity formation in Hele-Shaw bubbles

Physics of Fluids

Volumn 8, Issue 2, 1996, Pages 344-352

  Almgren, Robert ^a  

^a UNIVERSITY OF CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BUBBLES; HELE-SHAW; MODELLING-MATHEMATICAL; SURFACE TENSION;

EID: 0029656405     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.869102     Document Type: Article

References (30)

1. D. Bensimon, L. P. Kadanoff, S. Liang, B. I. Shraiman, and C. Tang, "Viscous flows in two dimensions," Rev. Mod. Phys. 58, 977 (1986).
2. D. A. Kessler, J. Koplik, and H. Levine, "Pattern selection in fingered growth phenomena," Adv. Phys. 37, 255 (1988).
3. P. W. Voorhees, G. B. McFadden, R. F. Boisvert, and D. I. Meiron, "Numerical simulation of morphological development during Ostwald ripening," Acta Metall. 36, 207 (1988).
4. P. Constantin and M. Pugh, "Global solutions for small data to the Hele-Shaw problem," Nonlinearity 6, 393 (1993).
5. X. Chen, "The Hele-Shaw problem and area-preserving curve-shortening motions," Arch. Rat. Mech. Anal. 123, 117 (1993).
6. P. Fife, Seminar at University of Utah, 1991.
7. D. P. Jackson, R. E. Goldstein, and A. O. Cebers, "Hydrodynamics of fingering instabilities in dipolar fluids," Phys. Rev. E 50, 298 (1994).
8. J. E. Taylor and J. W. Cahn, "Linking anisotropic sharp and diffuse surface motion laws via gradient flows," J. Stat. Phys. 77, 183 (1994).
9. F. Almgren and J. E. Taylor, "Optimal geometry in equilibrium and growth," Fractals (in press).
10. J. E. Taylor, "Mean curvature and weighted mean curvature (Overview no. 98-2)," Acta Metall. Mater. 40, 1475 (1992).
11. M. A. Grayson, "The heat equation shrinks embedded plane curves to round points," J. Diff. Geom. 26, 285 (1987).
12. M. Gage and R. S. Hamilton, "The heat equation shrinking convex plane curves," J. Diff. Geom. 23, 69 (1986).
13. M. A. Grayson, "A short note on the evolution of a surface by its mean curvature," Duke Math. J. 58, 555 (1989).
14. M. Gage, "On an area-preserving evolution equation for plane curves," in Nonlinear Problems in Geometry, Vol. 51 of Contemporary Mathematics, edited by D. M. DeTurck (American Mathematical Society, Providence, 1986), pp. 51-62.
15. J. W. Cahn and J. E. Taylor, "Surface motion by surface diffusion (Overview no. 113)," Acta Metall. Mater. 42, 1045 (1994).
16. C. M. Elliott and H. Garcke, "Existence results for diffusive surface motion laws," preprint (1994).
17. A. Cameron, Principles of Lubrication (Longmans, London, 1966).
18. R. E. Goldstein, A. I. Pesci, and M. J. Shelley, "Topology transitions and singularities in viscous flows," Phys. Rev. Lett. 70, 3043 (1993).
19. P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, M. Shelley, and S.-M. Zhou, "Droplet breakup in a model of the Hele-Shaw cell," Phys. Rev. E 47, 4169 (1993).
20. T. Dupont, R. Goldstein, L. Kadanoff, and S.-M. Zhou, "Finite-time singularity formation in Hele-Shaw systems," Phys. Rev. E 47, 4182 (1993).
21. R. Almgren, A. Bertozzi, and M. P. Brenner, "Stable and unstable singularities in the unforced Hele-Shaw cell," submitted to Phys. Fluids (1995).
22. A. L. Bertozzi, M. P. Brenner, T F. Dupont, and L. P. Kadanoff, "Singularities and similarities in interface flows," in Trends and Perspectives in Applied Mathematics, edited by L. Sirovich (Springer-Verlag, Berlin, 1994), pp. 155-208.
23. W.-S. Dai and M. J. Shelley, "A numerical study of the effect of surface tension and noise on an expanding Hele-Shaw bubble," Phys. Fluids A 5, 2131 (1993).
24. G. Tryggvason and H. Aref, "Numerical experiments on Hele-Shaw flow with a sharp interface," J. Fluid Mech. 136, 1 (1983).
25. R. Almgren, W.-S. Dai, and V. Hakim, "Scaling behavior in anisotropic Hele-Shaw flow," Phys. Rev. Lett. 71, 3461 (1993).
26. L. Greengard and V. Rokhlin, "A fast algorithm for particle summations," J. Comput. Phys. 73, 325 (1987).
27. R. C. Brower, D. A. Kessler, J. Koplik, and H. Levine, "Geometrical models of interface evolution," Phys. Rev. A 29, 1335 (1984).
28. T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, "Removing the stiffness from interfacial flows with surface tension," J. Comput. Phys. 114, 312 (1994).
29. Y. Saad and M. H. Schultz, "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM J. Sci. Stat. Comput. 7, 856 (1986).
30. R. F. Almgren, "Computing Laplacian dendrites," in Free Boundary Problems: Theory and Applications, edited by J. I. Diaz, M. A. Herrero, A. Liñan, and J. L. Vazquez (Longman Scientific and Technical, Essex, 1995), pp. 1-12.