Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T04:39:23.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Complex variable methods in Hele–Shaw moving boundary problems

Published online by Cambridge University Press:  16 July 2009

S. D. Howison
S. D. Howison
Affiliation:
Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss the one-phase Hele–Shaw problem in two space dimensions. We review exact solutions in the zero-surface-tension case, giving a unified account of the Schwarz function and conformal mapping approaches. We discuss the extension of the former method to the cases in which surface tension or ‘kinetic undercooling’ terms apply on the moving boundary, and we give some conjectures on the resulting singularity structure. Finally, we give a new interpretation of the linear stability analysis of the zero-surface-tension problem, and we suggest a possible regularization of ill-posed problems by the imposition of a unilateral constraint on the moving boundary.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 3 , September 1992 , pp. 209 - 224
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aitchison, J. M., Elliott, C. M. & Ockendon, J. R. 1983 Percolation in gently sloping beaches. IMA J. Appl. Math. 30, 269–287.CrossRefGoogle Scholar
Clarkson, P., Fokas, A. S. & Ablowitz, M. J. 1989 Hodograph transformations of linearizable partial differential equations. SIAM J. Appl. Math. 49, 1188–1209.CrossRefGoogle Scholar
Davis, P. J. 1974 The Schwarz Function and its Applications. Carus Math. Monographs 17, Math. Assoc. of America.Google Scholar
DiBenedetto, E. & Friedman, A. 1984 The ill-posed Hele–Shaw model and the Stefan problem for supercooled water. Trans. Am. Math. Soc. 282, 183–204.CrossRefGoogle Scholar
Duchon, J. & Robert, R. 1984 Evolution d'une interface par capillarité et diffusion de volume, I: existence locale en temps. Ann. Inst. H. Poincaré. 1, 361–378.CrossRefGoogle Scholar
Elliott, C. M. & Ockendon, J. R. 1982 Weak and Variational Methods for Free and Moving Boundary Problems. London: Pitman.Google Scholar
Elliott, C. M. & Janovsky, V. 1981 A variational inequality approach to Hele–Shaw flow with a moving boundary. Proc. Roy. Soc. Edin. A88, 93–107.CrossRefGoogle Scholar
Galin, L. A. 1945 Unsteady filtration with a free surface. Dokl. Akad. Nauk. S.S.S.R. 47, 246–249 (in Russian).Google Scholar
Gustafsson, B. 1984 Existence of weak backward solutions to a generalised Hele–Shaw moving boundary problem. Nonlinear Analysis, T.M.A. 9, 203–215.CrossRefGoogle Scholar
Gustafsson, B. 1987 An ill-posed moving boundary problem for doubly-connected domains. Arkiv für Mathematik. 25, 231–253.CrossRefGoogle Scholar
Hohlov, Yu. E. 1990 Time-dependent free boundary problems: the explicit solutions. M.I.A.N. Preprint no. 14 (Steklov Institute, Moscow).Google Scholar
Hohlov, Yu. E. 1990 Private communication.Google Scholar
Hohlov, Yu. E. & Howison, S. D. 1992 A note on the classification of solutions to the zero-surface- tension model for Hele–Shaw flows. Preprint.CrossRefGoogle Scholar
Hopper, R. W. 1990 Plane Stokes flow driven by capillarity on a free surface. J. Fluid Mech. 213, 349–375.CrossRefGoogle Scholar
Howison, S. D. 1985 Bubble growth in porous media and Hele–Shaw cells. Proc. Roy. Soc. Edin. A102, 141–148.CrossRefGoogle Scholar
Howison, S. D. 1986 Fingering in Hele–Shaw cells. J. Fluid Mech. 167, 439–453.CrossRefGoogle Scholar
Howison, S. D. 1986 Cusp development in Hele–Shaw flow with a free surface. SIAM J. Appl. Math. 46, 20–26.CrossRefGoogle Scholar
Howison, S. D., Lacey, A. A. & Ockendon, J. R. 1985 Singularity development in moving boundary problems. Q. J. Mech. Appl. Math. 38, 343–360.CrossRefGoogle Scholar
Howison, S. D., Lacey, A. A. & Ockendon, J. R. 1988 Hele–Shaw free boundary problems with suction. Q. J. Mech. Appl. Anal. 41, 184–193.Google Scholar
Ivantsov, G. P. 1947 Dokl. Akad. Nauk. S.S.S.R. 58, 567 (in Russian).Google Scholar
Kadanoff, L. 1990a Private communication.Google Scholar
Kadanoff, L. 1990b Exact solutions for the Saffman–Taylor problem with surface tension. Phys. Rev. Lett. 65, 2986–2988.CrossRefGoogle ScholarPubMed
Kufarev, P. P., Astafiev, P. P. & Boletski, K. A. 1952 Uchen. Zapiski Tomsk Univ. 17, 129–140 (in Russian).Google Scholar
Kufarev, P. P. 1948 Dokl. Akad. Nauk. S.S.S.R. 60, 1333–1334 (in Russian).Google Scholar
Kufarev, P. P. 1950 Dokl. Akad. Nauk. S.S.S.R. 75, 353–355 (in Russian).Google Scholar
Lacey, A. A. 1982 Moving boundary problems in the flow of liquid through porous media. J. Austral. Math. Soc. B24, 171–193.CrossRefGoogle Scholar
Lamé, G. & Clapeyron, B. P. 1831 Memoire sur la solidification per refroidissement d'un globe liquide. Ann. Chim. Phys. 47, 250–256.Google Scholar
Millar, R. F. 1989a The steady motion and shape of a bubble in a Hele–Shaw cell. In: Proc. Can. Appl. Math. Soc. Conf. on Continuum Mechanics & Applications. Hemingway.Google Scholar
Millar, R. F. 1989b An inverse moving boundary problem for Laplace's equation. In: Proc. Workshop on Inverse Problems and Imaging, (ed. Roach, G. F.). London: Pitman-Longman.Google Scholar
Ockendon, J. R. 1991 A class of moving boundary problems arising in industry. In: Applied and Industrial Maths, (ed. Spigler, R.). Amsterdam: Kluwer, pp. 141–150.CrossRefGoogle Scholar
Polubarinova-Kochina, P. Ya. 1945a Dokl. Akad. Nauk. S.S.S.R. 47, 254–257 (in Russian).Google Scholar
Polubarinova-Kochina, P. Ya. 1945b Prikl. Matem. Mech. 9, 79–90 (in Russian).Google Scholar
Richardson, S. 1972 Hele–Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56, 609–618.CrossRefGoogle Scholar
Richardson, S. 1981 Some Hele–Shaw flows with time-dependent free boundaries. J. Fluid Mech. 102, 263–278.CrossRefGoogle Scholar
Romero, L. 1982 Ph.D. Thesis, Caltech.Google Scholar
Saffman, P. G. 1959 Exact solutions for the growth of fingers from a flat interface between two fluids in a porous medium or Hele–Shaw cell. Q. J. Mech. Appl. Maths. 12, 146–150.CrossRefGoogle Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous liquid. Proc. Roy. Soc. A245, 312–329.Google Scholar
Schaefer, R. J. & Glicksman, M. E. 1969 Fully time-dependent theory for the growth of spherical crystal nuclei. J. Crystal Growth. 5, 44–58.CrossRefGoogle Scholar
Shraiman, B. & Bensimon, D. 1984 Singularities in nonlocal interface dynamics. Phys. Rev. A30, 2840–2842.CrossRefGoogle Scholar
Tanveer, S. A. 1991 Evolution of Hele–Shaw interface for small surface tension. Preprint.Google Scholar
Taylor, G. I. & Saffman, P. G. 1959 A note on the motion of bubbles in a Hele–Shaw cell and porous medium. Q. J. Mech. Appl. Math. 12, 265–279.CrossRefGoogle Scholar
Vinogradov, Yu. P. & Kufarev, P. P. 1947 Dokl. Akad. Nauk. S.S.S.R. 57, 335–338 (in Russian).Google Scholar
Vinogradov, Yu. P. & Kufarev, P. P. 1948 Prikl. Matem. Mech. 12, 181–198 (in Russian).Google Scholar
Worster, M. G. 1990 Private communication.Google Scholar
Zhuravlev, P. A. 1956 Zap. Leningr. Com. Inst. 33, 54–61 (in Russian).Google Scholar