Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T06:37:20.861Z Has data issue: false hasContentIssue false
  • English
  • Français

Hele–Shaw flows with a free boundary produced by multipoles

Published online by Cambridge University Press:  26 September 2008

Vladimir M. Entov ,
Pavel I. Etingof  and
Dmitry Ya. Kleinbock
Vladimir M. Entov
Affiliation:
Institute for Problems in Mechanics, Russian Academy of Sciences, prosp. Vernadskogo, 101 Moscow, Russia
Pavel I. Etingof
Affiliation:
Yale University, Department of Mathematics, 2155 Yale Station, New Haven, CT 06520USA
Dmitry Ya. Kleinbock
Affiliation:
Yale University, Department of Mathematics, 2155 Yale Station, New Haven, CT 06520USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We study Hele–Shaw flows with a moving boundary and multipole singularities. We find that such flows can be defined only on a finite time interval. Using a complex variable approach, we construct a family of explicit solutions for a single multipole. These solutions turn out to have the maximal possible lifetime in a certain class of solutions.

We also discuss the generalized Hele-Shaw model in which surface tension at the moving boundary is considered, and develop a method of finding steady shapes. This method yields new one-parameter families of stationary solutions. In the Appendix we discuss a connection between these solutions and a variational problem of potential theory.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 4 , Issue 2 , June 1993 , pp. 97 - 120
Copyright
Copyright © Cambridge University Press 1993

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

[1]Kufarev, P. P. 1948 The solution of the problems of oil contour shrinkage for the circle. Dolk. Akad. Nauk. SSSR 60, 1333–1334.Google Scholar
[2]Kufarev, P. P. 1950 The oil contour problem for the circle with any number of wells. Dolk. Akad. Nauk. SSSR 75 (4), 507–510.Google Scholar
[3]Kufarev, P. P., Astafiev, P. P. & Boletski, K. A. 1952 Uchenyezapiski. Tomskogo Universiteta 17, 129–140.Google Scholar
[4]Richardson, S. 1972 Hele-Shaw flows with a free boundary produced by injection of fluid into a narrow channel. J. Fluid Mech. 56, 609–618.CrossRefGoogle Scholar
[5]Richardson, S. 1981 Some Hele–Shaw flows with time-dependent free boundaries. J. Fluid Mech. 102, 263–278.CrossRefGoogle Scholar
[6]Etingof, P. I. 1990 Integrability of the problem of filtration with a moving boundary. Soviet Phys. Dokl. 35 (7), 625–628.Google Scholar
[7]Howison, S. D. 1992 Complex variable methods in Hele-Shaw moving boundary problems. Euro. J. Appl. Math. 3 (3), 209–224.CrossRefGoogle Scholar
[8]Lacey, A. A. 1982 Moving boundary problems in the flow of liquid through porous media. J. Austr. Math. Soc. B24, 171–193.CrossRefGoogle Scholar
[9]Millar, R. F. 1980 The analytic continuation of solutions to elliptic boundary value problems in two independent variables. J. Math. Anal. Appl. 76, 498–515.CrossRefGoogle Scholar
[10]Millar, R. F. 1989 An inverse moving boundary problem for Laplace's equation. In: Proc. Workshop on Inverse Problems and Imaging, Pitman-Longman.Google Scholar
[11]Richardson, S. 1991 Hele-Shaw flows with time-dependent free boundaries involving injection through slits. Preprint.CrossRefGoogle Scholar
[12]Varchenko, A. N. & Etingof, P. I. 1992 Why the Boundary of a Round Drop Becomes a Curve of Order Four. AMS, Providence (In Press).Google Scholar
[13]Entov, V. M. & Etingof, P. I. 1991 Bubble contraction in Hele–Shaw cells. Quart. J. Mech. Appl. Math. 44 (4), 507–535.CrossRefGoogle Scholar
[14]Howison, S. D. 1986 Bubble growth in porous media and Hele–Shaw cells. Proc. Royal Soc. Ed. 102, 141–148.CrossRefGoogle Scholar
[15]Howison, S. D. 1986 Cusp development in Hele–Shaw flow with a free surface. SIAM J. Appl. Math. 46 (1), 20–26.CrossRefGoogle Scholar
[16]Howison, S. D., Lacey, A. A. & Ockendon, J. R. 1985 Singularity development in moving boundary problems. Quart. J. Mech. Appl. Math. 38, 343–360.CrossRefGoogle Scholar
[17]Ahlfors, L. V. & Beurling, A. 1950 Conformal invariants and function-theoretic null sets. Acta Math. 83, 101–129.CrossRefGoogle Scholar
[18]Bensimon, D., Kadanoff, L. P., Liang, S., Shraiman, B. & Tang, C. 1986 Viscous flows in two dimensions. Rev. Mod. Phys. 58, 977–999.CrossRefGoogle Scholar
[19]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. Royal Society. Ser. A. 245, 312–329.Google Scholar
[20]Taylor, G. I. & Saffman, P. G. 1959 A note on the motion of bubbles in a Hele-Shaw cell and porous medium. Quart. J. Mech. Appl. Math. 12, 265–279.CrossRefGoogle Scholar
[21]Kadanoff, L. P. 1990 Exact solutions of Saffman-Taylor problem with surface tension. Phys. Rev. Lett. 65, 2989–2988.CrossRefGoogle ScholarPubMed
[22]Vasconcelos, G. L. & Kadanoff, L. P. 1991 Stationary solutions for the Saffman-Taylor problem with surface tension. Phys. Rev. A44 (10), 6490–6495.CrossRefGoogle Scholar
[23]Davis, P. J. 1974 Schwarz function and its applications. Springer-Verlag.CrossRefGoogle Scholar
[24]Millar, R. F. 1990 Inverse problems for a class of Schwarz functions. Compl. Var. Th. Appl. 15, 1–10.Google Scholar
[25]Goldstein, R. V. & Entov, V. M. 1989 Qualitative Methods in Continuum Mechanics. Nauka, Moskva.Google Scholar