Summary
The theory of algebraic curves and quadrature domains is used to construct exact solutions to the problem of the squeeze flow of multiply-connected fluid domains in a Hele-Shaw cell. The solutions are exact in that they can be written down in terms of a finite set of time-evolving parameters. The method is very general and applies to fluid domains of any finite connectivity. By way of example, the evolution of fluid domains with two and four air holes are calculated explicitly. For simply connected domains, the squeeze flow problem is well posed. In contrast, the squeeze flow problem for a multiply connected domain is not necessarily well-posed and solutions can break down in finite time by the formation of cusps on the boundaries of the enclosed air holes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Ablovitz & A. S. Fokas, Complex variables, Cambridge University Press, Cambridge (1997).
D. Aharonov & H. Shapiro, Domains on which analytic functions satisfy quadrature identities, J. Anal. Math., 30, 39–73 (1976).
D. G. Crowdy, Theory of exact solutions for the evolution of a fluid annulus in a rotating Hele-Shaw cell, Q. Appl. Math. (to appear).
D. G. Crowdy, On a class of geometry-driven free boundary problems, SIAM J. Appl. Math. (to appear).
D. G. Crowdy, On the construction of multiply-connected quadrature domains, SIAM J. Appl. Math. (to appear).
V. M. Entov, P. I. Etingof, & D. Ya Kleinbock, On nonlinear interface dynamics in Hele-Shaw flows, Eur. J. Appl. Math., 6, 399–420, (1995).
B. Gustafsson, Quadrature identities and the Schottky double, Acta. Appl. Math., 1, 209–240 (1983).
B. Gustafsson, Singular and special points on quadrature domains from an algebraic geometric point of view, J. Anal. Math., 51, 91–117 (1988).
P. P. Kufarev, The oil contour problem for the circle with any number of wells, Dokl. Akad. Nauk SSSR, 75, 507–510 (1950).
G. A. Jones & D. Singerman, Complex functions, Cambridge University Press, Cambridge (1987).
P. Ya. Polubarinova-Kochina, On the motion of the oil contour, Dokl. Akad. Nauk SSSR, 47, 254–257 (1945).
S. Richardson, Some Hele-Shaw flows with time-dependent free boundaries, J. Fluid Mech., 102, 263–278 (1981).
S. Richardson, Hele-Shaw flows with time-dependent free boundaries in which the fluid occupies a multiply connected region, Eur. J. Appl. Math., 5, 97–122 (1994).
S. Richardson, Hele-Shaw flows with time-dependent free boundaries involving a concentric annulus, Phil. Trans. R. Soc. Land., 354, 2513–2553 (1996).
M. Sakai, Quadrature domains, Lecture Notes in Mathematics, 934, Springer-Verlag, New York (1982).
S. Saks & A. Zygmund, Analytic functions, 3rd ed., Elsevier, Amsterdam (1971).
H. S. Shapiro, Unbounded quadrature domains, in Complex Analysis I, Proceedings, University of Maryland 1985–86, C. A. Bernstein (ed.), Lecture Notes in Mathematics, 1275, Springer-Verlag, Berlin (1987), pp. 287–331.
M. J. Shelley, F. R. Tian, & K. Wlodarski, Hele-Shaw flow and pattern formation in a timedependent gap, Nonlinearity, 10, 1471–1495 (1997).
M. Siegel, S. Tanveer, & W. S. Dai, Singular effects of surface tension in evolving Hele-Shaw cells, J. Fluid Mech., 323, pp. 201–236 (1996).
G. Valiron, Cours d’analyse mathematique: Theorie des fonctions, 2nd ed., Masson et Cie, Paris (1940).
A. N. Varchenko & P. I. Etingof, Why the boundary of a round drop becomes a curve of order four, University Lecture Series, American Mathematical Society, Providence, RI (1992).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Shelley
Rights and permissions
About this article
Cite this article
Crowdy, D., Kang, H. Squeeze flow of multiply-connected fluid domains in a Hele-Shaw cell. J. Nonlinear Sci. 11, 279–304 (2001). https://doi.org/10.1007/s00332-001-0397-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-001-0397-5