Theory of exact solutions for the evolution of a fluid annulus in a rotating Hele-Shaw cell
Author:
Darren Crowdy
Journal:
Quart. Appl. Math. 60 (2002), 11-36
MSC:
Primary 76D27; Secondary 30C20, 35R35, 76E17, 76M40, 76U05
DOI:
https://doi.org/10.1090/qam/1878257
MathSciNet review:
MR1878257
Full-text PDF Free Access
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Abstract: Motivated by a series of recent experiments on the evolution of fluid annuli in rotating Hele-Shaw cells, this paper presents a new class of exact time-dependent solutions to a mathematical model of this nonlinear free boundary problem. For a certain class of initial conditions, the free boundary problem is reduced to the solution of a finite set of coupled nonlinear ordinary differential equations. These solutions can be explicitly studied and, despite the fact that the model problem is mathematically ill-posed, display the same qualitative features as the recent experiments. It is discussed how the present exact solutions might form an important basis for further study of the appropriately regularized model problem.
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L. Carrillo, F. X. Magdeleno, J. Casademunt, and J. Ortin, Experiments in a rotating Hele-Shaw cell, Phys. Rev. E 54, 6260 (1996)
L. Carrillo, J. Soriano, and J. Ortin, Radial displacement of a fluid annulus in a rotating Hele-Shaw cell, Phys. Fluids 11 (4), 778 (1999)
L. Carrillo, J. Soriano, and J. Ortin, Interfacial instabilities of a fluid annulus in a rotating Hele-Shaw cell, Phys. Fluids 12 (7), 1685–1698 (2000)
D. Crowdy and S. Tanveer, A theory of exact solutions for annular viscous blobs, J. Nonlinear Sci. 8, 261–279 (1998)
D. G. Crowdy, A note on viscous sintering and quadrature identities, European J. Appl. Math. 10, 623–634 (1999)
D. G. Crowdy, Exact solutions for steady capillary waves on a fluid annulus, J. Nonlinear Sci. 9, 615–640 (1999)
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)
P. J. Davis, The Schwarz Function and its Applications, Carus Mathematical Monographs, No. 17, The Mathematical Association of America, Buffalo, NY, 1974
V. M. Entov, P. I. Etingof, and D. Ya. Kleinbock, On nonlinear interface dynamics in Hele-Shaw flows, European J. Appl. Math. 6, 399–420 (1995)
B. Gustafsson, Quadrature domains and the Schottky double, Acta Appl. Math. 1, 209–240 (1983)
M. Heins, Complex Function Theory, Academic Press, New York, 1968
S. Richardson, Hele-Shaw flows with time-dependent free boundaries involving a concentric annulus, Philos. Trans. Roy. Soc. London Ser. A 354, 2513–2553 (1996)
P. G. Saffman and G. I. Taylor, The penetration of a fluid into a porous medium of Hele-Shaw cell containing a more viscous fluid, Proc. Roy. Soc. London A, 245, 312–329 (1958)
S. Saks and A. Zygmund, Analytic Functions, Elsevier Publishing Company, Amsterdam, London, New York, 1971
L. W. Schwartz, Instability and fingering in a rotating Hele-Shaw cell or porous medium, Phys. Fluids 1 (2), 167–169 (1989)
M. Siegel, S. Tanveer, and W.-S. Dai, Singular effects of surface tension in evolving Hele-Shaw flows, J. Fluid Mech. 323, 201–236 (1996)
S. Tanveer, Evolution of a Hele-Shaw interface for small surface tension, Philos. Trans. Roy. Soc. London A, 343, 155–204 (1993)
S. Tanveer, Surprises in viscous fingering, J. Fluid Mech. 409, 273–308 (2000)
G. Valiron, Cours d’Analyse Mathématique: Théorie des Fonctions, 2nd Edition, Masson et Cie, Paris, 1947
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© Copyright 2002
American Mathematical Society
