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A geometric approach to the study of stationary free surface flows for viscous liquids

Published online by Cambridge University Press:  14 November 2011

Frédéric Abergel
Frédéric Abergel
Affiliation:
C.N.R.S. etUniversité Paris–Sud, Laboratoire d'Analyse Numérique, Bâtiment 425, 91405 Orsay, France
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Synopsis

We use a direct, geometric approach to study the free surface boundary conditions for stationary flows of viscous liquids. The free surface problem is characterised by a mapping on smooth variations of a given configuration; this mapping has a simple structure, which we determine by computing its differential, and studying it in terms of the space dimension and the surface tension coefficient. Applications are given to problems of existence, uniqueness and regularity in free surface flows.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 2 , 1993 , pp. 209 - 229
Copyright
Copyright © Royal Society of Edinburgh 1993

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References

1Abergel, F. and Bona, J. L.. A mathematical theory for viscous, free surface flows over a perturbed plane. Arch. Rational Mech. Anal. 118 (1992), 71–93. Abergel, F.. Some recent results on the stationary Navier-Stokes flows with a free surface, Mémoire d'habilitation, Université Paris XI, 1992.CrossRefGoogle Scholar
2Abergel, F. and Bona, J. L.. Free surface flows of a viscous fluid in a rotating cylinder (to appear).Google Scholar
3Alinhac, S. and Gérard, P.. Opérateurs pseudodifférentiels et théorème de Nash-Moser (Publications Universitaires Scientifiques, Université Paris XI, 1989).Google Scholar
4Allain, G.. Rôle de la tension superflcielle dans la convection de Bénard. Math. Modelling Numerical Anal. 24 (1990), 153–176.CrossRefGoogle Scholar
5Auchmuty, J. F.G.. Existence of axisymmetric equilibrium figures. Arch. Rational Mech. Anal. 65 (1977), 248–261.CrossRefGoogle Scholar
6Bemelmans, J.. On a free boundary problem for the stationary Navier–Stokes equations. Ann. Inst. H. Poincaré, Anal. Non Linéaire 4 (1987), 517–547.CrossRefGoogle Scholar
7Bemelmans, J.. Liquid drops in a viscous fluid under the influence of gravity and surface tension. Manuscripta Math. 36 (1981), 105–123.CrossRefGoogle Scholar
8Bemelmans, J.. Gleichgewichtsfiguren zäher Flissigkeiten mit Overflächenspaning. Analysis 1 (1981), 241–282.CrossRefGoogle Scholar
9Bemelmans, J. and Friedman, A.. Analyticity for the Navier–Stokes equations governed by surface tension on the free boundary. J. Differential Equations 55 (1984), 135–150CrossRefGoogle Scholar
10Benjamin, T. B. and Pathak, S. K.. Cellular flows of a viscous liquid that partly fills a horizontal rotating cylinder. J. Fluid Mech. 183 (1987), 399–420.CrossRefGoogle Scholar
11Doubrovine, B., Novikov, S. and Fomenko, A.. Géométrie contemporaine, Méthodes et Applications, vol. I (Moscow; Editions Mir, 1982).Google Scholar
12Finn, R.. Equilibrium Capillary Surfaces (Berlin: Springer, 1986).CrossRefGoogle Scholar
13Hadamard, J.. Miméire sur le problème d'Analyse relatif à l'équilibre des plaques élastiques encastrées, 1907. Hadamard, Oeuvres de J., vol. 2 (Paris: CNRS, 1968).Google Scholar
14Hörmander, L.. The boundary problem of physical geodesy. Arch. Rational Mech. Anal. 62 (1976), 1–52.CrossRefGoogle Scholar
15Jean, M.. Free surface of the steady flow of a Newtonian fluid in a finite channel. Arch. Rational Mech. Anal. 74 (1980), 197–217.CrossRefGoogle Scholar
16Lions, J. L. and Magenes, E.. Problèmes aux limites non homogènes et applications, vol. 1 (Paris: Dunod, 1968).Google Scholar
17Murat, F. and Simon, J.. Sur le contrôle par un domaine géométrique (Rapport du Laboratoire d'analyse Numérique 189, Université Paris VI, 1976).Google Scholar
18Nazarov, S. and Pileckas, K.. On noncompact free boundary problems for the plane stationary equations. J. Reine Angew. Math, (submitted).Google Scholar
19Oleinik, O. and Kondrat'ev, V. A.. Hardy's and Korn's type inequalities and their applications, Rend. Nat. Appl. (7)10 (1990), no 3, 641–66.Google Scholar
20Simon, J.. Variations with respect to domain for Neumann condition. Proceedings of the 1986 IFAC Congress Control of Distributed Parameter Systems, Pasadena.Google Scholar
21Solonnikov, V. A.. General boundary-value problem for Doublis–Nirenberg elliptic systems II. Proc. Steklov Inst. Math. 92 (1966) (Providence, R.I.: American Mathematical Society, 1968).Google Scholar
22Solonnikov, V.A., Solvability of a problem on the motion of a viscous, incompressible fluid bounded by a free surface. Math. USSR Izvestija 11 (1977), 1323–1358.CrossRefGoogle Scholar
23Solonnikov, V. A., On the Stokes equations in domains with non smooth boundaries and on viscous incompressible flows with a free surface. Nonlinear Partial Differential Equations and their Applications, vol. III, Collège de France Seminar 1980–1981, eds Brezis, H., Lions, J. L., Research Notes in Mathematics 70, pp. 340–423 (Boston: Pitman, 1982).Google Scholar
24Solonnikov, V. A.. Solvability of the problem of effluence of a viscous incompressible fluid into an infinite open basin. Proc. Steklov Inst. Math. 179 (1989), 192–225.Google Scholar
25Zehnder, E.. Generalized implicit function theorems with applications to some small divisor problems I. Comm. Pure Appl. Math. 28 (1975), 91–140.CrossRefGoogle Scholar