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On two-dimensional slow viscous flows past obstacles in a half-plane*

Published online by Cambridge University Press:  14 November 2011

T. M. Fischer ,
G. C. Hsiao  and
W. L. Wendland
T. M. Fischer
Affiliation:
Institut für Theoretische Strömungsmechanik, DFVLR, D-3400 Göttingen, Federal Republic of Germany
G. C. Hsiao
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19711, U.S.A.
W. L. Wendland
Affiliation:
Mathematisches Institut A, Universität Stuttgart, D-7000 Stuttgart 80, Federal Republic of Germany
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Synopsis

We consider a cylinder with arbitrary cross section moving in a viscous incompressible fluid parallel to a plane wall. Formal asymptotic expansions of the solution for small Reynolds numbers are constructed by using boundary integral equations of the first kind. In contrast to the problem without a wall, we show that there exists a unique solution to the zeroth order problem. However, the problem considered here is still singular in the sense that we find the Stokes paradox in the next higher order problem. A justification of the formal asymptotic expansion for the first two terms is established rigorously.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 3-4 , 1986 , pp. 205 - 215
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Babenko, K. I.. Perturbation theory of stationary flows of a viscous incompressible fluid for small Reynolds numbers. Soviet Phys. Dokl. 21 (1976), 143–145.Google Scholar
2Finn, R. and Smith, D. R.. On the stationary solutions of the Navier–Stokes equations in two dimensions. Arch. Rational Mech. Anal. 25 (1976), 26–39.CrossRefGoogle Scholar
3Fischer, T. M.. An integral equation procedure for the exterior three-dimensional slow viscous flow. Integral Equations Operator Theory 5 (1982), 490–505.CrossRefGoogle Scholar
4Fischer, T. M.. Über die langsame Bewegung eines starren Kö;rpers in einer zähen, inkompressiblen Flüssigkeit längs einer ebenen Wand. Ph.D. thesis, Technische Hochschule Darmstadt, 1983.Google Scholar
5Fischer, T. M.. Wall effects on the slow steady motion of a particle in a viscous incompressible fluid. Math. Methods Appl. Sci. 8 (1986), 23–40.CrossRefGoogle Scholar
6Fischer, T. M., Hsiao, G. C. and Wendland, W. L.. Singular perturbations for the exterior three-dimensional slow viscous flow problem. J. Math. Anal. Appl. 110 (1985), 583–603.CrossRefGoogle Scholar
7Frazer, R. A.. On the motion of circular cylinders in a viscous fluid. Philos. Trans. Roy. Soc. London Ser. A225 (1926), 93–130.Google Scholar
8Hsiao, G. C. and MacCamy, R. C.. Solution of boundary value problems by integral equations of the first kind. SIAM Rev. 15 (1973), 687–705.CrossRefGoogle Scholar
9Hsiao, G. C. and MacCamy, R. C.. Singular perturbations for the two-dimensional viscous flow problem. Lecture Notes in Mathematics 942, pp. 229–244. (Berlin: Springer-Verlag, 1982).Google Scholar
10Hsiao, G. C. and Wendland, W. L.. A finite element method for some integral equations of the first kind. J. Math. Anal. Appl. 58 (1977), 449–481.CrossRefGoogle Scholar
11Hsiao, G. C., Kopp, P. and Wendland, W. L.. Some applications of a Galerkin-collocation method for integral equations of the first kind. Math. Methods Appl. Sci. 6 (1984), 280–325.CrossRefGoogle Scholar
12Ladyzhenskaya, O. A.. The Mathematical Theory of Viscous Incompressible Flow (New York: Gordon and Breach, 1969).Google Scholar
13Odqvist, F. K. G.. Über die Randwertaufgaben der Hydrodynamik zäher Flüssigkeiten. Math. Z. 32 (1930), 329–375.CrossRefGoogle Scholar
14Oseen, C. W.. Neuere Methoden und Ergebnisse in der Hydrodynamik (Leipzig: Akad. Verlagsgesellschaft, 1927).Google Scholar
15Smith, D. R.. Estimates at infinity for stationary solutions of the Navier–Stokes equations in two dimensions. Arch. Rational Mech. Anal. 20 (1965), 341–372.CrossRefGoogle Scholar
16Socolescu, D.. On the relation between the Leray class of solutions with finite Dirichlet integral and Finn's class of physically reasonable solutions of the Dirichlet problem for the steady plane Navier-Stokes equations (Bericht Nr. 23, Universitat Karlsruhe, 1984).Google Scholar
17Takaisi, Y.. Note on the drag on a circular cylinder moving with low speeds in a serai-infinite viscous liquid bounded by a plane wall. J. Phys. Soc. Japan 11 (1956), 1004–1008.CrossRefGoogle Scholar
18Vasil'ev, M. M.. Determination of the flow field in a region adjacent to a streamlined body at small Reynolds number. J. Appl. Math. Mech. 41 (1977), 1072–1078.CrossRefGoogle Scholar
19Wendland, W. L.. On some mathematical aspects of boundary element methods for elliptic problems. In The Mathematics of Finite Elements and Applications V (ed. Whiteman, J. R.), pp. 193–227 (London, New York: Academic Press, 1985).CrossRefGoogle Scholar