The Dirichlet problem for Stokes equations outside open arcs in a half-plane and creeping flow over thin profiles
Author:
P. A. Krutitskii
Journal:
Quart. Appl. Math. 68 (2010), 537-556
MSC (2000):
Primary 35Q30, 76D07, 31A10, 45F05, 45F15
DOI:
https://doi.org/10.1090/S0033-569X-2010-01162-3
Published electronically:
May 27, 2010
MathSciNet review:
2676975
Full-text PDF Free Access
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Abstract: We study the Dirichlet problem for Stokes equations outside curvilinear open arcs in a half-plane. We prove existence and uniqueness of a classical solution to this problem. We obtain integral representation for a solution in the form of potentials, densities in which can be found as a unique solution of the system of the Fredholm integral equations of the second kind and index zero. The creeping flow of viscous fluid over thin profiles is described by the Dirichlet problem studied in this paper.
References
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References
- Popov A.N. Application of potential theory to solving the linearized Navier-Stokes system of equations in a two-dimensional case. Trudy MIAN, 1971, t. 116, pp. 162–180. (In Russian). MR 0364909 (51:1163)
- Pozrikidis C. Boundary integral and singularity method for linearized viscous flow. Cambridge University Press, Cambridge, 1992. MR 1156495 (93a:76027)
- Varnhorn W. The Stokes equations. Akademie Verlag, Berlin, 1994. MR 1282728 (95e:35162)
- Power H. The completed double layer boundary integral equation method for two-dimensional Stokes flow. IMA Journal of Applied Mathematics, 1993, v. 51, pp. 123–145. MR 1244192 (94i:76020)
- Vladimirov V.S. Equations of Mathematical Physics. Nauka, Moscow, 1981. (In Russian; English translation of 1st edition: Marcel Dekker, N.Y., 1971.)
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- Krutitskii P.A. On Stokes potentials for velocities. Russian mathematical surveys, 2007, v. 62, No. 6, pp. 1212–1214. MR 2382807 (2008k:35378)
- Krutitskii P.A. On properties of some integrals related to potentials for Stokes equations. Quarterly of Applied Mathematics, 2007, v. 65, No. 3, pp. 549-569. MR 2354887 (2008i:31003)
- Muskhelishvili N.I. Singular integral equations. Nauka, Moscow, 1968. (In Russian; English translation: Noordhoff, Groningen, 1972.) MR 0355494 (50:7968)
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- Hsu R., Canatos P. The motion of a rigid body in a viscous fluid bounded by a plane wall. J. Fluid Mech., 1989, v. 207, pp. 29–72.
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Additional Information
P. A. Krutitskii
Affiliation:
KIAM, Miusskaya Sq. 4, Moscow 125047, Russia
Received by editor(s):
November 24, 2008
Published electronically:
May 27, 2010
Article copyright:
© Copyright 2010
Brown University