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A direct boundary integral equation formulation for the Oseen flow past a two-dimensional cylinder of arbitrary cross-section. (English) Zbl 0658.76025

The method of linearizing the Navier-Stokes equations to describe the motion of an infinitely long cylinder immersed in a viscous fluid leads to the paradoxical conclusion that solutions obtained in this way are logarithmically unbounded at infinity. To clarify this paradox, Oseen suggested a more rational approximation of the equations in which the convective terms are considered. Further progress was achieved by Imai who found the explicit form for the steady-stream function asymptotically far from the cylinder.
The present work gives a new formulation of the Oseen approximate equations in the form of a pair of linear integral equations extended to the boundary of the cylinder and subsequently discretized into a system of linear algebraic equations. Numerical calculations, performed when the boundary is a circle, are compared with other results available in the literature.
Reviewer: P.Villagio

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
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[1] Bush, M. B.: Modelling two-dimensional flow past arbitrary cylindrical bodies using boundary element formulations. Appl. Math. Modelling7, 386-394 (1983). · Zbl 0524.76041
[2] Filon, L. N. G.: The forces on a cylinder in a stream of viscous fluid. Proc. Roy. Soc.A 113, 7-27 (1926). · JFM 52.0867.01
[3] Filon, L. N. G.: On the second approximation to the Oseen solution for the motion of a viscous fluid. Phil. Trans.A 227, 93-135 (1928). · JFM 54.0908.01
[4] Imai, I.: On the asymptotic behaviour of viscous fluid flow at a great distance from a cylindrical body, with special reference to Filon’s paradox. Proc. Roy. Soc.A 208, 487-516 (1951). · Zbl 0043.19007
[5] Ingham, D. B.: Steady flow past a rotating cylinder. Comput. Fluids11, 351-366 (1983). · Zbl 0526.76041
[6] Kelmanson, M. A.: Integral equation solution of viscous flows with free surfaces. J. Engng. Math.17, 329-343 (1983). · Zbl 0527.76027
[7] Kida, T.: A new perturbation approach to the laminar fluid flow behind a two dimensional solid body. SIAM J. Appl. Math.44, 929-951 (1984). · Zbl 0575.76034
[8] Lean, M. H., Wexler, A.: Accurate numerical integration of singular boundary element kernels over boundaries with curvature. Int. J. Num. Meth. Eng.21, 211-228 (1985). · Zbl 0555.65091
[9] Milne-Thomson, L. M.: Theoretical hydrodynamics, 4th edn. London: Macmillan 1962. · Zbl 0164.55802
[10] Oseen, C. W.: Über die Stokes’sche Formel, und über eine verwandte Aufgabe in der Hydrodynamic. Arkiv für matematik, astronomi og fysik6, no. 29 (1910).
[11] Proudman, I., Pearson, J. R. A.: Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid. Mech.2, 237-262 (1957). · Zbl 0077.39103
[12] Tomotika, S., Aoi, T.: The steady flow of viscous fluid past a sphere and circular cylinder at small Reynolds numbers. Q. J. Mech. Appl. Math.3, 140-161 (1950). · Zbl 0040.40401
[13] Tomotika, S., Aoi, T.: An expansion formula for the drag on a circular cylinder moving through a viscous fluid at small Reynolds numbers. Q. J. Mech. Appl. Math.4, 401-406 (1951). · Zbl 0043.19106
[14] Van Dyke, M.: Perturbation methods in fluid mechanics. Stanford: Parabolic Press 1975. · Zbl 0329.76002
[15] Yano, H., Kieda, A.: An approximate method for solving two dimensional low Reynolds number flow past arbitrary cylindrical bodies. J. Fluid Mech.97, 157-179 (1980). · Zbl 0421.76027
[16] Zauderer, E.: Partial differential equations of applied mathematics. New York: Wiley 1983. · Zbl 0551.35002
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