Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-18T18:35:58.239Z Has data issue: false hasContentIssue false
  • English
  • Français

Low-Reynolds-number flow past cylindrical bodies of arbitrary cross-sectional shape

Published online by Cambridge University Press:  21 April 2006

S. H. Lee  and
L. G. Leal
S. H. Lee
Affiliation:
Chemical Engineering Department, California Institute of Technology, Pasadena, CA 91125, USA Permanent address: Chevron Oil Field Research Co., PO Box 446, La Habra, CA 90631.
L. G. Leal
Affiliation:
Chemical Engineering Department, California Institute of Technology, Pasadena, CA 91125, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A numerical implementation of the method of matched asymptotic expansions is proposed to analyse two-dimensional uniform streaming flow at low Reynolds number past a straight cylinder (or cylinders) of arbitrary cross-sectional shape. General solutions for both the Stokes and Oseen equations in two dimensions are expressed in terms of a boundary distribution of fundamental single- and double-layer singularities. These general solutions are then converted to integral equations for the unknown distributions of singularity strengths by application of boundary conditions at the cylinder surface, and matching conditions between the Stokes and Oseen solutions. By solving these integral equations, using collocation methods familiar from three-dimensional application of ‘boundary integral’ methods for solutions of Stokes equation, we generate a uniformly valid approximation to the solution for the whole domain.

We demonstrate the method by considering, as numerical examples, uniform flow past an elliptic cylinder, uniform flow past a cylinder of rectangular cross-section, and uniform flow past two parallel cylinders which may be either equal in radius, or of different sizes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 164 , March 1986 , pp. 401 - 427
Copyright
© 1986 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chang, I.-D. & Finn, R. 1961 On the solutions of a class of equations occurring in continuum mechanics with application to the Stokes paradox. Arch. Rat. Mech. Anal. 7, 388.Google Scholar
Dorrepaal, J. M. & O'Neil, M. E. 1979 The existence of free eddies in a streaming Stokes flow. Q. J. Mech. Appl. Maths 32, 95.Google Scholar
Finn, R. 1959 On steady-state solutions of the Navier—Stokes partial differential equations. Arch. Hat. Mech. Anal. 9, 381.Google Scholar
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics. Noordhoff.
Jeffery, G. B. 1912 On a form of the solution of Laplace's equation suitable for problems relating to two spheres. Proc. R. Soc. A 87, 109.Google Scholar
Jeffery, G. B. 1922 The rotation of two circular cylinders in a viscous fluid. Proc. R. Soc. A 101, 169.Google Scholar
Kaplun, S. 1957 Low Reynolds number flow past a circular cylinder. J. Math. Mech. 6, 595.Google Scholar
Ladyzhenskaya, O. A. 1963 The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach, New York.
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Lee, S. H. & Leal, L. G. 1982 The motion of a sphere in the presence of deformable interface. II. A numerical study of the translation of a sphere normal to an interface. J. Colloid Interface Sci. 87, 81.Google Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1.Google Scholar
Oberbeck, A. J. 1876 über stationäre Flüssigkeitsbewegungen mit Berücksichtigung der inner Reibung. J. reine angew. Math. 81, 62.Google Scholar
O'Neill, M. E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematica 11, 67.Google Scholar
Oseen, C. W. 1927 Neuere Methoden und Ergbnisse in der Hydrodynamik. Leipzig, Akademische Verlag.
Protjdman, I. & Pearson, J. R. A. 1957 Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2, 237.Google Scholar
Rallison, J. M. 1981 A numerical study of the deformation and burst of a viscous drop in general shear flows. J. Fluid Mech. 90, 465.Google Scholar
Robertson, C. R. & Acrivos, A. 1970 Low Reynolds number shear flow past a rotating circular cylinder. Part 1. Momentum transfer. J. Fluid Mech. 40, 685.Google Scholar
Shintani, K., Umemura, A. & Takano, A. 1983 Low Reynolds number flow past an elliptic cylinder. J. Fluid Mech. 1367, 277.Google Scholar
Umemura, A. 1982 Matched-asymptotic analysis of low Reynolds number flow past two equal circular cylinders. J. Fluid Mech. 121, 345.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. Parabolic.
Yano, H. & Kieda, A. 1980 An approximate method for solving two-dimensional low Reynolds number flow past arbitrary cylindrical bodies. J. Fluid Mech. 97, 157.Google Scholar
Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69, 377.Google Scholar