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Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows

Published online by Cambridge University Press:  29 March 2006

Allen T. Chwang  and
T. Yao-Tsu Wu
Allen T. Chwang
Affiliation:
Engineering Science Department, California Institute of Technology, Pasadena
T. Yao-Tsu Wu
Affiliation:
Engineering Science Department, California Institute of Technology, Pasadena
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Abstract

The present study further explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson.

These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyper-bolic profiles), while the body shapes cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 67 , Issue 4 , 25 February 1975 , pp. 787 - 815
Copyright
© 1975 Cambridge University Press

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References

Batchelor, G. K. 1970a The stress system in a suspension of force-free particles. J. Fluid Mech., 41, 545–570.Google Scholar
Batchelor, G. K. 1970b Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech., 44, 419–440.Google Scholar
Blake, J. R. & Chwanc, A. T. 1974 Fundamental singularities of viscous flow. Part I. The image system in the vicinity of a stationary no-slip boundary. J. Engng Math., 8, 23–29.Google Scholar
Broersma, S. 1960 Viscous force constant for a closed cylinder. J. Chem. Phys., 32, 1632–1635.Google Scholar
Burgers, J. M. 1938 On the motion of small particles of elongated form suspended in a viscous liquid. Chap. I11 of Second Report on Viscosity and Plasticity. Kon. Ned. Akad. Wet., Verhand. 16, 113–184.Google Scholar
Chwang, A. T. & Wu, T. Y. 1974a Hydromechanics of low-Reynolds-number flow. Part 1. Rotation of axisymmetric prolate bodies. J. Fluid Mech., 63, 607–622.Google Scholar
Chwanc, A. T. & Wu, T. Y. 1974b A note of potential flow involving prolate spheroids. Schiffstech., 21, 19–31.Google Scholar
Cox, R. G. 1970 The motion of long slender bodies in a viscous fluid. Part 1. General theory. J. Fluid Alech., 44, 791–810.Google Scholar
Cox, R. G. 1971 The motion of long slender bodies in a viscous fluid. Part 2. Shear flow. J. Fluid Mech., 45, 625–657.Google Scholar
Edwardes, D. 1892 Steady motion of a viscous liquid in which an ellipsoid is constrained to rotate about a principal axis. Quart. J. Math., 26, 70–78.Google Scholar
Gans, R. 1928 Zur Theorie der Brownschen Molekularbewegung. Ann. Phys., 86, 628–656.Google Scholar
Gray, J. & Hancock, G. J. 1955 The propulsion of sea-urchin spermatozoa. J. Exp. Biol., 32, 302–814.Google Scholar
Hancock, G. J. 1953 The self-propulsion of microscopic organisms through liquids. Proc. Roy. Soc. A 217, 96–121.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. Roy. SOC. A 102, 161–179.Google Scholar
Lamb, H. 1932 Hydrcdynmics. Cambridge University Press.
Landau, L. D. & Lifshitz, E. M. 1959 Flulid Mechanics. Pergamon.
Landweber, L. 1951 The axially symmetric potential flow about elongated bodies of revolution. Rep. Taylor Model Basin, no. 761.Google Scholar
Lorentz, H. A. 1897 A general theorem concerning the motion of a viscous fluid and a few consequences derived from it. Versl. Kon. Akad. Wet Amst., 5, 168–175.Google Scholar
Oberbeck, A. 1876 Ueber stationäre Flussigkeitsbewegungen mit Berucksichtigung der inneren Reibung. J. reine angew. Math., 81, 62–80.Google Scholar
Oseen, C. W. 1927 Hydrodynamik. Leipzig: Akad. Verlagsgesellschaft.
Simha, R. 1936 Untersuchungen uber die Viskositiit von Suspensionen und Lösungen. Kolloid Z., 76, 16–19.Google Scholar
Taylor, G. I. 1969 Motion of axisymmetric bodies in viscous fluids. In Problems of Hydro-dynamics and Continuum Mechanics, pp. 718–724. S.I.A.M. Publ.Google Scholar
Tillett, J. P. K. 1970 Axial and transverse Stokes flow past slender axisymmetric bodies. J. Fluid Mech., 44, 401–417.Google Scholar
Tuck, E. O. 1964 Some methods for flows past blunt slender bodies. J. Fluid Mech., 18, 619–635.Google Scholar
Tuck, E. O. 1970 Toward the calculation and minimization of Stokes drag on bodies of arbitrary shape. Proc. 3rd. Austr. Conf. on Hydraulics & Fluid Mech. pp. 29–32.Google Scholar