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Stokes flow about a sphere attached to a slender body

Published online by Cambridge University Press:  29 March 2006

N. J. De Mestre  and
D. F. Katz
N. J. De Mestre
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge Present address: Department of Mathematics, Royal Military College, Duntroon, Australia.
D. F. Katz
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

Stokes flow is analysed for a combination body, consisting of a sphere attached to a slender body, translating along its axis in an infinite and otherwise un-disturbed fluid. The cross-section of the after-body, or tail, is circular; the radius, while not necessarily constant, is small compared with the radius of the spherical head. The tail is represented by a distribution of Stokeslets of strength per unit length F(z), located and directed along its axis. The interactive effect of head-tail attachment is manifested by the presence of image singularities located within the sphere. The image system for a single tail Stokeslet must be such that the no-slip condition is satisfied on the surface of the sphere. It is shown that this system consists of a Stokeslet, a Stokes doublet (stresslet only) and a source doublet located a t the image point. The strength F(z) is obtained by applying the no-slip condition to the combination body. The solution follows the lines of traditional slender-body theory, an expansion being performed in ascending powers of the reciprocal of the logarithm of the aspect ratio. The integral force parameters and F(z) are obtained to second order. The interactive effect is assessed, and the results are discussed in the context of a sedimenting micro-organism, such as a spermatozoon. The drag on the combination body is shown to be less by around 10% than the sum of the drags on an isolated sphere and tail. This drag, for a sperm-shaped body, is divided approximately equally between head and tail.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 64 , Issue 4 , 24 July 1974 , pp. 817 - 826
Copyright
© 1974 Cambridge University Press

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References

Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419–440.Google Scholar
Blake, J. R. 1971 A note on the image system for a Stokeslet in a no-slip boundary. Proc. Camb. Phil. Soc. 70, 303–310.Google Scholar
Collins, W. D. 1954 A note on Stokes's stream-function for the slow steady motion of viscous fluid before plane and spherical boundaries. Mathematika, 1, 125–130.Google Scholar
COX, R. G. 1970 The motion of long slender bodies in a viscous fluid. Part 1. General theory. J. Fluid Mech. 44, 791–810.Google Scholar
Katz, D. F. & Mestre, N. J. DE 1974 On the analysis of spermatozoan sedimentation. In preparation.
Mestre, N. J. DE 1973 Low-Reynolds-number fall of slender cylinders near boundaries. J. Fluid Mech. 58, 641–656.Google Scholar
Tillet, J. P. K. 1970 Axial and transverse Stokes flow past slender bodies. J. Fluid Mech. 44, 401.Google Scholar