Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T01:59:00.355Z Has data issue: false hasContentIssue false
  • English
  • Français

Hydromechanics of low-Reynolds-number flow. Part 1. Rotation of axisymmetric prolate bodies

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
Get access Rights & Permissions [Opens in a new window]

Abstract

The present series of studies is concerned with low-Reynolds-number flow in general; the main objective is to develop an effective method of solution for arbitrary body shapes. In this first part, consideration is given to the viscous flow generated by pure rotation of an axisymmetric body having an arbitrary prolate form, the inertia forces being assumed to have a negligible effect on the flow. The method of solution explored here is based on a spatial distribution of singular torques, called rotlehs, by which the rotational motion of a given body can be represented.

Exact solutions are determined in closed form for a number of body shapes, including the dumbbell profile, elongated rods and some prolate forms. In the special case of prolate spheroids, the present exact solution agrees with that of Jeffery (1922), this being one of very few cases where previous exact solutions are available for comparison. The velocity field and the total torque are derived, and their salient features discussed for several representative and limiting cases. The moment coefficient CM = M/(8πμω0ab2) (M being the torque of an axisymmetric body of length 2a and maximum radius b rotating at angular velocity ω0 about its axis in a fluid of viscosity μ) of various body shapes so far investigated is found to lie between $\frac{2}{3}$ and 1, usually very near unity for not extremely slender bodies.

For slender bodies, an asymptotic relationship is found between the nose curvature and the rotlet strength near the end of its axial distribution. It is also found that the theory, when applied to slender bodies, remains valid at higher Reynolds numbers than was originally intended, so long as they are small compared with the (large) aspect ratio of the body, before the inertia effects become significant.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 63 , Issue 3 , 29 April 1974 , pp. 607 - 622
Copyright
© 1974 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

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Batchelor, G. K. 1970 Stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.Google Scholar
Blake, J. R. & Chwang, A. T. 1974 Fundamental singularities of viscous flow. Part 1. The image system in the vicinity of a stationary no-slip boundary. J. Eng. Math. 8, 23–29.Google Scholar
Chwang, A. T. & Wu, T. Y. 1971 A note on the helical movement of micro-organisms. Proc. Roy. Soc. B 178, 327–346.Google Scholar
Chwang, A. T. & WU, T. Y. 1974 Helical movement of cylindrical tails in self-propelling micro-organisms. To be published.
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 Thoorie der Brownschen Molekularbewegung. Ann. Phys. 86, 628–656.Google Scholar
Gray, J. & Hancock, G. J. 1955 The propulsion of sea-urchin spermatozoa. J. Exp. Riol. 32, 802–814.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.
Jzffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Pyoc. Roy. Soc. A 102, 161–179.Google Scholar
Kellogg, O. D. 1929 Foundations of Potential Theory. New York: Murray Printing Co.
Lagerstrom, P. A. 1964 Laminar flow theory. In Theory of Laminar Flowa (ed. F. K. Moore), vol. IV, High Speed Aerodynamics and Jet Propulsion, 3B. Princeton University Press.
Lamb, H. 1932 Hydrodynamics Cambridge University Press.
Lighthill, M. J. 1969 Hydromechanics of aquatic animal propulsion. Ann. Rev. Fluid Mech. 1, 413–445.Google Scholar
Oseen, C. W. 1927 Neuere Methoden und Ergebnisse in der Hydrodynamik. Leipzig: Akad. Verlagsgesellschaft.