Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-19T02:54:15.566Z Has data issue: false hasContentIssue false
  • English
  • Français

The force on an axisymmetric body in linearized, time-dependent motion: a new memory term

Published online by Cambridge University Press:  21 April 2006

C. J. Lawrence  and
S. Weinbaum
C. J. Lawrence
Affiliation:
Department of Mechanical Engineering, The City College of The City University of New York, NY 10031, USA Present address: Department of Chemical Engineering, University of Wisconsin, 1415 Johnson Drive, Madison, WI 53706, USA, and Mathematics Research Center, University of Wisconsin, 610 Walnut Street, Madison, WI 53705, USA.
S. Weinbaum
Affiliation:
Department of Mechanical Engineering, The City College of The City University of New York, NY 10031, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In contrast to the steady Stokes equations for creeping motion, the time-dependent linearized Navier–Stokes equations have only been solved for very restricted geometries, the solution for the sphere being the sole solution for an isolated finite body. In the present paper, the linearized Navier–Stokes equations are further explored and a simple expression is derived which relates the force on an arbitrary axisymmetric body in oscillatory motion to the solution for the stream function in the far field. This result is applied to the case of a slightly eccentric spheroid and it is shown that the total hydrodynamic force contains four terms, three of which correspond to the classical solutions for the Stokes drag, added mass and Basset force on the perturbed sphere; the fourth term is only present when the body is non-spherical. In contrast to the three classical forces, the new term is not a simple power of the dimensionless frequency parameter iL2ω/ν, in which L is a length-scale, ω is the frequency of oscillation and ν is the kinematic viscosity of the fluid. A Laplace superposition is then used to find the force on the spheroid in an arbitrary axisymmetric motion with velocity U(t). The new memory term decays faster than the Basset force at large times and is bounded at short times.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 171 , October 1986 , pp. 209 - 218
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

Abramowitz, M. & Stegun I. A.1965 Handbook of Mathematical Functions, pp. 297–298. Dover.
Basset A. B.1888 A Treatise on Hydrodynamics, vol. 2, ch. 21, 22. Cambridge: Deighton Bell.
Batchelor G. K.1967 An Introduction to Fluid Dynamics, p. 354. Cambridge University Press.
Felderhof B. U.1976a Force density induced on a sphere in linear hydrodynamics. I. Fixed sphere, stick boundary conditions. Physica 84 A, 557.Google Scholar
Felderhof B. U.1976b Force density induced on a sphere in linear hydrodynamics. II. Moving sphere mixed boundary conditions. Physica 84 A, 569.Google Scholar
Green G.1833 Researches on the vibration of pendulums in fluid media. Trans. R. Soc. Edin. Reprinted in Mathematical Papers (ed. N. M. Ferrers). New York: Chelsea Publishing (1970).
Happel, J. & Brenner H.1965 Low Reynolds Number Hydrodynamics, ch. 4. Prentice-Hall.
Leichtberg S., Weinbaum S., Pfeffer, R. & Gluckman M. J.1976 A study of unsteady forces at low Reynolds number: a strong interaction theory for the coaxial settling of three or more spheres Phil. Trans. R. Soc. Lond. A 282, 585.Google Scholar
Lin W. H.1986 Hydrodynamic forces on multiple circular cylinders oscillating in a viscous incompressible fluid. J. Fluid Mech. (submitted).Google Scholar
Mazur, P. & Bedeaux D.1974 A generalization of Faxén's Theorem to nonsteady motion of a sphere through an incompressible fluid in arbitrary flow. Physica 76, 235.Google Scholar
Payne, L. E. & Pell W. H.1960 The Stokes flow problem for a class of axially symmetric bodies. J. Fluid Mech. 7, 529.Google Scholar
Stokes G. G.1851 On the effect of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8. Reprinted in Mathematical and Physical Papers III. Cambridge University Press (1922).Google Scholar