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On a submerged sphere in a viscous fluid excited by small-amplitude periodic motions

Published online by Cambridge University Press:  20 April 2006

S. A. Jenkins  and
D. L. Inman
S. A. Jenkins
Affiliation:
Center for Coastal Studies, University of California, San Diego, La Jolla, CA 92093
D. L. Inman
Affiliation:
Center for Coastal Studies, University of California, San Diego, La Jolla, CA 92093
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Abstract

Higher-order nonlinear corrections to the Stokes pendulum problem are calculated in perturbation schemes for small values of Reynolds numbers R(2) = umd0/2ν. Here the controlling lengthscale ½d0 is the displacement amplitude of the undisturbed periodic motion, um is the velocity amplitude and ν is the kinematic viscosity. Solutions for two general types of periodic motion are found; namely orbital motion as under deep water waves and oscillatory motion as under shallow water or acoustic waves. These solutions are found by matched asymptotic expansions using the fundamental irrotational oscillation to drive a thin Stokes a.c. boundary layer over the surface of the sphere. From the boundary layer several secondary motions are excited which die away in the neighbouring fluid. Among these are an orthogonal system of steady, rotational Eulerian streaming currents, and two outwardly radiating non-dispersive waves, one having the frequency of the fundamental but with a phase shift, the other appearing at the second harmonic.

With these solutions the forces and torques on a fixed sphere were computed. One of the orthogonal components of the rotational streaming field was found to produce a rotary lift force which opposed virtual-mass forces and diminished the resultant force component in quadrature to the fundamental oscillation. The other streaming component contributed damping terms which, unlike leading-order Stokes drag, vary nonlinearly with the displacement amplitude. Steady and second-harmonic torques were found to act on the sphere about the horizontal axis transverse to the fundamental oscillation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 157 , August 1985 , pp. 199 - 224
Copyright
© 1985 Cambridge University Press

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