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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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References

Andrews, D. G. & McIntyre M. E.1978 An exact theory of nonlinear waves on a Lagrangianmean flow. J. Fluid Mech. 89, 609–647.Google Scholar
Batchelor G. K.1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bickley W. G.1938 The secondary flow due to a sphere rotating in a viscous fluid Phil. Mag. (7) 25, 746–752.Google Scholar
Duck, P. W. & Smith F. T.1979 Steady streaming induced between oscillating cylinders. J. Fluid Mech. 91, 93–107.Google Scholar
Eckart C.1948 Vortices and streams caused by sound waves. Phys. Rev. 73, 68–76.Google Scholar
Goldstein, S. (ed.) 1938 Modern Developments in Fluid Dynamics. Oxford University Press.
Havelock T. H.1954 The forces on a submerged body moving under waves. The Collected Papers of Sir Thomas Havelock on Hydrodynamics; ONR/ACR-103, pp. 590–596.
Hildebrand F. B.1963 Advanced Calculus for Applications. Prentice-Hall.
Howarth L.1951 Note on the boundary layer on a rotating sphere Phil. Mag. (7) 42, 1308–1315.Google Scholar
Hunt, J. N. & Johns B.1963 Currents induced by tides and gravity waves. Tellus 15, 343–354.Google Scholar
Keulegan, G. H. & Carpenter L. H.1956 Forces on cylinders and plates in an oscillating fluid. Nat. Bur. Stand. 4821.Google Scholar
Lamb H.1932 Hydrodynamics. Dover.
Lamoure, J. & Mei C. C.1977 Effects of horizontally two-dimensional bodies on the mass transport near the sea bottom. J. Fluid Mech. 83, 415–433.Google Scholar
Longuet-Higgins M. S.1953 Mass transport in water waves Phil. Trans. R. Soc. Lond. A 245, 535–581.Google Scholar
Longuet-Higgins M. S.1970 Steady currents induced by oscillations around islands. J. Fluid Mech. 42, 701–720.Google Scholar
Morse, P. M. & Feshbach H.1953 Methods of Theoretical Physics, vols. 1 and 11. McGraw-Hill.
Prandtl, L. & Tietjens O. G.1934 Applied Hydro and Aeromechanics. McGraw-Hill.
Rayleigh Lord1876 On waves Phil. Mag. (5) 1, 257–259.Google Scholar
Riley N.1967 Oscillatory viscous flows, review and extension. J. Inst. Maths Applics 3, 419–434.Google Scholar
Rubinow, S. I. & Keller J. B.1961 The transverse force on a spinning sphere moving in a viscous fluid. J. Fluid Mech. 11, 447–459.Google Scholar
Schlichting H.1932 Berechnung ebener periodischer Grenzschichtströmungen. Phys. Z. 33, 327–335.Google Scholar
Stokes G. G.1851 On the effect of internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 25–106.Google Scholar
Van Dyke M.1975 Perturbation Methods in Fluid Mechanics. Parabolic.