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On stability of the flow around an oscillating sphere

Published online by Cambridge University Press:  26 April 2006

S. R. Otto
S. R. Otto
Affiliation:
Mathematics Department, North Park Road, University of Exeter, Exeter EX4 4QE, UK
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Abstract

The stability of the flow resulting from the oscillations of a sphere in a viscous fluid is investigated. The calculation for the transverse oscillations of the sphere is performed in a linear regime and the result in the weakly nonlinear regime is described; the stability in the case of torsional oscillations is considered in the linear regime, where we take torsional oscillations to mean oscillations about a fixed axis through the centre of the sphere. In both cases we assume that the frequency of the oscillations is large, so that the unsteady boundary layer that results is thin. In the transverse case, the linear stability problem depends only on the radial variable and time. Employing Floquet theory we may reduce the system to a coupled infinite system of ordinary differential equations, with homogeneous boundary conditions, the eigenvalues of this system being found numerically. In the torsional case, the linear stability problem again depends only on the radial variable and time, although the angular variation is retained in a parametric form and is determined at higher order. A WKBJ perturbation solution is constructed and the evolution of the amplitude of the vortex is found. The solution is determined by finding a saddle point in the complex plane of the angular coordinate, and thus the critical Taylor number is derived.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 239 , June 1992 , pp. 47 - 63
Copyright
© 1992 Cambridge University Press

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