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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
Affiliation:
Mathematics Department, North Park Road, University of Exeter, Exeter EX4 4QE, UK

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
Copyright
© 1992 Cambridge University Press

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References

Carrier, G. F. & Diprima, R. C., 1956 On the torsional oscillations of a solid sphere in a viscous fluid. Trans. ASME E: J. Appl. Mech. 23, 601.Google Scholar
Cowley, S. J.: 1986 High frequency Rayleigh instability of Stokes layers. In Proc. ICASE Workshop on the Instability of Time Dependent and Spatially Varying Flows.Google Scholar
Eagles, P. M.: 1971 On stability of Taylor vortices by fifth-order amplitude expansions. J. Fluid Mech. 49, 529.Google Scholar
Hall, P.: 1978 The linear stability of time dependent Poiseuille Flow. Proc. R. Soc. Lond. A 359, 181.Google Scholar
Hall, P.: 1982 Taylor–Görtler vortices in fully developed or boundary layer flows: linear theory. J. Fluid Mech. 124, 475.Google Scholar
Hall, P.: 1984 On the stability of the unsteady boundary layer on a cylinder oscillating transversely in a viscous fluid. J. Fluid Mech. 146, 347.Google Scholar
Hall, P.: 1986 Instability of time periodic flows. In Proc. ICASE Workshop on the Instability of Time Dependent and Spatially Varying Flows.Google Scholar
Honji, H.: 1981 Streaked flow around an oscillating circular cylinder. J. Fluid Mech. 107, 509.Google Scholar
Keller, H. B.: 1968 Numerical Methods for Two Point Boundary Value Problems. Waltham, MA: Blaisdell.
Papageorgiou, D.: 1987 Stability of the unsteady viscous flow in a curved pipe. J. Fluid Mech. 182, 209.Google Scholar
Riley, N.: 1967 Oscillatory viscous flows: review and extension. J. Inst. Maths Applics. 3, 419.Google Scholar
Schlichting, H.: 1932 Berechnung ebener periodischer Grenzschichströmungen. Phys. Z. 33, 327.Google Scholar
Seminara, G. & Hall, P., 1976 Centrifugal instability of a Stokes layer: linear theory. Proc. R. Soc. Lond. A 350, 29.Google Scholar
Soward, A. M. & Jones, C. A., 1983 The linear stability of the flow in the narrow gap between two concentric rotating spheres. Q. J. Mech. Appl. Maths 36, 19.Google Scholar
Stuart, J. T.: 1966 Double boundary layers in oscillating viscous flow. J. Fluid Mech. 24, 673.Google Scholar
Taylor, G. I.: 1923 Stability of a viscous liquid contained between two rotating cylinders. Phils. Trans. R. Soc. Lond. A 223, 289.Google Scholar
Tromans, P. S.: 1979 Stability and transition of periodic pipe flows. Ph.D. thesis, University of Cambridge.