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The first instability in spherical Taylor-Couette flow

Published online by Cambridge University Press:  21 April 2006

Géza Schrauf
Applied Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Present address: MBB Transport- und Verkehrsflugzeuge, Abt. TE 213, Huenefeldstr. 1-5, 2800 Bremen 1, West Germany.


In this paper continuation methods are applied to the axisymmetric Navier-Stokes equations in order to investigate how the stability of spherical Couette flow depends on the gap size σ. We find that the flow loses its stability due to symmetry-breaking bifurcations and exhibits a transition with hysteresis into a flow with one pair of Taylor vortices if the gap size is sufficiently small, i.e. if σ [les ] σB.

In wider gaps, i.e. for σB < σ [les ] σF, both flows, the spherical Couette flow and the flow with one pair of Taylor vortices, are stable. We predict that the latter exists in much wider gaps than previous experiments and calculations showed. Taylor vortices do not exist if σ > σF. The numbers σB and σF are computed by calculating the instability region of the spherical Couette flow and the region of existence of the flow with one pair of Taylor vortices.

Research Article
© 1986 Cambridge University Press

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Belyaev, Yu. N., Monakhov, A. A., Khlebutin, G. N. & Yavorskaya, I. M. 1980 Issledovanie ustoichivosti i needinstvennosti techenii vo vraschayuschikhsya sfericheskikh sloyakh (Investigation of stability and nonuniqueness of the flow between rotating spheres). Rep. No. 567, Space Research Institute of the Academy of Science, USSR.
Bartels, F. 1982 Taylor vortices between two concentric rotating spheres. J. Fluid Mech. 19, 1.Google Scholar
Bonnet, J.-P. & Alziary de Roquefort, T. 1976 Écoulement entre deux sphères concentriques en rotation. J. Mec. 13, 373.Google Scholar
Keller, H. B. 1977 Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. P. Rabinowitz). Academic.
Keller, H. B. 1982 Practical procedures in path following near limit points. In Proc. 5th Intl Symp. on Computing Methods in Applied Sciences and Engineering, Versailles (eds. R. Glowinsky & J. L. Lions), p. 177. North-Holland.
Khlebtin, G. N. 1968 Stability of fluid motion between a rotating and a concentric sphere. Fluid Dyn. 3, 31.Google Scholar
Krause, E. & Bartels, F. 1980 Finite-difference solutions of the Navier-Stokes equations for axially symmetric flows in spherical gaps. In Approximation Methods for Navier-Stokes Problems, Proc. IUTAM Symp., Paderborn (eds. E. Dold & B. Eckermann), p. 313.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Munson, B. R. & Joseph, D. D. 1971 Viscous incompressible flow between concentric rotating spheres. Part 1. Basic flow. J. Fluid Mech. 49, 289.Google Scholar
Roache, P. J. 1972 Computational Fluid Dynamics. Hermosa.
Rosenhead, L. 1963 Laminar Boundary Layers. Oxford University Press.
Sawatzki, O. & Zierep, J. 1970 DasStromfeld im Spalt zwischen zwei konzentrischen Kugelflächen, von denen die innere rotiert. Acta Mechanica 9, 13.Google Scholar
Schrauf, G. 1982 Branching of Navier-Stokes equation in a spherical gap. In 8th Intl Conf. on Numerical Methods in Fluid Dynamics (ed. E. Krause). Lecture Notes in Physics, vol. 170, p. 474. Springer.
Schrauf, G. 1983 Lösungen der Navier-Stokes Gleichungen für stationäre Strömungen im Kugelspalt. Ph.D. thesis, University of Bonn.
Schrauf, G., Fier, J. M. & Keller, H. B. 1985 Fold continuation in the Taylor Problem (in preparation).
Schrauf, G. & Krause, E. 1984 Symmetric and asymmetric Taylor vortices in a spherical gap. In The 2nd IUTAM Symp. on Laminar-Turbulent Transition (ed. V. V. Kozlov), p. 659. Springer.
Stokes, G. G. 1842 On the steady motion of incompressible fluids. Trans. Camb. Phil. Soc. 7, 439.Google Scholar
Tuckerman, L. S. 1983 Formation of Taylor vortices in spherical Couette flow. Ph.D. thesis, Massachusetts Institute of Technology.
Wimmer, M. 1976 Experiments on a viscous fluid flow between concentric rotating spheres. J. Fluid Mech. 78, 317.Google Scholar
Yavorskaya, I. M., Belyaev, Yu. N., Monakhov, A. A., Astaf'eva, N. M., Scherbakov, S. A. & Vvedenskaya, N. D.1980 Stability, non-uniqueness and the transition to turbulence in the flow between two rotating spheres. Rep. No. 595, Space Research Institute of the Academy of Science, USSR.