Primary and secondary bifurcations in the Couette-Taylor problem
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
We study the preturbulent transitions for the Couette-Taylor flows via bifurcation theory in the presence of symmetry. The difficulty is that the linearized stability analysis leads to multiple eigenvalues for the most simple flows. Only a consideration of the symmetry-group action on the critical eigenvectors allows us to derive and to solve the bifurcation equations. We recover through this analysis the different patterns which are observed in experiments as the Reynolds number is increased: Steady Taylor vortices and bifurcation of either wavy, or twisted vortices from the Taylor vortex flow in the case of co-rotating cylinders; spiral vortices in the case of (strongly) counterrotating cylinders, and ribbon-cells, which have not yet been observed in experiments. Then we show that, under natural assumptions on the loss of stability of these oscillatory flows, the next bifurcation leads to quasi-periodic flows without frequency locking, whose different patterns are studied.
- C. D. Andereck, R. Dickman and H. L. Swinney, New flows in a circular Couette system with corotating cylinders. Phys. Fluds,26 (1983), 1395–1401. CrossRef
- C. D. Andereck, S. S. Liu and H. L. Swinney, to be published.
- P. Chossat, Bifurcation and stability of convective flows in a rotating or not rotating spherical shell. SIAM J. Appl. Math.,37 (1979), 624–647. CrossRef
- G. Cognet, A. Bouabdallah and A. Aitaider, Laminar-Turbulent Transition (ed. R. Eppler, H. Fasel), Springer, 1982, 330.
- R. C. Di Prima and R. N. Grannick, A nonlinear investigation of the stability of flow between counter-rotating cylinders. IUTAM Symp. on Instability of Continuous Systems (ed. H. Leipholz), Springer, 1971.
- R. C. Di Prima and G. J. Habetler, A completeness theorem for non-selfadjoint eigenvalue problems in hydrodynamic stability. Arch. Rational Mech. Anal.,34 (1969), 218–227. CrossRef
- R. C. Di Prima and H. L. Swinney Instabilities and transition in flow between concentric cylinders. Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. Swinney, J. P. Gollub), Topics in Applied Physics, Vol. 45, Springer, 1981, 139–180.
- P. R. Fenstermacher, H. L. Swinney and J. P. Gollub, Dynamical instabilities and the transition to chaotic Taylor vortex flow. J. Fluid Mech.,94 (1979), 103–128. CrossRef
- P. Germain, Cours de Mécanique des Milieux Continus, Tome 1. Masson, 1973.
- M. Gorman, L. A. Reith and H. L. Swinney, Modulation patterns, multiple frequencies, and other phenomena in circular Couette flow. Nonlinear Dynamics (ed. R. Helleman), Ann. New York, Acad. Sci.,357 (1980), 10–21.
- J. K. Hale, Ordinary Differential Equations. Wiley Interscience, 1969.
- G. Iooss, Théorie non linéaire de la stabilité des écoulements laminaires dans le cas de l’échange des stabilités, Arch Rational Mech. Anal.,40 (1971), 166–208. CrossRef
- G. Iooss, Bifurcation and Transition to Turbulence in Hydrodynamics. CIME session on bifurcation theory and applications (ed. L. Salvadori), Lecture Notes in Maths. Vol. 1057, Springer, 1984, 152–201.
- G. Iooss, Bifurcation of Maps and Applications. Math. Studies,36, North Holland, 1979.
- D. D. Joseph, Stability of Fluid Motions I. Springer Tracts in Natural Philosophy,27, 1976.
- T. Kato, Perturbation Theory for Linear Operators. Springer, 1966.
- K. Kirchgassner and P. Sorger, Branching analysis for the Taylor problem, Quart. J. Mech. Appl. Math.,22 (1969), 183. CrossRef
- V. S. L’Vov, A. A. Predtechensky and A. I. Chernykh, Bifurcations and chaos in the system of Taylor vortices-laboratory and numerical experiments. Nonlinear Dynamics and Turbulence (eds. G. I. Barenblatt, G. Iooss and D. D. Joseph), I.M.M. Series, Pitman, London, 1983, 238.
- J. E. Marsden and M. Mc Cracken, The Hopf Bifurcation and Its Applications, Appl. Math. Sci.,19, Springer, 1976.
- W. Miller, Symmetry Groups and their Applications. Academic Press, New York, 1972.
- D. Rand, Dynamics and symmetry: Predictions for modulated waves in rotating fluids. Arch. Rational Mech. Anal.,79 (1982), 1–38. CrossRef
- M. Renardy, Bifurcation from rotating waves, Arch. Rational Mech. Anal.,79 (1982), 49–84.
- D. Sattinger, Branching in the Presence of Symmetry. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1983.
- R. S. Shaw, C. D. Andereck, L. A. Reith and H. L. Swinney, Superposition of travelling waves in the circular Couette system. Phys. Rev. Lett.,48 (1982), 1172–1175. CrossRef
- G. I. Taylor, Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Roy. Soc. A,223 (1923), 289. CrossRef
- S. A. Van Gils, Some studies in Dynamical Systems. Thesis, Amsterdam, 1984.
- Y. Demay and G. Iooss, Calcul des solutions bifurquées pour le problème de Couette-Taylor avec les deux cylindres en rotation. J. Méc. Th. Appl. (to appear in 1985).
- Primary and secondary bifurcations in the Couette-Taylor problem
Japan Journal of Applied Mathematics
Volume 2, Issue 1 , pp 37-68
- Cover Date
- Print ISSN
- Additional Links
- fluid mechanics
- bifurcation theory
- dynamical system
- nonlinear stability theory
- symmetry breaking