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Pitchfork bifurcations in small aspect ratio Taylor-Couette flow

  • Tom Mullin
  • Doug Satchwell
  • Yorinobu Toya
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 549)

Abstract

We present a discussion of steady bifurcation phenomena in Taylor-Couette flow. The emphasis is on the role of pitchfork bifurcations in mathematical models and their relevance to the physical problem. The general features of such bifurcations are reviewed before we discuss the numerical and experimental techniques used to ex- plore their properties. New results are then presented for a wide-gap small aspect ratio version of Taylor-Couette flow. We find good agreement between numerical and exper- imental results and show that the qualitative features of the bifurcation sequence are the same as those found with other radius ratios.

Keywords

Bifurcation Point Radius Ratio Pitchfork Bifurcation Small Aspect Ratio Taylor Vortex 
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References

  1. 1.
    T.B. Benjamin 1978a Bifurcation phenomena in steady flows of a viscous liquid. Part 1. Theory. Proc.R.Soc.Lond.A 359, 1–26.Google Scholar
  2. 2.
    T.B. Benjamin 1978b Bifurcation phenomena in steady flows of a viscous liquid. Part 2. Experiments. Proc.R.Soc.Lond.A 359, 27–43.Google Scholar
  3. 3.
    T.B. Benjamin and T. Mullin 1981 Anomalous modes in the Taylor experiment. Proc.R.Soc.Lond. A 377, 221–249.Google Scholar
  4. 4.
    T.B. Benjamin and T. Mullin 1982 Notes on the multiplicity of flows in the Taylor-Couette experiment.Google Scholar
  5. 5.
    J.E. Burkhalter and E.L. Koschmieder 1973 Steady supercritical Taylor vortex flows. J.Fluid Mech. 58, 547–560.Google Scholar
  6. 6.
    K.A. Cliffe 1983 Numerical calculations of two-cell and single-cell Taylor flows.J. Fluid Mech. 135, 219–233.Google Scholar
  7. 7.
    K.A. Cliffe and T. Mullin 1985A numerical and experimental study of anomalous modes in the Taylor experiment.J.Fluid Mech. 153, 243–258.Google Scholar
  8. 8.
    K.A. Cliffe, and Spence A.1986 Numerical calculations of bifurcations in the finite Taylor problem. In Numerical Methods for Bifurcation Problems (ed.T. Kupper, H.D. Mittleman and H. Weber), pp 129–144. Birkhauser:ISNM..Google Scholar
  9. 9.
    K.A. Cliffe 1988 Primary-flow exchange process in the Taylor problem.J.Fluid Mech. 197, 57–79.Google Scholar
  10. 10.
    K.A. Cliffe, J.J. Kobine and T. Mullin 1992 The role of anomalous modes in Taylor-Couette flow.Proc.R.Soc.Lond. A 439, 341–357.Google Scholar
  11. 11.
    D. Coles 1965 Transition in circular Couette flow.J.Fluid Mech.21, 385–425.Google Scholar
  12. 12.
    M. Golubitsky and D.G. Schaeffer 1985 Singularities and Groups in Bifurcation Theory.Vol.1. Applied Mathematical Sciences 51. Springer.Google Scholar
  13. 13.
    A. Jepson and A. Spence 1985 Folds in solutions of two parameter systems.SIAM J.Numer.Anal. 22, 347–368.Google Scholar
  14. 14.
    H.B. Keller 1977 Numerical solutions of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed.P.H. Rabinowitz), pp.359–384, Academic.Google Scholar
  15. 15.
    G. Moore and A. Spence 1980 The calculation of turning points of nonlinear equa-tions.SIAM J.Numer.Anal. 17, 567–576.Google Scholar
  16. 16.
    T. Mullin 1991 Finite-dimensional dynamics in Taylor-Couette flow. IMA J.App. Math. 46, 109–120.Google Scholar
  17. 17.
    T. Mullin 1993 The Nature of Chaos. Oxford University Press.Google Scholar
  18. 18.
    G. Pfister, H. Schmidt, K.A. Cliffe and T. Mullin 1988 Bifurcation phenomena in Taylor-Couette flow in a very short annulus.J.Fluid Mech. 191, 1–18.Google Scholar
  19. 19.
    D.G. Schaeffer 1980 Analysis of a model in the Taylor problem.Math.Proc. Camb.Phil.Soc. 87, 307–337.Google Scholar
  20. 20.
    H.L. Swinney and J.P. Gollub 1981 Hydrodynamic Instabilities and the Transition to Turbulence Topics in Applied Physics, Vol.45. Springer.Google Scholar
  21. 21.
    R. Tagg 1994 The Couette-Taylor problem.Nonlinear Science Today 4, 1–25.Google Scholar
  22. 22.
    G.I. Taylor 1923 Stability of a viscous liquid contained between two rotating cylinders.Phil.Trans.R.Soc.Lond. A 223, 289–343.Google Scholar
  23. 23.
    Y. Toya, L. Nakamura, S. Yamashita and Y. Ueki 1994 An experiment on a Taylor vortex flow in a gap with small aspect ratio.Acta Mechanica 102, 137–148.Google Scholar
  24. 24.
    B. Werner and A. Spence 1984 The computation of symmetry-reaking bifurcation points.SIAM J.Numer.Anal. 21, 388–399.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Tom Mullin
    • 1
  • Doug Satchwell
    • 1
  • Yorinobu Toya
    • 2
  1. 1.Department of Physics and AstronomyThe University of ManchesterManchesterUK
  2. 2.Department of Mechanical EngineeringNagano National College of TechnologyNaganoJapan

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