Applied Mechanics pp 1037-1044 | Cite as

Nonlinear aspects of instability in flow between rotating cylinders

  • R. C. DiPrima
  • J. T. Stuart
Conference paper

Abstract

Since Taylor’s (1923) original investigation of the stability of a viscous flow between concentric rotating cylinders, this particular flow has received considerable attention. Taylor observed that, at sufficiently high speeds of the inner cylinder, the laminar circumferential flow (Couette flow) becomes unstable, the instability yielding a steady secondary motion in the form of toroidal vortices (Taylor vortices) spaced regularly along the axis of the cylinders. The linearized problem for the stability of the flow with respect to axisymmetric disturbances leads to an eigenvalue problem for the critical speed of the inner cylinder, which appears in the form of a Taylor number T. It is a function of the parameters μ = Ω21 and η = R 1/R 2 which describe the basic velocity and the geometry up to scale factors, and the dimensionless wave number X of the disturbance. Here Ω12 and R 1/R 2 are the angular velocities and radii of the inner and outer cylinders respectively.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bisshopp, F.: Flow Between Concentric Rotating Cylinders — A Note on the Narrow Gap Approximation. Technical Report 51, ONR Contract Nonr 562(07), Brown University, 1963.Google Scholar
  2. [2]
    Coles, D.: Paper presented at the Tenth International Congress of Applied Mechanics, Stresa, Italy, 1960 (see Proceedings of that Congress, Amsterdam: Elsevier 1962, p. 163).Google Scholar
  3. [3]
    Coles, D.: Interfaces and Intermittency in Turbulent Shear Flows, see “Mécanique de la Turbulence” (Proceedings of Marseilles meeting 1961), Centre National de la Recherche Scientifique, Paris 1962, pp. 229-250.Google Scholar
  4. [4]
    Davey, A.: The Growth of Taylor Vortices in Flow Between Rotating Cylinders. J. Fluid Mech. 14 (1962) 336–368.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    DiPrima, R. C.: Stability of Nonrotationally-Symmetric Disturbances for Viscous Flow between Rotating Cylinders. Phys. Fluids 4 (1961) 751–755.CrossRefMATHGoogle Scholar
  6. [6]
    Donnelly, R. J., P. H. Roberts and K. W. Schwarz: Experiments on the Stability of Viscous Flow Between Rotating Cylinders. Pt. 6, Finite Amplitude Experiments. Proc. Roy. Soc. A 283 (1965) 531–546.CrossRefGoogle Scholar
  7. [7]
    Krueger, E. R.: The Stability of Couette Flow and Spiral Flow. Ph. D. Thesis, Rensselaer Polytechnic Institute, Troy, N. Y. 1962.Google Scholar
  8. [8]
    Krueger, E. R., A. Gross and R. C. DiPrima: J. Fluid Mech. 1966 (to be published).Google Scholar
  9. [9]
    Landau, L. D.: On the Problem of Turbulence, C. R. Acad. Sci. U.R.S.S. 44(1944)311-314. See also L.D. Landau and E. M. Lifshitz: Fluid Mechanics, Addison-Wesley Publishing Co. or Pergamon Press 1959.MATHGoogle Scholar
  10. [10]
    Nissan, A. H., J. L. Nardacci and C. Y. Ho: The Onset of Different Modes of Instability in Flow between Rotating Cylinders. A.I. Ch.E. J. 9 (1963) 620–624.CrossRefGoogle Scholar
  11. [11]
    Schultz-Grunow, F., and H. Hein: Beitrag zur Couetteströmung. Z. Flugwiss. 4 (1956) 28–30.Google Scholar
  12. [12]
    Schwarz, K. W., B. E. Springett and R. J. Donnelly: Modes of Instability in Spiral Flow Between Rotating Cylinders. J. Fluid Mech. 20 (1964) 281–289.CrossRefGoogle Scholar
  13. [13]
    Segel, L. A.: The Nonlinear Interaction of a Finite Number of Disturbances to a Layer of Fluid Heated from Below. J. Fluid Mech. 21 (1965) 359–284.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Stuart, J. T.: On the Nonlinear Mechanics of Wave Disturbances in Stable and Unstable Parallel Flows, I. J. Fluid Mech. 9 (1960) 353–370.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Stuart, J. T.: On Three-Dimensional Non-Linear Effects in the Stability of Parallel Flows. Advances in Aeronautical Sciences, Vols. 3-4, Pergamon Press 1961, pp. 121-142.Google Scholar
  16. [16]
    Stuart, J. T.: Contribution to “Stability of Systems”, Discussion at I. Mech. E. London, May 21/22, 1964. I. Mech. E. Proc. 178, Pt.3M, 1965.Google Scholar
  17. [17]
    Taylor, G.I.: Stability of a Viscous Liquid Contained Between Two Rotating Cylinders, Phil. Trans. Roy. Soc. London A 223 (1923) 289–343.CrossRefMATHGoogle Scholar
  18. [18]
    Watson, J.: On the Nonlinear Mechanics of Wave Disturbances in Stable and Unstable Parallel Flows, II. J. Fluid Mech. 9 (1960) 371–389.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1966

Authors and Affiliations

  • R. C. DiPrima
    • 1
  • J. T. Stuart
    • 2
  1. 1.Rensselaer Polytechnic InstituteTroyUSA
  2. 2.National Physical LaboratoryTeddingtonUK

Personalised recommendations