Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T02:17:01.332Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcations and instabilities in sliding Couette flow

Published online by Cambridge University Press:  19 April 2011

K. DEGUCHI  and
M. NAGATA
K. DEGUCHI
Affiliation:
Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan
M. NAGATA*
Affiliation:
Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We carry out linear and nonlinear analyses on a flow between two infinitely long concentric cylinders with the radii a and b subject to a sliding motion of the inner cylinder in the axial direction. We confirm the linear stability result of Gittler (Acta Mechanica, vol. 101, 1993, p. 1) for the axisymmetric case, namely the flow is linearly stable against axisymmetric perturbations when the radius ratio η = a/b is greater than 0.1415. We extend his analysis to the non-axisymmetric case and find that the stability of the flow is still determined by axisymmetric perturbations. Our nonlinear analysis exhibits that (i) finite-amplitude axisymmetric solutions exist far below the linear critical Reynolds number for η < 0.1415 and (ii) non-axisymmetric travelling wave solutions appear abruptly at a finite Reynolds number even for η > 0.1415 where the linear critical state is absent.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Instability: Nonlinear instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 678 , 10 July 2011 , pp. 156 - 178
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Arney, M. S., Bai, R., Guevara, E., Joseph, D. D. & Liu, K. 1993 Friction factor and holdup studies for lubricated pipelining. I. Experiments and correlations. Intl J. Multiphase Flow 19, 1061–1076.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.CrossRefGoogle ScholarPubMed
Frei, Ch., Lüscher, P. & Wintermantel, E. 2000 Thread-annular flow in vertical pipes. J. Fluid Mech. 410, 185–210.CrossRefGoogle Scholar
Gibson, J. F., Halcrow, J. & Cvitanovic, P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243–266.CrossRefGoogle Scholar
Gittler, P. 1993 Stability of Poiseuille–Couette flow between concentric cylinders. Acta Mechanica. 101, 1–13.CrossRefGoogle Scholar
Joseph, D. D. 1976 Stability of Fluid Motions, Vols I and II. Springer.CrossRefGoogle Scholar
Marques, F. 1990 On boundary conditions for velocity potentials in confined flows: application to Couette flow. Phys. Fluids A A2, 729–737.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.CrossRefGoogle Scholar
Okino, S., Nagata, M., Wedin, H. & Bottaro, A. 2010 A new nonlinear vortex state in square duct flow. J. Fluid Mech. 657, 413–429.CrossRefGoogle Scholar
Okino, S. & Nagata, M. 2011 Asymmetric traveling waves in a square duct. J. Fluid Mech. (submitted).CrossRefGoogle Scholar
Panoliaskos, A., Hallett, W. L. H. & Garis, I. 1985 Prediction of optical fiber coating thickness. Appl. Opt. 24, 2309–2312.CrossRefGoogle ScholarPubMed
Preziosi, L. & Rosso, F. 1990 Stability of a viscous liquid between sliding pipes. Phys. Fluids A 2 (7), 1158–1162.CrossRefGoogle Scholar
Pringle, C. C. T., Duguet, Y. & Kerswell, R. R. 2009 Highly symmetric travelling waves in pipe flow. Phil. Trans. R. Soc. A 367, 457–472.CrossRefGoogle ScholarPubMed
Pringle, C. C. T. & Kerswell, R. R. 2007 Asymmetric, helical and mirror-symmetric traveling waves in pipe flow. Phys. Rev. Lett. 99 (2), 074502.CrossRefGoogle ScholarPubMed
Sadeghi, V. M. & Higgins, B. G. 1991 Stability of sliding Couette–Poiseuille flow in an annulus subject to axisymmetric and asymmetric disturbances. Phys. Fluids A 3 (9), 2092–2104.CrossRefGoogle Scholar
Shands, J., Alfredsson, H. & Lindgren, E. R. 1980 Annular pipe flow subject to axial motion of the inner boundary. Phys. Fluids 23 (10), 2144–2145.CrossRefGoogle Scholar
Tadmor, Z. & Bird, R. B. 1974 Rheological analysis of stabilizing forces in wire-coating dies. Polym. Engng Sci. 14 (2), 124–136.CrossRefGoogle Scholar
Uhlmann, M., Kawahara, G. & Pinelli, A. 2010 Traveling-waves consistent with turbulence-driven secondary flow in a square duct. Phys. Fluids 22, 084102.CrossRefGoogle Scholar
Vaskopulos, T., Polymeropoulos, C. E. & Zebib, A. 1993 Heat transfer from optical fibre during the draw process. J. Mat. Proc. Manuf. Sci. 1, 261–271.Google Scholar
Walton, A. G. 2003 The nonlinear instability of thread-annular flow at high Reynolds number. J. Fluid Mech. 477, 227–257.CrossRefGoogle Scholar
Walton, A. G. 2004 Stability of circular Poiseuille–Couette flow to axisymmetric disturbances. J. Fluid Mech. 500, 169–210.CrossRefGoogle Scholar
Walton, A. G. 2005 The linear and nonlinear stability of thread-annular flow. Phil. Trans. R. Soc. A 363, 1223–1233.CrossRefGoogle ScholarPubMed
Wang, J., Gibson, J. F. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98, 204501.CrossRefGoogle ScholarPubMed
Wedin, H., Bottaro, A. & Nagata, M. 2009 Three-dimensional travelling waves in a square duct. Phys. Rev. E. 79, 065305.CrossRefGoogle Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar