Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-17T14:59:50.876Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear instability of annular Poiseuille flow

Published online by Cambridge University Press:  08 August 2008

C. J. HEATON
C. J. HEATON*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear stability of flow along an annular pipe formed by two coaxial circular cylinders is considered. We find that the flow is unstable above a critical Reynolds number for all 0 < η ≤ 1, where η is the ratio between the radii of the inner and outer cylinders. This contradicts a recent claim that the flow is stable at all Reynolds numbers for radius ratio η less than a finite critical value. We find that non-axisymmetric disturbances become stable at all Reynolds numbers for η < 0.11686215, and we are able to study this ‘bifurcation from infinity’ asymptotically. However, axisymmetric disturbances remain unstable, with critical Reynolds number tending to infinity as η → 0. A second asymptotic analysis is performed to show that the critical Reynolds number Rec ∝ η−1 log(η−1) as η → 0, with the form of the mean flow profile causing the appearance of the logarithm. The stability of Hagen–Poiseuille flow (η = 0) at all Reynolds numbers is therefore interpreted as a limit result, and there are no annular pipe flows which share this stability.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 610 , 10 September 2008 , pp. 391 - 406
Copyright
Copyright © Cambridge University Press 2008

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

Burridge, D. M. & Drazin, P. G. 1969 Comments on ‘Stability of pipe Poiseuille flow’. Phys. Fluids 12, 264–265.CrossRefGoogle Scholar
Corcos, G. M. & Sellars, J. R. 1959 On the stability of fully developed flow in a pipe. J. Fluid Mech. 5, 97–112.CrossRefGoogle Scholar
Cotrell, D. L. & Pearlstein, A. J. 2006 Linear stability of spiral and annular Poiseuille flow for small radius ratio. J. Fluid Mech. 547, 1–20.CrossRefGoogle Scholar
Cowley, S. J. & Smith, F. T. 1985 On the stability of Poiseuille–Couette flow: a bifurcation from infinity. J. Fluid Mech. 156, 83–100.CrossRefGoogle Scholar
Davey, A. & Salwen, H. 1994 On the stability of flow in an elliptic pipe which is nearly circular. J. Fluid Mech. 281, 357–369.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Garg, V. K. 1980 Spatial stability of concentric annular flow. J. Phys. Soc. Japan 49, 1577–1583.CrossRefGoogle Scholar
Kerswell, R. R. & Davey, A. 1996 On the linear instability of elliptic pipe flow. J. Fluid Mech. 316, 307–324.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn. Pergamon.Google Scholar
Le Dizès, S. & Fabre, D. 2007 Large-Reynolds-number asymptotic analysis of viscous centre modes in vortices. J. Fluid Mech. 585, 153–180.CrossRefGoogle Scholar
Mahadevan, R. & Lilley, G. M. 1977 The stability of axial flow between concentric cylinders to asymmetric disturbances. AGARD CP 224, pp. 9-1–9-10.Google Scholar
Mott, J. E. & Joseph, D. D. 1968 Stability of parallel flow between concentric cylinders. Phys. Fluids 11, 2065–2073.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Smith, F. T. 1979 a Instability of flow through pipes of general cross-section. Part I. Mathematika 26, 197–210.Google Scholar
Smith, F. T. 1979 b Instability of flow through pipes of general cross-section. Part II. Mathematika 26, 211–223.CrossRefGoogle Scholar
Smith, F. T. & Bodonyi, R. J. 1980 On the stability of the developing flow in a channel or circular pipe. Q. J. Mech. Appl. Maths 33, 293–320.CrossRefGoogle Scholar
Smith, F. T. & Bodonyi, R. J. 1982 Amplitude-dependent neutral modes in the Hagen–Poiseuille flow through a circular pipe. Proc. R. Soc. Lond. A 384, 463–489.Google Scholar
Stewartson, K. & Brown, S. N. 1984 Inviscid centre-modes and wall-modes in the stability theory of swirling Poiseuille flow. IMA J. Appl. Maths 32, 311–333.CrossRefGoogle Scholar
Walton, A. G. 2005 The linear and nonlinear stability of thread-annular flow. Phil. Trans. R. Soc. Lond. 363, 1223–1233.Google ScholarPubMed