Skip to main content
SpringerLink
Log in

Stability of Flow in a Diverging Channel

  • Chapter
Stability and Wave Propagation in Fluids and Solids

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 344))

Abstract

The linear, weakly nonlinear, and nonlinear stability of flows of a viscous incompressible fluid in a diverging channel will be treated theoretically. The results will be related to observations of flows, to computational fluid dynamics, to bifurcations of the solutions describing the flows as the Reynolds number increases, and to transition towards chaos. This will also serve as a case study of the concepts and methods of the theory of stability of all flows.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Batchelor, G.K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.

    MATH  Google Scholar 

  2. Nakayama, Y. ed. 1988 Visualized Flow. Oxford: Pergamon Press.

    Google Scholar 

  3. Durst, F., Melling, A, & Whitelaw, J.H. 1974 Low Reynolds number flow over a plane symmetric sudden expansion. J. Fluid Mech. 64, 111–128.

    Article  Google Scholar 

  4. Cherdron, W., Durst, F. & Whitelaw, J.H. 1978 Asymmetric flows and instabilities in symmetric ducts with sudden expansions. J. Fluid Mech. 84, 13–31.

    Article  Google Scholar 

  5. Sobey, I.J. 1985 Observations of waves during oscillatory channel flows. J. Fluid Mech. 151, 395–426.

    Article  Google Scholar 

  6. Sobey, I.J. & Drazin, P.G. 1986 Bifurcations of two-dimensional channel flows. J. Fluid Meci. 171, 263–287.

    Article  MathSciNet  MATH  Google Scholar 

  7. Fearn, R.M., Mullin, T. & Cliffe, K.A. 1990 Nonlinear flow phenomena in a symmetric sudden expansion. J. Fluid Mech. 211, 595–608.

    Article  Google Scholar 

  8. Durst, F., Pereira, J.C.F. & Tropea, C. 1993 The plane symmetric sudden-expansion at low Reynolds number. J. Fluid Mech. 248, 567–581.

    Article  Google Scholar 

  9. Drazin, P.G. & Reid, W.H. 1981 Hydrodynâmic Stability. Cambridge University Press.

    MATH  Google Scholar 

  10. Drazin, P.G. 1992 Non Hnear Systems. Cambridge University Press.

    Book  Google Scholar 

  11. Benjamin, T.B. 1976 Applications of Leray-Schauder degree theory to problems of hydrodynamic stability. Math. Proc. Camb. Phil. Soc. 79, 373–392,

    Article  MathSciNet  MATH  Google Scholar 

  12. Crawford, J.D. & Knobloch, E. 1991 Symmetry and symmetry-breaking bifurcations in fluid dynamics. Ann. Rev. Fluid Mech. 23, 341–387.

    Article  MathSciNet  Google Scholar 

  13. Eagles, P.M. 1966 The stability of a family of Jeffery-Hamel solutions for divergent channel flow. J. Fluid Mech. 24, 191–207.

    Article  MathSciNet  Google Scholar 

  14. Allmen, M.J. & Eagles, P.M. 1984 Stability of divergent channel ûows: a numerical approach. Proc. R. Soc. Lond. A 392, 359–372.

    Article  MATH  Google Scholar 

  15. Georgiou, G.A. & Eagles, P.M. 1985 The stability of flows in channels with small wall curvature. J. Fluid Mech. 159, 259–287.

    Article  MathSciNet  MATH  Google Scholar 

  16. Drazin, P.G. 1988 Perturbations of Jeffery-Hamel flows. On pp. 129–133 of A Symposium to Honor C.C. Lin, eds, D.J. Benney, F.H. Shu & C. Yuan, Singapore: World Scientific Press.

    Google Scholar 

  17. Jeffery, G.B. 1915 The two-dimensional steady motion of a viscous fluid. Phil. Mag. (6) 29, 455–465.

    Article  MATH  Google Scholar 

  18. Hamel, G. 1916 Spiralförmigen Bewegungen zäher Flüssigkeiten. Jahresbericht der Deutschen Math. Vereinigung 25, 34–60.

    MATH  Google Scholar 

  19. Gol’dshtik, M.A. & Shtern, V.N. 1989 Loss of symmetry in viscous flow from a line source. Fiuid Dynamics 24, 191–199. Also Izv. Akad. Nauk SSR Mekh. Zhid. i Gaza 24, 35–44.

    Article  MathSciNet  Google Scholar 

  20. Gol’dshtik, M.A., Hussain, F. & Shtern, V.N. 1991 Symmetry breaking in vortex-source and Jeffery-Hamel flows. J. Fluid Mech. 232, 521–566.

    Article  MathSciNet  Google Scholar 

  21. Fraenkel, L.E. 1962 Laminar flow in symmetrical channels with slightly curved walls. I. On the Jeffery-Hamel solutions for flaw between plane walls. Proc. R. Soc. Load. A 267, 119–138.

    Article  MathSciNet  MATH  Google Scholar 

  22. Dean, W.R. 1934 Note on the divergent flow of fluid. Phil. Mag. (7) 18, 759–777.

    MathSciNet  MATH  Google Scholar 

  23. Buitrago, S.E. 1983 Detailed analysis of the higher Jeffery-Hamel solutions. M.Phil, thesis, University of Sussex.

    Google Scholar 

  24. Banks, W.H.H., Drazin, P.G. & Zaturska, M.B. 1988 On perturbations of Jeffery-Hamel flow. J. Fluid Mech. 186, 559–581.

    Article  MathSciNet  MATH  Google Scholar 

  25. Dean, W.R. & Montagnon, P.E. 1949 On the steady motion of a viscous liquid in a corner. Proc. Camb. Phil Soc. 45, 389–394.

    Article  MathSciNet  MATH  Google Scholar 

  26. Sternberg, E. & Koiter, W.T. 1958 The wedge under a concentrated couple: a paradox in the two-dimensional theory of elasticity. Trans. ASMEE: J. Appl. Mech. 25, 575–581.

    MathSciNet  MATH  Google Scholar 

  27. Lugt, H.J. & Schwiderski, E.W. 1965 Flows around dihedral angles. I. Eigen-motion and analysis. Proc. R. Soc. Lond. A 285, 382–399.

    Article  MathSciNet  Google Scholar 

  28. Fraenkel, L.E. 1963 Laminar flow in symmetrical channels with slightly curved walls. II. An asymptotic series for the stream function. ProcR. Soc. Lond. A 272, 406–428.

    Article  MathSciNet  MATH  Google Scholar 

  29. Sobey, I.J. & Mullin, T. 1992 Calculation of multiple solutions for the two-dimensional Navier-Stokes equations. Proc. ICFD Conference, Reading.

    Google Scholar 

  30. Cliffe, K.A. & Greenfield, A.C. 1982 Some comments on laminar flow in symmetric two-dimensional channels. Rep. TP 939. AERE, Harwell.

    Google Scholar 

  31. Dennis, S.C.R., Banks, W.H.H., Drazin, P.G. & Zaturska, M.B. 1994 Flow along a diverging channel. (To be published.)

    Google Scholar 

  32. Shapira, M., Degani, D. & Weihs, D. 1991 Stability and existence of multiple solutions for viscous flow in suddenly enlarged channels. Computers & Fluids 18, 239–258.

    Article  Google Scholar 

  33. King, G.P. & Stewart, I.N. 1992 Symmetric chaos. On pp. 257–315 of Nonlinear Equations in the Applied Sciences, eds. W.F. Ames & C.F. Rogers. New York: Academic Press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Bristol, Bristol, UK

    Ph. G. Drazin

Authors
  1. Ph. G. Drazin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Ferrara, Italy

    G. P. Galdi

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Wien

About this chapter

Cite this chapter

Drazin, P.G. (1995). Stability of Flow in a Diverging Channel. In: Galdi, G.P. (eds) Stability and Wave Propagation in Fluids and Solids. CISM International Centre for Mechanical Sciences, vol 344. Springer, Vienna. https://doi.org/10.1007/978-3-7091-3004-9_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-3004-9_2

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-82687-4

  • Online ISBN: 978-3-7091-3004-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation