Transition to Chaos in Non-Parallel Two-Dimensional Flow in a Channel

  • P. G. Drazin
  • W. H. H. Banks
  • M. B. Zaturska
Part of the NATO ASI Series book series (NSSB, volume 225)


This paper is a review of some new work on an old problem. We shall first describe the origin of the problem, and then present a few snapshots of some of the solutions. Please forgive us for taking the snapshots from a parochial point of view, because this review is mostly of work done at Bristol.


Periodic Solution Hopf Bifurcation Stagnation Point Pitchfork Bifurcation Homoclinic Connection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Berman, A.S., 1953, Laminar flow in channels with porous walls, J. Appl. Phys., 24:1232.MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. Brady, J.F. and Acrivos, A., 1981, Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier-Stokes equations with reverse flow, J. Fluid Mech., 112: 127.MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. Calogero, F., 1984, A solvable nonlinear wave equation, Studies Appl. Math., 70:189.MathSciNetzbMATHGoogle Scholar
  4. Childress, S., Ierley, G.R., Spiegel, E.A. and Young, W.R., 1989, Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form, J. Fluid Mech., 203:1.MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. Cox, S.M., 1989, A similarity solution of the Navier-Stokes equations for two-dimensional flow in a porous-walled channel, Ph.D. dissertation, University of Bristol.Google Scholar
  6. Glendinning, P., 1988, Global bifurcations in flows, in “New Directions in Dynamical Systems”, T. Bedford and J. Swift ed., Cambridge University Press.Google Scholar
  7. Hiemenz, K., 1911, Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder, Dinglers J., 326:321.Google Scholar
  8. Homann, F., 1936, Der Einfluss grosser Zähigkeit bei der Strömung um den Zylinder und um die Kugel, Z. angew. Math. Mech., 16:153.zbMATHCrossRefGoogle Scholar
  9. Howarth, L., 1951, The boundary layer in three-dimensional flow. Part II: The flow near a Stagnation point. Phil. Mag., (7) 42:1433.MathSciNetzbMATHGoogle Scholar
  10. Newell, A.C., 1906, Side-roll instability — a new solution of the Guiness-Löwenbräu problem, Proc. Irish Narrative Soc, 8:1.Google Scholar
  11. Proudman, I. and Johnson, K., 1962, Boundary-layer growth near a stagnation point, J. Fluid Mech., 12:161.MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. Raithby, G.D. and Knudsen, D.C., 1974, Hydrodynamic development in a duct with suction and blowing, A.S.M.E. J. Appl. Mech. 41:896.ADSCrossRefGoogle Scholar
  13. Stuart, J.T., 1988, Nonlinear Euler partial differential equations: singularities in their solution, in “A Symposium to Honor C.C. Lin”, D.J. Benney, F.H. Shu and C. Yuan ed., World Scientific Publishing, Singapore.Google Scholar
  14. Taylor, C.L., Banks, W.H.H., Zaturska, M.B. and Drazin, P.G., 1990, Three-dimensional flow in a porous channel (to be published).Google Scholar
  15. Terrill, R.M., 1964, Laminar flow in a uniformly porous channel, Aero. Quart., 15:299.MathSciNetGoogle Scholar
  16. Watson, E.B.B., Banks, W.H.H., Zaturska, M.B. and Drazin, P.G., 1990, On transition to chaos in a two-dimensional channel flow driven symmetrically by accelerating walls, J. Fluid Mech. 212: (in the press).Google Scholar
  17. Watson, P., 1989, Symmetry breaking in a laminar channel flow driven by accelerating walls, M.Sc. dissertation, University of Bristol.Google Scholar
  18. Zaturska, M.B., Drazin, P.G. and Banks, W.H.H., 1988, On the flow of a viscous fluid driven along a channel by suction at porous walls, Fluid Dynamics Res., 4:151.ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • P. G. Drazin
    • 1
  • W. H. H. Banks
    • 1
  • M. B. Zaturska
    • 1
  1. 1.School of MathematicsUniversity WalkBristolUK

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