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Bifurcations of two-dimensional channel flows

Published online by Cambridge University Press:  21 April 2006

Ian J. Sobey  and
Philip G. Drazin
Ian J. Sobey
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK Present address: Schlumberger Cambridge Research, P.O. Box 153, Cambridge CB3 0HG, UK.
Philip G. Drazin
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
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Abstract

In this paper we study some instabilities and bifurcations of two-dimensional channel flows. We use analytical, numerical and experimental methods. We start by recapitulating some basic results in linear and nonlinear stability and drawing a connection with bifurcation theory. We then examine Jeffery–Hamel flows and discover new results about the stability of such flows. Next we consider two-dimensional indented channels and their symmetric and asymmetric flows. We demonstrate that the unique symmetric flow which exists at small Reynolds number is not stable at larger Reynolds number, there being a pitchfork bifurcation so that two stable asymmetric steady flows occur. At larger Reynolds number we find as many as eight asymmetric stable steady solutions, and infer the existence of another seven unstable solutions. When the Reynolds number is sufficiently large we find time-periodic solutions and deduce the existence of a Hopf bifurcation. These results show a rich and unexpected structure to solutions of the Navier–Stokes equations at Reynolds numbers of less than a few hundred.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 171 , October 1986 , pp. 263 - 287
Copyright
© 1986 Cambridge University Press

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References

Benjamin T. B.1978 Proc. R. Soc. Lond. A 359, 1.
Buitrago S. E.1983 M. Phil, thesis, University of Sussex.
Cherdron W., Durst, F. & Whitelaw J. H.1978 J. Fluid Mech. 84, 13.
Cliffe, K. A. & Greenfield A. C.1982 Report TP 939 AERE, Harwell.
Davis S. H.1969 Proc. R. Soc. Lond. A 310, 341.
Diprima, R. C. & Hall P.1984 Proc. R. Soc. Lond. A 396, 75.
Drazin P. G.1961 J. Fluid Mech. 10, 571.
Drazin, P. G. & Reid W. H.1981 Hydrodynamic Stability. Cambridge University Press.
Eagles P. M.1966 J. Fluid Mech. 24, 191.
Eagles P. M.1973 J. Fluid Mech. 57, 149.
Eagles, P. M. & Smith F. T.1980 J. Engng Maths 14, 219.
Eagles, P. M. & Weissman M. A.1975 J. Fluid Mech. 69, 241.
Fraenkel L. E.1962 Proc. R. Soc. Lond. A 267, 119.
Fraenkel L. E.1963 Proc. R. Soc. Lond. A 272, 406.
Fraenkel L. E.1973 Proc. Camb. Phil. Soc. 73, 61.
Georgiou, G. A. & Eagles P. M.1985 J. Fluid Mech. 159, 259.
Gillani, N. V. & Swanson W. M.1976 J. Fluid Mech. 78, 99.
Hamel G.1916 Jahresbericht der Deutschen Math. Vereinigung. 25, 34.
Hooper A., Duffy, B. R. & Moffatt H. K.1982 J. Fluid Mech. 117, 283.
Jeffery G. B.1915 Phil. Mag. 29 (6), 455.
Moffatt, H. K. & Duffy B. R.1980 J. Fluid Mech. 96, 299.
Serrin J.1959 Handbuch der Physik 8 (1), 253.
Sobey I. J.1980 J. Fluid Mech. 96, 1.
Sobey I. J.1982 J. Fluid Mech. 125, 359.
Sobey I. J.1985 J. Fluid Mech. 151, 395.
Wille, R. & Fernholz H.1965 J. Fluid Mech. 23, 801.
Wilson S.1969 J. Fluid Mech. 38, 793.
Woods L. C.1954 Aero. Quart. 5, 176.