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On perturbations of Jeffery-Hamel flow

Published online by Cambridge University Press:  21 April 2006

W. H. H. Banks ,
P. G. Drazin  and
M. B. Zaturska
W. H. H. Banks
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
P. G. Drazin
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
M. B. Zaturska
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
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Abstract

We examine various perturbations of Jeffery-Hamel flows, the exact solutions of the Navier-Stokes equations governing the steady two-dimensional motions of an incompressible viscous fluid from a line source at the intersection of two rigid plane walls. First a pitchfork bifurcation of the Jeffery-Hamel flows themselves is described by perturbation theory. This description is then used as a basis to investigate the spatial development of arbitrary small steady two-dimensional perturbations of a Jeffery-Hamel flow; both linear and weakly nonlinear perturbations are treated for plane and nearly plane walls. It is found that there is strong interaction of the disturbances up- and downstream if the angle between the planes exceeds a critical value 2α2, which depends on the value of the Reynolds number. Finally, the problem of linear temporal stability of Jeffery-Hamel flows is broached and again the importance of specifying conditions up- and downstream is revealed. All these results are used to interpret the development of flow along a channel with walls of small curvature. Fraenkel's (1962) approximation of channel flow locally by Jeffery-Hamel flows is supported with the added proviso that the angle between the two walls at each station is less than 2α2.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 186 , January 1988 , pp. 559 - 581
Copyright
© 1988 Cambridge University Press

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References

Allmen, M. J. & Eagles, P. M. 1984 Stability of divergent channel flows: a numerical approach. Proc. R. Soc. Lond. A 392, 359–372.Google Scholar
Brady, J. F. 1984 Fluid development in a porous channel and tube. Phys. Fluids 27, 1061–1067.Google Scholar
Bramley, J. S. & Dennis, S. C. R. 1982 The calculation of eigenvalues for the stationary perturbation of Poiseuille flow. J. Comp. Phys. 47, 179–198.Google Scholar
Buitrago, S. E. 1983 Detailed analysis of the higher Jeffery-Hamel solutions. M.Phil. thesis, University of Sussex.
Cherdron, W., Durst, F. & Whitelaw, J. H. 1978 Asymmetric flows and instabilities in symmetric ducts with sudden expansions. J. Fluid Mech. 84, 13–31.Google Scholar
Cliffe, K. A. & Greenfield, A. C. 1982 Some comments on laminar flow in symmetric two-dimensional channels. Rep. TP 939. AERE, Harwell.
Dean, W. R. & Montagnon, P. E. 1949 On the steady motion of viscous liquid in a corner. Proc. Camb. Phil. Soc. 45, 389–394.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Fraenkel, L. E. 1962 Laminar flow in symmetrical channels with slightly curved walls. I. On the Jeffery-Hamel solutions for flow between plane walls. Proc. R. Soc. Lond. A 267, 119–138.Google Scholar
Fraenkel, L. E. 1963 Laminar flow in symmetrical channels with slightly curved walls. II. An asymptotic series for the stream function. Proc. R. Soc. Lond. A 272, 406–428.Google Scholar
Fraenkel, L. E. 1973 On a theory of laminar flow in channels of a certain class. Proc. Camb. Phil. Soc. 73, 361–390.Google Scholar
Georgiou, G. A. & Eagles, P. M. 1985 The stability of flows in channels with small wall curvature. J. Fluid Mech. 159, 259–287.Google Scholar
Hamel, G. 1916 Spiralförmige Bewegungen zäher Flüssigkeiten. Jahresbericht der Deutschen Math. Vereinigung 25, 34–60.Google Scholar
Jeffery, G. B. 1915 The two-dimensional steady motion of a viscous fluid. Phil. Mag. (6) 29, 455–465.Google Scholar
Lcgt, H. J. & Schwiderski, E. W. 1965 Flows around dihedral angles. I. Eigenmotion and analysis. Proc. R. Soc. Lond. A 285, 382–399.Google Scholar
Moffatt, H. K. & Duffy, B. R. 1980 Local similarity solutions and their limitations. J. Fluid Mech. 96, 299–313.Google Scholar
Sobey, I. J. 1985 Observation of waves during oscillatory channel flow. J. Fluid Mech. 151, 395–426.Google Scholar
Sobey, I. J. & Drazin, P. G. 1986 Bifurcation of two-dimensional channel flows. J. Fluid Mech. 171, 263–287.Google Scholar
Sternberg, E. & Koiter, W. T. 1958 The wedge under a concentrated couple: a paradox in the two-dimensional theory of elasticity. Trans. ASME E: J. Appl. Mech. 25, 575–581.Google Scholar