Skip to main content

The Jeffery–Hamel similarity solution and its relation to flow in a diverging channel

  • P. E. Haines (a1), R. E. Hewitt (a1) and A. L. Hazel (a1)

We explore the relevance of the idealized Jeffery–Hamel similarity solution to the practical problem of flow in a diverging channel of finite (but large) streamwise extent. Numerical results are presented for the two-dimensional flow in a wedge of separation angle , bounded by circular arcs at the inlet/outlet and for a net radial outflow of fluid. In particular, we show that in a finite domain there is a sequence of nested neutral curves in the plane, each corresponding to a midplane symmetry-breaking (pitchfork) bifurcation, where is a Reynolds number based on the radial mass flux. For small wedge angles we demonstrate that the first pitchfork bifurcation in the finite domain occurs at a critical Reynolds number that is in agreement with the only pitchfork bifurcation in the infinite-domain similarity solution, but that the criticality of the bifurcation differs (in general). We explain this apparent contradiction by demonstrating that, for , superposition of two (infinite-domain) eigenmodes can be used to construct a leading-order finite-domain eigenmode. These constructed modes accurately predict the multiple symmetry-breaking bifurcations of the finite-domain flow without recourse to computation of the full field equations. Our computational results also indicate that temporally stable, isolated, steady solutions may exist. These states are finite-domain analogues of the steady waves recently presented by Kerswell, Tutty, & Drazin (J. Fluid Mech., vol. 501, 2004, pp. 231–250) for an infinite domain. Moreover, we demonstrate that there is non-uniqueness of stable solutions in certain parameter regimes. Our numerical results tie together, in a consistent framework, the disparate results in the existing literature.

Corresponding author
Email address for correspondence:
Hide All

Present address: School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia.

Hide All
1. Akulenko, L. D. & Kumakshev, S. A. 2008 Bifurcation of multimode flows of a viscous fluid in a plane diverging channel. J. Appl. Math. Mech. 72 (3), 296302.
2. Banks, W. H. H., Drazin, P. G. & Zaturska, M. B. 1988 On perturbations of Jeffery–Hamel flow. J. Fluid Mech. 186, 559581.
3. Cliffe, K. A. & Greenfield, A. C. 1982 Some comments on laminar flow in symmetric two dimensional channels. Harwell Report AERE-TP 939.
4. Cliffe, K. A., Spence, A. & Tavener, S. J. 2000 The numerical analysis of bifurcation problems with application to fluid mechanics. Acta Numerica 9, 39131.
5. Dean, W. R. 1934 Note on the divergent flow of fluid. Phil. Mag. 18 (7), 759777.
6. Dennis, S. C. R., Banks, W. H. H., Drazin, P. G. & Zaturska, M. B. 1997 Flow along a diverging channel. J. Fluid Mech. 336, 183202.
7. Drazin, P. G. 1995 Stability of flow in a diverging channel. In Stability and Wave Propagation in Fluids and Solids (ed. Galdi, G. P. ). pp. 3965. Springer.
8. Fearn, R. M., Mullin, T. & Cliffe, K. A. 1990 Nonlinear flow phenomena in a symmetric sudden expansion. J. Fluid Mech. 211, 595608.
9. 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. Ser. A, Math. Phys. Sci. 267 (1328), 119138.
10. Goldshtik, M., Hussain, F. & Shtern, V. 1991 Symmetry breaking in vortex-source and Jeffery–Hamel flows. J. Fluid Mech. 232, 521566.
11. Gresho, P. M. & Sani, R. L. 1998 Incompressible Flow and the Finite Element Method: Volume 1, Advection–Diffusion and Isothermal Laminar Flow. John Wiley & Sons.
12. Haines, P. E. 2010 The Jeffery–Hamel similarity solution and its relation to flow in a diverging channel. PhD thesis, The University of Manchester.
13. Hamadiche, M., Scott, J. & Jeandel, D. 1994 Temporal stability of Jeffery–Hamel flow. J. Fluid Mech. 268, 7188.
14. Hamel, G. 1916 Spiralförmige Bewegungen zäher Flüssigkeiten. Jahresbericht der Deutschen Mathematiker Vereinigung 25, 3460.
15. Heil, M. & Hazel, A. L. 2006 oomph-lib– An object-oriented multi-physics finite-element library. In Fluid-Structure Interaction (ed. Schafer, M. & Bungartz, H.-J. ). Lecture Notes in Computational Science and Engineering 53. pp. 1949. Springer.
16. Hewitt, R. E. & Hazel, A. L. 2007 Midplane-symmetry breaking in the flow between two counter-rotating disks. J. Engng Maths 57 (3), 273288.
17. Jeffery, G. B. 1915 The two-dimensional steady motion of a viscous fluid. Phil. Mag. 6 (29), 455465.
18. Kerswell, R. R., Tutty, O. R. & Drazin, P. G. 2004 Steady nonlinear waves in diverging channel flow. J. Fluid Mech. 501, 231250.
19. Kuznetsov, Y. A. 1998 Elements of Applied Bifurcation Theory. Springer.
20. Putkaradze, V. & Vorobieff, P. 2006 Instabilities, bifurcations, and multiple solutions in expanding channel flows. Phys. Rev. Lett. 97 (14), 144502.
21. Rosenhead, L. 1940 The steady two-dimensional radial flow of viscous fluid between two inclined plane walls. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. (1934–1990) 175 (963), 436467.
22. Salinger, A. G., Bou-Rabee, N. M., Pawlowski, R. P., Wilkes, E. D., Burroughs, E. A., Lehoucq, R. B. & Romero, L. A. 2002 LOCA 1.0 Library of continuation algorithms: theory and implementation manual. Tech. Rep. SAND2002-0396. Sandia National Laboratories.
23. Sobey, I. J. & Drazin, P. G. 1986 Bifurcations of two-dimensional channel flows. J. Fluid Mech. 171, 263287.
24. Sobey, I. J. & Mullin, T. 1993 Calculation of multiple solutions for the two-dimensional Navier–Stokes equations. Numer. Meth. Fluid Dyn. 4, 417.
25. Tutty, O. R. 1996 Nonlinear development of flow in channels with non-parallel walls. J. Fluid Mech. 326, 265284.
26. Yu, Z., Shao, X. & Lin, J. 2010 Numerical computations of the flow in a finite diverging channel. J. Zhejiang Univ.-Science A 11 (1), 5060.
27. Zienkiewicz, O. C. & Zhu, J. Z. 1992 The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique. Intl J. Numer. Meth. Engng 33 (7), 13311364.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 41 *
Loading metrics...

Abstract views

Total abstract views: 240 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th July 2018. This data will be updated every 24 hours.