DOI: 10.1017/s0022112080002133. Local similarity solutions and their limitations

Local similarity solutions and their limitations

10.1017/s0022112080002133

Crossref journal-article

Cambridge University Press (CUP)

Journal of Fluid Mechanics (56)

Abstract

Two problems exhibiting breakdown in local similarity solutions are discussed, and the appropriate asymptotic form of the exact solution is determined in each case. The first problem is the very elementary problem of pressure driven flow along a duct whose cross-section has a sharp corner of angle 2α. When 2α < ½π, a local similarity solution is valid, whereas, when 2α > ½π, the solution near the corner depends on the global geometry of the cross-section. The transitional behaviour when 2α = ½π is determined.The second problem concerns low-Reynolds-number flow in the wedge-shaped region |θ| < α when either a normal velocity proportional to distance from the vertex is imposed on both boundaries, or a finite flux 2Q is introduced or extracted at the vertex (the Jeffery–Hamel problem). It is shown that the similarity solution in either case is valid only when 2α < 2αc ≈ 257·5°; a modified problem is solved exactly revealing the behaviour when α > αc, and the transitional behaviour when α = αc (when the similarity solutions become infinite everywhere). An interesting conclusion for the Jeffery–Hamel problem is that when α > αc, inertia forces are of dominant importance throughout the flow field no matter how small the source Reynolds number 2Q/ν may be.

Bibliography

Moffatt, H. K., & Duffy, B. R. (1980). Local similarity solutions and their limitations. Journal of Fluid Mechanics, 96(2), 299â313.

Authors 2

H. K. Moffatt (first)
B. R. Duffy (additional)

References 13 Referenced 85

Fraenkel, L. E. 1973 On a theory of laminar flow in channels of a certain class.Proc. Camb. Phil. Soc. 73,361–390. (10.1017/S0305004100076933)
Allen, R. F. & Biggin, C. M. 1974 Longitudinal flow of a lenticular liquid filament down an inclined plane.Phys. Fluids 17,287–291. (10.1063/1.1694713)
Fraenkel, L. E. 1961 On corner eddies in plane inviscid shear flow.J. Fluid Mech. 11,400–406. (10.1017/S0022112061000603)
Collins, W. M. & Dennis, S. C. R. 1976 Viscous eddies near a 90° and a 45° corner in flow through a curved tube of triangular cross-section.J. Fluid Mech. 76,417–432. (10.1017/S0022112076000712)
Sternberg, E. & Koiter, W. T. 1958 The wedge under a concentrated couple: a paradox in the two-dimensional theory of elasticity.J. Appl. Mech. 25,575–581.
Hardy, G. H. 1902 On the zeroes of the integral function x—sin x = Σ1 ∞ (−1) n−1 x 2n+1/(2n + 1)! Messenger of Math. 31,161–165.
Jeffrey, D. J. & Sherwood, J. D. 1980 Streamline patterns and eddies in low-Reynolds-number flow.J. Fluid Mech. 96,315. (10.1017/S0022112080002145)
Moffatt, H. K. 1964a Viscous and resistive eddies near a sharp corner.J. Fluid Mech. 18,1–18. (10.1017/S0022112064000015)
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. Roy. Soc. A 267,119–138.
Moffatt, H. K. 1964b Viscous eddies near a sharp corner.Arch. Mech. Stosowanej 16,365–372.
Batchelor, G. K. 1956 On steady laminar flow with closed streamlines at large Reynolds number.J. Fluid Mech. 1,177–190. (10.1017/S0022112056000123)
Batchelor, G. K. 1967 Introduction to Fluid Dynamics. Cambridge University Press.
Tranter, C. J. 1948 The use of the Mellin transform in finding the stress distribution in an infinite wedge.Quart. J. Mech. Appl. Math. 1,125–130. (10.1093/qjmam/1.1.125)

Dates

Type	When
Created	19 years, 4 months ago (April 19, 2006, 10:16 a.m.)
Deposited	6 years, 4 months ago (April 5, 2019, 7:01 p.m.)
Indexed	1 month ago (July 30, 2025, 8:25 p.m.)
Issued	45 years, 7 months ago (Jan. 29, 1980)
Published	45 years, 7 months ago (Jan. 29, 1980)
Published Online	19 years, 4 months ago (April 19, 2006)
Published Print	45 years, 7 months ago (Jan. 29, 1980)

Funders 0

None

BibTeX

@article{Moffatt_1980, title={Local similarity solutions and their limitations}, volume={96}, ISSN={1469-7645}, url={http://dx.doi.org/10.1017/s0022112080002133}, DOI={10.1017/s0022112080002133}, number={2}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Moffatt, H. K. and Duffy, B. R.}, year={1980}, month=jan, pages={299–313} }

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