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On backward boundary layers and flow in converging passages

Published online by Cambridge University Press:  28 March 2006

Sydney Goldstein
Sydney Goldstein
Affiliation:
The Division of Engineering and Applied Physics, Pierce Hall, Harvard University, Cambridge, Massachusetts, U.S.A.
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Abstract

In a backward boundary layer the fluid has, in the mathematical model, been flowing along a solid wall through an infinite distance. The co-ordinate distance x along the boundary is measured upstream, and the velocity U of the flow outside the boundary layer is taken as negative. The main application is to flow in converging passages.

The existence of similar solutions is considered, with emphasis on the correct asymptotic behaviour for large values of the stretched co-ordinate normal to the wall. This emphasis is shown to be necessary in considering backward boundary layers.

For two-dimensional flow in converging passages the requirement that a boundary layer should be possible for vanishingly small viscosity with a potential core flow is shown to lead directly to Hamel's spirals as the shape of the boundary streamlines.

Flow in axisymmetric converging passages is considered. For flow in a cone there is no limit as the viscosity tends to zero, and no potential core flow with a boundary layer is possible. The nature of a solution of the Navier-Stokes equations for laminar flow is considered.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 21 , Issue 1 , January 1965 , pp. 33 - 45
Copyright
© 1965 Cambridge University Press

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References

Ackerberg, R. C. 1962 Harvard Ph.D. Thesis (to be published).
Ackerberg, R. C. 1965 J. Fluid Mech. 21, 47.
Coppel, W. A. 1960 Phil. Trans. A, 253, 101–36.
Falkner, V. M. & Skan, S. W. 1930 Aeronautical Research Committee, R. & M. no. 1314.
Goldstein, S. 1939 Proc. Camb. Phil. Soc. 35, 338–40.
Hamel, G. 1916 Jahresbericht der Deutschen Mathematiker-Vereinigung, 35, 34–60.
Hardy, G. H. 1939 Proc. Camb. Phil. Soc. 35, 652–3.
Harrison, W. J. 1920 Proc. Camb. Phil. Soc. 19, 307–12.
Hartree, D. R. 1937 Proc. Camb. Phil. Soc. 33, 223–39.
Iglisch, R. 1953 Z.A.M.M. 33, 143–7.
Iglisch, R. 1954 Z.A.M.M. 34, 441–3.
Kaplun, S. 1954 J. Appl. Math. Phys. 5, 111–35.
Lagerstrom, P. A. & Cole, J. D. 1955 J. Rat. Mech. Anal. 4, 817–82.
Mills, R. H. 1938 J. Aero. Sci. 5, 325–7.
Serrin, J. 1962 Department of Mathematics, University of Minnesota, Lecture Notes on Mathematical Aspects of Boundary Layer Theory.
Stewartson, K. 1954 Proc. Camb. Phil. Soc. 50, 454–65.
Weyl, H. 1942 Ann. Math. 43, 381–407.
Whittaker, E. T. & Watson, G. N. 1920 Modern Analysis (3rd ed.). Cambridge University Press.