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On a theory of laminar flow in channels of a certain class

Published online by Cambridge University Press:  24 October 2008

L. E. Fraenkel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Cambridge

Extract

This paper is concerned with the steady plane flow of a viscous fluid in symmetrical channels with slowly curving walls. The product of local channel half-width and local wall curvature is bounded by a small parameter ∈. We review the essentials of the formal approximation, in powers of ∈, proposed in (4) and (5); resolve a question, left open there, regarding the existence of the approximate series for the stream function for any value of the Reynolds number and to arbitrary order in ∈ and prove that, under certain restrictions on the Reynolds number and the divergence angle of the channel walls, this formal series is in fact a strict asymptotic expansion (for ∈ → 0) of an exact solution of the Navier–Stokes equations. As a result, the traditional picture of laminar separation, due to Prandtl, emerges as part of the steady flow field predicted by an exact solution that is known explicitly to arbitrary asymptotic order.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

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