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

Published online by Cambridge University Press:  24 October 2008

L. E. Fraenkel  and
P. M. Eagles
L. E. Fraenkel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Cambridge
P. M. Eagles
Affiliation:
Mathematics Department, The City University, London
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Extract

This paper continues (and concludes) the mathematical analysis begun in (8) of a formal theory of viscous flow in channels with slowly curving walls. In that paper, the theory was shown to yield strict asymptotic expansions, in powers of the small curvature parameter, of exact solutions of the Navier-Stokes equations, but the proofs were restricted to a set of Reynolds numbers and wall divergence angles that is distinctly smaller than the set on which the formal approximation is defined. In the present paper, we study in more detail a certain linear, partial differential operator TN, the invertibility of which is essential to the proofs. This operator is shown to be invertible (and the formal theory is thereby justified) on a parameter domain that is much larger than and may well be the whole of . A key step is to associate with TN a family of operators that approximate TN locally and have much simpler coefficients.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 77 , Issue 1 , January 1975 , pp. 199 - 224
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

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