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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
L. E. Fraenkel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Cambridge
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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
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 73 , Issue 2 , March 1973 , pp. 361 - 390
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

REFERENCES

(1)Berger, M. S. and Fraenkel, L. E.On singular perturbations of nonlinear operator equations. Indiana Univ. Math. J. 20 (1971), 623–631.CrossRefGoogle Scholar
(2)Buchwald, V. T.Eigenfunctions of plane elastostatics. I. The strip. Proc. Roy. Soc. Ser. A 277 (1964), 385–400.Google Scholar
(3)Coddington, E. A. and Levinson, N.Theory of ordinary differential equations (McGraw-Hill, 1955).Google Scholar
(4)Fraenkel, L. E.Laminar flow in symmetrical channels with slightly curved walls. I. On the Jeffery-Hamel solutions for flow between plane walls. Proc. Roy. Soc. Ser. A 267 (1962), 119–138.Google Scholar
(5)Fraenkel, L. E.Laminar flow in symmetrical channels with slightly curved walls. II. An asymptotic series for the stream function. Proc. Roy. Soc. Ser. A 272 (1963), 406–428.Google Scholar
(6)Fraenkel, L. E.On a class of linear partial differential equations with slowly varying coefficients. J. London Math. Soc. Ser. 2 5 (1972), 169–181.CrossRefGoogle Scholar
(7)Friedman, A.On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations. J. Math. Mech. 7 (1958), 43–59.Google Scholar
(8)Friedman, A.Partial differential equations (Holt, Rinehart and Winston, 1969).Google Scholar
(9)GÜNter, N. M.Potential theory (Ungar, 1967).Google Scholar
(10)Ladyzhenskaya, O. A.The mathematical theory of viscous incompressible flow (Gordon and Breach, 1963).Google Scholar
(11)Morrey, C. B.Multiple integrals in the calculus of variations (Springer, 1966).Google Scholar
(12)Nirenberg, L.On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115–162.Google Scholar
(13)Prandtl, L. and Tietjens, O. G.Applied hydro- and aeromechanics (McGraw-Hill, 1934; Dover, 1957).Google Scholar
(14)Rosenbead, L. (Editor) Laminar boundary layers (Oxford University Press, 1963).Google Scholar
(15)Sobolev, S. L.Applications of functional analysis in mathematical physics (American Mathematical Society, 1963).CrossRefGoogle Scholar