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On a linear, partly hyperbolic model of viscoelastic flow past a plate

Published online by Cambridge University Press:  14 November 2011

L. E. Fraenkel
L. E. Fraenkel
Affiliation:
School of Mathematics, University of Bath, Bath BA2 7AY, U.K.
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Synopsis

The paper presents solutions, for a class of shear relaxation functions, ofa linear problem formulated by Joseph [8] to elucidate the steady, supercritical flow of a viscoelastic fluid past a semi-infinite flat plate. The velocity (U, 0) at infinity is parallel to the plate, and‘supercritical flow ‘means that U is greater than the propagation speed C of shear waves. As a result, the vorticity satisfies a hyperbolic equation and is confined to the region downstream of a shock wave from the leading edge of the plate. The disturbance velocity fieldextends upstream of the shock and is continuous across it. In contrast to the case of a Newtonian fluid, the solutions are unique under the condition that the functions representing the vorticity on the two sides of the platebelong to a certain Banach space.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 3-4 , 1990 , pp. 299 - 354
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Butzer, P. L. and Nessel, R. J.. Fourier analysis and approximation, vol. I (New York: Academic Press, 1971).CrossRefGoogle Scholar
2Dunford, N. and Schwarz, J. T.. Linear operators, part I (New York: Wiley-Interscience, 1958).Google Scholar
3Fraenkel, L. E.. Some results for a linear, partly hyperbolic model of viscoelastic flow past a plate. In Material instabilities in continuum mechanics, ed. Ball, J. M., pp. 137–146 (Oxford: Oxford University Press, 1988).Google Scholar
4Fraenkel, L. E.. Examples of supercritical, linearized, viscoelastic flow past a plate (to appear).Google Scholar
5Fraenkel, L. E.. A space of slowly decreasing functions with pleasant Fourier transforms (to appear).Google Scholar
6Friedlander, F. G.. Introduction to the theory of distributions (Cambridge: Cambridge University Press, 1982).Google Scholar
7Hörmander, L.. The analysis of linear partial differential operators I (Berlin: Springer, 1983).Google Scholar
8Joseph, D. D.. Hyperbolic phenomena in the flow of viscoelastic fluids. In Viscoelasticity and rheology, eds. Lodge, A. S., Renardy, M. & Nohel, J. A., pp. 235–321 (New York: Academic Press, 1985).CrossRefGoogle Scholar
9Joseph, D. D. and Renardy, M.. Stokes' first problem for linear viscoelastic fluids with finite memory. Rheol. Acta 28 (1989), 453–456.CrossRefGoogle Scholar
10Kaplun, S.. The role of coordinate systems in boundary-layer theory. Z. Angew. Math. Phys. 5 (1954), 111–135.CrossRefGoogle Scholar
11Lewis, J. A. and Carrier, G. F.. Some remarks on the flat plate boundary layer. Quart. Appl. Math. 7 (1949), 228–234.CrossRefGoogle Scholar
12Noble, B.. The Wiener-Hopf technique (Oxford: Pergamon Press, 1958).Google Scholar
13Olmstead, W. E.. A homogeneous solution for viscous flow around a half-plane. Quart. Appl. Math. 33 (1975), 165–169.CrossRefGoogle Scholar
14Olmstead, W. E. and Hector, D. L.. On the nonuniqueness of Oseen flow pasta half plane. J. Math. Phys. 45 (1966), 408–417.CrossRefGoogle Scholar
15Paley, R. E. A. C. and Wiener, N.. Fourier transforms in the complex domain. American Mathematical Society, Colloquium Publication 19 (New York: American Mathematical Society, 1934).Google Scholar
16Protter, M. H. and Weinberger, H. F.. Maximum principles in differential equations (Englewood Cliffs, N.J.: Prentice-Hall, 1967).Google Scholar
17Renardy, M.. Some remarks on the propagation and non-propagation of discontinuities in linearly viscoelastic liquids. Rheol. Acta 21 (1982), 251–254.CrossRefGoogle Scholar
18Rudin, W.. Real and complex analysis (New York: McGraw-Hill, 1970).Google Scholar
19Schwartz, L.. Théorie des distributions, vols. I and II (Paris: Hermann, 1950).Google Scholar
20Schwartz, L.. Transformation de Laplace des distributions. Comm. Seminaire Math. Univ. Lund, tome suppl. dedié à M. Riesz (1952), 196–206.Google Scholar
21Titchmarsh, E. C.. The theory of functions (Oxford: Clarendon Press, 1932).Google Scholar
22Titchmarsh, E. C.. Introduction to the theory of Fourier integrals (Oxford: Clarendon Press, 1948).Google Scholar
23Triebel, H.. Interpolation theory, function spaces, and differential operators (Amsterdam: North-Holland, 1978).Google Scholar
24Dyke, M. Van. Perturbation methods in fluid mechanics (Stanford: Parabolic Press, 1975).Google Scholar
25Widder, D. V.. The Laplace transform (Princeton: University Press, 1946).Google Scholar
26Zemanian, A. H.. Distribution theory and transform analysis (New York: McGraw-Hill, 1965)Google Scholar