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On a self-similar solution for the decay of turbulent bursts

Published online by Cambridge University Press:  16 July 2009

S. P. Hastings  and
L. A. Peletier
S. P. Hastings
Affiliation:
Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15205, USA
L. A. Peletier
Affiliation:
Department of Mathematics and Computer Science, Leiden University, Postbus 9512, 2300 RA Leiden, The Netherlands
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Abstract

We discuss the self-similar solutions of the second kind associated with the propagation of turbulent bursts in a fluid at rest. Such solutions involve an eigenvalue parameter μ, which cannot be determined from dimensional analysis. Existence and uniqueness are established and the dependence of μ on a physical parameter λ in the problem is studied: estimates are obtained and the asymptotic behaviour as λ → ∞ is established.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 4 , December 1992 , pp. 319 - 341
Copyright
Copyright © Cambridge University Press 1992

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References

[A]Aronson, D. G. 1980 Density-dependent interaction - diffusion systems. In Dynamics and modelling of reactive systems (eds. Stewart, W. E., Harmon Ray, W. & Conley, C. C.). Academic Press, pp. 161–176.CrossRefGoogle Scholar
[B]Barenblatt, G. I. 1983 Selfsimilar turbulence propagation from an instantaneous plane source. In Nonlinear dynamics and turbulence (eds. Barenblatt, G. I., loos, G. & Joseph, D. D.). Pitman, pp. 48–60.Google Scholar
[BER]Barenblatt, G. I., Entov, V. M. & Ryzhik, V. M. 1990 Theory of fluid flow through natural rocks. Kluwer, p. 182.Google Scholar
[BGL]Barenblatt, G. I., Galerkina, N. L. & Luneva, M. V. 1987 Evolution of turbulent bursts. Inzhenerno-Fizicheskii Zh. (J. Engg. Phys.), 53, 733–740 (in Russian).Google Scholar
[CG]Chen, L.-Y. & Goldenfeld, N. 1992 Renormalisation group theory for the propagation of a turbulent burst. Phys. Rev. A., 45, 5572–5577.CrossRefGoogle Scholar
[DS]Diaz, J. I. & Saa, J. 1992 Uniqueness of very singular self-similar solution of the parabolic p Laplacian operator with absorption. Preprint.CrossRefGoogle Scholar
[GP]Gilding, B. H. & Peletier, L. A. 1976 On a class of similarity solutions of the porous media equation. J. Math. Anal. Appl. 55, 351–368.CrossRefGoogle Scholar
[KPV1]Kamin, S., Peletier, L. A. & Vazquez, J. L. 1991 On the Barenblatt equation of elastoplastic filtration. Indianna University Math. J., 40, 1333–1362.CrossRefGoogle Scholar
[KPV2]Kamin, S., Peletier, L. A. & Vazquez, J. L. 1989 Classification of singular solutions of a nonlinear diffusion equation. Duke Math. J. 58, 601–615.CrossRefGoogle Scholar
[KVa]Kamin, S. & Vazquez, J. L. 1991 The propagation of turbulent bursts. IMA preprint Series no. 843, 08.Google Scholar
[KVe]Kamin, S. & Veron, L. 1988 Existence and uniqueness of the very singular solution for the porous media equation with absorption. J. d'Anal. Math. 51, 245–258.CrossRefGoogle Scholar
[K]Kolmogorov, A. N. 1942 Equations of turbulent motion of an incompressible fluid. Izvestia Akademii Nauk SSSR, Serija Fizicheskaya 6, 56–58.Google Scholar
[MY]Monin, A. S. & Yaglom, A. M. 1971 Statistical fluid mechanics: mechanics of turbulence, Vol. I. MIT Press.Google Scholar
[P]Peletier, L. A. 1981 The porous media equation. In Applications of nonlinear analysis in the physical sciences (eds. Amann, H., Bazley, N. & Kirchgässner, K.). Pitman, 229–241.Google Scholar
[PT]Peletier, L. A. & Terman, D. 1986 A very singular solution of the porous media equation with absorption. J. Diff. Equ. 65, 396–410.CrossRefGoogle Scholar