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Instantaneous point source solutions in nonlinear diffusion with nonlinear loss or gain

Published online by Cambridge University Press:  17 February 2009

J. R. Philip
J. R. Philip
Affiliation:
CSIRO Center for Environmental Mechanics, Canberra, ACT 2601, Australia
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Abstract

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Exact solutions are developed for instantaneous point sources subject to nonlinear diffusion and loss or gain proportional to nth power of concentration, with n > 1. The solutions for the loss give, at large times, power-law decrease to zero of slug central concentration and logarithmic increase of slug semi-width. Those for gain give concentration decreasing initially, going through a minimum, and then increasing, with blow-up to infinite concentration in finite time. Slug semi-width increases with time to a finite maximum in finite time at a blow-up. Taken in conjunction with previous studies, these new results provide an overall schema for instantaneous nonlinear diffusion point sources with nonlinear loss or gain for the total range n ≥ 0. Six distinct regimes of behaviour of slug semi-width and concentration are identified, depending on the range of n, 0 ≤ n < 1, n = 1, or n > 1. Three of them are for loss, and three for gain. The classical Barenblatt-Pattle nonlinear instantaneous point-source solutions with material concentration occupy a central place in the total schema.

Type
Research Article
Information
The ANZIAM Journal , Volume 41 , Issue 3 , January 2000 , pp. 281 - 300
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Ames, W. F., Anderson, R. L., Dorodnitsyn, V. A., Ferapontov, E. V., Gasinov, R. V., Ibragimov, N. H. and Svirshchevskii, S. R., Lie Group Analysis of Differential Equations. Vol. 1, Symmetries, Exact Solutions and Conservation Laws (CRC Press, Boca Raton, 1994).Google Scholar
[2]Arrigo, D. J. and Hill, J. M., “Nonclassical symmetries for nonlinear diffusion and absorption”, Studies in Appl. Maths. 94 (1995) 21–39.CrossRefGoogle Scholar
[3]Barenblatt, G. I., “On some unsteady motions of a liquid and a gas in a porous medium”, Acad. Nauk SSSR Prikl. Math. i Mekh. 16 (1952) 67–78.Google Scholar
[4]Bertsch, M., Kersner, R. and Peletier, L. A., “Positivity versus localization in degenerate diffusion equations”, Nonlinear Anal: Theory, Methods Appl. 9 (1985) 977–1008.CrossRefGoogle Scholar
[5]Hill, J. M., Avagliano, A. A. and Edwards, M. P., “Some exact results for nonlinear diffusion with absorption”, IMA J. Appl. Math. 40 (1992) 283–304.CrossRefGoogle Scholar
[6]Kersner, R., “On the behaviour of temperature fronts in media with nonlinear heat conduction under absorption”, Vestnik Moskovskogo Universiteta Matematika 33 (1978) 44–51.Google Scholar
[7]Miller, C. A. and van Duijn, C. J., “Similarity solutions for gravity-dominated spreading of a lens of organic contaminant”, in Environmental Studies, IMA Volumes in Mathematics and its Applications (Springer, Heidelberg) (in press).Google Scholar
[8]Pattle, R. E., “Diffusion from an instantaneous point source with a concentration-dependent diffusion coefficient”, Q. J. Mech. Appl. Math. 12 (1959) 407–409.CrossRefGoogle Scholar
[9]Philip, J. R., “Exact solutions for nonlinear diffusion with first-order loss”, Int. J. Heat Mass Transfer 37 (1994) 479–484.CrossRefGoogle Scholar
[10]Philip, J. R., “Exact solutions for nonlinear diffusion with nonlinear loss”, ZAMP 45 (1994) 387–398.Google Scholar
[11]Zmitrenko, N. V., Kurdyumov, S. P., Mikhailov, A. P. and Samarskii, A. A., “Occurrence of structures in nonlinear media and the nonstationary thermodynamics of blow-up regimes”, Preprint 74, Institute of Applied Mathematics, Academy of Sciences, U.S.S.R., Moscow, 1976.Google Scholar