J Austral Math Soc Ser B 33 pp363--383, 1992.

Exact solutions for redistribution by nonlinear convection-diffusion

J. R. Philip

(Received 25 July 1990; revised 11 October 1990)

Abstract

The nonlinear convection-diffusion equation has been studied for 40 years in the context of nonhysteretic water movement in unsaturated soil. We establish new similarity solutions for instantaneous sources of finite strength redistributed by nonlinear convection-diffusion obeying the dimensionless equation

¶q/¶t = ¶(q^m¶q/¶z)/¶z - q^{m + 1}¶q/¶z (m ³ 0).

For m = 0 (Burgers' equation) solutions involve the error function, and for m = 1 Airy functions. Problems 1, 2, and 3 relate, respectively, to the regions 0 £ z £ ¥, -¥ £ z £ ¥, and -¥ £ z £ 0. Solutions for m = 0 have infinite tails, but for m > 0 and finite t, q > 0 inside, and q = 0 outside, a finite interval in z. At the slug boundary, q(z) is tangential to the z-axis for 0 < m < 1; and it meets the axis obliquely for m = 1 and normally for m > 1. Illustrative results are presented. For Problems 1 and 2 (but not 3) finiteness of source strength sets an upper bound on Q₀, the similarity "concentration" at z = 0. The magnitude of convection relative to diffusion increases with Q₀, and apparently the dynamic equilibrium between the two processes, implied by the similarity solutions, ceases to be possible when Q₀ is large enough.

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Author

J. R. Philip: CSIRO Centre foe Environmental Mechanics, Canberra, ACT 2601, Australia.

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Last Modified: Sun Jan 27 17:57:08 2002