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Exact solutions for redistribution by nonlinear convection-diffusion

Published online by Cambridge University Press:  17 February 2009

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

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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 ∂θ/∂t = ∂ (θm ∂θ/∂z) − θm+1 ∂θ/∂z (m ≥ 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, θ > 0 inside, and θ = 0 outside, a finite interval in z. At the slug boundary, θ(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 Θ0, the similarity “concentration” at z = 0. The magnitude of convection relative to diffusion increases with Θ0; and apparently the dynamic equilibrium between the two processes, implied by the similarity solutions, ceases to be possible when Θ0 is large enough.

Type
Research Article
Information
The ANZIAM Journal , Volume 33 , Issue 3 , January 1992 , pp. 363 - 383
Copyright
Copyright © Australian Mathematical Society 1992

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