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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

References

[1] Antosiewicz, H. A., “10. Bessel functions of fractional order”, in Handbook of mathematical functions, (eds. Abramowitz, M. and Stegun, I. A.), (U. S. Govt. Printing Office, Washington, 1964) 437–478.Google Scholar
[2] Buckingham, E., “Studies on the movement of soil moisture”, U.S. Dept. Agr. Bur. Soils Bull. 33 (1907).Google Scholar
[3] Burgers, J. M., “A mathematical model illustrating the theory of turbulence”, Adv. Applied Mechs. 1 (1948) 171–199.CrossRefGoogle Scholar
[4] Childs, E. C., The physical basis of soil water phenomena (Wiley, London, 1969).Google Scholar
[5] Gautschi, W., “7. Error function and Fresnel integrals”, in Handbook of mathematical functions, (eds. Abramowitz, M. and Stegun, I. A.), (U.S. Govt. Printing Office, Washington, 1964) 295–309.Google Scholar
[6] Hill, J. M. and Hill, D. M., “High-order nonlinear evolution equations”, IMA J. Appl. Math. 44 (1990) 243–265.CrossRefGoogle Scholar
[7] Hogarth, W. L., Parlange, J.-Y., and Braddock, R. D., “First integrals of the infiltration equation: 2. Nonlinear conductivity”, Soil Sci. 148 (1989) 165–171.CrossRefGoogle Scholar
[8] Parlange, J.-Y. and Fleming, J. F., “First integrals of the infiltration equation: 1. Theory”, Soil Sci. 137 (1984) 391–394.CrossRefGoogle Scholar
[9] Pattle, R. E., “Diffusion from an instantaneous point source with a concentration-dependent coefficient”, Q. J. Mech. Appl. Maths. 12 (1959) 407–409.CrossRefGoogle Scholar
[10] Philip, J. R., “The theory of infiltration: 3. Moisture profiles and relation to experiment”, Soil Sci. 84 (1957) 163–178.CrossRefGoogle Scholar
[11] Philip, J. R., “Theory of infiltration”, Adv. Hydrosci. 5 (1969) 215–296.CrossRefGoogle Scholar
[12] Philip, J. R., “Flow in porous media”, Ann. Rev. Fluid Mech. 2 (1970) 177–204.CrossRefGoogle Scholar
[13] Philip, J. R., “Flow in porous media”, in Theoretical and applied mechanics, (eds. Becker, E. and Mikhailov, G. K.), Proc. 13th Int. Congr. Theor. Appl. Mech. (Springer, Berlin, 1973) 279–294.CrossRefGoogle Scholar
[14] Philip, J. R., “Recent progress in the solution of nonlinear diffusion equations”, Soil Sci. 117 (1974) 257–264.CrossRefGoogle Scholar
[15] Philip, J. R., “Quasianalytic and analytic approaches to unsaturated flow”, in Flow and transport in the natural environment: advances and applications, (eds. Steffen, W. L. and Denmead, O. T.), (Springer, Heidelberg, 1988) 30–47.Google Scholar
[16] Philip, J. R., “Infiltration of water into soil”, ISI Atlas of Science: Animal and Plant Sciences 1 (1988) 231–235.Google Scholar
[17] Richards, L. A., “Capillary conduction of liquids through porous mediums”, Physics 1 (1931) 318–333.CrossRefGoogle Scholar
[18] Richardson, L. F., “The deferred approach to the limit, part 1” Phil. Trans. Roy. Soc. London A 226 (1927) 299–349.Google Scholar