Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-29T10:19:22.191Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact solutions to nonlinear diffusion-convection problems on finite domains

Published online by Cambridge University Press:  17 February 2009

G. C. Sander
G. C. Sander
Affiliation:
Division of Science and Technology, Griffith University, Nathan 4111.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

New exact solutions are presented for nonlinear diffusion and convection on a finite domain 0 ≤ z ≤ 1. These solutions are developed for the conditions of constant fluxes at both boundaries z = 0 and z = 1. In particular, solutions for the flux QL at the lower boundary z = 1, being a multiple of the flux Qs at the surface z = 0, (that is QL = aQs, where a = constant), are presented. Solutions for any constant, a, are given for an initial condition which is independent of space z. For the special cases (i) a = 1, and (ii) Qs = 0 and hence QL = 0, solutions are given for an initial condition which has an arbitrary dependence on z.

Type
Research Article
Information
The ANZIAM Journal , Volume 33 , Issue 3 , January 1992 , pp. 384 - 401
Copyright
Copyright © Australian Mathematical Society 1992

References

[1] Barry, D. A. and Sander, G. C., “Exact solutions for water infiltration with an arbitrary surface flux and nonlinear solute adsorption”, Water Resour. Res. (Accepted for publication).Google Scholar
[2] Bluman, G. and Kumei, S., “On the remarkable nonlinear diffusion equation ”, J Math. Phys. 21 (1980) 1019–1023.CrossRefGoogle Scholar
[3] Broadbridge, P., “Non-integrability of non-linear diffusion-convection equations in two spatial dimensions.” J. Phys. A: Math. Gen. 19 (1986) 1245–1257.CrossRefGoogle Scholar
[4] Broadbridge, P., Knight, J. H. and Rogers, C., “Constant rate rainfall infiltration in a bounded profile: solutions of a nonlinear model”, Soil Sci. Soc. Am. J. 52 (1988) 1526–1533.CrossRefGoogle Scholar
[5] Broadbridge, P. and White, I., “Constant rate rainfall infiltration: A versatile nonlinear model 1. Analytic solution”, Water Resour. Res. 24 (1988) 145–154.CrossRefGoogle Scholar
[6] Fokas, A. S. and Yortsos, Y. C., “On the exactly solvable equation occurring in two- phase flow in porous media”, SIAM J. Appl. Math. 42 (1982) 318–332.CrossRefGoogle Scholar
[7] Fujita, H., “The exact pattern of a concentration-dependent diffusion in a semi-infinite medium. Part 1”, Text. Res. 22 (1952) 757–760.CrossRefGoogle Scholar
[8] Fujita, H., “The exact pattern of a concentration-dependent diffusion in a semi-infinite medium. Part 2”, Text. Res. 22 (1952) 823–827.CrossRefGoogle Scholar
[9] Grundy, R. E., “The Cauchy problem for a nonlinear diffusion equation with absorption and convection”, IMA J. Appl. Math. 40 (1988) 183–204.CrossRefGoogle Scholar
[10] Hill, J. M., “Similarity solutions for nonlinear diffusion–a new integration procedure”, J. Engng. Math. 23 (1989) 141–155.CrossRefGoogle Scholar
[11] 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
[12] King, J. R., “Exact solutions to some nonlinear diffusion equations”, Q. J. Mech. Appl. Math. 42 (1989) 537–552.CrossRefGoogle Scholar
[13] King, M. J., “Immiscible two-phase flow in a porous medium: utilisation of a Laplace transform boost”, J. Math. Phys. 26 (1985) 870–877.CrossRefGoogle Scholar
[14] Kingston, J. G. and Rogers, C., “Reciprocal Bäcklund transformations of conservation laws”, Phys. Lett. 92A (1982) 261–264.CrossRefGoogle Scholar
[15] Knight, J. H. and Philip, J. R., “Exact solutions in nonlinear diffusion”, J. Engng. Math. 8 (1974) 219–227.CrossRefGoogle Scholar
[16] Parlange, J.-Y., Braddock, R. D. and Chu, B. T., “First integral of the diffusion equation; an extension of the Fujita solutions”, Soil Sci. Soc. Am. J. 44 (1980) 908–911.CrossRefGoogle Scholar
[17] Philip, J. R., “The theory of infiltration 1. The infiltration equation and its solution”, Soil Sci. 83 (1957), 345–357.CrossRefGoogle Scholar
[18] Rogers, C., “Application of reciprocal Bäcklund transformations to a class of nonlinear boundary value problems”, J. Phys. A: Math. Gen. 16 (1983) L493–L495.CrossRefGoogle Scholar
[19] Rogers, C., Stallybrass, M. P. and Clements, D. L., “On two phase filtration under gravity and with boundary infiltration: application of a Bäcklund transformation”, Nonlinear Anal. Theory Methods Appl. 7 (1983) 785–799.CrossRefGoogle Scholar
[20] Sander, G. C., Cunning, I. F., Hogarth, W. L. and Parlange, J.-Y., “Exact solution for nonlinear, nonhysteretic redistribution in vertical soil of finite depth”, Water Resour. Res. 27 (1991) 1529–1536.CrossRefGoogle Scholar
[21] Sander, G. C., Parlange, J.-Y., Kühnel, V., Hogarth, W. L., Lockington, D. and O'Kane, J. P. J., “Exact nonlinear solution for constant flux infiltration”, J. Hydrol. 97 (1988) 341–346.CrossRefGoogle Scholar
[22] Sander, G. C., Parlange, J.-Y., Kuhnel, V., Hogarth, W. L., and O'Kane, J. P. J., “Comment on ‘Constant rate rainfall infiltration: a versatile nonlinear model 1. Analytic solution’ by P. Broadbridge and I. White”, Water Resour. Res. 24 (1988) 2107–2108.CrossRefGoogle Scholar
[23] Storm, M. L., “Heat conduction in simple metals”, J. Appl. Phys. 22 (1951) 940–951.CrossRefGoogle Scholar
[24] Tuck, B., Atomic diffusion in III-V semiconductors, (Adam Hilger, Bristol, 1988).Google Scholar
[25] Warrick, A. W., Lomen, D. O. and Islas, A., “An analytical solution to Richards’ equation for a draining soil profile”, Water Resour. Res. 26 (1990) 253–258.CrossRefGoogle Scholar
[26] Watson, K. K., Reginato, R. J. and Jackson, R. D., “Soil water hysteresis in a field soil”, Soil Sci. Am. Proc. 39 (1975) 242–246.CrossRefGoogle Scholar
[27] Watson, K. K. and Sardana, V., “Numerical study of the effect of hysteresis on post infiltration redistribution, Infiltration Development and Application”, Proceedings of the International Conf. on Infiltration Development and Application in Hawaii, 241–250, 1987.Google Scholar