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Groups defined on images in fluid diffusion

Published online by Cambridge University Press:  17 February 2009

A. J. Bracken ,
H. S. Green  and
L. Bass
A. J. Bracken
Affiliation:
Department of Mathematics, University of Queensland, St Lucia, Queensland, 4067.
H. S. Green
Affiliation:
Department of Mathematics, University of Queensland, St Lucia, Queensland, 4067. Department of Mathematical Physics, University of Adelaide, South Australia, 5001.
L. Bass
Affiliation:
Department of Mathematics, University of Queensland, St Lucia, Queensland, 4067.
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Abstract

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A method based on the method of images is described for the solution of the linear equation modelling diffusion and elimination of substrate in a fluid flowing through a chemical reactor of finite length, when the influx of substrate is prescribed at the point of entry and Danckwerts' zero-gradient condition is imposed at the point of exit. The problem is shown to be transformable to an equivalent problem in heat conduction. Associated with the images appearing in the method of solution is a sequence of functions which form a vector space carrying a representation of the Lie group SO(2, 1) generated by three third-order differential operators. The functions are eigenfunctions of one of these operators, with integer-spaced eigenvalues, and they satisfy a third-order recurrence relation which simplifies their successive determination, and hence the determination of the Green's function for the problem, to any desired degree of approximation. Consequently, the method has considerable computational advantages over commonly used methods based on the use of Laplace and related transforms. Associated with these functions is a sequence of polynomials satisfying the same third-order differential equation and recurrence relation. The polynomials are shown to bear a simple relationship to Laguerre polynomials and to satisfy the ordinary diffusion equation, for which SO(2, 1) is therefore revealed as an invariance group. These diffusion polynomials are distinct from the well-known heat polynomials, but a relationship between them is derived. A generalised set of diffusion polynomials, based on the associated Laguerre polynomials, is also described, having similar properties.

Type
Research Article
Information
The ANZIAM Journal , Volume 30 , Issue 1 , July 1988 , pp. 101 - 119
Copyright
Copyright © Australian Mathematical Society 1988

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