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Approximate solutions for the Couette viscometry equation

Published online by Cambridge University Press:  17 April 2009

F. R. de Hoog  and
R. S. Anderssen
F. R. de Hoog
Affiliation:
CSIRO Mathematical and Information Sciences, GPO Box 664, Canberra ACT 2601, Australia
R. S. Anderssen
Affiliation:
CSIRO Mathematical and Information Sciences, GPO Box 664, Canberra ACT 2601, Australia
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The recovery of flow curves for non-Newtonian fluids from Couette rheometry measurements involves the solution of a quite simple first kind Volterra integral equation with a discontinuous kernel for which the solution, as a summation of an infinite series, has been known since 1953. Various methods, including an Euler-Maclaurin sum formula, have been proposed for the estimation of the value of the summation. They all involve the numerical differentiation of the observational data. In this paper, the properties of Bernoulli polynomials, in conjunctions with the special structure of the integral equation, are exploited to derive a parametric family of representations for its solution. They yield formulas similar to, but more general than, the previously published Euler-Maclaurin sum formula representations. The parameterisation is then utilised to derive two new classes of approximations. The first yields a family of finite difference approximations, which avoids the direct numerical differentiation of the observational data, while the second generates a framework for the construction of improved power law approximations.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 3 , December 2005 , pp. 461 - 470
Copyright
Copyright © Australian Mathematical Society 2005

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