Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-06-12T19:43:18.158Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

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.

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

References

[1]Abramowitz, M. and Stegun, I.A., Handbook of mathematical functions (Dover Publications, New York, 1965).Google Scholar
[2]Ancey, C., ‘Solving the Couette inverse problem by using a wavelet-vaguelette decomposition’, J. Rheol. 49 (2005), 441–460.Google Scholar
[3]Baudez, J.C. and Coussot, P., ‘Abrupt transition from viscoelastic solidlike to liquidlike behaviour in jammed materials’, Phys. Review Let. 93 (2004), #128302(4).CrossRefGoogle ScholarPubMed
[4]Code, R.K. and Raal, J.D., ‘Rates of shear in coaxial cylinder viscometers,’ Rheol. Acta 12 (1973), 578–587.Google Scholar
[5]Couette, M., ‘Études sur le frottement des liquides’, Ann. Chim. Phys. 21 (1890), 433–510.Google Scholar
[6]de Hoog, F.R. and Anderssen, R.S., ‘Regularization of first kind integral equations with application to Couette viscometry’, J. Integral Eqations Appl. (to appear).Google Scholar
[7]Donoho, D., ‘Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition’, Appl. Comput. Harmon. Anal. 2 (1995), 101–1126.CrossRefGoogle Scholar
[8]Elliott, D., ‘The Euler-Maclaurin formula revisited’, J. Austral. Math. Soc. Ser. B 40 (1998), E27–E76.Google Scholar
[9]Farrow, F.D., Lowe, G.M. and Neale, S.M., ‘The flow of starch pastes flow at high and low rates of shear’, J. Textile Inst. 19 (1928), T18–T31.Google Scholar
[10]Hart, J., Nonparametric smoothing and lack-of-fit tests (Springer, New York, 1999).Google Scholar
[11]Krieger, I.M., ‘Shear rate in the Couette viscometer’, Trans. Soc. Rheol. 12 (1968), 5–11.CrossRefGoogle Scholar
[12]Krieger, I.M. and Elrod, H., ‘Direct determination of the flow curves of non-Newtonian fluids. II. Shearing rate in the concentric cylinder viscometer’, J. Appl. Phys. 24 (1953), 134–136.CrossRefGoogle Scholar
[13]Krieger, I.M. and Maron, S.H., ‘Direct determination of the flow curves of non-Newtonian fluids’, J. Appl. Phys. 23 (1952), 147–149.CrossRefGoogle Scholar
[14]Leong, Y.K. and Yeow, Y.L., ‘Obtaining the shear stress shear rate relationship and yield stress of liquid foods from Couette viscometry data,’ Rheol. Acta 42 (2003), 365–371.Google Scholar
[15]Mooney, M., ‘Explicit formulas for slip and fluidity’, J. Rheol. 2 (1931), 210–222.Google Scholar
[16]Pawlowski, J., ‘Bestimmung des Reibungsgesetzes der nicht-Newtonschen Flüssigkeiten aus den Viskositätsmessungen mit Hilfe eines Rotationsviskosimeters’, Kolloid Zeit. 10 (1953), 129–131.CrossRefGoogle Scholar
[17]Piau, J.M., Bremond, M., Couette, J.M. and Piau, M., ‘Maurice Couette, one of the founders of rheology,’ Rheol. Acta 33 (1994), 357–368.CrossRefGoogle Scholar
[18]Picart, C., Piau, J.M., Galliard, H. and Carpenter, P., ‘Human blood yield stress and its hematorit dependence’, J. Rheol. 42 (1998), 1–12.CrossRefGoogle Scholar