Bull. Austral. Math. Soc. 72(3) pp.461--470, 2005.

Approximate solutions for the Couette viscometry equation

F.R. de Hoog

R.S. Anderssen

Received: 2nd August, 2005



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.

Click to download PDF of this article (free access until July 2006)

or get the no-frills version

[an error occurred while processing this directive]
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2199647 Z'blatt-MATH: 1127.76006


  1. M. Abramowitz and I.A. Stegun;
    Handbook of mathematical functions (Dover Publications, New York, 1966). MR208797
  2. C. Ancey;
    Solving the Couette inverse problem by using a wavelet-vaguelette decomposition,
    J. Rheol. 49 (2005), pp. 441--460.
  3. J.C. Baudez and P. Coussot;
    Abrupt transition from viscoelastic solidlike to liquidlike behaviour in jammed materials,
    Phys. Review Let. 93 (2004), pp. #128302(4).
  4. R.K. Code and J.D. Raal;
    Rates of shear in coaxial cylinder viscometers,
    Rheol. Acta 12 (1973), pp. 578--587.
  5. M. Couette;
    Études sur le frottement des liquides,
    Ann. Chim. Phys. 21 (1890), pp. 433--510.
  6. F.R. de Hoog and R.S. Anderssen;
    Regularization of first kind integral equations with application to Couette viscometry,
    J. Integral Equations Appl. (to appear).
  7. D. Donoho;
    Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition,
    Appl. Comput. Harmon. Anal. 2 (1995), pp. 101--126. MR1325535
  8. D. Elliott;
    The Euler--Maclaurin formula revisited,
    J. Austral. Math. Soc. Ser. B 40 (1998), pp. E27--E76. MR1659938
  9. F.D. Farrow, G.M. Lowe and S.M. Neale;
    The flow of starch pastes flow at high and low rates of shear,
    J. Textile Inst. 19 (1928), pp. T18--T31.
  10. J. Hart;
    Nonparametric smoothing and lack-of-fit tests (Springer, New York, 1999). MR1461272
  11. I.M. Krieger;
    Shear rate in the Couette viscometer,
    Trans. Soc. Rheol. 12 (1968), pp. 5--11.
  12. I.M. Krieger and H. Elrod;
    Direct determination of the flow curves of non-Newtonian fluids. II. Shearing rate in the concentric cylinder viscometer,
    J. Appl. Phys. 24 (1953), pp. 134--136.
  13. I.M. Krieger and S.H. Maron;
    Direct determination of the flow curves of non-Newtonian fluids,,
    J. Appl. Phys. 23 (1952), pp. 147--149.
  14. Y.K. Leong and Y.L. Yeow;
    Obtaining the shear stress shear rate relationship and yield stress of liquid foods from Couette viscometry data,
    Rheol. Acta 42 (2003), pp. 365--371.
  15. M. Mooney;
    Explicit formulas for slip and fluidity,
    J. Rheol. 2 (1931), pp. 210--222.
  16. J. Pawlowski;
    Bestimmung des Reibungsgesetzes der nicht-Newtonschen Flüssigkeiten aus den Viskositätsmessungen mit Hilfe eines Rotationsviskosimeters,
    Kolloid Zeit. 10 (1953), pp. 129--131.
  17. J.M. Piau, M. Bremond, J.M. Couette and M. Piau;
    Maurice Couette, one of the founders of rheology,
    Rheol. Acta 33 (1994), pp. 357--368.
  18. C. Picart, J.M. Piau, H. Galliard and P. Carpenter;
    Human blood yield stress and its hematorit dependence,
    J. Rheol. 42 (1998), pp. 1--12.

ISSN 0004-9727