Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 3 10.wxyz/CV72P3 http://www.austms.org.au/Publ/Bulletin/V72P3/ F.R. de Hoog R.S. Anderssen 13 February 2006 2006 2 13 461 470 723-5220-deHoAn-2005 10.wxyz/C2005V72P3p461 http://www.austms.org.au/Publ/Bulletin/V72P3/723-5220-deHoAn/ 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. 65B15, 76A05 M. Abramowitz and I.A. Stegun Dover Publications MR208797 M. Abramowitz and I.A. Stegun; \textit{Handbook of mathematical functions} (Dover Publications, New York, 1966). C. Ancey J. Rheol. C. Ancey; Solving the Couette inverse problem by using a wavelet-vaguelette decomposition, \textit{J. Rheol.} \textbf{49} (2005), pp.~441--460. J.C. Baudez and P. Coussot Phys. Review Let. J.C. Baudez and P. Coussot; Abrupt transition from viscoelastic solidlike to liquidlike behaviour in jammed materials, \textit{Phys. Review Let.} \textbf{93} (2004), pp.~#128302(4). R.K. Code and J.D. Raal Rheol. Acta R.K. Code and J.D. Raal; Rates of shear in coaxial cylinder viscometers, \textit{Rheol. Acta} \textbf{12} (1973), pp.~578--587. M. Couette Ann. Chim. Phys. M. Couette; \'{E}tudes sur le frottement des liquides, \textit{Ann. Chim. Phys.} \textbf{21} (1890), pp.~433--510. F.R. de Hoog and R.S. Anderssen J. Integral Equations Appl. F.R. de Hoog and R.S. Anderssen; Regularization of first kind integral equations with application to Couette viscometry, \textit{J. Integral Equations Appl.} (to appear). D. Donoho Appl. Comput. Harmon. Anal. MR1325535 D. Donoho; Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition, \textit{Appl. Comput. Harmon. Anal.} \textbf{2} (1995), pp.~101--126. D. Elliott J. Austral. Math. Soc. Ser. B MR1659938 D. Elliott; The Euler--Maclaurin formula revisited, \textit{J. Austral. Math. Soc. Ser. B} \textbf{40} (1998), pp.~E27--E76. F.D. Farrow, G.M. Lowe and S.M. Neale J. Textile Inst. F.D. Farrow, G.M. Lowe and S.M. Neale; The flow of starch pastes flow at high and low rates of shear, \textit{J. Textile Inst.} \textbf{19} (1928), pp.~T18--T31. J. Hart Springer MR1461272 J. Hart; \textit{Nonparametric smoothing and lack-of-fit tests} (Springer, New York, 1999). I.M. Krieger Trans. Soc. Rheol. I.M. Krieger; Shear rate in the Couette viscometer, \textit{Trans. Soc. Rheol.} \textbf{12} (1968), pp.~5--11. I.M. Krieger and H. Elrod J. Appl. Phys. I.M. Krieger and H. Elrod; Direct determination of the flow curves of non-Newtonian fluids. II. Shearing rate in the concentric cylinder viscometer, \textit{J. Appl. Phys.} \textbf{24} (1953), pp.~134--136. I.M. Krieger and S.H. Maron J. Appl. Phys. I.M. Krieger and S.H. Maron; Direct determination of the flow curves of non-Newtonian fluids,, \textit{J. Appl. Phys.} \textbf{23} (1952), pp.~147--149. Y.K. Leong and Y.L. Yeow Rheol. Acta Y.K. Leong and Y.L. Yeow; Obtaining the shear stress shear rate relationship and yield stress of liquid foods from Couette viscometry data, \textit{Rheol. Acta} \textbf{42} (2003), pp.~365--371. M. Mooney J. Rheol. M. Mooney; Explicit formulas for slip and fluidity, \textit{J. Rheol.} \textbf{2} (1931), pp.~210--222. J. Pawlowski Kolloid Zeit. J. Pawlowski; Bestimmung des Reibungsgesetzes der nicht-Newtonschen Fl\"{u}ssigkeiten aus den Viskosit\"{a}tsmessungen mit Hilfe eines Rotationsviskosimeters, \textit{Kolloid Zeit.} \textbf{10} (1953), pp.~129--131. J.M. Piau, M. Bremond, J.M. Couette and M. Piau Rheol. Acta J.M. Piau, M. Bremond, J.M. Couette and M. Piau; Maurice Couette, one of the founders of rheology, \textit{Rheol. Acta} \textbf{33} (1994), pp.~357--368. C. Picart, J.M. Piau, H. Galliard and P. Carpenter J. Rheol. C. Picart, J.M. Piau, H. Galliard and P. Carpenter; Human blood yield stress and its hematorit dependence, \textit{J. Rheol.} \textbf{42} (1998), pp.~1--12.