Non-Newtonian laminar flow in pipes using radius, stress, shear rate or velocity as the independent variable

Abstract

The flow of a non-Newtonian fluid in a circular pipe is a classic introductory transport phenomena problem, familiar to readers of Robert Byron Bird textbooks. A characteristic of Bird's work was taking the time to explore alternative ways to describe a problem and refine the results into elegant and readable formulas. Inspired by that approach, we compare methods for pipe flow solutions that differ on the independent variable used (radius, stress, shear rate) to obtain flow rate and residence time distributions for generalized Newtonian fluids. We highlight cases where using the shear rate as the independent variable has advantages for analytical and numerical solutions. We describe a method to use velocimetry experimental data coupled with a pressure drop measurement to directly construct a curve of flow rate vs pressure drop without the need of fitting the data to any rheological models. We present a geometrical interpretation of velocity profiles as areas in the stress–shear rate plane and derive analytical solutions for a three-parameter model of soft glassy materials [Caggioni et al., “Variations of the Herschel–Bulkley exponent reflecting contributions of the viscous continuous phase to the shear rate-dependent stress of soft glassy materials,” J. Rheol. 64, 413 (2020)] and a four-parameter model for chocolate melts [H. D. Tscheuschner, “Rheologische eigenschaften von lebensmittelsystemen,” in Rheologie Der Lebensmittel, edited by D. Weipert, H. Tscheuschner, and E. Windhab (Behr's Verlag, Hamburg, 1993), pp. 101–172]. We also compare the speed of various numerical approaches for a fractional viscoelastic model [A. Jaishankar and G. H. McKinley, “A fractional K-BKZ constitutive formulation for describing the nonlinear rheology of multiscale complex fluids,” J. Rheol. 58, 1751 (2014)].