Reflections on inflections

Abstract

In plastics processing, the single most important rheological property is the steady shear viscosity curve: the logarithm of the steady shear viscosity versus the logarithm of the shear rate. This curve governs the volumetric flowrate through any straight channel flow, and thus governs the production rate of extruded plastics. If the shear rate is made dimensionless with a characteristic time for the fluid (called the Weissenberg number, Wi), then we can readily identify the end of the Newtonian plateau of a viscosity curve with the value Wi≈1. Of far greater importance, however, is the slope at the point where the viscosity curve inflects, (n-1), where n is called the shear power-law index. This paper explores the physics of this point and related inflections, in the first and second normal stress coefficients. We also discuss the first and second inflection pairing times, λ′B and λ″B. First, we examine the generalized Newtonian fluid (Carreau model). Then, we analyze the more versatile model, the corotational Oldroyd 8-constant model, which reduces to many simpler models, for instance, the corotational Maxwell and Jeffreys models. We also include worked examples to illustrate the procedure for calculating inflection points and power-law coefficients for all three viscometric functions, \(\eta \left( {\dot \gamma } \right)\), \({\Psi _1}\left( {\dot \gamma } \right)\) and \({\Psi _2}\left( {\dot \gamma } \right)\).