Chaimongkol Saengow

Exact analytical solution for large-amplitude oscillatory shear flow from Oldroyd 8-constant framework: Shear stress

The Oldroyd 8-constant model is a continuum framework containing, as special cases, many important constitutive equations for elastic liquids. When polymeric liquids undergo large-amplitude oscillatory shear flow, the shear stress responds as a Fourier series, the higher harmonics of which are caused by the fluid nonlinearity. We choose this continuum framework for its rich diversity of special cases (we tabulate 14 of these). Deepening our understanding of this Oldroyd 8-constant framework thus at once deepens our understanding of every one of these special cases. Previously [C. Saengow et al., Macromol. Theory Simul. 24, 352 (2015)], we arrived at an exact analytical solution for the corotational Maxwell model. Here, we derive the exact analytical expression for the Oldroyd 8-constant framework for the shear stress response in large-amplitude oscillatory shear flow. Our exact solution reduces to our previous solution for the special case of the corotational Maxwell model, as it should. Our worked exampl...

Padé approximants for large-amplitude oscillatory shear flow

Analytical solutions for either the shear stress or the normal stress differences in large-amplitude oscillatory shear flow, both for continuum or molecular models, often take the form of the first few terms of a power series in the shear rate amplitude. Here, we explore improving the accuracy of these truncated series by replacing them with ratios of polynomials. Specifically, we examine replacing the truncated series solution for the corotational Maxwell model with its Padé approximants for the shear stress response and for the normal stress differences. We find these Padé approximants to agree closely with the corresponding exact solution, and we learn that with the right approximants, one can nearly eliminate the inaccuracies of the truncated expansions.

Reflections on Inflections

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,

Padé approximant for normal stress differences in large-amplitude oscillatory shear flow

\eta \left( {\dot \gamma } \right)

Nonlinear core deflection in injection molding

,

Plastic pipe solidification in extrusion

{\Psi _1}\left( {\dot \gamma } \right)