Capillarity-driven thinning and breakup of weakly rate-thickening fluids

A number of commercial fluids, including synthetic automotive oils, food and consumer products containing polymer additives exhibit weakly rate-thickening responses in the final stages of capillarity-driven thinning, where a large accumulated strain and high extensional strain rate alter the thinning dynamics of the slender liquid filament. Consequently, measurements of capillarity-driven thinning dynamics typically feature two distinct regions at the early and late stages of the filament breakup process, each dominated by distinct mechanisms. These features have been incorporated in a simple Inelastic Rate-Thickening (IRT) model with linear and quadratic contributions to the constitutive stress–strain rate relationship, in which the apparent extensional viscosity slowly thickens at high strain rates. We numerically compute the thinning dynamics of the IRT model assuming an axially-slender axisymmetric filament and no fluid inertia. The computational results motivate a similarity transformation and we obtain a new self-similar solution in which the second-order stress is balanced by capillarity. The new asymptotic solution leads to a self-similar filament shape that is more slender than the Newtonian counterpart and, close to singularity, results in a quadratic dependence of the mid-point radius of the filament with time to breakup. A new and distinct asymptotic geometric correction factor, $X\approx 0.5827$ is derived and we show that a more accurate value of the true extensional viscosity in a rate-thickening fluid can be recovered from an interpolated time-varying geometric correction factor based on the magnitudes of different stress components. Finally, we propose a statistically data-driven protocol to select the best-fit constitutive model using a parameter-free information criterion. This enables us to more accurately quantify the extensional rheological behavior of complex rate-thickening viscoelastic fluids using capillarity-driven thinning dynamics.