Rheological Chaos in a Scalar Shear-Thickening Model

TL;DR

This paper models a shear-thickening material using a scalar equation that exhibits instability and chaos due to nonlinear and retarded stress decay processes, revealing complex flow behavior.

Contribution

It introduces a simple scalar model capturing shear-thickening instability and chaos, highlighting the role of retarded stress relaxation in flow dynamics.

Findings

Steady state flow curve can be unstable despite being monotonic.

The model exhibits a period-doubling route to chaos.

Instability persists even when one decay process is negligible.

Abstract

We study a simple scalar constitutive equation for a shear-thickening material at zero Reynolds number, in which the shear stress \sigma is driven at a constant shear rate \dot\gamma and relaxes by two parallel decay processes: a nonlinear decay at a nonmonotonic rate R(\sigma_1) and a linear decay at rate \lambda\sigma_2. Here \sigma_{1,2}(t) = \tau_{1,2}^{-1}\int_0^t\sigma(t')\exp[-(t-t')/\tau_{1,2}] {\rm d}t' are two retarded stresses. For suitable parameters, the steady state flow curve is monotonic but unstable; this arises when \tau_2>\tau_1 and 0>R'(\sigma)>-\lambda so that monotonicity is restored only through the strongly retarded term (which might model a slow evolution of material structure under stress). Within the unstable region we find a period-doubling sequence leading to chaos. Instability, but not chaos, persists even for the case \tau_1\to 0. A similar generic mechanism might also arise in shear thinning systems and in some banded flows.