Rheological Chaos in a Scalar Shear-Thickening Model

Abstract

We study a simple scalar constitutive equation for a shear-thickening material at zero Reynolds number, in which the shear stress σis driven at a constant shear rate \dotγand relaxes by two parallel decay processes: a nonlinear decay at a nonmonotonic rate R(σ_1) and a linear decay at rate λσ_2. Here σ_{1,2}(t) = τ_{1,2}^{-1}\int_0^tσ(t')\exp[-(t-t')/τ_{1,2}] {\rm d}t' are two retarded stresses. For suitable parameters, the steady state flow curve is monotonic but unstable; this arises when τ_2>τ_1 and 0>R'(σ)>-λso 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 τ_1\to 0. A similar generic mechanism might also arise in shear thinning systems and in some banded flows.