Rheological Chaos in a Scalar Shear-Thickening Model
Résumé
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.