Rheological chaos in a scalar shear-thickening model

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 (gamma) over dot and relaxes by two parallel decay processes: a nonlinear decay at a nonmonotonic rate R(sigma(1)) and a linear decay at rate lambdasigma(2). Here sigma(1,2)(t)=tau(1,2)(-1)integral(0)(t)sigma(t('))exp[-(t-t('))/tau(1,2)]dt(') 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 the 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)-->0. A similar generic mechanism might also arise in shear thinning systems and in some banded flows.