Rheological instability in a simple shear thickening model

– We study the strain response to steady imposed stress in a spatially homogeneous, scalar model for shear thickening, in which the local rate of yielding Γ(l) of mesoscopic ‘elastic elements’ is not monotonic in the local strain l. Despite this, the macroscopic, steady-state flow curve (stress vs. strain rate) is monotonic. However, for a broad class of Γ(l), the response to steady stress is not in fact steady flow, but spontaneous oscillation. We discuss this finding in relation to other theoretical and experimental flow instabilities. Within the parameter ranges we studied, the model does not exhibit rheo-chaos. The flow behaviour of shear-thickening materials such as dense colloidal suspensions can be complex [1,2]. For example, imposition of a steady mean strain rate can lead to large, possibly chaotic, variations in the mean stress [1]. The same occurs in some types of shear-thickening micellar surfactant solutions, where true temporal chaos seems now to be established [3] (and also in shear thinning systems; see [4]). Other unexpected behaviour, such as a bifurcation to an oscillatory state, has also been seen in shear-thickening ‘onion’ phases of surfactant [5]. It is not yet known to what extent such unsteady flow is generic in shear-thickening systems; in this letter we attempt to shed some light on the issue by studying a much-simplified, generic model. In this model we find, for a wide range of parameters, spontaneous rheological oscillation of the strain rate at fixed stress. Rheo-chaos is, however, not found for the parameters studied so far. A feature that distinguishes the rheological instabilities encountered in shear-thickening from those arising in Newtonian fluids is that the nonlinearity is not inertial (not from the advective term of the Navier Stokes equation): the Reynolds number is essentially zero [6]. Instead it arises from anharmonic elastic responses at large deformations, complicated and perhaps strongly enhanced by the presence of memory effects. Flow instabilities leading to chaos have been studied recently by Grosso et al. [7] in a model for suspended rodlike particles. As that work shows, and our work confirms, temporal instabilities can arise even in a model where macroscopic spatial inhomogeneity is disallowed altogether. This is a strong demarcation from the familiar shear-banding instabilities that arise whenever the steady-state flow curve is nonmonotonic (the flow curve is the function σ(γ̇), where σ is shear stress and γ̇