From shear-thickening and periodic flow behavior to rheo-chaos in nonlinear Maxwell-model fluids

The nonlinear Maxwell model equation for the stress tensor as introduced previously in [O. Hess and S. Hess, Physica A 207 (1994) 517] to treat the shear-thickening and shear-thinning behavior of fluids can also be applied for temperatures and densities where a substance shows a yield stress. The basic equations are discussed. Analytic and mainly numerical results are presented for the plane Couette flow geometry. Depending on the model parameters and on the imposed shear rate, a stationary state can be reached or not. In the second case periodic solutions of stick-slip like motions or irregular chaotic behavior is found. For some typical cases the shear stress, the first and second normal stress differences as well as the stress components which break the Couette symmetry are displayed as functions of time. Different types of time dependent solutions can be distinguished by the rheological phase portraits. Some remarks are made on the entropy production associated with the viscous flow and the stress relaxation.