Shear banding of the fluid with a nonmonotonic dependence of flow stress upon strain rate

The problem of the pressure flow of a fluid in a flat channel with the counter motion of one of the walls is considered. The fluid is characterized a non-monotonic flow curve consisting of three segments: left segment (ascending branch), middle segment (descending branch) and right segment (ascending branch). The rheological properties of the fluid are described by a modified model of Vinogradov-Pokrovsky. The constants of the model are determined using the results of rheological tests of high-density polyethylene melt performed with a laser Doppler viscometer. All exact analytical solutions of this problem are obtained in parametric form. The profiles of velocity, effective viscosity and velocity gradient along the channel height are constructed for different parameters of the rheological model. It is shown that at the same prescribed stress field, in the range of shear rates corresponding to the descending branch of the flow curve, there are three solutions, of which one is unstable and not physically realizable and the other two are stable; which of them is realized depends on the loading prehistory. One of these solutions, corresponding to the left branch of the flow curve, is monotone, and the solution corresponding to the right branch of the curve demonstrates the stratification of the flow into strips with different physico-mechanical properties and at different strain rates. At the same time, the dependence of the effective viscosity on the strain rate is a monotonically decreasing function. The same problem is solved for a two-dimensional case by the finite element method using a weak Galerkin formulation. Comparison of the numerical results with the analytical solution shows that the results coincide with a sufficient degree of accuracy. In either case, as the counter pressure drop approaches zero, the limiting transition to the Couette flow is impossible.