Shear Banding of a Fluid Flow with a Nonmonotonic Dependence of the Flow Stress on the Strain Rate

Abstract

The problem of an enforced fluid flow in a flat channel with the counter motion of one of the walls was considered. The fluid was characterized by a nonmonotonic flow curve consisting of three segments; a left segment (an ascending branch), a middle segment (a descending branch), and a right segment (an ascending branch). The rheological properties of the fluid were described by the modified Vmogradov-Pokrovsky model. The constants of the model were determined using the results of rheological tests of a high-density polyethylene melt performed with a laser Doppler viscometer. All exact analytical solutions of this problem were obtained in a parametric form for the one-dimensional case. The profiles of the velocity, effective viscosity, and velocity gradient along the channel height are constructed for different values of the parameters of the rheological model. Three solutions exist for the same given stress field in the range of shear rates corresponding to the middle branch of the flow curve. One of them is unstable and physically unrealizable, while the other two solutions are stable; however, the loading prehistory determines which of them is observed. The solution corresponding to the left branch is monotonic, while the solution corresponding to the right branch of the curve demonstrates the flow stratification into “bands” with different strain rates and different physical and mechanical properties. At the same time, the dependence of the effective viscosity on the strain rate, which is a monotonically decreasing function, allows its representation in the form of an exponential series. The same pressure-flow problem is solved in the two-dimensional formulation by the finite-element method using the semiweak Galerkin formulation and an approximation function for the viscosity. The comparison of the numerical results and the analytical solution shows that they are similar with a sufficient degree of accuracy. In both cases, as the counter pressure difference approaches zero, a limiting transition to the Couette flow is impossible.