Quadratic Dynamical Systems Describing Shear Flow of Non-Newtonian Fluids

Abstract

Phase-plane techniques are used to analyze a quadratic system of ordinary differential equations that approximates a single relaxation-time system of partial differential equations used to model transient behavior of highly elastic non-Newtonian liquids in shear flow through slit dies. The latter one-dimensional model is derived from three-dimensional balance laws coupled with differential constitutive relations well-known by rheologists. The resulting initial-boundary-value problem is globally well-posed and possesses the key feature: the steady shear stress is a non-monotone function of the strain rate. Results of the global analysis of the quadratic system of ode’s lead to the same qualitative features as those obtained recently by numerical simulation of the governing pde’s for realistic data for polymer melts used in rheological experiments. The analytical results provide an explanation of the experimentally observed phenomenon called spurt; they also predict new phenoinena discovered in the numerical simulation; these phenomena should also be observable in experiments.

Supported by the U. S. Army Research Office under Grant DAAL03–87-K-0036, the National Science Foundation under Grants DMS-8712058 and DMS-8620303, and the Air Force Office of Scientific Research under Grants AFOSR-87–0191 and AFOSR-85–0141.