A singular perturbation study of the Rolie-Poly model

Abstract

A singular perturbation analysis for the Rolie-Poly model is presented for the limit of large relaxation time. The model describes entangled linear polymer melts and time scales for a Rouse time and reptation time. First, the case of the Rouse time approaching zero, is named the nonstretching model, and the addition of a solvent model gives the small parameter to be the ratio of the retardation time to the slow reptation time. This analysis is instructive in formulating the perturbation analysis for the full Rolie-Poly model without the solvent, where the small parameter is the ratio of Rouse time to reptation time. The resulting formulae elucidate the characteristic features of thixotropic yield stress fluids, including yield stress hysteresis, delayed yielding and long term persistence of a decreased viscosity after cessation of flow. Two fundamental initial value problems are solved: startup of shear flow and cessation of shear ow. The scalings lead naturally to simplified dynamical systems that describe regimes of fast, slow and yielded dynamics, and show how the combination of these regimes can be used to describe the entire flow evolution.