Analysis of a viscosity model for concentrated polymers

TL;DR

This paper develops and analyzes a complex mathematical model for polymeric fluids, incorporating polymerization effects into the viscosity and proving the existence of global solutions.

Contribution

It introduces a coupled Navier-Stokes and integro-differential system with shear-rate dependent viscosity, proving global existence of weak solutions for the first time.

Findings

Existence of global-in-time weak solutions.

Viscosity depends on shear-rate and polymer chain length.

Model captures polymerization effects on fluid viscosity.

Abstract

The paper is concerned with a class of mathematical models for polymeric fluids, which involves the coupling of the Navier-Stokes equations for a viscous, incompressible, constant-density fluid with a parabolic-hyperbolic integro-differential equation describing the evolution of the polymer distribution function in the solvent, and a parabolic integro-differential equation for the evolution of the monomer density function in the solvent. The viscosity coefficient appearing in the balance of linear momentum equation in the Navier-Stokes system includes dependence on the shear-rate as well as on the weight-averaged polymer chain length. The system of partial differential equations under consideration captures the impact of polymerization and depolymerization effects on the viscosity of the fluid. We prove the existence of global-in-time, large-data weak solutions under fairly general hypotheses.