The steady state configurational distribution diffusion equation of the standard FENE dumbbell polymer model: existence and uniqueness of solutions for arbitrary velocity gradients

TL;DR

This paper proves the existence and uniqueness of solutions for the steady state configurational distribution equation of the FENE polymer model, regardless of flow velocity gradients, which is fundamental for predicting polymer stress in viscoelastic flows.

Contribution

It provides a rigorous proof of the existence and uniqueness of solutions for the steady state FENE model under arbitrary flow conditions.

Findings

Unique solutions exist for all flow velocities.

The solutions are valid for both slow and fast flows.

The results underpin accurate stress calculations in polymer dynamics.

Abstract

The configurational distribution function, solution of an evolution (diffusion) equation of the Fokker-Planck-Smoluchowski type, is (at least part of) the corner stone of polymer dynamics: it is the key to calculating the stress tensor components. This can be reckoned from \cite{bird2}, where a wealth of calculation details is presented regarding various polymer chain models and their ability to accurately predict viscoelastic flows. One of the simplest polymer chain idealization is the Bird and Warner's model of finitely extensible nonlinear elastic (FENE) chains. In this work we offer a proof that the steady state configurational distribution equation has unique solutions irrespective of the (outer) flow velocity gradients (i.e. for both slow and fast flows).