New Temperature Dependent Configurational Probability Diffusion Equation For Diluted FENE Polymer Fluids: Existence of Solution Results

TL;DR

This paper derives a nonlinear temperature-dependent configurational probability density equation for FENE polymer fluids without linear gradient approximations and proves the existence of positive solutions, advancing the theoretical understanding of non-isothermal polymer rheology.

Contribution

It extends previous models by removing linear gradient approximations and establishes the existence of solutions for the nonlinear equation in FENE polymer fluids.

Findings

Derived a nonlinear temperature-dependent probability density equation

Proved the existence of positive variational solutions

Enhanced theoretical framework for non-isothermal polymer rheology

Abstract

The theory for the non-isothermal rheology of polymer fluids proposed in [14] used several approximations including the so-called linear gradient approximations for the temperature field and Brownian forces. While it had the significant advantage of dealing with linear equations, the approximations involved may have led to several non-physical predictions. This work is a continuation of [14] in that it obtains the corresponding non-linear configurational probability density equation in dimensionless form without the linear gradient approximations for the temperature field and Brownian forces. It does so for incompressible diluted polymer solutions with polymer molecules being modeled as FENE (F initely E xtensible N onlinear E lastic) chains. Next we prove the existence of temperature dependent, positive variational solutions for the probability density equation of the FENE model.