Existence of global weak solutions to the kinetic Peterlin model

TL;DR

This paper proves the existence of global weak solutions for a kinetic model of dilute polymer solutions, combining Navier-Stokes and Fokker-Planck equations with nonlinear spring laws.

Contribution

It establishes the first proof of global weak solutions for the kinetic Peterlin model in two dimensions, extending previous work to more complex nonlinear spring laws.

Findings

Existence of global weak solutions in 2D

Extension of previous models to nonlinear spring laws

Mathematical validation of the kinetic Peterlin model

Abstract

We consider a class of kinetic models for polymeric fluids motivated by the Peterlin dumbbell theories for dilute polymer solutions with a nonlinear spring law for an infinitely extensible spring. The polymer molecules are suspended in an incompressible viscous Newtonian fluid confined to a bounded domain in two or three space dimensions. The unsteady motion of the solvent is described by the incompressible Navier-Stokes equations with the elastic extra stress tensor appearing as a forcing term in the momentum equation. The elastic stress tensor is defined by the Kramers expression through the probability density function that satisfies the corresponding Fokker-Planck equation. In this case, a coefficient depending on the average length of polymer molecules appears in the latter equation. Following the recent work of Barrett and S\"uli we prove the existence of global-in-time weak solutions to the kinetic Peterlin model in two space dimensions.