An incompressible polymer fluid interacting with a Koiter shell

TL;DR

This paper proves the existence of weak solutions for a complex coupled system modeling an incompressible polymer fluid interacting with a flexible Koiter shell, combining fluid dynamics, mesoscopic polymer behavior, and shell elasticity.

Contribution

It introduces a novel coupled model integrating Navier-Stokes, Fokker-Planck, and Koiter shell equations, and establishes the existence of solutions for this system.

Findings

Existence of weak solutions until shell degenerates or self-intersects.

Mathematical framework for fluid-polymer-shell interaction.

Handling of nonlinear fourth-order shell equation.

Abstract

We study a mutually coupled mesoscopic-macroscopic-shell system of equations modeling a dilute incompressible polymer fluid which is evolving and interacting with a flexible shell of Koiter type. The polymer constitutes a solvent-solute mixture where the solvent is modelled on the macroscopic scale by the incompressible Navier-Stokes equation and the solute is modelled on the mesoscopic scale by a Fokker-Planck equation (Kolmogorov forward equation) for the probability density function of the bead-spring polymer chain configuration. This mixture interacts with a nonlinear elastic shell which serves as a moving boundary of the physical spatial domain of the polymer fluid. We use the classical model by Koiter to describe the shell movement which yields a fully nonlinear fourth order hyperbolic equation. Our main result is the existence of a weak solution to the underlying system which exists until the Koiter energy degenerates or the flexible shell approaches a self-intersection.