Existence of a local strong solution to the beam-polymeric fluid interaction system

TL;DR

This paper proves the existence of a unique local strong solution for a complex fluid-structure interaction model involving polymer fluids and elastic shells, extending previous results to arbitrary smooth 3D domains.

Contribution

It establishes local well-posedness for a coupled polymer fluid and elastic shell system on general 3D domains, including the first global solutions in 2D for the co-rotational Fokker-Planck model.

Findings

Existence of a unique local strong solution in 3D domains.

Global-in-time strong solutions in 2D for the co-rotational Fokker-Planck model.

Development of higher-order well-posedness techniques for solvent-structure systems.

Abstract

We construct a unique local strong solution to the finitely extensible nonlinear elastic (FENE) dumbbell model of Warner-type for an incompressible polymer fluid (described by the Navier-Stokes-Fokker-Planck equations) interacting with a flexible elastic shell. The latter occupies the flexible boundary of the polymer fluid domain and is modeled by a beam equation coupled through kinematic boundary conditions and the balance of forces. In the 2D case for the co-rotational Fokker-Planck model we obtain global-in-time strong solutions. A main step in our approach is the proof of local well-posedness for just the solvent-structure system in higher-order topologies which is of independent interest. Different from most of the previous results in the literature, the reference spatial domain is an arbitrary smooth subset of $\mathbb{R}^3$, rather than a flat one. That is, we cover viscoelastic shells rather than elastic plates. Our result also supplements the existing literature on the Navier-Stokes-Fokker-Planck equations posed on a fixed bounded domain.