Navier-Stokes-Fourier fluids interacting with elastic shells

TL;DR

This paper proves the existence of weak solutions for a complex fluid-structure interaction system involving a compressible heat-conducting fluid and an elastic shell, ensuring thermodynamic consistency and handling non-linearities.

Contribution

It establishes the first existence results for the Navier-Stokes-Fourier fluid interacting with a non-linear Koiter elastic shell, including thermodynamic laws.

Findings

Existence of weak solutions until boundary self-intersection

Solutions satisfy energy conservation and entropy inequality

Handles non-linear elastic shell with non-convex energy

Abstract

We study the motion of a compressible heat-conducting fluid in three dimensions interacting with a non-linear flexible shell. The fluid is described by the full Navier--Stokes--Fourier system. The shell constitutes an unknown part of the boundary of the physical domain of the fluid and is changing in time. The solid is described as an elastic non-linear shell of Koiter type; in particular it possesses a non-convex elastic energy. Its deformation is modelled by a (non-linear) elastic Koiter shell. We show the existence of a weak solution to the corresponding system of PDEs which exists until the moving boundary approaches a self-intersection or the non-linear elastic energy of the shell degenerates. It is achieved by compactness results (in highest order spaces) for the solid-deformation and fluid-density. Our solutions comply with the first and second law of thermodynamics: the total energy is preserved and the entropy balance is understood as a variational inequality.