On the vanishing rigid body problem in a viscous compressible fluid

TL;DR

This paper investigates the behavior of a small rigid body in a viscous compressible fluid, demonstrating convergence to the compressible Navier-Stokes system as the body's size diminishes, advancing understanding of fluid-structure interactions.

Contribution

It provides the first homogenization result for fluid-structure interaction in a compressible setting and improves existing results on obstacle influence for certain adiabatic indices.

Findings

System converges to compressible Navier-Stokes as body size tends to zero.

Established homogenization in fluid-structure interaction for compressible fluids.

Improved obstacle influence results for γ ≥ 6.

Abstract

In this paper we study the interaction of a small rigid body in a viscous compressible fluid. The system occupies a bounded three dimensional domain. The object it allowed to freely move and its dynamics follows the Newton's laws. We show that as the size of the object converges to zero the system fluid plus rigid body converges to the compressible Navier-Stokes system under some mild lower bound on the mass and the inertia momentum. It is a first result of homogenization in the case of fluid-structure interaction in the compressible situation. As a corollary we slightly improved the result on the influence of a vanishing obstacle in a compressible fluid for $ \gamma \geq 6$ .