On the motion of a nearly incompressible viscous fluid containing a small rigid body

TL;DR

This paper demonstrates that in a nearly incompressible viscous fluid, a small rigid body's influence is negligible, regardless of shape or boundary conditions, under low Mach number conditions in a planar domain.

Contribution

It establishes that the effect of a small rigid body on a nearly incompressible viscous fluid is negligible, independent of shape or boundary conditions, in a planar domain.

Findings

The influence of the rigid body is negligible under specified conditions.

Shape and boundary conditions of the body are irrelevant.

Collisions with the boundary are permitted.

Abstract

We consider the motion of a compressible viscous fluid containing a moving rigid body confined to a planar domain $\Omega \subset R^2$. The main result states that the influence of the body on the fluid is negligible if (i) the diameter of the body is small and (ii) the fluid is nearly incompressible (the low Mach number regime). The specific shape of the body as well as the boundary conditions on the fluid--body interface are irrelevant and collisions with the boundary $\partial \Omega$ are allowed. The rigid body motion may be enforced externally or governed solely by its interaction with the fluid.