On the trajectory of a light small rigid body in an incompressible viscous fluid

TL;DR

This paper analyzes the motion of a tiny rigid body in a viscous incompressible fluid, showing that as the body shrinks and loses mass, its trajectory aligns with the fluid flow, extending understanding in fluid-structure interaction.

Contribution

It demonstrates the limiting behavior of a small rigid body's trajectory in viscous fluids as its size and mass tend to zero, including cases with bounded density.

Findings

Rigid body's velocity matches fluid velocity in the limit.

Trajectory convergence as size and mass tend to zero.

Applicable to bodies with bounded density relative to size.

Abstract

In this paper we study the dynamics of a small rigid body in a viscous incompressible fluid in dimension two and three. More precisely we investigate the trajectory of the rigid body in the limit when the its mass and its size tend to zero. We show that the velocity of the center of mass of the rigid body coincides with the background fluid velocity in the limit. We are able to consider the case where the density of the small rigid body is uniformly bounded respect to its size.