On the velocity of a small rigid body in a viscous incompressible fluid in dimension two and three

TL;DR

This paper investigates how a small rigid body's velocity evolves in a viscous incompressible fluid in 2D and 3D, showing that as the body's size shrinks to zero, it is not accelerated by the fluid if its mass remains bounded below.

Contribution

It introduces a new a priori estimate on the velocities of rigid bodies that can decrease to zero mass, extending understanding of fluid-body interactions at small scales.

Findings

Small particles are not accelerated by the fluid as their size approaches zero.

New a priori velocity estimates for rigid bodies with decreasing mass.

Results apply in both two and three dimensions.

Abstract

In this paper we study the evolution of a small rigid body in a viscous incompressible fluid, in particular we show that a small particle is not accelerated by the fluid in the limit when its size converges to zero under a lower bound on its mass. This result is based on a new a priori estimate on the velocities of the centers of mass of rigid bodies that holds in the case when their masses are also allowed to decrease to zero.