Small moving rigid body into a viscous incompressible fluid

TL;DR

This paper investigates the behavior of a small rigid disk moving in a 2D viscous incompressible fluid as its size approaches zero, establishing convergence to Navier-Stokes solutions using decay estimates and fixed point methods.

Contribution

It introduces a novel approach combining decay estimates and fixed point arguments to analyze the zero-size limit of a moving rigid body in a viscous fluid.

Findings

Established uniform estimates for solid velocity as size tends to zero

Proved convergence of the fluid-solid system to Navier-Stokes solutions in 2D

Extended understanding of fluid-structure interaction in the zero-size limit

Abstract

We consider a single disk moving under the influence of a 2D viscous fluid and we study the asymptotic as the size of the solid tends to zero.If the density of the solid is independent of $\varepsilon$, the energy equality is not sufficient to obtain a uniform estimate for the solid velocity. This will be achieved thanks to the optimal $L^p-L^q$ decay estimates of the semigroup associated to the fluid-rigid body system and to a fixed point argument. Next, we will deduce the convergence to the solution of the Navier-Stokes equations in $\R^2$.