The motion of a fluid-rigid disc system at the zero limit of the rigid disc radius

TL;DR

This paper proves that as the radius of a rigid disc in a coupled fluid-rigid system approaches zero, the system's behavior converges to pure fluid flow, with the disc's center following a fluid particle trajectory.

Contribution

It establishes the zero-radius limit of a fluid-rigid disc system, showing convergence to Navier-Stokes solutions and particle-like motion of the disc's center.

Findings

System converges to Navier-Stokes solution as radius approaches zero.

Disc's center trajectory matches fluid particle path in the limit.

Rigid disc's rotation is not considered in the limit analysis.

Abstract

We consider the two-dimensional motion of the coupled system of a viscous incompressible fluid and a rigid disc moving with the fluid, in the whole plane. The fluid motion is described by the Navier-Stokes equations and the motion of the rigid body by conservation laws of linear and angular momentum. We show that, assuming that the rigid disc is not allowed to rotate, as the radius of the disc goes to zero, the solution of this system converges, in an appropriate sense, to the solution of the Navier-Stokes equations describing the motion of only fluid in the whole plane. We also prove that the trajectory of the centre of the disc, at the zero limit of its radius, coincides with a fluid particle trajectory.