Remote trajectory tracking of a rigid body in an incompressible fluid at low Reynolds number

TL;DR

This paper demonstrates that it is possible to precisely control the trajectory of a rigid body immersed in a stationary incompressible Stokes flow using boundary controls, enabling exact trajectory tracking under certain conditions.

Contribution

The paper introduces a method for exact remote trajectory control of a rigid body in a Stokes flow using boundary data, combining density arguments and an ODE reformulation.

Findings

Exact trajectory tracking is achievable for smooth paths away from the boundary.

Control is implemented via Dirichlet boundary conditions on the fluid domain.

The approach relies on density methods and reformulation of the solid's motion into an ODE.

Abstract

In this paper we study the motion of a rigid body driven by Newton's law immersed in a stationary incompressible Stokes flow occupying a bounded simply connected domain. The aim is that of trajectory tracking of the solid by the means of a control in the form of Dirichlet boundary data on the outside boundary of the fluid domain. We show that it is possible to exactly achieve any smooth trajectory for the solid that stays away from the external boundary, by the means of such a remote control. The proof relies on some density methods for the Stokes system, as well as a reformulation of the solid equations into an ODE.