Lagrangian controllability at low Reynolds number

TL;DR

This paper proves that at low Reynolds numbers, fluid flow can be controlled to move specific regions by boundary inputs, using advanced approximation techniques related to harmonic functions.

Contribution

It introduces a novel Lagrangian controllability result for Stokes flows, employing a weak Runge-Walsh approximation, with variants suitable for numerical implementation.

Findings

Establishes Lagrangian controllability for low Reynolds number fluids.

Develops a weak approximation theorem for harmonic functions related to Stokes flows.

Provides two variants of the controllability result, one optimized for numerical simulations.

Abstract

In this paper, we establish a result of Lagrangian controllability for a fluid at low Reynolds number, driven by the stationary Stokes equation. This amounts to the possibility of displacing a part of a fluid from one zone to another by suitably using a boundary control. This relies on a weak variant of the Runge-Walsh's theorem (on approximation of harmonic functions) concerning the Stokes equation. We give two variants of this result, one of which we believe to be better adapted to numerical simulations.