Metric Geometry Governs Optimal Control in Driven Stokes Flows: Magnetic Driving and Beyond

TL;DR

This paper demonstrates that in driven Stokes flows, optimal particle control paths are geodesics of an emergent metric, with applications to magnetic driving and general driven flows.

Contribution

It introduces a geometric framework linking energy-optimal control paths to Riemannian geodesics in driven Stokes flows, extending to 3D cases.

Findings

Energy-optimal control paths are geodesics of an emergent Riemannian metric.

Particle diffusion becomes anisotropic and metric-governed under boundary forcing.

The geometric control concepts generalize beyond magnetic driving to other driven Stokes flows.

Abstract

In a canonical Stokes flow geometry, the Hele-Shaw cell, we show that tunable circulations induced by Lorentz forces in a conducting fluid enable particle control. We reveal that energy-optimal control paths correspond to geodesics of an emergent Riemannian metric defined over the fluid domain, which are time-optimal under a maximum-power constraint. Subject to random boundary forcing, particle paths exhibit metric-governed anisotropic diffusion. Our geometric concepts governing optimal control, though developed explicitly for circulation-driven flows, generalize to generic driven Stokes flows and so elucidate recent observations in a three-dimensional context.