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

2025/11/18 by McKee, Kyle
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Geometric Analysis and Curvature Flows #Micro and Nano Robotics

paper · doi:10.48550/arxiv.2511.14479

openalex publication_date 2025/11/18 · openalex created_date 2025/11/20 · openalex updated_date 2026/07/28

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.

Citations

• Data-driven control of a magnetohydrodynamic flow
• Contactless Precision Steering of Particles in a Fluid inside a Cube with Rotating Walls
• Optimal Navigation in Microfluidics via the Optimization of a Discrete Loss
• Potential Flows with Electromagnetically-Induced Circulation in a Hele-Shaw Cell

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