Emergent Geometry Governs Optimal Control in 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, which are time-optimal under a maximum-power constraint. Particle paths exhibit metric-governed anisotropic diffusion under random boundary forcing. Our geometric concepts, though developed explicitly for circulation-driven Hele-Shaw flows, generalize to generic driven Stokes flows and so elucidate recent observations in a three-dimensional context.

Figures (3)

• Figure 1: Control of an a priori specified trajectory. (a) A star-shaped trajectory (magenta) is controlled by varying the circulation around four conductors. Instantaneous streamlines (black), potential contours (grey), and power-optimal circulations are displayed. (b) Electrical power of pseudo-inverse and the power-optimal controls.
• Figure 2: Metric visualization. (a),(b),(c) For three conductor geometries, $\log{|\sigma_{\mathrm{max}}\left(g\right)|}$ is depicted in colour, revealing how expensive control is across the domain. Each overlaid ellipse shows the locus of points a particle may be displaced to, from its center, using a fixed amount of energy, as is determined by the local metric $g$. In (b) circular conductors of radius $r=0.7$ are centered on a circle of radius $R=\sqrt{2}$ (solid). A dashed circle of radius $\sqrt{R^2-r^2}$ corresponds to the singularity of $\sigma_{\mathrm{max}}\left(g\right)$ predicted by \ref{['eq:mobius']}. (c) The addition of a central circle breaks the conformal equivalence to a geometry of the type depicted in (a), thus eliminating singularities of $\sigma_{\mathrm{max}}$. (d) Approximate isometric immersion in $\mathbb{R}^3$ of the metric $g$ over an annular region from (c). As the inner annular radius is approached, lengths (red, green, magenta) become dramatically stretched in the normal direction, such that the immersion resembles a smoothed funnel.
• Figure 3: (A) Four circular conductors are centered on the real axis. Geodesics between a square and four target points (stars) are depicted and labelled with $E_r$, the ratio of geodesic energy to that of a straight-line trajectory. Two green and two red trajectories connect the same endpoints. (B) Geodesics around five circular conductors, centered on the corners and center of a square. (C) Particles are released from five initial points (stars) in the presence of random circulation forcing. Each colour corresponds to 100 overlaid equal-time trajectories starting from the corresponding star.