Differential geometry of particle motion in Stokesian regime

TL;DR

This paper develops a differential geometric framework for particle motion in Stokes flow, revealing that physical trajectories are geodesics of a conformally scaled metric related to energy dissipation, not just the resistance tensor.

Contribution

It introduces a unified geometric formalism showing that particle trajectories are geodesics of a conformally scaled resistance metric, incorporating energy dissipation effects.

Findings

Trajectories are geodesics of a conformally scaled metric $ ilde{g}_{ij} = ext{D}( extbf{x}) R_{ij}$.

Physical trajectories experience geometric drift due to manifold curvature.

Application to spherical particle scattering recovers known trajectories from curvature analysis.

Abstract

We present a differential geometric framework for the motion of a non-Brownian particle in the presence of fixed obstacles in a quiescent fluid, in the deterministic Stokesian regime. While the Helmholtz Minimum Dissipation Theorem suggests that the hydrodynamic resistance tensor $R_{ij}$ acts as the natural Riemannian metric of the fluid domain, we demonstrate that particle trajectories driven by constant external forces are \emph{not} geodesics of this pure resistance metric. Instead, they experience a geometric drift perpendicular to the geodesic path due to the manifold's curvature. To reconcile this, we introduce a unified geometric formalism, proving that physical trajectories are geodesics of a conformally scaled metric, $\tilde{g}_{ij} = \mathcal{D}(\mathbf{x})R_{ij}$, where $\mathcal{D}$ is the local power dissipation. This framework establishes that the affine parameter along the trajectory corresponds to the cumulative energy dissipated. We apply this theory to the scattering of a spherical particle by a fixed obstacle, showing that the previously derived trajectory of the particle is recovered as a direct consequence of the curvature of this dissipation-scaled manifold.