Differential Geometry of Deterministic Stokesian Dynamics

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 one may think (owing to the Helmholtz Minimum Dissipation Theorem) that the hydrodynamic resistance tensor could be the natural candidate for a Riemannian metric shaping the particle trajectories, we demonstrate that the trajectories driven by constant external forces are \emph{not} geodesics of this bare resistance metric. Instead, they experience a geometric drift due to the curvature induced by the presence of obstacles. By rigorously incorporating translational-rotational coupling through a modified effective resistance tensor $\tilde{\mathbf{R}}={\mathbf{R}}_{\mathbf{FU}}-{\mathbf{R}}_{\mathbf{F}\omega }{\mathbf{R}}_{\mathbf{T}\omega }^{-1}{\mathbf{R}}_{\mathbf{F}\omega }^T$, which accounts for the hidden dissipation savings afforded by particle rotation, we introduce a unified geometric formalism. We prove that the physical particle trajectories are exact geodesics of a conformally scaled metric, ${\tilde{g}}_{ij}=\mathcal{D}(\mathbf{x}){\tilde{R}}_{ij}$, where $\mathcal{D}$ is the instantaneous power dissipation. This framework establishes that the affine parameter along the trajectory corresponds strictly to the cumulative energy dissipated. We apply this theory to the scattering of a spherical particle by a fixed spherical obstacle, showing that the exact analytical trajectory derived previously by the author is recovered as a geodesic of the dissipation-scaled manifold.