Differential geometry of particle motion in Stokesian regime

TL;DR

This work reframes Stokesian particle motion under constant external forces near obstacles as geodesic motion on a conformally scaled dissipation metric $\tilde{g}_{ij} = \mathcal{D}(\mathbf{x}) R_{ij}$, where the local dissipation $\mathcal{D} = \mathbf{U} \cdot \mathbf{R} \cdot \mathbf{U}$ sets the scale. By transforming the problem into a Jacobi-Maupertuis-type variational principle, the authors show that physical trajectories minimize the dissipative arc length $S = \int \sqrt{\tilde{g}_{ij} \dot{x}^i \dot{x}^j} \, dt$ with the affine parameter $s$ satisfying $ds = \mathcal{D} dt$, and that the gradient of $\mathcal{D}$ cancels the curvature-induced drift. In the two-sphere application, the Risbud-Drazer trajectory $b_{in} = y \exp(H(r))$ emerges naturally as a geodesic on the $\tilde{g}$ manifold, with $H(r) = \int_r^{\infty} \frac{R_B(s) - R_A(s)}{s R_B(s)} ds$, illustrating how anisotropic curvature governs the path. The framework provides a general geometric variational principle for dissipative Stokesian dynamics and suggests broad applicability to electrophoresis, porous media, and microfluidic design via curvature engineering of the dissipation metric.

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.