Differential geometry of particle motion in Stokesian regime
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911549443014656 |
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| author | Risbud, Sumedh R. |
| author_facet | Risbud, Sumedh R. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11377 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Differential geometry of particle motion in Stokesian regime Risbud, Sumedh R. Fluid Dynamics Mathematical Physics 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. |
| title | Differential geometry of particle motion in Stokesian regime |
| topic | Fluid Dynamics Mathematical Physics |
| url | https://arxiv.org/abs/2601.11377 |