Dynamics of a two-dimensional microswimmer near a slippery boundary
Abstract
Microswimmers near walls and interfaces can switch between qualitatively different modes of motion depending on the slip properties of the neighbouring boundaries. To study this theoretically, a two-dimensional point-singularity model of a microswimmer, which does not self-propel in unbounded fluid, is placed near a flat viscosity discontinuity between two fluids with viscosity ratio σ, and also near a partial-slip wall with Navier-slip length λ. Using complex-variable methods, the dynamical systems governing the swimmer motion are derived explicitly in both scenarios. Near the viscosity discontinuity, below a universal critical viscosity ratio σc=1/2, the swimmer is trapped at a fixed height with its stresslet axis normal to the boundary. When σ=σc it glides parallel to the boundary at a speed selected by its orientation, with a terminal speed determined by its initial state. For σ>σc it oscillates periodically in height and orientation while drifting along the boundary. The dynamics near a partial-slip wall depend on the ratio of the slip length to the length scale of the singularity and are similar in exhibiting an analogous transition at a critical slip length. It is found, however, that the dynamics near a viscosity discontinuity for σ above σc are richer than those near the partial-slip boundary, including the existence of a hydrodynamic bound state with zero net displacement over one period. Although the study is limited to two dimensions, these results show how a single boundary parameter organises qualitatively distinct modes of locomotion, in close analogy with slip-induced transitions reported for three-dimensional swimmers.
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BibTeX
@misc{kanazawa2026dynamics,
title = {{Dynamics of a two-dimensional microswimmer near a slippery boundary}},
author = {Takahiro Kanazawa and Darren Crowdy},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03243},
eprint = {2611.03243},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03243},
}
RIS
TY - UNPB
TI - Dynamics of a two-dimensional microswimmer near a slippery boundary
AU - Takahiro Kanazawa
AU - Darren Crowdy
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03243
UR - https://arxiv.org/abs/2611.03243
AB - Microswimmers near walls and interfaces can switch between qualitatively different modes of motion depending on the slip properties of the neighbouring boundaries. To study this theoretically, a two-dimensional point-singularity model of a microswimmer, which does not self-propel in unbounded fluid, is placed near a flat viscosity discontinuity between two fluids with viscosity ratio σ, and also near a partial-slip wall with Navier-slip length λ. Using complex-variable methods, the dynamical systems governing the swimmer motion are derived explicitly in both scenarios. Near the viscosity discontinuity, below a universal critical viscosity ratio σ_c=1/2, the swimmer is trapped at a fixed height with its stresslet axis normal to the boundary. When σ=σ_c it glides parallel to the boundary at a speed selected by its orientation, with a terminal speed determined by its initial state. For σ>σ_c it oscillates periodically in height and orientation while drifting along the boundary. The dynamics near a partial-slip wall depend on the ratio of the slip length to the length scale of the singularity and are similar in exhibiting an analogous transition at a critical slip length. It is found, however, that the dynamics near a viscosity discontinuity for σ above σ_c are richer than those near the partial-slip boundary, including the existence of a hydrodynamic bound state with zero net displacement over one period. Although the study is limited to two dimensions, these results show how a single boundary parameter organises qualitatively distinct modes of locomotion, in close analogy with slip-induced transitions reported for three-dimensional swimmers.
ER -