Dynamics of a two-dimensional microswimmer near a slippery boundary

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.