Squirmer dynamics near a boundary

The boundary behavior of axisymmetric microswimming squirmers is theoretically explored within an inertialess Newtonian fluid for a no-slip interface and also a free surface in the small capillary number limit, preventing leading-order surface deformation. Such squirmers are commonly presented as abridged models of ciliates, colonial algae, and Janus particles and we investigate the case of low-mode axisymmetric tangential surface deformations with, in addition, the consideration of a rotlet dipole to represent torque-motor swimmers such as flagellated bacteria. The resulting boundary dynamics reduces to a phase plane in the angle of attack and distance from the boundary, with a simplifying time-reversal duality. Stable swimming adjacent to a no-slip boundary is demonstrated via the presence of stable fixed points and, more generally, all types of fixed points as well as stable and unstable limit cycles occur adjacent to a no-slip boundary with variations in the tangential deformations. Nonetheless, there are constraints on swimmer behavior---for instance, swimmers characterized as pushers are never observed to exhibit stable limit cycles. All such generalities for no-slip boundaries are consistent with observations and more geometrically faithful simulations to date, suggesting the tangential squirmer is a relatively simple framework to enable predications and classifications for the complexities associated with axisymmetric boundary swimming. However, in the presence of a free surface, with asymptotically small capillary number, and thus negligible leading-order surface deformation, no stable surface swimming is predicted across the parameter space considered. While this is in contrast to experimental observations, for example, the free-surface accumulation of sterlet sperm, extensive surfactants are present, most likely invalidating the low capillary number assumption. In turn, this suggests the necessity of surface deformation for stable free-surface three-dimensional finite-size microswimming, as previously highlighted in a two-dimensional mathematical study of singularity swimmers [Crowdy et al., J. Fluid Mech. 681, 24 (2011)].