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)].

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We present a generalisation of efficient numerical frameworks for modelling fluid-filament interactions via the discretisation of a recently-developed, non-local integral equation formulation to incorporate regularised Stokeslets with half-space boundary conditions, as motivated by the importance of confining geometries in many applications. We proceed to utilise this framework to examine the drag on slender inextensible filaments moving near a boundary, firstly with a relatively-simple example, evaluating the accuracy of resistive force theories near boundaries using regularised Stokeslet segments. This highlights that resistive force theories do not accurately quantify filament dynamics in a range of circumstances, even with analytical corrections for the boundary. However, there is the notable and important exception of movement in a plane parallel to the boundary, where accuracy is maintained. In particular, this justifies the judicious use of resistive force theories in examining the mechanics of filaments and monoflagellate microswimmers with planar flagellar patterns moving parallel to boundaries. We proceed to apply the numerical framework developed here to consider how filament elastohydrodynamics can impact drag near a boundary, analysing in detail the complex responses of a passive cantilevered filament to an oscillatory flow. In particular, we document the emergence of an asymmetric periodic beating in passive filaments in particular parameter regimes, which are remarkably similar to the power and reverse strokes exhibited by motile 9+2 cilia. Furthermore, these changes in the morphology of the filament beating, arising from the fluid-structure interactions, also induce a significant increase in the hydrodynamic drag of the filament. †

How does a sperm find its way? The study of guidance cues has fascinated sperm biologists and in particular the prospect of rheotaxis, that is a fluid flow orienting the direction of sperm swimming, has been the subject of extensive recent study, as readily motivated by the prospect that such guidance may be active in the mammalian female reproductive tract. For instance, it has been hypothesized that helical sperm flagellar beating is necessary for such guidance, whereas there is an extensive diversity of flagellar beating patterns, with planar sperm beating readily observed in human cells for example. In particular, such cells will not be guided by fluid flow according to hypothesized mechanisms for rheotaxis presented thus far. Here, using simulation methods, we investigate rheotaxis for a wide range of flagellar beat patterns. Providing the virtual sperm firstly does not possess a tightly circling trajectory in the absence of a background flow and secondly, remains within a region of low shear to prevent being washed away by the background flow, rheotaxis is generally observed with the sperm swimming into the flow together with a possible transverse velocity. Tight circling sperm motility, as observed in select hyperactivated sperm and CatSper mutants, is predicted to disrupt the rheotactic response, whereas confinement to low shear regions generally requires boundary accumulation, thus introducing subtleties in the relationship between rheotactic behaviours and the flagellar waveform and sperm characteristics. Nonetheless, such predictions suggest such rheotactic guidance may be more common and robust than previously thought, and we document simple criteria for the presence of rheotaxis that are consistent with our simulations and understanding, as well as reported observations to date.

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