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\char22{}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)].

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.

Remarkably, mammalian sperm maintain a substantive proportion of their progressive swimming speed within highly viscous fluids, including those of the female reproductive tract. Here, we analyse the digital microscopy of a human sperm swimming in a highly viscous, weakly elastic mucus analogue. We exploit principal component analysis to simplify its flagellar beat pattern, from which boundary element calculations are used to determine the time-dependent flow field around the sperm cell. The sperm flow field is further approximated in terms of regularised point forces, and estimates of the mechanical power consumption are determined, for comparison with analogous low viscosity media studies. This highlights extensive differences in the structure of the flows surrounding human sperm in different media, indicating how the cell-cell and cell-boundary hydrodynamic interactions significantly differ with the physical microenvironment. The regularised point force decomposition also provides cell-level information that may ultimately be incorporated into sperm population models. We further observe indications that the core feature in explaining the effectiveness of sperm swimming in high viscosity media is the loss of cell yawing, which is related with a greater density of regularised point force singularities along the axis of symmetry of the flagellar beat to represent the flow field. In turn this implicates a reduction of the wavelength of the distal beat pattern - and hence dynamical wavelength selection of the flagellar beat - as the dominant feature governing the effectiveness of sperm swimming in highly viscous media.

The motions of an unsteady circular-disk squirmer and a spherical squirmer have been investigated in the presence of a no-slip infinite wall and a background shear flow in order to clarify the similarities and differences between two- and three-dimensional motions. Despite the similar bifurcation structure of the dynamical system, the stability of the fixed points differs due to the Hamiltonian structure of the disk squirmer. Once the unsteady oscillating surface velocity profile is considered, the disk squirmer can behave in a chaotic manner and cease to be confined in a near-wall region. In contrast, in an unsteady spherical squirmer, the dynamics is well attracted by a stable fixed point. Additional wall contact interactions lead to stable fixed points for the disk squirmer, and, in turn, the surface entrapment of the disk squirmer can be stabilized, regardless of the existence of the background flow. Finally, we consider spherical motion under a background flow. The separated time scales of the surface entrapment (thigmotaxis) and the turning toward the flow direction (rheotaxis) enable us to reduce the dynamics to two-dimensional phase space, and simple weather-vane mechanics can predict squirmer rheotaxis. The analogous structure of the phase plane with the wall contact in two and three dimensions implies that the two-dimensional disk swimmer successfully captures the nonlinear interactions, and thus two-dimensional approximation could be useful in designing microfluidic devices for the guidance of microswimmers and for clarifying the locomotions in a complex geometry.

We theoretically investigate self-oscillating waves of an active material, which were recently introduced as a nonsymmetric part of the elastic moduli, termed odd elasticity. Using Purcell's three-link swimmer model, we reveal that an odd-elastic filament at low Reynolds number can swim in a self-organized manner and that the time-periodic dynamics are characterized by a stable limit cycle generated by elastohydrodynamic interactions. Also, we consider a noisy shape gait and derive a swimming formula for a general elastic material in the Stokes regime with its elasticity modulus being represented by a nonsymmetric matrix, demonstrating that the odd elasticity produces biased net locomotion from random noise.