AbstractThe swimming trajectories of self-propelled organisms or synthetic devices in a viscous fluid can be altered by hydrodynamic interactions with nearby boundaries. We explore a multipole description of swimming bodies and provide a general framework for studying the fluid-mediated modifications to swimming trajectories. A general axisymmetric swimmer is described as a linear combination of fundamental solutions to the Stokes equations: a Stokeslet dipole, a source dipole, a Stokeslet quadrupole, and a rotlet dipole. The effects of nearby walls or stress-free surfaces on swimming trajectories are described through the contribution of each singularity, and we address the question of how accurately this multipole approach captures the wall effects observed in full numerical solutions of the Stokes equations. The reduced model is used to provide simple but accurate predictions of the wall-induced attraction and pitching dynamics for model Janus particles, ciliated organisms, and bacteria-like polar swimmers. Transitions in attraction and pitching behaviour as functions of body geometry and propulsive mechanism are described. The reduced model may help to explain a number of recent experimental results.

Motivated by recent experiments, we consider the hydrodynamic capture of a microswimmer near a stationary spherical obstacle. Simulations of model equations show that a swimmer approaching a small spherical colloid is simply scattered. In contrast, when the colloid is larger than a critical size it acts as a passive trap: the swimmer is hydrodynamically captured along closed trajectories and endlessly orbits around the colloidal sphere. In order to gain physical insight into this hydrodynamic scattering problem, we address it analytically. We provide expressions for the critical trapping radius, the depth of the "basin of attraction," and the scattering angle, which show excellent agreement with our numerical findings. We also demonstrate and rationalize the strong impact of swimming-flow symmetries on the trapping efficiency. Finally, we give the swimmer an opportunity to escape the colloidal traps by considering the effects of Brownian, or active, diffusion. We show that in some cases the trapping time is governed by an Ornstein-Uhlenbeck process, which results in a trapping time distribution that is well-approximated as inverse-Gaussian. The predictions again compare very favorably with the numerical simulations. We envision applications of the theory to bioremediation, microorganism sorting techniques, and the study of bacterial populations in heterogeneous or porous environments.

The motion of a rotating helical body in a viscoelastic fluid is considered. In the case of force-free swimming, the introduction of viscoelasticity can either enhance or retard the swimming speed and locomotive efficiency, depending on the body geometry, fluid properties, and the body rotation rate. Numerical solutions of the Oldroyd-B equations show how previous theoretical predictions break down with increasing helical radius or with decreasing filament thickness. Helices of large pitch angle show an increase in swimming speed to a local maximum at a Deborah number of order unity. The numerical results show how the small-amplitude theoretical calculations connect smoothly to the large-amplitude experimental measurements.

To better understand the role of wing and fin flexibility in flapping locomotion, we study through experiment and numerical simulation a freely moving wing that can "pitch" passively as it is actively heaved in a fluid. We observe a range of flapping frequencies corresponding to large horizontal velocities, a regime of underperformance relative to a clamped ͑nonpitching͒ flapping wing, and a surprising, hysteretic regime in which the flapping wing can move horizontally in either direction ͑despite left/right symmetry being broken by the specific mode of pitching͒. The horizontal velocity is shown to peak when the flapping frequency is near the immersed system's resonant frequency. Unlike for the clamped wing, we find that locomotion is achieved by vertically flapped symmetric wings with even the slightest pitching flexibility, and the system exhibits a continuous departure from the Stokesian regime. The phase difference between the vertical heaving motion and consequent pitching changes continuously with the flapping frequency, and the direction reversal is found to correspond to a critical phase relationship. Finally, we show a transition from coherent to chaotic motion by increasing the wing's aspect ratio, and then a return to coherence for flapping bodies with circular cross section.

The dynamics of a flexible filament sedimenting in a viscous fluid are explored analytically and numerically. Compared to the well-studied case of sedimenting rigid rods, the introduction of filament compliance is shown to cause a significant alteration in the long-time sedimentation orientation and filament geometry. A model is developed by balancing viscous, elastic, and gravitational forces in a slender-body theory for zero-Reynolds-number flows, and the filament dynamics are characterized by a dimensionless elasto-gravitation number. Filaments of both non-uniform and uniform cross-sectional thickness are considered. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations, and shapes. These are shown to match excellently with full numerical simulations. Furthermore, we show that trajectories of sedimenting flexible filaments, unlike their rigid counterparts, are restricted to a cloud whose envelope is determined by the elasto-gravitation number. In the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. A linear stability analysis provides a dispersion relation, illustrating clearly the competing effects of the compressive stress and the restoring elastic force in the buckling process. The instability travels as a wave along the filament opposite the direction of gravity as it grows and the predicted growth rates are shown to compare favorably with numerical simulations. The linear eigenmodes of the governing equation are also studied, which agree well with the finite-amplitude buckled shapes arising in simulations

Motile eukaryotic cells propel themselves in viscous fluids by passing waves of bending deformation down their flagella. An infinitely long flagellum achieves a hydrodynamically optimal low-Reynolds number locomotion when the angle between its local tangent and the swimming direction remains constant along its length. Optimal flagella therefore adopt the shape of a helix in three dimensions (smooth) and that of a sawtooth in two dimensions (non-smooth). Physically, biological organisms (or engineered micro-swimmers) must expend internal energy in order to produce the waves of deformation responsible for the motion. Here we propose a physically-motivated derivation of the optimal flagellum shape. We determine analytically and numerically the shape of the flagellar wave which leads to the fastest swimming while minimizing an appropriately-defined energetic expenditure. Our novel approach is to define an energy which includes not only the work against the surrounding fluid, but also (1) the energy stored elastically in the bending of the flagellum, (2) the energy stored elastically in the internal sliding of the polymeric filaments which are responsible for the generation of the bending waves (microtubules), and (3) the viscous dissipation due to the presence of an internal fluid. This approach regularizes the optimal sawtooth shape for two-dimensional deformation at the expense of a small loss in hydrodynamic efficiency. The optimal waveforms of finite-size flagella are shown to depend upon a competition between rotational motions and bending costs, and we observe a surprising bias towards half-integer wave-numbers. Their final hydrodynamic efficiencies are above 6%, significantly larger than those of swimming cells, therefore indicating available room for further biological tuning

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Most bacteria swim through fluids by rotating helical flagella which can take one of twelve distinct polymorphic shapes. The most common helical waveform is the "normal" form, used during forward swimming runs. To shed light on the prevalence of the normal form in locomotion, we gather all available experimental measurements of the various polymorphic forms and compute their intrinsic hydrodynamic efficiencies. The normal helical form is found to be the most hydrodynamically efficient of the twelve polymorphic forms by a significant margin -a conclusion valid for both the peritrichous and polar flagellar families, and robust to a change in the effective flagellum diameter or length. The hydrodynamic optimality of the normal polymorph suggests that, although energetic costs of locomotion are small for bacteria, fluid mechanical forces may have played a significant role in the evolution of the flagellum.