Cell motility in viscous fluids is ubiquitous and affects many biological processes, including reproduction, infection and the marine life ecosystem. Here we review the biophysical and mechanical principles of locomotion at the small scales relevant to cell swimming, tens of micrometers and below. At this scale, inertia is unimportant and the Reynolds number is small. Our emphasis is on the simple physical picture and fundamental flow physics phenomena in this regime. We first give a brief overview of the mechanisms for swimming motility, and of the basic properties of flows at low Reynolds number, paying special attention to aspects most relevant for swimming such as resistance matrices for solid bodies, flow singularities and kinematic requirements for net translation. Then we review classical theoretical work on cell motility, in particular early calculations of swimming kinematics with prescribed stroke and the application of resistive force theory and slender-body theory to flagellar locomotion. After examining the physical means by which flagella are actuated, we outline areas of active research, including hydrodynamic interactions, biological locomotion in complex fluids, the design of small-scale artificial swimmers and the optimization of locomotion strategies.

Near a solid boundary, Escherichia coli swims in clockwise circular motion. We provide a hydrodynamic model for this behavior. We show that circular trajectories are natural consequences of force-free and torque-free swimming and the hydrodynamic interactions with the boundary, which also leads to a hydrodynamic trapping of the cells close to the surface. We compare the results of the model with experimental data and obtain reasonable agreement. In particular, the radius of curvature of the trajectory is observed to increase with the length of the bacterium body.

Cells swimming in confined environments are attracted by surfaces. We measure the steady-state distribution of smooth-swimming bacteria (Escherichia coli) between two glass plates. In agreement with earlier studies, we find a strong increase of the cell concentration at the boundaries. We demonstrate theoretically that hydrodynamic interactions of the swimming cells with solid surfaces lead to their reorientation in the direction parallel to the surfaces, as well as their attraction by the closest wall. A model is derived for the steady-state distribution of swimming cells, which compares favorably with our measurements. We exploit our data to estimate the flagellar propulsive force in swimming E. coli.

The no-slip boundary condition at a solid-liquid interface is at the center of our understanding of fluid mechanics. However, this condition is an assumption that cannot be derived from first principles and could, in theory, be violated. In this chapter, we present a review of recent experimental, numerical and theoretical investigations on the subject. The physical picture that emerges is that of a complex behavior at a liquid/solid interface, involving an interplay of many physico-chemical parameters, including wetting, shear rate, pressure, surface charge, surface roughness, impurities and dissolved gas.

Nano-bubbles have recently been observed experimentally on smooth hydrophobic surfaces; cracks on a surface can likewise be the site of bubbles when partially wetting fluids are used. Because these bubbles may provide a zero shear stress boundary condition and modify considerably the friction generated by the solid boundary, it is of interest to quantify their influence on pressure-driven flow, with particular attention given to small geometries. We investigate two simple configurations of steady pressuredriven Stokes flow in a circular pipe whose surface contains periodically distributed regions of zero surface shear stress. In the spirit of experimental studies probing slip at solid surfaces, the effective slip length of the resulting flow is evaluated as a function of the degrees of freedom describing the surface heterogeneities, namely the relative width of the no-slip and no-shear stress regions and their distribution along the pipe. Comparison of the model with experimental studies of pressure-driven flow in capillaries and microchannels reporting slip is made and a possible interpretation of the experimental results is offered which is consistent with a large number of distributed slip domains such as nano-size and micron-size nearly flat bubbles coating the solid surface. Further, the possibility is suggested of a shear-dependent effective slip length, and an explanation is proposed for the seemingly paradoxical behaviour of the measured slip length increasing with system size, which is consistent with experimental results to date.

Microorganisms move in challenging environments by periodic changes in body shape. By contrast, current artificial microrobots cannot actively deform, exhibiting at best passive bending under external fields. Here, by taking advantage of the wireless, scalable and spatiotemporally selective capabilities that light allows, we show that soft microrobots consisting of photoactive liquid-crystal elastomers can be driven by structured monochromatic light to perform sophisticated biomimetic motions. We realized continuum yet selectively addressable artificial microswimmers that generate travelling-wave motions to self-propel without external forces or torques, as well as microrobots capable of versatile locomotion behaviours on demand. Both theoretical predictions and experimental results confirm that multiple gaits, mimicking either symplectic or antiplectic metachrony of ciliate protozoa, can be achieved with single microrobots. The principle of using structured light can be extended to other applications that require microscale actuation with sophisticated spatiotemporal coordination for advanced microrobotic technologies. 3Mobile micro-scale robots are envisioned to navigate within the human body to perform minimally invasive diagnostic or therapeutic tasks 1,2 . Biological microorganisms represent the natural inspiration for this vision. For instance, microorganisms successfully swim and move through a variety of fluids and tissues.Locomotion in this regime, where viscous forces dominate over inertia (low Reynolds number), is only possible through non-reciprocal motions demanding spatiotemporal coordination of multiple actuators 3 . A variety of biological propulsion mechanisms at different scales, from the peristalsis of annelids (Fig. 1a) to the metachrony of ciliates (Fig. 1b), are based on the common principle of travelling waves (Fig. 1c). These emerge from the distributed and self-coordinated action of many independent molecular motors 4,5 .Implementing travelling wave propulsion in an artificial device would require many discrete actuators, each individually addressed and powered in a coordinated fashion (Fig. 1d). The integration of actuators into microrobots that are mobile poses additional hurdles, since power and control need to be distributed without affecting the microrobots' mobility. Existing microscale actuators generally rely on applying external magnetic 6-10 , electric 11 , or optical 12,13 fields globally over the entire workspace. However, these approaches do not permit the spatial selectivity required to independently address individual actuators within a micro-device. Nevertheless, complex non-reciprocal motion patterns have been achieved by carefully engineering the response of different regions in a device to a spatially uniform external field 13,14 .The drawback is that this complicates the fabrication process, inhibits down-scaling and constrains the device to a single predefined behaviour. These challenges mean that most artificial microrobots actually have no actuators. Rather...

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.

Suspended colloidal particles interacting chemically with a solute can self-propel by autophoretic motion when they are asymmetrically patterned (Janus colloids). Here we demonstrate theoretically that such anisotropy is not necessary for locomotion and that the nonlinear interplay between surface osmotic flows and solute advection can produce spontaneous, and self-sustained motion of isotropic particles. Solving the classical autophoretic framework for isotropic particles, we show that, for given material properties, there exists a critical particle size (or Péclet number) above which spontaneous symmetry-breaking and autophoretic motion occur. A hierarchy of instabilities is further identified for quantized critical Péclet numbers.The locomotion of microorganisms has long been used as a motivation and a practical inspiration for the design of synthetic self-propelled particles. Typically, biological cells generate propulsion by deforming their slender appendages, termed flagella or cilia, in a non-time-reversible fashion 1 . However, and perhaps not surprisingly given the numerous microfabrication challenges, no genuinely self-propelled micro-swimmer has been manufactured in the lab so far. Instead, man-made biomimetic propellers are driven by external torques or forces. That actuation allows either to deform soft propellers whose deformed shape induce propulsion 2-4 , to continuously generate propulsion in chiral shapes 5,6 , or to exploit interactions with surfaces 7-9 .An alternative route for the production of artificial small-scale swimmers has proven to be much more successful. It consists in making miniaturized chemically powered "engines" with no moving parts, typically made of reactive Janus beads or rods 10-12 . The reaction products released by these chemically-asymmetric particles create concentration gradients which induce a net phoretic fluid motion near their surface leading to locomotion. Theoretically, the interest in these so-called autophoretic swimmers was triggered by a theoretical model which accounted for such a novel propulsion mechanism in a simple and generic fashion 13,14 . This model was then further elaborated to include the nonlinear interplay between the colloid motion and the advection of the reactants 15-17 or a more complex kinetic route for the surface chemistry 18 , to deal with the rotational Brownian motion of the swimmers 19 , and to detail the microscopic coupling between the concentration gradients and the fluid flows at small scales 20 .In order to self-propel, autophoretic swimmers are chemically patterned, and it is the asymmetries in the chemical reactions on their surfaces which are responsible for locomotion in the first place. This requirement would make it thus difficult to achieve high-throughput production. An ingenious solution to such an engineering issue was recently offered with the production of isotropic self-propelled Marangoni droplets 21 . In a mechanism akin to the one responsible for the spontaneous motion of reactive droplets surfing on fluid inter...