Eric Lauga

The hydrodynamics of swimming microorganisms

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. (Some figures in this article are in colour only in the electronic version) This article was invited by Christoph Schmidt.

Effective slip in pressure-driven Stokes flow

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 pressure-driven 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.

Hydrodynamic attraction of swimming microorganisms by surfaces

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.

Cargo-towing fuel-free magnetic nanoswimmers for targeted drug delivery.

Fuel-free nanomotors are essential for future in-vivo biomedical transport and drug-delivery applications. Herein, the first example of directed delivery of drug-loaded magnetic polymeric particles using magnetically driven flexible nanoswimmers is described. It is demonstrated that flexible magnetic nickel-silver nanoswimmers (5-6 μm in length and 200 nm in diameter) are able to transport micrometer particles at high speeds of more than 10 μm s(-1) (more than 0.2 body lengths per revolution in dimensionless speed). The fundamental mechanism of the cargo-towing ability of these magnetic (fuel-free) nanowire motors is modelled, and the hydrodynamic features of these cargo-loaded motors discussed. The effect of the cargo size on swimming performance is evaluated experimentally and compared to a theoretical model, emphasizing the interplay between hydrodynamic drag forces and boundary actuation. The latter leads to an unusual increase of the propulsion speed at an intermediate particle size. Potential applications of these cargo-towing nanoswimmers are demonstrated by using the directed delivery of drug-loaded microparticles to HeLa cancer cells in biological media. Transport of the drug carriers through a microchannel from the pick-up zone to the release microwell is further illustrated. It is expected that magnetically driven nanoswimmers will provide a new approach for the rapid delivery of target-specific drug carriers to predetermined destinations.

A smooth future

Research on superhydrophobic materials has mostly focused on their extreme non-wettability. However, the implications of superhydrophobicity beyond wetting, in particular for transport phenomena, remain largely unexplored.

Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations

The 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.

Propulsion in a viscoelastic fluid

Flagella beating in complex fluids are significantly influenced by viscoelastic stresses. Relevant examples include the ciliary transport of respiratory airway mucus and the motion of spermatozoa in the mucus-filled female reproductive tract. We consider the simplest model of such propulsion and transport in a complex fluid, a waving sheet of small amplitude free to move in a polymeric fluid with a single relaxation time. We show that, compared to self-propulsion in a Newtonian fluid occurring at a velocity UN, the sheet swims (or transports fluid) with velocity U∕UN=(1+De2ηs∕η)∕(1+De2), where ηs is the viscosity of the Newtonian solvent, η is the zero-shear-rate viscosity of the polymeric fluid, and De is the Deborah number for the wave motion, product of the wave frequency by the fluid relaxation time. Similar expressions are derived for the rate of work of the sheet and the mechanical efficiency of the motion. These results are shown to be independent of the particular nonlinear constitutive equations ...

Experimental investigations of elastic tail propulsion at low Reynolds number

A simple way to generate propulsion at low Reynolds number is to periodically oscillate a passive flexible filament. Here we present a macroscopic experimental investigation of such a propulsive mechanism. A robotic swimmer is constructed and both tail shape and propulsive force are measured. Filament characteristics and actuation are varied, and the resulting data are quantitatively compared with existing linear and nonlinear theories.

Geometric transition in friction for flow over a bubble mattress

Laminar flow over a bubble mattress is expected to experience a significant reduction in friction since the individual surfaces of the bubbles are shear-free. However, if the bubbles are sufficiently curved, their protrusion into the fluid and along the flow direction can lead to an increase in friction as was recently demonstrated experimentally and computationally. We provide in this paper a simple model for this result. We consider a shear flow at low Reynolds number past a two-dimensional array of bubbles and calculate analytically the effective slip length of the surface as a function of the bubble geometry in the dilute limit. Our model is able to reproduce quantitatively the relationship between effective friction and bubble geometry obtained in numerical computations and, in particular, (a) the asymmetry in friction between convex and concave bubbles and (b) the existence of a geometric transition from reduced to enhanced friction at a critical bubble protrusion angle.

High-speed propulsion of flexible nanowire motors: Theory and experiments

Micro/nano-scale propulsion has attracted considerable recent attention due to its promise for biomedical applications such as targeted drug delivery. In this paper, we report on a new experimental design and theoretical modelling of high-speed fuel-free magnetically-driven propellers which exploit the flexibility of nanowires for propulsion. These readily prepared nanomotors display both high dimensional propulsion velocities (up to ≈ 21 μm s−1) and dimensionless speeds (in body lengths per revolution) when compared with natural microorganisms and other artificial propellers. Their propulsion characteristics are studied theoretically using an elastohydrodynamic model which takes into account the elasticity of the nanowire and its hydrodynamic interaction with the fluid medium. The critical role of flexibility in this mode of propulsion is illustrated by simple physical arguments, and is quantitatively investigated with the help of an asymptotic analysis for small-amplitude swimming. The theoretical predictions are then compared with experimental measurements and we obtain good agreement. Finally, we demonstrate the operation of these nanomotors in a real biological environment (human serum), emphasizing the robustness of their propulsion performance and their promise for biomedical applications.