Active matter exhibits various forms of non-equilibrium states in the absence of external forcing, including macroscopic steady-state currents. Such states are often too complex to be modelled from first principles and our understanding of their physics relies heavily on minimal models. These have mostly been studied in the case of "dry" active matter, where particle dynamics are dominated by friction with their surroundings. Significantly less is known about systems with long-range hydrodynamic interactions that belong to "wet" active matter. Dilute suspensions of motile bacteria, modelled as self-propelled dipolar particles interacting solely through long-ranged hydrodynamic fields, are arguably the most studied example from this class of active systems. Their phenomenology is well-established: at sufficiently high density of bacteria, there appear large-scale vortices and jets comprising many individual organisms, forming a chaotic state commonly known as bacterial turbulence. As revealed by computer simulations, below the onset of collective motion, the suspension exhibits very strong correlations between individual microswimmers stemming from the long-ranged nature of dipolar fields. Here we demonstrate that this phenomenology is captured by the minimal model of microswimmers. We develop a kinetic theory that goes beyond the commonly used mean-field assumption, and explicitly takes into account such correlations. Notably, these can be computed exactly within our theory. We calculate the fluid velocity variance, spatial and temporal correlation functions, the fluid velocity spectrum, and the enhanced diffusivity of tracer particles. We find that correlations are suppressed by particle self-propulsion, although the mean-field behaviour is not restored even in the limit of very fast swimming.
The ability of microswimmers to deploy optimal propulsion strategies is of paramount importance for their locomotory performance and survival at low Reynolds numbers. Although for perfectly spherical swimmers minimum dissipation requires a neutral type swimming, any departure from the spherical shape may lead the swimmer to adopt a new propulsion strategy, namely those of puller- or pusher-type swimming. In this study, by using the minimum dissipation theorem for microswimmers, we determine the flow field of an optimal nearly spherical swimmer, and show that indeed depending on the shape profi…
Some microorganisms, such as spermatozoa, synchronize their flagella when swimming in close proximity. Using a simplified model (two infinite, parallel, two-dimensional waving sheets), we show that phase-locking arises from hydrodynamics forces alone, and has its origin in the front-back asymmetry of the geometry of their flagellar waveform. The time-evolution of the phase difference between co-swimming cells depends only on the nature of this geometrical asymmetry, and microorganisms can phase-lock into conformations which minimize or maximize energy dissipation.
Self-propulsion of cellular microswimmers generates flow signatures, commonly classified as pusher- and puller-type, which characterize hydrodynamic interactions with other cells or boundaries. Using experimentally measured beat patterns, we compute that flagellated alga and sperm oscillate between pusher and puller. Beyond a typical distance of 100 um from the swimmer, inertia attenuates oscillatory micro-flows. We show that hydrodynamic interactions between swimmers oscillate in time and are of similar magnitude as stochastic swimming fluctuations.
Helical propulsion is used by many microorganisms to swim in viscous-dominated environments. Their swimming dynamics are relatively well understood, but detailed study of the flow fields and actuation mechanisms are still needed to realize wall effects and hydrodynamic interactions. In this letter, we describe the development of an autonomous swimming robot with a helical tail that operates in the Stokes regime. The device uses a battery-based power system with a miniature motor that imposes a rotational speed to a helical tail. The speed, direction, and activation are controlled electronicall…
We present the first time-resolved measurements of the oscillatory velocity field induced by swimming unicellular microorganisms. Confinement of the green alga C. reinhardtii in stabilized thin liquid films allows simultaneous tracking of cells and tracer particles. The measured velocity field reveals complex time-dependent flow structures, and scales inversely with distance. The instantaneous mechanical power generated by the cells is measured from the velocity fields and peaks at 15 fW. The dissipation per cycle is more than four times what steady swimming would require.
We make remarks on Elfring and Lauga's [{\it Phys. Rev. Lett.} {\bf 103}, 088101 (2009)] paper. The energy dissipation or viscous dissipation plays an important role in the phase-locked state.
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 re-orientation 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 fav…
Classical Taylor-Aris dispersion theory is extended to describe the transport of suspensions of self-propelled dipolar cells in a tubular flow. General expressions for the mean drift and effective diffusivity are determined exactly in terms of axial moments, and compared with an approximation a la Taylor. As in the Taylor-Aris case, the skewness of a finite distribution of biased swimming cells vanishes at long times. The general expressions can be applied to particular models of swimming microorganisms, and thus be used to predict swimming drift and diffusion in tubular bioreactors, and to el…
Swimming microorganisms create flows that influence their mutual interactions and modify the rheology of their suspensions. While extensively studied theoretically, these flows have not been measured in detail around any freely-swimming microorganism. We report such measurements for the microphytes Volvox carteri and Chlamydomonas reinhardtii. The minute ~0.3% density excess of V. carteri over water leads to a strongly dominant Stokeslet contribution, with the widely-assumed stresslet flow only a correction to the subleading source dipole term. This implies that suspensions of V. carteri have …
Recently, an active microswimmer was constructed where a micron-sized droplet of bromine water was placed into a surfactant-laden oil phase. Due to a bromination reaction of the surfactant at the interface, the surface tension locally increases and becomes non-uniform. This drives a Marangoni flow which propels the squirming droplet forward. We develop a diffusion-advection-reaction equation for the order parameter of the surfactant mixture at the droplet interface using a mixing free energy. Numerical solutions reveal a stable swimming regime above a critical Marangoni number M but also stopp…
We study a swimming undulating sheet in the isotropic phase of an active nematic liquid crystal. Activity changes the effective shear viscosity, reducing it to zero at a critical value of activity. Expanding in the sheet amplitude, we find that the correction to the swimming speed due to activity is inversely proportional to the effective shear viscosity. Our perturbative calculation becomes invalid near the critical value of activity; using numerical methods to probe this regime, we find that activity enhances the swimming speed by an order of magnitude compared to the passive case.
Interaction of motile microrganisms with a nearby solid substrate is a well studied phenomenon. However, the effects of hydrodynamic slippage on the substrate have received a little attention. In the present study, within the framework of the squirmer model, we impose a tangential velocity at the swimmer surface as a representation of the ciliatory propulsion and subsequently obtain exact solution of the Stokes equation based on a combined analytical-numerical approach. We illustrate how the near-wall swimming velocities are non-trivially altered by the interaction of wall slip and hydrodynami…
One approach to quantifying biological diversity consists of characterizing the statistical distribution of specific properties of a taxonomic group or habitat. Microorganisms living in fluid environments, and for whom motility is key, exploit propulsion resulting from a rich variety of shapes, forms, and swimming strategies. Here, we explore the variability of swimming speed for unicellular eukaryotes based on published data. The data naturally partitions into that from flagellates (with a small number of flagella) and from ciliates (with tens or more). Despite the morphological and size diff…
Microorganisms often encounter anisotropy, for example in mucus and biofilms. We study how anisotropy and elasticity of the ambient fluid affects the speed of a swimming microorganism with a prescribed stroke. Motivated by recent experiments on swimming bacteria in anisotropic environments, we extend a classical model for swimming microorganisms, the Taylor swimming sheet, actuated either by transverse or longitudinal traveling waves in a three-dimensional nematic liquid crystal without twist. We calculate the swimming speed and entrained volumetric flux as a function of the swimmer's stroke p…
Microorganisms are rarely found in Nature swimming freely in an unbounded fluid. Instead, they typically encounter other organisms, hard walls, or deformable boundaries such as free interfaces or membranes. Hydrodynamic interactions between the swimmer and nearby objects lead to many interesting phenomena, such as changes in swimming speed, tendencies to accumulate or turn, and coordinated flagellar beating. Inspired by this class of problems, we investigate locomotion of microorganisms near deformable boundaries. We calculate the speed of an infinitely long swimmer close to a flexible surface…
The experiments of Leptos et al. [Phys. Rev. Lett. 103, 198103 (2009)] show that the displacements of small particles affected by swimming microorganisms achieve a non-Gaussian distribution, which nevertheless scales diffusively -- the 'diffusive scaling.' We use a simple model where the particles undergo repeated 'kicks' due to the swimmers to explain the shape of the distribution as a function of the volume fraction of swimmers. The net displacement is determined by the inverse Fourier transform of a single-swimmer characteristic function. The only adjustable parameter is the strength of the…
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 microns and below). The focus is on the fundamental flow physics phenomena occurring in this inertia-less realm, and the emphasis is on the simple physical picture. We review the basic properties of flows at low Reynolds number, paying special attention to aspects most relevant for swimming, such as resistance matrices fo…
Biological swimmers frequently navigate in geometrically restricted media. We study the prescribed-stroke problem of swimmers confined to a planar viscous membrane embedded in a bulk fluid of different viscosity. In their motion, microscopic swimmers disturb the fluid in both the membrane and the bulk. The flows that emerge have a combination of two-dimensional (2D) and three-dimensional (3D) hydrodynamic features, and such flows are referred to as quasi-2D. The cross-over from 2D to 3D hydrodynamics in a quasi-2D fluid is controlled by the Saffman length, a length scale given by the ratio of …
Enhanced diffusion of passive tracers immersed in active fluids is a universal feature of active fluids and has been extensively studied in recent years. Similar to microrheology for equilibrium complex fluids, the unusual enhanced particle dynamics reveal intrinsic properties of active fluids. Nevertheless, previous studies have shown that the translational dynamics of spherical tracers are qualitatively similar, independent of whether active particles are pushers or pullers---the two fundamental classes of active fluids. Is it possible to distinguish pushers from pullers by simply imaging th…
It has been shown that a nanoliter chamber separated by a wall of asymmetric obstacles can lead to an inhomogeneous distribution of self-propelled microorganisms. Although it is well established that this rectification effect arises from the interaction between the swimmers and the non-centrosymmetric pillars, here we demonstrate numerically that its efficiency is strongly dependent on the detailed dynamics of the individual microorganism. In particular, for the case of run-and-tumble dynamics, the distribution of run lengths, the rotational diffusion and the partial preservation of run orient…
Many microorganisms live and evolve in complex fluids. Examples include mammalian spermatozoa in cervical mucus, worms (e.g., \textit{C. elegans}) in wet soil, and bacteria (e.g., \textit{H. pylori}) in our stomach lining. Due to the presence of (bio)polymers and/or solids, such fluids often display nonlinear response to (shear) stresses including viscoelasticity and shear-rate dependent viscosity. The successful interaction between these microorganisms and their fluid environment is critical to the function of many biological processes including human reproduction, ecosystem dynamics, and the…
Microorganisms living in microfluidic environments often form multi-species swarms, where they can leverage collective motions to achieve enhanced transport and spreading. Nevertheless, there is a general lack of physical understandings of the origins of the multiscale unstable dynamics observed within these systems. Here, we build a computational model to study binary suspensions of rear- and front-actuated microswimmers, or respectively the so-called "pusher" and "puller" particles, that have different populations and swimming speeds. We perform direct particle simulations to reveal that col…
We present a lattice Boltzmann study of the hydrodynamics of a fully resolved squirmer, radius R, confined in a slab of fluid between two no-slip walls. We show that the coupling between hydrodynamics and short-range repulsive interactions between the swimmer and the surface can lead to hydrodynamic trapping of both pushers and pullers at the wall, and to hydrodynamic oscillations in the case of a pusher. We further show that a pusher moves significantly faster when close to a surface than in the bulk, whereas a puller undergoes a transition between fast motion and a dynamical standstill accor…
Many microorganisms take a chiral path while swimming in an ambient uid. In this paper, we study the combined behavior of two chiral swimmers using the well-known squirmer model taking into account chiral asymmetries. In contrast to the simple squirmer model, which has an axisymmetric distribution of slip velocity, the chiral squirmer has additional asymmetries in the surface slip, which contribute to both translations and rotations of the motion. As a result, swimming trajectories can become helical and chiral asymmetries arise in the ow patterns. We study the swimming trajectories of a pair …
The dynamic interactions between pairs of swimming microorganisms underpin the collective behaviour of larger suspensions, but accurately calculating pairwise collisions has typically required the use of numerical simulations in which hydrodynamic interactions are fully resolved. In this paper, we utilise analytical expressions for forces and torques acting on two closely separated spherical squirmers -- accurate to second order in the ratio of cell-cell spacing to squirmer radius -- in order to calculate their scattering dynamics. Attention is limited to squirmers whose orientation vectors li…
In addition to enabling movement towards environments with favourable living conditions, swimming by microorganisms has also been linked to enhanced mixing and improved nutrient uptake by their populations. Experimental studies have shown that Brownian tracer particles exhibit enhanced diffusion due to the swimmers, while theoretical models have linked this increase in diffusion to the flows generated by the swimming microorganisms, as well as collisions with the swimmers. In this study, we perform detailed simulations based on the force-coupling method and its recent extensions to the swimmin…
We present a fast, high-throughput method for characterizing the motility of microorganisms in 3D based on standard imaging microscopy. Instead of tracking individual cells, we analyse the spatio-temporal fluctuations of the intensity in the sample from time-lapse images and obtain the intermediate scattering function (ISF) of the system. We demonstrate our method on two different types of microorganisms: bacteria, both smooth swimming (run only) and wild type (run and tumble) Escherichia coli, and the bi-flagellate alga Chlamydomonas reinhardtii. We validate the methodology using computer sim…
We present a study of the hydrodynamics of an active particle, a model squirmer, in an envi- ronment with a broken rotational symmetry: a nematic liquid crystal. By combining simulations with analytic calculations, we show that the hydrodynamic coupling between the squirmer flow field and liquid crystalline director can lead to re-orientation of the swimmers. The preferred orientation depends on the exact details of the squirmer flow field. In a steady state, pushers are shown to swim parallel with the nematic director while pullers swim perpendicular to the nematic director. This behaviour ar…
Microorganisms can preferentially orient and move along gradients of a chemo-attractant (i.e., chemotax) while colonies of many microorganisms can collectively undergo complex dynamics in response to chemo-attractants that they themselves produce. For colonies or groups of micro-swimmers we investigate how an "auto-chemotactic" response that should lead to swimmer aggregation is affected by the non-trivial fluid flows that are generated by collective swimming. For this, we consider chemotaxis models based upon a hydrodynamic theory of motile suspensions that are fully coupled to chemo-attracta…
Marine microorganisms must cope with complex flow patterns and even turbulence as they navigate the ocean. To survive they must avoid predation and find efficient energy sources. A major difficulty in analysing possible survival strategies is that the time series of environmental cues in non-linear flow is complex, and that it depends on the decisions taken by the organism. One way of determining and evaluating optimal strategies is reinforcement learning. In a proof-of-principle study, Colabrese et al. [Phys. Rev. Lett. (2017)] used this method to find out how a micro-swimmer in a vortex flow…
Populations of swimming microorganisms produce fluid motions that lead to dramatically enhanced diffusion of tracer particles. Using simulations of suspensions of swimming particles in a periodic domain, we capture this effect and show that it depends qualitatively on the mode of swimming: swimmers ``pushed'' from behind by their flagella show greater enhancement than swimmers that are ``pulled'' from the front. The difference is manifested by an increase, that only occurs for pushers, of the diffusivity of passive tracers and the velocity correlation length with the size of the periodic domai…
Interactions between microorganisms and their complex flowing environments are essential in many biological systems. We develop a model for microswimmer dynamics in non-Newtonian Poiseuille flows. We predict that swimmers in shear-thickening (-thinning) fluids migrate upstream more (less) quickly than in Newtonian fluids and demonstrate that viscoelastic normal stress differences reorient swimmers causing them to migrate upstream at the centreline, in contrast to well-known boundary accumulation in quiescent Newtonian fluids. Based on these observations, we suggest a sorting mechanism to selec…
Swimming microorganisms often self propel in fluids with complex rheology. While past theoretical work indicates that fluid viscoelasticity should hinder their locomotion, recent experiments on waving swimmers suggest a possible non-Newtonian enhancement of locomotion. We suggest a physical mechanism, based on fluid-structure interaction, leading to swimming in a viscoelastic fluid at a higher speed than in a Newtonian one. Using Taylor's two-dimensional swimming sheet model, we solve for the shape of an active swimmer as a balance between the external fluid stresses, the internal driving mome…
Flagella and cilia are cellular appendages that inherit essential functions of microbial life including sensing and navigating the environment. In order to propel a swimming microorganism they displace the surrounding fluid by means of periodic motions, while precisely-timed modulations of their beating patterns enable the cell to steer towards or away from specific locations. Characterizing the dynamic forces, however, is challenging and typically relies on indirect experimental approaches. Here, we present direct in vivo measurements of the dynamic forces of motile Chlamydomonas reinhardtii …
The incessant activity of swimming microorganisms has a direct physical effect on surrounding microscopic objects, leading to enhanced diffusion far beyond the level of Brownian motion with possible influences on the spatial distribution of non-motile planktonic species and particulate drifters. Here we study in detail the effect of eukaryotic flagellates, represented by the green microalga Chlamydomonas reinhardtii, on microparticles. Macro- and micro-scopic experiments reveal that microorganism-colloid interactions are dominated by rare close encounters leading to large displacements through…
In dilute suspensions of swimming microorganisms the local fluid velocity is a random superposition of the flow fields set up by the individual organisms, which in turn have multipole contributions decaying as inverse powers of distance from the organism. Here we show that the conditions under which the central limit theorem guarantees a Gaussian probability distribution function of velocities are satisfied when the leading force singularity is a Stokeslet, but are not when it is any higher multipole. These results are confirmed by numerical studies and by experiments on suspensions of the alg…
We analyze the dynamics of a microswimmer in pressure-driven Poiseuille flow, where fluid inertia is small but non-negligible. Using perturbation theory and the reciprocal theorem, we show that in addition to the classical inertial lift of passive particles, the active nature generates a `swimming lift', which we evaluate for neutral and pusher/puller-type swimmers. Accounting for fluid inertia engenders a rich spectrum of novel complex dynamics including bistable states, where tumbling coexists with stable centerline swimming or swinging. The dynamics is sensitive to the swimmer's hydrodynami…
Concentrated suspensions of swimming microorganisms and other forms of active matter are known to display complex, self-organized spatio-temporal patterns on scales large compared to those of the individual motile units. Despite intensive experimental and theoretical study, it has remained unclear the extent to which the hydrodynamic flows generated by swimming cells, rather than purely steric interactions between them, drive the self-organization. Here we utilize the recent discovery of a spiral-vortex state in confined suspensions of \textit{B. subtilis} to study this issue in detail. Those …
Suspensions of motile cells are model systems for understanding the unique mechanical properties of living materials which often consist of ensembles of self-propelled particles. We present here a quantitative comparison of theory against experiment for the rheology of such suspensions. The influence of motility on viscosities of cell suspensions is studied using a novel acoustically-driven microfluidic capillary-breakup extensional rheometer. Motility increases the extensional viscosity of suspensions of algal pullers, but decreases it in the case of bacterial or sperm pushers. A recent model…
Swimming microorganisms can influence the diffusion of passive particles. The effect of this swimmer-particle interaction depends on different properties, such as the hydrodynamic field of the swimmer and the relative sizes of microorganisms and particles. We investigated an enhancement of the diffusion of silica doublets in a suspension of microalgae Chlamydomonas reinhardtii in a flat capillary. Depending on the concentration of microswimmers, the translational and rotational diffusion constants increase by several orders of magnitude in the presence of the swimming algae. For low concentrat…
Many eukaryotic cells undergo frequent shape changes (described as amoeboid motion) that enable them to move forward. We investigate the effect of confinement on a minimal model of amoeboid swimmer. Complex pictures emerge: (i) The swimmer's nature (i.e., either pusher or puller) can be modified by confinement, thus suggesting that this is not an intrinsic property of the swimmer. This swimming nature transition stems from intricate internal degrees of freedom of membrane deformation. (ii) The swimming speed might increase with increasing confinement before decreasing again for stronger confin…
The motility of microorganisms is influenced greatly by their hydrodynamic interactions with the fluidic environment they inhabit. We show by direct experimental observation of the bi-flagellated alga Chlamydomonas reinhardtii that fluid elasticity and viscosity strongly influence the beating pattern - the gait - and thereby control the propulsion speed. The beating frequency and the wave speed characterizing the cyclical bending are both enhanced by fluid elasticity. Despite these enhancements, the net swimming speed of the alga is hindered for fluids that are sufficiently elastic. The origin…
Swimming eukaryotic microorganisms such as spermatozoa, algae and ciliates self-propel in viscous fluids using travelling wave-like deformations of slender appendages called flagella. Waves are predominant because Purcell's scallop theorem precludes time-reversible kinematics for locomotion. Using the calculus of variations on a periodic long-wavelength model of flagellar swimming, we show that the planar flagellar kinematics maximising the time-averaged propulsive force for a fixed amount of energy dissipated in the surrounding fluid correspond for all times to waves travelling with constant …
Many motile microorganisms adjust their swimming motion relative to the gravitational field and thus counteract sedimentation to the ground. This gravitactic behavior is often the result of an inhomogeneous mass distribution which aligns the microorganism similar to a buoy. However, it has been suggested that gravitaxis can also result from a geometric fore-rear asymmetry, typical for many self-propelling organisms. Despite several attempts, no conclusive evidence for such an asymmetry-induced gravitactic motion exists. Here, we study the motion of asymmetric self-propelled colloidal particles…
Efficient uptake of nutrients from the environment is an important component in the fitness of all microorganisms, and its dependence on size may reveal clues to the origins of evolutionary transitions to multicellularity. Because potential benefits in uptake rates must be viewed in the context of other costs and benefits of size, such as varying predation rates and the increased metabolic costs associated with larger and more complex body plans, the uptake rate itself is not necessarily that which is optimized by evolution. Uptake rates can be strongly dependent on local organism geometry and…
Swimming cells and microorganisms are a critical component of many biological processes. In order to better interpret experimental studies of low Reynolds number swimming, we combine experimental and numerical methods to perform an analysis of the flow-field around the swimming nematode Caenorhabditis elegans. We first use image processing and particle tracking velocimetry to extract the body shape, kinematics, and flow-fields around the nematode. We then construct a three-dimensional model using the experimental swimming kinematics and employ a boundary element method to simulate flow-fields,…
We investigate the hydrodynamic interactions between microorganisms swimming at low Reynolds number. By considering simple model swimmers, and combining analytic and numerical approaches, we investigate the time-averaged flow field around a swimmer. At short distances the swimmer behaves like a pump. At large distances the velocity field depends on whether the swimming stroke is invariant under a combined time-reversal and parity transformation. We then consider two swimmers and find that the interaction between them consists of two parts; a dead term, independent of the motion of the second s…
