The effects of fluid viscosity on the kinematics of a small swimmer at low Reynolds number are investigated in both experiments and in a simple model. The swimmer is the nematode Caenorhabditis elegans, which is an undulating roundworm approximately 1 mm long. Experiments show that the nematode maintains a highly periodic swimming behavior as the fluid viscosity is varied from 1.0 mPa-s to 12 mPa-s. Surprisingly, the nematode's swimming speed (~0.35 mm/s) is nearly insensitive to the range of fluid viscosities investigated here. However, the nematode's beating frequency decreases to an asymptotic value (~1.7 Hz) with increasing fluid viscosity. A simple model is used to estimate the nematode's Young's modulus and tissue viscosity. Both material properties increase with increasing fluid viscosity. It is proposed that the increase in Young's modulus may be associated with muscle contraction in response to larger mechanical loading while the increase in effective tissue viscosity may be associated with the energy necessary to overcome increased fluid drag forces.
Undulatory locomotion, as seen in the nematode \emph{Caenorhabditis elegans}, is a common swimming gait of organisms in the low Reynolds number regime, where viscous forces are dominant. While the nematode's motility is expected to be a strong function of its material properties, measurements remain scarce. Here, the swimming behavior of \emph{C.} \emph{elegans} are investigated in experiments and in a simple model. Experiments reveal that nematodes swim in a periodic fashion and generate traveling waves which decay from head to tail. The model is able to capture the experiments' main features…
Biological organisms swimming at low Reynolds number are often influenced by the presence of rigid boundaries and soft interfaces. In this paper we present an analysis of locomotion near a free surface with surface tension. Using a simplified two-dimensional singularity model, and combining a complex variable approach with conformal mapping techniques, we demonstrate that the deformation of a free surface can be harnessed to produce steady locomotion parallel to the interface. The crucial physical ingredient lies in the nonlinear hydrodynamic coupling between the disturbance flow created by th…
At the millimeter scale and in the intermediate Reynolds number (Re) regime, the midge and mosquito larvae can reach swimming speeds of more than one body length per cycle performing a "figure-of-8" gait, in which their elongated bodies periodically bend nearly into circles and then fully unfold. To elucidate the propulsion mechanism of this cycle of motion, we conducted a 3D numerical study which investigates the hydrodynamics of undergoing the prescribed kinematics. Novel propulsion mechanisms, such as modulating the body deformation rate to dynamically increase the maximum net propulsion fo…
Many microorganisms swim through gels, materials with nonzero zero-frequency elastic shear modulus, such as mucus. Biological gels are typically heterogeneous, containing both a structural scaffold (network) and a fluid solvent. We analyze the swimming of an infinite sheet undergoing transverse traveling wave deformations in the "two-fluid" model of a gel, which treats the network and solvent as two coupled elastic and viscous continuum phases. We show that geometric nonlinearities must be incorporated to obtain physically meaningful results. We identify a transition between regimes where the …
Reciprocal movement cannot be used for locomotion at low-Reynolds number in an infinite fluid or near a rigid surface. Here we show that this limitation is relaxed for a body performing reciprocal motions near a deformable interface. Using physical arguments and scaling relationships, we show that the nonlinearities arising from reciprocal flow-induced interfacial deformation rectify the periodic motion of the swimmer, leading to locomotion. Such a strategy can be used to move toward, away from, and parallel to any deformable interface as long as the length scales involved are smaller than int…
We use the boundary element method to study the low-Reynolds number locomotion of a spherical model microorganism in a circular tube. The swimmer propels itself by tangen- tial or normal surface motion in a tube whose radius is on the order of the swimmer size. Hydrodynamic interactions with the tube walls significantly affect the average swimming speed and power consumption of the model microorganism. In the case of swimming parallel to the tube axis, the locomotion speed is always reduced (resp. increased) for swimmers with tangential (resp. normal) deformation. In all cases, the rate of wor…
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,…
The effects of fluid elasticity on the swimming behavior of the nematode \emph{Caenorhabditis elegans} are experimentally investigated by tracking the nematode's motion and measuring the corresponding velocity fields. We find that fluid elasticity hinders self-propulsion. Compared to Newtonian solutions, fluid elasticity leads to 35% slower propulsion speed. Furthermore, self-propulsion decreases as elastic stresses grow in magnitude in the fluid. This decrease in self-propulsion in viscoelastic fluids is related to the stretching of flexible molecules near hyperbolic points in the flow.
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…
Inspired by recent experiments using synthetic microswimmers to manipulate droplets, we investigate the low-Reynolds-number locomotion of a model swimmer (a spherical squirmer) encapsulated inside a droplet of comparable size in another viscous fluid. Meditated solely by hydrodynamic interactions, the encaged swimmer is seen to be able to propel the droplet, and in some situations both remain in a stable co-swimming state. The problem is tackled using both an exact analytical theory and a numerical implementation based on boundary element method, with a particular focus on the kinematics of th…
The motility behavior of the nematode Caenorhabditis elegans in polymeric solutions of varying concentrations is systematically investigated in experiments using tracking and velocimetry methods. As the polymer concentration is increased, the solution undergoes a transition from the semi-dilute to the concentrated regime, where these rod-like polymers entangle, align, and form networks. Remarkably, we find an enhancement in the nematode's swimming speed of approximately 65% in concentrated solutions compared to semi-dilute solutions. Using velocimetry methods, we show that the undulatory swimm…
We develop a qualitative geometric approach to swimming at low Reynolds number which avoids solving differential equations and uses instead landscape figures of two notions of curvatures: The swimming curvature and the curvature derived from dissipation. This approach gives complete information for swimmers that swim on a line without rotations and gives the main qualitative features for general swimmers that can also rotate. We illustrate this approach for a symmetric version of Purcell's swimmer which we solve by elementary analytical means within slender body theory. We then apply the theor…
The physical and bio-mechanical principles that govern undulatory movement on wet surfaces have important applications in physiology, physics, and engineering. The nematode {\it C. elegans}, with its highly stereotypical and functionally distinct sinusoidal locomotory gaits, is an excellent system in which to dissect these properties. Measurements of the main forces governing the {\it C. elegans} crawling gait on lubricated surfaces have been scarce, primarily due to difficulties in estimating the physical features at the nematode-gel interface. Using kinematic data and a hydrodynamic model ba…
We computationally study the kinematics of a simple model reciprocal swimmer (asymmetric dumbbell) as a function of the Reynolds number (Re) and investigate how the onset and gradual increase of inertia impacts the swimming behavior: a reversal in the swim direction, flow directions, and the swim stroke. We divide the swim stroke into the expansion and compression of the two spheres and relate them to power and recovery strokes. We find that the switch in swim direction also corresponds to a switch in power and recovery strokes. We obtain expressions for the mean swimming velocity by collapsin…
Swimming cells often have to self-propel through fluids displaying non-Newtonian rheology. While past theoretical work seems to indicate that stresses arising from complex fluids should systematically hinder low-Reynolds number locomotion, experimental observations suggest that locomotion enhancement is possible. In this paper we propose a physical mechanism for locomotion enhancement of microscopic swimmers in a complex fluid. It is based on the fact that micro-structured fluids will generically phase-separate near surfaces, leading to the presence of low-viscosity layers which promote slip a…
We describe experiments and simulations demonstrating the propulsion of a neutrally-buoyant swimmer that consists of a pair of spheres attached by a spring, immersed in a vibrating fluid. The vibration of the fluid induces relative motion of the spheres which, for sufficiently large amplitudes, can lead to motion of the center of mass of the two spheres. We find that the swimming speed obtained from both experiment and simulation agree and collapse onto a single curve if plotted as a function of the streaming Reynolds number, suggesting that the propulsion is related to streaming flows. There …
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…
The locomotion of swimming bacteria in simple Newtonian fluids can successfully be described within the framework of low Reynolds number hydrodynamics. The presence of polymers in biofluids generally increases the viscosity, which is expected to lead to slower swimming for a constant bacterial motor torque. Surprisingly, however, several experiments have shown that bacterial speeds increase in polymeric fluids, and there is no clear understanding why. Therefore we perform extensive coarse-grained simulations of a bacterium swimming in explicitly modeled solutions of macromolecular polymers of …
Free-living nematode Caenorhabditis elegans is a powerful genetic model, essential for investigations ranging from behavior to neuroscience to aging, and locomotion is a key observable used in these studies. However, despite the fact that in its natural environment C. elegans moves in three-dimensional (3D) complex media (decomposing organic matter and water), quantitative in vestigations of its locomotion have been limited to two-dimensional (2D) motion. Based on our recent quantitative analysis of 2D turning maneuvers [Phys. Fluids 25, 081902 (2013)] we follow with the first quantitative des…
Low Reynolds number direct simulations of large populations of hydrodynamically interacting swimming particles confined between planar walls are performed. The results of simulations are compared with a theory that describes dilute suspensions of swimmers. The theory yields scalings with concentration for diffusivities and velocity fluctuations as well as a prediction of the fluid velocity spatial autocorrelation function. Even for uncorrelated swimmers, the theory predicts anticorrelations between nearby fluid elements that correspond to vortex-like swirling motions in the fluid with length s…
Emergent swimming strategies of a smart three-bead swimmer
Institute for Theoretical Physics, TU Wien（维也纳工业大学理论物理研究所）；Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）；Institute of Computer Engineering, TU Wien（维也纳工业大学计算机工程研究所）；Allen Discovery Center, Tufts University（塔夫茨大学艾伦发现中心）
In this fluid dynamics video, we show that low Reynolds number swimmers, such as Caenorhabditis (C.) elegans, synchronize their gait when swimming in close proximity to maximize utilization of space. Synchronization most likely results from steric hindrance and enhances the propulsive speed only marginally.
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…
Cells swimming in viscous fluids create flow fields which influence the transport of relevant nutrients, and therefore their feeding rate. We propose a modeling approach to the problem of optimal feeding at zero Reynolds number. We consider a simplified spherical swimmer deforming its shape tangentially in a steady fashion (so-called squirmer). Assuming that the nutrient is a passive scalar obeying an advection-diffusion equation, the optimal use of flow fields by the swimmer for feeding is determined by maximizing the diffusive flux at the organism surface for a fixed rate of energy dissipati…
We propose minimal models of one-, two- and three-dimensional micro-swimmers at low Reynolds number with a periodic non-reciprocal motion. These swimmers are either "pushers" or "pullers" of fluid along the swimming axis, or combination of the two, depending on the history of the swimming motion. We show this with a linear three-bead swimmer by analytically evaluating the migration speed and the strength of the dipolar flow induced by its swimming motion. It is found that the distance traveled per cycle and the dipolar flow can be obtained from an integral over the area enclosed by the traject…
The millimeter-long soil-dwelling nematode {\it C. elegans} propels itself by producing undulations that propagate along its body and turns by assuming highly curved shapes. According to our recent study [PLoS ONE \textbf{7}, e40121 (2012)] all these postures can be accurately described by a piecewise-harmonic-curvature (PHC) model. We combine this curvature-based description with highly accurate hydrodynamic bead models to evaluate the normalized velocity and turning angles for a worm swimming in an unconfined fluid and in a parallel-wall cell. We find that the worm moves twice as fast and na…
The drag anisotropy of slender filaments is a critical physical property allowing swimming in low-Reynolds number flows, and without it linear translation is impossible. Here we show that, in contrast, net rotation can occur under isotropic drag. We first demonstrate this result formally by considering the consequences of the force- and torque-free conditions on swimming bodies and we then illustrate it with two examples (a simple swimmers made of three rods and a model bacterium with two helical flagellar filaments). Our results highlight the different role of hydrodynamic forces in generatin…
There is increasing evidence that fish gain energetic benefits when they swim in a school. The most recent indications of such benefits are a lower tail (or fin) beat at the back of a school and reduced oxygen consumption in schooling fish versus solitary ones. How such advantages may arise is poorly understood. Current hydrodynamic theories concern either fish swimming side by side or in a diamond configuration and they largely ignore effects of viscosity and interactions among wakes and individuals. In reality, however, hydrodynamic effects are complex and fish swim in many configurations. S…
We investigate a self-organized swimmer at low Reynolds numbers. The microscopic swimmer is composed of three spheres that are connected by two identical active linker arms. Each linker arm contains molecular motors and elastic elements and can oscillate spontaneously. We find that such a system immersed in a viscous fluid can self-organize into a state of directed swimming. The swimmer provides a simple system to study important aspects of the swimming of micro-organisms.
We use numerical simulations to address locomotion at zero Reynolds number in viscoelastic (Giesekus) fluids. The swimmers are assumed to be spherical, to self-propel using tangential surface deformation, and the computations are implemented using a finite element method. The emphasis of the study is on the change of the swimming kinematics, energetics, and flow disturbance from Newtonian to viscoelastic, and on the distinction between pusher and puller swimmers. In all cases, the viscoelastic swimming speed is below the Newtonian one, with a minimum obtained for intermediate values of the Wei…
External gradients can strongly influence the collective behavior of microswimmers. In this paper, we study the behavior of two hydrodynamically interacting self-propelled chiral swimmers, in the low-Reynolds number regime, under the influence of an external linear chemical gradient. We use the generalized squirmer model called the chiral squirmer, a spherically shaped body with an asymmetric surface slip velocity, to represent the swimmer. We find that the external gradient favors the attraction between the swimmers and, in some situations, leads to a bounded state in which the swimmers move …
It has been known for some time that some microorganisms can swim faster in high-viscosity gel-forming polymer solutions. These gel-like media come to mimic highly viscous heterogeneous environment that these microorganisms encounter in-vivo. The qualitative explanation of this phenomena first offered by Berg and Turner [Nature (London) 278, 349 (1979)], suggests that propulsion enhancement is a result of flagellum pushing on quasi-rigid loose polymer network formed in some polymer solutions. Inspired by these observations, inertia-less propulsion in a heterogeneous viscous medium composed of …
Unicellular microscopic organisms living in aqueous environments outnumber all other creatures on Earth. A large proportion of them are able to self-propel in fluids with a vast diversity of swimming gaits and motility patterns. In this paper we present a biophysical survey of the available experimental data produced to date on the characteristics of motile behaviour in unicellular microswimmers. We assemble from the available literature empirical data on the motility of four broad categories of organisms: bacteria (and archaea), flagellated eukaryotes, spermatozoa and ciliates. Whenever possi…
Geometric confinements are frequently encountered in soft matter systems and in particular significantly alter the dynamics of swimming microorganisms in viscous media. Surface-related effects on the motility of microswimmers can lead to important consequences in a large number of biological systems, such as biofilm formation, bacterial adhesion and microbial activity. On the basis of low-Reynolds-number hydrodynamics, we explore the state diagram of a three-sphere microswimmer under channel confinement in a slit geometry and fully characterize the swimming behavior and trajectories for neutra…
Obtaining a detailed understanding of the physical interactions between a cell and its environment often requires information about the flow of fluid surrounding the cell. Cells must be able to effectively absorb and discard material in order to survive. Strategies for nutrient acquisition and toxin disposal, which have been evolutionarily selected for their efficacy, should reflect knowledge of the physics underlying this mass transport problem. Motivated by these considerations, in this paper we discuss the results from an undergraduate research project on the advection-diffusion equation at…
A variety of swimming microorganisms, called ciliates, exploit the bending of a large number of small and densely-packed organelles, termed cilia, in order to propel themselves in a viscous fluid. We consider a spherical envelope model for such ciliary locomotion where the dynamics of the individual cilia are replaced by that of a continuous overlaying surface allowed to deform tangentially to itself. Employing a variational approach, we determine numerically the time-periodic deformation of such surface which leads to low-Reynolds locomotion with minimum rate of energy dissipation (maximum ef…
Most classical work on the hydrodynamics of low-Reynolds-number swimming addresses deterministic locomotion in quiescent environments. Thermal fluctuations in fluids are known to lead to a Brownian loss of the swimming direction. As most cells or synthetic swimmers are immersed in external flows, we consider theoretically in this paper the stochastic dynamics of a model active particle (a self-propelled sphere) in a steady general linear flow. The stochasticity arises both from translational diffusion in physical space, and from a combination of rotary diffusion and run-and-tumble dynamics in …
In several biologically relevant situations, cell locomotion occurs in polymeric fluids with Weissenberg {number} larger than one. Here we present results of three-dimensional numerical simulations for the steady locomotion of a self-propelled body in a model polymeric (Giesekus) fluid at low Reynolds number. Locomotion is driven by steady tangential deformation at the surface of the body (so-called squirming motion). In the case of a spherical squirmer, we show that the swimming velocity is systematically less than that in a Newtonian fluid, with a minimum occurring for Weissenberg numbers of…
Self-propulsion at low Reynolds number is notoriously restricted, a concept that is commonly known as the "scallop theorem". Here we present a truly self-propelled swimmer (force- and torque- free) that, while unable to swim in a Newtonian fluid due to the scallop theorem, propels itself in a non-Newtonian fluid as a result of fluid elasticity. This propulsion mechanism is demonstrated using a robotic swimmer, comprised of a "head" sphere and a "tail" sphere, whose swimming speed is shown to have reasonable agreement with a microhydrodynamic asymptotic theory and numerical simulations. Schlier…
We analyse a simple 'Stokesian squirmer' model for the enhanced mixing due to swimming micro-organisms. The model is based on a calculation of Thiffeault & Childress [Physics Letters A, 374, 3487 (2010), arXiv:0911.5511], where fluid particle displacements due to inviscid swimmers are added to produce an effective diffusivity. Here we show that for the viscous case the swimmers cannot be assumed to swim an infinite distance, even though their total mass displacement is finite. Instead, the largest contributions to particle displacement, and hence to mixing, arise from random changes of directi…
Swimming fish and flying insects use the flapping of fins and wings to generate thrust. In contrast, microscopic organisms typically deform their appendages in a wavelike fashion. Since a flapping motion with two degrees of freedom is able, in theory, to produce net forces from a time-periodic actuation at all Reynolds number, we compute in this paper the optimal flapping kinematics of a rigid spheroid in a Stokes flow. The hydrodynamics for the force generation and energetics of the flapping motion is solved exactly. We then compute analytically the gradient of a flapping efficiency in the sp…
Swimming at a micrometer scale demands particular strategies. Indeed when inertia is negligible as compared to viscous forces (i.e. Reynolds number $Re$ is lower than unity), hydrodynamics equations are reversible in time. To achieve propulsion at low Reynolds number, swimmers must then deform in a way that is not invariant under time reversal. Here, we investigate dispersal properties of self propelled organisms by means of microscopy and cell tracking. Our system of interest is the micro-alga \textit{Chlamydomonas Reinhardtii}, a motile single celled green alga about 10 micrometers in diamet…
We experimentally and theoretically study the dynamics of a low-Reynolds number helical swimmer moving across viscosity gradients. Experimentally, a double-layer viscosity is generated by superposing two miscible fluids with similar densities but different dynamic viscosities. A synthetic helical magnetically-driven swimmer is then made to move across the viscosity gradients along four different configurations: either head-first (pusher swimmer) or tail-first (puller), and through either positive (i.e. going from low to high viscosity) or negative viscosity gradients. We observe qualitative di…
We seek the best stroke sequences of a finite-size swimming predator chasing a non-motile point or finite--size prey at low Reynolds number. We use optimal control to seek the globally-optimal solutions for the former and RL for general situations. The predator is represented by a squirmer model that can translate forward and laterally, rotate and generate a stresslet flow. We identify the predator's best squirming sequences to achieve the time-optimal (TO) and efficiency-optimal (EO) predation. For a point prey, the TO squirmer executing translational motions favours a two-fold L-shaped traje…
In a recent letter (Friedrich et al., Phys. Rev. Lett. 109:138102, 2012), a minimal model swimmer was proposed that propels itself at low Reynolds numbers by a revolving motion of a pair of spheres. The motion of the two spheres can synchronize by virtue of a hydrodynamic coupling that depends on the motion of the swimmer, but is rather independent of direct hydrodynamic interactions. This novel synchronization mechanism could account for the synchronization of a pair of flagella, e.g. in the green algae Chlamydomonas. Here, we discuss in detail how swimming and synchronization depend on the g…
The natural habitats of microorganisms in the human microbiome and ocean and soil ecosystems are full of colloids and macromolecules, which impart non-Newtonian flow properties drastically affecting the locomotion of swimming microorganisms. Although the low-Reynolds-number hydrodynamics of the swimming of flagellated bacteria in simple Newtonian fluids has been well developed, our understanding of bacterial motility in complex non-Newtonian fluids is still primitive. Even after six decades of research, fundamental questions about the nature and origin of bacterial motility enhancement in poly…
Locomotion and transport of microorganisms in fluids is an essential aspect of life. Search for food, orientation toward light, spreading of off-spring, and the formation of colonies are only possible due to locomotion. Swimming at the microscale occurs at low Reynolds numbers, where fluid friction and viscosity dominates over inertia. Here, evolution achieved propulsion mechanisms, which overcome and even exploit drag. Prominent propulsion mechanisms are rotating helical flagella, exploited by many bacteria, and snake-like or whip-like motion of eukaryotic flagella, utilized by sperm and alga…
The induced diffusion of tracers in a bacterial suspension is studied theoretically and experimentally at low bacterial concentrations. Considering the swimmer-tracer hydrodynamic interactions at low-Reynolds number and using a kinetic theory approach, it is shown that the induced diffusion coefficient is proportional to the swimmer concentration, their mean velocity and a coefficient $β$, as observed experimentally. The coefficient $β$ scales as the tracer-swimmer cross section times the mean square displacement produced by single scatterings. The displacements depend on the swimmer propulsio…
Brown University（布朗大学）；Universidad Nacional Autónoma de México（墨西哥国立自治大学）
Marine exploration is essential to understanding ocean processes and organisms. While the use of current unmanned underwater vehicles has enabled many discoveries, there are still plenty of limitations toward exploring complex environments. Bio-inspired robots are a promising solution for highly maneuverable underwater swimming at moderate speeds. Krill, especially, are efficient swimmers in the intermediate Reynolds number regime and can inform engineering solutions for ocean exploration. In this paper, we present the design, manufacture, and validation of a new krill-inspired, metachronal, d…
Both natural and artificial small-scale swimmers may often self-propel in environments subject to complex geometrical constraints. While most past theoretical work on low-Reynolds number locomotion addressed idealised geometrical situations, not much is known on the motion of swimmers in heterogeneous environments. As a first theoretical model, we investigate numerically the behaviour of a single spherical micro-swimmer located in an infinite, periodic body-centred cubic lattice consisting of rigid inert spheres of the same size as the swimmer. Running a large number of simulations we uncover …
