We illustrate a concept for shape-changing microswimmers, which exploits the hysteresis of a shape transition of an elastic object, by an elastic disk undergoing cyclic localized swelling. Driving the control parameter of a hysteretic shape transition in a completely time-reversible manner gives rise to a non-time-reversible shape sequence and a net swimming motion if the elastic object is immersed into a viscous fluid. We prove this concept with a microswimmer which is a flat circular elastic disk that undergoes a transition into a dome-like shape by localized swelling of an inner disk. The control parameter of this shape transition is a scalar swelling factor of the disk material. With a fixed outer frame with an additional attractive interaction in the central region, the shape transition between flat and dome-like shape becomes hysteretic and resembles a hysteretic opening and closing of a scallop. Employing Stokesian dynamics simulations of a discretized version of the disk we show that the swimmer is effectively moving into the direction of the opening of the dome in a viscous fluid if the swelling parameter is changed in a time-reversible manner. The swimming mechanism can be qualitatively reproduced by a simple 9-bead model.
Microswimmers (planktonic microorganisms or artificial active particles) immersed in a fluid interact with the ambient flow, altering their trajectories. In surface gravity waves, a common goal for microswimmers is vertical migration (e.g., to reach the free surface or to dive to deeper depths). By modelling microswimmers as spheroidal bodies with an intrinsic swimming velocity that supplements advection and reorientation by the flow, we investigate how shape and swimming affect vertical transport of microswimmers in waves. We find that it is possible for microswimmers to be initially swimming…
In this paper we study swimming of a model organism, the so-called Taylor's swimming sheet, in a viscoelastic fluid close to a solid boundary. This situation comprises natural habitats of many swimming microorganisms, and while previous investigations have considered the effects of both swimming next to a boundary and swimming in a viscoelastic fluid, seldom have both effects been considered simultaneously. We re-visit the small wave amplitude result obtained by Elfring and Lauga (Gwynn J. Elfring and Eric Lauga, in Saverio E. Spagnolie, editor, Complex Fluids in Biological Systems, Springer N…
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…
Physics governing the locomotion of microorganisms and other microsystems is dominated by viscous damping. An effective swimming strategy involves the non-reciprocal and periodic deformations of the considered body. Here, we show that a magnetocapillary-driven self-assembly, composed of three soft ferromagnetic beads, is able to swim along a liquid-air interface when powered by an external magnetic field. More importantly, we demonstrate that trajectories can be fully controlled, opening ways to explore low Reynolds number swimming. This magnetocapillary system spontaneously forms by self-asse…
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…
Microswimmers in nature often experience spatial gradients of viscosity. In this work we develop theoretical results for the dynamics of active particles, biological or otherwise, swimming through viscosity gradients. We model the active particles using the squirmer model, and show how viscosity gradients lead to viscotaxis for squirmers, and how the effects of viscosity gradients depend on the swimming gait of the microswimmers. We also show how such gradients in viscosity can be used to control active particles and suggest a mechanism to sort them based on their swimming style.
As a natural and functional behavior, various microorganisms exhibit gravitaxis by orienting and swimming upwards against gravity. Swimming autophoretic nanomotors described herein, comprising bimetallic nanorods, preferentially orient upwards and swim up along a wall, when tail-heavy (i.e. when the density of one of the metals is larger than the other). Through experiment and theory, two mechanisms were identified that contribute to this gravitactic behavior. First, a buoyancy or gravitational torque acts on these rods to align them upwards. Second, hydrodynamic interactions of the rod with t…
We present a realization of a fast interfacial Marangoni microswimmer by a half-spherical alginate capsule at the air-water interface, which diffusively releases water-soluble spreading molecules (weak surfactants such as polyethylene glycol (PEG)), which act as "fuel" by modulating the air-water interfacial tension. For a number of different fuels, we can observe symmetry breaking and spontaneous propulsion although the alginate particle and emission are isotropic. The propulsion mechanism is similar to soap or camphor boats, which are, however, typically asymmetric in shape or emission to se…
We consider the active Brownian particle (ABP) model for a two-dimensional microswimmer with fixed speed, whose direction of swimming changes according to a Brownian process. The probability density for the swimmer evolves according to a Fokker-Planck equation defined on the configuration space, whose structure depends on the swimmer's shape, center of rotation and domain of swimming. We enforce zero probability flux at the boundaries of configuration space. We derive a reduced equation for a swimmer in an infinite channel, in the limit of small rotational diffusivity, and find that the invari…
The measurement of a quantitative and macroscopic parameter to estimate the global motility of a large population of swimming biological cells is a challenge Experiments on the rheology of active suspensions have been performed. Effective viscosity of sheared suspensions of live unicellular motile micro-algae (\textit{Chlamydomonas Reinhardtii}) is far greater than for suspensions containing the same volume fraction of dead cells and suspensions show shear thinning behaviour. We relate these macroscopic measurements to the orientation of individual swimming cells under flow and discuss our res…
We analyse the effect of confining rigid and elastic boundaries on the motility of a model dipolar microswimmer. Flexible boundaries are deformed by the velocity field of the swimmer in such a way that the motility of both extensile and contractile swimmers is enhanced. The magnitude of the increase in swimming velocity is controlled by the ratio of the swimmer-advection and elastic timescales, and the dipole moment of the swimmer. We explain our results by considering swimming between inclined rigid boundaries.
Oscillatory flows are commonly experienced by swimming microorganisms in the environment, industrial applications and rheological investigations. We experimentally characterise the response of the alga {\it Dunaliella salina} to oscillatory shear flows, and report the surprising discovery that algal swimming trajectories orient perpendicular to the flow-shear plane. The ordering has the characteristics of a resonance in the driving parameter space. The behaviour is qualitatively reproduced by a simple model and simulations accounting for helical swimming, providing the mechanism for ordering a…
We discuss the dynamics of a generalized three-sphere microswimmer in which the spheres are connected by two elastic springs. The natural length of each spring is assumed to undergo a prescribed cyclic change. We analytically obtain the average swimming velocity as a function of the frequency of the cyclic change in the natural length. In the low-frequency region, the swimming velocity increases with the frequency and its expression reduces to that of the original three-sphere model by Najafi and Golestanian. In the high-frequency region, conversely, the average velocity decreases with increas…
A model of three-disk micromachine swimming in a quasi two-dimensional supported membrane is proposed. We calculate the average swimming velocity as a function of the disk size and the arm length. Due to the presence of the hydrodynamic screening length in the quasi two-dimensional fluid, the geometric factor appearing in the average velocity exhibits three different asymptotic behaviors depending on the microswimmer size and the hydrodynamic screening length. This is in sharp contrast with a microswimmer in a three-dimensional bulk fluid that shows only a single scaling behavior. We also find…
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…
It has long been known that some microswimmers seem to swim counter-intuitively faster when the viscosity of the surrounding fluid is increased, whereas others slow down. This conflicting dependence of the swimming velocity on the viscosity is poorly understood theoretically. Here we explain that any mechanical microswimmer with an elastic degree of freedom in a simple Newtonian fluid can exhibit both kinds of response to an increase in the fluid viscosity for different viscosity ranges, if the driving is weak. The velocity response is controlled by a single parameter $\varGamma$, the ratio of…
Scallop Theorem for Swimming in Anisotropic Fluids
University of Pennsylvania（宾夕法尼亚大学）；Kent State University（肯特州立大学）；University of Ljubljana（卢布尔雅那大学）；University of Warwick（华威大学）；Jožef Stefan Institute（约瑟夫·斯特凡研究所）
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 microalga Chlamydomonas Reinhardtii (CR) is used here as a model system to study the effect of complex environments on the swimming of micro-organisms. Its motion can be modelled by a run and tumble mechanism so that it describes a persistent random walk from which we can extract an effective diffusion coefficient for the large-time dynamics. In our experiments, the complex medium consists in a series of pillars that are designed in a regular lattice using soft lithography microfabrication. The cells are then introduced in the lattice, and their trajectories within the pillars are tracked …
The field of active matter in general and microswimming in particular has experienced a rapid and ongoing expansion over the last decade. A particular interesting aspect is provided by artificial autonomous microswimmers constructed from individual active and inactive functional components into self-propelling complexes. Such modular microswimmers may exhibit directed motion not seen for each individual component. In this review, we focus on the establishment and recent developments in the modular approach to microswimming. We introduce the bound and dynamic prototypes, show mechanisms and typ…
Microswimmers exhibit more diverse behavior in quasi-two dimensions than in three dimensions. Such behavior remains elusive due to the analytical difficulty of dealing with two parallel solid boundaries. The existence of additional obstacles in quasi-two dimensional systems further complicates the analysis. Combining experiments and hydrodynamic simulations, we investigate how the spatial dimension affects the interactions between microswimmers and obstacles. We fabricated microscopic pillars in quasi-two dimensions by etching glass coverslips and observed bacterial swimming among the pillars.…
We discuss the locomotion of a three-sphere microswimmer in a viscoelastic structured fluid characterized by typical length and time scales. We derive a general expression to link the average swimming velocity to the sphere mobilities. In this relationship, a viscous contribution exists when the time-reversal symmetry is broken, whereas an elastic contribution is present when the structural symmetry of the microswimmer is broken. As an example of a structured fluid, we consider a polymer gel, which is described by a "two-fluid" model. We demonstrate in detail that the competition between the s…
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 …
Small organisms (e.g., bacteria) and artificial microswimmers move due to a combination of active swimming and passive Brownian motion. Considering a simplified linear three-sphere swimmer, we study how the swimmer size regulates the interplay between self-driven and diffusive behavior at low Reynolds number. Starting from the Kirkwood-Smoluchowski equation and its corresponding Langevin equation, we derive formulas for the orientation correlation time, the mean velocity and the mean square displacement in three space dimensions. The validity of the analytical results is illustrated through nu…
The design of artificial microswimmers is often inspired by the strategies of natural microorganisms. Many of these creatures exploit the fact that elasticity breaks the time-reversal symmetry of motion at low Reynolds numbers, but this principle has been notably absent from model systems of active, self-propelled microswimmers. Here we introduce a class of microswimmer that spontaneously self-assembles and swims without using external forces, driven instead by surface phase transitions induced by temperature variations. The swimmers are made from alkane droplets dispersed in aqueous surfactan…
For natural microswimmers, the interplay of swimming activity and external flow can promote robust motion, e.g. propulsion against ("upstream rheotaxis") or perpendicular to the direction of flow. These effects are generally attributed to their complex body shapes and flagellar beat patterns. Here, using catalytic Janus particles as a model experimental system, we report on a strong directional response that occurs for spherical active particles in a channel flow. The particles align their propulsion axes to be nearly perpendicular to both the direction of flow and the normal vector of a nearb…
Typically, in the description of active Brownian particles, a constant effective propulsion force is assumed, which is then subjected to fluctuations in orientation and translation leading to a persistent random walk with an enlarged long-time diffusion coefficient. Here, we generalize previous results for the swimming path statistics to a time-dependent and thus in many situations more realistic propulsion which is a prescribed input. We analytically calculate both the noise-free and the noise-averaged trajectories for time-periodic propulsion under the action of an additional torque. In the …
The locomotion of microorganisms and spermatozoa in complex viscoelastic fluids is of critical importance in many biological processes such as fertilization, infection, and biofilm formation. Depending on their propulsion mechanisms, microswimmers display various responses to a complex fluid environment: increasing or decreasing their swimming speed and efficiency, modifying their propulsion kinematics and swimming gaits, and experiencing different hydrodynamic interactions with their surroundings. In this article, we review the fundamental physics of locomotion of biological and synthetic mic…
We investigate the hydrodynamic interaction between two elastic swimmers which are composed of three spheres and two harmonic springs. In this model, the natural length of each spring is assumed to undergo a prescribed cyclic change, representing internal states of the swimmer [K. Yasuda et al., J. Phys. Soc. Jpn. 86, 093801 (2017)]. We obtain the average velocities of two identical elastic swimmers as a function of the distance between them both for structurally asymmetric and symmetric swimmers. We show that the mean velocity of the two swimmers is always smaller than that of a single elasti…
The effect of shape asymmetry of microswimmers on their adsorption capacity at confining channel walls is studied by a simple dumbbell model. For a shape polarity of a forward-swimming cone, like the stroke-averaged shape of a sperm, extremely long wall retention times are found, caused by a non-vanishing component of the propulsion force pointing steadily into the wall, which grows exponentially with the self-propulsion velocity and the shape asymmetry. A direct duality relation between shape asymmetry and wall curvature is proposed and verified. Our results are relevant for the design micros…
Hydrodynamic interactions are crucial for determining the cooperative behavior of microswimmers at low Reynolds numbers. Here we provide a comprehensive analysis of the scaling and strength of the interactions in the case of a pair of three-sphere swimmers with intrinsic elasticity. Both stroke-based and force-based microswimmers are analyzed using an analytic perturbative approach. Following a detailed analysis of the passive interactions, as well as active translations and rotations, we find that the mapping between the stroke-based and force-based swimmers is only possible in a low driving …
The optimal strategy for a microscopic swimmer to migrate across a linear shear flow is discussed. The two cases, in which the swimmer is located at large distance, and in the proximity of a solid wall, are taken into account. It is shown that migration can be achieved by means of a combination of sailing through the flow and swimming, where the swimming strokes are induced by the external flow without need of internal energy sources or external drives. The structural dynamics required for the swimmer to move in the desired direction is discussed and two simple models, based respectively on th…
We study the shear induced migration of microswimmers (primarily, active Brownian particles or ABP's) in a plane Poiseuille flow. For wide channels characterized by $U_b/HD_r \ll 1$, the separation between time scales characterizing the swimmer orientation dynamics (of O($D^{-1}_r$)) and those that characterize migration across the channel (of O($H^{2}D_r/U^{2}_b$)), allows for use of the method of multiple scales to derive a drift-diffusion equation for the swimmer concentration profile; here, $U_b$ is the swimming speed, $H$ is the channel half-width, and $D_r$ is the swimmer rotary diffusiv…
Despite their vast morphological diversity, many invertebrates have similar larval forms characterized by ciliary bands, innervated arrays of beating cilia that facilitate swimming and feeding. Hydrodynamics suggests that these bands should tightly constrain the behavioral strategies available to the larvae; however, their apparent ubiquity suggests that these bands also confer substantial adaptive advantages. Here, we use hydrodynamic techniques to investigate "blinking," an unusual behavioral phenomenon observed in many invertebrate larvae in which ciliary bands across the body rapidly chang…
Recent research has shown that motile cells can adapt their mode of propulsion depending on the environment in which they find themselves. One mode is swimming by blebbing or other shape changes, and in this paper we analyze a class of models for movement of cells by blebbing and of nano- robots in a viscous fluid at low Reynolds number. At the level of individuals, the shape changes comprise volume exchanges between connected spheres that can control their separation, which are simple enough that significant analytical results can be obtained. The goal is to understand how the efficiency of m…
Active systems comprised of self-propelled units show fascinating transitions from Brownian-like dynamics to collective coherent motion. Swirling of swimming bacteria is a spectacular example. This study demonstrates that a nematic liquid crystal environment patterned as a spiral vortex controls individual-to-collective transition in bacterial swirls and defines whether they expand or shrink. In dilute dispersions, the bacteria swim along open spiral trajectories, following the pre-imposed molecular orientation. The trajectories are nonpolar. As their concentration exceeds some threshold, the …
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…
Enhanced diffusion is an emergent property of many experimental microswimmer systems that usually arises from a combination of ballistic motion with random reorientations. A subset of these systems, autophoretic droplet swimmers that move as a result of Marangoni stresses, have additionally been shown to respond to local, self-produced chemical gradients that can mediate self-avoidance or self-attraction. Via this mechanism, we present a mathematical model constructed to encode experimentally observed self-avoidant memory and numerically study the effect of this particular memory on the enhanc…
