Periodic and quasiperiodic motion of an elongated microswimmer in Poiseuille flow

Enhancement of bacterial rheotaxis in non-Newtonian fluids

Many microorganisms exhibit upstream swimming, which is important to many biological processes and can cause contamination of biomedical devices and the infection of organs. This process, called rheotaxis, has been studied extensively in Newtonian fluids. However, most microorganisms thrive in non-Newtonian fluids that contain suspended polymers such as mucus and biofilms. Here, we investigate the rheotactic behavior of Escherichia coli near walls in non-Newtonian fluids. Our experiments demonstrate that bacterial upstream swimming is enhanced by an order of magnitude in shear-thinning (ST) polymeric fluids relative to Newtonian fluids. This result is explained by direct numerical simulations, revealing a torque that promotes the alignment of bacteria against the flow. From this analysis, we develop a theoretical model that accurately describes experimental rheotactic data in both Newtonian and ST fluids.

Exploring noisy Jeffery orbits: A combined Fokker-Planck and Langevin analysis in two and three dimensions

The behavior of nonspherical particles in a shear flow is of significant practical and theoretical interest. These systems have been the object of numerous investigations since the pioneering work of Jeffery a century ago. His eponymous orbits describe the deterministic motion of an isolated, rodlike particle in a shear flow. Subsequently, the effect of adding noise was investigated. The theory has been applied to colloidal particles, macromolecules, anisometric granular particles, and most recently to microswimmers, for example, bacteria. We study the Jeffery orbits of elongated (uniaxial, prolate) particles subject to noise using Langevin simulations and a Fokker-Planck equation. We extend the analytical solution for infinitely thin needles (β=1) obtained by Doi and Edwards to particles with arbitrary shape factor (0≤β≤1) and validate the theory by comparing it with simulations. We examine the rotation of the particle around the vorticity axis and study the orientational order matrix. We use the latter to obtain scalar order parameters s and r describing nematic ordering and biaxiality from the orientational distribution function. The value of s (nematic ordering) increases monotonically with increasing Péclet number, while r (measure of biaxiality) displays a maximum value. From perturbation theory, we obtain simple expressions that provide accurate descriptions at low noise (or large Péclet numbers). We also examine the orientational distribution in the v-grad v plane and in the perpendicular direction. Finally, we present the solution of the Fokker-Planck equation for a strictly two-dimensional (2D) system. For the same noise amplitude, the average rotation speed of the particle in 3D is larger than in 2D.

Active particle motion in Poiseuille flow through rectangular channels

We investigate the dynamics of a pointlike active particle suspended in fluid flow through a straight channel. For this particle-fluid system, we derive a constant of motion for a general unidirectional fluid flow and apply it to an approximation of Poiseuille flow through channels with rectangular cross- sections. We obtain a 4D nonlinear conservative dynamical system with one constant of motion and a dimensionless parameter describing the ratio of maximum flow speed to intrinsic active particle speed. Applied to square channels, we observe a diverse set of active particle trajectories with variations in system parameters and initial conditions which we classify into different types of swinging, trapping, tumbling, and wandering motion. Regular (periodic and quasiperiodic) motion as well as chaotic active particle motion are observed for these trajectories and quantified using largest Lyapunov exponents. We explore the transition to chaotic motion using Poincaré maps and show "sticky" chaotic tumbling trajectories that have long transients near a periodic state. We briefly illustrate how these results extend to rectangular cross-sections with a width-to-height ratio larger than one. Outcomes of this paper may have implications for dynamics of natural and artificial microswimmers in experimental microfluidic channels that typically have rectangular cross sections.

Collective dynamics of active dumbbells near a circular obstacle

In this article, we present the collective dynamics of active dumbbells in the presence of a static circular obstacle using Brownian dynamics simulation. The active dumbbells aggregate on the surface of a circular obstacle beyond a critical radius. The aggregation is non-uniform along the circumference, and the aggregate size increases with the activity (Pe) and the curvature radius (Ro). The dense aggregate of active dumbbells displays persistent rotational motion with a certain angular speed, which linearly increases with activity. Furthermore, we show a strong polar ordering of the active dumbbells within the aggregate. The polar ordering exhibits long-range correlation, with the correlation length corresponding to the aggregate size. Additionally, we show that the residence time of an active dumbbell on the obstacle surface increases rapidly with area fraction due to many-body interactions that lead to a slowdown of the rotational diffusion. This article further considers the dynamical behavior of a tracer particle in the solution of active dumbbells. Interestingly, the speed of the passive tracer particle displays a crossover from monotonically decreasing to increasing with the size of the tracer particle upon increasing the dumbbells' speed. Furthermore, the effective diffusion of the tracer particle displays non-monotonic behavior with the area fraction; the initial increase in diffusivity is followed by a decrease for a larger area fraction.

Microswimmers in vortices: dynamics and trapping

Biological and artificial microswimmers often self-propel in external flows of vortical nature; relevant examples include algae in small-scale ocean eddies, spermatozoa in uterine peristaltic flows and bacteria in microfluidic devices. A recent experiment has shown that swimming bacteria in model vortices are expelled from the vortex all the way to a well-defined depletion zone (A. Sokolov and I. S. Aranson, Rapid expulsion of microswimmers by a vortical flow. Nat. Commun., 2016, 7, 11114). In this paper, we propose a theoretical model to investigate the dynamics of elongated microswimmers in elementary vortices, namely active particles in two- and three-dimensional rotlets. A deterministic model first reveals the existence of bounded orbits near the centre of the vortex and unbounded orbits elsewhere. We further discover a conserved quantity of motion that allows us to map the phase space according to the type of the orbit (bounded vs unbounded). We next introduce translational and rotational noise into the system. Using a Fokker-Planck formalism, we quantify the quality of trapping near the centre of the vortex by examining the probability of escape and the mean time of escape from the region of deterministically bounded orbits. We finally show how to use these findings to formulate a prediction for the radius of the depletion zone, which compares favourably with the experiments (A. Sokolov and I. S. Aranson, Rapid expulsion of microswimmers by a vortical flow. Nat. Commun., 2016, 7, 11114).

Diffusion of chiral active particles in a Poiseuille flow

We study the diffusive behavior of chiral active (self-propelled) Brownian particles in a two-dimensional microchannel with a Poiseuille flow. Using numerical simulations, we show that the behavior of the transport coefficients of particles, for example, the average velocity v and the effective diffusion coefficient D_{eff}, strongly depends on flow strength u_{0}, translational diffusion constant D_{0}, rotational diffusion rate D_{θ}, and chirality of the active particles Ω. It is demonstrated that the particles can exhibit upstream drift, resulting in a negative v, for the optimal parameter values of u_{0}, D_{θ}, and Ω. Interestingly, the direction of v can be controlled by tuning these parameters. We observe that for some optimal values of u_{0} and Ω, the chiral particles aggregate near a channel wall and the corresponding D_{eff} are enhanced. However, for the nonchiral particles (Ω=0), D_{eff} is suppressed by the presence of Poiseuille flow. It is expected that these findings have a great potential for developing microfluidic and lab-on-a-chip devices for separating the active particles.

On the cross-streamline lift of microswimmers in viscoelastic flows

The current work studies the dynamics of a microswimmer in pressure-driven flow of a weakly viscoelastic fluid. Employing a second-order fluid model, we show that a self-propelling swimmer experiences a viscoelastic swimming lift in addition to the well-known passive lift that arises from its resistance to shear flow. Using the reciprocal theorem, we evaluate analytical expressions for the swimming lift experienced by neutral and pusher/puller-type swimmers and show that they depend on the hydrodynamic signature associated with the swimming mechanism. We find that, in comparison to passive particles, the focusing of neutral swimmers towards the centerline can be significantly accelerated, while for force-dipole swimmers no net modification in cross-streamline migration occurs.

Migration of active filaments under Poiseuille flow in a microcapillary tube

We present a comprehensive study of active filaments confined in a cylindrical channel under Poiseuille flow. The activity drives the filament towards the channel boundary, whereas external fluid flow migrates the filament away from the boundary. This migration further shifts towards the centre for higher flow strength. The migration behaviour of the filaments is presented in terms of the alignment order parameter that shows the alignment grows with shear and activity. Further, we have also addressed the role of length of filament on the migration behaviour, which suggests higher migration for larger filaments. Moreover, we discuss the polar ordering of filaments as a function of distance from the centre of channel that displays upstream motion near the boundary and downstream motion at the centre of the tube.

Change in rheotactic behavior patterns of dinoflagellates in response to different microfluidic environments

Plankton live in dynamic fluid environments. Their ability to change in response to different hydrodynamic cues is critical to their energy allocation and resource uptake. This study used a microfluidic device to evaluate the rheotactic behaviors of a model dinoflagellate species, Karlodinium veneficum, in different flow conditions. Although dinoflagellates experienced forced alignment in strong shear (i.e. “trapping”), fluid straining did not play a decisive role in their rheotactic movements. Moderate hydrodynamic magnitude (20 < |u_f| < 40 µm s⁻¹) was found to induce an orientation heading towards an oncoming current (positive rheotaxis), as dinoflagellates switched to cross-flow swimming when flow speed exceeded 50 µm s⁻¹. Near the sidewalls of the main channel, the steric mechanism enabled dinoflagellates to adapt upstream orientation through vertical migration. Under oscillatory flow, however, positive rheotaxis dominated with occasional diversion. The varying flow facilitated upstream exploration with directional controlling, through which dinoflagellates exhibited avoidance of both large-amplitude perturbance and very stagnant zones. In the mixed layer where water is not steady, these rheotactic responses could lead to spatial heterogeneity of dinoflagellates. The outcome of this study helps clarify the interaction between swimming behaviors of dinoflagellates and the hydrodynamic environment they reside in.

Rebound and scattering of motile Chlamydomonas algae in confined chambers

Motivated by recent experiments demonstrating that motile algae get trapped in draining foams, we study the trajectories of microorganisms confined in model foam channels (section of a Plateau border). We track single Chlamydomonas reinhardtii cells confined in a thin three-circle microfluidic chamber and show that their spatial distribution exhibits strong corner accumulation. Using empirical scattering laws observed in previous experiments (scattering with a constant scattering angle), we next develop a two-dimension geometrical model and compute the phase space of trapped and periodic trajectories of swimmers inside a three-circles billiard. We find that the majority of cell trajectories end up in a corner, providing a geometrical mechanism for corner accumulation. Incorporating the distribution of scattering angles observed in our experiments and including hydrodynamic interactions between the cells and the surfaces into the geometrical model enables us to reproduce the experimental probability density function of micro-swimmers in microfluidic chambers. Both our experiments and models demonstrate therefore that motility leads generically to trapping in complex geometries.

Erratum: Periodic Orbits of Active Particles Induced by Hydrodynamic Monopoles [Phys. Rev. Lett. 124, 088003 (2020)]

This corrects the article DOI: 10.1103/PhysRevLett.124.088003.

Chirality-induced bacterial rheotaxis in bulk shear flows

Interaction of swimming bacteria with flows controls their ability to explore complex environments, crucial to many societal and environmental challenges and relevant for microfluidic applications such as cell sorting. Combining experimental, numerical, and theoretical analysis, we present a comprehensive study of the transport of motile bacteria in shear flows. Experimentally, we obtain with high accuracy and, for a large range of flow rates, the spatially resolved velocity and orientation distributions. They are in excellent agreement with the simulations of a kinematic model accounting for stochastic and microhydrodynamic properties and, in particular, the flagella chirality. Theoretical analysis reveals the scaling laws behind the average rheotactic velocity at moderate shear rates using a chirality parameter and explains the reorientation dynamics leading to saturation at large shear rates from the marginal stability of a fixed point. Our findings constitute a full understanding of the physical mechanisms and relevant parameters of bacteria bulk rheotaxis.

Trapping of swimmers in a vortex lattice

We examine the motion of rigid, ellipsoidal swimmers subjected to a steady vortex flow in two dimensions. Numerical simulations of swimmers in a spatially periodic array of vortices reveal a range of possible behaviors, including trapping inside a single vortex and motility-induced diffusion across many vortices. While the trapping probability vanishes at a sufficiently high swimming speed, we find that it exhibits surprisingly large oscillations as this critical swimming speed is approached. Strikingly, at even higher swimming speeds, we find swimmers that swim perpendicular to their elongation direction can again become trapped. To explain this complex behavior, we investigate the underlying swimmer phase-space geometry. We identify the fixed points and periodic orbits of the swimmer equations of motion that regulate swimmer trapping inside a single vortex cell. For low to intermediate swimming speeds, we find that a stable periodic orbit surrounded by invariant tori forms a transport barrier to swimmers and can trap them inside individual vortices. For swimming speeds approaching the maximum fluid speed, we find instead that perpendicular swimmers can be trapped by asymptotically stable fixed points. A bifurcation analysis of the stable periodic orbit and the fixed points explains the complex and non-monotonic breakdown and re-emergence of swimmer trapping as the swimmer speed and shape are varied.

Active dipolar spheroids in shear flow and transverse field: Population splitting, cross-stream migration, and orientational pinning

We study the steady-state behavior of active, dipolar, Brownian spheroids in a planar channel subjected to an imposed Couette flow and an external transverse field, applied in the "downward" normal-to-flow direction. The field-induced torque on active spheroids (swimmers) is taken to be of magnetic form by assuming that they have a permanent magnetic dipole moment, pointing along their self-propulsion (swim) direction. Using a continuum approach, we show that a host of behaviors emerges over the parameter space spanned by the particle aspect ratio, self-propulsion and shear/field strengths, and the channel width. The cross-stream migration of the model swimmers is shown to involve a regime of linear response (quantified by a linear-response factor) in weak fields. For prolate swimmers, the weak-field behavior crosses over to a regime of full swimmer migration to the bottom half of the channel in strong fields. For oblate swimmers, a counterintuitive regime of reverse migration arises in intermediate fields, where a macroscopic fraction of swimmers reorient and swim to the top channel half at an acute "upward" angle relative to the field axis. The diverse behaviors reported here are analyzed based on the shear-induced population splitting (bimodality) of the swim orientation, giving two distinct, oppositely polarized, swimmer subpopulations (albeit very differently for prolate/oblate swimmers) in each channel half. In strong fields, swimmers of both types exhibit net upstream currents relative to the laboratory frame. The onsets of full migration and net upstream current depend on the aspect ratio, enabling efficient particle separation strategies in microfluidic setups.

Biofilms at interfaces: microbial distribution in floating films

Cellular motility is a key function guiding microbial adhesion to interfaces, which is the first step in the formation of biofilms. The close association of biofilms and bioremediation has prompted extensive research aimed at comprehending the physics of microbial locomotion near interfaces. We study the dynamics and statistics of microorganisms in a 'floating biofilm', i.e., a confinement with an air-liquid interface on one side and a liquid-liquid interface on the other. We use a very general mathematical model, based on a multipole representation and probabilistic simulations, to ascertain the spatial distribution of microorganisms in films of different viscosities. Our results reveal that microorganisms can be distributed symmetrically or asymmetrically across the height of the film, depending on their morphology and the ratio of the film's viscosity to that of the fluid substrate. Long-flagellated, elongated bacteria exhibit stable swimming parallel to the liquid-liquid interface when the bacterial film is less viscous than the underlying fluid. Bacteria with shorter flagella on the other hand, swim away from the liquid-liquid interface and accumulate at the free surface. We also analyze microorganism dynamics in a flowing film and show how a microorganism's ability to resist 'flow-induced-erosion' from interfaces is affected by its elongation and mode of propulsion. Our study generalizes past efforts on understanding microorganism dynamics under confinement by interfaces and provides key insights on biofilm initiation at liquid-liquid interfaces.

Alignment of Nonspherical Active Particles in Chaotic Flows

We study the orientation statistics of spheroidal, axisymmetric microswimmers, with shapes ranging from disks to rods, swimming in chaotic, moderately turbulent flows. Numerical simulations show that rodlike active particles preferentially align with the flow velocity. To explain the underlying mechanism, we solve a statistical model via the perturbation theory. We show that such an alignment is caused by correlations of fluid velocity and its gradients along particle paths combined with fore-aft symmetry breaking due to both swimming and particle nonsphericity. Remarkably, the discovered alignment is found to be a robust kinematical effect, independent of the underlying flow evolution. We discuss its possible relevance for aquatic ecology.

Frequency-dependent higher-order Stokes singularities near a planar elastic boundary: Implications for the hydrodynamics of an active microswimmer near an elastic interface

The emerging field of self-driven active particles in fluid environments has recently created significant interest in the biophysics and bioengineering communities owing to their promising future for biomedical and technological applications. These microswimmers move autonomously through aqueous media, where under realistic situations they encounter a plethora of external stimuli and confining surfaces with peculiar elastic properties. Based on a far-field hydrodynamic model, we present an analytical theory to describe the physical interaction and hydrodynamic couplings between a self-propelled active microswimmer and an elastic interface that features resistance toward shear and bending. We model the active agent as a superposition of higher-order Stokes singularities and elucidate the associated translational and rotational velocities induced by the nearby elastic boundary. Our results show that the velocities can be decomposed in shear and bending related contributions which approach the velocities of active agents close to a no-slip rigid wall in the steady limit. The transient dynamics predict that contributions to the velocities of the microswimmer due to bending resistance are generally more pronounced than those due to shear resistance. Bending can enhance (suppress) the velocities resulting from higher-order singularities whereas the shear related contribution decreases (increases) the velocities. Most prominently, we find that near an elastic interface of only energetic resistance toward shear deformation, such as that of an elastic capsule designed for drug delivery, a swimming bacterium undergoes rotation of the same sense as observed near a no-slip wall. In contrast to that, near an interface of only energetic resistance toward bending, such as that of a fluid vesicle or liposome, we find a reversed sense of rotation. Our results provide insight into the control and guidance of artificial and synthetic self-propelling active microswimmers near elastic confinements.

Effect of external shear flow on sperm motility

The trajectory of sperm in the presence of background flow is of utmost importance for the success of fertilization, as sperm encounter background flow of different magnitude and direction on their way to the egg. Here, we have studied the effect of an unbounded simple shear flow as well as a Poiseuille flow on the sperm trajectory. In the presence of a simple shear flow, the sperm moves on an elliptical trajectory in the reference frame advecting with the local background flow. The length of the major-axis of this elliptical trajectory decreases with the shear rate. The flexibility of the flagellum and consequently the length of the major axis of the elliptical trajectories increases with the sperm number. The sperm number is a dimensionless number representing the ratio of viscous force to elastic force. The sperm moves downstream or upstream depending on the strength of background Poiseuille flow. In contrast to the simple shear flow, the sperm also moves toward the centerline in a Poiseuille flow. Far away from the centerline, the cross-stream migration velocity of the sperm increases as the transverse distance of the sperm from the centerline decreases. Close to the centerline, on the other hand, the cross-stream migration velocity decreases as the sperm further approaches the center. The cross-stream migration velocity of the sperm also increases with the sperm number.

Oscillatory surface rheotaxis of swimming E. coli bacteria

Bacterial contamination of biological channels, catheters or water resources is a major threat to public health, which can be amplified by the ability of bacteria to swim upstream. The mechanisms of this 'rheotaxis', the reorientation with respect to flow gradients, are still poorly understood. Here, we follow individual E. coli bacteria swimming at surfaces under shear flow using 3D Lagrangian tracking and fluorescent flagellar labelling. Three transitions are identified with increasing shear rate: Above a first critical shear rate, bacteria shift to swimming upstream. After a second threshold, we report the discovery of an oscillatory rheotaxis. Beyond a third transition, we further observe coexistence of rheotaxis along the positive and negative vorticity directions. A theoretical analysis explains these rheotaxis regimes and predicts the corresponding critical shear rates. Our results shed light on bacterial transport and reveal strategies for contamination prevention, rheotactic cell sorting, and microswimmer navigation in complex flow environments.

Collective dynamics in a monolayer of squirmers confined to a boundary by gravity

We present a hydrodynamic study of a monolayer of squirmer model microswimmers confined to a boundary by strong gravity using the simulation method of multi-particle collision dynamics. The squirmers interact with each other via their self-generated hydrodynamic flow fields and thereby form a variety of fascinating dynamic states when density and squirmer type are varied. Weak pushers, neutral squirmers, and pullers have an upright orientation. With their flow fields they push neighbors away and thereby form a hydrodynamic Wigner fluid at lower densities. Furthermore, states of fluctuating chains and trimers, of kissing, and at large densities a global cluster exist. Finally, pushers at all densities can tilt against the wall normal and their in-plane velocities align to show swarming. It turns into chaotic swarming for strong pushers at high densities. We characterize all these states quantitatively.

Gyrotactic phytoplankton in laminar and turbulent flows: A dynamical systems approach

Gyrotactic algae are bottom heavy, motile cells whose swimming direction is determined by a balance between a buoyancy torque directing them upwards and fluid velocity gradients. Gyrotaxis has, in recent years, become a paradigmatic model for phytoplankton motility in flows. The essential attractiveness of this peculiar form of motility is the availability of a mechanistic description which, despite its simplicity, revealed predictive, rich in phenomenology, easily complemented to include the effects of shape, feedback on the fluid and stochasticity (e.g., in cell orientation). In this review we consider recent theoretical, numerical and experimental results to discuss how, depending on flow properties, gyrotaxis can produce inhomogeneous phytoplankton distributions on a wide range of scales, from millimeters to kilometers, in both laminar and turbulent flows. In particular, we focus on the phenomenon of gyrotactic trapping in nonlinear shear flows and in fractal clustering in turbulent flows. We shall demonstrate the usefulness of ideas and tools borrowed from dynamical systems theory in explaining and interpreting these phenomena.

State diagram of a three-sphere microswimmer in a channel

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 neutral swimmers, puller- and pusher-type swimmers. While pushers always end up trapped at the channel walls, neutral swimmers and pullers may further perform a gliding motion and maintain a stable navigation along the channel. We find that the resulting dynamical system exhibits a supercritical pitchfork bifurcation in which swimming in the mid-plane becomes unstable beyond a transition channel height while two new stable limit cycles or fixed points that are symmetrically disposed with respect to the channel mid-height emerge. Additionally, we show that an accurate description of the averaged swimming velocity and rotation rate in a channel can be captured analytically using the method of hydrodynamic images, provided that the swimmer size is much smaller than the channel height.

Collective sedimentation of squirmers under gravity

Active particles, which interact hydrodynamically, display a remarkable variety of emergent collective phenomena. We use squirmers to model spherical microswimmers and explore the collective behavior of thousands of them under the influence of strong gravity using the method of multi-particle collision dynamics for simulating fluid flow. The sedimentation profile depends on the ratio of swimming to sedimentation velocity as well as on the squirmer type. It shows closely packed squirmer layers at the bottom and a highly dynamic region with exponential density dependence towards the top. The mean vertical orientation of the squirmers strongly depends on height. For swimming velocities larger than the sedimentation velocity, squirmers show strong convection in the exponential region. We quantify the strength of convection and the extent of convection cells by the vertical current density and its current dipole, which are large for neutral squirmers as well as for weak pushers and pullers.

Swinging motion of active deformable particles in Poiseuille flow

Dynamics of active deformable particles in an external Poiseuille flow is investigated. To make the analysis general, we employ time-evolution equations derived from symmetry considerations that take into account an elliptical shape deformation. First, we clarify the relation of our model to that of rigid active particles. Then, we study the dynamical modes that active deformable particles exhibit by changing the strength of the external flow. We emphasize the difference between the active particles that tend to self-propel parallel to the elliptical shape deformation and those self-propelling perpendicularly. In particular, a swinging motion around the centerline far from the channel walls is discussed in detail.

Population splitting of rodlike swimmers in Couette flow

We present a quantitative analysis on the response of a dilute active suspension of self-propelled rods (swimmers) in a planar channel subjected to an imposed shear flow. To best capture the salient features of the shear-induced effects, we consider the case of an imposed Couette flow, providing a constant shear rate across the channel. We argue that the steady-state behavior of swimmers can be understood in the light of a population splitting phenomenon, occurring as the shear rate exceeds a certain threshold, initiating the reversal of the swimming direction for a finite fraction of swimmers from down- to upstream or vice versa, depending on the swimmer position within the channel. Swimmers thus split into two distinct, statistically significant and oppositely swimming majority and minority populations. The onset of population splitting translates into a transition from a self-propulsion-dominated regime to a shear-dominated regime, corresponding to a unimodal-to-bimodal change in the probability distribution function of the swimmer orientation. We present a phase diagram in terms of the swim and flow Péclet numbers showing the separation of these two regimes by a discontinuous transition line. Our results shed further light on the behavior of swimmers in a shear flow and provide an explanation for the previously reported non-monotonic behavior of the mean, near-wall, parallel-to-flow orientation of swimmers with increasing shear strength.

Flagella bending affects macroscopic properties of bacterial suspensions

To survive in harsh conditions, motile bacteria swim in complex environments and respond to the surrounding flow. Here, we develop a mathematical model describing how flagella bending affects macroscopic properties of bacterial suspensions. First, we show how the flagella bending contributes to the decrease in the effective viscosity observed in dilute suspension. Our results do not impose tumbling (random reorientation) as was previously done to explain the viscosity reduction. Second, we demonstrate how a bacterium escapes from wall entrapment due to the self-induced buckling of flagella. Our results shed light on the role of flexible bacterial flagella in interactions of bacteria with shear flow and walls or obstacles.

Hotspots of boundary accumulation: dynamics and statistics of micro-swimmers in flowing films

Biological flows over surfaces and interfaces can result in accumulation hotspots or depleted voids of microorganisms in natural environments. Apprehending the mechanisms that lead to such distributions is essential for understanding biofilm initiation. Using a systematic framework, we resolve the dynamics and statistics of swimming microbes within flowing films, considering the impact of confinement through steric and hydrodynamic interactions, flow and motility, along with Brownian and run-tumble fluctuations. Micro-swimmers can be peeled off the solid wall above a critical flow strength. However, the interplay of flow and fluctuations causes organisms to migrate back towards the wall above a secondary critical value. Hence, faster flows may not always be the most efficacious strategy to discourage biofilm initiation. Moreover, we find run-tumble dynamics commonly used by flagellated microbes to be an intrinsically more successful strategy to escape from boundaries than equivalent levels of enhanced Brownian noise in ciliated organisms.

Succeed escape: Flow shear promotes tumbling of Escherichia colinear a solid surface

Understanding how bacteria move close to a surface under various stimuli is crucial for a broad range of microbial processes including biofilm formation, bacterial transport and migration. While prior studies focus on interactions between single stimulus and bacterial suspension, we emphasize on compounding effects of flow shear and solid surfaces on bacterial motility, especially reorientation and tumble. We have applied microfluidics and digital holographic microscopy to capture a large number (>10⁵) of 3D Escherichia coli trajectories near a surface under various flow shear. We find that near-surface flow shear promotes cell reorientation and mitigates the tumble suppression and re-orientation confinement found in a quiescent flow, and consequently enhances surface normal bacterial dispersion. Conditional sampling suggests that two complimentary hydrodynamic mechanisms, Jeffrey Orbit and shear-induced flagella unbundling, are responsible for the enhancement in bacterial tumble motility. These findings imply that flow shear may mitigate cell trapping and prevent biofilm initiation.

Taylor line swimming in microchannels and cubic lattices of obstacles

Microorganisms naturally move in microstructured fluids. Using the simulation method of multi-particle collision dynamics, we study in two dimensions an undulatory Taylor line swimming in a microchannel and in a cubic lattice of obstacles, which represent simple forms of a microstructured environment. In the microchannel the Taylor line swims at an acute angle along a channel wall with a clearly enhanced swimming speed due to hydrodynamic interactions with the bounding wall. While in a dilute obstacle lattice swimming speed is also enhanced, a dense obstacle lattice gives rise to geometric swimming. This new type of swimming is characterized by a drastically increased swimming speed. Since the Taylor line has to fit into the free space of the obstacle lattice, the swimming speed is close to the phase velocity of the bending wave traveling along the Taylor line. While adjusting its swimming motion within the lattice, the Taylor line chooses a specific swimming direction, which we classify by a lattice vector. When plotting the swimming velocity versus the magnitude of the lattice vector, all our data collapse on a single master curve. Finally, we also report more complex trajectories within the obstacle lattice.

Motility, Force Generation, and Energy Consumption of Unicellular Parasites

Motility is a key factor for pathogenicity of unicellular parasites, enabling them to infiltrate and evade host cells, and perform several of their life-cycle events. State-of-the-art methods of motility analysis rely on a combination of optical tweezers with high-resolution microscopy and microfluidics. With this technology, propulsion forces, energies, and power generation can be determined so as to shed light on the motion mechanisms, chemotactic behavior, and specific survival strategies of unicellular parasites. With these new tools in hand, we can elucidate the mechanisms of motility and force generation of unicellular parasites, and identify ways to manipulate and eventually inhibit them.

Rapid expulsion of microswimmers by a vortical flow

Interactions of microswimmers with their fluid environment are exceptionally complex. Macroscopic shear flow alters swimming trajectories in a highly nontrivial way and results in dramatic reduction of viscosity and heterogeneous bacterial distributions. Here we report on experimental and theoretical studies of rapid expulsion of microswimmers, such as motile bacteria, by a vortical flow created by a rotating microparticle. We observe a formation of a macroscopic depletion area in a high-shear region, in the vicinity of a microparticle. The rapid migration of bacteria from the shear-rich area is caused by a vortical structure of the flow rather than intrinsic random fluctuations of bacteria orientations, in stark contrast to planar shear flow. Our mathematical model reveals that expulsion is a combined effect of motility and alignment by a vortical flow. Our findings offer a novel approach for manipulation of motile microorganisms and shed light on bacteria–flow interactions.

The control of microswimmers such as bacteria is important for emerging applications of active bioinspired materials. Here, the authors demonstrate the use of vortical shear to expel suspended motile bacteria from the vicinity of a rotating microparticle.

Getting drowned in a swirl: Deformable bead-spring model microswimmers in external flow fields

Deformability is a central feature of many types of microswimmers, e.g., for artificially generated self-propelled droplets. Here, we analyze deformable bead-spring microswimmers in an externally imposed solvent flow field as simple theoretical model systems. We focus on their behavior in a circular swirl flow in two spatial dimensions. Linear (straight) two-bead swimmers are found to circle around the swirl with a slight drift to the outside with increasing activity. In contrast to that, we observe for triangular three-bead or squarelike four-bead swimmers a tendency of being drawn into the swirl and finally getting drowned, although a radial inward component is absent in the flow field. During one cycle around the swirl, the self-propulsion direction of an active triangular or squarelike swimmer remains almost constant, while their orbits become deformed exhibiting an "egglike" shape. Over time, the swirl flow induces slight net rotations of these swimmer types, which leads to net rotations of the egg-shaped orbits. Interestingly, in certain cases, the orbital rotation changes sense when the swimmer approaches the flow singularity. Our predictions can be verified in real-space experiments on artificial microswimmers.

Upstream Swimming in Microbiological Flows

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 centerline, in contrast to well-known boundary accumulation in quiescent Newtonian fluids. Based on these observations, we suggest a sorting mechanism to select microbes by swimming speed.

Bimodal rheotactic behavior reflects flagellar beat asymmetry in human sperm cells

Rheotaxis, the directed response to fluid velocity gradients, has been shown to facilitate stable upstream swimming of mammalian sperm cells along solid surfaces, suggesting a robust physical mechanism for long-distance navigation during fertilization. However, the dynamics by which a human sperm orients itself relative to an ambient flow is poorly understood. Here, we combine microfluidic experiments with mathematical modeling and 3D flagellar beat reconstruction to quantify the response of individual sperm cells in time-varying flow fields. Single-cell tracking reveals two kinematically distinct swimming states that entail opposite turning behaviors under flow reversal. We constrain an effective 2D model for the turning dynamics through systematic large-scale parameter scans, and find good quantitative agreement with experiments at different shear rates and viscosities. Using a 3D reconstruction algorithm to identify the flagellar beat patterns causing left or right turning, we present comprehensive 3D data demonstrating the rolling dynamics of freely swimming sperm cells around their longitudinal axis. Contrary to current beliefs, this 3D analysis uncovers ambidextrous flagellar waveforms and shows that the cell's turning direction is not defined by the rolling direction. Instead, the different rheotactic turning behaviors are linked to a broken mirror symmetry in the midpiece section, likely arising from a buckling instability. These results challenge current theoretical models of sperm locomotion.

Hydrodynamic analysis of flagellated bacteria swimming in corners of rectangular channels

The influence of nearby solid surfaces on the motility of bacteria is of fundamental importance as these interactions govern the ability of the microorganisms to explore their environment and form sessile colonies. Reducing biofouling in medical implants and controlling the transport of bacterial cells in a microfluidic device are two applications that could benefit from a detailed understanding of swimming in microchannels. In this study, we investigate the self-propelled motion of a model bacterium, driven by rotating a single helical flagellum, in such an environment. In particular, we focus on the corner region of a large channel modeled as two perpendicular sections of no-slip planes joined with a rounded corner. We numerically solve the equations of Stokes flow using the boundary element method to obtain the swimming velocities at different positions and orientations relative to the channel corner. From these velocities, we construct many trajectories to ascertain the general behavior of the swimmers. Considering only hydrodynamic interactions between the bacterium and the channel walls, we show that some swimmers can become trapped near the corner while moving, on average, along the axis of the channel. This result suggests that such bacteria may be found at much higher densities in corners than in other parts of the channel. Another implication is that these corner accumulating bacteria may travel quickly through channels since they are guided directly along the corner and do not turn back or swim transversely across the channel.

Self-phoretic active particles interacting by diffusiophoresis: A numerical study of the collapsed state and dynamic clustering

Self-phoretic active colloids move and orient along self-generated chemical gradients by diffusiophoresis, a mechanism reminiscent of bacterial chemotaxis. In combination with the activity of the colloids, this creates effective repulsive and attractive interactions between particles depending on the sign of the translational and rotational diffusiophoretic parameters. A delicate balance of these interactions causes dynamic clustering and for overall strong effective attraction the particles collapse to one single cluster. Using Langevin dynamics simulations, we extend the state diagram of our earlier work (Phys. Rev. Lett. 112, 238303 (2014)) to regions with translational phoretic repulsion. With increasing repulsive strength, the collapsed cluster first starts to fluctuate strongly, then oscillates between a compact form and a colloidal cloud, and ultimately the colloidal cloud becomes static. The oscillations disappear if the phoretic interactions within compact clusters are not screened. We also study dynamic clustering at larger area fractions by exploiting cluster size distributions and mean cluster sizes. In particular, we identify the dynamic clustering 2 state as a signature of phoretic interactions. We analyze fusion and fission rate functions to quantify the kinetics of cluster formation and identify them as local signatures of phoretic interactions, since they can be measured on single clusters.

Flexibility of bacterial flagella in external shear results in complex swimming trajectories

Many bacteria use rotating helical flagella in swimming motility. In the search for food or migration towards a new habitat, bacteria occasionally unbundle their flagellar filaments and tumble, leading to an abrupt change in direction. Flexible flagella can also be easily deformed by external shear flow, leading to complex bacterial trajectories. Here, we examine the effects of flagella flexibility on the navigation of bacteria in two fundamental shear flows: planar shear and Poiseuille flow realized in long channels. On the basis of slender body elastodynamics and numerical analysis, we discovered a variety of non-trivial effects stemming from the interplay of self-propulsion, elasticity and shear-induced flagellar bending. We show that in planar shear flow the bacteria execute periodic motion, whereas in Poiseuille flow, they migrate towards the centre of the channel or converge towards a limit cycle. We also find that even a small amount of random reorientation can induce a strong response of bacteria, leading to overall non-periodic trajectories. Our findings exemplify the sensitive role of flagellar flexibility and shed new light on the navigation of bacteria in complex shear flows.

Microbes in flow

Microbes often live in moving fluids. Despite the multitude of implications that flow has on microbial ecology and environmental microbiology, only recently have experimental tools and conceptual frameworks from fluid physics been applied systematically to further our knowledge of the behavior of microbes in flow. This nascent research field, which truly straddles biology and physics, has already produced important contributions to our understanding of the physical interaction between microbes and flow, both in bulk fluid and close to surfaces, at the same time revealing the richness and complexity of the resulting dynamics.

Hydrodynamic analysis of flagellated bacteria swimming near one and between two no-slip plane boundaries

The motility of swimming bacteria near solid surfaces has implications in a wide range of scenarios, including water treatment facilities, microfluidics, and biomedical implants. Using the boundary element method to numerically solve the equations of low Reynolds number fluid flow, we investigate the dynamics of a model swimmer propelled by rotating a single helical flagellum. Building on previous simulation results for swimmers near a single plane boundary, we introduce a second, parallel boundary and show that the bacterial trajectories change as the two plates are brought closer together. Analysis of this dynamical system shows that the configuration in the center of the channel and parallel to the walls is an unstable equilibrium state for large plate separations, but it becomes the only stable position for swimmers when the plate separation is reduced to three to four times the cell width. Our model also predicts that transient trajectories, i.e., those not at steady states, can exhibit curvature in the opposite sense to that expected from the well-known explanation for circular bacterial paths near a single wall.

Impact of external flow on the dynamics of swimming microorganisms near surfaces

Swimming microorganisms have been previously observed to accumulate along walls in confined systems both experimentally and in computer simulations. Here, we use computer simulations of dilute populations for a simplified model of an organism to calculate the dynamics of swimmers between two walls with an external fluid flow. Simulations with and without hydrodynamic interactions (HIs) are used to quantify their influence on surface accumulation. We found that the accumulation of organisms at the wall is larger when HIs are included. An external fluid flow orients the organisms parallel to the fluid flow, which reduces the accumulation at the walls. The effect of the flow on the orientations is quantified and compared to previous work on upstream swimming of organisms and alignment of passive rods in flow. In pressure-driven flow, the zero shear rate at the channel center leads to a dip in the concentration of organisms in the center. The curvature of this dip is quantified as a function of the flow rate. The fluid flow also affects the transport of organisms across the channel from one wall to the other.

Chaotic dynamics of a microswimmer in Poiseuille flow

The chaotic dynamics of pointlike, spherical particles in cylindrical Poiseuille flow is theoretically characterized and numerically confirmed when their own intrinsic swimming velocity undergoes temporal fluctuations around an average value. Two dimensionless ratios associated with the three significant temporal scales of the problem are identified that fully determine the chaos scenario. In particular, small but finite periodic fluctuations of swimming speed result in chaotic or regular motion depending on the position and orientation of the microswimmer with respect to the flow center line. Remarkably, the spatial extension of chaotic microswimmers is found to depend crucially on the fluctuations' period and amplitude and to be highly sensitive to the Fourier spectrum of the fluctuations. This has implications for the design of artificial microswimmers.

Flow loading induces oscillatory trajectories in a bloodstream parasite

The dynamics of isolated microswimmers are studied in bounded flow using the African trypanosome, a unicellular parasite, as the model organism. With the help of a microfluidics platform, cells are subjected to flow and found to follow an oscillatory path that is well fit by a sine wave. The frequency and amplitudes of the oscillatory trajectories are dependent on the flow velocity and cell orientation. When traveling in such a manner, trypanosomes orient upstream while downstream-facing cells tumble within the same streamline. A comparison with immotile trypanosomes demonstrates that self-propulsion is essential to the trajectories of trypanosomes even at flow velocities up to ∼40 times higher than their own swimming speed. These studies reveal important swimming dynamics that may be generally pertinent to the transport of microswimmers in flow and may be relevant to microbial pathogenesis.