Suspensions Of Swimmers Research Articles

Defect turbulence in a dense suspension of polar, active swimmers.

We study the effects of inertia in dense suspensions of polar swimmers. The hydrodynamic velocity field and the polar order parameter field describe the dynamics of the suspension. We show that a dimensionless parameter R (ratio of the swimmer self-advection speed to the active stress invasion speed [Phys. Rev. X 11, 031063 (2021)2160-330810.1103/PhysRevX.11.031063]) controls the stability of an ordered swimmer suspension. For R smaller than a threshold R_{1}, perturbations grow at a rate proportional to their wave number q. Beyond R_{1} we show that the growth rate is O(q^{2}) until a second threshold R=R_{2} is reached. The suspension is stable for R>R_{2}. We perform direct numerical simulations to characterize the steady-state properties and observe defect turbulence for R<R_{2}. An investigation of the spatial organization of defects unravels a hidden transition: for small R≈0 defects are uniformly distributed and cluster as R→R_{1}. Beyond R_{1}, clustering saturates and defects are arranged in nearly stringlike structures.

The interplay between bulk flow and boundary conditions on the distribution of microswimmers in channel flow

While previous experimental and numerical studies of dilute microswimmer suspensions have focused on the behaviours of swimmers in the bulk flow and near boundaries, models typically do not account for the interplay between bulk flow and the choice of boundary conditions imposed in continuum models. In our work, we highlight the effect of boundary conditions on the bulk flow distributions, such as through the development of boundary layers or secondary peaks of cell accumulation in bulk-flow swimmer dynamics. For the case of a dilute swimmer suspension in Poiseuille flow, we compare the distribution (in physical and orientation space) obtained from individual-based stochastic models with those from continuum models, and identify under what conditions it is mathematically sensible to use specific continuum boundary conditions to capture different physical scenarios (i.e. specular reflection, uniform random reflection and absorbing boundaries). We identify that the spread of preferred cell orientations is dependent on the interplay between rotation driven by the shear flow (Jeffery orbits) and rotational diffusion. We find that in the absence of hydrodynamic wall interactions, swimmers preferentially approach the walls perpendicular to the surface in the presence of high rotational diffusion, and that the preferential approach of swimmers to the walls is shape-dependent at low rotational diffusion (when suspensions tend towards a fully deterministic case). In the latter case, the preferred orientations are nearly parallel to the surface for elongated swimmers and nearly perpendicular to the surface for near-spherical swimmers. Furthermore, we highlight the effects of swimmer geometries and shear throughout the bulk-flow on swimmer trajectories and show how the full history of bulk-flow dynamics affects the orientation distributions of microswimmer wall incidence.

Swarm of slender pusher and puller swimmers at finite Reynolds numbers

The study of the flow field resulting from suspensions of swimmers at moderate Reynolds numbers, along with hydrodynamic interactions, has received little attention until now despite being of great interest to researchers in the fields of marine ecology, biology, and engineering. By means of direct numerical simulations, employing a state-of-the-art fully resolved immersed boundary method, the suspensions of inertial slender pusher and puller swimmers are investigated in dilute volume fractions and swimming Reynolds numbers ranging from 1 to 50 with the objective to identify the existence of correlated flow motions and scales when inertia plays a crucial role. The properties of the flow field resulting from the collective motion of the swimmers, as well as the characteristics of their orientation along with their temporal correlation, have been analyzed. Results show nontrivial flow motions as the Reynolds number changes along with a complex swimmer dynamics.

Enhanced velocity fluctuations in interacting swimmer suspensions

Abstract

Characterization of active matter in dense suspensions with heterodyne laser Doppler velocimetry

We present a novel approach for characterizing the properties and performance of active matter in dilute suspension as well as in crowded environments. We use Super-Heterodyne Laser-Doppler-Velocimetry (SH-LDV) to study large ensembles of catalytically active Janus particles moving under UV illumination. SH-LDV facilitates a model-free determination of the swimming speed and direction, with excellent ensemble averaging. In addition, we obtain information on the distribution of the catalytic activity. Moreover, SH-LDV operates away from walls and permits a facile correction for multiple scattering contributions. It thus allows for studies of concentrated suspensions of swimmers or of systems where swimmers propel actively in an environment crowded by passive particles. We demonstrate the versatility and the scope of the method with a few selected examples. We anticipate that SH-LDV complements established methods and paves the way for systematic measurements at previously inaccessible boundary conditions.

Run-and-tumble bacteria slowly approaching the diffusive regime.

The run-and-tumble (RT) dynamics followed by bacterial swimmers gives rise first to a ballistic motion due to their persistence and later, through consecutive tumbles, to a diffusive process. Here we investigate how long it takes for a dilute swimmer suspension to reach the diffusive regime as well as what is the amplitude of the deviations from the diffusive dynamics. A linear time dependence of the mean-squared displacement (MSD) is insufficient to characterize diffusion and thus we also focus on the excess kurtosis of the displacement distribution. Four swimming strategies are considered: (i) the conventional RT model with complete reorientation after tumbling; (ii) the case of partial reorientation, characterized by a distribution of tumbling angles; (iii) a run-and-reverse model with rotational diffusion; and (iv) a RT particle where the tumbling rate depends on the stochastic concentration of an internal protein. By analyzing the associated kinetic equations for the probability density function and simulating the models, we find that for models (ii), (iii), and (iv) the relaxation to diffusion can take much longer than the mean time between tumble events, evidencing the existence of large tails in the particle displacements. Moreover, the excess kurtosis can assume large positive values. In model (ii) it is possible for some distributions of tumbling angles that the MSD reaches a linear time dependence but, still, the dynamics remains non-Gaussian for long times. This is also the case in model (iii) for small rotational diffusivity. For all models, the long-time diffusion coefficients are also obtained. The theoretical approach, which relies on eigenvalue and angular Fourier expansions of the van Hove function, is in excellent agreement with the simulations.

Swimmer Suspensions on Substrates: Anomalous Stability and Long-Range Order.

We present a comprehensive theory of the dynamics and fluctuations of a two-dimensional suspension of polar active particles in an incompressible fluid confined to a substrate. We show that, depending on the sign of a single parameter, a state with polar orientational order is anomalously stable (or anomalously unstable), with a nonzero relaxation (or growth) rate for angular fluctuations, not parallel to the ordering direction, at zero wave number. This screening of the broken-symmetry mode in the stable state does lead to conventional rather than giant number fluctuations as argued by Bricard etal., Nature 503, 95 (2013), but their bend instability in a splay-stable flock does not exist and the polar phase has long-range order in two dimensions. Our theory also describes confined three-dimensional thin-film suspensions of active polar particles as well as dense compressible active polar rods, and predicts a flocking transition without a banding instability.

Self-organization of swimmers drives long-range fluid transport in bacterial colonies

Motile subpopulations in microbial communities are believed to be important for dispersal, quest for food, and material transport. Here, we show that motile cells in sessile colonies of peritrichously flagellated bacteria can self-organize into two adjacent, centimeter-scale motile rings surrounding the entire colony. The motile rings arise from spontaneous segregation of a homogeneous swimmer suspension that mimics a phase separation; the process is mediated by intercellular interactions and shear-induced depletion. As a result of this self-organization, cells drive fluid flows that circulate around the colony at a constant peak speed of ~30 µm s−1, providing a stable and high-speed avenue for directed material transport at the macroscopic scale. Our findings present a unique form of bacterial self-organization that influences population structure and material distribution in colonies.

Effective viscosity of a suspension of flagellar-beating microswimmers: Three-dimensional modeling.

Micro-organisms usually can swim in their liquid environment by flagellar or ciliary beating. In this numerical work, we analyze the influence of flagellar beating on the orbits of a swimming cell in a shear flow. We also calculate the effect of the flagellar beating on the rheology of a dilute suspension of microswimmers. A three-dimensional model is proposed for Chlamydomonas Reinhardtii swimming with a breaststroke-like beating of two anterior flagella modeled by two counter-rotating fore beads. The active swimmer model reveals unusual angular orbits in a linear shear flow. Namely, the swimmer sustains orientations transiently across the flow. Such behavior is a result of the interplay between shear flow and the swimmer's periodic beating motion of flagella, which exert internal torques on the cell body. This peculiar behavior has some significant consequences on the rheological properties of the suspension. We calculate Einstein's viscosity of the suspension composed of such isolated modeled microswimmers (dilute case) in a shear flow. We use numerical simulations based on a Rotne-Prager-like approximation for hydrodynamic interaction between simplified flagella and the cell body. The results show an increased intrinsic viscosity for active swimmer suspensions in comparison to nonactive ones as well as a shear thinning behavior in accordance with previous experimental measurements [Phys. Rev. Lett. 104, 098102 (2010)10.1103/PhysRevLett.104.098102].

Non-Gaussian limit fluctuations in active swimmer suspensions.

We investigate the hydrodynamic fluctuations in suspensions of swimming microorganisms (Chlamydomonas) by observing the probe particles dispersed in the media. Short-term fluctuations of probe particles were superdiffusive and displayed heavily tailed non-Gaussian distributions. The analytical theory that explains the observed distribution was derived by summing the power-law-decaying hydrodynamic interactions from spatially distributed field sources (here, swimming microorganisms). The summing procedure, which we refer to as the physical limit operation, is applicable to a variety of physical fluctuations to which the classical central limiting theory does not apply. Extending the analytical formula to compare to experiments in active swimmer suspensions, we show that the non-Gaussian shape of the observed distribution obeys the analytic theory concomitantly with independently determined parameters such as the strength of force generations and the concentration of Chlamydomonas. Time evolution of the distributions collapsed to a single master curve, except for their extreme tails, for which our theory presents a qualitative explanation. Investigations thereof and the complete agreement with theoretical predictions revealed broad applicability of the formula to dispersions of active sources of fluctuations.

Modeling of active swimmer suspensions and their interactions with the environment.

In this article, we review mathematical models used to study the behaviour of suspensions of micro-swimmers and the accompanying biophysical phenomena, with specific focus on stimulus response. The methods discussed encompass a range of interactions exhibited by the micro-swimmers; including passive hydrodynamic (gyrotaxis) and gravitational (gravitaxis) effects, and active responses to chemical cues (chemotaxis) and light intensities (phototaxis). We introduce the simplest models first, and then build towards more sophisticated recent developments, in the process, identifying the limitations of the former and the new results obtained by the latter. We comment on the accuracy/validity of the models adopted, based on the agreement between theoretical results and experimental observations. We conclude by identifying some of the open problems and associated challenges faced by researchers in the realm of active suspensions.

Collective motion in a suspension of micro-swimmers that run-and-tumble and rotary diffuse

Recent experiments have shown that suspensions of swimming micro-organisms are characterized by complex dynamics involving enhanced swimming speeds, large-scale correlated motions and enhanced diffusivities of embedded tracer particles. Understanding this dynamics is of fundamental interest and also has relevance to biological systems. The observed collective dynamics has been interpreted as the onset of a hydrodynamic instability, of the quiescent isotropic state of pushers, swimmers with extensile force dipoles, above a critical threshold proportional to the swimmer concentration. In this work, we develop a particle-based model to simulate a suspension of hydrodynamically interacting rod-like swimmers to estimate this threshold. Unlike earlier simulations, the velocity disturbance field due to each swimmer is specified in terms of the intrinsic swimmer stress alone, as per viscous slender-body theory. This allows for a computationally efficient kinematic simulation where the interaction law between swimmers is knowna priori. The neglect of induced stresses is of secondary importance since the aforementioned instability arises solely due to the intrinsic swimmer force dipoles.Our kinematic simulations include, for the first time, intrinsic decorrelation mechanisms found in bacteria, such as tumbling and rotary diffusion. To begin with, we simulate so-called straight swimmers that lack intrinsic orientation decorrelation mechanisms, and a comparison with earlier results serves as a proof of principle. Next, we simulate suspensions of swimmers that tumble and undergo rotary diffusion, as a function of the swimmer number density$(n)$, and the intrinsic decorrelation time (the average duration between tumbles,${\it\tau}$, for tumblers, and the inverse of the rotary diffusivity,$D_{r}^{-1}$, for rotary diffusers). The simulations, as a function of the decorrelation time, are carried out with hydrodynamic interactions (between swimmers) turned off and on, and for both pushers and pullers (swimmers with contractile force dipoles). The ‘interactions-off’ simulations allow for a validation based on analytical expressions for the tracer diffusivity in the stable regime, and reveal a non-trivial box size dependence that arises with varying strength of the hydrodynamic interactions. The ‘interactions-on’ simulations lead us to our main finding: the existence of a box-size-independent parameter that characterizes the onset of instability in a pusher suspension, and is given by$nUL^{2}{\it\tau}$for tumblers and$nUL^{2}/D_{r}$for rotary diffusers; here,$U$and$L$are the swimming speed and swimmer length, respectively. The instability manifests as a bifurcation of the tracer diffusivity curves, in pusher and puller suspensions, for values of the above dimensionless parameters exceeding a critical threshold.

Spontaneous Flows in Suspensions of Active Cyclic Swimmers

Many swimming cells rely on periodic deformations to achieve locomotion. We introduce in this work a theoretical model and numerical simulations in order to elucidate the impact of these cyclic strokes on the emergence of mesoscale structures and collective motion in swimmer suspensions. The model extends previous kinetic theories for populations of identical swimmers to the case of self-propelled particles undergoing transitions between pusher and puller states, and is applied to quantify how the unsteadiness of the hydrodynamic velocity field, to which each swimmer population contributes, affects the onset and characteristics of spontaneous flows. A linear stability analysis reveals that the sign of the population-averaged dipole determines the stability of the uniform isotropic state, with suspensions dominated by pushers being subject to growing nematic bend fluctuations. Stochastic transitions, however, are also seen to provide an additional damping mechanism. To investigate the population dynamics above the instability threshold, we also perform direct particle simulations based on a slender-body model, where the growth or decay of the active power generated by the swimmers is found to be a robust measure of the structural and dynamical instability.

An introduction to the hydrodynamics of swimming microorganisms

This manuscript is a summary of a set of lectures given at the Geilo School 2013 Soft Matter Confinement: from Biology to Physics. It aims to provide an introduction to the hydrodynamics that underlies the way in which microorganisms, such as bacteria and algae, and fabricated microswimmers, swim. We focus on two features peculiar to bacterial swimming: the Scallop theorem and the dipolar nature of the far flow field. We discuss the consequences of these to the velocity field of a swimmer suspension and to the motion of passive tracers as a bacterium swims past.

Casimir effect in swimmer suspensions.

We show that the Casimir effect can emerge in microswimmer suspensions. In principle, two effects conspire against the development of Casimir effects in swimmer suspensions. First, at low Reynolds number, the force on any closed volume vanishes, but here the relevant effect is the drag by the flow produced by the swimmers, which can be finite. Second, the fluid velocity and the pressure are linear on the swimmer force dipoles, and averaging over the swimmer orientations would lead to a vanishing effect. However, being that the suspension is a discrete system, the noise terms of the coarse-grained equations depend on the density, which itself fluctuates, resulting in effective nonlinear dynamics. Applying the tools developed for other nonequilibrium systems to general coarse-grained equations for swimmer suspensions, the Casimir drag is computed on immersed objects, and it is found to depend on the correlation function between the rescaled density and dipolar density fields. By introducing a model correlation function with medium-range order, explicit expressions are obtained for the Casimir drag on a body. When the correlation length is much larger than the microscopic cutoff, the average drag is independent of the correlation length, with a range that depends only on the size of the immersed bodies.

Diffusion of colloidal particles in swimming suspensions

Previously, we have proposed to analyse the hydrodynamic interactions in a suspension of swimmers with respect to an effective hydrodynamic diffusion coefficient, which only considers the fluctuating motion caused by the stirring of the fluid. In this work, we study the diffusion of colloidal particles immersed in a bath of swimmers. To accurately resolve the many-body hydrodynamic interactions responsible for this diffusion, we use a direct numerical simulation scheme based on the smooth profile method. We consider a squirmer model for the self-propelled swimmers, as it accurately reproduces the flow field generated by real microorganisms, such as bacteria or spermatozoa. We show that the diffusion coefficients of the colloids are comparable with the effective diffusion coefficients of the swimmers, provided that the concentration of swimmers is high enough. At low concentrations, the difference in the way colloids and swimmers react to the flow leads to a reduction in the diffusion coefficient of the colloids. This is clearly seen in the appearance of a negative-correlation region for the velocity-correlation function of the colloids, which does not exist for the swimmers.

Active temperature and velocity correlations produced by a swimmer suspension

The agitation produced in a fluid by a suspension of microswimmers in the low Reynolds number limit is studied. In this limit, swimmers are modeled as force dipoles all with equal strength. The agitation is characterized by the active temperature defined, as in kinetic theory, as the mean square velocity, and by the equal-time spatial correlations. Considering the phase in which the swimmers are homogeneously and isotropically distributed in the fluid, it is shown that the active temperature and velocity correlations depend on a single scalar correlation function of the dipole-dipole correlation function. By making a simple medium-range order model, in which the dipole-dipole correlation function is characterized by a single correlation length k(0)(-1) it is possible to make quantitative predictions. It is found that the active temperature depends on the system size, scaling as L(4-d) at large correlation lengths L<<k(0)(-1), while in the opposite limit it saturates in three dimensions and diverges logarithmically with the system size in two dimensions. In three dimensions the velocity correlations decay as 1/r for small correlation lengths, while at large correlation lengths the transverse correlation function becomes negative at maximum separation r~L/2, an effect that disappears as the system increases in size.

The structure and rheology of sheared model swimmer suspensions

We analyze the rheological response of a suspension of simple model swimmers subject to a steady Couette flow. We consider the squirmer model as a means to control systematically the interplay between self-propulsion and active stress generation and analyze their relative impact both on the effective viscosity of a suspension and the microstructure the squirmers develop. We show how self-propulsion introduces an intrinsic contribution to the effective viscosity of the active suspension. Accordingly, apolar squirmers show shear thickening while polar ones develop a shear thinning response to the applied shear. We show that the detailed coupling of the squirmers to the bounding walls has a strong influence on the structure of the suspension and the development of shear bands.

Critical bacterial concentration for the onset of collective swimming

We examine the stability of a suspension of swimming bacteria in a Newtonian medium. The bacteria execute a run-and-tumble motion, runs being periods when a bacterium on average swims in a given direction; runs are interrupted by tumbles, leading to an abrupt, albeit correlated, change in the swimming direction. An instability is predicted to occur in a suspension of ‘pushers’ (e.g.E. Coli,Bacillus subtilis, etc.), and owes its origin to the intrinsic force dipoles of such bacteria. Unlike the dipole induced in an inextensible fibre subject to an axial straining flow, the forces constituting the dipole of a pusher are directed outward along its axis. As a result, the anisotropy in the orientation distribution of bacteria due to an imposed velocity perturbation drives a disturbance velocity field that acts to reinforce the perturbation. For long wavelengths, the resulting destabilizing bacterial stress is Newtonian but with a negative viscosity. The suspension becomes unstable when the total viscosity becomes negative. In the dilute limit (nL3≪ 1), a linear stability analysis gives the threshold concentration for instability as (nL3)crit= ((30/Cℱ(r))(DrL/U)(1 + 1/(6τDr)))/(1−(15𝒢(r)/Cℱ(r))(DrL/U)(1 + 1/(6τDr))) for perfectly random tumbles; here,LandUare the length and swimming velocity of a bacterium,nis the bacterial number density,Drcharacterizes the rotary diffusion during a run and τ−1is the average tumbling frequency. The function ℱ(r) characterizes the rotation of a bacterium of aspect ratiorin an imposed linear flow; ℱ(r) = (r2−1)/(r2+ 1) for a spheroid, and ℱ(r) ≈ 1 for a slender bacterium (r≫ 1). The function 𝒢(r) characterizes the stabilizing viscous response arising from the resistance of a bacterium to a deforming ambient flow; 𝒢(r) = 5π/6 for a rigid spherical bacterium, and 𝒢(r)≈ π/45(lnr) for a slender bacterium. Finally, the constantCdenotes the dimensionless strength of the bacterial force dipole in units of μU L2; forE. Coli,C≈ 0.57. The threshold concentration diverges in the limit ((15𝒢(r)/Cℱ(r)) (DrL/U)(1 + 1/(6τDr))) → 1. This limit defines a critical swimming speed,Ucrit= (DrL)(15𝒢(r)/Cℱ(r))(1 + 1/(6τDr)). For speeds smaller than this critical value, the destabilizing bacterial stress remains subdominant and a dilute suspension of these swimmers therefore responds to long-wavelength perturbations in a manner similar to a suspension of passive rigid particles, that is, with a net enhancement in viscosity proportional to the bacterial concentration.On the other hand, the stability analysis predicts that the above threshold concentration reduces to zero in the limitDr→ 0, τ → ∞, and a suspension of non-interacting straight swimmers is therefore always unstable. It is then argued that the dominant effect of hydrodynamic interactions in a dilute suspension of such swimmers is via an interaction-driven orientation decorrelation mechanism. The latter arises from uncorrelated pair interactions in the limitnL3≪ 1, and for slender bacteria in particular, it takes the form of a hydrodynamic rotary diffusivity (Dhr); forE. Coli, we findDhr= 9.4 × 10−5(nUL2). From the above expression for the threshold concentration, it may be shown that even a weakly interacting suspension of slender smooth-swimming bacteria (r≫ 1, ℱ(r) ≈ 1, τ → ∞) will be stable providedDhr&gt; (C/30)(nUL2) in the limitnL3≪ 1. The hydrodynamic rotary diffusivity ofE. Coliis, however, too small to stabilize a dilute suspension of these swimmers, and a weakly interacting suspension ofE. Coliremains unstable.

Dynamics of confined suspensions of swimming particles

Low Reynolds number direct simulations of large populations of hydrodynamicallyinteracting swimming particles confined between planar walls are performed. The results ofsimulations are compared with a theory that describes dilute suspensions of swimmers. Thetheory yields scalings with concentration for diffusivities and velocity fluctuations as well asa prediction of the fluid velocity spatial autocorrelation function. Even for uncorrelatedswimmers, the theory predicts anticorrelations between nearby fluid elements thatcorrespond to vortex-like swirling motions in the fluid with length scale set by the size of aswimmer and the slit height. Very similar results arise from the full simulationsindicating either that correlated motion of the swimmers is not significant at theconcentrations considered or that the fluid phase autocorrelation is not a sensitivemeasure of the correlated motion. This result is in stark contrast with resultsfrom unconfined systems, for which the fluid autocorrelation captures large-scalecollective fluid structures. The additional length scale (screening length) introducedby the confinement seems to prevent these large-scale structures from forming.