We analyse a simple 'Stokesian squirmer' model for the enhanced mixing due to swimming micro-organisms. The model is based on a calculation of Thiffeault & Childress [Physics Letters A, 374, 3487 (2010), arXiv:0911.5511], where fluid particle displacements due to inviscid swimmers are added to produce an effective diffusivity. Here we show that for the viscous case the swimmers cannot be assumed to swim an infinite distance, even though their total mass displacement is finite. Instead, the largest contributions to particle displacement, and hence to mixing, arise from random changes of direction of swimming and are dominated by the far-field stresslet term in our simple model. We validate the results by numerical simulation. We also calculate nonzero Reynolds number corrections to the effective diffusivity. Finally, we show that displacements due to randomly-swimming squirmers exhibit PDFs with exponential tails and a short-time superdiffusive regime, as found previously by several authors. In our case, the exponential tails are due to 'sticking' near the stagnation points on the squirmer's surface.
In addition to enabling movement towards environments with favourable living conditions, swimming by microorganisms has also been linked to enhanced mixing and improved nutrient uptake by their populations. Experimental studies have shown that Brownian tracer particles exhibit enhanced diffusion due to the swimmers, while theoretical models have linked this increase in diffusion to the flows generated by the swimming microorganisms, as well as collisions with the swimmers. In this study, we perform detailed simulations based on the force-coupling method and its recent extensions to the swimmin…
Many microorganisms take a chiral path while swimming in an ambient uid. In this paper, we study the combined behavior of two chiral swimmers using the well-known squirmer model taking into account chiral asymmetries. In contrast to the simple squirmer model, which has an axisymmetric distribution of slip velocity, the chiral squirmer has additional asymmetries in the surface slip, which contribute to both translations and rotations of the motion. As a result, swimming trajectories can become helical and chiral asymmetries arise in the ow patterns. We study the swimming trajectories of a pair …
We present a two dimensional model of hydrodynamic interaction between a circular swimmer and a circular post at low Reynolds number, using a point singularity description of the swimming activity. We derive a nonlinear dynamical system fully describing the motion and discuss the generic features of the phase portrait and typical trajectories for a variety of squirmer modes. Contractile swimmers exhibit stable bound orbits arising from the contrasting nature of monopolar and dipolar squirmer modes, which are robust with respect to swimmer size and the inclusion of higher squirmer modes. The be…
Three-dimensional simulations with fully resolved hydrodynamics are performed to study the dynamics of a single squirmer under gravity, in order to clarify its motion in the vicinity of a flat plate. Different dynamics emerge for different gravity strengths. In a moderate gravity regime, neutral squirmers and pullers eventually stop moving and reorient in a direction perpendicular to the plate; pushers, instead, exhibit continuous motion in a tilted direction. In the strong gravity regime, all types of squirmers sediment and reorient perpendicularly to the plate. In this study, the chirality i…
We consider the tracer diffusion $D_{rr}$ that arises from the run-and-tumble motion of low Reynolds number swimmers, such as bacteria. In unbounded dilute suspensions, where the dipole swimmers move in uncorrelated runs of length $λ$, an exact solution showed that $D_{rr}$ is independent of $λ$. Here we verify this result in numerical simulations for a particular model swimmer, the spherical squirmer. We also note that in confined microenvironments, such as microscopic droplets, microfluidic devices and bacterial microzones in marine ecosystems, the size of the system can be comparable to $λ$…
We run numerical simulations of strongly confined suspensions of model spherical swimmers called "squirmers". Because of the confinement, the Stokeslet dipoles generated by the particles are quickly screened and the far-field flow is dominated by the source dipole for all the different kinds of squirmers. However, we show that the collective behavior of the suspension still depends on the self-propelling mechanism of the swimmers as polar states can only be observed for neutral squirmers. We demonstrate that the near-field hydrodynamic interactions play a crucial role in the alignment of the o…
The behaviour of microscopic swimmers has previously been explored near large scale confining geometries and in the presence of very small-scale surface roughness. Here we consider an intermediate case of how a simple microswimmer, the tangential spherical squirmer, behaves adjacent to singly- and doubly-periodic sinusoidal surface topographies that spatially oscillate with an amplitude that is an order of magnitude less than the swimmer size and wavelengths within an order of magnitude of this scale. The nearest neighbour regularised Stokeslet method is used for numerical explorations after v…
The geometric phase techniques for swimming in viscous flows express the net displacement of a swimmer as a path integral of a field in configuration space. This representation can be transformed into an area integral for simple swimmers using Stokes theorem. Since this transformation applies for any loop, the integrand of this area integral can be used to help design these swimmers. However, the extension of this Stokes theorem technique to more complicated swimmers is hampered by problems with variables that do not commute and by how to visualise and understand the higher dimensional spaces.…
The ciliary locomotion and feeding of an axisymmetric micro-swimmer in a complex fluid whose viscosity depends on nutrient concentration are investigated numerically. The micro-swimmer is modeled as having spheroidal geometry, and ciliary beating is modeled by a slip velocity; i.e. a squirmer is adapted. Looking at the coupling between swimming and feeding of spheroidal squirmers, it is found that swimming speed and feeding are most affected by a non-uniform viscosity environment when the ratio of advection-forces to diffusion-transport, characterized by the nondimensional Peclét number, is mo…
We present a new development of the force-coupling method (FCM) to address the accurate simulation of a large number of interacting micro-swimmers. Our approach is based on the squirmer model, which we adapt to the FCM framework, resulting in a method that is suitable for simulating semi-dilute squirmer suspensions. Other effects, such as steric interactions, are considered with our model. We test our method by comparing the velocity field around a single squirmer and the pairwise interactions between two squirmers with exact solutions to the Stokes equations and results given by other numeric…
An axisymmetric squirmer in a Bingham viscoplastic fluid is studied numerically to determine the effect of a yield stress environment on locomotion. The nonlinearity of the governing equations necessitates numerical methods, which is accomplished by solving a variable-viscosity Stokes equation with a Finite Element approach. The effects of stroke modes, both pure and combined, are investigated and it is found that for the treadmill or "neutral" mode, the swimmer in a yield stress fluid has a lower swimming velocity and uses more power. However, the efficiency of swimming reaches its maximum at…
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…
We present a study of the hydrodynamics of an active particle, a model squirmer, in an envi- ronment with a broken rotational symmetry: a nematic liquid crystal. By combining simulations with analytic calculations, we show that the hydrodynamic coupling between the squirmer flow field and liquid crystalline director can lead to re-orientation of the swimmers. The preferred orientation depends on the exact details of the squirmer flow field. In a steady state, pushers are shown to swim parallel with the nematic director while pullers swim perpendicular to the nematic director. This behaviour ar…
The interaction between swimming microorganisms or artificial self-propelled colloids and passive (tracer) particles in a fluid leads to enhanced diffusion of the tracers. This enhancement has attracted strong interest, as it could lead to new strategies to tackle the difficult problem of mixing on a microfluidic scale. Most of the theoretical work on this topic has focused on hydrodynamic interactions between the tracers and swimmers in a bulk fluid. However, in simulations, periodic boundary conditions (PBCs) are often imposed on the sample and the fluid. Here, we theoretically analyze the e…
Using lattice Boltzmann simulations we study the hydrodynamics of an active spherical particle near a no-slip wall. We develop a computational model for an active Janus particle, by considering different and independent mobilities on the two hemispheres and compare the behaviour to a standard squirmer model.We show that the topology of the far-field hydrodynamic nature of the active Janus particle is similar to the standard squirmer model, but in the near-field the hydrodynamics differ. In order to study how the near-field effects affect the interaction between the particle and a flat wall, we…
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Brandeis University（布兰迪斯大学）；Center for Systems Biology Dresden（德累斯顿系统生物学中心）；TU Dresden（德累斯顿工业大学）
We consider the stirring of an inviscid fluid caused by the locomotion of bodies through it. The swimmers are approximated by non-interacting cylinders or spheres moving steadily along straight lines. We find the displacement of fluid particles caused by the nearby passage of a swimmer as a function of an impact parameter. We use this to compute the effective diffusion coefficient from the random walk of a fluid particle under the influence of a distribution of swimming bodies. We compare with the results of simulations. For typical sizes, densities and swimming velocities of schools of krill,…
In several biologically relevant situations, cell locomotion occurs in polymeric fluids with Weissenberg {number} larger than one. Here we present results of three-dimensional numerical simulations for the steady locomotion of a self-propelled body in a model polymeric (Giesekus) fluid at low Reynolds number. Locomotion is driven by steady tangential deformation at the surface of the body (so-called squirming motion). In the case of a spherical squirmer, we show that the swimming velocity is systematically less than that in a Newtonian fluid, with a minimum occurring for Weissenberg numbers of…
Mixing induced by microswimmers as probed by mutual information
Max Planck Institute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）；Institute of Science and Technology Austria（奥地利科学技术研究所）；Department of Mathematics, Kyoto University（京都大学数学系）；Jožef Stefan Institute（约瑟夫·斯特凡研究所）；Rudolf Peierls Centre for Theoretical Physics, University of Oxford（牛津大学鲁道夫·皮尔斯理论物理中心）
Micro-organisms expend energy moving through complex media. While propulsion speed is an important property of locomotion, efficiency is another factor that may determine the swimming gait adopted by a micro-organism in order to locomote in an energetically favorable manner. The efficiency of swimming in a Newtonian fluid is well characterized for different biological and artificial swimmers. However, these swimmers often encounter biological fluids displaying shear-thinning viscosities. Little is known about how this nonlinear rheology influences the efficiency of locomotion. Does the shear-t…
The experiments of Leptos et al. [Phys. Rev. Lett. 103, 198103 (2009)] show that the displacements of small particles affected by swimming microorganisms achieve a non-Gaussian distribution, which nevertheless scales diffusively -- the 'diffusive scaling.' We use a simple model where the particles undergo repeated 'kicks' due to the swimmers to explain the shape of the distribution as a function of the volume fraction of swimmers. The net displacement is determined by the inverse Fourier transform of a single-swimmer characteristic function. The only adjustable parameter is the strength of the…
We consider the enhanced mixing due to multiple cylinders organised in schools moving synchronously in a potential flow. Here simple interactions between cylinders are modelled by the method of image doublets. This is an extension to Thiffeault \& Childress's work [\emph{Physics Letters A} \textbf{374}, 3487 (2010)] where fluid particle displacements due to non-interacting swimmers were analysed to produce an effective diffusivity that may have a significant impact in ocean mixing. Our results show that schools of two cylinders induce nonlinearly boosted diffusivity compared with the non-inter…
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…
