Search

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 (Sokolov and Aranson (2016) "Rapid expulsion of microswimmers by a vortical flow." Nature Comm. 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 of Sokolov and Aranson (2016).

Control on microscopic scales depends critically on our ability to manipulate interactions with different physical fields. The creation of micro-machines therefore requires us to understand how multiple fields, such as surface capillary or electro-magnetic fields, can be used to produce predictable behaviour. Recently, a spinning micro-raft system was developed that exhibited both static and dynamic self-assembly [Wang et al., Sci. Adv., 2017, 3, e1602522]. These rafts employed both capillary and magnetic interactions and, at a critical driving frequency, would suddenly change from stable orbital patterns to static assembled structures. In this paper, we explain the dynamics of two interacting micro-rafts through a combination of theoretical models and experiments. This is first achieved by identifying the governing physics of the orbital patterns, the assembled structures, and the collapse separately. We find that the orbital patterns are determined by the short range capillary interactions between the disks, while the explanations of the other two behaviours only require the capillary far field. Finally we combine the three models to explain the dynamics of a new micro-raft experiment., European Union (European Union); European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme; Max Planck Society; Alexander von Humboldt Foundation

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., A mathematical model explains the experimental accumulation in corners of Chlamydomonas algae swimming in 2D models of foam channels. Combining experiments with theory shows that motility leads generically to trapping in complex geometries.

The rotary motor of bacteria is a natural nano-technological marvel that enables cell locomotion by powering the rotation of semi-rigid helical flagellar filaments in fluid environments. It is well known that the motor operates essentially at constant torque in counter-clockwise direction but past work have reported a large range of values of this torque. Focusing on Escherichia coli cells that are swimming and cells that are stuck on a glass surface for which all geometrical and environmental parameters are known (N. C. Darnton et al., J. Bacteriol., 2007, 189, 1756-1764), we use two validated numerical methods to compute the value of the motor torque consistent with experiments. Specifically, we use (and compare) a numerical method based on the boundary integral representation of Stokes flow and also develop a hybrid method combining boundary element and slender body theory to model the cell body and flagellar filament, respectively. Using measured rotation speed of the motor, our computations predict a value of the motor torque in the range 440 pN nm to 829 pN nm, depending critically on the distance between the flagellar filaments and the nearby surface.

A colloidal particle driven by externally actuated rotation can self-propel parallel to a rigid boundary by exploiting the hydrodynamic coupling that surfaces induce between translation and rotation. As such a roller moves along the boundary it generates local vortical flows, which can be used to trap and transport passive cargo particles. However, the details and conditions for this trapping mechanism have not yet been fully understood. Here, we show that the trapping of cargo is accomplished through time-irreversible interactions between the cargo and the boundary, leading to its migration across streamlines into a steady flow vortex next to the roller. The trapping mechanism is explained analytically with a two dimensional model, investigated numerically in three dimensions for a wide range of parameters and is shown to be analogous to the deterministic lateral displacement (DLD) technique used in microfluidics for the separation of differently sized particles. The several geometrical parameters of the problem are analysed and we predict that thin, disc-like rollers offer the most favourable trapping conditions.

Experiments involving the two-dimensional passive diffusion of colloidal boomerangs tracked off their centre of mobility have shown striking non-Gaussian tails in their probability distribution function [Chakrabarty et al., Soft Matter 12, 4318 (2016)]. This in turn can lead to anomalous diffusion characteristics, including mean drift. In this paper, we develop a general theoretical explanation for these measurements. The idea relies on calculating the two-dimensional probability densities at the centre of mobility of the particle, where all distributions are Gaussian, and then transforming them to a different reference point. Our model clearly captures the experimental results, without any fitting parameters, and demonstrates that the one-dimensional probability distributions may also exhibit strongly non-Gaussian tops. These results indicate that the choice of tracking point can cause a considerable departure from Gaussian statistics, potentially causing some common modelling techniques to fail., 7 pages, 4 figures, submitted. LK and ML contributed equally

Numerical study of the hydrodynamics of thin sheets and ribbons presents difficulties associated with resolving multiple length scales. To circumvent these difficulties, asymptotic methods have been developed to describe the dynamics of slender fibres and ribbons. However, such theories entail restrictions on the shapes that can be studied, and often break down in regions where standard boundary element methods are still impractical. In this paper we develop a regularised stokeslet method for ribbons and sheets in order to bridge the gap between asymptotic and boundary element methods. The method is validated against the analytical solution for plate ellipsoids, as well as the dynamics of ribbon helices and an experimental microswimmer. We then demonstrate the versatility of this method by calculating the flow around a double helix, and the swimming dynamics of a microscale "magic carpet".

In this paper, we study the dynamic response of 1D microfluidic-droplet streams to finite-amplitude longitudinal perturbations and demonstrate experimentally that the excitation of localized jams results in the propagation of shock waves. The shock velocity is shown to vanish as the average particle density approaches a critical value thereby leaving long-lived disturbance in the spatial organization of the streams. Using a gradient expansion of the hydrodynamic coupling between the advected particles, we then theoretically derive the non-linear constitutive equation relating particle current to particle density, and show that it leads to the Burgers equation for the droplet stream density.

Motivated by recent advances in vesicle engineering, we consider theoretically the locomotion of shape-changing bilayer vesicles at low Reynolds number. By modulating their volume and membrane composition, the vesicles can be made to change shape quasi-statically in thermal equilibrium. When the control parameters are tuned appropriately to yield periodic shape changes which are not time-reversible, the result is a net swimming motion over one cycle of shape deformation. For two classical vesicle models (spontaneous curvature and bilayer coupling), we determine numerically the sequence of vesicle shapes through an enthalpy minimization, as well as the fluid-body interactions by solving a boundary integral formulation of the Stokes equations. For both models, net locomotion can be obtained either by continuously modulating fore-aft asymmetric vesicle shapes, or by crossing a continuous shape-transition region and alternating between fore-aft asymmetric and fore-aft symmetric shapes. The obtained hydrodynamic efficiencies are similar to that of other low Reynolds number biological swimmers, and suggest that shape-changing vesicles might provide an alternative to flagella-based synthetic microswimmers.

Janus particles with the ability to move phoretically in self-generated chemical concentration gradients are model systems for active matter. Their motion typically consists of straight paths with rotational diffusion being the dominant reorientation mechanism. In this paper, we show theoretically that by a suitable surface coverage of both activity and mobility, translational and rotational motion can be induced arbitrarily in three dimensions. The resulting trajectories are in general helical, and their pitch and radius can be controlled by adjusting the angle between the translational and angular velocity. Building on the classical mathematical framework for axisymmetric self-phoretic motion under fixed-flux chemical boundary condition, we first show how to calculate the most general three-dimensional motion for an arbitrary surface coverage of a spherical particle. After illustrating our results on surface distributions, we next introduce a simple intuitive patch model to serve as a guide for designing arbitrary phoretic spheres., Comment: 12 pages, 5 figures, submitted for publication in Soft Matter

Phoretic mechanisms, whereby gradients of chemical solutes induce surface-driven flows, have recently been used to generate directed propulsion of patterned colloidal particles. When the chemical solutes diffuse slowly, an instability further provides active but isotropic particles with a route to self-propulsion by spontaneously breaking the symmetry of the solute distribution. Here we show theoretically that, in a mechanism analogous to B\'enard-Marangoni convection, phoretic phenomena can create spontaneous and self-sustained wall-driven mixing flows within a straight, chemically-uniform active channel. Such spontaneous flows do not result in any net pumping for a uniform channel but greatly modify the distribution of transport of the chemical solute. The instability is predicted to occur for a solute P\'eclet number above a critical value and for a band of finite perturbation wavenumbers. We solve the perturbation problem analytically to characterize the instability, and use both steady and unsteady numerical computations of the full nonlinear transport problem to capture the long-time coupled dynamics of the solute and flow within the channel., Comment: 15 pages, 11 figures, to appear in Soft Matter