Exact solutions for hydrodynamic interactions of two squirming spheres

We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head-tail reversal of the swimmer, in the near field microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest two squirming modes. By comparing these to asymptotic approximations of the interaction we find that the transition from near- to far-field behaviour occurs at a separation of about two swimmer diameters. These solutions are for the rotational velocity about the wall normal, or common diameter of two spheres, and the translational speed along that same direction, and are obtained using the Lorentz reciprocal theorem for Stokes flows in conjunction with known solutions for the conjugate Stokes drag problems, the derivations of which are demonstrated here for completeness. The analogous motions in the perpendicular directions, i.e. parallel to the wall, currently cannot be calculated exactly since the relevant Stokes drag solutions needed for the reciprocal theorem are not available.Comment: 27 pages, 7 figure

Particles held by springs in a linear shear flow exhibit oscillatory motion

The dynamics of small spheres, which are held by linear springs in a low Reynolds number shear flow at neighboring locations is investigated. The flow elongates the beads and the interplay of the shear gradient with the nonlinear behavior of the hydrodynamic interaction among the spheres causes in a large range of parameters a bifurcation to a surprising oscillatory bead motion. The parameter ranges, wherein this bifurcation is either super- or subcritical, are determined.Comment: 4 pages, 5 figure

Partial synchronisation of stochastic oscillators through hydrodynamic coupling

Holographic optical tweezers are used to construct a static bistable optical potential energy landscape where a Brownian particle experiences restoring forces from two nearby optical traps and undergoes thermally activated transitions between the two energy minima. Hydrodynamic coupling between two such systems results in their partial synchronisation. This is interpreted as an emergence of higher mobility pathways, along which it is easier to overcome barriers to structural rearrangement.Comment: 4 pages, 4 figures, submitted to Phys. Rev. Let

Brownian motion in a non-homogeneous force field and photonic force microscope

The Photonic Force Microscope (PFM) is an opto-mechanical technique based on an optical trap that can be assumed to probe forces in microscopic systems. This technique has been used to measure forces in the range of pico- and femto-Newton, assessing the mechanical properties of biomolecules as well as of other microscopic systems. For a correct use of the PFM, the force field to measure has to be invariable (homogeneous) on the scale of the Brownian motion of the trapped probe. This condition implicates that the force field must be conservative, excluding the possibility of a rotational component. However, there are cases where these assumptions are not fulfilled Here, we show how to improve the PFM technique in order to be able to deal with these cases. We introduce the theory of this enhanced PFM and we propose a concrete analysis workflow to reconstruct the force field from the experimental time-series of the probe position. Furthermore, we experimentally verify some particularly important cases, namely the case of a conservative or rotational force-field

Hydrodynamics of flagellated microswimmers near free-slip interfaces

The hydrodynamics of a flagellated microorganism is investigated when swimming close to a planar free-slip surface by means of numerical solu- tions of the Stokes equations obtained via a Boundary Element Method. Depending on the initial condition, the swimmer can either escape from the free-slip surface or collide with the boundary. Interestingly, the mi- croorganism does not exhibit a stable orbit. Independently of escape or attraction to the interface, close to a free-slip surface, the swimmer fol- lows a counter-clockwise trajectory, in agreement with experimental find- ings, [15]. The hydrodynamics is indeed modified by the free-surface. In fact, when the same swimmer moves close to a no-slip wall, a set of initial conditions exists which result in stable orbits. Moreover when moving close to a free-slip or a no-slip boundary the swimmer assumes a different orientation with respect to its trajectory. Taken together, these results contribute to shed light on the hydrodynamical behaviour of microorgan- isms close to liquid-air interfaces which are relevant for the formation of interfacial biofilms of aerobic bacteria

Instabilities and turbulence-like dynamics in an oppositely driven binary particle mixture

Using extensive particle-based simulations, we investigate out-of-equilibrium pattern dynamics in an oppositely driven binary particle system in two dimensions. A surprisingly rich dynamical behavior including lane formation, jamming, oscillation and turbulence-like dynamics is found. The ratio of two friction coefficients is a key parameter governing the stability of lane formation. When the friction coefficient transverse to the external force direction is sufficiently small compared to the longitudinal one, the lane structure becomes unstable to shear-induced disturbances, and the system eventually exhibits a dynamical transition into a novel turbulence-like phase characterized by random convective flows. We numerically construct an out-of-equilibrium phase diagram. Statistical analysis of complex spatio-temporal dynamics of the fully nonlinear turbulence-like phase suggests its apparent reminiscence to the swarming dynamics in certain active matter systems.Comment: 6 pages, 6 figures, accepted for publication in EP

Influence of Noise on Force Measurements

We demonstrate how the ineluctable presence of thermal noise alters the measurement of forces acting on microscopic and nanoscopic objects. We quantify this effect exemplarily for a Brownian particle near a wall subjected to gravitational and electrostatic forces. Our results demonstrate that the force measurement process is prone to artifacts if the noise is not correctly taken into account.Comment: 4 Pages, 4 Figures, Accepte

Colloid-colloid and colloid-wall interactions in driven suspensions

We investigate the non-equilibrium fluid structure mediated forces between two colloids driven through a suspension of mutually non-interacting Brownian particles as well as between a colloid and a wall in stationary situations. We solve the Smoluchowski equation in bispherical coordinates as well as with a method of reflections, both in linear approximation for small velocities and numerically for intermediate velocities, and we compare the results to a superposition approximation considered previously. In particular we find an enhancement of the friction (compared to the friction on an isolated particle) for two colloids driven side by side as well as for a colloid traveling along a wall. The friction on tailgating colloids is reduced. Colloids traveling side by side experience a solute induced repulsion while tailgating colloids are attracted to each other.Comment: 8 Pages, 8 figure

The short-time self-diffusion coefficient of a sphere in a suspension of rigid rods

The short--time self diffusion coefficient of a sphere in a suspension of rigid rods is calculated in first order in the rod volume fraction. For low rod concentrations the correction to the Einstein diffusion constant of the sphere is a linear function of the rod volume fraction with the slope proportional to the equilibrium averaged mobility diminution trace of the sphere interacting with a single freely translating and rotating rod. The two--body hydrodynamic interactions are calculated using the so--called bead model in which the rod is replaced by a stiff linear chain of touching spheres. The interactions between spheres are calculated numerically using the multipole method. Also an analytical expression for the diffusion coefficient as a function of the rod aspect ratio is derived in the limit of very long rods. We show that in this limit the correction to the Einstein diffusion constant does not depend on the size of the tracer sphere. The higher order corrections depending on the applied model are computed numerically. An approximate expression is provided, valid for a wide range of aspect ratios.Comment: 11 pages, 6 figure