Abstract. Some types of bacteria use rotating helical flagella to swim. The motion of such organisms takes place in the regime of low Reynolds numbers where viscous effects dominate and where the dynamics is governed by hydrodynamic interactions. Typically, rotating flagella form bundles, which means that their rotation is synchronized. The aim of this study is to investigate whether hydrodynamic interactions can be at the origin of such a bundling and synchronization. We consider two stiff helices that are modelled by rigidly connected beads, neglecting any elastic deformations. They are driven by constant and equal torques, and they are fixed in space by anchoring their terminal beads in harmonic traps. We observe that, for finite trap strength, hydrodynamic interactions do indeed synchronize the helix rotations. The speed of phase synchronization decreases with increasing trap stiffness. In the limit of infinite trap stiffness, the speed is zero and the helices do not synchronize.

We theoretically study in detail the hydrodynamic coupling of two equal-sized colloidal spheres at low Reynolds numbers assuming the particles to be harmonically trapped with respect to both their positions and orientations. By taking into account the rotational motion, we obtain a rich spectrum of collective eigenmodes whose properties we determine on the basis of pure symmetry arguments. Extending recent investigations on translational correlations [J.-C. Meiners and S. R. Quake, Phys. Rev. Lett. 82, 2211 (1999)], we derive the complete set of autocorrelation and cross-correlation functions emphasizing the coupling of rotation to translation which we illustrate in a few examples. An important feature of our system is the self-coupling of translation and rotation of one particle mediated by the neighboring particle that is clearly visible in the appropriate autocorrelation function. This coupling is a higher-order effect and therefore not included in the widely used Rotne-Prager approximation for the hydrodynamic mobilities.

Particles suspended in a viscous fluid circle in optical vortices generated by holographic optical-tweezer techniques [Curtis J E and Grier D G 2003 Phys. Rev. Lett. 90 133901]. We model this system and show that hydrodynamic interactions between the circling particles determine their collective motion. We perform a linear-stability analysis to investigate the stability of regular particle clusters and illustrate the limit cycle to which the unstable modes converge. We clarify that drafting of particle doublets is essential for the understanding of the limit cycle.

We experimentally and theoretically investigate the collective behavior of three colloidal particles that are driven by a constant force along a toroidal trap. Due to hydrodynamic interactions, a characteristic limit cycle is observed. When we additionally apply a periodic sawtooth potential, we find a novel caterpillar-like motional sequence that is dominated by hydrodynamic interactions and promotes the surmounting of potential barriers by the particles. PACS numbers: 82.70.Dd, 67.40.Hf, 83.80.Hj Hydrodynamic interactions (HI) play an important role whenever two or more particles move in a viscous fluid [1,2]. Due to their long-range nature, they govern the dynamics of colloidal suspensions, e.g., during self-and collective diffusion [3], sedimentation [4], and aggregation processes [5]. Furthermore, HI can lead to pattern formation of rotating motors [6] with a possible two-dimensional melting transition [7] and they are indispensable for the locomotion of microorganisms [8,9] or in the transport of fluid by beating cilia [10]. While in all these examples many colloids are involved, the effect of HI in few-particle systems has been investigated only recently. It has been demonstrated that HI mediate the correlated motion of a pair of colloids trapped in optical tweezers [11] and that they give rise to interesting collective behavior, e.g., periodic or almost periodic motions in time [12] or even transient chaotic dynamics in sedimenting three-particle clusters [13].In this Letter, we experimentally and theoretically demonstrate how HI lead to a novel motional behavior of a colloidal system comprised of at most three particles. In contrast to the aforementioned examples, where the colloids exhibit either deterministic drift or Brownian diffusion, in the following we concentrate on a non-equilibrium system where both deterministic and stochastic motions are of importance. This work is partially motivated by a recent theoretical analysis of particles driven by a constant tangential force along a toroidal trap [14]. Owing to HI, the particles first go through a transient regime and then enter a characteristic limit cycle. Here, we present the first experimental confirmation of these findings. Our main objective, however, is to investigate experimentally and theoretically how the collective motion of interacting particles changes when a sawtooth potential is added to the constant driving force. Sawtooth potentials are an important component for thermal ratchets studied, e.g., in connection with biological motors [15]. Here, we demonstrate that, due to HI, two-particle clusters exhibit an unexpected caterpillar-like motion which facilitates the surmounting of potential barriers. This motional sequence is largely dominated by hydrodynamic interactions in the system.Tangential driving forces were exerted on colloidal particles with a single three-dimensional laser tweezer that scans a circle inside our sample cell with the help of computer-controlled mirrors at a frequency f T . In contrast to high scanning speeds...

By combining optical tweezers with polarization microscopy, the hydrodynamic coupling between position and orientation fluctuations in a pair of colloidal spheres has been measured. Imaging of birefringent particles under crossed polarizers allows for the simultaneous determination of the positions and orientations of both particles. The temporal cross-correlation function between random displacements of one particle and orientation fluctuations of its neighbor allows for the quantification of the hydrodynamic rotation-translation coupling between the spheres. Our results are in good agreement with predictions for the hydrodynamic mobility tensors calculated in the creeping-flow limit of the Navier-Stokes equation.