Hydrodynamic interactions betweem many spheres

This paper is an introductory guide to many-particle hydrodynamic interactions. Basic concepts of the fluid dynamics are assumed to be known. Experience in the Stokes equations is useful but not necessary. The study is estimated to fit five sessions about three hours each.Comment: Latex, 17 page

A class of periodic and quasi-periodic trajectories of particles settling under gravity in a viscous fluid

We investigate regular configurations of a small number of particles settling under gravity in a viscous fluid. The particles do not touch each other and can move relative to each other. The dynamics is analyzed in the point-particle approximation. A family of regular configurations is found with periodic oscillations of all the settling particles. The oscillations are shown to be robust under some out-of-phase rearrangements of the particles. In the presence of an additional particle above such a regular configuration, the particle periodic trajectories are horizontally repelled from the symmetry axis, and flattened vertically. The results are used to propose a mechanism how a spherical cloud, made of a large number of particles distributed at random, evolves and destabilizes.Comment: 11 pages, 16 figure

Stable Configurations Of Charged Sedimenting Particles

The qualitative behavior of charged particles in a vacuum is given by Earnshaw's Theorem which states that there is no steady configuration of charged particles in a vacuum which is asymptotically stable to perturbations. In a viscous fluid, examples of stationary configurations of sedimenting uncharged particles are known, but they are unstable or neutrally stable - they are not attractors. In this paper, it is shown by example that two charged particles settling in a fluid may have a configuration which is asymptotically stable to perturbations, for a wide range of charges, radii and densities. The existence of such "bound states" is essential from a fundamental point of view and it can be significant for dilute charged particulate systems in various biological, medical and industrial contexts.Comment: 5 pages, 2 figures, supplemental materia

Dynamics of elastic dumbbells sedimenting in a viscous fluid: oscillations and hydrodynamic repulsion

Hydrodynamic interactions between two identical elastic dumbbells settling under gravity in a viscous fluid at low-Reynolds-number are investigated within the point-particle model. Evolution of a benchmark initial configuration is studied, in which the dumbbells are vertical and their centres are aligned horizontally. Rigid dumbbells and pairs of separate beads starting from the same positions tumble periodically while settling down. We find that elasticity (which breaks time-reversal symmetry of the motion) significantly affects the system's dynamics. This is remarkable taking into account that elastic forces are always much smaller than gravity. We observe oscillating motion of the elastic dumbbells, which tumble and change their length non-periodically. Independently of the value of the spring constant, a horizontal hydrodynamic repulsion appears between the dumbbells - their centres of mass move apart from each other horizontally. The shift is fast for moderate values of the spring constant k, and slows down when k tends to zero or to infinity; in these limiting cases we recover the periodic dynamics reported in the literature. For moderate values of the spring constant, and different initial configurations, we observe the existence of a universal time-dependent solution to which the system converges after an initial relaxation phase. The tumbling time and the width of the trajectories in the centre-of-mass frame increase with time. In addition to its fundamental significance, the benchmark solution presented here is important to understand general features of systems with larger number of elastic particles, at regular and random configurations.Comment: 12 pages, 7 figure

Translational and rotational Brownian displacements of colloidal particles of complex shapes

The exact analytical expressions for the time-dependent cross-correlations of the translational and rotational Brownian displacements of a particle with arbitrary shape were derived by us in [J. Chem. Phys. 142, 214902 (2015) and 144, 076101 (2016)]. They are in this work applied to construct a method to analyze Brownian motion of a particle of an arbitrary shape, and to extract accurately the self-diffusion matrix from the measurements of the cross-correlations, which in turn allows to gain some information on the particle structure. As an example, we apply our new method to analyze the experimental results of D. J. Kraft et al. for the micrometer-sized aggregates of the beads [Phys. Rev. E 88, 050301 (R) (2013)]. We explicitly demonstrate that our procedure, based on the measurements of the time-dependent cross-correlations in the whole range of times, allows to determine the self diffusion (or alternatively the friction matrix) with a much higher precision than the method based only on their initial slopes. Therefore, the analytical time-dependence of the cross-correlations serves as a useful tool to extract information about particle structure from trajectory measurements.Comment: 4 pages, 2 figure

Note: Brownian motion of colloidal particles of arbitrary shape

The analytical expressions for the time-dependent cross-correlations of the translational and rotational Brownian displacements of a particle with arbitrary shape are derived. The reference center is arbitrary, and the reference frame is such that the rotational-rotational diffusion tensor is diagonal.Comment: 2 page

Brownian motion of a particle with arbitrary shape

Brownian motion of a particle with an arbitrary shape is investigated theoretically. Analytical expressions for the time-dependent cross-correlations of the Brownian translational and rotational displacements are derived from the Smoluchowski equation. The role of the particle mobility center is determined and discussed.Comment: 11 page

Periodic and quasiperiodic motions of many particles falling in a viscous fluid

Dynamics of regular clusters of many non-touching particles falling under gravity in a viscous fluid at low Reynolds number are analysed within the point-particle model. Evolution of two families of particle configurations is determined: 2 or 4 regular horizontal polygons (called `rings') centred above or below each other. Two rings fall together and periodically oscillate. Four rings usually separate from each other with chaotic scattering. For hundreds of thousands of initial configurations, a map of the cluster lifetime is evaluated, where the long-lasting clusters are centred around periodic solutions for the relative motions, and surrounded by regions of the chaotic scattering,in a similar way as it was observed by Janosi et al. (1997) for three particles only. These findings suggest to consider the existence of periodic orbits as a possible physical mechanism of the existence of unstable clusters of particles falling under gravity in a viscous fluid.Comment: 11 pages, 10 figure

Hydrodynamic radius approximation for spherical particles suspended in a viscous fluid: influence of particle internal structure and boundary

Systems of spherical particles moving in Stokes flow are studied for a different particle internal structure and boundaries, including the Navier-slip model. It is shown that their hydrodynamic interactions are well described by treating them as solid spheres of smaller hydrodynamic radii, which can be determined from measured single-particle diffusion or intrinsic viscosity coefficients. Effective dynamics of suspensions made of such particles is quite accurately described by mobility coefficients of the solid particles with the hydrodynamic radii, averaged with the unchanged direct interactions between the particles.Comment: 7 pages, 2 figure

Dynamics of flexible fibers in shear flow

Dynamics of flexible non-Brownian fibers in shear flow at low-Reynolds-number are analyzed numerically for a wide range of the ratios A of the fiber bending force to the viscous drag force. Initially, the fibers are aligned with the flow, and later they move in the plane perpendicular to the flow vorticity. A surprisingly rich spectrum of different modes is observed when the value of A is systematically changed, with sharp transitions between coiled and straightening out modes, period-doubling bifurcations from periodic to migrating solutions, irregular dynamics and chaos.Comment: 5 pages, 7 figure