Institute of Fundamental Technological Research

The dynamics of groups of non-touching particles settling under gravity in a crowded fluid medium are studied at the zero Reynolds number. It is assumed that the fluid velocity satisfies the Brinkman–Debye–Büche equations, and the particle dynamics are described in terms of the point-force model. The systems of particles at vertices of two or four horizontal regular polygons are considered that in the Stokes flow for a very long time do not destabilize, i.e., all the particles stay close to each other, performing periodic or quasiperiodic motions. It is known that such motions, as invariant manifolds, are essential for groups of particles at random initial positions to survive for a very long time and not destabilize. This work demonstrates that when the medium permeability is decreased, periodic motions cease to exist, and groups of particles split into smaller subgroups, moving away from each other. This mechanism seems to facilitate particle transport in a permeable medium.

In this short review, we visualize the fluid velocity generated by a point force close to a plane free surface or a plane rigid wall. We present separately contributions from all the multipoles which form the corresponding classical systems of images. Such graphical images might be useful in the theoretical and numerical modeling of the dynamics of micro-objects moving close to an interface.

Hydrodynamic interactions between two identical elastic dumbbells settling under gravity in a viscous fluid at low Reynolds number are investigated using the point-particle model. The 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. We find that elasticity (which breaks the time-reversal symmetry of the motion) significantly affects the system dynamics. This is remarkable when 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. This motion 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 understanding general features of systems with a larger number of elastic particles, in regular and random configurations.

complex fluids, low-Reynolds-number flows, Stokesian dynamics

The dynamics of regular clusters of many nontouching particles falling under gravity in a viscous fluid at low Reynolds number are analyzed within the point-particle model. The evolution of two families of particle configurations is determined: two or four regular horizontal polygons (called “rings”) centered 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 in which the long-lasting clusters are centered around periodic solutions for the relative motions, and they are surrounded by regions of chaotic scattering in a similar way to what was observed by Janosi et al. [Phys. Rev. E. 56, 2858 (1997)] for three particles only. These findings suggest that we should 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.

Stokes equations, particle clusters, sedimentation, chaotic scattering, periodic orbits

Dynamics of a cluster of non-Brownian particles falling under gravity in a viscous fluid at low-Reynolds-numer regime has been extensively studied in the literature both for small and large number of particles and oscillating motions have been discovered. In this work we investigate dynamics of clusters of many non-Brownian particles in regular configurations settling under gravity in a viscous fluid. The point particle approximation is applied for the hydrodynamic interactions. We find out that a wide range of regular initial configurations of many particles leads to very long lifetime of the cluster with periodic and quasi-periodic relative motions of particles. We vary the relative distance between the particles and observe how does it affect the dynamics. Several types of periodic and quasi-periodic solutions are discovered. For broad range of initial configurations we show that a slight change of initial conditions has a large influence on the system lifetime – we observe chaotic scattering.

complex fluids, Stokesian dynamics, low-Reynolds-number regime

Dynamics of two identical elastic dumbbells, settling under gravity in a viscous fluid at low Reynolds number are analyzed within the point-particle model. Initially, the dumbbells are vertical, their centers are aligned horizontally, and the springs which connect the dumbbell's beads are at the equilibrium. The motion of the beads is determined numerically with the use of the Runge-Kutta method. After an initial relaxation phase, the system converges to a universal time-dependent solution. The elastic dumbbells tumble while falling, but their relative motion is not periodic (as in case of rigid dumbbells or pairs of separated beads). The elastic constraints break the time-reversal symmetry of the motion. As the result, the horizontal distance between the dumbbells slowly increases - they are hydrodynamically repelled from each other. This effect can be very large even though the elastic forces are always much smaller than gravity. [For the details, see M. Bukowicki, M. Gruca, M. L. Ekiel-Jezewska, J. Fluid Mech. 767, p. 95 (2015).]

The dynamics described above are equivalent to the motion of a single elastic dumbbell under a constant external force which is parallel to a flat free surface. The dumbbell migrates away from the interface and its tumbling time increases.

Stokes equations, dumbbells, point-particle model, hydrodynamic repulsion

We investigate the dynamics of many particles settling under gravity in a viscous fluid within a Stokes flow regime. We consider two families with a very wide range of regular initial configurations of many point-particles which lead to periodic and quasi-periodic motion. We vary the relative distance between the particles and observe how does it affect the dynamics. We observe the oscillations under some out-of-phase rearrangements of the particles and obtain several types of periodic motions for specified range of initial conditions. We also see a large influence of initial conditions on the cluster lifetime.

complex fluid, low-Reynolds-number regime

Periodic motion of several particles falling under gravity in a viscous fluid was theoretically and experimentally observed in a range of systems, including some four-particle configurations or a pair of rigid rods. In addition to its fundamental significance, such a motion is considered as important to understand general features of sedimenting random swarms, and suspensions. In this work, we consider a symmetric system of two elastic fibres, modeled as elastic dumbbells, sedimenting in a vertical plane. We focus on the problem how the elasticity (which breaks time-reversal symmetry of the motion) affects the system's dynamics. The point particle model is used. We observe oscillating, but non-periodic motion of the elastic particles. Independently of the value of the spring constant, the hydrodynamic repulsion appears between the dumbbells. The trajectory shift is slower when k tends to 0 or to infinity – in these limiting cases we recover the periodic dynamics reported in the literature. For a given finite but non-zero spring constant we observe existence of a universal time-dependent trajectory to which the system converge.

Stokesian dynamics, elastic dumbbells, hydrodynamics repulsion