Instytut Podstawowych Problemów Techniki

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The dynamics of two identical elastic filaments settling under gravity in a viscous fluid in the low Reynolds number regime is investigated numerically. A large family of initial configurations symmetric with respect to a vertical plane is considered, as well as their non-symmetric perturbations. The behaviour of the filaments is primarily governed by the elasto-gravitational number, which depends on the filament's length and flexibility, and the strength of the external force. Flexible filaments usually converge toward horizontal and parallel orientation. We explain this phenomenon and show that it occurs also for curved rigid particles of similar shapes. Once aligned, the two fibres either converge toward a stationary, flexibility-dependent distance, or tend to collide or continuously repel each other. Rigid and straight rods perform periodic motions while settling down. Apart from very stiff particles, the dynamics is robust to non-symmetric perturbations.

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The main goal of this paper is to examine theoretically and numerically the impact of a chosen bending model on the dynamics of elastic filaments settling in a viscous fluid under gravity at low-Reynolds-number. We use the bead-spring approximation of a filament and the Rotne–Prager mobility matrix to describe hydrodynamic interactions between the beads. We analyze the dynamics of trumbbells, for which bending angles are typically larger than for thin and long filaments. Each trumbbell is made of three beads connected by springs and it exhibits a bending resistance, described by the harmonic or - alternatively - by the 'cosine' (also called the Kratky–Porod) bending models, both often used in the literature. Using the harmonic bending potential, and coupling it to the spring potential by the Young's modulus, we find simple benchmark solutions: stable stationary configurations of a single elastic trumbbell and attraction of two elastic trumbbells towards a periodic long-lasting orbit. As the most significant result of this paper, we show that for very elastic trumbbells at the same initial conditions, the Kratky–Porod bending potential can lead to qualitatively and quantitatively different spurious dynamics, with artificially large bending angles and unrealistic shapes. We point out that for the bead models of an elastic filament, the range of applicability of the Kratky–Porod model might not go beyond bending angles smaller than π/2 for touching beads and beyond an even much lower value for beads well-separated from each other. The existence of stable stationary configurations of elastic trumbbells and a family of periodic oscillations of two elastic trumbbells are very important findings on their own.

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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.

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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.

Słowa kluczowe:

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

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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.

Słowa kluczowe:

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

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Driving on uneven roads causes vibrations of the vehicle. Modern smartphones are equipped with accelerometers and gyroscopes which allow for inexpensive acquisition of information about the shaking level. This study reports on field tests of a system, which measures vibrations of a driving vehicle using four smartphones as sensors. The collected data undergo a series of processing steps, namely synchronization, virtual reorientation, temporal and spectral filtering, and assignment of road localization. We use quarter-car and half-car suspension models to retrieve the longitudinal road profile and to compute unevenness indicators. The accuracy of the system was computed by comparison with traditional laser profilometer on an 18.6 km long test section using eight rides with different speeds. It averages to 71% and exceeds 80% for the best calibrated cases. Additionally, we report on the feedback obtained during tests of the system in a district road administration in Poland.

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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.

Słowa kluczowe:

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

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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.

Słowa kluczowe:

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

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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.

Słowa kluczowe:

complex fluid, low-Reynolds-number regime

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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.

Słowa kluczowe:

Stokesian dynamics, elastic dumbbells, hydrodynamics repulsion

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