Streaming Current for Surfaces Covered by Square and Hexagonal Monolayers of Spherical Particles.

The interface and particle contributions to the streaming current of flat substrates covered with ordered square or hexagonal monolayers of spherical particles were theoretically evaluated for particle coverage up to close packing. The exact numerical results were approximated using fitting functions that contain exponential and linear terms to account for hydrodynamic screening and charge convection from the particle surfaces exposed to external flow. According to our calculations, the streaming currents for the ordered and random particle arrangements differ within a typical experimental error. Thus, streaming-current measurements, supplemented with our fitting functions, can be conveniently used to evaluate the particle coverage without detailed knowledge of the particle distribution. Our results for equal interface and particle ζ-potentials indicate that roughness can reduce the streaming current by more than 30%, even in the limit of the small size of spherical roughness asperities.

Dynamics of ball chains and highly elastic fibres settling under gravity in a viscous fluid.

We study experimentally the dynamics of one and two ball chains settling under gravity in a highly viscous silicon oil at a Reynolds number much smaller than unity. We record the motion and shape deformation using two cameras. We demonstrate that single ball chains in most cases do not tend to be planar and often rotate, not keeping the ends at the same horizontal level. Shorter ball chains usually form shapes resembling distorted U. Longer ones in the early stage of the evolution form a shape resembling distorted W, and later deform non-symmetrically and significantly out of a plane. The typical evolution of shapes observed in our experiments with single ball chains is reproduced in our numerical simulations of a single elastic filament. In the computations, the filament is modelled as a chain of beads. Consecutive beads are connected by springs. Additional springs link consecutive pairs of beads. Elastic forces are assumed to be much smaller than gravity. As a result, the fibre is very flexible. We assume that the fluid sticks to the surfaces of the beads. We perform multipole expansion of the Stokes equations, with a lubrication correction. This method is implemented in the precise HYDROMULTIPOLE numerical codes. In our experiments, two ball chains, initially one above the other, later move away or approach each other, for a larger or smaller initial distance, respectively.

DNA supercoiling-induced shapes alter minicircle hydrodynamic properties.

DNA in cells is organized in negatively supercoiled loops. The resulting torsional and bending strain allows DNA to adopt a surprisingly wide variety of 3-D shapes. This interplay between negative supercoiling, looping, and shape influences how DNA is stored, replicated, transcribed, repaired, and likely every other aspect of DNA activity. To understand the consequences of negative supercoiling and curvature on the hydrodynamic properties of DNA, we submitted 336 bp and 672 bp DNA minicircles to analytical ultracentrifugation (AUC). We found that the diffusion coefficient, sedimentation coefficient, and the DNA hydrodynamic radius strongly depended on circularity, loop length, and degree of negative supercoiling. Because AUC cannot ascertain shape beyond degree of non-globularity, we applied linear elasticity theory to predict DNA shapes, and combined these with hydrodynamic calculations to interpret the AUC data, with reasonable agreement between theory and experiment. These complementary approaches, together with earlier electron cryotomography data, provide a framework for understanding and predicting the effects of supercoiling on the shape and hydrodynamic properties of DNA.

DNA supercoiling-induced shapes alter minicircle hydrodynamic properties.

DNA in cells is organized in negatively supercoiled loops. The resulting torsional and bending strain allows DNA to adopt a surprisingly wide variety of 3-D shapes. This interplay between negative supercoiling, looping, and shape influences how DNA is stored, replicated, transcribed, repaired, and likely every other aspect of DNA activity. To understand the consequences of negative supercoiling and curvature on the hydrodynamic properties of DNA, we submitted 336 bp and 672 bp DNA minicircles to analytical ultracentrifugation (AUC). We found that the diffusion coefficient, sedimentation coefficient, and the DNA hydrodynamic radius strongly depended on circularity, loop length, and degree of negative supercoiling. Because AUC cannot ascertain shape beyond degree of non-globularity, we applied linear elasticity theory to predict DNA shapes, and combined these with hydrodynamic calculations to interpret the AUC data, with reasonable agreement between theory and experiment. These complementary approaches, together with earlier electron cryotomography data, provide a framework for understanding and predicting the effects of supercoiling on the shape and hydrodynamic properties of DNA.

Correction: Stokesian dynamics of sedimenting elastic rings.

Correction for 'Stokesian dynamics of sedimenting elastic rings' by Magdalena Gruziel-Slomka et al., Soft Matter, 2019, 15, 7262-7274, https://doi.org/10.1039/C9SM00598F.

Flexible fibers in shear flow approach attracting periodic solutions.

The three-dimensional dynamics of a single non-Brownian flexible fiber in shear flow is evaluated numerically, in the absence of inertia. A wide range of ratios A of bending to hydrodynamic forces and hundreds of initial configurations are considered. We demonstrate that flexible fibers in shear flow exhibit much more complicated evolution patterns than in the case of extensional flow, where transitions to higher-order modes of characteristic shapes are observed when A exceeds consecutive threshold values. In shear flow, we identify the existence of an attracting steady configuration and different attracting periodic motions that are approached by long-lasting rolling, tumbling, and meandering dynamical modes, respectively. We demonstrate that the final stages of the rolling and tumbling modes are effective Jeffery orbits, with the constant parameter C replaced by an exponential function that either decays or increases in time, respectively, corresponding to a systematic drift of the trajectories. In the limit of C→0, the fiber aligns with the vorticity direction and in the limit of C→∞, the fiber periodically tumbles within the shear plane. For moderate values of A, a three-dimensional meandering periodic motion exists, which corresponds to intermediate values of C. Transient, close to periodic oscillations are also detected in the early stages of the modes.

Sedimenting pairs of elastic microfilaments.

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.

Stokesian dynamics of sedimenting elastic rings.

We consider elastic microfilaments which form closed loops. We investigate how the loops change shape and orientation while settling under gravity in a viscous fluid. Loops are circular at the equilibrium. Their dynamics are investigated numerically based on the Stokes equations for the fluid motion and the bead-spring model of the microfilament. The Rotne-Prager approximation for the bead mobility is used. We demonstrate that the relevant dimensionless parameter is the ratio of the bending resistance of the filament to the gravitation force corrected for buoyancy. The inverse of this ratio, called the elasto-gravitation number B, is widely used in the literature for sedimenting elastic linear filaments. We assume that B is of the order of 104-106, which corresponds to easily deformable loops. We find out that initially tilted circles evolve towards different sedimentation modes, depending on B. Very stiff or stiff rings attain almost planar, oval shapes, which are vertical or tilted, respectively. More flexible loops deform significantly and converge towards one of several characteristic periodic motions. These sedimentation modes are also detected when starting from various shapes, and for different loop lengths. In general, multi-stability is observed: an elastic ring converges to one of several sedimentation modes, depending on the initial conditions. This effect is pronounced for very elastic loops. The surprising diversity of long-lasting periodic motions and shapes of elastic rings found in this work gives a new perspective for the dynamics of more complex deformable objects at micrometer and nanometer scales, sedimenting under gravity or rotating in a centrifuge, such as red blood cells, ring polymers or circular DNA.

Periodic Motion of Sedimenting Flexible Knots.

We study the dynamics of knotted deformable closed chains sedimenting in a viscous fluid. We show experimentally that trefoil and other torus knots often attain a remarkably regular horizontal toroidal structure while sedimenting, with a number of intertwined loops, oscillating periodically around each other. We then recover this motion numerically and find out that it is accompanied by a very slow rotation around the vertical symmetry axis. We analyze the dependence of the characteristic timescales on the chain flexibility and aspect ratio. It is observed in the experiments that this oscillating mode of the dynamics can spontaneously form even when starting from a qualitatively different initial configuration. In numerical simulations, the oscillating modes are usually present as transients or final stages of the evolution, depending on chain aspect ratio and flexibility, and the number of loops.

Different bending models predict different dynamics of sedimenting elastic trumbbells.

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.

Dynamics of flexible fibers and vesicles in Poiseuille flow at low Reynolds number.

The dynamics of flexible fibers and vesicles in unbounded planar Poiseuille flow at low Reynolds number is shown to exhibit similar basic features, when their equilibrium (moderate) aspect ratio is the same and vesicle viscosity contrast is relatively high. Tumbling, lateral migration, accumulation and shape evolution of these two types of flexible objects are analyzed numerically. The linear dependence of the accumulation position on relative bending rigidity, and other universal scalings are derived from the local shear flow approximation.

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.

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

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.

Structure and dynamics of a layer of sedimented particles.

We investigate experimentally and theoretically thin layers of colloid particles held adjacent to a solid substrate by gravity. Epifluorescence, confocal, and holographic microscopy, combined with Monte Carlo and hydrodynamic simulations, are applied to infer the height distribution function of particles above the surface, and their diffusion coefficient parallel to it. As the particle area fraction is increased, the height distribution becomes bimodal, indicating the formation of a distinct second layer. In our theory, we treat the suspension as a series of weakly coupled quasi-two-dimensional layers in equilibrium with respect to particle exchange. We experimentally, numerically, and theoretically study the changing occupancies of the layers as the area fraction is increased. The decrease of the particle diffusion coefficient with concentration is found to be weakened by the layering. We demonstrate that particle polydispersity strongly affects the properties of the sedimented layer, because of particle size segregation due to gravity.

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.

Hydrodynamic interactions between a sphere and a number of small particles.

Exact expressions are derived for the pair and three-body hydrodynamic interactions between a sphere and a number of small particles immersed in a viscous incompressible fluid. The analysis is based on the Stokes equations of low Reynolds number hydrodynamics. The results follow by a combination of the solutions for flow about a sphere with no-slip boundary condition derived by Stokes and Kirchhoff and the result derived by Oseen for the Green tensor of Stokes equations in the presence of a fixed sphere.

Class of periodic and quasiperiodic trajectories of particles settling under gravity in a viscous fluid.

We investigate regular configurations of a small number of non-Brownian 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 of how a spherical cloud, made of a large number of particles distributed at random, evolves and destabilizes.

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 different particle internal structures 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.

Lateral migration of flexible fibers in Poiseuille flow between two parallel planar solid walls.

Dynamics of non-Brownian flexible fibers in Poiseuille flow between two parallel planar solid walls is evaluated from the Stokes equations which are solved numerically by the multipole method. Fibers migrate towards a critical distance from the wall zc, which depends significantly on the fiber length N and bending stiffness A. This effect can be used to sort fibers. Three types of accumulation are found, depending on a shear-to-bending parameter Γ. In the first type, stiff fibers deform only a little and accumulate close to the wall, where their tendency to drift away from the channel is balanced by the repulsive hydrodynamic interaction with the wall. In the second type, flexible fibers deform significantly and accumulate far from the wall. In both types, the fiber shapes at the accumulation positions are repeatable, while in the third type, they are very compact and non-repeatable. The difference between the second and third accumulation types is a special case of the difference between the regular and irregular modes for the dynamics of migrating fibers. At the regular mode, far from walls, the fiber tumbling frequency satisfies Jeffery's expression, with the local shear rate and the aspect ratio close to N.