Institute of Fundamental Technological Research

The dynamics of a single highly elastic fibre settling under gravity in a very viscous fluid is studied numerically. We employ the bead model and multipole expansion of the Stokes equations, corrected for lubrication that is implemented in the precise Hydromultipole numerical codes. Four attracting regular dynamical modes of highly elastic fibres are found: two stationary shapes (one translating and the other rotating and translating), and two periodic oscillations around such shapes. The phase diagram of these modes is presented. It illustrates that the existence of each mode depends not only on the elasto-gravitation number but also on the fibre aspect ratio. Characteristic time scales, fibre deformation patterns and motion in the different modes are determined.

stokesian dynamics, particle/fluid flows

We analyze the three-dimensional (3D) buckling of an elastic filament in a shear flow of a viscous fluid at low Reynolds number and high Péclet number. We apply the Euler-Bernoulli beam (elastica) theoretical model. We show the universal character of the full 3D spectral problem for a small perturbation of a thin filament from a straight position of arbitrary orientation. We use the eigenvalues and eigenfunctions for the linearized elastica equation in the shear plane, found earlier by Liu et al. [Phys. Rev. Fluids 9, 014101 (2024)] with the Chebyshev spectral collocation method, to solve the full 3D eigenproblem. We provide a simple analytic approximation of the eigenfunctions, represented as Gaussian wave packets. As the main result of the paper, we derive the square-root dependence of the eigenfunction wave number on the parameter χ˜ = −η sin 2φ sin2 θ, where

The long-time behavior of highly elastic fibers in a shear flow is investigated experimentally and numerically. Characteristic attractors of the dynamics are found. It is shown that for a small ratio of bending to hydrodynamic forces, most fibers form a spinning elongated double helix, performing an effective Jeffery orbit very close to the vorticity direction. Recognition of these oriented shapes, and how they form in time, may prove useful in the future for understanding the time history of complex microstructures in fluid flows and considering processing steps for their synthesis.

Stokes equations, shear flow, elastic fibers

We present the spectral analysis of three-dimensional dynamics of an elastic filament in a shear flow of a viscous fluid at a low Reynolds number in the absence of Brownian motion. The elastica model is used. The fiber initially is almost straight at an arbitrary orientation, with small perpendicular perturbations in the shear plane and out-of-plane. To analyze the stability of both perturbations, equations for the eigenvalues and eigenfunctions are derived and solved by the Chebyshev spectral collocation method. It is shown that their crucial features are the same as in the case of the two-dimensional elastica dynamics in shear flow [Becker and Shelley, Phys. Rev. Lett. 87, 198301 (2001)] and the three-dimensional elastica dynamics in the compressional flow [Chakrabarti et al., Nat. Phys. 16, 689 (2020)]. We find a similar dependence of the buckled shapes on the ratio of bending to hydrodynamic forces as in the simulations for elastic fibers of a nonzero thickness [Słowicka et al., New J. Phys. 24, 013013 (2022)].

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.

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.

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.

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.

Three-dimensional dynamics of flexible fibers in shear flow are studied numerically, with a qualitative comparison to experiments. Initially, the fibers are straight, with different orientations with respect to the flow. By changing the rotation speed of a shear rheometer, we change the ratio A of bending to shear forces. We observe fibers in the flow-vorticity plane, which gives insight into the motion out of the shear plane. The numerical simulations of moderately flexible fibers show that they rotate along effective Jeffery orbits, and therefore the fiber orientation rapidly becomes very close to the flow-vorticity plane, on average close to the flow direction, and the fiber remains in an almost straight configuration for a long time. This ``ordering'' of fibers is temporary since they alternately bend and straighten out while tumbling. We observe numerically and experimentally that if the fibers are initially in the compressional region of the shear flow, they can undergo a compressional buckling, with a pronounced deformation of shape along their whole length during a short time, which is in contrast to the typical local bending that originates over a long time from the fiber ends. We identify differences between local and compressional bending and discuss their competition, which depends on the initial orientation of the fiber and the bending stiffness ratio A. There are two main finding. First, the compressional buckling is limited to a certain small range of the initial orientations, excluding those from the flow-vorticity plane. Second, since fibers straighten out in the flow-vorticity plane while tumbling, the compressional buckling is transient - it does not appear for times longer than 1/4 of the Jeffery period. For larger times, bending of fibers is always driven by their ends.

Stokes flow, flexible fibers, bending, buckling, orientational order

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

We investigate qualitatively a uniform non-Brownian sedimenting suspension in a stationary state. As a base of our analysis we take the BBGKY hierarchy derived from the Liouville equation. We then show that assumption of the plasma-like screening relations can cancel some long-range terms in the hierarchy but it does not provide integrable solutions for correlation functions. This suggests breaking the translational symmetry of the system. Therefore a non-uniform structure can develop to suppress velocity fluctuations and make the range of correlations finite.

non-Brownian sedimentation, stability, BBGKY hierarchy, hydrodynamic screening, correlation functions, low-Reynolds-number hydrodynamics

We present a numerical study of the dynamics of an elastic fibre in a shear flow at low Reynolds number, and seek to understand several aspects of the fibre's motion using the equations for slender-body theory coupled to the elastica. The numerical simulations are performed in the bead-spring framework including hydrodynamic interactions in two theoretical schemes: the generalized Rotne-Prager-Yamakawa model and a multipole expansion corrected for lubrication forces. In general, the two schemes yield similar results, including for the dominant scaling features of the shape that we identify. In particular, we focus on the evolution of an initially straight fibre oriented in the flow direction and show that the time scales of fibre bending, curling and rotation, which depend on its length and stiffness, determine the overall motion and evolution of the shapes. We document several characteristic time scales and curvatures representative of the shape that vary as power laws of the bending stiffness and fibre length. The numerical results are further supported by an interpretation using an elastica model.

Stokesian dynamics, particle/fluid flow, slender-body theory

We study the dynamics of two charged point particles settling in a Stokes flow. We find what ranges of initial relative positions and what ranges of system parameters lead to formation of stable doublets. The system is parameterized by the ratio of radii, ratio of masses and the ratio of electrostatic to gravitational force. We focus on opposite charges. We find a new class of stationary states with the line of the particle centers inclined with respect to gravity and demonstrate that they are always locallyasymptotically stable. Stability properties of stationary states with the vertical line of the particle centers are also discussed. We find examples of systems with multiple stable stationary states. We show that the basin of attraction for each stable stationary state has infinite measure, so that particles can capture one another even when they are very distant, and even if their charge is very small. This behavior is qualitatively different from the uncharged case where there only exists a bounded set of periodic relative trajectories. We determine the range of ratios of Stokes velocities and ratio masses which give rise to non-overlapping stable stationary states (given the appropriate ratio of electrostatic to gravitational force). For non-overlapping stable inclined or vertical stationary states the larger particle is always above the smaller particle. The non-overlapping stable inclined stationary states existonly if the larger particle has greater Stokes velocity, but there are non-overlapping stable vertical stationary states where the larger particle has higher or lower Stokes velocity.

charged particles, stability, capturing orbits, Stokes equations, point-particle model

For flows in microchannels, a slip on the walls may be efficient in reducing viscous dissipation. A related issue, addressed in this article, is to decrease the effective viscosity of a dilute monodisperse suspension of spheres in Poiseuille flow by using two parallel slip walls. Extending the approach developed for no-slip walls in Feuillebois et al. (J. Fluid Mech., vol. 800, 2016, pp. 111–139), a formal expression is obtained for the suspension intrinsic viscosity [μ] solely in terms of a stresslet component and a quadrupole component exerted on a single freely suspended sphere. In the calculation of [μ], the hydrodynamic interactions between a sphere and the slip walls are approximated using either the nearest wall model or the wall-superposition model. Both the stresslet and quadrupole are derived and accurately calculated using bipolar coordinates. Results are presented for [μ] in terms of H/(2a) and ˜λ = λ/a ≤ 1, where H is the gap between walls, a is the sphere radius and λ is the wall slip length using the Navier slip boundary condition. As compared with the no-slip case, the intrinsic viscosity strongly depends on ˜λ for given H/(2a), especially for small H/(2a). For example, in the very confined case H/(2a) = 2 (a lower bound found for practical validity of single-wall models) and for ˜λ = 1, the intrinsic viscosity is three times smaller than for a suspension bounded by no-slip walls and five times smaller than for an unbounded suspension (Einstein, Ann. Phys., vol. 19, 1906, pp. 289–306). We also provide a handy formula fitting our results for [μ] in the entire range 2 ≤ H/(2a) ≤ 100 and ˜λ ≤ 1.

complex fluids, low-Reynolds-number flows

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.

We present a study of peptide reorientational dynamics in solution analyzed from the perspective of fluorescence anisotropy decay (FAD) experiments, and atomistic molecular dynamics (MD) and continuum hydrodynamics modeling. Earlier, FAD measurements and MD simulations of the model dipeptide N-acetyltryptophanamide (NATA) in explicit water and in aqueous solutions of urea, guanidinium chloride, and proline co-solvents identified excellent agreement of MD results with experimental data, indicating the presence of significant effects of peptide–solvent interactions, and the overall tumbling of the peptide could be well described by contributions from individual conformers, represented by dihedral-restrained MD. Here, we extend these studies by analyzing dynamic inhomogeneity in the solutions and by developing a hydrodynamic model (HM) of the conformer dynamics. The MD simulation data indicate the presence of markedly different dynamic microenvironments for the four studied solutions, with the average water reorientations being different in all systems, partly reflecting the bulk viscosities. Additionally, the water dynamics also exhibited a marked slowdown in the vicinity of the co-solvents, especially chloride and proline. To gain further insight, we applied the HM to predict rotational correlation times of tryptophan for the individual NATA conformers identified in MD. The hydrodynamic results were in very good agreement with MD simulations for the individual structures, showing that the HM model provides a realistic description of rotational diffusion for rigid peptide structures. Overall, our study generated new microscopic insights into the complex nature of the structure and dynamics of peptide solvation shells for systems containing water and denaturing and stabilizing co-solvents.

A detailed theoretical and experimental study is presented concerning the streaming current and the derivative effective ζ-potential for a planar surface covered by a monolayer of adsorbed particles. Precise simulation results are obtained for the equilibrium and random-sequential-adsorption (RSA) distributions of monodisperse spherical particles interacting via the excluded-volume potential. The streaming current is calculated in the thin-double-layer regime for all physically accessible particle area fractions. The results are expressed as a linear combination of the interface and particle contributions DI and DP weighted by the interface and particle ζ-potentials ζI and ζP. We find that in the area-fraction regime where both particle distributions exist, the equilibrium and RSA results for the streaming current are nearly indistinguishable. Our numerical data show that DI exponentially decays to zero when the particle area fraction θ is increased, whereas DP exponentially tends to a linear behavior. The results are described (with the accuracy better than 1.5% of the maximal value) by the exponential and linear plus exponential approximations, with only one fitting parameter. The numerical and theoretical predictions are in agreement with experimental data obtained for a wide range of ζ-potentials of the interface and the particles. Results obtained for a rough surface with spherical asperities indicate that the roughness can reduce the effective ζ-potential (as evaluated from the streaming current) by more than 25%; this prediction is also confirmed by experiments.

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 [scr B, script letter B], is widely used in the literature for sedimenting elastic linear filaments. We assume that [scr B, script letter B] is of the order of 10^4–10^6, which corresponds to easily deformable loops. We find out that initially tilted circles evolve towards different sedimentation modes, depending on [scr B, script letter 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.

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.

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.

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.

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.

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.

Motion of a particle with stick-slip boundary conditions towards a hard wall or free surface is investigated in the range of Reynolds numbers much smaller than unity, based on the multipole expansion of the Stokes equations. The slip parameter can be interpreted as a measure of a solid particle roughness or as the effect of a surfactant on the motion of a small spherical non-deformable bubble. The particle friction coefficient is evaluated as a function of the distance from its center to the wall, based on the inverse power series expansion, and the results are used to derive explicit lubrication expressions for the friction coefficient, in a wide range of the slip parameters. It is pointed out that for a very small thickness of the fluid film, the lubrication expressions are more accurate than the series expansion. The drainage time is calculated and analyzed, and estimated in terms of explicit lubrication expressions.

particle, boundary conditions, hard wall, free surface

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 [3, 4]. They are in this work applied to construct a method to analyze the Brownian motion of a particle of an arbitrary shape, and to extract accurately the self-diffusion matrix from the measurements of the crosscorrelations, 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 [8]. 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.

Brownian motion, Smoluchowski equation, hydrodynamic interactions, self-diffusion matrix, friction coefficients, cross-correlations of translational and rotational Brownian displacements

It is shown that the formal expression for the effective viscosity of a dilute suspension of arbitrary-shaped particles in Poiseuille flow contains a novel quadrupole term, besides the expected stresslet. This term becomes important for a very confined geometry. For a high-frequency flow field (in the sense used in Feuillebois et al. (J. Fluid Mech., vol. 764, 2015, pp. 133–147), the suspension rheology is Newtonian at first order in volume fraction. The effective viscosity is calculated for suspensions of N-bead rods and of prolate spheroids with the same length, volume and aspect ratio (up to 6), entrained by the Poiseuille flow between two infinite parallel flat hard walls. The numerical computations, based on solving the Stokes equations, indicate that the quadrupole term gives a significant positive contribution to the intrinsic viscosity [μ] if the distance between the walls is less than ten times the particle width, or less. It is found that the intrinsic viscosity in bounded Poiseuille flow is generally smaller than the corresponding value in unbounded flow, except for extremely narrow gaps when it becomes larger because of lubrication effects. The intrinsic viscosity is at a minimum for a gap between walls of the order of 1.5–2 particle width. For spheroids, the intrinsic viscosity is generally smaller than for chains of beads with the same aspect ratio, but when normalized by its value in the bulk, the results are qualitatively the same. Therefore, a rigid chain of beads can serve as a simple model of an orthotropic particle with a more complicated shape. The important conclusion is that the intrinsic viscosity in shear flow is larger than in the Poiseuille flow between two walls, and the difference is significant even for relatively wide channels, e.g. three times wider than the particle length. For such confined geometries, the hydrodynamic interactions with the walls are significant and should be taken into account.

low-Reynolds-number flows

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.

Poiseuille flow, Stokes equations, vesicles, flexible fibers

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.

Rotational correlation time, Tensor methods, Brownian motion, Colloidal systems, Matrix equations

A general expression for the effective viscosity of a dilute suspension of arbitrary-shaped particles in linear shear flow between two parallel walls is derived in terms of the induced stresslets on particles. This formula is applied to N-bead rods and to prolate spheroids with the same length, aspect ratio and volume. The effective viscosity of non-Brownian particles in a periodic shear flow is considered here. The oscillating frequency is high enough for the particle orientation and centre-of-mass distribution to be practically frozen, yet small enough for the flow to be quasi-steady. It is known that for spheres, the intrinsic viscosity [μ] increases monotonically when the distance H between the walls is decreased. The dependence is more complex for both types of elongated particles. Three regimes are theoretically predicted here: (i) a ‘weakly confined’ regime (for H>l, where l is the particle length), where [μ] is slightly larger for smaller H; (ii) a ‘semi-confined’ regime, when H becomes smaller than l, where [μ] rapidly decreases since the geometric constraints eliminate particle orientations corresponding to the largest stresslets; (iii) a ‘strongly confined’ regime when H becomes smaller than 2–3 particle widths d, where [μ] rapidly increases owing to the strong hydrodynamic coupling with the walls. In addition, for sufficiently slender particles (with aspect ratio larger than 5–6) there is a domain of narrow gaps for which the intrinsic viscosity is smaller than that in unbounded fluid.

complex fluids, low-Reynolds-number flows, suspensions

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

We use numerical simulations of a bead–spring model chain to investigate the evolution of the conformations of long and flexible elastic fibers in a steady shear flow. In particular, for rather open initial configurations, and by varying a dimensionless elastic parameter, we identify two distinct conformational modes with different final size, shape, and orientation. Through further analysis we identify slipknots in the chain. Finally, we provide examples of initial configurations of an 'open' trefoil knot that the flow unknots and then knots again, sometimes repeating several times.

knots, low Reynolds number flows, multipole method

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.

Hydrodynamics, Tensor methods, Stokes flows, Boundary value problems, Torque

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.

Shear flows, Chaotic dynamics, Vortex dynamics, Numerical solutions, Periodic solutions

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.

colloidal particles, gravity, horizontal wall, two-dimensional layers, diffusion, height distribution function

rownian 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.

Tensor methods, Brownian motion, Matrix theory, Rotational correlation time, Eigenvalues

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

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.

hydrodynamic radius, Stokes equations

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.

Stokes equations, sedimentation, particle cluster, suspension drop, periodic orbits

The authors report that there is a confusion in the definition of the friction factors, pffp, pccp in Pasol et al. (2011).

friction factors, Poiseuille flow, spherical particle, field-flow fractionation, hydrodynamic chromatotography

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.

Self-organisation, Supramolecular assemblies

Short-time dynamics and high-frequency rheology for suspensions of non-overlapping core–shell particles with thin shells were analyzed. In the thin-shell limit, the single-particle scattering coefficients were derived and shown to define a unique effective radius. This result was used to justify theoretically (in the thin-shell limit) the accuracy of the annulus approximation with the inner radius equal to the effective hydrodynamic radius of the core–shell particle. The two-particle virial expansion of the translational and rotational self-diffusion, sedimentation and viscosity was performed. The virial coefficients were evaluated and shown to be accurately approximated by the effective annulus model, in contrast to the imprecise effective hard sphere model.

Stokes equations, Core–shell particles, Permeable medium

Każdy z nas ma własne wspomnienia szkolne, a także słyszał opowieści o nauce rodziców i dziadków. Wyselekcjonujmy z nich takie, które dotyczą ważnych elementów kształcenia i pokażmy innym. Porozmawiajmy o tym, na czym polega takie uczenie się, jakiego potrzebujemy i za jakim tęsknimy. Przykładem może służyć rozmowa, która zaczęła się od pokazania osobistych pamiątek szkolnych kilku rodzin z regionu Mińska Mazowieckiego i przywołania wspomnień związanych z najlepszymi tradycjami i historią polskiej edukacji w I połowie XX wieku. Eksponaty te od 22 listopada 2012 do 10 marca 2013 r. można było zobaczyć w Muzeum Ziemi Mińskiej na wystawie „Nasi dziadkowie jako uczniowie i nauczyciele”. Artykuł opowiada o tej wystawie i relacjonuje niektóre zebrane przy tej okazji wspomnienia.

nauka, edukacja, tradycje

Short-time dynamic properties of concentrated suspensions of colloidal core-shell particles are studied using a precise force multipole method which accounts for many-particle hydrodynamic interactions. A core-shell particle is composed of a rigid, spherical dry core of radius a surrounded by a uniformly permeable shell of outer radius b and hydrodynamic penetration depth κ−1. The solvent flow inside the permeable shell is described by the Brinkman-Debye-Bueche equation, and outside the particles by the Stokes equation. The particles are assumed to interact non-hydrodynamically by a hard-sphere no-overlap potential of radius b. Numerical results are presented for the high-frequency shear viscosity, η∞, sedimentation coefficient, K, and the short-time translational and rotational self-diffusion coefficients, D t and D r. The simulation results cover the full three-parametric fluid-phase space of the composite particle model, with the volume fraction extending up to 0.45, and the whole range of values for κb, and a/b. Many-particle hydrodynamic interaction effects on the transport properties are explored, and the hydrodynamic influence of the core in concentrated systems is discussed. Our simulation results show that for thin or hardly permeable shells, the core-shell systems can be approximated neither by no-shell nor by no-core models. However, one of our findings is that for κ(b − a) ≳ 5, the core is practically not sensed any more by the weakly penetrating fluid. This result is explained using an asymptotic analysis of the scattering coefficients entering into the multipole method of solving the Stokes equations. We show that in most cases, the influence of the core grows only weakly with increasing concentration.

core-shell particles, suspension, diffusion, sedimentation, effective viscosity

A single Brownian particle of arbitrary shape is considered. The time-dependent translational mean square displacement W(t) of a reference point at this particle is evaluated from the Smoluchowski equation. It is shown that at times larger than the characteristic time scale of the rotational Brownian relaxation, the slope of W(t) becomes independent of the choice of a reference point. Moreover, it is proved that in the long-time limit, the slope of W(t) is determined uniquely by the trace of the translational-translational mobility matrix μtt evaluated with respect to the hydrodynamic center of mobility. The result is applicable to dynamic light scattering measurements, which indeed are performed in the long-time limit.

translational and rotational Brownian motion, mean square displacement, particle of arbitrary shape, mobility center

Dynamics of single flexible non-Brownian fibers, tumbling in a Poiseuille flow between two parallel solid plane walls, is studied with the use of the HYDROMULTIPOLE numerical code, based on the multipole expansion of the Stokes equations, corrected for lubrication. Fibers, which are closer to a wall, more flexible (less stiff) or longer, deform more significantly and, for a wide range of the system parameters, they faster migrate towards the middle plane of the channel. For the considered systems, fiber velocity along the flow is only slightly smaller than (and can be well approximated by) the Poseuille flow velocity at the same position. In this way, the history of a fiber migration across the channel is sufficient to determine with a high accuracy its displacement along the flow.

Stokes equations, flexible fiber, Poiseuille flow, solid walls

Hydrodynamic properties of fibrinogen molecules were theoretically calculated. Their shape was approximated by the bead model, considering the presence of flexible side chains of various length and orientation relative to the main body of the molecule. Using the bead model, and the precise many-multipole method of solving the Stokes equations, the mobility coefficients for the fibrinogen molecule were calculated for arbitrary orientations of the arms whose length was varied between 12 and 18 nm. Orientation averaged hydrodynamic radii and intrinsic viscosities were also calculated by considering interactions between the side arms and the core of the fibrinogen molecule. Whereas the hydrodynamic radii changed little with the interaction magnitude, the intrinsic viscosity exhibited considerable variation from 30 to 60 for attractive and repulsive interactions, respectively. These theoretical results were used for the interpretation of experimental data derived from sedimentation and diffusion coefficient measurements as well as dynamic viscosity measurements. Optimum dimensions of the fibrinogen molecule derived in this way were the following: the contour length 84.7 nm, the side arm length 18 nm, and the total volume 470 nm3, which gives 16% hydration (by volume). Our calculations enabled one to distinguish various conformational states of the fibrinogen molecule, especially the expanded conformation, prevailing for pH < 4 and lower ionic strength, characterized by high intrinsic viscosity of 50 and the hydrodynamic radius of 10.6 nm. On the other hand, for the physiological condition, that is, pH = 7.4 and the ionic strength of 0.15 M NaCl, the semi-collapsed conformation dominates. It is characterized by the average angle equal to = 55, intrinsic viscosity of 35, and the hydrodynamic radius of 10 nm. Additionally, the interaction energy between the arms and the body of the molecule was predicted to be 4 kT units, confirming that they are oppositely charged than the central nodule. Results obtained in our work confirm an essential role of the side chains responsible for a highly anisotropic charge distribution in the fibrinogen molecule. These finding can be exploited to explain anomalous adsorption of fibrinogen on various surfaces.

In a successfully developing system, goal, principles and processes are coupled with each other and with a continuous increase of knowledge. Analyzing these relations, Wilson, Barsky and Daviss proposed universal criteria to assess progress of systemic changes. These criteria can be applied to development of a modern technology or science, as well as to education reforms. The goal of this work is to apply these criteria to analyze effectiveness of the schooling system created by the Polish Commission of the National Education (KEN). We have shown how the overarching goal of education – economical and political development of Poland – was consistent with the KEN basic principles: common goals, freedom and autonomy, ownership of accumulated knowledge. We have demonstrated how KEN kept alive six processes inherent to successful system reforms: knowledge accumulation, sustained professional development within a recognized hierarchy of practitioner’s expertise and transmission of this expertise to others, design of complex entities, improvement of quality, diffusion of innovations, and redesign. In this way, we have provided an example how a similar assessment can be performed for modern educational initiatives, programs, systems and reforms, in contrast to common evaluation of individual teachers and students. Following the best Polish educational tradition, we have reminded that, nowadays, we need to rethink what does it mean to learn and teach productively, and what should be the overarching goal of schooling in the society of the XXI century.

Komisja Edukacji Narodowej, reforma edukacji, procesy charakteryzujące rozwój systemu, sprzężecie celu, procesów, zasad i wiedzy w skutecznym systemie

The motion of a solid spherical particle entrained in a Poiseuille flow between parallel plane walls has various applications to separation methods, like field-flow fractionation and hydrodynamic chromatography. Various handy formulae are presented here to describe the particle motion, with these applications in mind. Based on the assumption of a low Reynolds number, the multipole expansion method coupled to a Cartesian representation is applied to provide accurate results for various friction factors in the motion of a solid spherical particle embedded in a viscous fluid between parallel planes. Accurate results for the velocity of a freely moving solid spherical particle are then obtained. These data are fitted so as to obtain handy formulae, providing e.g. the velocity of the freely moving sphere with a 1% error. For cases where the interaction with a single wall is sufficient, simpler fitting formulae are proposed, based on earlier results using the bispherical coordinates method. It appears that the formulae considering only the interaction with a nearest wall are applicable for a surprisingly wide range of particle positions and channel widths. As an example of application, it is shown how in hydrodynamic chromatography earlier models ignoring the particle-wall hydrodynamic interactions fail to predict the proper choice of channel width for a selective separation. The presented formulae may also be used for modeling the transport of macromolecular or colloidal objects in microfluidic systems.

Creeping flow, Particle, Suspension, Interaction with walls, Separations, Selectivity

In our recent work on concentrated suspensions of uniformly porous colloidal spheres with excluded volume interactions, a variety of short-time dynamic properties were calculated, except for the rotational self-diffusion coefficient. This missing quantity is included in the present paper. Using a precise hydrodynamic force multipole simulation method, the rotational self-diffusion coefficient is evaluated for concentrated suspensions of permeable particles. Results are presented for particle volume fractions up to 45% and for a wide range of permeability values. From the simulation results and earlier results for the first-order virial coefficient, we find that the rotational self-diffusion coefficient of permeable spheres can be scaled to the corresponding coefficient of impermeable particles of the same size. We also show that a similar scaling applies to the translational self-diffusion coefficient considered earlier. From the scaling relations, accurate analytic approximations for the rotational and translational self-diffusion coefficients in concentrated systems are obtained, useful to the experimental analysis of permeable-particle diffusion. The simulation results for rotational diffusion of permeable particles are used to show that a generalized Stokes-Einstein-Debye relation between rotational self-diffusion coefficient and high-frequency viscosity is not satisfied.

self-diffusion, permeable particles, concentrated suspensions

The evolution of a cluster of three identical nontouching spheres settling in a vertical plane due to gravity in Stokes flow has been evaluated numerically for a two-parameter family of initial configurations, with the sphere centers close to each other and located on a horizontal line. A condition has been specified which relates the cluster lifetime to its geometry. The cluster lifetime and the label of the sphere left behind are very sensitive to initial conditions, as within the point-particle approximation of Janosi et al. [Phys. Rev. E 56, 2858 (1997)]. The finding is that an increase of the sphere radii always reduces the range of initial conditions leading to chaotic scattering. The hydrodynamic interactions between close spheres stabilize the cluster.

Stokes flow, particles settling under gravity, hydrodynamic interactions, chaotic scattering, cluster lifetime

The evolution of suspension drops sedimenting under gravity in a viscous fluid close to a vertical wall was studied experimentally and numerically with the use of the point-force model, in the Stokes flow regime. The fluid inside and outside the drop was identical. The initial distribution of the suspended solid heavy particles was uniform inside a spherical volume. In the experiments and in the simulations, the suspension drops evolved qualitatively in the same way as in an unbounded fluid. However, it was observed, both experimentally and numerically, that, on the average, the destabilization time T and the distance L traveled by the drop until break-up were smaller for a closer distance h of the drop center from the wall, with approximately linear dependence of T and L on D/h, for h larger or comparable to the drop diameter D. Destabilization times and lengths of individual drops with different random configurations of the particles were shown to differ significantly from each other, owing to the chaotic nature of the particle dynamics.

suspension drop, Stokes flow, point-particle model, experiment, destabilization time

For suspensions of permeable particles, the short-time translational and rotational self-diffusion coefficients, and collective diffusion and sedimentation coefficients are evaluated theoretically. An individual particle is modeled as a uniformly permeable sphere of a given permeability, with the internal solvent flow described by the Debye-Bueche-Brinkman equation. The particles are assumed to interact non-hydrodynamically by their excluded volumes. The virial expansion of the transport properties in powers of the volume fraction is performed up to the two-particle level. The first-order virial coefficients corresponding to two-body hydrodynamic interactions are evaluated with very high accuracy by the series expansion in inverse powers of the inter-particle distance. Results are obtained and discussed for a wide range of the ratio, x, of the particle radius to the hydrodynamic screening length inside a permeable sphere. It is shown that for x≥10, the virial coefficients of the transport properties are well-approximated by the hydrodynamic radius (annulus) model developed by us earlier for the effective viscosity of porous-particle suspensions.

Stokes equations, hydrodynamic interactions, diffusion, sedimentation, permeable particles, suspesnion, virial expansion

Recent developments in the electrokinetic determination of particle, protein and polyelectrolyte monolayers at solid/electrolyte interfaces, are reviewed. Illustrative theoretical results characterizing particle transport to interfaces are presented, especially analytical formulae for the limiting flux under various deposition regimes and expressions for diffusion coefficients of various particle shapes. Then, blocking effects appearing for higher surface coverage of particles are characterized in terms of the random sequential adsorption model. These theoretical predictions are used for interpretation of experimental results obtained for colloid particles and proteins under convection and diffusion transport conditions. The kinetics of particle deposition and the structure of monolayers are analyzed quantitatively in terms of the generalized random sequential adsorption (RSA) model, considering the coupling of the bulk and surface transport steps. Experimental results are also discussed, showing the dependence of the jamming coverage of monolayers on the ionic strength of particle suspensions. In the next section, theoretical and experimental results pertaining to electrokinetics of particle covered surfaces are presented. Theoretical models are discussed, enabling a quantitative evaluation of the streaming current and the streaming potential as a function of particle coverage and their surface properties (zeta potential). Experimental data related to electrokinetic characteristics of particle monolayers, mostly streaming potential measurements, are presented and interpreted in terms of the above theoretical approaches. These results, obtained for model systems of monodisperse colloid particles are used as reference data for discussion of experiments performed for polyelectrolyte and protein covered surfaces. The utility of the electrokinetic measurements for a precise, in situ determination of particle and protein monolayers at various interfaces is pointed out.

Colloid deposition, Nanoparticle deposition, Particle covered surfaces, Polyelectrolyte deposition, Protein deposition, Streaming potential of covered surfaces

In this paper we aim to create an experimental and numerical model of nano and micro filaments suspended in a confined Poiseuille flow. The experimental data obtained for short nanofibres will help to elucidate fundamental questions concerning mobility and deformation of biological macromolecules due to hydrodynamic stresses from the surrounding fluid motion. Nanofibres used in the experiments are obtained by electrospinning polymer solutions. Their typical dimensions are 100–1000 μm (length) and 0.1–1 μm (diameter). The nanofibre dynamics is followed experimentally under a fluorescence microscope. A precise multipole expansion method of solving the Stokes equations, and its numerical implementation are used to construct a bead-spring model of a filament moving in a Poiseuille flow between two infinite parallel walls. Simulations show typical behaviour of elongated macromolecules. Depending on the parameters, folding and unfolding sequences of a flexible filament are observed, or a rotational and translation motion of a shape-preserving filament. An important result of our experiments is that nanofibres do not significantly change their shape while interacting with a micro-flow. It appeared that their rotational motion is better reproduced by the shape-preserving Stokesian bead model with all pairs of beads connected by springs, omitting explicit bending forces.

Nanofibres suspension, Microchannels, Filament dynamics, Stokesian dynamics, Multipole expansion

We study short-time diffusion properties of colloidal suspensions of neutral permeable particles. An individual particle is modeled as a solvent-permeable sphere of interaction radius a and uniform permeability k, with the fluid flow inside the particle described by the Debye–Bueche–Brinkman equation, and outside by the Stokes equation. Using a precise multipole method and the corresponding numerical code HYDROMULTIPOLE that account for higher-order hydrodynamic multipole moments, numerical results are presented for the hydrodynamic function, H(q), the short-time self-diffusion coefficient, Ds, the sedimentation coefficient K, the collective diffusion coefficient, Dc, and the principal peak value H(qm), associated with the short-time cage diffusion coefficient, as functions of porosity and volume fraction. Our results cover the full fluid phase regime. Generic features of the permeable sphere model are discussed. An approximate method by Pusey to determine Ds is shown to agree well with our accurate results. It is found that for a given volume fraction, the wavenumber dependence of a reduced hydrodynamic function can be estimated by a single master curve, independent of the particle permeability, given by the hard-sphere model. The reduced form is obtained by an appropriate shift and rescaling of H(q), parametrized by the self-diffusion and sedimentation coefficients. To improve precision, another reduced hydrodynamic function, hm(q), is also constructed, now with the self-diffusion coefficient and the peak value, H(qm), of the hydrodynamic function as the parameters. For wavenumbers qa > 2, this function is permeability independent to an excellent accuracy. The hydrodynamic function of permeable particles is thus well represented in its q-dependence by a permeability-independent master curve, and three coefficients, Ds, K, and H(qm), that do depend on the permeability. The master curve and its coefficients are evaluated as functions of concentration and permeability.

Stokes equations, hydrodynamic interactions, self-diffusion, sedimentation, permeable particles, suspension

Evolution of a suspensiondrop entrained by Poiseuille flow is studied numerically at a low Reynolds number. A suspensiondrop is modeled by a cloud of many nontouching particles, initially randomly distributed inside a spherical volume of a viscous fluid which is identical to the host fluid outside the drop. Evolution of particle positions and velocities is evaluated by the accurate multipole method corrected for lubrication, implemented in the HYDROMULTIPOLE numerical code. Deformation of the drop is shown to be smaller for a larger volume fraction. At high concentrations, hydrodynamic interactions between close particles significantly decrease elongation of the suspensiondrop along the flow in comparison to the corresponding elongation of the pure-fluid drop. Owing to hydrodynamic interactions, the particles inside a dense-suspension drop tend to stay for a long time together in the central part of the drop; later on, small clusters occasionally separate out from the drop, and are stabilized by quasiperiodic orbits of the constituent nontouching particles. Both effects significantly reduce the drop spreading along the flow. At large volume fractions, suspensiondrops destabilize by fragmentation, and at low volume fractions, by dispersing into single particles.

Stokes equations, Poiseuille flow, suspension drop, hydrodynamic interactions

We determine the high-frequency limiting shear viscosity in colloidal suspensions of rigid, uniformly porous spheres of radius a as a function of volume fraction and inverse porosity parameter x. Our study covers the complete fluid-state regime. The flow inside the spheres is modeled by the Debye–Bueche–Brinkman equation using the boundary condition that fluid velocity and stress change continuously across the sphere surfaces. The many-sphere hydrodynamic interactions in concentrated systems are fully accounted for by a precise hydrodynamic multipole method encoded in our HYDROMULTIPOLE program extended to porous particles. A truncated virial expansion is used to derive an accurate and easy-to-use generalized Saitô formula for. The simulation data are used to test the performance of two simplifying effective particle models. The first model describes the effective particle as a nonporous sphere characterized by a single effective radius dependent on x. In the more refined second model, the porous spheres are modeled as spherical annulus particles with an inner hydrodynamic radius as a function of x, defining the nonporous dry core and characterizing hydrodynamic interactions, and an outer excluded volume radius a characterizing the unchanged direct interactions. Only the second model is in a satisfactory agreement with the simulation data.

Stokes flow, permeable particles, effective viscosity, lubrication, concentrated suspensions

Hydrodynamic coupling of a spherical particle to an undeformable planar fluid-fluidinterface under creeping-flow conditions is discussed. The interface can be either surfactant-free or covered with an incompressible surfactant monolayer. In the incompressible surfactant limit, a uniform surfactant concentration is maintained by Marangoni stresses associated with infinitesimal surfactant redistribution. Our detailed numerical calculations show that the effect of surface incompressibility on lateral particle motion is accurately accounted for by the first reflection of the flow from the interface. For small particle-interface distances, the remaining contributions are significant, but they are weakly affected by the surface incompressibility. We show that for small particle-wall gaps, the transverse and lateral particle resistance coefficients can be rescaled onto corresponding universal master curves. The scaling functions depend on a scaling variable that combines the particle-wall gap with the viscosity ratio between fluids on both sides of the interface. A logarithmic dependence of the contact value of the lateral resistance function on the viscosity ratio is derived. Accurate numerical calculations are performed using our Cartesian-representation method.

Viscosity, Friction, Lubrication, Liquid liquid interfaces, Surfactants

We have developed a new technique based on our Cartesian-representation method to describe hydrodynamic interactions of a spherical particle with an undeformable planar fluid-fluid interface under creeping-flow conditions. The interface can be either surfactant-free or covered with an incompressible surfactant monolayer. We consider the effect of surface incompressibility and surface viscosity on particle motion. The new algorithm allows to calculate particle mobility coefficients for hydrodynamically coupled particles, moving either on the same or on the opposite sides of the interface.

Stokes equations, hydrodynamic interactions, fluid-fluid interface, surfactant

We study the high-frequency limiting shear viscosity, η∞, of colloidal suspensions of uncharged porous particles. An individual particle is modeled as a uniformly porous sphere with the internal solvent flow described by the Debye–Bueche–Brinkman equation. A precise hydrodynamic multipole method with a full account of many-particle hydrodynamic interactions encoded in the HYDROMULTIPOLE program extended to porous particles, is used to calculate η∞ as a function of porosity and concentration. The second-order virial expansion for η∞ is derived, and its range of applicability assessed. The simulation results are used to test the validity of generalized Stokes–Einstein relations between η∞ and various short-time diffusion coefficients, and to quantify the accuracy of a simplifying cell model calculation of η∞. An easy-to-use generalized Saitˆo formula for η∞ is presented which provides a good description of its porosity and concentration dependence.

Stokes flow, hydrodynamic interactions, permeable particles, dense suspensions, effective viscosity

We calculate short-time diffusion properties of suspensions of porous colloidal particles as a function of their permeability, for the full fluid-phase concentration range. The particles are modeled as spheres of uniform permeability with excluded volume interactions. Using a precise multipole method encoded in the HYDROMULTIPOLE program, results are presented for the hydrodynamic function, H(q), sedimentation coefficient, and self-diffusion coefficients with a full account of many-body hydrodynamic interactions. While self-diffusion and sedimentation are strongly permeability dependent, the wave-number dependence of the hydrodynamic function can be reduced by appropriate shifting and scaling, to a single master curve, independent of permeability. Generic features of the permeable sphere model are discussed.Rychlewski

Stokes equations, hydrodynamic interactions, permeable particles, concentrated suspensions, self-diffusion, hydrodynamic function, collective diffusion

An experiment was performed to test the influence of walls on destabilization of a suspension drop settling under gravity in a viscous fluid in the low Reynolds number regime. The averaged distance the drop settled until destabilization was determined. In was found that in the presence of one or two parallel vertical walls close by, the destabilization length is shorter than in an unbounded fluid.

Low Reynolds number flow, Destabilization of suspension drop

The multiple expansion method was applied for calculating friction tensors and hydrodynamic radii RH of rigid molecules of various shape, composed of ns equal sized, touching spheres. The maximum value of ns studied was 450, which covers most situations met in practice. Calculations were performed for linear chains, half-circles, circles (cyclic molecules) and S-shaped aggregates. It was shown that our results agreed with previous theoretical data obtained for linear chains and cyclic aggregates, for ns < 100. For larger ns, studied exclusively in our work, interpolating analytical expressions were formulated for the hydrodynamic radii RH. These expressions, involving logarithmic function of the aspect ratio parameter (length to width ratio of the macromolecules), are the main finding of our work. Using these expressions, the ratio of the hydrodynamic radius of cyclic-to-linear aggregate qf was calculated, which is a parameter of vital significance. It was determined that qf attained values close to 0.95 for ns ∼450. This suggests that the previous analytical results derived by Tchen [19], in the slender body limit, who predicted that qf → 12/11 = 1.09, are not applicable for ns < 450. Using the RH values, the average translation diffusion coefficients and the sedimentation coefficients for these aggregate shapes were calculated. It was shown that our theoretical results are in good agreement with experimental data obtained for polyelectrolytes and for DNA fragments of various molecular mass. It was concluded that our results can be effectively used to determine the shape of macromolecules, in particular to discriminate between linear and cyclic DNA configurations.

Aggregates of particles of various shapes, Bead model of particle aggregates, Diffusion coefficients of particle aggregates, DNA bead model of, DNA fragment hydrodynamic radii, Hydrodynamic radius of aggregates, Linear chain aggregates, Sedimentation coefficients of aggregates and macromolecules

A fluid flow through a nonisotropic porous medium with an axial symmetry is considered. The Green tensors for the corresponding nonisotropic Debye–Büche–Brinkman equations are calculated in terms of single integrals. Short-distance and far-field limiting behavior is discussed. The exact solution for the Green tensors is found explicitly in the limiting case of an infinite shielding length along the symmetry axis of the system.

Tensor methods, Porous media, Fluid equations, Integral equations, Exact solutions

Un groupe polonais (dont les responsables ont été B. Cichocki durant la période 1996-1997 et M. L. Ekiel-Jeżewska depuis 1998 jusqu’à ce jour, en 2010) et un groupe français (dont le responsable est F. Feuillebois) collaborent depuis 1996 dans le cadre des échanges entre le CNRS (Laboratoires PMMH jusqu’à fin 2009, puis maintenant LIMSI) et l’Académie des Sciences de Pologne (IPPT PAN). Le domaine d’étude de cette collaboration en Mécanique des Fluides concerne les suspensions de particules dans des fluides visqueux et en particulier les interactions hydrodynamiques dans les suspensions.

suspensions, hydrodynamic interactions, viscous fluids, microscale

Na podstawie archiwaliów dotyczących oświaty i życia społecznego w powiecie mińskim odnalezionych w mieszkaniu nauczycieli Janiny i Józefa Sylwestrowiczów opowiedziana zostaje ciekawa historia Biblioteki Oddziału Powiatowego Związku Nauczycielstwa Polskiego w Mińsku Mazowieckim w okresie od lat trzydziestych po pięćdziesiąte. Opowieść ta ilustruje, jak ważną rolę może mieć udostępnienie nowych, nieznanych dotąd i nie opracowanych dokumentów w ramach nowej stałej wystawy w Muzeum Ziemi Mińskiej. Otóż analiza materiałów źródłowych przez młodzież szkolną i pedagogów w ramach warsztatów edukacyjnych wykorzystujących te zbiory może przyczynić się do głębszego poznania historii regionu, a zarazem stworzenia nowych atrakcyjnych form kształcenia.

biblioteka nauczycielska, muzeum, edukacja

W artykule na fotografiach pokazujemy fragment tworzonej właśnie w Muzeum Ziemi Mińskiej nowej wystawy stałej, a mianowicie skromnie wyposażoną kuchnię i łazienkę polskiego mieszkania nauczycieli w okresie międzywojennym, a obok analogiczne wnętrza stanowiące ekspozycję stałą Muzeum Nissim de Camondo w Paryżu.

Celem tworzonej wystawy stałej jest przedstawienie postaw obywatelskich, działań, dziejów i realiów życia Polaków w I połowie XX wieku, ze szczególnym podkreśleniem roli nauczycieli w budowaniu systemu polskiej państwowości po I wojnie światowej – tworzeniu kultury, samorządności i powszechnej edukacji. Wiodącą koncepcją wystawy jest pokazanie tych procesów na przykładzie jednej rodziny nauczycieli. Pokazanie realiów codzienności - m. in. poprzez wyposażenie kuchni i łazienki - zilustruje typową dla dwudziestolecia międzywojennego oszczędność i funkcjonalność, a ponadto pomoże zwiedzającym poczuć się w tym mieszkaniu jak „u siebie” a dzięki temu ułatwi identyfikację z pokazanymi na wystawie postawami życiowymi i pomoże w zrozumieniu procesów historycznych oraz kontekstu skutcznej edukacji.

edukacja, oszczędność, tradycje i wartości II RP

Wielowątkowa kolekcja oświatowa Sylwestrowiczów jest zalążkiem, który stwarza szansę na pokazanie w Muzeum Ziemi Mińskiej szerokim kręgom zainteresowanych, na czym polegało życie i praca przedwojennych polskich nauczycieli. Na nowej stałej wystawie można tu będzie usłyszeć barwną opowieść, ilustrowaną prawdziwymi rekwizytami tej właśnie rodziny, ich zdjęciami i dokumentami, odwiedzić pokoje, z których „jakby przed chwilą wyszli” dawni mieszkańcy, posłuchać wspomnień z tamtych lat. A studentów i pedagogów z pewnością zaciekawi wzięcie udziału w lekcji szkolnej i poeksperymentowanie zgodnie z treścią podręcznika czy zapisków nauczyciela, a także zapoznanie się na podstawie dokumentacji z przebiegiem kursów pedagogicznych i rolą bibliotek1 oraz przestudiowanie przedwojennych dokumentów.

Nauka, edukacja, muzeum, tradycja, pedagog

Streaming potential changes induced by deposition of particles at solid/liquid interfaces are considered theoretically. The solution is obtained in terms of a virial expansion of the streaming potential up to the third order of the surface coverage of particles, assumed to be distributed according to the hard sphere equilibrium distribution function. Theoretical methods, including the idea of cluster expansion, are adopted from statistical physics. In the cluster expansion, two-body and three-body hydrodynamic interactions are evaluated with a high precision using the multipole method. The multipole expansion algorithm is also used to perform numerical simulations of the streaming potential, valid for the entire surface coverage range met in practice. Results of our calculations are in good agreement with the experimental data for spherical latex particles adsorbed on a mica surface.

Streaming current, streaming potential, particle-covered wall, Stokes equations, hydrodynamic interactions, multiple expansion, viral exapnsion

Self-diffusion of a sphere in a network of rods is analyzed theoretically.Hydrodynamic interactions are taken into account according to the model of Dhont et al. [J. Chem. Phys.122, 044905 (2005); Dhont et al., J. Chem. Phys.124, 044907 (2006); Dhont et al., J. Chem. Phys.126, 214501 (2007)] based on the Debye–Bueche–Brinkman equation. The hydrodynamic screening length of the effective medium is assumed to be much larger than the sphere radius and the rod thickness. The self-diffusion coefficient, given by Dhont et al. in terms of four-dimensional integrals, is in this work expressed in terms of a single integral only and therefore evaluated numerically with a high precision. Moreover, simple expressions for the self-diffusion coefficient are derived and shown to be independent of the rod length. They can be useful for experimental verification of the model.

Hydrodynamics, Friction, Hydrological modeling, Self diffusion, Tensor methods

It is shown that non-symmetric microobjects orient while settling under gravity in a viscous fluid. To analyze this process, a simple shape is chosen: a non-deformable 'chain'. The chain consists of two straight arms, made of touching solid spheres. In the absence of external torques, the spheres are free to spin along the arms. The motion of the chain is evaluated by solving the Stokes equations with the use of the multipole method. It is demonstrated that the spinning beads speed up sedimentation by a small amount and increase the orientation rate significantly in comparison to the corresponding rigid chain. It is shown that chains orient towards the V-shaped stable stationary configuration. In contrast, rods and star-shaped microobjects do not rotate. The hydrodynamic orienting is relevant for efficient swimming of non-symmetric microobjects and for sedimenting suspensions.

Stokes flow, hydrodynamic interactions, particles settling under gravity, hydrodynamic orientinfg

Clusters made of a small number of close solid spherical particles at a random configuration, sedimenting through a viscous fluid at small Reynolds number, were experimentally investigated at a short-time scale. The cluster settling velocities were measured and shown to be well approximated by the ensemble-averaged formula derived earlier for the uniform distribution of the point particles inside a spherical volume. It was emphasized that the “effective radius” of this volume in general should be smaller than the actual radius of a cluster made of the spheres, and the relation between both radii was determined. The formula was also shown to account well for the gravitational settling of rigid conglomerates, measured and computed elsewhere.

cluster of sendimenting particles, small Reynolds number

An efficient procedure with a controlled high accuracy, the spherical-multipole method, is presented, adequate for evaluating Stokesian dynamics of non-deformable spherical particles suspended in a fluid, or hydrodynamic resistance of moving or motionless systems of such particles under low-Reynolds-number flows. We present a few examples of specific applicatins of this method. In this procedure, the relative motion of particles is corrected for lubrication, to achieve fast convergence with the multipole order of the truncation. The main advantage of this algorithm is that it is possible to perform computations with a very high multipole order of the truncation, controlling the accuracy. Moreover, the method is applicable to systems of various types of the particles in a fluid bounded by one or two parallel flat interfaces.

Stokes equations, hydrodynamic interactions, multipole expansion

A planar hard surface covered with elongated stiff rodlike particles in shear flow is considered in the low-Reynolds-number regime assuming low particle surface coverage. The particles are modeled as straight chains of spherical beads. Multipole expansion of the Stokes equations (the accurate HYDROMULTIPOLE algorithm) is applied to evaluate the hydrodynamic force exerted by the fluid on the rodlike particles, depending on their shape, i.e., on the number of beads and their orientation with respect to the wall and to the ambient shear flow.

Stokes flow, hydrodynamic interactions, rigid rod, solid wall, hydrodynamic friction

Lubrication expressions for the friction coefficients of a spherical particle moving in a fluid between and along two parallel solid walls are explicitly evaluated in the low-Reynolds-number regime. They are used to determine lubrication expression for the particle free motion under an ambient Poiseuille flow. The range of validity and the accuracy of the lubrication approximation are determined by comparing with the corresponding results of the accurate multipole procedure. The results are applicable for thin, wide, and long microchannels, or quasi-two-dimensional systems.

Lubrication, Friction, Poiseuille flow, Particle velocity, Fluid equations

Stokesian dynamics simulations of close particles are reported, taking into account lubrication forces and many-body hydrodynamic interactions between spheres. A periodic trajectory of three particles maintaining a permanent proximity to each other has been found and analyzed. This solution is used as a benchmark to study the accuracy and stability of various numerical integration schemes. In particular, different methods of preventing unphysical overlaps of the particles are considered and potential artifacts discussed.

Kinematics, Renormalization, Surface dynamics, Lubrication, Hydrodynamics

In daily life, at the macroscopic scale, inertia of water flows is commonly experienced and applied. However, for micro-objects moving in water environment, inertia is negligible. That is why sometimes the processes observed at the microscale are in contradiction with our intuition. The examples of such amazing effects can be seen on the famous G.I. Taylor’s movie. The aim of this paper is to analyze the most spectacular of the demonstrations presented there: apparently mixed fluid recovers its initial inhomogeneous structure when the direction of the mixing is reversed.

Lepki płyn, odwracalność mikroprzepływów, bezwładność

Praca zawiera opis badań zjawiska opadania cząstek kulistych w cieczy o dużej lepkości, a w szczególności wzajemnych oddzaiływań między poszczególnymi, opadającymi razem cząstkami. Opisano stanowisko badawcze, sposób przeprowadzenia pomiarów oraz przedstawiono wybrane wyniki badań.

gliceryna, kilka cząsteki szklanych, eksperyment, trajektorie cząstek, dynamika

Hydrodynamic interactions between spheres immersed in a low-Reynolds-number fluid flow close to a flat free surface or hard wall are investigated. The spheres may have different or equal radii, and may be separated from the boundary or at contact with the free surface. A simple and useful expression is derived for the propagator (Green operator) connecting centers of two spheres. In the derivation, the method of images and the displacement theorems are used. Symmetry of the displacement operators is explicitly shown. The significance of these results in efficient Stokesian and Brownian dynamics simulations is outlined. An example of an application is shown.

Free surface, Hydrodynamics, Friction, Fluid flows, Mirrors

Mathematical structure of the linearized Boltzmann equation is outlined. Physical interpretation of the assumed relativistic cross-sections is discussed.

relativistic Boltzmann equation, relativistic cross-section

Dynamics of three identical solid spheres falling under gravity in low-Reynolds-number fluid flow is investigated. Stationary solutions are found. Their stability is discussed. Phase portraits for two types of symmetric motions are calculated.

Stokes equations, hydrodynamic interactions, particles settling under gravity

Statistical mechanics is applied to calculate ensemble-averaged particle and fluid velocity fields of a spherical cloud of point particles sedimenting at a low Reynolds number. The analogy with the fall of a liquid drop in another lighter fluid is discussed.

Fluid drops, Tensor methods, Fluid equations, Hydrodynamics, Reynolds stress modeling

Exemplary dynamics of three point particles falling under gravity in Stokes flow with periodic boundary conditions is presented in a movie. Stable and unstable solutions of the equations of Stokesian dynamics are explicitly shown for two initial configurations: equilateral triangles with side lengths close to a critical size.

Kinematics, Suspensions, Boundary value problems, Viscosity, Anisotropy

An example system is studied to discuss precision of the multipole expansion, applied to determine forces exerted on particles by a viscous low-Reynolds-number fluid flow. A single sphere in an ambient flow (pure shear, quadratic, and modulated shear) parallel to a close plane wall is considered. Forces and torques exerted by the ambient flow on a motionless sphere are evaluated. Their precision is determined and related to a multipole order of the truncation. Similar analysis is performed for a moving sphere with no ambient flow and for a freely moving sphere. Relative motion of the sphere with respect to the wall gives rise to strong lubrication interactions. It is analysed how these interactions affect accuracy of the pure multipole expansion, and what are the smallest distances where it becomes insufficient. An alternative precise method is applied, in which lubrication expressions are subtracted from the hydrodynamic forces and torques, and the residue is evaluated as a fast-convergent series of inverse powers of the distance between the sphere centre and the wall. The accuracy of this procedure is carefully analysed.

Stokes flow, lubrication, multipole method

Evolution of three identical solid spheres falling under gravity in a low-Reynolds number flow is investigated for a symmetric initial configuration. Three spheres aligned horizontally at equal distances evolve towards an equilibrium relative configuration while the point particles collapse onto a single point in a finite time.

Stokes equations, hydrodynamic interactions, particles settling under gravity

Solutions of the equations of Stokesian dynamics for point particles are found for periodic boundary conditions with three particles per unit cell of a simple cubic lattice. Two particles per cell move with equal velocity, but three particles per cell usually lead to irregular motion. For a class of initial conditions with special symmetry motions are found that are periodic in time as well as in space. It is shown that there is a range of stability in which the motions are robust under perturbation.

Tensor methods, Boundary value problems, Eigenvalues, Periodic solutions, Equations of motion

General expressions for the frequency-dependent Brownian contribution to the intrinsic viscosity of arbitrary-shaped particles have been derived from the Smoluchowski equation.

intrinsic viscosity, Brownian motion, particle of arbitrary shape

A rigid polymer model consisting of two identical spherical particles, irreversibly adsorbed on a planar channel wall is considered. The linear dimensions of the polymer are assumed to be small enough to validate an expansion of the flow inside the channel to linear terms only. The polymer is therefore effectively immersed in a shearing flow. Total hydrodynamic force acting on the polymer is calculated. The dependence of that force on the polymers orientation is derived taking into account symmetries of the system. Large differences in the force are found for different polymer orientations. Applications of these results are sketched.

Stokes equations, hydrodynamic interactions, hydrodynamic friction, rigid rod, plane wall

A family of periodic in time solutions of the Stokesian dynamics has been found numerically for three identical spherical particles settling under gravity in a vertical plane. The particles stay close to each other and hydrodynamic interaction between their surfaces is essential to detect the oscillations, which have not been found in the point-particle simulations. The periodic orbits are responsible for 'chaotic scattering' of the particles, which come and stay for some time sufficiently close to these orbits before 'escaping'. The existence of such periodic solutions is of fundamental importance for understanding evolution of many-particle systems, and also as a benchmark for testing numerical procedures used to evaluate the Stokesian dynamics.

Stokes equations, hydrodynamic interactions, particles settling under gravity, periodic orbits

Stokes equations, Euler-Bernoulli beam, elastica, elastic filament, shear flow, buckling

Stokes equations, Euler-Bernoulli beam, elastica, elastic filament, shear flow, buckling

fluid dynamics, Stokes flows, particulate flows, hydrodynamic interactions, bending

Dynamics of a single non-Brownian flexible fiber in shear flow at low Reynolds number is investigated numerically. Initially, the fiber is straight and at the equilibrium. For different initial orientations and values of bending stiffness, three generic scenarios are observed: the fiber tends to: align along the vorticity direction, tumble within the plane perpendicular to vorticity, or perform a periodic motion superposed with translation along the flow.

Stokes equations, flexible fibers, shear flow

Dynamics of flexible fibers and vesicles in unbounded planar Poiseuille flow at the low-Reynolds-number are shown to exhibit similar basic features, when their equilibrium (moderate) aspect ratio is the same and vesicle viscosity contrast is relatively high. The lateral migration and accumulation of these two types of flexible objects are analyzed numerically.

Stokes equations, vesicles, flexible fibers

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

Modes of the dynamics of flexible fibers in shear flow in a plane perpendicular to vorticity are analyzed numerically.

Dynamics of flexible fibers in shear flow are analyzed numerically for a wide range of the ratios A of the fiber bending force to the hydrodynamic force. The Reynolds number is much smaller that unity, and the Péclet number much greater than one. A fiber is modeled as a chain of solid beads as in Ref. [1]. The centers of the consecutive beads are linked by springs. The equilibrium length of each spring is such that the consecutive beads almost touch each other. The spring constant is large, and the fiber’s length practically does not change with time. At the equilibrium, the fiber is straight. At a deformed configuration, there appear a bending force exerted on each bead, proportional to the bending parameter A. The details of the model and the numerical method can be found in Ref. [2]. Velocities of the beads at given positions are evaluated with the use of the HYDROMULTIPOLE numerical code, based on solving the Stokes equations by the multipole expansion [3]. The time-dependent positions of the beads are determined by the adaptive fourth order Runge-Kutta method.

Initially, the fiber is aligned with the flow, and the springs are at the equilibrium. Owing to symmetry, the centers of all the beads move in the plane perpendicular to the vorticity direction. The fiber end-to-end vector tumbles, in a similar way as a rigid elongated body [4]. While the fiber turns, its shape evolves accordingly, and the center-of-mass oscillates across the flow. A surprisingly rich spectrum of different modes is observed when the value of A is systematically changed, with regular and chaotic trajectories. (For the details, see Ref. [5].)

For some ranges of small and large values of A, the center-of-mass trajectories are periodic with a single tumbling time τ, and there is no migration across the flow. For a certain range of small values of A, evolutions during every second tumbling time are the same, but consecutive tumbling times differ from each other, what leads to regular migrating trajectories of the fiber center-of-mass. For moderate values of A, a chaotic behavior is observed - a large sensitivity of the dynamics to a small change of A, with (typically) many irregular, erratic trajectories or (exceptionally) some regular migrating trajectories. At a moderate value of A, a transition is observed between a ‘straightening out’ mode of more stiff fibers to the coiled mode of more flexible fibers. In the straightening out mode, the fiber significantly changes its shape while tumbling - from almost straight and aligned with the flow to S-shaped. In the coiled mode, the fiber is always compact, it never aligns with the flow.

References
[1] E. Gauger, H. Stark, Numerical study of a microscopic artificial swimmer, Phys. Rev. E 74, 021907 (2006).
[2] A. M. Słowicka, E. Wajnryb, M. L. Ekiel-Jeżewska, Lateral migration of flexible fibers in Poiseuille flow between two parallel planar solid walls, Eur. Phys. J. E 36, 1-12 (2013).
[3] B. Cichocki, M. L. Ekiel-Jeżewska, E. Wajnryb, Lubrication corrections for three-particle contribution to short-time self-diffusion coefficients in colloidal dispersions, J. Chem. Phys. 111, 3265 (1999).
[4] G. Jeffery, The Motion of Ellipsoidal Particles Immersed in a Viscous Fluid, Proc. R. Soc. Lond. A. 102, 161-179 (1922).
[5] A. M. Słowicka, E. Wajnryb, M. L. Ekiel-Jeżewska Dynamics of flexible fibers in shear flow, arXiv [cond-mat.soft] (2015).

Stokes equations, flexible fibers, shear flow

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

In this work, we consider a single non-Brownian mobile and flexible fiber immersed in Poiseuille flow in a channel consisting of two parallel infinite walls. The dynamics of the fiber is evaluated numerically from the Stokes equations by a multipole code HYDROMULTIPOLE. Investigating the fiber dynamics we found out that fibers migrate to a critical position across the channel. The distance between the wall and a limiting position depends on the fiber elongation and flexibility. For more stiff fibers the critical position results from the interplay between their tendency to drift away from the channel and the repulsive hydrodynamic interaction with the wall. For less stiff fibers the limiting position is not influenced by the presence of the wall. Differences between the critical position for different fibers can be used in the process of microfibers separation by the flow.

Stokes equations, hydrodynamic interactions, Poiseuille flow, flecible fiber

Using the point-force model, we analyze how evolution of a suspension drop settling under gravity in a viscous fluid is influenced by the presence of a vertical wall near by. In particular, we show that a close drop moves away from the wall while settling along.

Stokes equations, suspension drop, sedimentaiton, vertical wall, point particles, hydrodynamic interactions

Stokes equations, Poiseuille flow, flexible fibers, migration

Short-time dynamic properties of concentrated suspensions of colloidal core-shell particles have been recently studied [1] using a precise force multipole method which accounts for many-particle hydrodynamic interactions (HIs). A core-shell particle is composed of a rigid, spherical dry core of radius a surrounded by an uniformly permeable shell of outer radius b and hydrodynamic penetration depth κ-1. The solvent flow inside the permeable shell is described by the Brinkman-Debye-Bueche equation, and outside the particles by the Stokes equation. The particles are assumed to interact non-hydrodynamically by a hard-sphere no-overlap potential of radius b. Numerical results are presented for the high-frequency shear viscosity, sedimentation coefficient and the short-time translational and rotational self-diffusion coefficients. The simulation results cover the full three-parametric fluid-phase space of the composite particle model, with the volume fraction extending up to 0.45, and the whole range of values for κb, and a/b. Many-particle hydrodynamic interaction effects on the transport properties are explored, and the hydrodynamic influence of the core in concentrated systems is discussed.

Stokes equations, Brinkman-Debye-Bueche equations, permeable particles, translational and rotational self-diffusion, sedimentation, effective viscosity

W niniejszej prezentacji zostanie pokazany wpływ płaskiej pionowej ściany na ewolucję kropli zawiesiny opadającej pod wpływem siły grawitacyjnej w lepkim płynie dla liczb Reynoldsa znacznie mniejszych od jedności. Płyn wewnątrz i na zewnątrz opadającej kropli był identyczny, a cząstki początkowo były przypadkowo rozmieszczone wewnątrz sferycznej objętości. Przeanalizowana zostanie ewolucja opadającej kropli złożonej z cząstek. Krople podczas opadania „gubiły” cząstki, które jako wolniejsze pozostawały z tyłu, tworząc ponad kroplą „ogon”, powoli się spłaszczały i rozszerzały, a potem nagle dzieliły na dwa lub więcej fragmentów. Średnia odległość przebyta przez opadającą kroplę zawiesiny aż do rozpadu jest tym któtsza, im mniejsza jest odległość h od środka ściainy. Badanie ewolucji kropli wykonane zostało za pomocą modelu cząstek punktowych w oparciu o funkcje Greena dla równań Stokesa.

liczba Reynoldsa, model cząstek punktowych, równania Stokesa