Sedimentation of hard-sphere suspensions at low Reynolds number (original) (raw)
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Interactions between hard spheres sedimenting at low Reynolds number
European Journal of Mechanics - B/Fluids, 2005
The present article reports on experimental and numerical study for sedimentation of non-colloidal suspended particles in a viscous fluid. The hydrodynamic interactions between several particles sedimenting at low Reynolds numbers have been investigated. The numerical model is based on separation of the real velocity field into two parts. The first part is symmetrical and represents the conventional Stokes contribution. The second non-symmetrical part represents the net inertial contribution. The first contribution has been modelled using the Stokesian Dynamics method. Whereas the second one has been accounted for by assuming the validity of Faxen's first law. The numerical results agree with those from literature in the limit of the Stokes' flow regime (Re = 0). Unfortunately, when Re > 0, the simulation appears to quantitatively overestimate the influence of the real inertial effects. However, the accuracy of the results has been observed to be improved by introducing the empirical correction proposed by Happel et al. at the level of the individual particle. In this way, the proposed numerical tool becomes able to determine accurately the behaviour of several particles in different configurations. The agreement between the simulation results and the experimental ones is very satisfactory.
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Several results on sedimenting equal spheres obtained by resolved simulations with the Physalis method are presented. The volume fraction ranges from 8.7% to 34.9% and the particle Galilei number from 49.7 to 99.4. The results shown concern particle collisions, diffusivities, mean free path, particle pair distribution function and other features. It is found that many qualitative trends found in earlier studies continue to hold in the parameter range investigated here as well. The analysis of collisions reveals that particles interact prevalently via their flow fields rather than by direct contacts. A tendency toward particle clustering is demonstrated. The time evolution of the shape and size of particle tetrads is examined.
FLUIDS 4 , 014304 ( 2019 ) Resolved simulations of sedimenting suspensions of spheres
2019
Daniel P. Willen1,* and Andrea Prosperetti2,3,† 1Department of Mechanical Engineering, Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218, USA 2Department of Mechanical Engineering, University of Houston, 4726 Calhoun Road, Houston, Texas 77204-4006, USA 3Faculty of Science and Technology and J.M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 Enschede, Overijssel, Netherlands
Physics of Fluids, 2000
Non-Brownian sedimenting suspensions exhibit density and velocity fluctuations. We have performed experiments on a quasi-two-dimensional counter-flow stabilized suspension of 2000 spherical particles, namely a liquid-solid fluidized bed in a Hele-Shaw cell. This two-dimensional suspension displays a uniform concentration but the particle radial distribution function and the fluctuations of the particle number in a subvolume of the suspension suggest that the microstructure is far from being random. We have also measured the velocity fluctuations of a test particle and the fluctuations of the mean particle velocity in a subvolume. It happens that the relation between velocity and concentration fluctuations in a subvolume can be deduced from a balance between buoyancy and parietal friction forces.
Fluctuations and stratification in sedimentation of dilute suspensions of spheres
Physics of Fluids, 2009
We have tested in experiments and simulations whether stratification can control velocity fluctuations in suspensions of sedimenting spheres. The initial value and early decay of the velocity fluctuations are not affected by stratification. On the other hand, in the descending front where the stratification is strong and well defined, the velocity fluctuations are inhibited according to a previously proposed scaling. In between, after the initial decay and before the arrival of the front, the local value of the stratification does not always play a role.
The Effect of Many-Body Interactions on the Sedimentation of Monodisperse Particle Dispersions
Journal of Colloid and Interface Science, 1998
interactions include many-body thermodynamic interactions An experimental investigation was made of the sedimentation and many-body hydrodynamic interactions. Many-body rate of low-charged monodisperse silica and polystyrene latex parthermodynamic interactions are due to depletion and structicle dispersions as a function of the particle volume fraction. tural forces. It was found that the normalized sedimentation velocity U/U 0 , Since the ability of nonadsorbing polymers to promote corrected for the effect of the two-body hydrodynamic interaction, flocculation of a colloidal dispersion due to the many-body increases with the particle volume fraction, which indicates that thermodynamic interaction was first discovered 70 years ago, the degree of particle aggregation inside the dispersions increases numerous experiments and theories (1-15) have been conwith the particle volume fraction. This phenomenon results from ducted and developed to explain the depletion interaction attractive many-body hydrodynamic interactions between colloidal particles. It is reported for the first time that the many-body resulting from free (nonadsorbing) polymers, micelles, or hydrodynamic interaction becomes important at the particle consmall colloidal spheres. The depletion force arises whenever centration of 6.5 vol% in monodisperse dispersions, and the manythe concentration of nonadsorbing polymers, micelles, or body thermodynamic interaction is negligible at a low particle hard spheres in the gap region between two particles beconcentration, i.e., less than 15 vol%. The effect of many-body comes different from that in the bulk. At small gap widths, hydrodynamic interaction on the particle microstructure was also the nonadsorbing polymers, micelles, or hard spheres are experimentally examined by using a nondestructive Kossel diffracexcluded from the gap, resulting in an attractive force due tion technique based on the principle of back-light scattering. It to the difference in osmotic pressure. was found that the particle packing structure inside the dispersion Recently, it was found experimentally (16-24) and theoinitially becomes more ordered with the increase of the particle retically (21, 25-27) that free (nonadsorbing) polymers, volume fraction. However, there is less increase in the particle micelles, or small colloidal spheres not only can destabilize ordering structure after 6 vol%. Furthermore, after the particle concentration reaches 10 vol%, the particle packing structure de-a colloidal dispersion, but also can stabilize it by forming a creases to a value lower than that of 6 vol% due to the increased layering structure between colloidal particles under certain particle aggregation, as found in the sedimentation experiments. conditions: low concentrations of fine particles can destabi-Predictions of a statistical thermodynamic model were compared lize the system and enhance the fluid-solid phase transitions with the experimental data on structure factors. It is found that and separations. This effect has been rationalized in terms particle dimerization occurs around 10 vol%, which agrees with of the attractive depletion force. At higher concentrations the sedimentation results. ᭧ 1998 Academic Press of fine particles, surface-induced structural forces prevent
Anisotropic Velocity Fluctuations and Particle Diffusion in Sedimentation
Journal of the Physical Society of Japan, 2013
Direct numerical simulations were performed using a smooth profile method to investigate the steady-state sedimentation of monodisperse spherical particles in an incompressible fluid at finite Peclet numbers 0 Pe 115. Hydrodynamic interactions caused strong fluctuations in the instantaneous velocity of the particles around the mean settling velocity. We found that the amplitude of these velocity fluctuations increases in direct proportion with the square of the Stokes velocity at higher Peclet numbers, where sedimentation is dominated by non-equilibrium hydrodynamic fluctuations. The diffusive behaviour of the particles was observed to be in a steady state over long time scales, and the steady-state self-diffusion coefficient was found to increase linearly with the Peclet number. Our results provide new insights into the anisotropy of vertical and horizontal diffusion. This anisotropy increases with an increasing Peclet number, plateauing at a high Peclet number.
Three-particle contribution to sedimentation and collective diffusion in hard-sphere suspensions
Chemical Physics, 2002
The virial expansion of the collective mobility ͑sedimentation͒ coefficient is considered for hard sphere suspensions at equilibrium. The term of the second order in volume fraction, which involves three-particle hydrodynamic interactions, is calculated with high accuracy. To achieve that we represent the collective mobility coefficient as the sum of convergent integrals over particle configurations. In this way the short-wave-number limit k→0 is avoided. Moreover, an efficient numerical procedure is applied to evaluate the hydrodynamic interactions. The algorithm is based on the multipole expansion, corrected for lubrication. The method allows us to analyze contributions to the collective mobility coefficient from different configurations of three particles and to select the dominant part. This suggests a general approximation scheme.
Sedimentation dynamics and equilibrium profiles in multicomponent mixtures of colloidal particles
Journal of Physics: Condensed Matter, 2014
In this paper we give a general theoretical framework that describes the sedimentation of multicomponent mixtures of particles with sizes ranging from molecules to macroscopic bodies. Both equilibrium sedimentation profiles and the dynamic process of settling, or its converse, creaming, are modeled. Equilibrium profiles are found to be in perfect agreement with experiments. Our model reconciles two apparently contradicting points of view about buoyancy, thereby resolving a long-lived paradox about the correct choice of the buoyant density. On the one hand, the buoyancy force follows necessarily from the suspension density, as it relates to the hydrostatic pressure gradient. On the other hand, sedimentation profiles of colloidal suspensions can be calculated directly using the fluid density as apparent buoyant density in colloidal systems in sedimentation-diffusion equilibrium (SDE) as a result of balancing gravitational and thermodynamic forces. Surprisingly, this balance also holds in multicomponent mixtures. This analysis resolves the ongoing debate of the correct choice of buoyant density (fluid or suspension): both approaches can be used in their own domain. We present calculations of equilibrium sedimentation profiles and dynamic sedimentation that show the consequences of these insights. In bidisperse mixtures of colloids, particles with a lower mass density than the homogeneous suspension will first cream and then settle, whereas particles with a suspension-matched mass density form transient, bimodal particle distributions during sedimentation, which disappear when equilibrium is reached. In all these cases, the center of the distribution of the particles with the lowest mass density of the two, regardless their actual mass, will be located in equilibrium above the so-called isopycnic point, a natural consequence of their hard-sphere interactions. We include these interactions using the Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state. Finally, we demonstrate that our model is not limited to hard spheres, by extending it to charged spherical particles, and to dumbbells, trimers and short chains of connected beads.