Hindered settling and hydrodynamic dispersion in quiescent sedimenting suspensions

Ham, J.M.; Homsy, G. M.

doi:10.1016/0301-9322(88)90056-0

Cited by 173 publications

(126 citation statements)

References 11 publications

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“…In fact, Koch & Shaqfeh (1991) predict that long-range hydrodynamic interactions cause the hydrodynamic diffusivity to increase with decreasing concentration. Experiments with nearly monodisperse suspensions by Davis & Hassen (1988) and Ham & Homsy (1988) gave hydrodynamic diffusivities of O(aU,) which increase with increasing concentration for dilute suspensions, reach a maximum at a concentration of only a few percent by volume, and then decrease with further concentration increases. It may be that the initial increase at low concentrations is due to a change in mechanism from pairs separating due to the small differences in particle sizes to pairs separating because of encountering a third particle.…”

Section: Discussionmentioning

confidence: 99%

“…Of additional interest is the possibility of predicting the velocity variance and the coefficient of hydrodynamic diffusivity, which are measures of the fluctuations in the settling speeds of individual sedimenting particles. Hydrodynamic diffusion during batch sedimentation of nearly monodisperse suspensions has been observed experimentally by Davis & Hassen (1988), who measured the rate of spreading of the interface a t the top of the suspension, and by Ham & Homsy (1988), who measured the variance in the time for a marked sphere in the interior of the suspension to fall through a given distance. It should be emphasized that the phenomenon of hydrodynamic diffusion arises from hydrodynamic interactions between particles in the suspension fluid, and is unrelated to Brownian diffusion which arises from the thermal motion of the fluid molecules surrounding each particle.…”

Section: Introductionmentioning

confidence: 99%

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Hydrodynamic diffusion of a sphere sedimenting through a dilute suspension of neutrally buoyant spheres

Davis

Hill

1992

J. Fluid Mech.

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The motion of a heavy sphere sedimenting through a dilute background suspension of neutrally buoyant spheres is analysed for small Reynolds number and large PBclet number. For this particular problem, it is possible not only to calculate the mean velocity of the heavy particle, but also the variance of the velocity and the coefficient of hydrodynamic diffusivity . Pairwise, hydrodynamic interactions between the heavy sphere and the background sphere are considered exactly using volume integrals and a trajectory analysis. Explicit formulae are given for the two limiting cases when the radius of the heavy sphere is much greater and much less than that of the background spheres, and numerical results are given for moderate size ratios. The mean velocity is relatively insensitive to the ratio of the radius of the background spheres to that of the heavy sphere, unless this ratio is very large, whereas the hydrodynamic diffusivity increases rapidly as the radius ratio is increased. The predictions are in reasonable agreement with the results of falling-ball rheome try experiments.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hydrodynamic diffusion of a sphere sedimenting through a dilute suspension of neutrally buoyant spheres

Davis

Hill

1992

J. Fluid Mech.

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“…The predicted velocity fluctuation divergence received both theoretical and experimental scrutiny without consensus [5,[9][10][11][12][13][14][15][16]. Computer simulations support the conclusion, finding an increase in the amplitude of velocity fluctuations with system size [9,10].…”

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confidence: 89%

“…These simulations relax assumptions (ii) and (iii), since they approximate many particle hydrodynamics and the system may evolve via hydrodynamic interactions away from initially randomized configurations. On the other hand, experiments find no dependence on system size, either experiments measuring the diffusion of a test particle [11][12][13] or direct measurements of the velocity fluctuations [7]. Koch and Shaqfeh [14] gave the first explanation for the absence of divergent velocity fluctuations in terms of a violation of assumption (iii), where every particle has a net deficit of one particle surrounding it within a correlation range j.…”

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confidence: 99%

Settling Statistics of Hard Sphere Particles

2001

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Direct imaging of settling, non-Brownian, hard sphere, particles allows measurement of particle occupancy statistics as a function of time and sampling volume dimension. Initially random relative particle number fluctuations, ͗N 2 2 ͗N͘ 2 ͗͘͞N͘ 1, become suppressed, anisotropic, and ͗N͘ dependent. Fitting to a simple Gaussian pair correlation model suggests a minute long ranged correlation leads to strong if not complete suppression of number fluctuations. Calflisch and Luke predict a divergence in velocity fluctuations with increasing sample volume size based on random (Poisson) statistics. Our results suggest this is not a valid assumption for settling particles.

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“…Historically, the first was the three-parameter Markov model [107,112], which used the variance and autocorrelation of velocity as additional parameters. A decade later, a model was developed [55] (see also [37]) that combined the variance and autocorrelation in a coefficient of self-induced hydrodynamic diffusion. Thus, the two models are related, but not identical [106].…”

Section: Physical Explanation Of Lemma 61 Recall That Fmentioning

confidence: 99%

Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

Tory¹,

Karlsen²,

Bürger³

et al. 2003

SIAM J. Appl. Math.

140

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Abstract. We show how existing models for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions forming compressible sediments ("sedimentation with compression" or "sedimentation-consolidation process"). For N solid particle species, this theory reduces in one space dimension to an N × N coupled system of quasilinear degenerate convection-diffusion equations. Analyses of the characteristic polynomials of the Jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic-hyperbolic type for arbitrary N and particle size distributions. Bounds for the eigenvalues of both matrices are derived. The mathematical model for N = 3 is illustrated by a numerical simulation obtained by the Kurganov-Tadmor central difference scheme for convectiondiffusion problems. The numerical scheme exploits the derived bounds on the eigenvalues to keep the numerical diffusion to a minimum.Key words. polydisperse suspensions, sedimentation, systems of conservation laws, strongly degenerate parabolic-hyperbolic systems, central difference approximation AMS subject classifications. 35K65, 35L40, 35L65, 65M06, 76T05 DOI. 10.1137/S00361399024081631. Introduction. Mathematical models for the (controlled) sedimentation of polydisperse suspensions of small particles, which belong to a finite number of species differing in size or density and are suspended in a viscous fluid, are important to many applications such as the chemical engineering, ceramic, pulp and paper, and food industries, mineral processing, wastewater treatment, and medicine [3,50,88,89,101,122]. The characteristic behavior of such mixtures is differential sedimentation, which leads to areas of different composition if an initially homogeneous suspension is allowed to settle. In this paper, we consider the additional property that the solid particles possibly form a compressible sediment layer. A mathematical model for polydisperse suspensions forming compressible sediments is developed, analyzed, and simulated, focusing on three different aspects.First, we show how two existing sedimentation models-one for monodisperse flocculated suspensions, which are described by scalar strongly degenerate parabolichyperbolic equations, and one for polydisperse suspensions of rigid spheres differing

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Hindered settling and hydrodynamic dispersion in quiescent sedimenting suspensions

Cited by 173 publications

References 11 publications

Hydrodynamic diffusion of a sphere sedimenting through a dilute suspension of neutrally buoyant spheres

Hydrodynamic diffusion of a sphere sedimenting through a dilute suspension of neutrally buoyant spheres

Settling Statistics of Hard Sphere Particles

Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

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