Collision rates of permeable particles in creeping flows

Rebouças, R.B.; Loewenberg, Michael

doi:10.1063/5.0060018

Cited by 3 publications

(8 citation statements)

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“…Displacements resulting from collisions between particles of different sizes are usually smaller than displacements resulting from collisions between equal-size particles. This is true for particles that undergo contact interactions, as discussed in § 4.2, because collision rates diminish rapidly with size ratio (Adler 1981; Wang, Zinchenko & Davis 1994; Reboucas & Loewenberg 2021 b ). Thus, the superposition approximation may be expected to hold for particles with vastly different sizes.…”

Section: Stationary Particle Distributions: General Resultsmentioning

confidence: 99%

“…Accordingly, particles with short-range binary contact interactions have circular upstream collision cross-sections, defined by (2.27a,b), where r c depends on size ratio, and on the roughness, permeability, or drop-to continuous-phase viscosity ratio, respectively, for rough or permeable particles or drops. Collision cross-sections, or equivalently collision efficiencies E 12 = (r c /(2ā)) 3 , are available in the literature for rough and permeable particles and drops (Wang et al 1994;Reboucas & Loewenberg 2021b). Trajectories with offsets r −∞ > r c are reversible, i.e.…”

Section: Trajectories Of Particles With Contact Interactionsmentioning

confidence: 99%

“…The prototypical example of a short-range mechanism for contact interactions is small-amplitude surface roughness, , that prevents surface-to-surface particle separations less than (Smart & Leighton 1989; Smart, Beimfohr & Leighton 1993; da Cunha & Hinch 1996; Rampall, Smart & Leighton 1997; Wilson 2005; Ingber & Zinchenko 2012). Other examples of particles with short-range contact interactions include particles with weak permeability (Reboucas & Loewenberg 2021 a , b , 2022), particles stabilized by screened electrostatic interactions (Zinchenko 1984; Kremer, Robbins & Grest 1986) and emulsion drops under small-deformation conditions (Loewenberg & Hinch 1997; Ramachandran et al. 2010).…”

Section: Introductionmentioning

confidence: 99%

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A pairwise hydrodynamic theory for flow-induced particle transport in shear and pressure-driven flows

2022

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An exact pairwise hydrodynamic theory is developed for the flow-induced spatial distribution of particles in dilute polydisperse suspensions undergoing two-dimensional unidirectional flows, including shear and planar Poiseuille flows. Coupled diffusive fluxes and a drift velocity are extracted from a Boltzmann-like master equation. A boundary layer is predicted in regions where the shear rate vanishes with thickness set by the radii of the upstream collision cross-sections for pair interactions. An analysis of this region yields linearly vanishing drift velocities and non-vanishing diffusivities where the shear rate vanishes, thus circumventing the source of the singular particle distribution predicted by the usual models. Outside of the boundary layer, a power-law particle distribution is predicted with exponent equal to minus half the exponent of the local shear rate. Trajectories for particles with symmetry-breaking contact interactions (e.g. rough particles, permeable particles, emulsion drops) are analytically integrated to yield particle displacements given by quadratures of hard-sphere (or spherical drop) mobility functions. Using this analysis, stationary particle distributions are obtained for suspensions in Poiseuille flow. The scale for the particle distribution in monodisperse suspensions is set by the collision cross-section of the particles but its shape is almost universal. Results for polydisperse suspensions show size segregation in the central boundary layer with enrichment of smaller particles. Particle densities at the centreline scale approximately with the inverse square root of particle size. A superposition approximation reliably predicts the exact results over a broad range of parameters. The predictions agree with experiments in suspensions up to approximately 20 % volume fraction without fitting parameters.

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Section: Stationary Particle Distributions: General Resultsmentioning

confidence: 99%

Section: Trajectories Of Particles With Contact Interactionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A pairwise hydrodynamic theory for flow-induced particle transport in shear and pressure-driven flows

2022

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“…In a recent analysis of the near-contact motion between permeable spheres, we showed that the lubrication resistance between permeable particles is non-singular at contact, in contrast to the O(a/h 0 ) lubrication singularity that characterizes the relative motion of hard spheres (Reboucas & Loewenberg 2021a). This feature allows contact between particles in suspension, even without the presence of interparticle forces (Reboucas & Loewenberg 2021b).…”

Section: Introductionmentioning

confidence: 91%

Resistance and mobility functions for the near-contact motion of permeable particles

Rebouças

Loewenberg

2022

J. Fluid Mech.

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A lubrication analysis is presented for the resistances between permeable spherical particles in near contact, $h_0/a\ll 1$ , where $h_0$ is the minimum separation between the particles, and $a=a_1 a_2/(a_1+a_2)$ is the reduced radius. Darcy's law is used to describe the flow inside the permeable particles and no-slip boundary conditions are applied at the particle surfaces. The weak permeability regime $K=k/a^{2} \ll 1$ is considered, where $k=\frac {1}{2}(k_1+k_2)$ is the mean permeability. Particle permeability enters the lubrication resistances through two functions of $q=K^{-2/5}h_0/a$ , one describing axisymmetric motions, the other transverse. These functions are obtained by solving an integral equation for the pressure in the near-contact region. The set of resistance functions thus obtained provide the complete set of near-contact resistance functions for permeable spheres and match asymptotically to the standard hard-sphere resistances that describe pairwise hydrodynamic interactions away from the near-contact region. The results show that permeability removes the contact singularity for non-shearing particle motions, allowing rolling without slip and finite separation velocities between touching particles. Axisymmetric and transverse mobility functions are presented that describe relative particle motion under the action of prescribed forces and in linear flows. At contact, the axisymmetric mobility under the action of oppositely directed forces is $U/U_0=d_0K^{2/5}$ , where $U$ is the relative velocity, $U_0$ is the velocity in the absence of hydrodynamic interactions and $d_0=1.332$ . Under the action of a constant tangential force, a particle in contact with a permeable half-space rolls without slipping with velocity $U/U_0=d_1(d_2+\log K^{-1})^{-1}$ , where $d_1=3.125$ and $d_2=6.666$ ; in shear flow, the same expression holds with $d_1=7.280$ .

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“…This type of two-particle analysis has been extensively used in the literature to investigate particle collision and agglomeration (e.g. Zeichner & Schowalter 1977;Davis 1984;Rother & Davis 2001;Phan et al 2003;Roure & Cunha 2018;Reboucas & Loewenberg 2021;Rother, Stark & Davis 2022). More specifically, to model the different aspects involved in the novel agglomeration method by emulsion binders, the works by Davis & Zinchenko (2018) and Roure & Davis (2021b) respectively investigated the effects of permeability and osmotic swelling on particle capture.…”

Section: Introductionmentioning

confidence: 99%

Particle capture by expanding droplets: effects of inner diffusion

2022

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Fast agglomeration by emulsion binders to capture fine, hydrophobic particles has been developed in the past few years as an alternative to froth flotation by small air bubbles. This new method consists of mixing a particle suspension and saltwater-filled droplets covered with semi-permeable oil layers. These droplets expand due to an osmotic flux of water caused by the presence of salt inside the droplets. To better understand the physics underlying this novel particle capture method, we investigate binary interactions between droplets and particles. The current work examines the dynamics of a rigid spherical particle and a semi-permeable spherical drop that expands due to osmosis in an external, pure-extensional flow field. The droplet is governed by an expansion-diffusion problem, which is coupled to the set of dynamical equations governing the relative particle trajectory. By performing multiple trajectory simulations, we calculate transient collision efficiencies, which can be used to determine the collision kernel for population dynamics. We also use these simulations to better understand the evolution of the microstructure by determining the transient behaviour of the pair distribution function. Our results indicate that the presence of drop expansion increases the collision efficiency of the system, especially for very small particles, which are the most difficult to capture by froth flotation. Moreover, although the presence of slow salt diffusion inside the drops can mitigate this improvement, the contribution of expansion to the collision efficiency may still be considerable, even in the absence of hydrophobic or other attractive forces.

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Collision rates of permeable particles in creeping flows

Cited by 3 publications

References 83 publications

A pairwise hydrodynamic theory for flow-induced particle transport in shear and pressure-driven flows

A pairwise hydrodynamic theory for flow-induced particle transport in shear and pressure-driven flows

Resistance and mobility functions for the near-contact motion of permeable particles

Particle capture by expanding droplets: effects of inner diffusion

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