Ensemble-average versus suspension-scale Cauchy continuum-mechanical definitions of stress in polarized suspensions: Global homogenization of a dilute suspension of dipolar spherical particles

A method of evaluating the transfer terms appearing in the ensemble-average fluid transport equation is developed and applied to obtain the transport equations describing flow of a dilute particulate mixture containing solid spherical particles. The equations apply in the limit where interactions between phases are both quasi-steady and viscous, which is defined as flows that meet the three criterion Ref(a/ℒ)2 ≪ 1, vfτ/a2 ≫ 1, and Rep ≪ 1. In this limit, the terms describing transfer of momentum between the two phases of the mixture are evaluated to O(1) in the particle radius and O(γp) in the particle phase density. The continuity equations and consistency principle are exact. When the first two conditions are not met, the transport equations require the terms that describe virtual mass forces; when the third is not met, the transport equations require terms that describe Oseen corrections to the drag term.

Download Full-text

Electrophoretic Mobility of a Semi-dilute Suspension of Spherical Particles with Thick Double Layers and Low Zeta Potentials

Journal of Colloid and Interface Science ◽

10.1006/jcis.1997.4907 ◽

1997 ◽

Vol 189 (2) ◽

pp. 382-383 ◽

Cited By ~ 3

Author(s):

J. Ennis ◽

L.R. White

Keyword(s):

Electrophoretic Mobility ◽

Spherical Particles ◽

Double Layers ◽

Zeta Potentials ◽

Dilute Suspension

Download Full-text

The effect of Brownian motion on the bulk stress in a suspension of spherical particles

Journal of Fluid Mechanics ◽

10.1017/s0022112077001062 ◽

1977 ◽

Vol 83 (1) ◽

pp. 97-117 ◽

Cited By ~ 1352

Author(s):

G. K. Batchelor

Keyword(s):

Brownian Motion ◽

Probability Density ◽

Volume Fraction ◽

Joint Probability ◽

Small Degree ◽

Spherical Particles ◽

Stress System ◽

Bulk Stress ◽

Dilute Suspension ◽

Thermodynamic Forces

The effect of Brownian motion of particles in a statistically homogeneous suspension is to tend to make uniform the joint probability density functions for the relative positions of particles, in opposition to the tendency of a deforming motion of the suspension to make some particle configurations more common. This smoothing process of Brownian motion can be represented by the action of coupled or interactive steady ‘thermodynamic’ forces on the particles, which have two effects relevant to the bulk stress in the suspension. Firstly, the system of thermodynamic forces on particles makes a direct contribution to the bulk stress; and, secondly, thermodynamic forces change the statistical properties of the relative positions of particles and so affect the bulk stress indirectly. These two effects are analysed for a suspension of rigid spherical particles. In the case of a dilute suspension both the direct and indirect contributions to the bulk stress due to Brownian motion are of order ø2, where ø([Lt ] 1) is the volume fraction of the particles, and an explicit expression for this leading approximation is constructed in terms of hydrodynamic interactions between pairs of particles. The differential equation representing the effects of the bulk deforming motion and the Brownian motion on the probability density of the separation vector of particle pairs in a dilute suspension is also investigated, and is solved numerically for the case of relatively strong Brownian motion. The suspension has approximately isotropic structure in this case, regardless of the nature of the bulk flow, and the effective viscosity representing the stress system to order ϕ2 is found to be \[ \mu^{*} = \mu(1+2.5\phi + 6.2\phi^2). \] The value of the coefficient of ø2 for steady pure straining motion in the case of weak Brownian motion is known to be 7[sdot ]6, which indicates a small degree of ‘strain thickening’ in the ø2-term.

Download Full-text

Transverse shear-induced gradient diffusion in a dilute suspension of spheres

Journal of Fluid Mechanics ◽

10.1017/s0022112097008148 ◽

1998 ◽

Vol 357 ◽

pp. 279-287 ◽

Cited By ~ 30

Author(s):

Y. WANG ◽

R. MAURI ◽

A. ACRIVOS

Keyword(s):

Three Dimensional ◽

Spherical Particles ◽

Particle Interactions ◽

Simple Shearing ◽

Ambient Flow ◽

Gradient Diffusion ◽

Areal Fraction ◽

Dilute Suspension ◽

Neutrally Buoyant ◽

Shearing Motion

We study the shear-induced gradient diffusion of particles in an inhomogeneous dilute suspension of neutrally buoyant spherical particles undergoing a simple shearing motion, with all inertia and Brownian motion effects assumed negligible. An expansion is derived for the flux of particles due to a concentration gradient along the directions perpendicular to the ambient flow. This expression involves the average velocity of the particles, which in turn is expressed as an integral over contributions from all possible configurations. The integral is divergent when expressed in terms of three-particle interactions and must be renormalized. For the monolayer case, such a renormalization is achieved by imposing the condition of zero total macroscopic flux in the transverse direction whereas, for the three-dimensional case, the additional constraint of zero total macroscopic pressure gradient is required. Following the scheme of Wang, Mauri & Acrivos (1996), the renormalized integral is evaluated numerically for the case of a monolayer of particles, giving for the gradient diffusion coefficient 0.077γa2c¯2, where is the applied shear rate, a the radius of the spheres and c¯ their areal fraction.

Download Full-text

The Plane Parallel Flow of a Density-Stratified Fluid With and Without a Suspension of Particles

Journal of Applied Mechanics ◽

10.1115/1.3422982 ◽

1973 ◽

Vol 40 (2) ◽

pp. 326-330 ◽

Cited By ~ 1

Author(s):

J. Val. Healy

Keyword(s):

Particle Density ◽

Infinite Plate ◽

Axial Flow ◽

Wall Stress ◽

Spherical Particles ◽

Uniform Density ◽

Asymptotic Results ◽

Infinite Region ◽

Dilute Suspension ◽

Long Cylinder

The response of a viscous incompressible nondiffusive fluid containing a dilute suspension of small spherical particles and occupying a semi-infinite region bounded by an infinite plate, when the latter is slowly moved parallel to itself in a prescribed manner normal to the direction of stratification, is studied. In particular, when the system density decays exponentially with distance from the plate, the problem is equivalent to the axial flow of the same system in a long tube. When the particle density vanishes, the problem reduces to the unsteady diffusion of vorticity in a long cylinder, with the “diffusivity” a function of the stratification length, whose solution is well known. A degeneracy occurs in the solution as the density approaches uniformity and this is examined using asymptotic expansions and some exact and asymptotic results for an impulsively moved plate are given. It is found that, for a given stratification, the ratio of the wall stress with particles to that without, increases first and then, after the non-equilibrium overshoot, it continues to increase with time; this is in contrast to the case of uniform density, when the ratio goes asymptotically to a constant value. When the particle density vanishes and the kinematic viscosity of the fluid goes to infinity at very large distances from the plate, the fluid there acquires, in a finite time, a finite velocity that strongly depends on the stratification length.

Download Full-text

On the Interaction of Insoluble Spherical Particles with a Solidifying Interface in the Presence of Morphological Instabilities

MRS Proceedings ◽

10.1557/proc-529-33 ◽

1998 ◽

Vol 529 ◽

Author(s):

L. Hadji ◽

A.M.J. Davis

Keyword(s):

Stability Analysis ◽

Linear Stability ◽

Binary Alloy ◽

Linear Stability Analysis ◽

Control Parameter ◽

Critical Value ◽

Planar Interface ◽

Unidirectional Solidification ◽

Spherical Particles ◽

Dilute Suspension

AbstractThe interaction of a dilute suspension of spherical particles with a moving solidifying front is investigated analytically. The front emerges during the unidirectional solidification of a dilute binary alloy. A linear stability analysis of the planar interface reveals that the presence of the particles has a destabilizing effect, i.e. the critical value of the control parameter, taken here to be the concentration faraway from the interface, for the onset of the Mullins-Sekerka instability is lowered due to the particles' presence.

Download Full-text