Environmental Engineering Reference
In-Depth Information
suspensions in a Couette device [ 6 ]. In the presence of shear, particles undergo a
random walk that results in no net displacement. This source of diffusive flux is
called “shear-induced self or tracer diffusion” [ 12 , 13 ]. In the case of non-uniform
concentrations in shear, particles tend to drift from regions of high to low particle
concentrations due to particle collisions, which is referred to as “drift diffusion” [ 13 ].
While effective shear diffusivity consists of both drift diffusion and aforementioned
tracer-diffusion, drift diffusion dominates in the case of concentrated monodisperse
suspensions.
While both diffusive flux models and suspension balance models have been suc-
cessful in capturing the particle migration behaviour under shear, they differ substan-
tially in their derivation of particle flux. The diffusive flux phenomenology consists
of semi-empirical laws that describe particle migration based on irreversible particle
collisions and does not account for the non-Newtonian viscosity of the particle-
fluid mixture. The suspension balance approach, on the other hand, relies on the
non-Newtonian normal stresses induced by shear, which give rise to the particle
migration. Therefore, viscously generated normal stresses are crucial in the suspen-
sion balance approach. In particular, the anistropic normal stresses have been shown
to be important in predicting correct secondary flows in a pressure-driven tube flow
[ 20 ]. Thus, the neglect of normal stress differences in the diffusive fluxmodel is prob-
lematic especially in the non-dilute concentration limit, as Couturier and co-authors
[ 3 ] experimentally demonstrated the significance of normal stress differences for the
volume fraction greater than 0.17.
Despite the apparent limitations, the diffusive flux approach is “contained” within
the suspension balance model and can yield the same set of equations in the unidirec-
tional, fully-developed flows [ 17 ]. For instance, Timberlake and Morris [ 24 ] exper-
imentally and theoretically studied the gravity-driven, free-surface flow containing
neutrally buoyant particles. They observed the deformation of the free surface and
particle migration, which sufficiently matched their mathematical model. Although
their model was based on the suspension balance approach, the resultant equations
for the flux of particles were essentially identical to those derived based on diffusive
flux approach of [ 16 ]. More recently, [ 21 ] observed the accumulation and depletion
of the particles on the advancing meniscus and found that, based on the suspension
balance model, this depended on the balance between gravitational flux and shear-
induced migration. This particular result corresponds exactly to the findings of [ 15 ]
who identified different particle regimes at varying inclination angles and particle
volume fractions based on the diffusive flux approach, further demonstrating the
validity of the simpler diffusive flux model in primarily unidirectional flows.
Contrary to the monodisperse case, tracer diffusion becomes important in polydis-
perse suspensions. Reference [ 25 ] investigated the resuspension of heavy particles in
a Couette device, with the addition of neutrally buoyant particles of the equal size. At
a given shear rate, they found that an increasing concentration of neutrally buoyant
particles caused the heavy particles to rise higher to mix with neutrally buoyant ones
on the free surface. Based on diffusive flux phenomenology, Tripathi and Acrivos
derived a continuum model to match the experimental observations and found that
the tendency of particle species to mix is attributed to tracer diffusivity.
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