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SEDIMENTATION OF HARD SPHERE SUSPENSIONSAT LOW REYNOLDS NUMBER USING A LATTICE-BOLTZMANN METHOD
By
NHAN-QUYEN NGUYEN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2004
Copyright 2004
by
Nhan-Quyen Nguyen
This dissertation is for my parents, Peter and Hoa Thi Nguyen, for all their
support.
ACKNOWLEDGMENTS
I would like to thank my advisor, Tony Ladd, for his guidance and understanding
in giving me the opportunity to work on the sedimentation problem. The work was
long and tedious, but his professionalism and hard work were an inspiration that kept
me going. I would also like to thank my fellow colleagues in the Chemical Engineering
department. With such good people around, it made the long hours of doing research
bearable. I thank my lab members Jonghoon Lee, for always being helpful; Byoungjin
Chun, for his congeniality; Berk Usta, for his generosity; and Murali Rangarayan, for
being a good guy. Thanks also go to Piotr Szymczak and Rolf Verberg, two postdocs
whose infinite knowledge and wisdom I was fortunate to learn from. Finally, I thank my
Mom, Dad, brothers, and sister for being a part of my life.
iv
TABLE OF CONTENTSPage
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Properties of a Sedimenting Suspension . . . . . . . . . . . . . . . . . . 21.2 Fluid Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Hydrodynamics at Low Reynolds Number . . . . . . . . . . . . . . . . 5
1.3.1 Long-Range Interactions and the Divergence Problem . . . . . . 81.3.2 Multi-Particle Interactions . . . . . . . . . . . . . . . . . . . . . 101.3.3 Particles Near Contact . . . . . . . . . . . . . . . . . . . . . . . 11
2 DESCRIPTION OF THE LATTICE-BOLTZMANN MODEL . . . . . . . . . 13
2.1 Molecular Theory for Fluid Flow . . . . . . . . . . . . . . . . . . . . . 132.1.1 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . 142.1.2 Connection between Microscopic and Macroscopic Dynamics . . 15
2.2 Lattice-Boltzmann Method . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 Gas Kinetics to Fluid Dynamics: The Chapman-Enskog Ex-
pansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.2 Solid-Fluid Boundary Conditions . . . . . . . . . . . . . . . . . . 302.2.3 Hydrodynamic Radius of a Particle . . . . . . . . . . . . . . . . 332.2.4 Particle Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 INCORPORATING LUBRICATION INTO THELATTICE-BOLTZMANN METHOD . . . . . . . . . . . . . . . . . . . . . . . 40
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1.1 Surfaces Near Contact . . . . . . . . . . . . . . . . . . . . . . . . 423.1.2 Lubrication Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1.3 Particle Wall Lubrication . . . . . . . . . . . . . . . . . . . . . . 443.1.4 Cluster Implicit Method . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 SEDIMENTATION OF HARD-SPHERE SUSPENSIONSAT LOW REYNOLDS NUMBER . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1.1 Sedimentation Simulations . . . . . . . . . . . . . . . . . . . . . 56
v
4.1.2 Settling of Pairs of Particles . . . . . . . . . . . . . . . . . . . . 574.2 Velocity Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.1 Time-Dependent Velocity Fluctuations . . . . . . . . . . . . . . 594.2.2 Steady-State Settling . . . . . . . . . . . . . . . . . . . . . . . . 634.2.3 Numerical Errors . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.1 Particle Concentration . . . . . . . . . . . . . . . . . . . . . . . 694.3.2 Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.3.3 Mean Settling Speed . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 SEDIMENTATION OF A POLYDISPERSE SUSPENSION . . . . . . . . . . 77
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.1.1 Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.1.2 Velocity fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 795.1.3 Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.1.4 Stratification in Laboratory Experiments . . . . . . . . . . . . . 82
5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
vi
LIST OF TABLESPage
2–1 Variance in the computed drag force δF =√
< F 2 > − < F >2/ < F >for a particle of radius a moving along a random orientation with respectto the grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2–2 Hydrodynamic radius ahy (in units of ∆x) for various fluid viscosities; a isthe input particle radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3–1 Lubrication ranges for various kinematic viscosities, determined for a sphereof radius a = 8.2∆x. The ranges are determined separately for the nor-mal, hN , tangential, hT , and rotational, hR, motions. . . . . . . . . . . . . . 46
4–1 Steady state particle velocity fluctuations in a monodisperse suspension asa function of system width. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5–1 Steady-state particle velocity fluctuations in a 10% polydisperse suspen-sion as a function of system width. . . . . . . . . . . . . . . . . . . . . . . 79
vii
LIST OF FIGURESPage
2–1 The 18 velocity vectors (ci) in the lattice-Boltzmann model. . . . . . . . . 17
2–2 Location of boundary nodes for a curved surface (a) and two surfaces innear contact (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2–3 Moving boundary node update. The solid circles are fluid nodes, squaresare boundary nodes with the associated velocity vector, and open circlesare interior nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2–4 Drag force F as a function of time, normalized by the drag force on anisolated sphere, F0 = 6πηaU . Time is measured in units of the Stokestime, tS = a/U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2–5 Actual (solid lines) and hydrodynamic (dashed lines) surfaces for a parti-cle settling onto a wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3–1 Normal force on a particle of input radius a settling onto a horizontal pla-nar surface at zero Reynolds number, with F0=6πηahyU . Solid symbolsindicate: ν∗ = 1/6 (circles), ν∗ = 1/100 (triangles), and ν∗ = 1/1200(squares). Results including the lubrication correction are shown by theopen symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3–2 Tangential force on a particle settling next to a vertical planar surface atzero Reynolds number. Simulation data for drag force divided by F0=6πηahyU ,is indicated by solid symbols. Results including the lubrication correctionare shown by the open symbols. . . . . . . . . . . . . . . . . . . . . . . . . 46
3–3 Torque on a particle settling next to a vertical planar surface at zero Reynoldsnumber. Simulation data for torque divided by T0=8πηa2
hyU , is indicatedby solid symbols. Results including the lubrication correction are shownby the open symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3–4 Torque on a particle rotating next to a vertical planar surface at zero Reynoldsnumber. Simulation data for torque divided by T0=8πηa3
hyΩ, is indicatedby solid symbols. Results including the lubrication correction are shownby the open symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3–5 Settling of a sphere (a=4.8) onto a horizontal wall. The gap between theparticle surface and wall, h, relative to the hydrodynamic radius, is plot-ted as a function of the non-dimensional time (open circles). . . . . . . . . 49
3–6 Illustration of the algorithm to determine the list of clusters. . . . . . . . . 50
3–7 The maximum cluster size as a function of the cluster cutoff gap hs/a atvarying volume fractions, φ. . . . . . . . . . . . . . . . . . . . . . . . . . . 51
viii
3–8 Number of clusters as a function of the cluster cutoff gap hs/a at varyingvolume fractions, φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4–1 Settling of a horizontal pair of particles at different Reynolds numbers;Re ∼ 0.1 (circles), 0.05 (squares), 0.025 (triangles). . . . . . . . . . . . . . 58
4–2 Settling of a pair particles at various orientations: θ = 0 (circles), 45
(squares), 68 (triangles), 90 (diamonds). The Reynolds number Re =0.1 in each case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4–3 Particle velocity fluctuations as a function of time with a with W = 48a. . 60
4–4 Particle velocity fluctuations as a function of time for ‘Box” and “Bounded”geometries at various widths: W/a = 16 (circles), 32 (squares), and 48(triangles) respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4–5 Mean settling velocity,⟨U‖
⟩, and fluctuations in settling velocity,
⟨∆U2
‖⟩,
as a function of time for different system heights, H. . . . . . . . . . . . . . 63
4–6 Particle velocity fluctuations at steady state. Profiles are shown for the“Box” system (solid symbols) at different widths: W/a = 16 (circles), 32(squares), and 48 (triangles). . . . . . . . . . . . . . . . . . . . . . . . . . 64
4–7 Steady state particle velocity fluctuations as a function of system widthfor simulation versus experimental fit (solid line). . . . . . . . . . . . . . . 65
4–8 Effect of inertia on the mean settling velocity and fluctuations in settlingvelocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4–9 Effect of grid resolution on the mean settling velocity and fluctuations insettling velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4–10 Effect of compressibility on the spatial variation of the mean settling ve-locity. The data is for Reynolds number of Re = 0.06. . . . . . . . . . . . . 68
4–11 Particle volume fraction as a function of height, z. The dots indicate theinstantaneous average, and the steady-state (1000 < t/tS < 1400) densityprofile in the viewing window is shown in the adjacent plots. . . . . . . . . 69
4–12 Structure factor describing horizontal density fluctuations, S(k⊥), for var-ious system sizes: W = 16a, 32a, and 48a. . . . . . . . . . . . . . . . . . . 71
4–13 Structure factor at different angles; vertical [0,0,1] direction (circles), 45o
[1,0,1] direction (squares), and horizontal [1,0,0] direction (triangles). . . . 73
4–14 Time evolution of structure factor for horizontal density fluctuations. . . . 74
5–1 Profiles of the particle volume fractions as a function of the system height. 78
5–2 Comparison of particle volume fraction profiles for monodisperse and poly-disperse suspensions at steady-state, 1000 < t/tS < 1400. . . . . . . . . . 79
5–3 Particle velocity fluctuation as a function of time for a 10% polydispersesuspension. The data is taken for three system widths: W/a = 16 (cir-cles), 32 (squares), 48 (triangles). . . . . . . . . . . . . . . . . . . . . . . . 80
ix
5–4 Comparison of particle velocity fluctuations for monodisperse and 10%polydisperse suspensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5–5 Comparison of structure factors for different degrees of polydispersity: a)monodisperse (circles) and 10% polydisperse (squares) suspensions (W=48a,N=72000); b) 2% (circles), 5% (squares), and 10% (triangles) polydis-perse suspensions (W=32a, N=32000). . . . . . . . . . . . . . . . . . . . . 82
x
Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
SEDIMENTATION OF HARD SPHERE SUSPENSIONSAT LOW REYNOLDS NUMBER USING A LATTICE-BOLTZMANN METHOD
By
Nhan-Quyen Nguyen
August 2004
Chair: Anthony J. LaddMajor Department: Chemical Engineering
Particle motion in a settling suspension is directly influenced by hydrodynamic
interactions, such that the particle velocities will exhibit a random component, char-
acterized by a hydrodynamically induced diffusion. Statistically, this means that the
particles tend to be uniformly distributed within the bulk. Theoretical predictions for
the sedimentation velocity in homogeneous suspensions are in good agreement with ex-
periments. On the other hand, a calculation of the particle velocity fluctuations, based
on the same assumptions, is in conflict with experimental results. Our main objective
was to understand this discrepancy.
Numerical simulations have shown that dynamics within the bulk are not indepen-
dent of the size of the container, even when the boundaries are located far away from
the observation window. We hypothesized that density fluctuations in the bulk drain
away to the suspension-sediment and suspension-supernatent interfaces. This generates
a continual decay in velocity fluctuations, which is replenished by hydrodynamic diffu-
sion due to short range multiparticle interactions. A balance between convective and
diffusive transport of density fluctuations leads to a correlation length beyond which
the velocity fluctuations are screened. Numerical results were found to support this
hypothesis.
An examination of the microstructure as the suspension settled showed that
it evolves toward a statistically nonrandom state, so that particles are distributed
xi
very uniformly at large length scales. We identified this result based on the observed
damping of the structure factor, S(k) → k2 as k goes to zero.
Our numerical simulations show that the long-range correlations found in monodis-
perse suspensions are destroyed by small amounts of polydispersity, which are found in
typical laboratory experiments. The variation in particle size also leads to a segregation
of different sizes as the suspension settles. The net effect is a pronounced stratification
of particle concentration at the supernatent front that persists into the bulk. Because
of the stratification, the velocity fluctuations within the bulk do not have to convect
toward the density gradients at the front; rather, the gradient is strong enough inside
the bulk to create a sink for the velocity fluctuations.
xii
CHAPTER 1INTRODUCTION
We studied gravity driven non-Brownian particles within a viscous fluid. We
deal with low-Reynolds number flow, such that the dynamics of particles in the
dispersion is strongly affected by hydrodynamic interactions generated by the relative
motion of particles and fluid. Owing to the complexity of suspension dynamics, in
1985, a paradox for steady state sedimentation was pointed out by Caflisch and Luke
(CL) (1). Consider a steadily sedimenting suspension of hard spheres. A concentration
fluctuation gives rise to a velocity field decaying as 1/r with distance r from the origin
of the fluctuations. The linearity of Stokes flow implies that the velocity field from
many spatially distributed concentration fluctuations is simply a sum Σiui of the
individual contributions. If these concentration fluctuations take place in a random,
spatially uncorrelated manner throughout the suspension, the resulting variance Σu2
in the velocity at any point in the suspension would be the sum of the squares of the
individual contributions. This sum Σiu2i has N ∼ L3 terms with N solute particles in a
container of linear dimension L in all directions. The 1/r contribution mentioned above
leads to a velocity fluctuation of < u2i >∼ L−2, so that Σiu
2 ∼ L.
The CL prediction is based on two simple physical arguments:
• The fluid flow is slow so that the Reynolds number Re ¿ 1.
• The concentration fluctuations are statistically independent from one point to
another in space.
However, most experiments do not find the size dependence predicted by CL; the
reason for the disagreement is unclear. One hypothesis is that a sufficiently strong
anticorrelation of density fluctuations develops at large length scales, which suppress
the CL divergence. We present findings from lattice-Boltzmann simulations to verify
this hypothesis.
Simulations of the sedimentation problem require large-scale numerical calcula-
tions, for which the lattice-Boltzmann method is a useful tool. The simulation is based
1
2
on the development of microscopic interactions that lead to macroscopic fluid flow. This
is done by applying the Boltzmann equation developed from gas kinetic theory. The
simulation uses a spatial grid for the fluid phase, making it an inherently parallizeable
algorithm where simulations of large-scale systems spread over multiple processors can
be performed. With boundary conditions at the surface of the particles, the lattice-
Boltzmann method becomes applicable to multi-phase flows. We are able follow the
trajectories of each particle in the suspension over time, giving a detailed picture of the
particle dynamics.
1.1 Properties of a Sedimenting Suspension
Sedimentation deals with systems that exist in a non-equilibrium steady state. The
particles are driven through the solvent by a density mismatch, and have a downward
velocity resulting from a balance between gravity and viscous drag force. This leads to
a phase separation over time, which develops a region of pure fluid on top, and a layer
of sediment at the bottom. The shrinking domain of bulk suspension is separated from
the pure fluid by a horizontal interface, moving downward, and from the sediment by
another horizontal interface, moving upward. Ultimately, these interfaces meet, the
motion ceases, and separation is complete.
Further observation of a settling suspension shows that particles exhibit random
motion even when the particles are large and Brownian motion is negligible. The
flow developed by each particle influencing the other makes the dynamics chaotic,
and highly sensitive to initial conditions. The resulting chaos implies that the time-
evolution of coarse-grained quantities must be described using diffusion coefficients and
noise sources, even though the microscopic dynamics in the absence of thermal Brown-
ian motion are entirely deterministic (2, 3). This induced large -scale diffusive behavior,
in the absence of thermal noise, is called hydrodynamic diffusion or hydrodynamic
dispersion.
Important macroscopic questions arise in the study of this ”simple” prototype
system. For instance: what is the time of separation? What is the distribution of
velocities and concentration? How is this process affected by the container boundaries
and dimensions?
3
1.2 Fluid Dynamic Equations
The equations of motion of a fluid are in essence a reformulation of Newton’s
second law (which states that the force exerted on a body of mass m equals m times
the accelerations a) (4). For a fluid volume V , the mass density and acceleration can
vary from place to place. The volume V is therefore divided into infinitesimally small
volumes dV over which ρ and a are constant. Hence for an arbitrary volume ∆V
F = ma =
∫
∆V
ρadV. (1.1)
The force F exerted on a fluid volume ∆V consists of two distinct contributions:
• A body force exerted on the total volume.
• A surface force exerted on the boundary A of volume ∆V .
Hence
F = Fbody + Fsurf . (1.2)
Common body forces are gravity and external electric or magnetic fields. Defining fext
as the body force per unit volume, we can write
Fbody =
∫
∆V
fextdV. (1.3)
The surface forces can be found by considering the stresses exerted on the boundary A
of the volume ∆V :
Fsurf =
∫
A
Π · dA (1.4)
where Π is the stress tensor that specifies the local normal and tangential forces per
unit area exerted on a fluid element; and dA is a vector pointing perpendicular to the
surface, its magnitude determined by the surface area of the infinitesimal element dA.
Using the divergence theorem, Eq. 1.4 can be written as
Fsurf =
∫
∆V
∇ ·ΠdV. (1.5)
From Eqs. 1.1, 1.3, and 1.5 into 1.2, the equation of motion for a fluid can be written as
ρa = ∇ ·Π + fext. (1.6)
4
In specifying the acceleration term, an Eulerian description of the velocity u(r, t) is
presented. Hence,
a =Du
Dt=
∂u
∂t+ u · ∇u. (1.7)
The two terms on the right -hand side can be interpreted as the local rate of change
in velocity (∂u/∂t) due to changes at position r and a convective rate of change in
velocity (u · ∇u) due to the transport of the element to a different position (4). The
value ∇u is a second-order tensor describing the gradient of the velocity field. It can be
split into symmetric and antisymmetric parts as
∇u =1
2
(∇u +∇uT)
+1
2
(∇u−∇uT)
(1.8)
where ∇uT is the transpose of ∇u. Equation 1.8 can then be rewritten as
∇u = S + Λ (1.9)
where S is the rate-of-strain tensor that describes fluid deformation, and Λ is the
vorticity tensor that corresponds to a solid-body rotation (4).
The stress tensor Π must still be defined. In the absence of fluid flow, the stresses
in a fluid are caused by the static fluid pressure, which is isotropic (i.e.,
Π = −PI (1.10)
where P is the hydrostatic pressure and I is the unit tensor). However, when fluid
flow occurs, the stress tensor becomes anisotropic, and can be expressed as the sum of
an isotropic pressure P (usually different from the static pressure) and a nonisotropic
contribution. For incompressible Newtonian liquids, this nonisotropic contribution is
given by 2µS, where µ is the viscosity of the fluid. This proportionality between the
stress and the rate of strain is a phenomenological relationship, which many fluids have
been found to obey (4). Hence
Π = −PI + 2µS. (1.11)
5
Taking the divergence (and using ∇uT =0 for an incompressible fluid) gives
∇ ·Π = −∇P + µ∇2u. (1.12)
Substituting Eq. 1.7 and 1.12 into Eq. 1.6 gives the equation of motion for the velocity
field
ρ
(∂u
∂t+ u · ∇u
)= −∇P + µ∇2u + fext, (1.13)
which together with the continuity equation for an incompressible fluid
∇ · u = 0, (1.14)
are referred to as the Navier-Stokes equations.
The Navier-Stokes equations can be written in nondimensional form by introducing
a characteristic length scale l and velocity u0. Defining the dimensionless variables
t = u0t/l, rα = rα/l, u = u/u0, P = lP/µu0 and fext = fextl2/µu0 (1.15)
then substituting them into Eq. 1.13 gives
lu0ρ
µ·(
∂u
∂t+ u · ∇u
)= −∇P + ∇2u + fext, (1.16)
where the tildes represent nondimensional parameters and operators. The constant
developed on the LHS is defined as the Reynolds number,
Re =lu0ρ
µ, (1.17)
and can be interpreted as a measure of the ratio of inertial to viscous forces.
1.3 Hydrodynamics at Low Reynolds Number
The standard problem in hydrodynamics is to find the velocity of a particle on
which forces are acting. The simple example of a sphere moving under the action
of a constant external force through a quiescent fluid can be considered as a model
problem. As the isolated object settles in the fluid, it creates a disturbance flow in its
surrounding neighborhood which becomes zero at large distances. After a short time,
the sphere will reach a steady-state velocity, U. To determine U, we must find the fluid
6
flow field u around the sphere, and calculate the stress exerted on the sphere surface by
the fluid. When the Re ¿ 1, the terms on the LHS of Eq. 1.16 are neglected, and the
Navier-Stokes equation reduces to the Stokes or creeping-flow equation. In dimensional
form, Eq. 1.13 can be re-written as
−∇P + µ∇2u = fext (1.18)
with ∇ · u = 0. (1.19)
A solution to the Stokes equation, Eq. 1.18, can be developed using Fourier
transforms (5, 6). First, the Fourier-transform pair is defined in real space r and
Fourier space k as
u(k) =
∫eik·ru(r)dr (1.20)
u(r) =1
(2π)3
∫e−ik·ru(k)dk. (1.21)
Taking the Fourier transform of Eq. 1.18 yields
− ikP + µk2u = F(k), (1.22)
k · u = 0. (1.23)
Taking the dot product of k on the transformed equation of motion, Eq. 1.22, elimi-
nates the u term to give
P(k) = iF · kk2
. (1.24)
Substituting P into Eq. 1.22 gives
u(k) =1
µk2
(F− F · kk
k2
). (1.25)
To invert the transformed solutions (Eqs. 1.24 and 1.25) we use the following
relation
i
(2π)3
∫F · kk2
e−ik·rdk = F · ∇ 1
4πr, (1.26)
7
1
(2π)3
∫F
k2e−ik·rdk = F · 1
4πr(1.27)
and1
(2π)3
∫Fkk
k4e−ik·rdk = F ·
(1
8πr− rr
8πr3
). (1.28)
From Eqs. 1.24 and 1.26, the pressure field is determined to be
P(r) =1
4πr3r · fext, (1.29)
with fext representing the force from a point source. The velocity field is determined by
taking the inverse Fourier transform of Eq. 1.25 with Eqs. 1.27 and 1.28 to give
u(r) =1
8πµ
(I
r+
rr
r3
)· fext. (1.30)
Hence, the Oseen tensor is given as
O(r) =1
8πµ
(I
r+
rr
r3
). (1.31)
We now look at the case when the full velocity field is not needed, but only
the force on a particle is desired. Consider a rigid sphere of radius a and surface S
immersed in a fluid with velocity field u∞(r) in the absence of the sphere. The sphere
is centered at r0, translates with velocity U0 and rotates about its center with angular
velocity Ω. Owing to its motion, the sphere affects the fluid through forces distributed
over its surface, Π(r) per unit area. From Eq. 1.30 the velocity field at r is now u∞(r)
plus that generated by the superposition of forces Π(r′)dr′ (5), or
u(r) = u∞(r)−∫
S
O(r− r′) ·Π(r′)dr′. (1.32)
At the surface of the sphere, this must match the sphere velocity given by U =
U0 + Ω× (r− r0). Thus,
U0 + Ω× (r− r0) = u∞(r)−∫
S
O(r− r′) ·Π(r′)dr′. (1.33)
Integration of Eq. 1.33 over the sphere surface results in
∫
S
Π(r′)dr′ = F =3µ
2a
∫
S
[u∞(r)−U0] dr. (1.34)
8
Expanding u∞(r) about the sphere center for creeping flow, and integrating over S
yields Faxen’s first law
F = 6πµa
[(u∞ +
1
6a2∇2u∞)0 −U0
]. (1.35)
The subscript zero indicates evaluation at r0, the particle center. This is an important
result, showing that in a nonuniform flow, the particle does not move with the mean
fluid velocity (u∞) in the absence of an external force, but that there is a correction
proportional to a2∇2u∞.
Faxen’s second law can be derived by taking the vector product of Eq. 1.33 with
(r − r0) before carrying out the second integration, and then separating the equation
into antisymmetric and symmetric parts, which must be satisfied individually (5). In
this way the torque on a sphere (7) is obtained as
T = 8πµa3
[1
2(∇× u∞)0 −Ω
](1.36)
and the force dipole (stresslet) as
S =10
3πµa3
(1 +
1
10a2∇2
)(∇u∞ +∇uT
∞). (1.37)
From Eq. 1.35, Stokes’s law for the force exerted on a sphere settling in a infinite
medium is obtained. By setting U0 = 0 or u∞ = 0 we get
F = 6πµau∞ or = −6πµaU0, (1.38)
and the torque is
T = −8πµa3Ω. (1.39)
1.3.1 Long-Range Interactions and the Divergence Problem
To obtain the flow disturbance caused by the presence of a sphere, Eq. 1.32 must
be evaluated. By setting Π = F/4πa2 = −3µU0/2a and expanding the Oseen tensor
9
about the center of the sphere (5) leaves
u(r) =3µ
2aU0 ·
∫
S
[O(R) + (r′ − r0) · ∇O(R) +
1
2(r′ − r0)(r
′ − r0) : ∇∇O(R) + . . .
]dr′
= 6πµaU0 ·(
1 +1
6a2∇2
)O(R) (1.40)
with R = r− r0, and U0 representing the Stokes settling velocity for an isolated sphere.
Equation 1.40 is derived from the identities
∫
S
dr′ = 4πa2 (1.41)
∫
S
(r′ − r0)(r′ − r0)dr
′ =4πa4
3I, (1.42)
and the remaining terms integrate to zero, since
∫
S
(r′ − r0)ndr′ = 0 for n odd, (1.43)
∇ · O = 0, and ∇2∇2O = 0. Substituting the Oseen tensor (Eq. 1.31) into Eq. 1.40
gives the velocity field (5)
u(R) =3a
4R
(1 +
a2
3R2
)U0 +
3a
4R
(1− a2
R2
)r ·U0r
R2, (1.44)
which satisfies the boundary conditions
u = U0, R = a (1.45)
u → 0, R →∞. (1.46)
Therefore, the disturbance is long range, decaying as 1/R.
Now we add one more sphere at a position r2 from r1 so that |r1 − r2| = R12 À a.
To first order from Eq. 1.44, Sphere 1 feels the flow caused by Sphere 2 with the
following strength
U12 =3a
4R12
(1 + R12R12) ·U0 + . . . (1.47)
where the vector R12 is the unit vector R12/R12. The velocity of Sphere 1 is now
increased with respect to the single sphere sedimentation velocity
U1 = U0 +3a
4R12
(1 + R12R12) ·U0 + . . . (1.48)
10
If the same reasoning is applied to the sedimentation of a cloud of N identical
spheres, then to first order in the inverse of the interparticle distance, we get
Ui = U0 +N∑j
3a
4Rij
(1 + RijRij) ·U0 + . . . (1.49)
Because the disturbance in the fluid due to particle interactions is long range, the
summation in Eq. 1.49 will diverge as the volume of the system becomes large.
The divergence problem was solved by Batchelor (8), who pointed out that there is
an upcurrent of displaced fluid that had not been considered. This backflow cancels the
long-range hydrodynamic interactions, and causes the settling velocity of a suspension
to be less than that of an isolated particle.
1.3.2 Multi-Particle Interactions
With the linearity of the Stokes equation presented in Sect. 1.3, the development
of the resistance and mobility relations for a single particle moving in a fluid can be
obtained, where
F = −ζTT ·U, (1.50)
T = −ζRR ·Ω. (1.51)
The forces F and torques T exerted on the particle by the fluid are related to the
particle velocities U and angular velocities Ω via the configuration dependent friction-
coefficients ζ (6, 9), which are developed from the solution of the Stokes equations.
For multi-particle interactions, the above equations become
Fi = −N∑
j=1
[ζTT
ij ·Uj + ζTRij ·Ωj
], (1.52)
Ti = −N∑
j=1
[ζRT
ij ·Uj + ζRRij ·Ωj
], (1.53)
where the summation extends over all N particles in the suspension, and ζ is the
resistance matrix. The superscripts T and R refer to the translational and rotational
components of ζ, with TR representing the coupling between the two components.
11
Equations 1.52 and 1.53 can be inverted to find the corresponding mobility
matrices µ, relating the particle velocities to the hydrodynamic forces and torques,
Ui = −N∑
j=1
[µTT
ij · Fj + µTRij ·Tj
], (1.54)
Ωi = −N∑
j=1
[µRT
ij · Fj + µRRij ·Tj
], (1.55)
Thus, the resistance and mobility matrices completely characterize the linear relations
between the force and torque with the translational and rotational velocities for particle
motion at low Reynolds number flow.
1.3.3 Particles Near Contact
For the case when particles come in close proximity with one another, lubrication
theory is applied to provide the leading order terms in the resistance and mobility
matrix. Lubrication theory is calculated based on the motion of two surfaces near
contact and the forces are pairwise additive in this limit,
F1
F2
T1
T2
= −µ
A11 A12 B11 B12
A21 A22 B21 B22
B11 B12 C11 C12
B21 B22 C21 C22
U1
U2
Ω1
Ω2
. (1.56)
The square resistance matrix contains second rank tensors A,B, and C.
The elements of the resistance matrix obey a number of symmetry conditions. The
reciprocal theorem states that the resistance matrix is symmetric (6, 9), so that
Aαβij = Aβα
ji , (1.57)
Bαβij = Bβα
ji , (1.58)
Cαβij = Cβα
ji . (1.59)
With e = R/R being the unit vector along the line of centers, we can write (6)
Aαβij = XA
αβeiej + Y Aαβ(δij − eiej), (1.60)
Bαβij = Bβα
ji = Y Bαβεijkek, (1.61)
Cαβij = XC
αβeiej + Y Cαβ(δij − eiej). (1.62)
12
The form of these scalar functions are developed, which is a function of the dimension-
less separation a/R. The dimensionless functions for diagonal elements of the resistance
matrix will approach unity for large R because the scales were chosen by considering
the single sphere result (6). From Eqs. 1.57 - 1.62, we obtain 10 independent non-
dimensional scalar functions to describe the resistance matrix. Analytic approximations
to these scalar functions can then be derived from lubrication theory for particles in
near contact.
CHAPTER 2DESCRIPTION OF THE LATTICE-BOLTZMANN MODEL
The lattice-Boltzmann model is based on the Boltzmann equation developed
from gas kinetic theory. Kinetic theory is the branch of statistical physics dealing with
the dynamics of non-equilibrium processes and their relaxation to thermodynamic
equilibrium. The Boltzmann equation, established by Ludwig Boltzmann in 1872, is its
cornerstone.
2.1 Molecular Theory for Fluid Flow
According to the molecular theory of matter, a macroscopic volume of gas (say,
1 cm3) is a system of a very large number (say, 1020) of molecules moving in a rather
irregular way. In principle, we may assume that the molecules are particles obeying
the laws of classical mechanics. It is also assumed that the laws of interaction between
the molecules are perfectly known so that, in principle, the evolution of the system is
computable, provided suitable initial data are given.
However, solving the initial value problem for a number of particles of a realistic
order of magnitude (say, N ' 1020) is an impossible task. As a result, only averages
can be computed and are related to macroscopic quantities as pressure, temperature,
stresses, heat flow, etc.; this is the basic idea of statistical mechanics.
From statistical mechanics, we talk about probabilities instead of certainties; that
is, a given particle will not have a definite position and velocity, but only probabilities
of having different positions and velocities. Under suitable assumptions, the information
required to compute averages for these systems can be reduced to the solution of
an equation, the so-called Boltzmann equation. With a solution to the Boltzmann
equation, we can then deduce macroscopic fluid flow from the microscopic model.
13
14
2.1.1 Boltzmann Equation
The kinematic equation for the one-body distribution function (10) reads as
follows:
Dtf =[∂t +
p
m· ∂r + F · ∂p
]f(r,p, t) = C12, (2.1)
where the left-hand side represents the streaming motion of the molecules along the
trajectories associated with the external force field F and C12 represents the effect of
intermolecular (two-body) collisions taking molecules in/out of the streaming trajectory.
Here r is the position of the particle, p ≡ mc its linear momentum, and F is the
force experienced by the particle as a result of intermolecular interactions and possibly
external fields.
The collision operator involves the two-body distribution function f12 that ex-
presses the probability of finding a molecule, say 1, around r1 with velocity c1 and
a second molecule, say 2, around r2 with velocity c2, both at time t. The dynamic
equation for C12 can be developed, but because the collisions are coupled this equation
calls into play the three-body function C123, which in turn depends on C1234 and so on
down an endless line known as the BBGKY hierarchy, after Bogoliubov, Born, Green,
Kirkwood, and Yvon (11).
To simplify Eq. 2.1, Boltzmann assumed that the system is a dilute gas of point-
like structureless molecules interacting via a short-range two-body potential. Under
such conditions, intermolecular interactions can be described solely in terms of localized
binary collisions, where molecules spends most of their time on free trajectories
perfectly unaware of each other. With this picture, the collision term splits into gain
(G) and loss (L) components:
C12 ≡ G− L =
∫(f1′2′ − f12)gσ(g, Ω)dΩdp2 (2.2)
corresponding to direct/inverse collisions taking molecules out/in the volume ele-
ment (10). The parameter σ is the collision cross section, which expresses the number
of molecules with relative speed ~g = g~Ω around the solid angle ~Ω.
15
Then comes Boltzmann’s closure assumption:
f12 = f1f2. (2.3)
This closure assumes no correlations between molecules entering a collision (molecular
chaos or Stosszahlansatz) and is the essential component that describes the Boltzmann
equation. The Boltzmann equation then takes the form:
[∂t +
p
m· ∂r + Fext · ∂p
]f(r,p, t) =
∫(f1′f2′ − f1f2)gσ(g, Ω)dΩdp2. (2.4)
The left hand side is Newton’s equation for single particle dynamics, while the right
hand side describes intermolecular collisions under the Stosszahlansatz approximation.
2.1.2 Connection between Microscopic and Macroscopic Dynamics
The mass density ρ(r, t) is the integral of the density in the one-particle phase
space f(r, c, t) with respect to all possible velocities
ρ(r, t) =
∫mfdc, (2.5)
where m is the molecular mass. Here, the probability distribution function f(r, c, t)
describes the number of molecules at time t, with position r lying within the velocity
space c. Because of the probabilistic meaning of f , the density ρ is the expected mass
per unit volume at (r, t) or the product of the molecular mass m by the probability
density of finding a molecule at (r, t), which turns out to be the number density n(r, t)
n(r, t) = ρ(r, t)/m. (2.6)
From Eq. 2.5, the momentum ρu can be written as follows
ρu =
∫cmfdc (2.7)
or, using the following components:
ρui =
∫cimfdc. (2.8)
The velocity u can be perceived as the macroscopic observation developed from
molecular motion; it is zero for the steady state of a gas enclosed in a box at rest.
16
Each molecule has its own velocity c which can be decomposed into the sum of u and
another velocity
v = c− u, (2.9)
which describes the random deviation of the molecule from the ordered motion with
velocity u. The velocity v is usually called the peculiar velocity or the random velocity;
it coincides with c when the gas is macroscopically at rest (12). Thus, we have from
Eqs. 2.5, 2.8, and 2.9,
∫vimfdc =
∫cimfdc− ui
∫mfdc = ρui − ρui = 0. (2.10)
Another quantity needed for developing macroscopic hydrodynamics is the mo-
mentum flux. Since momentum is a vector, we therefore consider the flow of the jth
component of momentum in the ith direction; given by
∫ci(cjmf)dc =
∫cicjmfdc. (2.11)
Eq. 2.11 shows that the momentum flux is described by a symmetric tensor of second
order. It is known that in a macroscopic description only a part of the microscopically
evaluated momentum flux will be identified as convective, because the integral in
Eq. 2.11 will be in general different from zero even if the gas is macroscopically at rest.
In order to find out how the momentum flux will appear in a macroscopic description,
we have to use the splitting of c into mean velocity u and peculiar velocity v. This is
written as
∫cicjmfdc =
∫(ui + vi)(uj + vj)mfdc = uiuj
∫mfdc + ui
∫vjmfdc
+uj
∫vimfdc +
∫vivjmfdc = ρuiuj +
∫vivjmfdc, (2.12)
where Eq. 2.5 and 2.10 have been applied. The momentum flux therefore breaks into
two parts, one of which is recognized as the macroscopic momentum flux, while the
second part is a hidden momentum flux due to the random motion of the molecules.
For example, in a fixed region of gas, we observe that the change in momentum is only
attributed to the matter that enters and leaves the region, which means the second part
has no macroscopic explanation unless it is attributed to the action of a force exerted
17
x
z
y
Figure 2–1: The 18 velocity vectors (ci) in the lattice-Boltzmann model.
on the boundary of the region of interest by the contiguous region of the gas. In other
words, the integral of∫
vivjmfdc appears as the contribution to the stress tensor (12)
pij =
∫vivjmfdc. (2.13)
Here, the trace pii ≡ Σ3i=1pii is the isotropic part of the stress tensor. More
specifically, the fluid pressure can be written as P = pii/3 for the case at equilibrium.
2.2 Lattice-Boltzmann Method
The computational utility of lattice-Boltzmann models depends on the fact that
only a small set of velocities are necessary to simulate the Navier-Stokes equations (13).
Thus the more general kinetic equations in Sec. 2.1 are simplified by considering a
discrete set of velocities only. The lattice used in this simulation is a three dimensional
cubic lattice for which there are 18 velocities, Fig. 2–1. This model uses the [100] and
[110] directions, with twice the density of particles moving in the [100] directions as
in [110] directions. This gives the symmetry condition necessary to ensure that the
hydrodynamic transport coefficients are isotropic. In this work the 18-velocity model is
augmented with stationary particles, which enables it to account for small deviations
from the incompressible limit, although in simulations of stationary flows it is found
that the numerical differences are small (14).
18
In the lattice-Boltzmann method (LBM), probabilities substitute for discrete
particles in the particle velocity function ni(r, t). The LBM then describes the time
evolution of the discretized velocity distribution
ni(r + ci∆t, t + ∆t) = ni(r, t) + ∆i [n(r, t)] , (2.14)
where ∆i is the change in ni due to molecular collisions at the lattice nodes, and ∆t
is the time step. This one-particle distribution function describes the mass density of
particles with velocity ci, at a lattice node r, and at a time t; r, t, and ci are discrete,
whereas ni is continuous. The hydrodynamic fields, mass density ρ, momentum density
j = ρu (where u is the fluid flow velocity), and momentum flux Π, are moments of this
velocity distribution:
ρ =∑
i
ni, j = ρu =∑
i
nici, Π =∑
i
nicici. (2.15)
The post-collision distribution, ni(r, t) + ∆i [n(r, t)], is propagated for one time-
step, in the direction ci. The collision operator ∆i(n) depends on all the ni’s at the
node represented collectively by n(r, t), with the constraints of mass and momentum
conservation. Exact expressions for the Boltzmann collision operator have been
derived for several lattice-gas models (15, 16) by making use of the “molecular chaos”
assumption (that the effect of the collision operator depends only on the instantaneous
state of a node). However, such collision operators are complex and ill-suited to
numerical simulation. A computationally more useful form for the collision operator
∆i(n) can be constructed by linearizing about the local equilibrium(17) neq
∆i(n) = ∆i(neq) +
∑j
Lijnj, (2.16)
where Lij are the matrix elements of the linearized collision operator, and ∆i(neq) = 0.
L is calculated by considering the general principles of conservation and symmetry and
the construction of the eigenvalues and eigenvectors of L.
The population density associated with each velocity has a weight aci that de-
scribes the fraction of particles with velocity ci in a system at rest; these weights
depend only on the speed ci and are normalized so that Σiaci = 1. The velocities ci are
19
chosen such that all particles move from node to node simultaneously. For any cubic
lattice,∑
i
acicici = C2c2I, (2.17)
where c = ∆x/∆t, ∆x is the grid spacing, and C2 is a numerical coefficient that
depends on the choice of weights. However, in order for the viscous stresses to be
independent of direction, the velocities must also satisfy the isotropy condition;
∑i
aciciαciβciγciδ = C4c4 δαβδγδ + δαγδβδ + δαδδβγ . (2.18)
The isotropy condition requires that a multi-speed model be applied, which in this case
is the 18-velocity model for a simple cubic lattice.
As a first step, the correct form of the equilibrium distribution must be established.
To determine the equilibrium distribution, the velocity distribution function is split into
a local equilibrium part and a non-equilibrium part
ni = neqi + nneq
i . (2.19)
The equilibrium distribution is a collisional invariant (i.e., ∆i(neq) = 0). The form
of the equilibrium distribution is constrained by the moment conditions required to
reproduce the inviscid (Euler) equations with non-dissipative hydrodynamics on large
length and time scales. In particular, the second moment of the equilibrium distribution
should be equal to the inviscid momentum flux PI + ρuu:
ρ =∑
i
neqi (2.20)
j =∑
i
neqi ci = ρu (2.21)
Πeq =∑
i
neqi cici = ρc2
sI + ρuu (2.22)
The equilibrium distribution can be used in Eqs. 2.20 and 2.21 (c.f. Eq. 2.15)
because mass and momentum are conserved during the collision process, thus
∑i
nneqi =
∑i
nneqi ci = 0. (2.23)
20
The pressure in Eq. 2.22, P= ρc2s, takes the form of an ideal gas equation of state
with adiabatic sound speed cs. This is also adequate for the liquid phase if the density
fluctuations are small (i.e. the Mach number M = u/cs ¿ 1), so that ∇P = c2s∇ρ.
For small Mach numbers, the equilibrium distribution can be expanded as a power
series in u · ci/c2s, and it then follows from Eq. 2.22 that the weights must be chosen so
that C2 = (cs/c)2; i.e. from Eq. 2.17
∑i
acicici = c2sI. (2.24)
A suitable form for the equilibrium distribution of the 19-velocity model that satisfies
Eqs. 2.20- 2.22, as well as the isotropy condition (18) (Eq. 2.18), is
neqi = aci
[ρ +
j · ci
c2s
+ρuu : (cici − c2
sI)
2c4s
], (2.25)
where cs =√
c2/3, c = ∆x/∆t, P = ρc2s, and the coefficients of the three speeds are
a0 =1
3, a1 =
1
18, a
√2 =
1
36. (2.26)
In this case the coefficient in Eq. 2.18 is C4 = (cs/c)4.
From Eqs. 2.25 and 2.26, the inviscid hydrodynamic equations are correctly repro-
duced. The viscous stresses come from moments of the non-equilibrium distribution,
as in the Chapman-Enskog solution of the Boltzmann equation. The fundamental
limitation of this class of lattice-Boltzmann model is that the Mach number be small,
less than 0.3; our suspension simulations always keep M < 0.1.
Having constructed the local equilibrium distribution, we come to the relaxation
of the non-equilibrium component of the local distribution. Beyond the conservation of
mass and momentum inherent in neq, the eigenvalue equations of the collision operator
for nneq develop the isotropic relaxation of the stress tensor.
The linearized collision operator must satisfy the following eigenvalue equations;
∑i
Lij = 0,∑
i
ciLij = 0,∑
i
ciciLij = λcjcj,∑
i
c2iLij = λvc
2j , (2.27)
where cici, indicates the traceless part of cici. The first two equations follow from
conservation of mass and momentum (c.f. Eq. 2.23), and the last two equations
21
describe the isotropic relaxation of the stress tensor; the eigenvalues λ and λv are
related to the shear and bulk viscosities and lie in the range −2 < λ < 0. Equation 2.27
accounts for only 10 of the eigenvectors of L. The remaining 9 modes are higher-order
eigenvectors of L that are not relevant to the Navier-Stokes equations, but which do
affect the boundary conditions at the solid-fluid interfaces. In general the eigenvalues
of these kinetic modes are set to -1, which both simplifies the simulation and ensures a
rapid relaxation of the non-hydrodynamic modes (19).
The collision operator can be further simplified by taking a single eigenvalue for
both the viscous and kinetic modes (20, 18). This exponential relaxation time (ERT)
approximation, ∆i = −nneqi /τ , has become the most popular form for the collision
operator because of its simplicity and computational efficiency. However, the absence
of a clear time scale separation between the kinetic and hydrodynamic modes can
sometimes cause significant errors at solid-fluid boundaries, and thus the more flexible
collision operator described by Eq. 2.27 is preferred.
A 3-parameter collision operator is used in the suspension simulations, allowing
for separate relaxation of the 5 shear modes, 1 bulk mode, and 9 kinetic modes. The
post-collision distribution n?i = ni + ∆i is written as a series of moments (Eq. 2.15), as
seen with the equilibrium distribution (Eq. 2.25),
n?i = aci
(ρ +
j · ci
c2s
+(ρuu + Πneq,?) : (cici − c2
sI)
2c4s
). (2.28)
The zeroth (ρ) and first (j = ρu) moments are the same as in the equilibrium distri-
bution (Eq. 2.25), but the non-equilibrium second moment Πneq is modified by the
collision process, according to Eq. 2.27:
Πneq,? = (1 + λ)Πneq
+1
3(1 + λv)(Π
neq : I)I, (2.29)
where Πneq = Π − Πeq. The kinetic modes can also contribute to the post-collision
distribution, but the eigenvalues of these modes are set at −1, so that they have no
effect on n?i . Equation 2.29 with λ = λv = −1 is equivalent to the ERT model with
τ = 1; for λ < −1, the kinetic modes relax more rapidly than the viscous modes, which
is the proper limit for hydrodynamics.
22
The macrodynamical behavior arises from the lattice-Boltzmann equation with the
multi-time-scale analysis applied (15, 14). The Navier-Stokes equations,
∂tρ + ∇ · (ρu) = 0
∂t(ρu) + ∇ · (ρuu) + ∇ρc2s = µ∇2u + (µv + µ)∇(∇ · u),
(2.30)
are recovered in the low velocity limit, with viscosities
µ = −ρc2s∆t
(1
λ+
1
2
)and µv = −ρc2
s∆t
(2
3λv
+1
3
). (2.31)
The factors of 1/2 and 1/3, which appear in the definitions of the shear and bulk
viscosities serve to correct for numerical diffusion, so that viscous momentum diffuses at
the expected speed.
In the presence of an externally imposed force density f , for example a pressure
gradient or a gravitational field, the time evolution of the lattice-Boltzmann model
includes an additional contribution fi(r, t),
ni(r + ci∆t, t + ∆t) = ni(r, t) + ∆i [n(r, t)] + fi(r, t). (2.32)
This forcing term can be expanded in a power series in the velocity,
fi = aci
[f · ci
c2s
+(uf + fu) : (cici − c2
sI)
2c4s
]∆t. (2.33)
The expression relating the force density to the change in velocity distribution was
determined by matching the macroscopic dynamics from Eq. 2.32 to the Navier-Stokes
equations (14). More accurate solutions to the velocity field are obtained if it includes a
portion of the momentum (14) added to each node,
j′ = ρu′ =∑
i
nici +1
2f∆t. (2.34)
2.2.1 Gas Kinetics to Fluid Dynamics: The Chapman-Enskog Expansion
In order to solve the Boltzmann equation introduced in Sec. 2.1 for realistic
nonequilibrium situations, we must rely upon approximation methods, in particular,
perturbation procedures. In order to do this, we look for a parameter ε which we
consider to be small. Therefore, in obtaining macroscopic hydrodynamic equations, we
23
introduce the nondimensional Knudsen number denoted by Kn
Kn = ε = l/d, (2.35)
where l is the mean free path between molecular collisions, and d is a typical length
scale. Thus, Kn → 0 corresponds to a fairly dense gas and Kn → ∞ to a free
molecular flow (i.e., a flow where molecules have negligible interactions with each
other).
The Chapman-Enskog expansion, then, considers multiple time scales, of orders ε
and ε2, to recover the Navier-Stokes equation. In this sense, the Navier-Stokes equation
describes two kinds of processes, convection and diffusion, which act on two different
time scales (21). If we consider only the first scale, we obtain the compressible Euler
equation; if we move on to the second one we obtain the Navier-Stokes equation while
losing the compressibility effect. If we want both compressibility and diffusion, we
have to keep both scales at the same time and think of n in terms of time and space
derivatives
∂n
∂t= ε
∂n
∂t1+ ε2 ∂n
∂t2(2.36)
∂n
∂r= ε
∂n
∂r1
. (2.37)
This enables us to introduce two different time variables t1 = εt, t2 = ε2t and a new
space variable r1 = εr, such that the fluid dynamical variables are functions of r1, t1, t2,.
The compatibility conditions at the first order give that the time derivatives of
the fluid dynamic variables with respect to t1 is determined by the Euler equation but
the derivatives with respect to t2 are determined only at the next level and are given
by the terms of the Navier-Stokes equation describing the effect of viscosity and heat
conductivity (21). The two contributions are to be added as indicated in Eq. 2.36 to
obtain the full time derivative and thus write the Navier-Stokes equation.
While the Chapman-Enskog expansion was originally applied to the continuous
Boltzmann equation in Eq. 2.1, here we derive the Navier-Stokes equation using the
24
simplified lattice Boltzmann equation represented as follows:
ni(r + ci∆t, t + ∆t)− ni(r, t) = Ωi(r, t), (2.38)
with
Ωi(r, t) = −1
τ[ni(r, t)− neq
i (r, t)]. (2.39)
The collision operator here is from the single relaxation time Bhatnagar-Gross-Krook
(BGK) model where neqi represents the equilibrium distribution and τ is a typical
time-scale associated with collisional relaxation to the local equilibrium.
To evaluate the LBGK model, first a Taylor series expansion for ni(r+ ci∆t, t+∆t)
is performed,
ni(r + ci∆t, t + ∆t) = ni(r, t) + ∆t∂ni
∂t+ ciα∆t
∂ni
∂rα
+
(∆t)2
2
[∂2ni
∂t2+ 2ciα
∂2ni
∂t∂rα
+ ciαciβ∂2ni
∂rα∂rβ
]. (2.40)
Next we introduce the two time scales and one spatial scale as noted in Eq. 2.36
and 2.37. The distribution function is also expanded accordingly:
ni = n(0)i + εn
(1)i + ε2n
(2)i + O(ε3). (2.41)
Substitute Eq. 2.40 into the LHS of Eq. 2.38 with Eq. 2.39 gives
∆t∂ni
∂t+ ciα∆t
∂ni
∂rα
+(∆t)2
2
[∂2ni
∂t2+ 2ciα
∂2ni
∂t∂rα
+ ciαciβ∂2ni
∂rα∂rβ
]
+1
τ(ni − neq
i ) = 0. (2.42)
Substitute Eqs. 2.36, 2.37, and 2.41 into Eq. 2.42 gives
ε∆t
[∂n
(0)i
∂t1+ ciα
∂n(0)i
∂r1α
]+ ε2∆t
[∂n
(1)i
∂t1+
∂n(0)i
∂t2+ ciα
∂n(1)i
∂r1α
]+ (2.43)
ε2 (∆t)2
2
[∂2n
(0)i
∂t1∂t1+ 2ciα
∂2n(0)i
∂t1∂r1α
+ ciαciβ∂2n
(0)i
∂r1α∂r1β
]+ (2.44)
1
τ(n
(0)i − neq
i ) + ε1
τn
(1)i + ε2 1
τn
(2)i = 0. (2.45)
25
Arranging terms in ascending orders of ε gives
O(ε0) : n(0)i = neq
i (2.46)
O(ε1) : n(1)i = −τ∆t
[∂n
(0)i
∂t1+ ciα
∂n(0)i
∂r1α
](2.47)
O(ε2) : n(2)i = −τ∆t
[∂n
(1)i
∂t1+
∂n(0)i
∂t2+ ciα
∂n(1)i
∂r1α
]
−τ(∆t)2
2
[∂2n
(0)i
∂t1∂t1+ 2ciα
∂2n(0)i
∂t1∂r1α
+ ciαciβ∂2n
(0)i
∂r1α∂r1β
]. (2.48)
It is noted that the equilibrium distribution, along with Eq. 2.46, should satisfy the
following constraints:
ρ =m∑
i=0
ni =m∑
i=0
neqi =
m∑i=0
n(0)i (2.49)
ρuα =m∑
i=0
ciαni =m∑
i=0
ciαneqi =
m∑i=0
ciαn(0)i , (2.50)
where m is the total number of directions in the lattice model, and the molecular mass
is set to unity. With these constraints applied to Eq. 2.41, the constraints for the 1st
and 2nd order components in ε become
0 =m∑
i=0
n(1)i =
m∑i=0
n(2)i (2.51)
0 =m∑
i=0
ciαn(1)i =
m∑i=0
ciαn(2)i . (2.52)
Using the above relation in Eq. 2.47 gives the following reduction
m∑i=0
[∂n
(0)i
∂t1+ ciα
∂n(0)i
∂r1α
]= 0 → ∂
∂t1
m∑i=0
n(0)i +
∂
∂r1α
m∑i=0
ciαn(0)i = 0
→ ∂ρ
∂t1+
∂ρuα
∂r1α
= 0. (2.53)
Furthermore, the second moment is obtained by
m∑i=0
ciβ
[∂n
(0)i
∂t1+ ciα
∂n(0)i
∂r1α
]= 0 → ∂
∂t1
m∑i=0
ciβn(0)i +
∂
∂r1α
m∑i=0
ciαciβn(0)i = 0
→ ∂ρuβ
∂t1+
∂P(0)αβ
∂r1α
= 0. (2.54)
26
Recall from the derivation in Eq. 2.12 using the peculiar velocity, the viscous stress in
terms of the distribution function n can be determined to be
m∑i=0
ciαciβn(0)i = P
(0)αβ = ρuαuβ + Pδαβ. (2.55)
Thus, to first order in ε, from Eq. 2.53 the continuity equation is obtained and from
Eqs. 2.54 and 2.55 the Euler equation is determined
ρ∂uβ
∂t+ ρuα
∂uβ
∂r1α
= − ∂P
∂r1α
. (2.56)
Next, the O(ε2) terms are analyzed. From Eq. 2.48, the relations in Eqs. 2.49, 2.51,
2.53, and 2.54 are applied to get a relation in terms of t2, and
m∑i=0
n(2)i = 0 → 0 =
m∑i=0
[∂n
(1)i
∂t1+
∂n(0)i
∂t2+ ciα
∂n(1)i
∂r1α
]+ (2.57)
m∑i=0
∆t
2
[∂2n
(0)i
∂t1∂t1+ 2ciα
∂2n(0)i
∂t1∂r1α
+ ciαciβ∂2n
(0)i
∂r1α∂r1β
].
Then
∂
∂t1
m∑i=0
n(1)i = 0, (2.58)
∂
∂t2
m∑i=0
n(0)i =
∂ρ
∂t2, (2.59)
∂
∂r1α
m∑i=0
ciαn(1)i = 0. (2.60)
Substitution of Eqs. 2.53 and 2.54 into the 4th, 5th, and 6th terms of Eq. 2.57 gener-
ates and overall value of zero. Thus, the zeroth moment of O(ε2) becomes
∂ρ
∂t2= 0. (2.61)
For the momentum conservation, Eq. 2.52 is applied to give
m∑i=0
ciαn(2)i = 0 → 0 =
m∑i=0
ciα
[∂n
(1)i
∂t1+
∂n(0)i
∂t2+ ciα
∂n(1)i
∂r1α
]+ (2.62)
m∑i=0
ciα∆t
2
[∂2n
(0)i
∂t1∂t1+ 2ciα
∂2n(0)i
∂t1∂r1α
+ ciαciβ∂2n
(0)i
∂r1α∂r1β
].
27
Here, applying Eqs. 2.50 and 2.52 gives
∂
∂t1
m∑i=0
ciαn(1)i = 0, (2.63)
∂
∂t2
m∑i=0
ciαn(0)i =
∂ρuα
∂t2. (2.64)
Applying Eq. 2.47 into the 3rd component of Eq. 2.62 gives
∂
∂r1α
m∑i=0
ciαciβn(1)i = −τ∆t
∂
∂r1α
m∑i=0
ciαciβ
[∂n
(0)i
∂t1+ ciγ
∂n(0)i
∂r1γ
]. (2.65)
The 4th component in Eq. 2.62 can be evaluated further by substitution to give
∆t
2
m∑i=0
ciα∂2n
(0)i
∂t1∂t1=
∆t
2
∂2ρuα
∂t1∂t1=
∆t
2
∂
∂t1
(−∂P
(0)αβ
∂riβ
)(2.66)
= −∆t
2
∂
∂t1
∂
∂r1β
(ρuαuβ + ρc2
sδαβ
)(2.67)
≈ −∆t
2c2s
∂
∂r1β
∂ρ
∂t1
δαβ, (2.68)
where O(u2) terms are neglected due to low Mach number flows. Here cs respresents
the speed of sound which is characterized by the pressure at equilibrium (i.e P=ρRT =
ρc2s). Substitution from Eq. 2.53 gives
=∆t
2c22
∂
∂r1α
∂
∂r1β
(ρuα) =∆t
2c2s∇∇ · (ρu). (2.69)
The 5th component of Eq. 2.62 is considered in conjunction with the 1st term in
Eq. 2.65 to get
(1− τ)∆t∂
∂t1
∂
∂r1α
m∑i=0
ciαciβn(0)1 = (1− τ)∆t
∂
∂t1
∂
∂r1β
P(0)αβ
≈ −(1− τ)∆tc2s
∂
∂r1α
∂
∂r1β
(ρuα) = −(1− τ)∆tc2s∇∇ · (ρu) (2.70)
28
and the 6th component of Eq. 2.62 with the second term in Eq. 2.65 gives
−(
τ − 1
2
)∆t
∂
∂r1α
∂
∂r1β
m∑i=0
ciαciβciγn(0)i (using defn. from Eq. 2.25)
≈ −(
τ − 1
2
)∆t
c2s
∂
∂r1α
∂
∂r1β
m∑i=0
aciciαciβciγ(ciκρuκ) (2.71)
≈ −(
τ − 1
2
)∆tc2
s
[∂
∂r1α
∂
∂r1α
(ρuβ) + 2∂
∂r1α
∂
∂r1β
(ρuβ)
]
= −(
τ − 1
2
)∆tc2
s
[∇2(ρu) + 2∇∇ · (ρu)]. (2.72)
The new term in Eq. 2.71 is taken from the istropic condition for this lattice model.
Finally combining Eqs. 2.64, 2.69, 2.70, and 2.72 into Eq. 2.62 gives
∂
∂t2(ρu) =
(τ − 1
2
)∆tc2
s
[∇2(ρu) +∇∇ · (ρu)]. (2.73)
Thus the dynamic shear and bulk viscosities for the lattice BGK model are
µS = µB =
(τ − 1
2
)∆tc2
s. (2.74)
The end result is to sum over all orders of ε. With Eq. 2.53 added to Eq. 2.61, the
continuity equation is determined:
∂ρ
∂t+
∂ρuα
∂rα
= 0. (2.75)
Eqs. 2.54, with Eq. 2.55, added to Eq. 2.73 gives
∂ρuα
∂t+ uβ
∂ρuα
∂rβ
= − ∂P
∂rα
+
(τ − 1
2
)∆tc2
s
[∇2(ρu) +∇∇ · (ρu)]. (2.76)
In the incompressible limit,
∂uβ
∂rβ
= 0 (2.77)
∂ρuα
∂t+ uβ
∂ρuα
∂rβ
= − ∂P
∂rα
+ µ∇2(ρu). (2.78)
Thus, the continuum Navier-Stokes equation has been derived. In the end, the basic
result of the Chapman-Enskog procedure is that the macroscopic description of a fluid
is recovered by a suitable expansion of the Boltzmann equation.
29
a
b
Figure 2–2: Location of boundary nodes for a curved surface (a) and two surfaces innear contact (b).
30
2.2.2 Solid-Fluid Boundary Conditions
Boundary conditions in the lattice-Boltzmann model are straightforward to
implement, even for non-planar surfaces (19). Solid particles are defined by a surface,
which cuts some of the links between lattice nodes. Fluid particles moving along these
links interact with the solid surface at boundary nodes placed halfway along the links.
Thus we obtain a discrete representation of the particle surface, which becomes more
and more precise as the particle gets larger (Fig. 2–2).
The motion of a particle is determined by the forces and torques acting on it.
These forces consist of external, non-hydrodynamic forces, and hydrodynamic forces.
Non-hydrodynamic forces, such as attraction and/or repulsion between surfaces due
to an inter-particle potential, can be calculated separately and incorporated into the
equations of motion. In contrast, the hydrodynamic forces are calculated from the
resultant velocity distributions, ni.
Stationary solid objects are introduced into the simulation by the “bounce-back”
collision rule at the specified boundary nodes, in which incoming distributions are
reflected back towards the nodes they came from. Surface forces are calculated from the
momentum transfer at each boundary node and summed to give the force and torque
on each object (19). While more accurate methods that require knowledge of the shape
of the particle surface have been investigated (22, 23, 24, 25, 26, 27), the bounce-back
rule can be applied to surfaces of arbitrary shape, without additional complications.
However, for second-order convergence the bounce-back rule requires a calibration of
the hydrodynamic radius, which is not always convenient.
The boundary nodes are located midway between interior (solid) and exterior
(fluid) nodes (28, 29). The normal collision rules are carried out at all fluid nodes, and
augmented by bounce-back rules at the midpoints of links connecting the lattice nodes.
This is depicted by the arrows in Fig. 2–2a linking the normal distribution between
fluid (circles) and boundary nodes (squares). In Poiseuille flow the “link bounce-back”
rule gives velocity fields that deviate from the exact solution by a constant slip velocity,
us = uLBE − uExact, proportional to L−2 (30) (L being the channel width);
us/uc = β∆x2/L2, (2.79)
31
ub
t
t + 1
Figure 2–3: Moving boundary node update. The solid circles are fluid nodes, squaresare boundary nodes with the associated velocity vector, and open circles are interiornodes.
where uc = L2|∇p|/8ν is the exact velocity at the center of the channel and the
coefficient β depends on the eigenvalues of the collision operator.
In the past the lattice nodes were treated on either side of the boundary surface
in an identical fashion, so that fluid fills the whole volume of space, both inside and
outside the solid particles. Because of the relatively small volume inside each particle,
the interior fluid quickly relaxes to rigid-body motion, characterized by the particle
velocity and angular velocity, and closely approximates a rigid body of the same
mass (14). However, at short times the inertial lag of the fluid is noticeable, and the
contribution of the interior fluid to the particle force and torque reduces the stability
of the particle velocity update. For these reasons, with the suggestions in Ref. (31),
the interior fluid is removed from the particle representation. A somewhat different
implementation of the same idea is described in Ref. (32).
The moving boundary condition (19) without interior fluid (31) is implemented
as shown in Fig. 2–3. We take the set of fluid nodes r just outside the particle surface,
and for each node all the velocities cb such that r + cb∆t lies inside the particle surface.
Each of the corresponding population densities is then updated according to a simple
rule which takes into account the motion of the particle surface (19);
nb′(r, t + ∆t) = n∗b(r, t)−2acbρ0ub · cb
c2s
, (2.80)
32
where n∗b(r, t) is the post-collision distribution at (r, t) in the direction cb, and cb′ =
−cb. For a stationary boundary the population is simply reflected back in the direction
it came from with ub = 0 (15, 33). The local velocity of the particle surface,
ub = U + Ω×(rb −R), (2.81)
is determined by the particle velocity U, angular velocity Ω, and center of mass R;
rb = r + 12cb∆t is the location of the boundary node. The mean density ρ0 in Eq. 2.80
is used instead of the local density ρ(r, t) since it simplifies the update procedure. The
differences between the two methods are small, of the same order (ρu3) as the error
terms in the lattice-Boltzmann model. Test calculations show that even large variations
in fluid density (up to 10%) have a negligible effect on the force (less than 1 part in
104).
As a result of the boundary node updates, momentum is exchanged locally between
the fluid and the solid particle, but the combined momentum of solid and fluid is
conserved. The forces exerted at the boundary nodes can be calculated from the
momentum transferred in Eq. 2.80,
f(rb, t + 12∆t) =
∆x3
∆t
[2n∗b(r, t)−
2acbρ0ub · cb
c2s
]cb, (2.82)
The particle forces and torques are then obtained by summing f(rb) and rb × f(rb) over
all the boundary nodes associated with a particular particle. It can be shown analyt-
ically that the force on a planar wall in a linear shear flow is exact (19), and several
numerical examples of lattice-Boltzmann simulations of hydrodynamic interactions are
given in Ref. (34).
To understand the physics of the moving boundary condition, one can imagine
an ensemble of particles, moving at constant speed cb, impinging on a massive wall
oriented perpendicular to the particle motion. The wall itself is moving with velocity
ub ¿ cb. The velocity of the particles after collision with the wall is −cb + 2ub
and the force exerted on the wall is proportional to cb − ub. Since the velocities in
the lattice-Boltzmann model are discrete, the desired boundary condition cannot be
implemented directly, but we can instead modify the density of returning particles so
33
that the momentum transferred to the wall is the same as in the continuous velocity
case. It can be seen that this implementation of the no-slip boundary condition leads
to a small mass transfer across a moving solid-fluid interface. This is physically correct
and arises from the discrete motion of the solid surface. Thus during a time step ∆t
the fluid is flowing continuously, while the solid particle is fixed in space. If the fluid
cannot flow across the surface there will be large artificial pressure gradients, arising
from the compression and expansion of fluid near the surface. For a uniformly moving
particle, it is straightforward to show that the mass transfer across the surface in a time
step ∆t (Eq. 2.80) is exactly recovered when the particle moves to its new position.
For example, each fluid node adjacent to a planar wall has 5 links intersecting the
wall. If the wall is advancing into the fluid with a velocity U, then the mass flux across
the interface (from Eq. 2.80) is ρ0U. Apart from small compressibility effects, this is
exactly the rate at which fluid mass is absorbed by the moving wall. For sliding motion,
Eq. 2.80 correctly predicts no net mass transfer across the interface.
2.2.3 Hydrodynamic Radius of a Particle
Although the link-bounce-back rule at the solid-fluid interface is second order
accurate for planar walls oriented along lattice symmetry directions, it is only first
order accurate for channels oriented at arbitrary angles (35, 36). Thus for large
channels, the hydrodynamic boundary is displaced by an amount ∆ from the physical
boundary, where ∆ is independent of channel width but depends on the orientation of
the channel with respect to the underlying lattice. Convex bodies sample a variety of
boundary orientations, so that it is not possible to make an analytical determination
of the displacement of the hydrodynamic boundary from the solid particle surface.
Nevertheless, the displacement of the boundary can be determined numerically from
simulations of flow around isolated particles. By considering the size of the particles
to be the hydrodynamic radius, ahy = a + ∆, rather than the physical radius a,
approximate second-order convergence can be obtained, even for dense suspensions (34).
The hydrodynamic radius can be determined from the drag on a fixed sphere in
a pressure-driven flow (34). Galilean invariance can be demonstrated by showing that
the same force is obtained for a moving particle in a stationary fluid (34). Since the
34
1.320
1.360
1.400
1.440
100 120 140 160 180 200
F/F
0
a = 2.7a = 2.5
1.415
1.420
100 120 140 160 180 200
F/F
0
t/tS
a = 8.2a = 8.5
Figure 2–4: Drag force F as a function of time, normalized by the drag force on anisolated sphere, F0 = 6πηaU . Time is measured in units of the Stokes time, tS = a/U .
particle samples different boundary node maps as it moves on the grid, it is important
to sample different particle positions when determining the hydrodynamic radius,
especially when the particle radius is small (< 5∆x). Rather than averaging over
many fixed configurations, we chose to have the particle move slowly across the grid,
at constant velocity, sampling different boundary node maps as it goes. The changing
boundary node maps lead to fluctuations in the drag force, as shown in Fig. 2–4. The
force has been averaged over a Stokes time, tS, so that the relative fluctuations in force
are comparable to the relative fluctuations in velocity of a neutrally buoyant particle
in a constant force field. The force fluctuations, δF =√
(< F 2 > − < F >2)/ < F >,
are of the order of one percent for particles of radius 2 − 3∆x, and are considerably
smaller for larger particles (Table 2–1). More sophisticated boundary conditions have
been developed using finite-volume methods (37, 38) and interpolation (23, 25, 39).
Both methods reduce the force fluctuations by at least an order of magnitude from
those observed here, but even with the simple bounce-back scheme, the fluctuations
in force can be reduced by an appropriate choice of particle radius. We have noticed
35
Table 2–1: Variance in the computed drag force δF =√
< F 2 > − < F >2/ < F > for aparticle of radius a moving along a random orientation with respect to the grid.
a/∆x 2.7 2.5 8.2 8.5
δF 5.738e-03 1.208e-02 4.332e-04 5.674e-04
that fluctuations in particle force are strongly correlated with fluctuations in particle
volume. Thus we choose the radius of the boundary node map so as to minimize
fluctuations in particle volume for random locations on the grid. It can be seen
from Table 2–1 that a two fold reduction in the force fluctuations is possible by this
procedure, although for sufficiently large particles the difference is minimal. A set of
optimal particle radii is given in Table 2–2.
The bounce-back rule leads to a velocity field in the region of the boundary
that is first-order accurate in the grid spacing ∆x. The hydrodynamic boundary
(the surface where the fluid velocity field matches the velocity of the particle) is
displaced from the particle surface by a constant, ∆ (Fig. 2–5), that depends on the
viscosity of the fluid (34). For the range of kinematic viscosities used in this work,
1/6 ≤ ν∗ ≤ 1/1200, ∆ varies from 0 to 0.5∆x (Table 2–2); the dimensionless kinematic
viscosity ν∗ = ν∆t/∆x2. For small particles (a < 5∆x), ∆ also depends weakly on
the particle radius (Table 2–2). Although the difference between the hydrodynamic
boundary and the physical boundary is small, it is important in obtaining accurate
results with computationally useful particle sizes. Calibration of the hydrodynamic
radius leads to approximately second-order convergence from an inherently first-order
boundary condition; it will not be necessary when more accurate boundary conditions
are implemented.
The hydrodynamic radii (ahy) in Table 2–2 were determined from the drag force on
a single sphere in a periodic cubic cell (34). The Reynolds number in these simulations
Table 2–2: Hydrodynamic radius ahy (in units of ∆x) for various fluid viscosities; a isthe input particle radius.
a/∆x 1.24 2.7 4.8 6.2 8.2
ν∗ = 1/6 1.10 2.66 4.80 6.23 8.23ν∗ = 1/100 1.42 2.92 5.04 6.45 8.46ν∗ = 1/1200 1.64 3.18 5.31 6.72 8.75
36
∆
∆
ahy
h
a
Figure 2–5: Actual (solid lines) and hydrodynamic (dashed lines) surfaces for a particlesettling onto a wall.
(< 0.2) was sufficiently small to have a negligible effect on the drag force. The time
averaged force was used to determine the effective hydrodynamic radius for three
different kinematic viscosities: ν∗ = 1/6, ν∗ = 1/100, and ν∗ = 1/1200. In each case
the cell dimensions were 10 times the particle radius and the corrections for periodic
boundary conditions (about 40%) were made as described in Ref. (34).
The difference between the hydrodynamic radius and the input radius, ∆ =
ahy − a, is independent of particle size for radii a > 5∆x, and its magnitude increases
with decreasing kinematic viscosity (14). The kinematic viscosity ν∗ = 1/6 gives
a hydrodynamic radius that is the same as the input radius (for sufficiently large
particles), and is the natural choice for simulations at low Reynolds number. At higher
Reynolds numbers a lower viscosity is necessary to maintain incompressibility (14, 34),
and for accurate results it is then essential to use the calibrated hydrodynamic radius.
In order to implement the hydrodynamic radius in a multi-particle suspension, all
distance calculations are based on the hydrodynamic radius (as shown in Fig. 2–5);
the input radius a is only used to determine the location of the boundary nodes. It
should be noted that not all combinations of particle radius and viscosity can be used.
Table 2–2 indicates that particle radii less than 3∆x cannot be used with a kinematic
viscosity ν∗ = 1/6, since the hydrodynamic radius is then less than the input radius.
37
2.2.4 Particle Motion
An explicit update of the particle velocity
U(t + ∆t) = U(t) +∆t
mF(t) (2.83)
Ω(t + ∆t) = Ω(t) +∆t
IT(t) (2.84)
has been found to be unstable (34) unless the particle radius is large or the particle
mass density is much higher than the surrounding fluid. In previous work (34) the
instability was reduced, but not eliminated, by averaging the forces and torques over
two successive time steps. Subsequently, an implicit update of the particle velocity was
proposed (40) as a means of ensuring stability. Here we present a generalized version of
that idea, which can be adapted to situations where two particles are in near contact.
The particle force and torque can be separated into a component that depends
on the incoming velocity distribution and a component that depends, via ub, on the
particle velocity and angular velocity (Eqs. 2.81 and 2.82);
F = F0 − ζFU ·U− ζFΩ ·Ω (2.85)
T = T0 − ζTU ·U− ζTΩ ·Ω. (2.86)
The velocity independent forces and torques are determined at the half-time step
F0(t +1
2∆t) =
∆x3
∆t
∑
b
2n∗b(r, t)cb (2.87)
T0(t +1
2∆t) =
∆x3
∆t
∑
b
2n∗b(r, t)(rb × cb), (2.88)
where the sum is over all the boundary nodes, b, describing the particle surface, with cb
representing the velocity associated with the boundary node b and pointing towards the
38
particle center. The matrices
ζFU =2ρ0∆x3
c2s∆t
∑
b
acbcbcb (2.89)
ζFΩ =2ρ0∆x3
c2s∆t
∑
b
acbcb(rb × cb) (2.90)
ζTU =2ρ0∆x3
c2s∆t
∑
b
acb(rb × cb)cb (2.91)
ζTΩ =2ρ0∆x3
c2s∆t
∑
b
acb(rb × cb)(rb × cb) (2.92)
are high-frequency friction coefficients, and describe the instantaneous force on a
particle in response to a sudden change in velocity.
The magnitude of the friction coefficients can be readily estimated, thereby
establishing bounds on the stability of an explicit update. Apart from irregularities in
the discretized surface, ζFU and ζTΩ are diagonal matrices, while ζFΩ = ζTU = 0. For
a node adjacent to a planar wall∑
i acic2
i = 518
c2 where the sum is over the 5 directions
that cross the wall. The number of such nodes is approximately 4πa2/∆x2, so that
ζFU ∼ 20π
9
ρ0∆xa2
∆t. (2.93)
Similarly,
ζTΩ ∼ 8π
9
ρ0∆xa4
∆t. (2.94)
These estimates of the translational and rotational friction coefficients are within 20%
and 50% of numerically computed values, respectively. The stability criterion for an
explicit update ζFU∆t/m < 2 then reduces to a simple condition involving the particle
radius and mass density:
5
3
ρf∆x
ρsa< 2. (2.95)
The corresponding condition for the torque leads to the same stability criterion
ζTΩ∆t
I∼ 5
3
ρf∆x
ρsa< 2, (2.96)
whereas with interior fluid the numerical factors were 6 and 10 respectively (34),
showing that interior fluid destabilizes an explicit update.
39
The friction coefficients in Eqs. 2.85 and 2.86 are essentially constant, fluctuating
slowly in time as the particle moves on the underlying grid; thus the particle velocities
can be updated assuming these friction matrices are constant. The equations of motion
can then be written in matrix form as
U(t + ∆t)
Ω(t + ∆t)
=
U(t)
Ω(t)
+
m∆t
1 + αζFU αζFΩ
αζTU I∆t
1 + αζTΩ
−1
·
F0 − ζFUU(t)− ζFΩΩ(t)
T0 − ζTUU(t)− ζTΩΩ(t)
(2.97)
where α is a parameter that controls the degree of implicitness. An explicit update (34)
corresponds to α = 0, an implicit update (40) corresponds to α = 1, and a second order
semi-implicit update corresponds to α = 12. The explicit, implicit, and semi-implicit
updates evaluate the velocity-dependent force at t, t + ∆t, and t + 12∆t respectively. In
practice we find only small differences between semi-implicit and fully implicit methods
and we use the fully implicit method (α = 1) in this work. The boundary node map
is updated infrequently (every 10-100 time steps) and the 6 × 6 matrix inversion need
only be done when the map is updated. We note that in the limit of ζFU∆t/m >> 1
only the fully implicit (α = 1) version of Eq. 2.97 reduces to the steady state force and
torque balance, F0− ζFUU(t+∆t) = T0− ζTΩΩ(t+∆t) = 0. The semi-implicit method
(α = 12) produces an oscillating solution and the explicit method (α = 0) a diverging
solution.
An implicit update of the particle velocities requires two passes through the
boundary nodes. On the first pass the population densities are used to calculate F0 and
T0. The implicit equations (Eq. 2.97) are then solved for U(t + ∆t) and Ω(t + ∆t)
for the given implicit parameter α. These velocities are then used to calculate the new
population densities in a second sweep through the boundary nodes. The computational
overhead incurred by the boundary node updates is typically less than 100%, even at
high concentrations.
CHAPTER 3INCORPORATING LUBRICATION INTO THE
LATTICE-BOLTZMANN METHOD
3.1 Introduction
Lattice-Boltzmann simulations (19, 34) are being increasingly used to calculate
hydrodynamic interactions (31, 41, 42, 43, 44, 45, 46), by direct numerical simulation
of the fluid motion generated by the moving interfaces. However, as two particles
approach each other the calculation of the hydrodynamic force breaks down in the gap
between the particles, typically when the minimum distance between the two surfaces
is of the order of the grid spacing. For example, it is impractical to use sufficient mesh
resolution to resolve the flow in the small gaps that can result from an imposed shear
flow. At high shear rates the rheology of a suspension of spheres is qualitatively affected
by lubrication forces between particles separated by gaps less than 0.01a, where a is the
particle radius (47, 48). A simulation at this resolution (≈ 107 grid points per particle)
is unfeasible for more than a few particles. The number of grid points can be reduced
by using body fitted coordinates (49) or adaptive meshes (23, 25), but the small zone
size in the gap reduces the time step that can be used to integrate the flow field (50).
The problem is exacerbated by the uniform grid used in lattice-Boltzmann simulations,
but it should be noted that similar techniques, using particles embedded in regular
grids, are becoming increasingly popular in computational fluid dynamics (51, 52).
Some aspects of this work may therefore be applicable to such methods as well.
Simulations of hydrodynamically interacting particles typically use multipole
expansions of the Stokes equations (53, 54). Again the calculations break down
when the particles are close to contact, unless an impractical number of multipole
moments are used (55). A solution to this problem is to calculate the lubrication
forces between pairs of particles for a range of small interparticle gaps, and then patch
these solutions on to the friction coefficients calculated from the multipole expansion.
The method exploits the fact that lubrication forces can be added pair-by-pair, and
40
41
has been shown to work well in practice (53). A simplified version of this approach
has already been adopted to include normal lubrication forces in lattice-Boltzmann
simulations (56). Here we extend our previous work to include all components of the
lubrication force and torque, and test the numerical scheme for the interactions between
a spherical particle and a planar wall. We find that the hydrodynamic interactions
can be well represented over the entire range of separations by patching only the most
singular components of the lubrication force onto the force calculated from the lattice-
Boltzmann model. This is simpler than the Stokesian dynamics approach, where the
patch is calculated at every separation.
In this chapter, we propose a comprehensive solution to the technical difficulties
that arise when particles are close to contact, in particular the loss of mass conserva-
tion. The bulk of our numerical results are a series of detailed tests of the hydrody-
namic interactions between two surfaces in near contact. We demonstrate that after
including corrections for the lubrication forces we obtain accurate results over a wide
range of fluid viscosities. Finally, we describe an efficient implicit algorithm for updat-
ing the particle velocities even in the presence of stiff lubrication forces. An explicit
solution of these differential equations requires either that the particles are much denser
than the surrounding fluid (34), or that very small time steps are used to update the
particle velocities. On the other hand, if the particle velocities are updated implicitly,
the resulting system of linear equations severely limits the number of particles that can
be simulated. We describe what we call a “cluster-implicit” method, whereby groups
of closely interacting particles are grouped in individual clusters and interactions be-
tween particles in the same cluster are updated implicitly, while all other lubrication
forces are updated explicitly. In fluidized suspensions clusters typically contain less
than 10 particles, even at high concentrations, so that the implicit updates are a small
overhead. Our simulations efficiently cope with clusters of several hundred particles by
using conjugate-gradient methods, and only slow down if the number of particles in the
cluster exceeds 103.
42
3.1.1 Surfaces Near Contact
When two particle surfaces come within 1 grid spacing, fluid nodes are excluded
from regions between the solid surfaces (Fig 2–2b), leading to a loss of mass conserva-
tion. This happens because boundary updates at each link produces a mass transfer
(2acbρ0ubcb/c2s)∆x3 across the solid-fluid interface, which is necessary to accommodate
the discrete motion of the particle surface (see Section 2.2.2). The total mass transfer
in or out of an isolated particle,
∆M = −2∆x3ρ0
c2s
[U ·
∑
b
acbcb + Ω ·∑
b
acbrb × cb
]= 0, (3.1)
regardless of the particle’s size or shape.
Although the sums∑
b acbcb and∑
b acbrb × cb are zero for any closed surface,
when two particles are close to contact some of the boundary nodes are missing and
the surfaces are no longer closed. In this case ∆M 6= 0 and mass conservation is no
longer ensured. Two particles that remain in close proximity never reach a steady state,
no matter how slowly they move, since fluid is constantly being added or removed,
depending on the particle positions and velocities. If the two particles move as a rigid
body mass conservation is restored, but in general this is not the case.
The accumulation or loss of mass occurs slowly, and in many dynamical simula-
tions it fluctuates with changing particle configuration but shows no long-term drift.
However, we enforce mass conservation, particle-by-particle, by redistributing the excess
mass among the boundary nodes (c.f . Eq. 2.80)
ni′(r, t + ∆t) = n∗i (r, t)−2acbρ0ub · cb
c2s
− acbρ0∆M
A(3.2)
where A = ∆x3 ∑b
acbρ0.
The force and torque arising from this redistribution of mass is small, but not
exactly zero;
∆F =∆x3ρ0
∆t
[−∆M
A
∑
b
acbcb
](3.3)
∆T =∆x3ρ0
∆t
[−∆M
A
∑
b
acbrb × cb
]. (3.4)
43
They can be compactly included by a redefinition of the friction coefficients, Eqs. 2.89-
2.92, replacing cb and rb × cb by their deviation from the mean,
cb =
∑b
acbcb
∑b
acband rb × cb =
∑b
acbrb × cb
∑b
acb, (3.5)
so that cb → cb − cb and rb × cb → rb × cb − rb × cb. Then the force and torque are
correctly calculated from Eqs. 2.85 and 2.86, even when mass is redistributed.
3.1.2 Lubrication Forces
When two particles are in near contact, the fluid flow in the gap cannot be
resolved. For particle sizes that are typically used in multiparticle simulations (a <
5∆x), the lubrication breakdown in the calculation of the hydrodynamic interaction
occurs at gaps less than 0.1a. However, in some flows, notably the shearing of a dense
suspension, qualitatively important physics occurs at smaller separations, typically
down to 0.01a. Here we describe a method to implement lubrication corrections into a
lattice-Boltzmann simulation.
For particles close to contact, the lubrication force, torque, and stresslet can be
calculated from a sum of pairwise-additive contributions (53), and if we consider only
singular terms, they can be calculated from the particle velocities alone (57). In the
grand-resistance-matrix (58) formulation
F1
T1
T2
S1
S2
= −
A11 −B11 B22
B11 C11 C12
−B22 C12 C22
G11 H11 H12
−G22 H21 H22
U12
Ω1
Ω2
, (3.6)
where F2 = −F1, U12 = U1 −U2, and the friction matrices are as given in Ref. (58).
We have made full use of the symmetries inherent in the friction matrices, but without
assuming that the particle radii are the same. Most importantly, the external flow field
does not enter into the lubrication calculation, which removes the need for estimates of
the local flow field.
44
We have noted in previous lattice-Boltzmann simulations (14, 56) that the calcu-
lated forces follow the Stokes flow results down to a fixed separation, around 0.5∆x,
and remain roughly constant thereafter. The solid symbols in Fig. 3–1, for example,
show this behavior for the normal force between a spherical particle and a plane wall.
This suggests a lubrication correction of the form of a difference between the lubrication
force at h and the force at some cut off distance hN ; i.e.
Flub = −6πηa21a2
2
(a1+a2)2
(1h− 1
hN
)U12 · R12 h < hN
= 0 h > hN ,(3.7)
where U12 = U1 −U2, h = |R12| − a1 − a2 is the gap between the two surfaces, and the
unit vector R12 = R12/|R12|.The friction coefficients in Eq. 3.6 are all products of a scalar function of the gap
between the particles, either 1/h or ln(1/h), multiplied by a polynomial of the unit
vector connecting the particle centers (58). For two spheres of arbitrary size, there are
a total of 15 independent scalar coefficients, which fall into 3 classes. Again using the
notation of Ref. (58) these are XA11, XG
11, XG22 (normal force); YA
11, YB11, YB
22, YG11, YG
22
(tangential force); and YC11, YC
12, YC22, YH
11, YH12, YH
21, YH22 (rotation). We implement our
lubrication correction by calculating a modified form of each scalar coefficient as in
Eq. 3.7; for example
XA11(h) = XA
11(h)− XA11(hN) h < hN
XA11(h) = 0 h > hN
(3.8)
which vanishes at the cut-off distance h = hN . We allow for different cut off distances
for each of the 3 types of lubrication interaction.
3.1.3 Particle Wall Lubrication
The hydrodynamic interactions between two moving surfaces have been calculated
for the simplest geometry, namely a spherical particle adjacent to a planar wall. We
used 3 different particle sizes, with input radii a/∆x = 2.7, 4.5, and 8.2, chosen to
minimize volume fluctuations (see Section 2.2.2) with the exception of the results
for a/∆x = 4.5, which were generated before the optimum radius (4.8∆x) was
determined. The hydrodynamic radii, ahy(a, ν∗), which are used to determine the
45
1
10
100
1000
0.001 0.01 0.1 1
F/F
0
a = 2.7
1
10
100
1000
0.001 0.01 0.1 1
F/F
0
a = 4.5
1
10
100
1000
0.001 0.01 0.1 1
F/F
0
h/ahy
a = 8.2
Figure 3–1: Normal force on a particle of input radius a settling onto a horizontalplanar surface at zero Reynolds number, with F0=6πηahyU . Solid symbols indicate:ν∗ = 1/6 (circles), ν∗ = 1/100 (triangles), and ν∗ = 1/1200 (squares). Results includingthe lubrication correction are shown by the open symbols.
positions of the lubricating surfaces, were taken from Table 2–2. The location of the
planar wall was shifted by ∆(ν∗), corresponding to the a → ∞ limit in Table 2–2
(Fig. 2–5). In this way we ensure that the lubricating surfaces are in the same position
as the hydrodynamic boundaries in the lattice-Boltzmann simulations. The unit cell
is periodic in four directions with a top and bottom wall, and cell length of 10 times
the particle radius, which is sufficiently large that the effects of periodic images were
negligible. The simulation determines the steady state force and torque on the particle
for a given velocity and angular velocity, which was then used to calculate the friction
coefficient as a function of the gap h from the wall. Simulation results for the frictional
forces and torques are shown in Figs. 3–1 - 3–4 for three different fluid viscosities
ν∗ = 1/6, 1/100, and 1/1200.
The normal force shows the trend discussed in section 3.1.2 for each particle size
and fluid viscosity (Fig. 3–1). The simulated force (solid symbols) follows the exact
result (solid line) down to an interparticle gap, hN < ∆x, that is independent of particle
46
1
2
3
4
5
0.001 0.01 0.1 1
F/F
0
a = 2.7
1
2
3
4
5
0.001 0.01 0.1 1
F/F
0
a = 4.5
1
2
3
4
5
0.001 0.01 0.1 1
F/F
0
h/ahy
a = 8.2
Figure 3–2: Tangential force on a particle settling next to a vertical planar surface atzero Reynolds number. Simulation data for drag force divided by F0=6πηahyU , is indi-cated by solid symbols. Results including the lubrication correction are shown by theopen symbols.
size. For larger particles the lattice-Boltzmann method captures progressively more
of the lubrication force, but even for a = 8.2∆x there are noticeable deviations for
h/ahy < 0.01. The simulation reproduces more of the lubrication force at smaller
viscosities because the shift in the hydrodynamic radius delays the contact of the par-
ticle surfaces. The data obtained for a particle radius of 8.2∆x was used to determine
the lubrication cutoff hN(ν∗) for each viscosity, and the numerical values are recorded
in Table 3–1. These lubrication corrections bring the simulated normal force into
agreement with theory for all the particle sizes, interparticle gaps, and fluid viscosities
studied (open symbols in Fig. 3–1). The corresponding result for the force and torque
Table 3–1: Lubrication ranges for various kinematic viscosities, determined for a sphereof radius a = 8.2∆x. The ranges are determined separately for the normal, hN , tangen-tial, hT , and rotational, hR, motions.
hN/∆x hT /∆x hR/∆x
ν∗ = 1/6 0.67 0.50 0.43ν∗ = 1/100 0.24 0.50 0.15ν∗ = 1/1200 0.10 0.50 0.00
47
00.10.20.30.40.50.6
0.001 0.01 0.1 1
T/T
0
a = 2.7
00.10.20.30.40.50.6
0.001 0.01 0.1 1
T/T
0
a = 4.5
00.10.20.30.40.50.6
0.001 0.01 0.1 1
T/T
0
h/ahy
a = 8.2
Figure 3–3: Torque on a particle settling next to a vertical planar surface at zeroReynolds number. Simulation data for torque divided by T0=8πηa2
hyU , is indicatedby solid symbols. Results including the lubrication correction are shown by the opensymbols.
on a sphere sliding along the wall is shown in Figs. 3–2 & 3–3. Again we see that the
lubrication correction gives consistently accurate forces and torques, though not quite
as accurate as the normal force. The lubrication ranges for tangential motion were
found to be independent of the fluid viscosity, as shown in Table 3–1. We also noticed
that the reciprocal relations are obeyed; the force on a rotating sphere is similar to the
data in Fig. 3–3. The calculated torque on a rotating sphere (Fig. 3–4) is in agreement
with theory for the higher viscosities ν∗ = 1/6 and ν∗ = 1/100, but not at the lowest
viscosity ν∗ = 1/1200. Here the lattice-Boltzmann method over predicts the torque on
a rotating sphere by 20-30%. We think that the error is caused by the large difference
between the hydrodynamic and input radii (∆ = 0.55∆x) and it implies that viscosities
ν∗ less than 0.01 are not suitable for suspension simulations, at least with bounce-back
boundary conditions. In practice this is not a serious limitation: a viscosity ν∗ = 0.01
48
1
2
3
4
0.001 0.01 0.1 1
T/T
0
a = 2.7
1
2
3
4
0.001 0.01 0.1 1
T/T
0
a = 4.5
1
2
3
4
0.001 0.01 0.1 1
T/T
0
h/ahy
a = 8.2
Figure 3–4: Torque on a particle rotating next to a vertical planar surface at zeroReynolds number. Simulation data for torque divided by T0=8πηa3
hyΩ, is indicatedby solid symbols. Results including the lubrication correction are shown by the opensymbols.
allows simulations with a Reynolds number up to 10 per grid point (with a Mach num-
ber ∼ 0.1), which is at or beyond the limit of resolution of the flow. In other words,
there is little practical value in viscosities less than 0.01.
Finally, we examined the dynamic motion of a particle (a = 4.8∆x) settling
onto a solid wall (Fig. 3–5). The lubrication force was calculated using the ranges
given in Table 3–1. The particle was given a finite mass and placed with an initial gap
of h = 0.2ahy between the particle and wall. The simulations were performed at a
Reynolds number Re ∼ 0.02 by applying a constant force to the particle. The results
show the expected exponential decay of the gap between the particle and wall over
time, in quantitative agreement with lubrication theory (59) (shown by the solid line).
3.1.4 Cluster Implicit Method
The lubrication forces complicate the update of the particle velocity because they
involve interactions between many particles, especially at higher concentrations. For
simplicity we update the particle velocities in two steps; first the lattice-Boltzmann
49
10-4
10-3
10-2
10-1
10 0
0.1 1 10 100 1000h/
a hy
ν∗= 1/6
10-4
10-3
10-2
10-1
10 0
0.1 1 10 100 1000
h/a hy
νt/a2
hy
ν∗= 1/100
Figure 3–5: Settling of a sphere (a=4.8) onto a horizontal wall. The gap between theparticle surface and wall, h, relative to the hydrodynamic radius, is plotted as a func-tion of the non-dimensional time (open circles).
forces and torques (Eq. 2.97), followed by the lubrication forces. The overall procedure
is still first order accurate, but the lubrication forces can cause instabilities whenever
the particles are in near contact. The instability is driven by the normal forces, and the
stability criteria
ξ∆t
m=
6πηa2∆t43πρsa3h
=9
2
η∆t
ρsah< 2 (3.9)
is violated when h is less than ∼ 0.1∆x.
It is impractical to solve all the equations implicitly, so we implemented an
algorithm which uses an implicit update only where necessary. In our simulations
we used the conservative criteria ξ∆t/m < 0.1. Schematically, we solve the coupled
differential equations
x = −A · x + b (3.10)
by splitting the dissipative matrix A into regular and singular components, A =
AR + AS. In our context AS only contains components of the normal friction coefficient
50
1
16
16
1 9
9
1 99
9
c
9
1
16
16
1 10
32
b
9
99
9
11
1
17
16
27 10
33
3235
13
a
9
11
Figure 3–6: Illustration of the algorithm to determine the list of clusters.
when the gap between particles is less than the stability cut off, hs, determined from
Eq. 3.9. Thus AR contains all the non-zero components of the lubrication correction
but with the interparticle separation in the normal force regularized by hs so that
the larger of hij and hs is used to calculate the force. The remaining normal force is
included in AS, with the form of Eq. 3.7, but with hN replaced by hs. Using a mixed
explicit-implicit differencing,
x(t + ∆t)− x(t)
∆t= −AR · x(t)−AS · x(t + ∆t) + b, (3.11)
we obtain the first order update
(1 + AS∆t) · x(t + ∆t) = x(t)−AR∆t · x(t) + b∆t. (3.12)
The important point is that, by a suitable relabeling of the particle indices, AS can
be cast into a block diagonal form, with the potential for an enormous reduction
in the computation time for the matrix inversion. The relabeling is achieved by a
51
1
10
100
1000
0 0.02 0.04 0.06 0.08 0.1 0.12M
axim
um C
lust
er S
ize
hs / a
φ = 0.50φ = 0.25φ = 0.10
Figure 3–7: The maximum cluster size as a function of the cluster cutoff gap hs/a atvarying volume fractions, φ.
cluster analysis. First, all pairs of particles that are closer than the stability cut off are
identified, and a list is made of all such pairs. An illustration is shown in Fig. 3–6a,
where pairs of particles with separations less than hs are indicated by the solid lines.
The cluster labels are initialized to the particle index; each particle is then relabeled by
giving it the smallest label of all the particles it is connected to. After 1 pass the labels
are as shown in Fig. 3–6b and after 2 passes 3 distinct clusters have been identified,
each with a unique label (Fig. 3–6c). The algorithm stops when no further relabeling
takes place. Although more efficient schemes are possible, this simple scheme is more
than adequate for our purposes. Once the clusters have been identified, the implicit
equations can be solved for each cluster. We use conjugate gradients to exploit the
sparseness of AS, which is extensive even within each diagonal block.
1
10
100
1000
0 0.02 0.04 0.06 0.08 0.1 0.12
Num
ber
of C
lust
ers
hs / a
φ = 0.50φ = 0.25φ = 0.10
Figure 3–8: Number of clusters as a function of the cluster cutoff gap hs/a at varyingvolume fractions, φ.
52
The computational cost of the cluster-implicit algorithm depends primarily on
the maximum cluster size, which is shown in Fig. 3–7 as a function of the cluster
cutoff gap hs. A random distribution of 1000 particles was sampled in a periodic box
at volume fractions 0.10, 0.25, and 0.50. At low and moderate volume fractions the
cluster size is only weakly dependent on hs/a, ranging from 2-7, and clusters of this size
impose a negligible computational overhead. However, at high volume fractions there
is a percolation threshold at hs/a ∼ 0.02 beyond which a single cluster more or less
spans the whole volume. In this case the cluster will grow to encompass almost all the
particles in the system. Thus at high densities computational efficiency requires that
hs/a < 0.02. When combined with the stability criteria, which implies hs/a ≈ 1/a2,
we find a minimum radius of a = 10∆x to keep ξ∆t/m < 0.5. A simulation of several
hundred such particles is possible on a personal computer or desktop workstation.
In Fig. 3–8 we show the corresponding number of clusters. In general there is a
steep rise in the number of clusters with increasing hs/a, leveling off to around 100
clusters. The sharp drop in the number of clusters at the highest volume fraction is
associated with the percolation transition, as seen in Fig. 3–7.
3.2 Conclusions
In this work we have described and tested several extensions to the lattice-
Boltzmann method for particle suspensions, which enable reasonably accurate force
calculations to be made even for particles in near contact. In particular we have
shown how to deal with problems of mass conservation when two particles are in near
contact, and how to account for the lubrication forces between closely-spaced particles.
Numerical tests show that the forces and torques between a particle and a plane wall
can be computed to within a few percent of the exact result for Stokes flow. We note
that the torque on a rotating sphere adjacent to a plane wall is seriously in error (30%)
when the fluid viscosity is very small (1/1200). This suggests that the calibration
procedure may break down when the hydrodynamic boundary is displaced by more
than ∆x/2 from the physical one.
Inclusion of the lubrication forces leads to large forces and stiff differential equa-
tions for the particle velocities. We have developed a mixed explicit-implicit velocity
53
update, which minimizes the number of linear equations that must be solved, while
maintaining absolute stability.
CHAPTER 4SEDIMENTATION OF HARD-SPHERE SUSPENSIONS
AT LOW REYNOLDS NUMBER
4.1 Introduction
Particles larger than a few microns tend to settle out of suspension, because grav-
itational forces then dominate over the diffusive flux arising from gradients in particle
concentration. The detailed dynamics of the most idealized flow, the sedimentation of
hard spheres in the absence of inertia and Brownian motion is still controversial. When
a suspension settles, each particle experiences a different shielding of the fluid drag,
due to the fluctuating arrangements of its neighbors. These hydrodynamic interactions
are long range, decaying asymptotically as 1/R, and drive large fluctuations in particle
velocity, which, for particles more than about 10µm in diameter, completely dominate
the thermal Brownian motion. A relatively simple calculation shows that, if the par-
ticle positions are independently and uniformly distributed, the velocity fluctuations
are proportional to the linear dimension of the container (1). However, two different
sets of experiments found that the velocity fluctuations converge to a fixed value for
sufficiently large systems (60, 61). The qualitative discrepancy between theory and
experiment has generated considerable attention, focusing on possible mechanisms
for screening the hydrodynamic interactions (62) and thereby saturating the velocity
fluctuations when the system size is larger than the hydrodynamic screening length.
The key theoretical idea (62) is that hydrodynamic interactions can be screened by
changes in suspension microstructure, analogous to the screening of electrostatic inter-
actions in charged systems. Hydrodynamic screening occurs when a test particle and
its neighbors are, collectively, neutrally buoyant with respect to the bulk suspension; in
other words, when density fluctuations at length scales larger than the screening length
are suppressed. This requires that a rather delicate long-range correlation develop in
the distribution of particle pairs, which is resistant to the randomizing effects of hydro-
dynamic dispersion. Several different bulk mechanisms for the microstructural changes
54
55
have been proposed, including three-body hydrodynamic interactions (62), a convective
instability (63), and a coupled convection-diffusion model (2, 64). However, these the-
ories are all inconsistent with the results of numerical simulations (42, 56, 65), which
have shown that, in homogeneous suspensions (with periodic boundary conditions),
particles are distributed randomly at separations beyond a few particle diameters and
the velocity fluctuations remain proportional to container size.
More recently, the idea that the container boundaries play a critical role in
determining the amplitude of the velocity fluctuations has been explored in de-
tail (46, 66, 67). The primary goal has been to discover if container walls can quali-
tatively change the hydrodynamic interactions in the bulk suspension. Brenner (66)
analyzed the effects of vertical container walls on the particle velocity fluctuations, but
found that it does not eliminate the dependence on system size. However, numerical
simulations (46) subsequently showed that rigid boundaries at the top and bottom of
the vessel cause a strong time-dependent damping of the velocity fluctuations even in
bulk regions far from the walls. These boundaries introduce interfaces between sedi-
ment, suspension, and supernatent fluid, which are absent in homogenous systems with
periodic boundary conditions. The time-dependent damping of the velocity fluctuations
was explained by convective draining of large-scale fluctuations in particle density to
these interfaces (46), a suggestion made earlier by Hinch (68). A different picture of the
effect of container walls was proposed independently, and more or less simultaneously
by Tee (67), where it was suggested that hydrodynamic dispersion at the suspension-
supernatent interface leads to a stratification in the particle concentration. A stratified
suspension introduces another length scale, beyond which the hydrodynamic interac-
tions are screened (69). However, a model based on convection of density fluctuations
leads to qualitatively different conclusions from a stratified suspension, and we will
compare these ideas to the results of our simulations in Secs. 4.3.1 and 5.1.1.
Recent experimental results have cast doubt on the conclusion that the velocity
fluctuations are necessarily independent of container size (67, 70). Brenner (70) found
that the velocity fluctuations in very dilute suspensions (volume fraction φ < 1%) did
not saturate, even for container dimensions as large as 800a, where a is the particle
56
radius. Tee et al. (67) found that for large cells and dilute suspensions they could not
even obtain a steady state. Instead the velocity fluctuations decayed for the duration
of the experiment, no matter what the height of the container; these observations
could be explained by an increasing stratification of the suspension with time. On the
other hand some experiments found that the velocity fluctuations were steady and
independent of system size, even in very large vessels (71). To make further progress,
it will be necessary to reconcile the apparently conflicting experimental results, and
to develop the correct physical picture underlying the screening of the hydrodynamic
interactions in settling suspensions.
The focus of the present work is on the microstructure of a settling suspension, as
characterized by the distribution of pairs of particles in the bulk suspension. At low
Reynolds number the instantaneous particle velocities are completely determined by the
particle positions, and it can be shown that the dominant contribution to the velocity
fluctuations can be calculated from the structure factor
S(k) = N−1
N∑ij=1
exp(−ik · rij), (4.1)
which is the Fourier transform of the pair correlation function (64, 65). The effects
of hydrodynamic screening show up in the long-wavelength (small k) behavior of the
S(k); an asymptotic k2 dependence of the structure factor indicates screening and an
eventual saturation of the velocity fluctuations with increasing container size. It has
not proven possible to measure S(k) directly by light scattering, due to the large size of
the particles, but direct imaging has been used to calculate related spatial fluctuations
in particle concentration (72). In our simulations we have been able to determine the
structure factor directly. Here we present a detailed account of our results for both the
velocity fluctuations and the microstructure, including effects of polydispersity, which
may be important in interpreting experimental results.
4.1.1 Sedimentation Simulations
Simulations of sedimentation are computationally demanding because of the large
number of particles necessary to capture the length scales involved in the development
of hydrodynamic screening. The simulation parameters are therefore a carefully chosen
57
compromise between several competing factors, limited by the requirement that a
calculation complete in a reasonable amount of time, of the order of one month. Experi-
ments (61) suggests that the characteristic length scale is the mean interparticle spacing
l = aφ−1/3 and that the screening length is of the order of 10l. The computational time
required for our simulations is, to a first approximation, proportional to the number of
fluid grid points, and therefore the total volume is an important limitation. We have
limited our calculations to a single volume fraction φ = 0.13 (l ≈ 2a), which was
chosen as a reasonable compromise between keeping l small and avoiding the additional
complications of a highly concentrated suspension. We used a cell with a square cross
section, and studied a range of widths from W = 8l to W = 24l. In most instances
the cells were bounded at the top and bottom by rigid impermeable walls and we have
found from experience that a cell height H = 1000a is necessary to allow time for the
suspension to reach a steady state. We have varied the cell height in some instances, as
reported in Sec. 4.2.2. At the chosen volume fraction the suspensions contains between
8000 and 72000 solid particles.
A key component of this work is to assess the effects of the macroscopic boundary
conditions on the suspension microstructure and dynamics. A no-slip boundary breaks
the macroscopic translational invariance in the direction normal to the boundary
surface, and because of the long-range character of hydrodynamic interactions, this
symmetry breaking can have a significant effect far from the boundary. We have
therefore compared results for three different sets of boundary conditions: systems
bounded in all directions by rigid walls, which we denote as “Box”, systems bounded
by walls at the top and bottom, while the vertical faces are periodic (“Bounded”),
and systems with periodic boundary conditions on all faces (“PBC”). For systems
with periodic boundaries on all faces, the height to width ratio was set to 4:1, as
previous work showed no change occurred with taller cells (56). In the simulations
with no-slip boundaries at the top and bottom (“Box” and “Bounded”), data was
taken in a window, typically of height 200a, centered around 400a from the bottom
of the vessel, as is typical in experimental measurements. We have checked that the
results are insensitive to the position of the viewing window as long as it is located
58
in a region of uniform particle concentration. Fully periodic systems (“PBC”) are
spatially homogeneous, and data is averaged over the entire volume in this case. When
considering the system dimensions, it should be noted that an extra 2a has been added
to all dimensions bounded by no-slip walls. This is to allow for the excluded volume
with the walls and keep the particle concentration as close as possible to 13%. We will
identify the system dimensions without any excluded volume correction.
A particle radius of two grid spacings, a = 2∆x, was used in all these simulations,
as compared with a = 1.5∆x or a = 1.25∆x in previous work (46, 56). Test calculations
with small number of particles (see Sec. 4.2.3) suggest that there are no major changes
in the results if larger particles are used. The largest systems studied in this work
contains 72000 particles and approximately 20 million grid points. The simulations
were run for 2000 Stokes times, where the Stokes time tS = a/U0 is the time it takes an
isolated particle to settle one radius. The gravitational force was set so that the Stokes
time corresponded to 250 time steps and the complete settling simulation therefore
required half a million steps. The calculation takes about 1 month on a cluster of 8
processors. The typical particle Reynolds number, based on the mean settling velocity,
was Re = 2ρ 〈U〉 a/µ = 0.06, whereas in laboratory experiments it is usually one to
two orders of magnitude smaller. Although the residual inertia is a potential source
of complication, tests described in Sec. 4.2.3 and laboratory experiments (73) suggest
effects of inertia are negligible at Reynolds numbers Re < 1.
4.1.2 Settling of Pairs of Particles
Here we examine the settling of pairs of particles to assess the effects of grid
artifacts and the small residual fluid inertia. Two spheres settling side-by-side repel
each other when the spacing is of the order of the particle diameter (74, 75). The
rotation of the particles gives rise to a lift force at finite Reynolds number, which
tends to drive the particles apart. At very small Reynolds numbers, Re < 0.1, the
repulsion between the particles become negligible and the separation distance does not
change appreciably over time (76). We examine the settling of two particles at a range
of particle Reynolds numbers 0.0025 < Re < 0.1. Fig. 4–1 shows that the particles
maintain their orientation, and slowly drift apart with a velocity proportional to Re.
59
0 200 400 600 8001
2
3
4
5
∆r /
a
0 200 400 600 800
t/tS
0
20
40
60
80
100θ
Figure 4–1: Settling of a horizontal pair of particles at different Reynolds numbers;Re ∼ 0.1 (circles), 0.05 (squares), 0.025 (triangles).
Fig. 4–2 shows the separation distance ∆r/a between particle centers as they
settle with different initial orientations to the horizontal. Particles oriented in the
horizontal (θ = 0) and vertical (θ = 90) directions maintain their initial orientations
over time, although the particles drift away from each other (horizontal) or towards
each other (vertical), due to the residual fluid inertia. At low Reynolds numbers, a
pair of exactly spherical particles oriented at some intermediate angle should settle as
a unit, maintaining the separation and orientation of the pair. Our results show that
particles at off-axis orientations have slightly different velocities so the pair rotates with
time, even at very low Re. This is a numerical artifact caused by the discrete lattice.
It can only be reduced by using larger particles with more grid points to represent the
particle surface, or by a more complicated boundary condition that smooths out the
ragged representation of the surface shown in Fig. 2–2. Nevertheless the grid artifacts
cause much smaller variations in particle velocity than the polydispersity inherent in
laboratory experiments.
4.2 Velocity Fluctuations
In this section we examine the transient and steady-state fluctuations in particle
velocity under different macroscopic boundary conditions. This issue has already been
60
0 200 400 600 8001
2
3
4
5
∆r /
a
0 200 400 600 800
t/tS
0
20
40
60
80
100θ
Figure 4–2: Settling of a pair particles at various orientations: θ = 0 (circles), 45
(squares), 68 (triangles), 90 (diamonds). The Reynolds number Re = 0.1 in each case.
examined from several different perspectives; here we present a more complete and
up-to-date account of our own investigations.
4.2.1 Time-Dependent Velocity Fluctuations
The initial particle configuration is sampled from the most-probable (or maximum
entropy) distribution of hard spheres, which was generated by an equilibrium Monte-
Carlo simulation. At moderate concentrations this distribution deviates from a random
distribution because of the excluded volume. Thus the initial distribution does not obey
Poisson statistics, but the pair probability is nevertheless random for separations of
more than 5-10 particle radii. The particle velocities are initialized by a run of 200tS
where the particle positions are frozen. Thus the initial configuration for the dynamical
simulation (t = 0) is an idealized well-stirred distribution, without large-scale spatial
inhomogeneities.
Simulations have shown that there is a qualitative difference in the time evolution
of the velocity fluctuations depending on the macroscopic boundary conditions (46).
The velocity fluctuations shown in Fig. 4–3 illustrate the different time dependence
of “PBC” and “Box” geometries. With periodic boundary conditions the velocity
fluctuations initially increase with time, reaching a plateau after approximately 200tS.
61
0 500 1000 1500 20000
0.1
0.2
0.3
0.4
0.5
<∆U
||2 > /
U02
PeriodicBox
0 500 1000 1500 2000
t/tS
0
0.02
0.04
0.06
0.08<
∆U⊥2 >
/ U
02
Figure 4–3: Particle velocity fluctuations as a function of time with a with W = 48a.
This has previously been explained in terms of an increasing number of particle
contacts as the settling proceeds (56, 65). This is an example of a change in suspension
microstructure, i.e. how pairs of particles are distributed, even though the particle
concentration is exactly the same. However, when the container is bounded by no-
slip walls, the fluctuations decay with time, over a period of approximately 1000tS.
The initial fluctuations are comparable to the values in the homogeneous suspension
(“PBC”), but in the “Box” geometry they decay over time to a much smaller value. A
similar decay in the velocity fluctuations has been observed experimentally (70, 71),
over comparable time scales.
Simulations also demonstrate that the boundary conditions on the vertical faces of
the container do not play a qualitatively important role in determining the magnitude
of the velocity fluctuations. The data illustrated in Fig. 4–4 show that velocity fluctua-
tions decay in a similar way for both “Box” and “Bounded” geometries, and therefore
the amplitude of the velocity fluctuations is largely controlled by the boundary con-
ditions at the top and bottom of the container (46). As an additional test, we ran a
few simulations with no-slip vertical walls and periodic boundary conditions at the
top and bottom; these results were similar to the fully periodic case. Vertical no-slip
boundaries may play an important role in determining the fluctuations in containers
62
0 500 1000 1500 20000
0.2
0.4
<∆U
||
2 > /
U02
0 500 1000 1500 2000
t/tS
0
0.2
0.4<
∆U||
2 > /
U02
Box
Bounded
Figure 4–4: Particle velocity fluctuations as a function of time for ‘Box” and“Bounded” geometries at various widths: W/a = 16 (circles), 32 (squares), and 48(triangles) respectively.
with large aspect ratios in the horizontal plane (66), but our results show that there is
no qualitative effect in containers with a square cross-section.
The decay of the velocity fluctuations in the bulk of the suspension cannot be
explained by direct screening of the long-range interactions by the no-slip boundaries.
Although a no-slip boundary does screen the hydrodynamic interactions of particles
near the boundary, the screening does not extend to particles that are farther from
the wall than they are to each other (66). Thus in the measurement window, which is
typically ∼ 5W from the bottom, the hydrodynamic interactions are not significantly
affected by the container walls. Moreover, the time-dependent decay of the fluctuations
suggests that there is a rearrangement of the typical particle configurations during
the settling process. The most likely mechanism is that the container bottom and the
suspension-supernatent interface act as sinks of fluctuation energy, as suggested earlier
by Hinch (68). A horizontal density fluctuation is idealized as two regions (or blobs)
side by side, one slightly heavier than the average and the other slightly lighter. These
density fluctuations convect to one of these two interfaces and are absorbed by the
density gradient at the interface. Scaling arguments suggest that a horizontal density
63
fluctuation of length l convects with a velocity vl ≈ U0(φl/a)1/2 with respect to the
mean (68), so that fluctuations drain away on a time scale
tc(l) ≈ l/vl ≈ (l/aφ)1/2tS. (4.2)
In the absence of no-slip boundaries at the top and bottom of the container, large-scale
density fluctuations recirculate through the suspension, and the fluctuations in density
and velocity are therefore time independent.
To account for the velocity fluctuations reaching steady state, the model must
include some mechanism for replenishing the large-scale density fluctuations; otherwise
the velocity fluctuations will continually decrease. We will assume that small-scale (of
order a) density fluctuations are generated by conversion of gravitational potential
energy, and then spread out by hydrodynamic diffusion resulting from short-range
multi-particle interactions. The characteristic length of the density fluctuation will grow
to order l in a time of order
td(l) ≈ l2/D, (4.3)
where D ≈ U0a is the hydrodynamic dispersion coefficient. By balancing the convection
time tc with the diffusion time td we obtain a critical length scale, lc ≈ aφ−1/3, beyond
which fluctuations drain away more rapidly than they can be replenished by diffusion.
Thus the system can reach a steady state with a correlation length that is independent
of system width and proportional to the mean interparticle spacing aφ−1/3, as observed
experimentally (61, 71). This is the physical idea underlying the convection-diffusion
model proposed by Levine et al.(64), which will be discussed in detail in Sec. 4.3.2. An
alternative explanation, based on the development of a stratified density profile in the
suspension (77), will be discussed in Sec. 4.3.1.
Within a viewing window far from the sediment and supernatent interfaces the
velocity fluctuations reach a quasi-steady state after a time period of the order of
1000tS. The duration of the steady state, prior to the supernatent interface reaching
the viewing window, depends on the system height, as shown in Fig. 4–5. However,
the mean settling velocity and the velocity fluctuations within the viewing window are
otherwise unaffected by system height. For the height used in most of our simulations
64
0 500 1000 1500
t/tS
0
0.04
0.08
<∆U
||
2 > /
U02
0 500 1000 15000
0.2
0.4
<U
||>
/ U
0
H/a = 1000H/a = 1500H/a = 2000
Figure 4–5: Mean settling velocity,⟨U‖
⟩, and fluctuations in settling velocity,
⟨∆U2
‖⟩,
as a function of time for different system heights, H.
(H = 1000a) we typically have an upper time limit to the simulations of about 1500tS,
before the supernatent interface starts to interfere with the measurements in the
viewing window. We find that the time taken to reach steady state depends on the
container width (Fig. 4–4), ranging from 600− 700tS for the smallest system (W = 16a)
to about 1000tS for the largest system (W = 48a).
4.2.2 Steady-State Settling
A settling particle produces a disturbance to the fluid flow that decays as r−1 at
low Reynolds number, where r is the distance from the disturbance. This disturbance
produces a fluctuation in the velocity of another particle that decays as R−2, where R
is the interparticle separation. Assuming the particles are distributed uniformly, then
integration over volume leads to the well-known result:
⟨U2
⟩ ∝ L, (4.4)
where L is the container dimension (1). Precise results have been worked out in the
low-concentration limit for several different geometries (65, 77, 78). The data in Fig. 4–
4 shows that the initial velocity fluctuations follow the Caflisch and Luke (1) scaling
65
0 10 20 30 40 500
0.02
0.04
0.06
0.08
<∆U
||
2 > /
U02
0 10 20 30 40 50
x/a
0
0.01
0.02
0.03
0.04<
∆U⊥2 >
/ U
02
Figure 4–6: Particle velocity fluctuations at steady state. Profiles are shown for the“Box” system (solid symbols) at different widths: W/a = 16 (circles), 32 (squares), and48 (triangles).
(Eq. 4.4), while the steady state velocity fluctuations grow more slowly with system
size.
Table 4–1: Steady state particle velocity fluctuations in a monodisperse suspension as afunction of system width.
W/a Box Center Bounded Time (tS) Space (a)
< ∆U2‖ > /U2
0 16 0.033 0.036 0.027 Box 600-1000 150-550
32 0.048 0.052 0.053 800-1200 200-50048 0.059 0.063 0.076 1100-1500 250-450
< ∆U2⊥ > U2
0 16 0.012 0.016 0.007 Bounded 700-1100 200-65032 0.018 0.022 0.013 800-1200 250-60048 0.022 0.028 0.018 1000-1400 250-500
The steady state fluctuations across the viewing window are shown in Fig. 4–6.
The time window for averaging was determined separately for each system, depending
on the time to reach a quasi-steady state (Fig. 4–4); the duration of the time window
was 400tS in all cases. The results were also averaged over a range of vertical position,
chosen so that the density and velocity fluctuations were constant apart from statistical
errors (Fig. 5–4). The spatial and temporal windows used in Fig. 4–6 are listed in
Table 4–1. For the “Bounded” geometry the velocity fluctuations were averaged
66
over the whole horizontal plane, while for the “Box” geometry the fluctuations were
measured in a thin slice of width 2a located in the center of the container. Figure 4–6
shows that even though the velocity fluctuations grow less than linearly with width,
they do not converge to a width-independent value for the range of system sizes
studied.
0 2 4 6 8 10 12 140
1
2√
<∆U
||2 > /
<U
||>ø
1/3
Mono : BoundedMono : BoxPoly : Box
0 2 4 6 8 10 12 14
Wø1/3
/a
0
0.5
1
√
<
∆U⊥2 >
/ <
U||>
ø1/
3
Figure 4–7: Steady state particle velocity fluctuations as a function of system width forsimulation versus experimental fit (solid line).
The steady state velocity fluctuations are summarized in Table 4–1. Results for the
“Box” geometry are averaged across the horizontal plane and along the centerline of
the system (the central position in Fig. 4–6); results for the “Bounded” geometry are
shown averaged over the horizontal plane. The results for the “Box” geometry show
more tendency to saturation than the “Bounded” geometry, but it is not clear that this
is a qualitative as opposed to a quantitative difference. It will take larger system sizes
than are computationally feasible at present to resolve this issue. However, Fig. 4–7
shows that the simulation data agrees rather well with experimental measurements for
comparable container sizes. The solid lines are fits (61) to a variety of experiments (61,
79, 80). The experimental data suggests that somewhat larger system sizes are needed
for the velocity fluctuations to saturate and become independent of system size. We
note that inclusion of polydispersity improves the agreement with experiment, but we
67
will see in Sec. 5.1.3 that a different screening mechanism is operating in polydisperse
suspensions. It should be noted that in using the analytic fit to experimental data in
Fig. 4–7, we have taken the smallest container dimension as the controlling length,
whereas Segre (61) used the larger lateral dimension in fitting their experiments.
They claimed a better fit from this choice, but it cannot be justified physically or
theoretically.
4.2.3 Numerical Errors
0 500 1000 1500
t/tS
0
0.04
0.08
<∆U
||
2 > /
U02
0 500 1000 15000.2
0.3
0.4
0.5
<U
||>
/ U
0
Re = 0.015Re = 0.03Re = 0.06Re = 0.12
Figure 4–8: Effect of inertia on the mean settling velocity and fluctuations in settlingvelocity.
The simulations described in this paper are not expected to be a completely
quantitative description of Stokes-flow hydrodynamics. In particularly, we cannot
disregard the possibility that inertial effects play a role on scales larger than the
particle radius. At the volume fraction used in this work, we can expect that particle
velocity correlations persist for distances of the order of 40a (61), and the Reynolds
number based on this distance is 1.2. Nevertheless the data in Fig. 4–8 shows that, in
a system of width W = 16a (N = 8000), inertia plays no role for particle Reynolds
numbers Re < 0.1. We have verified that this conclusion holds in a larger systems
as well (W = 32a, N = 32000); no significant difference in velocity fluctuations was
observed between Re = 0.06 and 0.03. Again computational limitations prevent a study
68
0 500 1000 1500
t/tS
0
0.05
0.1
<∆U
||
2 > /
U02
0 500 1000 15000
0.2
0.4
<U
||>
/ U
0
a = 1.25a = 2.0a = 2.7
Figure 4–9: Effect of grid resolution on the mean settling velocity and fluctuations insettling velocity.
of the effects of inertia in larger systems, but experimental results (73) suggest that it is
small whenever Re < 1.
An important concern with lattice-Boltzmann simulations are artifacts introduced
by the motion of particles across the grid. It was shown in Sec. 4.1.2 that these
grid artifacts cause a weak but measurable dispersion in the trajectories of pairs of
particle, and the same random noise may serve to weaken the microstructural changes
that lead to suppression of velocity fluctuations. However, simulations with larger
particles, which have smoother trajectories (81), are not noticeable different. The
effects of grid resolution are shown in Fig. 4–9; the larger particles correspond to a
more refined mesh. There are quantitative differences in the velocity fluctuations
calculated with a = 1.25∆x (46), and the present results (a = 2∆x) over the time range
250 < t/tS < 1000. A further increase in particle size (a = 2.7∆x) does not lead to
statistically significant changes.
Finally, we have noticed that the mean settling velocity is not always independent
of vertical position in the bulk region at Re = 0.06, although at lower Re the mean
settling velocity was essentially constant. It turns out that this is an artifact of the
finite compressibility of the lattice-Boltzmann model, rather than an inertial effect.
69
200 400 600
z/a
0.2
0.25
0.3
0.35
0.4
<U
||> /
U0
Ma = 0.004Ma = 0.001
Figure 4–10: Effect of compressibility on the spatial variation of the mean settlingvelocity. The data is for Reynolds number of Re = 0.06.
Under the conditions of the simulations, with a kinematic viscosity ν = 0.167∆x2/∆t,
a Reynolds number Re ≈ 0.06 corresponds to a Mach number Ma = 0.004. By
reducing the viscosity, we can reduce the Mach number, while keeping the Reynolds
number constant. Figure 4–10 shows that at constant Reynolds number, Re ∼ 0.06, the
mean velocity is independent of position at sufficiently small Mach numbers. However,
we could not detect significant differences in the fluctuations at either smaller Re or
smaller Ma, although the non-uniform settling velocity does lead to a small reduction in
particle concentration (∼ 0.01) during the course of the simulation.
4.3 Microstructure
The dynamics of suspensions at low Reynolds numbers are controlled by the
distribution of particle positions, which is sufficient to determine the particle velocities
at any instant of time. In a settling suspension, subtle shifts in the pair correlation
function can have a dramatic effect on the macroscopic behavior. Specifically, it has
been established that the amplitude of the velocity fluctuations are largely determined
by the structure factor (64, 65), and in particular its low-k limiting behavior:
< ∆Uα∆Uβ >=χU2
0 φ
a
∫S(k)
k4
[δαβ − kαkβk2
z
k4
]dk, (4.5)
where S(k) was defined in Eq. 4.1 and χ is a numerical factor of order 1. It has
not been possible to measure the structure factor of a settling suspension of non-
Brownian particles experimentally, since the particles are too large for light scattering
70
measurements. Fluctuations in particle concentration have been measured within a
cylindrical or rectangular window (72), but this only gives an angle average of the pair
distribution. In numerical simulations it is possible to calculate S(k) as a function
of both wavelength and direction. This is important since some theories predict that
the structure factor becomes highly anisotropic at long-wavelengths (64), with the
horizontal fluctuations vanishing as k2 while the vertical fluctuations remain finite at all
wavelengths. Here we report on the structure factor in a steadily settling suspension,
and investigate the possibility of stratification of the particle concentration (77).
4.3.1 Particle Concentration
0 200 400 6000
0.1
0.2
0.3
0.4
0.5
0.6
ø
0 200 400 600
z/a
0
0.1
0.2
0.3
0.4
0.5
0.6
ø
200 300 400 5000.110
0.115
0.120
0.125
200 300 400 500
z/a
0.110
0.115
0.120
0.125
Bounded
Box
Figure 4–11: Particle volume fraction as a function of height, z. The dots indicate theinstantaneous average, and the steady-state (1000 < t/tS < 1400) density profile in theviewing window is shown in the adjacent plots.
Figure 4–11 shows that the density profile is uniform in the bulk, with a sharp
interface between the suspension and supernatent fluid. A sharp interface has also been
observed experimentally (82), but recently it has been suggested that hydrodynamic
dispersion at the suspension-supernatent interface could lead to a weak stratification of
the particles and a non-uniform density in the bulk (3, 67, 77). The time evolution of
the concentration profile, φ(z, t), can be described by a convection-diffusion equation,
which, in a Lagrangian frame moving with the mean settling velocity −U(φ), can be
71
written as
dφ
dt+ U ′∂zφ = D∂2
zφ; (4.6)
U ′ = −φdU/dφ is the velocity of a density perturbation with respect to the mean
settling speed and D is the hydrodynamic diffusion coefficient. The solution to Eq. 4.6
is qualitatively similar to the profiles shown in Fig. 4–11; in particular there is a bulk
region where the concentration is constant. However, if the diffusion coefficient is very
large, the suspension-supernatent interface spreads into the bulk leading to a weakly
stratified suspension, with a small density gradient even in the bulk (77).
Stratification has been suggested as a mechanism for suppressing velocity fluctu-
ations (69), by making it possible for fluctuations in particle concentration to reach a
neutrally buoyant position in the suspension without draining to the interfaces. How-
ever, the density profiles shown in Fig. 4–11 are not stratified; instead the concentration
profile is uniform in the bulk and the suspension-supernatent interface is sharp. The
plots of average concentration show that any mean variation in density is well below
the statistical noise, even after averaging over several hundred Stokes times. Our data
suggests that any residual density gradient must have a characteristic length of at least
104a, or 10 times the container height.
Hydrodynamic dispersion does cause a spreading of the interface, but this is com-
pensated by hindered settling, which convects the less dense regions at a higher velocity
than the high density regions and thereby sets up a convective flux in opposition to
the diffusive flux. Balancing the convective and diffusive concentration fluxes with
respect to a frame moving with the mean settling velocity, we estimate an interface
thickness, D‖/U ′ ≈ 5a, at this volume fraction. Here the vertical dispersion coefficient,
D‖ = 4⟨U‖
⟩a, was taken from our simulations and is comparable to experimental
measurements (80). Our simulations do not rule out the possible significance of in-
terfacial diffusion in very dilute suspensions (φ < 1%), such as are commonly used
in Particle-Image-Velocimetry measurements (67, 70, 71), but we do exclude it as a
general mechanism for hydrodynamic screening.
72
0 1 20
0.5
1
1.5
S(k ⊥)
W/a = 16
0 1 20
0.5
1
1.5
S(k ⊥)
W/a = 32
0 1 2
k⊥a/π
0
0.5
1
1.5
S(k ⊥)
W/a = 48
Figure 4–12: Structure factor describing horizontal density fluctuations, S(k⊥), forvarious system sizes: W = 16a, 32a, and 48a.
4.3.2 Structure Factor
The structure factors were calculated directly from particle positions within the
viewing window in simulations with the “Bounded” geometry. Periodic boundary
conditions in the horizontal plane allow for more accurate Fourier analysis and this
is reflected in the varying quality of data for fluctuations in vertical and horizontal
directions shown in Fig. 4–13. Figure 4–12 shows that, in comparison with the initial
equilibrium distribution, horizontal density fluctuations are strongly suppressed by no-
slip boundaries at the top and bottom of the container. On the other hand it is already
known that suspensions with periodic boundary conditions do not undergo significant
changes in microstructure during settling (56, 83). In particular the structure factor
remains finite at all wavelengths. The significance of this result is that it demonstrates
that the macroscopic boundaries have a profound effect on the distribution of particles
in the bulk suspension. It may also be the mechanism by which hydrodynamic interac-
tions are screened during the settling process. A physical interpretation of the data in
Fig. 4–12 is that, if the viewing window was divided into vertical slices of sufficient size,
73
there would be essentially the same number of particles in each slice, rather than the
expected variation of order√
Ns, where Ns is the number of particles in the slice.
The structure factor in a settling suspension develops a strong anisotropy, as
shown in Fig. 4–13. Although the horizontal density fluctuations are proportional
to k2⊥, the vertical fluctuations tend to a non-zero constant at small k‖. Damping of
horizontal density fluctuations (at least as fast as k2⊥) is the minimum requirement
for hydrodynamic screening, as was shown by Levine et al. (64). They proposed that
there are two qualitatively distinct non-equilibrium phases for settling suspensions, an
unscreened phase characterized by a random microstructure and a screened phase where
the horizontal density fluctuations are damped out at long wavelengths. They derived
an expression for the non-equilibrium structure-factor,
S(k) =N⊥k2
⊥ + N‖k2‖
D⊥k2⊥ + D‖k2
‖ + γk2⊥/k2
, (4.7)
which is consistent in functional form with the structure factor obtained in our numer-
ical simulations (Fig 4–13). The renormalized parameters, the fluctuations in particle
flux Ni and the diffusion coefficients Di, together with the damping coefficient, γ, were
calculated from coupled field equations describing the evolution of the particle concen-
tration and fluid velocity. According to the theory, the phase boundary is determined
by the anisotropy in the renormalized noise, N⊥/N‖, and diffusivity, D⊥/D‖.
The structure factor data can be used to extract ratios of the parameters that
appear in Eq. 4.7; namely N⊥/γ = 0.4a2, and N‖/D‖ = 0.17. We obtained N⊥/γ
and N‖/D‖ from the low-k behavior of the horizontal and vertical density fluctuations,
but since our data is rather noisy, it is impossible to extract a meaningful value of
D⊥/γ, which appears as a quartic correction to the asymptotic k2 dependence of
the structure factor. Nevertheless, for the sake of completeness we will use our best
estimate, D⊥/γ = 0.5a2, to determine the ratio N⊥/N‖ ≈ 0.7. When combined with
tracer-diffusion measurements of D⊥/D‖ = 0.16, this suggests we are near the transition
between screened and unscreened phases (64). Unfortunately our data is not sufficiently
precise to enable a definitive conclusion to be drawn. Significantly larger systems sizes
74
0 0.2 0.4
ka/π
0
0.2
0.4
S(k)
[0,0,1][1,0,1][1,0,0]
Figure 4–13: Structure factor at different angles; vertical [0,0,1] direction (circles), 45o
[1,0,1] direction (squares), and horizontal [1,0,0] direction (triangles).
will be necessary for a quantitative comparison with the predictions of the theory, with
greatly increased computational requirements.
Density fluctuations have been measured at other low-index directions; for example
Fig. 4–13 also shows data for the 45o [1,0,1] direction. Unfortunately the system is not
periodic for any k-vectors that lie outside the horizontal plane, so this data is inherently
noisier. Further complicating the analysis is the effect of renormalization, which
produces a shoulder that is best seen for the horizontal [1,0,0] direction at ka ≈ π/4.
For ka > π/4 the structure factor is similar to the equilibrium distribution (Fig. 4–
12) but the convective effects described qualitatively in Sec. 4.2.1 produce significant
damping when ka < π/4. The data for the 45o may also be showing a shoulder at a
smaller k, ka ≈ π/6, but a wider cell is necessary to confirm this. This is an important
point, because hydrodynamic screening requires that the density fluctuations are
damped in all directions except the vertical. In practice this means the shoulder is
expected to move to smaller and smaller k as the direction rotates from horizontal to
vertical.
Although the structure factors measured in the simulations are consistent with
the predictions of Levine (64), their theory postulates that all the important dynamics
occurs in the bulk, independent of the macroscopic boundary conditions. Although
this is a logical assumption, our numerical simulations have shown that it is incorrect.
Simulations with periodic boundary conditions (42, 56) do not exhibit any damping
75
of the horizontal density fluctuations, as would be expected if the model were correct
in all essentials. Instead, our simulations suggest that the container bottom and the
suspension-supernatent interface act as sinks of fluctuation energy, as suggested earlier
by Hinch (68). Random density fluctuations convect to one of these two interfaces and
are absorbed by the density gradient at the interface. The data shown in Fig. 4–14
supports this conclusion, albeit not conclusively. Here we show the structure factor in
the viewing window during its evolution from an equilibrium state to the steady state.
Despite the limited time averaging (a total of 4 × 200tS = 800tS for each plot), the
implication is that the long wavelength fluctuations decay fastest. If so, this is evidence
for the immediate convection of large-scale density fluctuations (68), rather than the
establishment of a density gradient by hydrodynamic diffusion (77).
The screening by Mucha (77) is generated by transient vertical variations in
particle concentration, rather than by a steady-state change in pair correlations. The
key difference is that the theory by Levine (64) adds strongly anisotropic concentration
fluctuations, which are an empirical representation of the supposed effects of the many-
body hydrodynamic interactions. To obtain a hydrodynamically screened phase, the
theory requires that small-scale concentration fluctuations are largest in the horizontal
plane, N0⊥ À N0
‖ (64). By contrast, in a random suspension the density fluctuations are
isotropic on all length scales. Our simulations show that there is a pronounced change
0 0.5 1 1.5 2k⊥a/π
0
0.5
1
1.5
S(k ⊥
)
Equilibriumt/t
s = 0-200
t/ts = 200-400
t/ts = 400-600
0 0.2 0.4k⊥a/π
0
0.2
0.4
0.6
S(k ⊥)
Figure 4–14: Time evolution of structure factor for horizontal density fluctuations.
76
in the anisotropy of the density fluctuations as the settling proceeds, so the character of
the noise may change as the suspension evolves to steady state.
4.3.3 Mean Settling Speed
Our results suggest that the mean settling velocity decreases during the settling
process by about 25% as shown in Fig. 4–8. This is a generic result whenever there
is a no-slip wall bounding the top and bottom of the container, and is independent of
system height (Fig. 4–5), Reynolds number (Fig. 4–8) and grid resolution (Fig. 4–9).
To our knowledge a systematic variation of settling velocity with time has not been
reported previously, but it is consistent with a time-dependent reduction in number-
density fluctuations, which has been observed experimentally by Lei et al.(72). In that
work it was shown that the number-density fluctuations in a fixed volume decreases
as the suspension settles. We have observed a similar decrease in number-density
fluctuations in our simulations, but changes in microstructure show up more clearly in
the structure factor (Sec. 4.3.2).
If the change in settling velocity is due to changes in the microstructure at long
wavelengths, then it can be estimated from the relation between the settling velocity
and structure factor (65), which is quite precise at long wavelengths:
< ∆U >=6πU0a
(2π)3
∫[S(k)− Seq(k)]
[k2
x + k2y
k4
]dk, (4.8)
The steady-state structure factor was taken from Eq. 4.7 with the parameters deter-
mined from the simulation data (Sec. 4.3.2), and the isotropic equilibrium structure
factor was fitted with a quadratic polynomial; the integration was taken up to a maxi-
mum ka = π/2, beyond which it was assumed that the structure factors were similar.
This gave a predicted decrease in the mean settling speed < ∆U >= −0.20U0, while the
simulation result was around −0.13U0.
4.4 Conclusions
The focus of this work has been to address the role of macroscopic boundary
conditions on the microstructure of a settling suspension. In monodisperse suspensions
we have found that there is a rearrangement of the pair distribution in the bulk region
of the suspension, which suppresses the long-wavelength density fluctuations, especially
77
in the horizontal plane where a clear k2 dependence was observed. Simulations with
different macroscopic boundary conditions on the container walls have demonstrated
that the boundary conditions at the top and bottom of the container play a crucial
role in determining the distribution of particle pairs. The measured structure factor is
consistent with the key qualitative features predicted theoretically by Levine et al. (64)
using a renormalized convection-diffusion model. However, we note that this theory
cannot yet explain why the macroscopic boundary conditions should play a crucial role.
Our simulations predict that the mean settling velocity decreases during the
settling process. While this has been shown to be consistent with the observed changes
in suspension microstructure there is no experimental confirmation at the present time.
We have found no evidence for stratification in monodisperse suspensions at
moderate concentrations. Our results show that the interface is sharp and that the
concentration in the bulk is very uniform. The suggestion by Mucha (77) that our
observations (46) could be explained in terms of stratification is incorrect. We con-
clude that there is a mechanism for microstructural rearrangement in the bulk of a
monodisperse suspension during settling, which leads to a substantial reduction in the
amplitude of the velocity fluctuations from the predictions for randomly distributed
particles. Based on our simulations and the experimental measurements of Bernard-
Michel (70), we believe that the question of saturation of the velocity fluctuations
in monodisperse suspensions is still open. Both our simulations and these experi-
ments show the steady-state velocity fluctuations growing less rapidly than linearly in
container size, W , but not yet fully saturating.
CHAPTER 5SEDIMENTATION OF A POLYDISPERSE SUSPENSION
5.1 Introduction
Particles used in laboratory measurements have a typical polydispersity in the
range 5 − 10% (67, 71, 80) although some experiments use significantly more monodis-
perse particles (61, 70). In a fluidized bed it is well known that particles segregate
according to size, with a denser suspension of larger particles at the bottom and less
dense suspension of lighter particles at the top. The larger particles have an inherently
higher settling velocity and therefore match the fluidization velocity at a higher con-
centration than the smaller particles. Thus a fluidized bed of polydisperse particles is
inevitably stratified in both concentration and volume fraction. We have studied the
settling of polydisperse suspensions to find out if there is significant segregation during
the settling process, and what effects that may have on the velocity fluctuations and
microstructure.
5.1.1 Stratification
Figure 5–1 shows the instantaneous density profiles in a 10% polydisperse suspen-
sion at steady state (t = 1200tS). In contrast to a monodisperse suspension (Fig. 4–11),
the suspension supernatent interface is considerably broader, as shown in Fig. 5–1a, and
there is a weak underlying stratification of the particle volume fraction. The volume
fraction profile is broken down in Fig. 5–1b into contributions from three different
size ranges. It can be seen that there is considerable segregation by this time and the
suspension-supernatent interface is dominated by the smallest particles. Small particles
tend to drift into the upper region of the front, while the large particles settle faster
and are dominant in the region nearest the dense pack; medium size particles are dis-
tributed throughout the suspension. We emphasize that the interface spreading seen
in Fig. 5–1a is a result of differential convection of different particle sizes rather than
hydrodynamic diffusion (77), which we have seen does not produce a broad interface at
78
79
0 200 400 600
z/a
0
0.1
0.2
0.3
0.4
0.5
0.6
ø
0 200 400 600
z/a
0
0.1
0.2
0.3
LargeMediumSmall
a) b)
Figure 5–1: Profiles of the particle volume fractions as a function of the system height.
this volume fraction. Even at concentrations less than 1%, convective spreading due to
polydispersity is often more important than hydrodynamic diffusion (84).
Our simulations, Fig. 5–2, show that there is a stratification of the particle mass
density (or volume fraction) in polydisperse suspensions, which persists deep into the
bulk. In comparison with the monodisperse suspension, here there is a clear decrease
of the particle mass density (or volume fraction) with height that is visible over the
statistical noise. Segregation and stratification can be explained by a one-dimensional
convection-diffusion equation analogous to Eq. 4.6:
∂tφi + ∂z(Uiφi) = D∂2zφi, (5.1)
where the particle sizes have been divided into a number of different size ranges, with
volume fraction φi and settling velocity Ui = Ui,0f(φ). The settling velocity was
constructed from the usual approximation that takes the settling velocity of the isolated
particle and a hindrance function, f(φ), based on the total volume fraction φ =∑
i φi.
In the absence of diffusion, D = 0, we find an interface spreading and segregation
that is qualitatively similar to the simulation data, confirming that polydispersity is
the dominant mechanism for stratification at this volume fraction. More complicated
explanations of stratification based on interface diffusion (77) or convection of density
fluctuations (85) have not considered the effects of polydispersity, which may partly or
even fully account for the experimental observations (see Sec. 5.1.4).
80
200 300 400 500
z/a
0.105
0.110
0.115
0.120
0.125
ø
200 300 400 500
z/a
0.105
0.110
0.115
0.120
0.125
Mono Poly
Figure 5–2: Comparison of particle volume fraction profiles for monodisperse and poly-disperse suspensions at steady-state, 1000 < t/tS < 1400.
5.1.2 Velocity fluctuations
Our simulations indicate that the concentration profile at the interface between
a polydisperse suspension and the supernatent fluid is continually evolving during
settling, so the designation of a steady-state may be more arbitrary than in the
monodisperse case. Nevertheless there are time windows of about 400tS duration
when the velocity fluctuations are essentially stationary, as shown in Fig. 5–3. The
time dependence and steady-state values of the velocity fluctuations are comparable
to the monodisperse case (Fig. 4–7), although the anisotropy between vertical and
horizontal velocity fluctuations is about 4 for the polydisperse suspension, comparable
to experiment, while it is somewhat less for monodisperse suspensions. The velocity
fluctuations in the different geometries are summarized in Table 5–1 in the same format
as in Table 4–1.
Table 5–1: Steady-state particle velocity fluctuations in a 10% polydisperse suspensionas a function of system width.
W/a Box Center Bounded Time (tS) Space (a)
< ∆U2‖ > /U2
0 16 0.053 0.061 0.053 Box 600-1000 150-400
32 0.083 0.094 0.095 800-1200 200-40048 0.104 0.115 0.144 1100-1500 250-400
< ∆U2⊥ > U2
0 16 0.011 0.015 0.009 Bounded 800-1200 200-50032 0.021 0.028 0.018 800-1200 250-50048 0.029 0.037 0.029 1000-1400 200-400
81
0 500 1000 1500 20000
0.1
0.2
0.3
0.4
<∆U
||
2 > /
U02
0 500 1000 1500 2000
t/tS
0
0.02
0.04
0.06
0.08<
∆U⊥2 >
/ U
02
Figure 5–3: Particle velocity fluctuation as a function of time for a 10% polydis-perse suspension. The data is taken for three system widths: W/a = 16 (circles), 32(squares), 48 (triangles).
The profiles of the velocity fluctuations as a function of height are qualitatively
different in monodisperse and polydisperse suspensions. Figure 5–4 shows that the
velocity fluctuations in monodisperse suspensions are independent of vertical position
in the bulk suspension, while in polydisperse suspensions they decay with increasing
height. It has been pointed out by Mucha et al. (77) that a stratified suspension
is expected to lead to a reduction of the velocity fluctuations near the suspension-
supernatent interface, because the density gradient is larger there (Fig. 5–1a) and so is
more effective in damping the velocity fluctuations. The decreasing fluctuations shown
in Fig. 5–4 is further evidence that the polydisperse suspension is stratified, while the
monodisperse suspension is not. The data in Figs. 5–2 and 5–4 suggest that the velocity
fluctuations are controlled by stratification for z > 400a. The gradient in volume
fraction at this point (z = 400a), βa = −d(log φ)/dz ≈ 2.5 × 10−4, is consistent with a
scaling argument (77) that suggests velocity fluctuations are controlled by stratification
when β > βcritical, which for our system is roughly 2× 10−4a−1.
82
0 200 400 600 800
z/a
0
0.02
0.04
0.06
0.08
0.1
<∆U
2>
/ U
02
0 200 400 600 800
z/a
0
0.02
0.04
0.06
0.08
0.1
Mono Poly
Figure 5–4: Comparison of particle velocity fluctuations for monodisperse and 10%polydisperse suspensions.
5.1.3 Structure Factor
Our simulations show that small amounts of polydispersity destroy the delicate mi-
crostructural rearrangements observed in strictly monodisperse suspensions. Horizontal
density fluctuations in 10% polydisperse suspensions are not damped at long wave-
lengths as they are for monodisperse suspensions, as shown in Fig. 5–5a. This suggests
that different mechanisms for damping the velocity fluctuations exist. In monodisperse
suspensions, the distribution of particle pairs adjusts itself during settling so that
the density fluctuations are damped as k2 at long wavelengths. This leads to a time-
dependent damping of the velocity fluctuations and the possibility of a size-independent
saturation. In polydisperse suspensions the microstructure is apparently randomized by
the varying settling speeds and this mechanism no longer holds. However in this case
the particle velocity fluctuations may be damped by stratification due to segregation of
the different particle sizes.
The structure factors for different degrees of polydispersity are compared in Fig. 5–
5b. The systems are smaller in this case and so there are fewer k-vectors. The structure
factor for 2% polydispersity is similar to the monodisperse case, apparently vanishing as
k2⊥ at long wavelengths. For higher degrees of polydispersity, 5% and 10%, the structure
factor tends to a non-zero value at low k⊥. It appears that only very small degrees of
polydispersity can be tolerated if laboratory experiments are to mimic the properties of
a monodisperse suspension.
83
0 0.2 0.4
k⊥a/π0
0.2
0.4
S(k ⊥
)
MonodispersePolydisperse
0 0.2 0.4
k⊥a/π0
0.2
0.4
σ = 2%σ = 5%σ =10%
a) b)
Figure 5–5: Comparison of structure factors for different degrees of polydispersity:a) monodisperse (circles) and 10% polydisperse (squares) suspensions (W=48a,N=72000); b) 2% (circles), 5% (squares), and 10% (triangles) polydisperse suspensions(W=32a, N=32000).
5.1.4 Stratification in Laboratory Experiments
Our investigations suggest that there may be more than one mechanism at work
in determining the velocity fluctuations in a settling suspension. In moderately dense
suspensions, interface diffusion is prevented by the stronger convective effects of
hindered settling, leading to a sharp interface and a uniform particle concentration
in the bulk. On the other hand, moderate to high concentration experiments (79, 80)
have used suspensions with significant polydispersity, in excess of 5%. In this case the
particle density becomes stratified over the duration of the experiment, Fig. 5–2, even
deep into the bulk where measurements are made. The development of long-range
correlations (Fig. 4–12) are hindered by even small amounts of polydispersity, so that
stratification may be the dominant mechanism for controlling the velocity fluctuations
in these experiments. We have solved the convection-diffusion equation (Eq. 5.1) at
13% concentration and 10% polydispersity; we find a stratification βa = 3 × 10−4 at
a point H/3 of the way up the container and at a time when the suspension front has
fallen approximately H/2. The calculated stratification is comparable to what was
found in our polydisperse simulations (H = 1000a) and is sufficiently strong to control
the velocity fluctuations (Fig. 5–4).
Many experiments (61, 67, 70, 71, 84) are done at very low particle concentra-
tions, to permit the use of Particle-Image-Velocimetry measurements. In this case
84
hindered settling is negligible and the interface spreads by both diffusion and con-
vective segregation of different particle sizes. It is not computationally feasible to
simulate suspensions at these low concentrations by the lattice-Boltzmann method
(Sec. 4.1.1), but it is interesting to consider whether stratification can be important
in these systems. Mucha (77) has considered this question in detail for monodisperse
suspensions; here we compare the contributions of segregation and dispersion, via solu-
tions of Eq. 5.1. Again we consider the stratification β at a location H/3 after a time
H/2U0 as a reference point. Hindered settling leads to the development of a traveling
concentration profile of constant shape (82), but in dilute suspensions the shape of the
profile is constantly evolving. In the absence of hindered settling, the natural length
scale is the height of the container and in these units there are only two parameters; the
degree of polydispersity, σ, and the dimensionless diffusion coefficient, D∗ = D/U0H;
the dimensionless stratification is now βH. In the absence of polydispersity (σ = 0),
large diffusion coefficients D∗ > 0.01 are needed to produce significant stratification.
In our opinion this is a weakness of the analysis of Mucha (77); in order for interface
diffusion to produce significant stratification, numerical values of the diffusion coeffi-
cient must exceed all experimental measurements. However, even modest amounts of
polydispersity can produce significant stratification. For σ = 0.1, stratification is in fact
dominated by segregation; a reduced diffusion coefficient D∗ = 0.01 is insufficient to
contribute significantly to the overall stratification. Even at 5% polydispersity, segre-
gation produces significant stratification and can be assisted by a small hydrodynamic
dispersion, D∗ = 0.001. At 2% polydispersity or less, segregation is insignificant,
and the convection-diffusion model suggests that only interface diffusion can produce
stratification.
Of the experiments listed at the beginning of the previous paragraph, only one
set (70), systematically examines the dependence on container size using nearly
monodisperse particles, with polydispersity as low as 1%. In these experiments a clear
saturation with container size was not observed, particularly at the lowest volume
fraction. It also took significantly longer for the suspension to reach a steady-state than
in other experiments. The polydispersity was about 7% in some experiments (67, 71,
85
84), which is sufficient for significant stratification by segregation. Segre et al. (61) used
nearly monodisperse particles (σ = 0.01), but primarily varied the volume fraction
rather than the container size. We conclude that the issue of the saturation of the
velocity fluctuations in a strictly monodisperse suspension is still an open question.
5.2 Conclusions
Most experiments use particles with significant polydispersity in size. In this
case our results show that the pair correlations are noticeably more random than
in the monodisperse case and for σ > 0.02, it seems likely that the driving force
for microstructural rearrangements is no longer sufficient to combat the additional
randomization from variations in particle velocity. However increasing polydispersity
leads to stratification of the particle concentration even at moderate concentrations.
Since in this case the stratification is driven by convective segregation of different
particle sizes it can spread over large distances, even in the presence of hindered
settling. We suggest that segregation-induced stratification may have a profound effect
on the interpretation of experimental measurements, even in dense suspensions.
Several recent experiments have been carried out at very low particle concentra-
tions, while our simulations are limited by computational efficiency to much larger
volume fractions, in this case φ = 0.13. Our results do not therefore discount the pos-
sibility that interface diffusion causes significant stratification at low volume fractions
as claimed by Mucha (77). Nevertheless the convection-diffusion model shows that
segregation is often more important than interface diffusion in promoting stratification.
Only in dilute suspensions of nearly monodisperse particles would we expect interface
diffusion to be important, although stratification by segregation can occur under a
broader range of circumstances.
CHAPTER 6SUMMARY
We have applied the lattice-Boltzmann method to examine particle dynamics in a
gravity driven suspension. Initially, the particles exist in a randomly distributed state
that exhibits large particle velocity fluctuations and no long range pair correlations
within the microstructure. With periodic boundaries applied, the fluctuations are found
to increase over short times and persist thereafter. However, no-slip boundaries intro-
duced at the top and bottom of the vessel lead to a decay in the velocity fluctuation
over time within the bulk, which is comparable with experimental findings. Here, the
steady-state microstructure exhibits an ordered configuration at long wavelengths,
which signifies that screening is present when no-slip boundaries are applied.
The numerical results are shown to compare well with experimental results, but
due to computational limitations, a complete analysis of the sedimentation problem
has not yet been attained. In experiments, the velocity fluctuation are found to
become independent of system size beyond a certain point (61, 80). Our results do not
unambiguously show this effect. The velocity fluctuations are found to grow less than
linearly with system width, but they do not converge to a width independent value for
the system sizes evaluated. However, quantitative results for the systems studied agree
well with a fit to experimental data (61). Because experiments were carried out with
larger containers before the screening effect became pronounced, simulations of larger
system sizes should be evaluated for a more conclusive comparison.
Luke (69) proposed a mechanism due to stratification by hydrodynamic diffusion
at the suspension-supernatent front leading to a damping of the velocity fluctuations
within the bulk. We studied the concentration profile for a monodisperse suspension at
13% volume fraction, but found a sharp front develop at the interface. Here, hindered
settling, because of particle crowding interfering with the motion of the individual
particles, has a stronger net effect than a diffusive spreading. The concentration
was found then to be uniformly distributed within the bulk. This is contrary to
86
87
simulations (77) and experiments (61, 71) which were performed at low volume
fractions, where stratification from diffusive spreading occurs more readily. As we
see it, the stratification effect does suppress velocity fluctuations, but only at low
volume fractions where hindered settling is not present.
Without stratification, the mechanism for density fluctuation decay is then
attributed to convective transport. We hypothesize that density fluctuations in the
bulk can drain away at the suspension-sediment and suspension-supernatent interfaces.
This generates a continual decay in velocity fluctuations, which is replenished by
hydrodynamic diffusion due to short range multiparticle interactions. A balance
between convective and diffusive transport of density fluctuations leads to a correlation
length scale beyond which the velocity fluctuations are screened.
Further examination of the suspension microstructure showed that an evolution
towards a statistically non-random state developed, where particles are distributed
uniformly at large length scales. We identified this result based on the damping of the
long range pair correlations in the structure factor, S(k⊥) → k2⊥ as k⊥ goes to zero.
Moreover, the pair correlations are found to be anisotropic between the vertical and
horizontal planes, such that the vertical fluctuations were finite at all wavelengths.
This correlates with a convection-diffusion model (64), verifying that the transport of
fluctuations due to convection is the dominant mechanism present leading to the decay
in particle velocity fluctuations.
Monodisperse suspensions represent an idealized situation where there is no varia-
tions in particle size. Typical experiments at moderate to high volume fractions (79, 80)
exhibit size polydispersity of 5% or greater. We find that for polydispersity in excess
of 5%, a different mechanism for the decay in velocity fluctuations is observed. The
variation in particle size leads to a segregation of different sizes as the suspension set-
tled. Smaller particles tend to settle slower and stay closer to the supernatent front,
while large particles settle faster accumulating more into the bulk. The net effect was a
pronounced stratification of particle concentration at the suspension-supernatent front
that persists into the bulk. Because of the stratification, the velocity fluctuations within
the bulk do not have to convect towards the density gradients at the front, rather the
88
gradient is strong enough inside the bulk to create a sink for the velocity fluctuations to
decay.
The structure factor in the polydisperse suspension does not exhibit the same
damping at long wavelengths seen in monodisperse systems, indicating a more ran-
domly distributed microstructure. However, velocity fluctuations are still damped, but
this is due to stratification induced by segregation of different particle sizes. We suggest
then that polydispersity may have a larger effect in experimental measurements than
previously supposed.
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BIOGRAPHICAL SKETCH
Nhan-Quyen Nguyen was born in the Binh Thuan province of Vietnam on August
10, 1977. He came to America when he was 3 years old, living in New Orleans, LA.
until he turned 22. He attended Tulane University in New Orleans, where he received
his bachelor’s degree in chemical engineering and math in May 1999. He went on to
graduate school at the University of Florida, chemical engineering department, where he
received his doctorate degree in August 2004.
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