1, 2, 3 . . .
spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Sedimentation
Élisabeth Guazzelli and Jeffrey F. Morris
with illustrations by Sylvie Pic
Adapted from Chapter 6 of A Physical Introduction to Suspension Dynamics
Cambridge Texts in Applied Mathematics
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
1 1, 2, 3 . . . spheres
2 Clusters and clouds
3 Settling of a suspension of spheres
4 Intrinsic convection
5 Velocity fluctuations and hydrodynamic diffusion
6 Fronts
7 Boycott effect
8 More on polydispersity and anisotropy
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
1 1, 2, 3 . . . spheres
2 Clusters and clouds
3 Settling of a suspension of spheres
4 Intrinsic convection
5 Velocity fluctuations and hydrodynamic diffusion
6 Fronts
7 Boycott effect
8 More on polydispersity and anisotropy
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Sedimentation of a single sphere
Stokes velocity
US = 2(ρp − ρf )a2 g/9µ
Slow-decay disturbance ∼ O( aUr S )
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Sedimentation of a pair of identical spheres
Udoublet 3a
= 1+ for θ = 0,
US 2r
Udoublet 3a π
Two identical spheres fall at the same
= 1+ for θ= velocity and therefore do not change their
US 4r 2
orientation and separation
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Sedimentation of a triplet
case (a):
UA UC 3 a a 9a
= = 1+ ( + )= 1+
US US 2 r 2r 4r
UB 3 a a 3a
= 1+ ( + ) = 1+
US 2 r r r
case (b):
UA UC 3 a a 9a
= = 1+ ( + )= 1+
US US 4 r 2r 8r
The particles do not maintain UB 3 a a 3a
= 1+ ( + ) = 1+
constant separation: the middle US 4 r r 2r
particle B falls faster
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Stokeslet simulation of a triplet
Sensitivity to initial configurations: signature of chaotic behavior originating in the
long-range and many-body character of the hydrodynamic interactions
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
1 1, 2, 3 . . . spheres
2 Clusters and clouds
3 Settling of a suspension of spheres
4 Intrinsic convection
5 Velocity fluctuations and hydrodynamic diffusion
6 Fronts
7 Boycott effect
8 More on polydispersity and anisotropy
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Settling of a spherical cloud of particles
Cloud velocity:
N 34 πa3 (ρp − ρ)g
Ucloud =
2πµ 2+3λ
λ+1
R
6a
= N US
2+3λ
2 λ+1 R
Collective motion: toroidal
circulation of the particles inside the
cloud
But chaotic fluctuations leading to
particle leakage
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Instability of a settling cloud of particles
Evolution of the cloud into a torus and subsequent breakup
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
1 1, 2, 3 . . . spheres
2 Clusters and clouds
3 Settling of a suspension of spheres
4 Intrinsic convection
5 Velocity fluctuations and hydrodynamic diffusion
6 Fronts
7 Boycott effect
8 More on polydispersity and anisotropy
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Summing the effects between pairs of particles
Velocity of a pair of spheres at a separation r :
US + ∆U where ∆U(r ) incremental velocity due to a second particle
Averaging over all possible separations which occur with conditional
probability P1|1 (r ) Z
US + ∆U P1|1 (r ) dV
|{z}
r≥2a | {z }
aUS
r ng(r)=n
Divergence with the size L of the vessel as
Z L
r −1 r 2 dr ∼ L2
2a
Strong divergence due to long-range hydrodynamic interactions
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Hindered settling
Mean velocity:
huip = US f (φ)
Richardson-Zaki 1954: f (φ) = (1 − φ)n
with n ≈ 5 at low Re
Main effect = Back-flow
Batchelor 1972:
f (φ) = 1 + Sφ + O(φ2 ) with S = −6.55
assuming uniformly dispersed rigid spheres
Results depend on microstructure in turn
determined by hydrodynamics
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
1 1, 2, 3 . . . spheres
2 Clusters and clouds
3 Settling of a suspension of spheres
4 Intrinsic convection
5 Velocity fluctuations and hydrodynamic diffusion
6 Fronts
7 Boycott effect
8 More on polydispersity and anisotropy
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Influence of the lateral walls of the vessel
Intrinsic convection = global
convection of the suspension
superimposed on the settling of the
particles relative to the suspension
Intrinsic convection originates in the
buoyancy of the particle-depleted
layer next to the side walls
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Intrinsic convection
Particle-depleted layer next to the
side walls: the centers of the spheres
cannot come closer than a radius a
to the cell wall
This buoyant particle-depleted layer
located at one particle radius from
the wall drives an upward flow near
the wall
Because no net flux condition across
any horizontal section, downward
return flow in the center
Boundary layer formulation:
Poiseuille flow with a slip velocity at
the wall w∗ = 49 φUS
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
1 1, 2, 3 . . . spheres
2 Clusters and clouds
3 Settling of a suspension of spheres
4 Intrinsic convection
5 Velocity fluctuations and hydrodynamic diffusion
6 Fronts
7 Boycott effect
8 More on polydispersity and anisotropy
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Velocity fluctuations
Random walk through the
suspension after a large enough
number of hydrodynamic interactions
Diffusive nature of the long-time
fluctuating particle motion
Anisotropic hydrodynamic
self-diffusivities
Large velocity fluctuations of the
same order as the mean particle
velocity
Anisotropic fluctuations with a larger
value in the direction of gravity
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Divergence of velocity fluctuations?
Random mixing of the suspension
√ creates
statistical fluctuations of O( Nl )
Balance
√ 4 of the fluctuations in the weight
Nl 3 πa3 (ρp − ρ)g by Stokes drag on the
blob 6πµlwp′
Convection currents, also called ‘swirls,’ on
all length-scales l
√
Nl 34 πa3 (ρp − ρ)g
r
l
wp′ (l) ∼ ∼ US φ
6πµl a
Large-scale fluctuations are dominant
r
Blob of size l (aφ−1/3 < l < L) L
wp′ ∼ US φ diverge with L
containing Nl = φl 3 /a3 particles a
BUT no such divergence seen in experiments
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
1 1, 2, 3 . . . spheres
2 Clusters and clouds
3 Settling of a suspension of spheres
4 Intrinsic convection
5 Velocity fluctuations and hydrodynamic diffusion
6 Fronts
7 Boycott effect
8 More on polydispersity and anisotropy
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Kinematic wave equation
Conservation of particles
∂φ ∂(wp φ)
+ =0
∂t ∂z
Hyperbolic wave equation
∂φ ∂φ
+ c(φ) =0
∂t ∂z
Wave speed
d(wp φ)
c(φ) = = US [f (φ) + φf ′ (φ)]
dφ
Only variation in the direction of using wp = US f (φ)
gravity z
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Kinematic wave speed
f (φ): decreasing function of φ thus f ′ (φ) < 0
c(φ) 6 wp (φ)
c(φ) ≡ US at φ = 0 and then decreases
rapidly to negative values before increasing to
a small negative value at maximum packing
Lower values of φ propagate faster than
larger values
Self-sharpening
∴ Formation of sharp shocks
d(wp φ)
c(φ) = = US [f (φ)+φf ′ (φ)]
dφ
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Shock speed
Conservation of particle flux across the shock
(subscript 1 ahead of the shock and 2 behind)
[wp φ]21 wp (φ2 )φ2 − wp (φ1 )φ1
Ushock = =
[φ]21 φ2 − φ1
Velocity of the sedimentation front
(φ2 = 0, φ1 = φ0 )
Usedimentation = wp (φ0 ) = US f (φ0 )
Velocity of the growing-sediment front
(φ2 = φ0 , φ1 ≈ φmax )
φ0 wp (φ0 ) φ0 f (φ0 )
Usediment = = −US
φ0 − φmax φmax − φ0
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Front spreading
Diffusive spreading of the sedimentation front
Nonlinear convection-diffusion equation
∂φ ∂φ ∂ ∂φ
+ c(φ) = (D c )
∂t ∂z ∂z ∂z
with D c (φ) gradient diffusivity
But also convective spreading
Polydispersity in particle size leading to a
distribution of sedimentation velocity
Differences in settling velocity of the
density fluctuations created by the mixing
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
1 1, 2, 3 . . . spheres
2 Clusters and clouds
3 Settling of a suspension of spheres
4 Intrinsic convection
5 Velocity fluctuations and hydrodynamic diffusion
6 Fronts
7 Boycott effect
8 More on polydispersity and anisotropy
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Sedimentation in an inclined channel
Enhancement in settling rate: (H/b) sin θ
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
1 1, 2, 3 . . . spheres
2 Clusters and clouds
3 Settling of a suspension of spheres
4 Intrinsic convection
5 Velocity fluctuations and hydrodynamic diffusion
6 Fronts
7 Boycott effect
8 More on polydispersity and anisotropy
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Sedimentation of a suspension of bidisperse spheres
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Instability of a sedimenting suspension of fibers
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
Movie references
Taylor, G. I. 1966. Low Reynolds Number Flows, The U.S.
National Committee for Fluid Mechanics Films.
http://media.efluids.com/galleries/ncfmf?medium=305
Guazzelli, É., and Hinch, E. J. 2011. Fluctuations and
instability in sedimentation. Ann. Rev. Fluid Mech., 43,
87–116.
SUPPLEMENTAL MATERIALS
http://www.annualreviews.org/doi/suppl/10.1146/annurev-
fluid-122109-160736
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
1, 2, 3 . . . spheres Clouds Mean settling Intrinsic convection Fluctuations Fronts Boycott effect Polydispersity and anisotropy
General references
Davis, R. H., and Acrivos, A. 1985. Sedimentation of
noncolloidal particles at low Reynolds numbers. Ann. Rev.
Fluid Mech., 17, 91–118.
Guazzelli, É., and Hinch, E. J. 2011. Fluctuations and
instability in sedimentation. Ann. Rev. Fluid Mech., 43,
87–116.
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics
Sedimentation
