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Particles in Fluids

H. J. Herrmann

IfB, ETH Zurich, Honggerberg, 8093 Zurich, Switzerland and Depr. de Fısica, Univ. Fed. do

Ceara, 60451- 970 Fortaleza, Ceara, Brazil.

J. S. Andrade Jr., A. D. Araujo, M. P. Almeida

Departamento de Fısica, Universidade Federal do Ceara, 60451-970 Fortaleza, Ceara, Brazil.

V. Komiwes

Laboratoire LIP6 104, Paris 6, Av. du President Kennedy 75016 , Paris, France

J. Harting

Technische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven, Netherlands

D. Kadau

IfB, ETH Zurich, Honggerberg, 8093 Zurich, Switzerland.

c© Societa Italiana di Fisica 1

2H. J. Herrmann, J. S. Andrade Jr., A. D. Araujo, M. P. Almeida, V. Komiwes, D. Kadau, J. Harting

Summary. — For finite Reynolds numbers the interaction of moving fluids withparticles is still only understood phenomenologically. We will present various nu-merical studies. First we will introduce a numerical technique to simulate granularmotion in a fluid. Particle trajectories are calculated by Newton’s law and colli-sions are described by soft-sphere potentials. The fluid flow is calculated solvingthe Navier-Stokes equation. The momentum transfer is directly calculated from thestress tensor around particles. This scheme is validated calculating the drag coef-ficient, finding the limitations on the Reynolds number in mesh size and computertime. Then we discuss sedimentation of two particles and reproduce the ”Draft, Kissand Tumbled” effect showing that we can reproduce hydrodynamic interactions onthe scale of the particle. The terminal velocity of particles is in good agreementwith experiments and we recover the Richardson and Zaki law. We also will use theLattice Boltzmann Method and the solver “Fluent” which elucidates this issue fromdifferent points of view. We show that the distribution of particle velocities insidea sheared fluid can be obtained over many orders of magnitude. We also considerthe case of fixed particles, i.e. a porous medium and present the distribution ofchannel openings and fluxes. These distributions show a scaling law in the densityof particles and for the fluxes follow an unexpected stretched exponential behaviour.The next issue will be filtering, i.e. the release of massive tracer particles within thisfluid. Interestingly a critical Stokes number below which no particles are capturedand which is characterized by a critical exponent of 1/2. Finally we will also showdata on saltation, i.e. the motion of particles on a surface which are dragged by thefluid performs jumps. This is the classical aeolian transport mechanism responsi-ble for dune formation. The empirical relations between flux and wind velocity arereproduced. Finally we also briefly discuss quicksand and its collapse.

1. – Introduction

Particles in fluids (liquids or gases) appear in many applications in chemical engineer-

ing, fluid echanics, geology and biology [1, 2, 3]. Also fluid flow through a porous medium

is of importance in many practical situations ranging from oil recovery to chemical reac-

tors and has been studied experimentally and theoretically for a long time [4, 5, 6].

Several industrial processes use sedimentation, pneumatic transport or fluidized cat-

alyst beds. Their correct operation requires the understanding of the behavior of a

granular bed interacting with a fluid. However, the hydrodynamics of such processes is

still badly known, bringing into play physical phenomena that act on the scale of the

particle. Numerical simulation makes it possible to investigate these phenomena. Several

models can be used to simulate two phase flows of fluid-solid type:

1. Continuous models for the phases liquid and solid [7, 8, 9, 10, 11] are popular in

the engineering sciences. In these models, questions remain open concerning the

Particles in Fluids 3

determination of the proper constitutive equation for the solid and the momentum

transfer between the solid and fluid phases. In general, the momentum transfer is

modelled by a local drag force depending on the local relative velocity between the

solid and the fluid, the Reynolds number and the local volume fraction of the solid

and the fluid. This drag law is based on Ergun’s law [12] which yields the pressure

drop in a granular media.

2. Eulerian-Lagrangian models for the fluid and solid phases respectively [13, 14, 15]

implemented using a finite element method and no-slip boundary conditions on the

surface of each particle have been employed for very few particles.

3. Eulerian models for the fluid phase coupled to a discrete description of the solid

phase. [16, 17, 18, 19, 20, 21, 22, 23] use a drag law, [24] following an idea of

Fogelson and Peskin [25] and [26, 27] integrate the stress tensor around the surface

of the particles.

We will present in the following a model of the third type and show some examples of

its applications. Later we will also discuss results obtained with the Lattice Boltzmann

Method and with commercial codes.

2. – Description of the Model

2.1. Particle motion. – The particles are described as discrete objects [28, 29, 30].

Their trajectories are calculated through Newton’s equation and collisions are described

by a soft-sphere potentials as follows:

fg + fn + ft + fh = mip

d2xip

dt2, (ip = 1, . . . , np).(1)

Here fg, fn, ft and fh are respectively the gravitational force, the collision force in the

normal and the tangential direction and the hydrodynamic force from the fluid acting

on the particle.

The gravitational force fg is given by

fg = −mipgux2 ,(2)

with

mip =4

3πr3ipρp ,(3)

the gravitational acceleration g and the direction of the gravitational force ux2 .

The collision force in the normal direction is given by

fn = fel + ffrn(4)


where fel and ffrn are the elastic force and the frictional force in the normal direction.

The elastic force between particel ip and jp is given by

fel =

{

−knδr n δr > 0

0 δr ≤ 0,(5)

where

n =xip − xjp

|xip − xjp |,(6)

δr = rip + rjp − (xip − xjp) · n ,(7)

as well as kn, xip , xjp , rip and rjp are respectively the elastic modulus and the positions

and radii of particle ip and jp. The frictional force in the normal direction is given by

ffrn = −2γnmredvn ,(8)

where γn is the normal dynamic friction coefficient,

vn = ((vip − vjp) · n) n ,(9)

the relative velocity in the normal direction between particle ip and jp with their respec-

tive velocities vip and vjp , and

mred =mipmjp

mip +mjp

,(10)

the reduced mass with mip and mjp the masses of particle ip and jp.

The collision force in the tangential direction is given by

ft = −min(µc|fn|, |ffrt + fs|) t,(11)

where µc is the Coulomb friction coefficient, fn the collision force in the normal direction

given by eq. (4), ffrt the frictional force in the tangential direction and fs the static

friction. With the relative velocity in the tangential direction

vt = (vip − vjp)− vn ,(12)

we get

t =vt

|vt|.(13)


t=tc0<t<tc

t=0t<0

Fig. 1. – Collision event

The frictional force in the tangential direction is given by

ffrt = −2γtmredvt ,(14)

where γt is the tangential dynamic friction coefficient. Finally

fs = −ks

∫ tc

0

vt dt(15)

where ks and tc are respectively the static friction coefficient and the collision time.

2.1.1. Dimensionless equations for particle motion. The dimensionless variables are

obtained by normalizing with the average radius r for the length, the Stokes velocity Vs

for the velocity and the fluid density ρf :

xip = rx∗ip,

t =r

Vst∗,

vip = Vsv∗ip.

By replacing these quantities, the dimensionless equation of motion is written as:

f∗g + f∗n + f∗t + f∗h = m∗ip

d2x∗ip

dt2, (ip = 1, . . . , np).(16)

f∗totip = f∗g + f∗n + f∗t + f∗h(17)


f∗g = −m∗i Frux2(18)

f∗n = f∗el + f∗frn(19)

f∗t = −min(µc|f∗n |, |f∗frt + f∗s |) t(20)

f∗el = −k∗n δr∗ n(21)

f∗frn = −2γ∗nm

∗v∗n(22)

f∗frt = −2γ∗t m

∗v∗t(23)

f∗s = −k∗s

∫ t∗c

0

v∗t dt(24)

with:

r∗ip =ripr,(25)

δr∗ =δr

r,(26)

m∗ip

=4

3π(rip

r

)3 ρpρf

,(27)

m∗ =1

np

np∑

i=1

m∗ip,(28)

v∗n =

vn

Vs,(29)

v∗t =

vt

Vs,(30)

kn∗ =

knrV 2

s ρf,(31)

k∗s =ks

rV 2s ρf

,(32)

γ∗n =

γnrVs

mred

mip

m,(33)

γ∗t =

γtrVs

mred

mip

m,(34)

Fr =rg

V 2s

.(35)

2.1.2. Numerical scheme. The integration of the equations of motion is carried out

using the explicit leap-frog scheme: Let

xnip

= xip(n∆tp)

vnip

= vip(n∆tp)

fntotip = ftotip (n∆tp)


anip = aip(n∆tp)

where n = 0, 12 , 1, . . ..

The leap-frog scheme is derived by writing the following Taylor expansion:

vn+ 1

2

ip= vn

ip+

∆tp2

anip +∆tp

2

4

d2vip

dt2+O(∆tp

3)(36)

vn− 1

2

ip= vn

ip−

∆tp2

anip +∆tp

2

4

d2vip

dt2+O(∆tp

3)(37)

xn+1ip

= xnip+∆tpv

nip+

∆tp2

2anip +O(∆tp

3)(38)

The velocity at time step (n+ 12 )∆tp and the position at time step (n+1)∆tp are obtained

by subtracting eq.(37) from eq.(36):

vn+ 1

2

ip= v

n− 12

ip+∆tpa

nip+O(∆tp

3)(39)

with

anip =fntotip

mip

.(40)

Finally

xn+1ip

= xnip+∆tpv

n+ 12

ip+O(∆tp

3).(41)

2.1.3. Linked-Cell algorithm. The integration of the equations of motion requires for

a given particle in principle the calculation of the interaction forces with all the (np − 1)

other particles. The resulting algorithm goes like O(np2).

We use a search algorithm of linked-cell type to find the possible neighbors of collisions

to reduce this cost by dividing space in cells containing the particles. The particles in

a cell can only interact with the other particles in the cell or with the particles in the

neighboring cells. Once the space is divided in cells and the particles indexed by cells,

the calculation of the interaction forces becomes simple. This algorithm goes like O(np)

and has the advantage of being easily adapted to parallelisation [29, 31].

Let the number of cells in every direction be C1, C2 and C3. The size of the cells is

two times larger than the maximum radius of the particles. The linked-cell algorithm is

well described in Ref. [32].

The calculation of the interaction forces is executed cell by cell. In the three-dimensio-

nal case, every cell contains on average the following number of particles:

nc =np

C1C2C3(42)


8

10

11

9

3

j

i

5

1

7

2 4

6

Fig. 2. – Illustration of the linked cell algorithm.

As every cell has 26 neighbor cells, the number of interactions is on average 26npnc.

Using the principle actio = reactio, only the lower neighbors need to be checked (see Fig.

3). The cost of the algorithm for the calculation of the interaction forces is thus divided

by two, i.e. is of the order of 13npnc.

2.2. Fluid flow . – The fluid is considered incompressible and Newtonian. The di-

mensionless variables are obtained by dividing through: the average radius r for the

length, the Stokes velocity Vs for the velocity and the fluid density ρf . The dimensionless

Navier-Stokes equations describe the motion of the interstitial fluid on a resolution scale

h smaller than the particle diameter dp as shown in Fig. 4:

∂vi∂t

= −∂(vivj)

∂xj

−∂p

∂xi

+1

Re

∂2vi∂x2

i

+ fi(43)

with

f = fiuxi= −Frux2 ,(44)


Fig. 3. – Only the cells’ neighbor on the lower part are checked.

pd

h

Fig. 4. – The resolution scale h is smaller than the particle diameter dp.


where uxiis a unit vector in the direction of the gravitational force. The conservation

of the fluid mass together with the incompressibility gives

∂vi∂xi

= 0.(45)

The Reynolds number is defined as

Re =ρfVsr

µf(46)

and the Froude number is given by

Fr =rg

V 2s

.(47)

2.2.1. Numerical scheme. An explicit operator splitting, fractional-time-step method,

first order in time proposed by Chorin [33] is used in the following to solve the Navier-

Stokes equation:

vn+1i − vni∆tf

= −∂pn+1

∂xi

−∂(vni v

nj )

∂xj

+1

Re∆(vni ).(48)

The introduction of the quantity vn+ 1

2

i

vn+1i − vni∆tf

=(vn+1

i − vn+12 )− (vni − vn+

12 )

∆tf(49)

gives

vn+ 1

2i − vni

∆tf= −

∂(vni vnj )

∂xj

+1

Re∆(vni )(50)

and

vn+1i − v

n+ 12

i

∆tf=

∂pn+1

∂xi

.(51)

Taking the divergence of equation (51) gives a Poisson equation for the pressure:

∂2pn+1

∂xj∂xj

=1

∆tf

∂vn+ 1

2i

∂xi

.(52)

A staggered marker and cell (MAC) mesh as base for a second order spatial finite

difference centered discretization is used to discretize each term [34] (see Fig. 5). The


discrete Poisson equation for the pressure is given by:

1

h2

(

pn+1(k+1,l,m) + pn+1

(k−1,l,m) + pn+1(k,l+1,m) + pn+1

(k,l−1,m)

+ pn+1(k,l,m+1) + pn+1

(k,l,m−1) − 6pn+1(k,l,m)

)

=1

h∆tf

(

vn+ 1

21(k+1

2,l,m)

− vn+ 1

21(k−

12,l,m)

+ vn+ 1

22(k,l+1

2,m)

− vn+ 1

22(k,l− 1

2,m)

+ vn+ 1

23(k,l,m+ 1

2)− v

n+ 12

3(k,l,m−

12)

)

(53)

The discretization produces external nodes, like v1(k+ 1

2,−1,m)

, v2(−1,l+1

2,m)

, v3(−1,l,m+1

2)

or p(−1,l,m). The velocity value at a node is determined through linear extrapolation.

For example at node (−1, l+ 12 ,m), we have

v2(−1,l+1

2,m)

= 2v2Γ − v2(0,l+ 1

2,m)

.(54)

To fix the value of v1(− 1

2,l,m)

, we write

v1(− 1

2,l,m)

= v1Γ(− 1

2,l,m)

.(55)

The boundary condition for the pressure is determined by calculating the scalar prod-

uct of the discretized equation (51) with the normal unit vector of the domain boundary:

pn+1(0,l,m) − pn+1

(−1,l,m)

h= −

1

∆tf

(

vn+11(− 1

2,l,m)

− vn+ 1

21(− 1

2,l,m)

)

.(56)

Replacing eq.(56) in eq.(53), the quantity vn+ 1

21(− 1

2,l,m)

cancels and gives for the boundary

node:

vn+ 1

21(− 1

2,l,m)

= vn+11(− 1

2,l,m)

(= v1Γ(− 1

2,l,m)

)(57)

and

pn+1(−1,l,m) = pn+1

(0,l,m).(58)

The Poisson equation (53) for the pressure with its boundary condition (58) is solved

using the conjugate gradient method [35].


Γ

k

l

Fig. 5. – Staggered marker and cell (MAC) mesh.

2.3. Momentum transfer: The particle-fluid coupling. – The momentum transfer be-

tween the fluid and the particle is directly calculated by integrating the stress over the

entire surface of the particle obtained from the hydrodynamic force:

(fhip)i=

∫

Sip

σijnj dSip =

∫

Sip

pni dSip +2

Re

(

∂vi∂xj

+∂vj∂xi

)

nj(59)

The discrete expression for the hydrodynamic force is given by :

(fhip)i=

∑

node(k,l,m) of the particle surface

(fh(k,l,m))i

(60)

where

(fh(k,l,m)) =

(

−p(k,l,m) +1

Re

1

hσ(k,l,m)

)

h2 n(k,l,m) .(61)

Here n(k,l,m) is the unit normal vector of the surface of the particle ip at node (k, l,m)

and σ(k,l,m) is given by

σ(k,l,m) = 2(v1(k+3

2,l,m)

− v1(k+1

2,l,m)

)


+v1(k+1

2,l+1,m)

− v1(k+ 1

2,l,m)

+ v2(k+1,l+ 1

2,m)

− v2(k,l+ 1

2,m)

+v1(k+1

2,l,m+1)

− v1(k+ 1

2,l,m)

+ v3(k+1,l,m+ 1

2)− v3

(k,l,m+ 12)

+v1(k+1

2,l+1,m)

− v1(k+ 1

2,l,m)

+ v2(k+1,l+ 1

2,m)

− v2(k,l+ 1

2,m)

+2(v2(k,l+3

2,m)

− v2(k,l+ 1

2,m)

)

+v2(k,l+1

2,m+1)

− v2(k,l+ 1

2,m)

+ v3(k,l+1,m+ 1

2)− v3

(k,l,m+ 12)

+v1(k+1

2,l,m+1)

− v1(k+ 1

2,l,m)

+ v3(k+1,l,m+ 1

2)− v3

(k,l,m+ 12)

+v2(k,l+1

2,m+1)

− v2(k,l+ 1

2,m)

+ v3(k,l+1,m+ 1

2)− v3

(k,l,m+ 12)

+2(v3(k,l,m+3

2)− v3

(k,l,m+ 12).

We calculate the quantities

Vs =2

9

r2(ρp − ρf)g

µf(62)

and

St =2

9

ρpVsr

µf,(63)

where n is the index of a time step and nmax is the maximum number of time steps of

the simulation.

The time steps ∆tp and ∆tf for particle and fluid update is given by

∆tf = min

(

h2Re

6,

2

(v21 + v22 + v23)Re

)

,(64)

where equation (64) is the stability condition [33].

∆tp =tc25

≪ tc(65)

mmax =∆tf∆tp

+ 1(66)

mmax is the necessary number of fluid time steps per particle time step.

3. – Drag law

3.1. Empirical expression of the drag law . – The drag coefficient Cd for a spherical

particle of radius r fixed in a fluid flow is defined by:

Cd =fh

12ρfSFUs

2 .(67)


Here fh is the norm of the hydrodynamic force acting on the spherical particle, Us is the

superficial mean velocity and SF = πr2 is the cross section of the particle.

With the Stokes approximation Re ≪ 1, the hydrodynamic force is calculated through

fh = 6πrµfUs,(68)

and gives the following drag coefficient

Cd =24

Re′,(69)

with Re′ = 2rUsρf

µf. Schiller and Nauman measured experimentally an empirical drag

coefficient given by

Cd =24

Re′

(

1 + 0.15Re′0.687

)

.(70)

3.2. Parameters and boundary conditions . – The dimensions of the simulation box

are:

Lx1 = 20 r, Lx2 = 40 r and Lx3 = 20 r.

We see no change in the results using a bigger box (relative error less than 1%). The

particle is placed and fixed on the position (see Fig. 6.):

xp =

(

Lx1

2,Lx2

4,Lx3

2

)

.

We choose two different mesh sizes: h1 = 0.4 r and h2 = 0.2 r at a moderate Reynolds

number. The time step is calculated from the stability condition (64). A uniform dimen-

sionless velocity Us = 1.0 is imposed as boundary condition on x2 = 0. The pressure is

set to zero on x2 = Lx2.

3.3. Results . – Fig. 7 shows the drag coefficient calculated as a function of the Reynolds

number Re′.

Relation (70) for the drag coefficient for Re′ ≤ 20 is reproduced with a maximal error

of 14% with a mesh size of h1 = 0.4 r and a maximal error of 10% with a mesh size of

h2 = 0.2 r. So, the simulation is in fair agreement with the empirical relation of Schiller

and Nauman for Re′ ≤ 20. Larger Reynolds numbers require a smaller mesh size given

by the relation

h

r= O(Re−

94 )(71)


��

��

x

Lx1 = 20r

Lx2 = 40r

p=0

v=Us1

2x

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

Fig. 6. – Simulation box at x3 =Lx32

.

and consume too much computer time because the computer time goes like O(k5), where

k is the reduction factor of the mesh size (O(k2) for the stability condition (64) and

O(k3) for the number of nodes).

For example, when a smaller mesh size h2 = 0.2 r is used instead of h1 = 0.4 r, the

reduction factor is k = 2 and the CPU time is multiplied by 25 = 32.

The CPU time is about 6 µs per node and per time step on a Silicon Graphics

Workstation Octane with a R 10 000 processor.

4. – Draft, Kiss and Tumbled Effect

Let us now simulate the sedimentation of two particles and reproduce the Draft, Kiss

and Tumbled effect.

4.1. Description of the effect . – The Draft, Kiss and Tumbled effect may be observed

when two particles fall under the action of gravity in a viscous Newtonian fluid. It

consists of the three following steps named “Draft”, “Kiss” and “Tumbled”: Particle 1


0.1 1.0 10.0 100.0

Reynolds number

1*10-1

5*10-1

1*100

5*100

1*101

5*101

1*102

5*102

1*103

Dra

g co

effic

ient

h1

h2

Stokes

Experiment

Fig. 7. – Drag coefficient as function of the Reynolds number Re′.

is first placed within the hydrodynamic drag above particle 2. As particle 2 produces

a depression zone, particle 1 is attracted: The Draft. Particle 1 increases its vertical

velocity until it touches particle 2: The Kiss. The horizontal velocity of particle 2

increases and its vertical velocity decreases below that of particle 1. Particle 1 having the

same horizontal velocity and higher vertical velocity than particle 2, overtakes particle 2:

The Tumbled.

4.2. Parameters and boundary conditions . – We use the following physical parameters

ρf = 1000 kgm−3, ρp = 2000 kgm−3, µf = 10−3 kg sm−1, dp = 200 µm

and the three dimensionless parameters

Re = 2.2, St = 0.97, Fr = 2.1.

The mesh size is h = 0.4 r and the stability condition gives the time step ∆tf =


1-0.024-0.016-0.009-0.0030.0010.0060.0130.0190.025

2

-0.045-0.032-0.018-0.0070.0040.0160.0260.0340.045

Fig. 8. – Velocity and pressure fields of the fluid at x3 =Lx3

2and times t = 14 and 77.

5.8 · 10−2. The dimensions of the simulation box are:

Lx1 = 12 r, Lx2 = 24 r, Lx3 = 12 r.

We use periodic boundary conditions for velocity and pressure of the fluid in each spatial

direction.

4.3. Results . – The simulation reproduces the Draft, Kiss and Tumbled effect of the

sedimentation of two particles under gravity. Figs. 8 and 9 show the velocity and pressure

fields of the fluid at x3 =Lx3

2 and times t = 14, 77, 88, 126 corresponding to the Draft,

Kiss and Tumbled situations.

Fig. 10 shows the time evolution of the horizontal and vertical velocities of the two

1-0.049-0.035-0.022-0.0120.0020.0140.0240.0330.041

1-0.049-0.031-0.016-0.0040.0070.0150.0210.0290.038


2and times t = 88 and 126 .


0 50 100 150 200

time

-0.15

-0.1

-0.05

0.0

0.05

horiz

onta

l vel

ocity

of p

artic

les

particle1

particle2

0 50 100 150 200

time

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

Ver

tical

vel

ocity

of p

artic

les

particle1

particle2

isolated

Schiller

Fig. 10. – Horizontal and vertical velocity (vp)1 of particles 1 and 2 .

particles. The simulations are in agreement with the experimental description of the

Draft, Kiss and Tumbled effect: Particle 1 increases its horizontal velocity more than

particle 2 until it touches particle 2: The Kiss happens at time t = 70. After this time,

the vertical velocity of particle 2 decreases and its horizontal velocity increases. The

terminal velocities of the two particles are in good agreement with the terminal velocity

of a single particle found by an independent simulation and calculated using the empirical

Schiller and Nauman formula (70).

5. – Sedimentation of many particles

5.1. The empirical law of Richarson and Zaki . – The mean velocity v of a system

of particles sedimenting in a fluid depends on the volume concentration of the solid Φ

through the Richardson and Zaki [36] law:

v(Φ) = vs(1− Φ)n(72)

where vs is the Stokes velocity of an isolated particle. The exponent n is a function of the

Reynolds number and is close to 5 for Reynolds numbers Re ≤ 1. This law is confirmed

by many experiments and in particular by Nicolai et al. [37].

5.2. Parameters . – We use the following physical parameters:

ρf = 1000 kgm−3, ρp = 2500 kgm−3, µf = 10−3 kg sm−1, dp = 100 µm

giving

Re = 0.4, Fr = 7.4, St = 0.23


0.0 0.1 0.2 0.3 0.4 0.5

Volumic concentration of solid

0.0

0.2

0.4

0.6

0.8

1.0

V/V

s

h1

h2

Richardson

Fig. 11. – Mean sedimentation velocity as function of volume concentration.

and a dimensionless time step of ∆tf = 10−2. The dimensions of the simulation box are:

Lx1 = 40 r, Lx2 = 80 r and Lx3 = 40 r

with resolution h1 = 0.4 r. Periodic boundary conditions are used in each spatial direc-

tion.

5.3. Results . – The simulations carried out for mesh sizes h1 = 0.4 r and h2 = 0.2 r

yield relation (72) for different volume concentrations Φ (see Fig. 11). These simulations

show the limitations due to the mesh size: Direct simulations in a diluted medium can

be carried out with a grid of mesh size h1 = 0.4 r and in a dense medium (Φ < 0.2) with

h2 = 0.2 r. These limitations are due to the fact that the denser the medium, the smaller

are the pores through which the fluid can pass requiring a finer resolution.

6. – Velocity distribution in sheared suspensions

Some common particle-fluid mixtures are ubiquitous in our daily life and include

the cacao drink, tooth paste and wall paint which are mixtures of finely ground solid

ingredients in fluids. Of particular interest for industrial applications are suspensions

under the influence of external forces causing sedimentation or shear flows. Detailed

experiments have been performed for more than a hundred years, but questions about



2with a mesh size h1 = 0.4 r.

the microstructure during sedimentation or structural relaxations of the sediment are still

not well understood. Statistically averaged data like the probability distribution function

of particle velocities P (v) can be of use for a better understanding of suspensions.

Naively, one might expect P (v) to be of similar shape as for an ideal gas, i.e., like

a Maxwellian. However, it has been found by numerous authors that the probability of

high velocities is substantially larger than predicted by a Gaussian shaped distribution

function [38, 39]. Experimentally, Rouyer et al. [40] studied quasi 2D hard-sphere sus-

pensions and found a stretched exponential form of P (v) with concentration dependent

exponents between 1 and 2 corresponding to purely exponential distributions for high

concentrations and a Gaussian for small particle counts. These experimental results are

in contrast to theoretical predictions of a transition from exponential to Gaussian with

increasing volume concentration [38, 39]. However, both experimental and theoretically

studied systems are not able to obtain valuable statistics over more than 2-4 decades.

If one does not have enough data for high quality PDFs, a final answer on the nature

of the function cannot be given since for high velocities the variation of the data points

is too large. Indeed, we have found [41] that even stretched exponentials can fit PDFs

with purely exponential tails and Gaussian centers if only 2-4 decades of probability are

covered. But as soon as one adds more data points, the exponential nature of the tails

becomes more distinct and it becomes impossible to fit the whole PDF with a single

stretched exponential function.


6.1. Simulation method . – The system under consideration is a three-dimensional

Couette setup as shown in Fig. 13. At the top and bottom closed sheared walls are

applied and the shear rate is γ = 2vshear/Nz, with Nz being the distance between the

shear planes and vshear the shear velocities of the planes. All other boundaries are

periodic. A body force f can be added to mimic gravity. We consider 384 up to 1728

initially randomly placed suspended particles of equal radius corresponding to a particle

volume concentration Φ between 6.8% and 30.7%.

vshear

yx

z

f

−vshearFig. 13. – Schematic sketch of the simulation setup.

For our simulations [41], we apply a hybrid method composed of a Lattice-Boltzmann

solver (LB) for the fluid solvent and a molecular dynamics (MD) algorithm for the

motion of suspended particles. With appropriate boundary conditions being imposed

at solid/fluid interfaces, colloidal suspensions can be modeled. This approach and re-

cent additions for the calculation of lubrication forces and stability improvements of

the MD were originally introduced by A.J.C. Ladd and coworkers [42, 43, 44, 45]. The

algorithm has been applied by numerous groups and is well established in the litera-

ture [42, 43, 44, 45, 46, 47]. For our simulations we found a particle radius of a=1.25

lattice sites being sufficient since for larger particles P (vz) does not change significantly

anymore, but the computational effort increases substantially. The simulation volume

has dimensions 64a × 8a × 48a and the shear rate γ is varied between 2.3 · 10−4 and

1 · 10−3 (in lattice units). The fluid density is kept constant and the kinematic viscosity


is set to ν = 0.05 if not specified otherwise. Additionally, we add a backflow force that

reduces the pressure gradient due to the weight of the particles by an equivalent force

density. We have checked that the initial condition does not have an influence on the

final results by comparing the data of simulations that have been performed utilizing

different starting configurations.

A single simulation runs for 6.25 million Lattice-Boltzmann time steps, where during

the last 5 million time steps the z component of the velocity of every individual particle is

gathered in a histogram in order to obtain P (vz). Up to 1010 data points per histogram

allow us to obtain good statistics over six orders of magnitude. All distributions are

normalized such that∫

P (vz) dvz = 1 and∫

v2zP (vz) dvz = 1 with the RMS velocity

vRMSz =

√

〈v2z〉 = 1.

-0.001 0 0.00110-6

10-4

10-2

100

-8 -6 -4 -2 0 2 4 6 8 10vz

1×10-6

1×10-5

1×10-4

1×10-3

1×10-2

1×10-1

1×100

P(v

z) 0 5×10-4

1×10-3

·γ

0

1×10-4

2×10-4

v RM

S

Fig. 14. – Probability distribution function of vz for f = 0.72 · 10−4 corresponding to a Stokesvelocity of vs=1.4 · 10−3, Φ = 13.6% and different shear rates 3.3 · 10−4, 6.7 · 10−4, and 1 · 10−3.The full line corresponds to an approximative theory. The lower inset shows the unscaled data,where higher γ relate to wider P (vz). The upper inset depicts the linear relation between γ andvRMSz for the PDFs presented in the main figure as well as four additional data sets. Symbolscorrespond to the data and the line is given by a linear fit with slope 0.21.

6.2. Results . – First, we consider suspensions with constant Φ and various shear rates

under the influence of a body force f . The dependence of P (vz) on the shear rate

for three representative γ is depicted in Fig. 14. Distributions in other directions are

essentially identical if one deducts the shear velocity and are therefore not shown. P (vz)

is symmetric and 〈vz〉 = 0 for all cases considered in this section. As shown in the

lower inset, the not normalized distributions widen for higher shear rates. However, a

very good scaling is observed: All normalized curves collapse onto a single one. In the

upper inset we show the influence of γ on vRMSz : As expected from the theory, the shear


rate only sets a scale for the velocity corresponding to a linear relation between γ and

vRMSz . To obtain an insight into the properties of P (vz), we compute the cumulant κ

from our data and find that for all simulation parameters studied in this section it varies

between 3.8 and 4.6. Knowing κ, we can compute η =√

6/κ− 1. Due to the large

number of data points in our histograms, we calculate κ for periods of 1 million time

steps each and then compute the arithmetic average of the last 5 million time steps of a

simulation run. We find that κ varies by up to 10% within a single simulation which is

of the same order as the difference of the individual PDFs in Fig. 14. For the collapse

in Fig. 14 we get η = 0.73. As shown in the figure, the solid line given by the theory

and the simulation data show an excellent agreement over the full range of six orders of

magnitude of probability.

Next, we consider neutrally-buoyant suspended hard-spheres under shear. The shear

rate is kept fixed at γ = 6.7·10−4 and the particle volume concentration is varied between

Φ = 6.8% and 30.7%. Due to hydrodynamic interactions, the particles tend to move to

the center of the system, i.e., to an area where the shear is low creating a depleted region

close to the sheared walls.

-8 -4 0 4 8vz

1×10-6

1×10-5

1×10-4

1×10-3

1×10-2

1×10-1

1×100

P(v

z)

-8 -4 0 4 8vz

a) b)

Fig. 15. – a) P (vz) for f = 0, γ = 6.7 · 10−4 and Φ = 6.8%, 13.6%, 20.5%, 23.9%, and 27.3%.b) Φ is kept at 13.6% and the kinematic viscosity is set to ν = 0.017, 0.05 and 0.1. In bothfigures, all data sets collapse onto a single curve and the lines are given by the theory [41] withη = 0.69.

The corresponding normalized P (vz) are presented in Fig. 15a. As depicted in the

figure, all PDFs except for the lowest particle concentration Φ = 6.8% (circles) collapse

onto a single curve. At very low concentrations, the tails of P (vz) are still not fully

converged due to the limited number of particle-particle interactions taking place within

the simulation time frame. Again, the full line in Fig. 15a is an approximate theory [41].


As before, a very good agreement between simulation and theory is observed. The full

circles in Fig. 16 depict the dependence of vRMSz on Φ. For concentrations of at least

Φ = 13.6% vRMSz (Φ) can be fitted by a line with slope 1.8 · 10−4. The disagreement

of the linear fit for low Φ is consistent with the not fully converged PDFs as shown in

Fig. 15a. By keeping all simulation parameters except the kinematic viscosity ν constant,

the dependence of ν on P (vz) can be studied. As demonstrated by the squares depicting

the dependence of vRMSz on ν in Fig. 16, vRMS

z and thus the probability distribution

functions are independent of the viscosity. Thus, the steady state curve obtained for

different volume concentrations is identical to the one for different ν as shown in Fig. 15b.

0 0.1 0.2 0.3φ

2×10-5

4×10-5

6×10-5

8×10-5

v zRM

S

vRMS(φ)

0 0.05 0.1 0.15ν

vRMS(ν)

Fig. 16. – vRMSz in dependence of Φ (circles) and ν (squares). Data corresponds to P (vz) as in

Fig. 15, but covers a wider range of Φ and ν. Note the different x-axes.

It would be of interest to study the influence of the body force f on the shape of the

probability distribution. However, f and the shear forces are in a subtle interplay since

the height of the steady state sediment depends on both parameters and thus influences

the local particle volume concentration.

While different authors report on transitions between Gaussian and (stretched) ex-

ponential tails [38, 39, 40], we have shown [41] that there is no such transition and that

such findings are due to insufficient statistics.

7. – Porous media

7.1. Fluids in Porous media. – Due to disorder, porous media display many interesting

properties that are however difficult to handle even numerically. One important feature

is the presence of heterogeneities in the flux intensities due the varying channel widths.

They are crucial to understand stagnation, filtering, dispersion and tracer diffusion.

In the porous space the fluid mechanics is based on the assumption that a Newtonian

and incompressible fluid flows under steady-state conditions. We consider the Navier-

Stokes and continuity equations for the local velocity u and pressure fields p, being ρ the

density of the fluid. No-slip boundary conditions are applied along the entire solid-fluid

interface, whereas a uniform velocity profile, ux(0, y) = V and uy(0, y) = 0, is imposed


at the inlet of the channel. For simplicity, we restrict our study to the case where the

Reynolds number, defined here as Re ≡ ρV Ly/µ, is sufficiently low (Re < 1) to ensure

a laminar viscous regime for fluid flow. We use FLUENT [48], a computational fluid

dynamic solver, to obtain the numerical solution on a triangulated grid of up to hundred

thousand points adapted to the geometry of the porous medium.

The investigation of single-phase fluid flow at low Reynolds number in disordered

porous media is typically performed using Darcy’s law [4, 6], which assumes that a ma-

croscopic index, the permeability K, relates the average fluid velocity V through the

pores with the pressure drop ∆P measured across the system,

V = −K

µ

∆P

L,(73)

where L is the length of the sample in the flow direction and µ is the viscosity of the

fluid. In previous studies [49, 50, 51, 52, 53, 54, 55], computational simulations based on

detailed models of pore geometry and fluid flow have been used to predict permeability

coefficients.

Here we present numerical calculations for a fluid flowing through a two-dimensional

channel of width Ly and length Lx filled with randomly positioned circular obstacles [56].

For instance, this type of model has been frequently used to study flow through fibrous

filters [57]. Here the fluid flows in the x-direction at low but non-zero Reynolds number

and in the y-direction we impose periodic boundary conditions. We consider a particular

type of random sequential adsorption (RSA) model [58] in two dimensions to describe

the geometry of the porous medium. As shown in Fig. 17, disks of diameter D are

placed randomly by first choosing from a homogeneous distribution between D/2 and

Lx −D/2 (Ly −D/2) the random x-(y-)coordinates of their center. If the disk allocated

at this position is separated by a distance smaller than D/10 or overlaps with an already

existing disk, this attempt of placing a disk is rejected and a new attempt is made. Each

successful placing constitutes a decrease in the porosity (void fraction) ǫ by πD2/4LxLy.

One can associate this filling procedure to a temporal evolution and identify a successful

placing of a disk as one time step. By stopping this procedure when a certain value of ǫ

is achieved, we can produce in this way systems of well controlled porosity. We study in

particular configurations with ǫ = 0.6, 0.7, 0.8 and 0.9.

7.2. Various distributions in porous medium. – The geometry of our random config-

urations can be analyzed making a Voronoi construction of the point set given by the

centers of the disks [59, 60]. We define two disks to be neighbors of each other if they are

connected by a bond of the Voronoi tessellation. These bonds constitute therefore the

openings or pore channels through which a fluid can flow when it is pushed through our

porous medium, as can be seen in the close-up of Fig. 17. We measure he channel widths

l as the length of these bonds minus the diameter D and plot in Fig. 18 the (normalized)

distributions of the normalized channel widths l∗ = l/D for the four different porosities.

Clearly one notices two distinct regimes: (i) for large widths l∗ the distribution decays

seemingly exponentially with l∗, and (ii) for small l∗ it has a strong dependence on the


Fig. 17. – Velocity magnitude for a typical realization of a pore space with porosity ǫ = 0.7subjected to a low Reynolds number and periodic boundary conditions applied in the y-direction.The fluid is pushed from left to right. The colors ranging from blue (dark) to red (light)correspond to low and high velocity magnitudes, respectively. The close-up shows a typical poreopening of length l across which the fluid flows with a line average velocity v. The local flux atthe pore opening is given by q = vl cos θ, where θ is the angle between v and the vector normalto the line connecting the two disks.

porosity, increasing with decreasing porosity dramatically at the origin. Between the two

regimes a crossover is visible as a peak which shifts between ǫ = 0.9 and 0.8 and then

stays for smaller porosities at about l∗ = 1, i.e., l = D.

Let us now analyse the distribution of fluxes throughout the porous medium. Each

local flux q crossing its corresponding pore opening l is given by q = vl cos θ, where θ is the

angle between v and the vector normal to the cross section of the channel (see Fig. 17). In

Fig. 19 we show that the distributions of normalized fluxes φ = q/qt, where qt = V Ly is

the total flux, have a stretched exponential form, P (φ) ∼ exp(−√

φ/φ0), with φ0 ≈ 0.005

being a characteristic value. This simple form is quite unexpected considering the rather

complex dependence of P (l) on ǫ. Moreover, all flux distributions P (φ) collapse on top of

each other when rescaled by the corresponding value of 〈l∗〉−1

ǫ2. This collapse for distinct

porous media results from the fact that mass conservation is imposed at the microscopic

level of the geometrical model adopted here, which is microscopically disordered, but

is macroscopically homogeneous at a larger scale [6]. As also shown in Fig. 19, it is

possible to reconstruct the distribution of fluxes using a convolution of the distribution

of velocities v and the distribution of oriented channel widths, namely l cos θ. Indeed, if

we calculate the integral

P (φ) =

∫

P (v)P (l cos θ)δ(φ− vl cos θ) dv d(l cos θ)(74)


0 1 2 3 4 5 6 7

l*

0

0.5

1

1.5

2

P(l

*)

ε=0.6

ε=0.7

ε=0.8

ε=0.9

Fig. 18. – The distributions of the normalized channel widths l∗ = l/D for different values ofporosity ǫ. From left to right, the two vertical dashed lines indicate the values of the minimumdistance between disks l∗ = 0.1 and the size of the disks l∗ = 1.

we find that the original distribution P (φ) is approximately retrieved, as can also be

seen in Fig. 19 (solid line). Finally, the inset of Fig. 19 shows that the permeability

of the two-dimensional porous media closely follows the semi-empirical Kozeny-Carman

equation [4]

K

K0= κ

ǫ3

(1− ǫ)2,(75)

where K0 = h2/12 is a reference value taken as the permeability of an empty channel

between two walls separated by a distance h. The proportionality constant κ is given by

the following expression:

κ =

(

D

2h

)21

τα,(76)

where τ = (Le/L)2 is the hydraulic tortuosity of the porous medium, α corresponds to

the pore shape factor and Le is an effective flow length [4].

7.3. Results on Filtration. – Filtration is typically used to get clean air or water and

also plays a crucial role in the chemical industry. For this reason it has been studied

extensively in the past [61]. In particular, we will focus here on deep bed filtration where

the particles in suspension are much smaller than the pores of the filter which they

penetrate until being captured at various depths. For non-Brownian particles, at least


−7 −6 −5 −4 −3 −2 −1 0 1log10(φ

∗)

−6

−4

−2

0

2lo

g 10P

(φ∗ )

ε=0.6ε=0.7ε=0.8ε=0.9

100

101

102

ε3/(1−ε)2

10−5

10−4

10−3

10−2

K/K0

Fig. 19. – The log-log plot of the distributions of the normalized local fluxes φ = q/qt for different

porosities ǫ. The (red) dashed line is a fit of the form exp(−√

φ/φ0), where φ0 ≈ 0.005. Inthe inset we see a double-logarithmic plot of the global flux and the straight line verifies theKozeny-Carman equation (75).

four capture mechanisms can be distinguished, namely, the geometrical, the chemical,

the gravitational and the hydrodynamical one [61].

Very carefully controlled laboratory experiments were conducted by Ghidaglia et

al. [62, 63] evidencing a sharp transition in particle capture as function of the dimension-

less ratio of particle to pore diameter characterized by the divergence of the penetration

depth. Subsequently, Lee and Koplik [64] found a transition from an open to a clogged

state of the porous medium that is function of the mean particle size. Much less effort,

however, has been dedicated to quantify the effect of inertial impact on the efficiency of

a deep bed filter.

Here we will concentrate on the inertial effects in capture which constitute an impor-

tant mechanism in most practical cases and, despite much effort, are quantitatively not

yet understood, as reviewed in Ref. [65]. The effect of inertia on the suspended particles

is usually quantified by the dimensionless Stokes number, St ≡ V d2pρp/18ℓµ, where dpand ρp are the diameter and density of the particle, respectively, ℓ is a characteristic

length of the pores, µ is the viscosity and V is the velocity of the fluid. Inertial capture

by fixed bodies has already been described since 1940 by Taylor and proven to happen

for inviscid fluids above a critical Stokes number [66]. It is our aim to present a detailed

hydrodynamic calculation of the inertial capture of particles in a porous medium. We


δ D

x

y

u

Fig. 20. – The trajectories of particles released from different positions at the inlet of the periodicporous medium cell. St = 0.25 and the flow field u is calculated from the analytical solution ofMarshall et al. [57]. The thick solid lines correspond to the limiting trajectories that determineδ.

will disclose novel scaling relations.

Let us first consider the case of an infinite ordered porous medium composed of a

periodic arrangement of fixed circular obstacles (e.g., cylinders) [67]. This system can

then be completely represented in terms of a single square cell of unitary size and porosity

given by ǫ ≡ (1− πD2/4), where D is the diameter of the obstacle, as shown in Fig. 20.

Assuming Stokesian flow through the void space an analytical solution has been provided

by Marshall et al. [57]. Here we use this solution to obtain the velocity flow field u and

study the transport of particles numerically. For simplicity, we assume that the influx of

suspended particles is so small that (i) the fluid phase is not affected by changes in the

particle volume fraction, and (ii) particle-particle interactions are negligible. Moreover,

we also consider that the movement of the particles does not impart momentum to the

flow field. Finally, if we assume that the drag force and gravity are the only relevant

forces acting on the particles, their trajectories can be calculated by integration of the

following equation of motiondu∗

p

dt∗=

(u∗−u∗

p)

St + Fgg

|g| , where Fg ≡ (ρp − ρ)ℓ|g|/(V 2ρp) is

a dimensionless parameter, g is gravity, t∗ is a dimensionless time, and u∗p and u∗ are

the dimensionless velocities of the particle and the fluid, respectively.

We show in Fig. 20 some trajectories calculated for particles released in the flow for

St = 0.25. Once a particle touches the boundary of the obstacle, it gets trapped. Our

objective here is to search for the position y0 of release at the inlet of the unit cell (x0 = 0)

and above the horizontal axis (the dashed line in Fig. 20), below which the particle is

always captured and above which the particle can always escape from the system. As

depicted in Fig. 20, the particle capture efficiency can be straightforwardly defined as

δ ≡ 2y0. In the limiting case where St → ∞, since the particles move ballistically towards

the obstacle, the particle efficiency reaches its maximum, δ = D. For St → 0, on the

other hand, the efficiency is smallest, δ = 0. In this last situation, the particles can be

considered as tracers that follow exactly the streamlines of the flow and are therefore not


−4 −3 −2 −1 0 1

log10[St (ε−ε

min)]

−4

−3

−2

−1

0

log

10(δ/D)

ε=0.85

ε=0.90

ε=0.95

−4 −3 −2 −1 0

log10(St−St

c)

−2

−1

0

log

10( δ/D)

0.5

1.0

/

Fig. 21. – Log-log plot of the dependence of the capture efficiency δ on the rescaled Stokesnumber St/(ǫ− ǫmin) for periodic porous media in the presence of gravity. The inset shows thatthe behavior of the system without gravity and can be characterized as a second order transition,δ ∼ (St− Stc)

α, with α ≈ 0.5 and Stc = 0.2679 ± 0.0001, 0.2096 ± 0.0001 and 0.1641 ± 0.0001,for ǫ = 0.85, 0.9 and 0.95, respectively.

trapped.

We show in Fig. 21 the log-log plot of the variation of δ/D with the rescaled Stokes

number in the presence of gravity for three different porosities. In all cases, the variable δ

increases linearly with St to subsequently reach a crossover at St×, and finally approach

its upper limit (δ = D). The results of our simulations also show that St× ∼ (ǫ − ǫmin),

where ǫmin corresponds to the minimum porosity below which the distance between inlet

and obstacle is too small for a massive particle to deviate from the obstacle. The collapse

of all data shown in Fig. 21 confirms the validity of the scaling law.

In Fig. 22 we see that the behavior of the system in terms of particle capture becomes

significantly different in the absence of gravity. The efficiency δ remains equal to zero

up to a certain critical Stokes number, Stc, that corresponds to the maximum value of

St below which particles cannot be captured, regardless of the position y0 at which they

have been released. Right above Stc, the variation of δ can be described in terms of a

power-law, δ ∼ (St− Stc)α, with an exponent α ≈ 0.5, which can be seen in the inset of

Fig. 21. Our results show that, while the exponent α is practically independent of the

porosity for ǫ > 0.8, the critical Stokes number decreases with ǫ, and therefore with the

distance from the obstacle where the particle is released (see Fig. 21). To our knowledge,

this behavior, that is typical of a second order transition, has never been reported before


0.0 0.5 1.0 1.5 2.0

St

0.0

0.2

0.4

0.6

0.8

1.0

δ/D g

g/10

g/100

g/1000

zero gravity

Fig. 22. – Normalized capture efficiency δ as function of the Stokes number St for differentvalues of gravity. Here we use Fg = 16, a value that is compatible with the experimental setupdescribed in Ref. [62, 63].

for inertial capture of particles. The same result is obtained when, instead of flow in a

square unit cell, we use the circular setup proposed by Kuwabara [68].

In order to have a more realistic model for the porous structure we did also include

disorder [67]. Here we adopted a random pore space geometry [58] shown in Fig. 17. For

compatibility between periodic and disordered descriptions, we take the characteristic

pore size to be ℓ ≡ D/20 (i.e., half of the minimum distance between any two obstacles

of the disordered system). To reduce finite-size effects, periodic boundary conditions are

applied in the y direction. Finally, end effects of the flow field are reduced by attaching

a header (inlet) and a recovery (outlet) region to the two opposite faces of the channel.

The trajectories of the particles are calculated by numerical integration of the equations

of motion, but now considering a drag coefficient which is based on the empirical relation

proposed by Morsi and Alexander [69]. For the capture efficiency we obtained the same

results as for the regular case, i.e. a critical Stokes number.

For a fixed value of St, we consider up to 1000 particles to determine (i) whether or

not these particles get trapped and (ii) the precise position at the surface of the porous

matrix where their capture takes place. From these positions, we obtain the profiles of

the fraction of non-captured particles φ as function of the longitudinal distance x along

the channel. In the limiting case of a very dilute system (ǫ ≈ 1) with particles being

transported in the ballistic regime (St → ∞), it is easy to show that φ(x) = exp(−x/λ),


with a penetration length given by λ = πD/4(1 − ǫ). For low and moderate values of

St, the behavior of φ(x) is still exponential, but λ now being a function of the Stokes

number. It seems that the previous result can be generalized to any combination of ǫ

and St as

λ = πD2/4(1− ǫ)δ ,(77)

where the length δ is the capture efficiency analogously defined as for the periodic porous

medium. The penetration length follows a power-law λ ∼ St−α, with a scaling exponent

α ≈ 1 that is, within the numerical error bars, the same for the three values of porosity

investigated. Simulations performed for a different realization of the disordered porous

medium resulted in the same exponent. This value is also consistent with the exponent

found for the periodic case with gravity. The random porous system can be described

very closely by the relation

λ

D=

β

St(1 − ǫ),(78)

with a prefactor β ≈ 0.058. For all practical purposes, this value is a constant for the

physical model of porous geometry, flow and phenomenology of particle capture studied

here. Indeed the data collapse of all profiles of φ in terms of the rescaled variable

St(1 − ǫ)x/L providing strong evidence that, under conditions of viscous flow and drag

transport, Eq. (78) should remain valid for any value of ǫ and St.

In summary, the phenomenon of inertial capture of particles can be studied in two-

dimensional periodic as well as random porous media. For the periodic model in the

absence of gravity, there exists a finite Stokes number below which particles never get

trapped. Furthermore, our results indicate that the transition from non-trapping to

trapping with the Stokes number is of second order with a scaling exponent α ≈ 0.5.

In the presence of gravity, we show that (i) this non-trapping regime is suppressed (i.e.,

Stc = 0) and (ii) the scaling exponent changes to α ≈ 1. Finally, the behavior of the

random porous medium subject to gravity is shown to be consistent with this last picture,

and a new way to rescale the relevant variables of the filtering system is presented. As a

future work, we intend to investigate the effect on the capture efficiency of simultaneous

multiple particle release, periodic systems with multiple cells, and the possibility of non-

trapping at first contact.

8. – Saltation

8.1. The Mechanism of Saltation. – Aeolian of sand is a major factor in sand encroach-

ment, dune motion and the formation of coastal and desert landscapes. The dominating

transport mechanism is saltation as first described by Bagnold [70] which consists of

grains being ejected upwards, accelerated by the wind and finally impacting onto the

ground producing a splash of new ejected particles. Reviews are given in Refs. [71, 72].

A quantitative understanding of this process is however not achieved.


θ>

up

>>

ux(y) y

x

Fig. 23. – Setup showing the mobile wall at the top, the velocity field at different positions inthe y-direction and the trajectory of a particle stream (dashed line).

The wind loses more momentum with increasing number of airborne particles due to

Newton’s second law until a saturation is reached. The maximum number of grains a

wind of given strength can carry through a unit area per unit time defines the saturated

flux of sand qs. This quantity has been measured by many authors in wind tunnel

experiments and on the field, and numerous empirical expressions for its dependence on

the strength of the wind have been proposed [73, 74]. In previous studies theoretical forms

have also been derived using approximations for the drag in turbulent flow [75, 76, 77].

All these relations are expressed as polynomials in the wind shear velocity u∗ which

are of third order, under the assumption that the grain hopping length scales with u∗

[73, 74, 75, 76, 77, 78]. The velocity profile in a particle laden layer has also been the

object of measurement [79, 80] and modellization [81]. The complete analytical treatment

of this problem remains out of reach not only because of the turbulent character of the

wind, but also due to the underlying moving boundary conditions in the equations of

motion. More recently, a deterministic model for aeolian sand transport without height

dependency in the feedback has been proposed [82]. Despite much research in the past

[83] there remain many uncertainties about the trajectories of the particles and their

feedback with the velocity field of the wind.

Here we present the first numerical study of saltation which solves the turbulent wind

field and its feedback with the dragged particles [84]. As compared to real data, our

values have no experimental fluctuations neither in the wind field nor in the particle size.

As a consequence, we can determine all quantities with higher precision than ever before,

and therefore with a better resolution close to the critical velocity at which grains start

to be transported.

8.2. The numerical wind channel . – To get a quantitative understanding of the layer

of airborne particle transport above a granular surface, we simulate the situation inside a

two-dimensional channel with a mobile top wall as shown in Fig. 23 schematically. Here

a pressure gradient is imposed between the left and the right side. Gravity points down,

i.e., in negative y-direction. The y-dependence of the pressure drop is adjusted in such a

way as to insure a logarithmic velocity profile along the entire channel in the case without


particles, as it is expected in fully developed turbulence [85]. More precisely, this profile

follows the classical form

ux(y) =u∗

κln

(

y

y0

)

,(79)

where ux is the component of the wind velocity in the x-direction, u∗ is the shear velocity,

κ = 0.4 is the von Karman constant and y0 is the roughness length which is typically

between 10−4 and 10−2 m. The upper wall of the channel is moved with a velocity

equal to the velocity of the wind at that height in order to insure a non-slip boundary

condition.

Inside the channel we have air flowing under steady-state and homogeneous turbulent

conditions. The Reynolds-averaged Navier-Stokes equations with the standard k-ǫ model

are used to describe turbulence. The solution for the velocity and pressure fields is

obtained through discretization by means of the control volume finite-difference technique

[48].

Having produced a steady-state turbulent flow, we proceed with the simulation of

the particle transport along the channel. Assuming that drag and gravity are the only

relevant forces acting on the particles, their trajectory can be obtained by integrating

the following equation of motion:

dup

dt= FD(u− up) + g

ρp − ρ

ρp,(80)

where up is the particle velocity, g is gravity and ρp = 2650 kgm−3 is a typical value for

the density of sand particles. The term FD(u − up) represents the drag force per unit

particle mass where

FD =18µ

ρpd2p

CDRe

24,(81)

dp = 2.5 · 10−4 m is a typical particle diameter, Re ≡ ρdp |up − u| /µ is the particle

Reynolds number, and the drag coefficient CD is taken from empirical relations. Each

particle in our calculation represents in fact a stream of real grains. It is necessary to

take into account the feedback on the local fluid velocity due to the momentum transfer

to and from the particles.

8.3. Results for the saturated flux . – We see in Fig. 23 the trajectory of one particle

stream and the velocity vectors along the y-direction. Each time a particle hits the

ground it loses a fraction r of its energy and a new stream of particles is ejected at

that position with an angle θ. The parameters r = 0.84 and θ = 36◦ are chosen from

experiments [86, 87].

If u∗ is below a threshold value ut the energy loss at each impact prevails over the

energy gain during the acceleration through drag and particle transport comes to a halt.

If the particle has an initial energy this one decreases at each impact so that the jumps


0.0 0.5 1.0 1.5 2.0 2.5

0.08

0.06

0.04

0.02

0.00

y (m

)

x (m)

u <u* t

u >u* t

Fig. 24. – Typical trajectories of fixed number of particles computed for u∗ < ut (full line) andu∗ > ut (dashed line).

become lower each time until the trajectory ends on the ground as illustrated in Fig. 24.

Only for u∗ > ut steady sand motion can be achieved. The resulting flux is given by

q = mpnp ,(82)

np being the number of particle streams released. In fact, if the initial energy is below

the one of steady state the first added particle streams are strongly accelerated in the

channel and their jumping amplitude increases after each ejection until a maximum is

reached as seen in Fig. 24.

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

0 1 2 3 4 5

x (m)

y (m

)

q=10gm-

s-1 1

q=43gm-

s-1 1

q=20gm-

s-1 1

Fig. 25. – Particle trajectories for three different values of mass flow rate q = 10 gm−1 s−1

(dotted line), q = 20 gm−1 s−1 (dashed line), and q = qs = 43 gm−1 s−1 (full line), in asaturated flow which has u∗ = 0.5064 when in the absence of particles.


One can also at fixed wind strength u∗ vary the sand flux by changing the amount of

mass - or equivalently the number of particles - in a trajectory. The more particles are

injected the smaller is the final height of the trajectories. Beyond a certain number np

of particle streams, the trajectories however start to loose energy and the overall flux is

reduced. This critical value np characterizes the saturated flux qs through Eq. (82). We

see this situation in Fig. 25. When the flux is saturated the trajectory attains a perfectly

periodic motion after a short transient of several jumps. If less mass is transported the

trajectory keeps slowly increasing in height. Let us note that the fact that in Fig. 25

some trajectories don’t seem to touch the ground is only a graphic artifact due to the

discretization.

We see in Fig. 26 the plot of qs as function of the wind velocity u∗. Clearly, there

exists a critical wind velocity threshold ut below which no sand transport occurs at all.

This agrees well with experimental observations [70, 74]. Also shown in Fig. 26 is the

best fit to the numerical data using the classical expression proposed by Lettau and

Lettau [74],

qs = CLρ

gu2∗(u∗ − ut) ,(83)

where CL is an adjustable parameter. We find rather good agreement using fit parameters

of the same order as those of the original work [74] and a threshold value of ut =

0.35 ± 0.02. This is in fact, to our knowledge, the first time a numerical calculation is

able to quantitatively reproduce this empirical expression and it confirms the validity of

our simulation procedure. Other empirical relations from the literature [75, 76, 77, 78]

can also be used to fit these results. In Fig. 26 we also show that for large values of u∗

asymptotically one recovers Bagnold’s cubic dependence. Close to the critical velocity

ut interestingly we find that a parabolic expression of the form

qs = a(u∗ − ut)2(84)

fits the data better than Eq. (83), as can be seen in Fig. 26 and in particular in the inset.

In the limit u∗ ≫ ut one obtains the classical behavior of Bagnold [73], as verified by

the dash-dotted line in Fig. 26 and which is consistent with Refs. [74, 75, 76, 77, 78].

The limit u∗ ≈ ut, however, yields the quadratic relation for the flux given in Eq. (84).

Physically this is due to the fact that close to ut the laminar component is relevant.

8.4. Results for the wind profile. – Because of the feedback of particle motion on the

wind velocity field, i.e. the momentum loss of the fluid due to Newton’s second law, the

wind strength is substantially weakened when carrying a saturated flux. One can actually

define the wind strength u∗(q = qs) at the saturated mass flow condition by fitting its

profile ux(y) for large heights y to a logarithm as given by Eq. 79. Interestingly this new

weakened wind strength u∗(q = qs) depends linearly on the undisturbed wind strength

u∗(q = 0) as seen in Fig. 27 but is less than half as strong. It would be very interesting

to check this prediction experimentally in a wind tunnel.


0.5 1.0 1.5 2.0

u* (m/s)

0.001

0.01

0.1

1.0

qs (k

g m

−1s−

1)

simulation

Eq. (84)

Lettau−Lettau

Bagnold

0.01 0.1 1.0 10u*−u

t

0.001

0.01

0.1

1.0

10

qs

2.0

Fig. 26. – Logarithmic plot of the saturated flux qs as function of u∗. The dashed line is thefit using the expression proposed by Lettau and Lettau [74], qs ∝ u2

∗(u∗ − ut), with ut =0.35 ± 0.02. The full line corresponds to Eq. (84) and the dashed-dotted line to Bagnold’srelation, qs ∝ u3

∗ [73]. The results shown in the inset confirm the validity of the the power-lawrelation Eq. (84), qs ∝ (u∗ − ut)

2. The critical point is ut = 0.33 ± 0.01.

0.8

0.7

0.6

0.5

0.4

0.30.8 1 1.2 1.4 1.6 1.8 2

u(q

=q

)*

s

u (q=0)*

Fig. 27. – Plot of u∗ at the saturated mass flow condition (q = qs) against its value in a pureflow without particle q = 0.


0.0 0.5 1.0 1.5

1.0

0.8

0.6

0.4

0.2

0.0

y (m

)

y (m

)

u (0)-u (q) (m/s)x x

[ux(0)-ux

(q)]/q

0 10 20 30 40

1.0

0.8

0.6

0.4

0.2

0.0

q=0.010

q=0.043

q=0.030

q=0.020

q=0.015

Fig. 28. – Profile of the velocity difference ux(0)−ux(q) for different values of the flux q at fixedu∗ = 0.51 m/s. The flux is changed by changing the mass per trajectory. Only the right-mostcurve corresponds to the saturated flux qs. The inset shows the data collapse of these dataobtained by rescaling the velocity difference with the corresponding q.

The velocity profile of the wind within the layer of grain transport is experimentally

much more difficult to access than the sand flux. Close to the ground this profile clearly

deviates very much from the undisturbed logarithmic form of Eq. (79) because of the

momentum that the fluid must locally yield to the particles. In Fig. 28 we show the height

(y) dependent loss of velocity with respect to the logarithmic profile without particles of

Eq. (79) for different values of q.

As seen clearly in Fig. 28, for a given wind strength u∗ the loss of velocity is maximal

at the same height ymax, regardless of the value of the flux q. Except for large values

of the flux, dividing the velocity axis by q one can collapse all the profiles quite well on

top of each other as can be verified in the inset of Fig. 28. This shows that the loss of

momentum of the wind is as expected proportional to the amount of grains it carries.

The difference between the disturbed and the undisturbed profile at the saturated

flux qs(u∗) is shown in Fig. 29 for different wind strengths u∗. The shapes of the profiles

are similar to each but a good collapse cannot be achieved by rescaling of the horizontal

axis dividing by the value of qs(u∗) as seen in the inset of Fig. 29. This is due to the non-

linearity of the function qs(u∗) (see Fig. 26). The deviation of a collapse gets stronger

close to the threshold ut. The height ymax at which the loss of momentum is largest does

clearly increase with u∗.


11

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0 0.5 1 1.5 2 2.5 3

u (0)-u (q ) (m/s)x x s

[ux

(0)-ux

(qs

)]/qs

y (m

)

y (m

)

0 50 100

u*

=0.3562

u*

=0.7167

u*

=0.4377

u*

=0.3863

u*

=0.5665

u*

=0.5064

u*

=0.6208

Fig. 29. – Difference between the velocity profiles of a flow without particle (q = 0) and withmaximum particle flow mass (q = qs), for different flow conditions expressed by the value ofu∗ in the flow with q = qs. In the inset we have these differences divided by their respectivesaturated mass flow rate qs.

The height ymax at which the momentum loss is maximized depends essentially linearly

on u∗ as shown in Fig. 30. This is consistent with the observation that the saltation

jump length is proportional to u∗ [76, 77] because the height of a saltation trajectories

is proportional to its length. We get from Fig. 30:

ymax = 0.35 s · (u∗ − ut)(85)

Quantitatively the results in Fig. 30 also agree very well with the experimental data

of Ref. [88] and are consistent with analytical arguments given by Sørensen [76, 77].

Extrapolating to ymax = 0, we obtain an alternative estimate for the threshold velocity,

ut = 0.35 m/s, that is consistent with the values calculated before by fitting our data to

Eqs. (83) and (84) in Fig. 26.

If one goes to the desert or to a beach during a very windy day one realizes that the

saltation process in nature looks like a sheet of particles floating above the ground at

a certain height ys which strongly depends on the wind velocity. This height seems to

correspond to the position of the largest likelihood to find a particle as obtained from the

maximum of the density profile of particles as function of height y. Fig. 30 implies that

the profile of velocity difference of the wind has a minimum at a similar height, which

is consistent with the maximal loss of momentum. Within the error bars our results in

fact yield that ys coincides with the values of ymax in Fig. 30. It is important to note

that both heights, ymax and ys, also have the same linear dependence on u∗.


0.15

0.10

0.05

0.000.3 0.4 0.5 0.6 0.7 0.8

u*

(m/s)

y(m

)m

ax

0.35

ut

Fig. 30. – Height ymax of the maximum loss of velocity as function of u∗. The height ys of thelargest probability to find a particle coincides with ymax. The solid line corresponds to the bestlinear fit to the data with a slope equal to 0.35. By extrapolation, the intercept with the x-axisprovides an alternative estimate for the critical point, ut = 0.35 m/s, that is consistent with theother calculations.

9. – Quicksand

The image of quicksand merciless swallowing a victim has inspired the fantasy of kids

and helped writers and moviemakers to get rid of evil figures [89, 90, 91]. Is this really

possible? [92, 93] This is still disputed since till today it is not even clear what quicksand

exactly is [94, 95, 96, 97, 98].

The fluidization of a soil due to an increase in ground water pressure which in fact is

often responsible for catastrophic failures at construction sites is called by engineers the

“quick-condition” and can theoretically happen to any soil [97, 99, 100]. Is this condi-

tion, which can be reproduced on rather short time scales, equivalent to the legendary

quicksands? Another way of fluidization can be vibrations either from an engine [101] or

through an earthquake [90]. Recently Khaldoun et al. [96] have studied natural quick-

sand brought from a salt lake close to Qom in Iran. They found strong shear thinning

behavior, and claim that the presence of salt is crucial. Their samples behaved simi-

larly to artificial quicksand produced in the lab and no strong memory effects have been

reported.


Fig. 31. – Typical quicksand bed at the shore of a drying lagoon in Lencois Maranhenses(Maranhao state, North-East of Brazil). Beyond a threshold pressure pc, the crust of thequicksand breaks in a brittle way leaving a network of mode I cracks, and the material collapses.The maximum penetration depth for the human body was not greater than one meter.

9.1. Experiments on quicksand . – We investigated quicksand [102] in the Lencois

Maranhenses, a natural park in the state of Maranhao in the North-East of Brazil con-

sisting of barchanoid dunes separated by lagoons that are pushed inland by strong winds

with a velocity of 4 to 8 meters per year [103]. It is well known for its beauty as well as

the presence of quicksands in which vehicles have often been trapped and oil companies

have lost equipment. This quicksand appears at the shore of drying lagoons after the

rain season. These lagoons, placed amidst very clean sand, have no inlet or outlet and

are exclusively filled by rain water. Their bottom is covered by a soft brown or green

sheet of algae and cyanobacteria.

Precisely, we performed our investigations at Lat: 2◦28.76′ S, Long: 43◦03.53′ W.

Provided one does not exert on the surface a pressure higher than pc = 10 − 20 kPa,

it is possible to step on it and the surface will elastically deform in a very similar way

to what happens when one walks on a waterbed. These deformations visibly range over

tens of meters. If at some point the pressure pc is exceeded, the surface cracks in a brittle

way producing a network of tensile (mode I) cracks as seen in Fig. 31. Out of the cracks

pours water. The object or person rapidly sinks inside, until reaching the bottom of the

basin, which in our case could be up to one meter deep, and is then trapped within a

consolidated soil. Objects less than one meter long but lighter than water like tables

of wood are easily drawn inside and become nearly impossible to retrieve. We conclude

that, if the basin is deeper than two meters which could possibly happen, a human being

might perish.

Once the crust has broken, water and solid phase segregate. This explosion of excess


0 20 40 60 80h(cm)

0

20

40

60

80

100

120

τ(K

Pa)

field 1 (unpert.)field 2 (unpert.)field 3 (unpert.)field 1 (pert.)field 2 (pert.)field 3 (pert.)

Fig. 32. – Shear strength as a function of depth before (empty symbols) and after (filled symbols)the collapse of the quicksand. The least squares fit to data of a linear function gives τ = ahwith a = 1.2± 0.1 kPa/cm after the collapse. The shear strength of the unperturbed quicksand(before the collapse) follows an approximately constant behavior τ ≈ 5 kPa until reaching thebottom of the basin.

pore-water and repacking of sand grains has been discussed by several authors (see [104]

and refs. therein). The remaining soil shows pronounced shear thinning behavior similar

to the one reported in Ref. [96] and releases a gas when strongly agitated. The original

status of a crust with waterbed motion can not be recovered neither artificially nor

after waiting a long time. The collapse of the quicksand is irreversible. We conclude

that it is not possible to understand this quicksand by only investigating samples in

the lab. One has to study it in situ because the sampling itself does already destroy

the metastable quicksand condition. By placing light plates on the surface we could

walk on the quicksand without visually modifying it and made various measurements

before and then after the collapse. The most striking result concerns the shear strength

τ measured for three different fields with a vane rheometer [105] as shown in Fig. 32.

Before destroying the crust, τ is essentially constant up to the bottom of the basin and

then it rapidly increases. After the system collapsed and the water came out, τ linearly

increases with depth h:

τ(h) = ah ,(86)

with a = 1.2 ± 0.1 kPa/cm. We conclude from our measurements that this specific


quicksand is essentially a metastable granular suspension with depth independent static

viscosity. Once the collapse takes place, it becomes a soil dominated by the Mohr-

Coulomb friction criterion for its shear strength.

Two questions arise: What produces the impermeable crust enclosing the fluid bubble

and how is this crust formed? We investigated the material of the bottom of the lake,

forming the crust and constituting the interior of the bubble physically, chemically and

biologically. The most visible finding was the huge amount of living merismopedia,

cylindrospernopsis and other cyanobacteria as well as of diatomacea of various types

(e.g., frustulia) and other eukaryotes [106]. They constitute the largest fraction of mass

besides the silicates of the sand. Still water and tropical weather conditions provide

them an ideal environment. When the lake dries, they form the quite elastic and rather

impermeable crust which hinders further water from evaporating and which therefore just

stays below in the bubble. The cementing of soils by cyanobacteria and other algae has

in fact already been reported in previous studies [107, 108]. We can therefore conclude

that this quicksand is a living structure. We also would like to point out that we found

no salt in the water which means that the presence of salt is not a necessary condition

to get quicksand, as opposed to the finding of Ref. [96].

9.2. Simulations of quicksand crumbling. – We perform computational simulations

with a model specially built to represent the physics of an object being pushed inside

and subsequently removed from a fragile granular structure.

In our model simulation, we consider a system with width and initial height of 51 and

180 particle diameters, respectively, where periodic boundary conditions are applied in

the horizontal direction. The unperturbed quicksand is modeled as a granular network

consisting of cohesive disks put together through the contact dynamics technique [109,

110] and a ballistic deposition process driven by gravity, as shown in Fig. 33a. After the

settlement of all particles, the cohesive forces between them are tuned to the point in

which a barely stable structure of grains is assured. This accounts for the slowly drying

process of the lakes that results in a tenuous network of grains, like in a house of cards.

In our model the surrounding pore water is not explicitly considered but is taken into

account as a buoyant medium, thus reducing the effective gravity acting onto the grains.

We then proceed with the simulation by pushing a large disk of low density (half of the

grain density) at constant force into the granular structure. In Fig. 33 we show a typical

simulation of this process. As depicted, the penetration of the disk causes the partial

destruction of the porous network and the subsequent compaction of the disassembled

material. We observe the creation of a channel (Fig. 33b) which finally collapses over the

descending intruder. At the end of the penetration process (Fig. 33c), the larger disk is

finally buried under the loose debris of small particles. Since the collapse takes place in

a rather short time scale compared to the formation of the quicksand, we assume that

no new cohesive bonds are build up instantaneously and that broken cohesive bonds will

not have time to recover during penetration.

In our simulations the density of the original packing is roughly two thirds that of

the compacted material below which the intruder remains trapped. One should also


(a) (b) (c)

Fig. 33. – Snapshots from computational simulations [102] showing a typical realization of thepenetration process of a lighter intruder into a very loose cohesive packing. The (unperturbed)quicksand shown in (a) is modeled as a tenuous granular network of cohesive disks assembledby a gravity driven process of ballistic deposition and contact dynamics [109, 110]. As shown in(b), the movement of the disk is responsible for the partial destruction of the granular structurealong its trajectory. At the end of the penetration process shown in (c), the intruder rests undera much more compact mass of (perturbed) quicksand.

note that the constant force applied to the disk must exceed a certain value to allow

for penetration, otherwise the object will stay above the surface, in agreement with our

field experiments. The snapshots shown in Figs. 33a-c have been obtained from model

calculations with an applied force that is slightly above the penetration threshold. The

further increase of the force does not lead to any substantial changes in these pictures.

Our results indicate that, if we allow for the cohesive bonds in the material to be

completely restored after penetration, the force strength needed to remove the intruder

disk after the penetration process can be up to three times higher than the pushing force.

10. – Conclusion and Outlook

We have presented a large spectrum of techniques to treat particles in fluids and

given many examples where interesting results could be obtained numerically using these

techniques.


We have found [56] that although the distribution of channel widths in a porous

medium made by a two-dimensional RSA process is rather complex and exhibits a

crossover at l ∼ D, the distribution of fluxes through these channels shows an aston-

ishingly simple behavior, namely a square-root stretched exponential distribution that

scales in a simple way with the porosity. Future tasks consist in generalizing these stud-

ies to higher Reynolds numbers, other types of disorder and three dimensional models of

porous media.

We also presented results for the inertial capture of particles in two-dimensional pe-

riodic as well as andom porous media [67]. For the periodic model in the absence of

gravity, there exists a finite Stokes number below which particles never get trapped.

Furthermore, our results indicate that the transition from non-trapping to trapping with

the Stokes number is of second order with a scaling exponent α ≈ 0.5. In the presence

of gravity, we show that (i) this non-trapping regime is suppressed (i.e., Stc = 0) and

(ii) the scaling exponent changes to α ≈ 1. We intend to investigate in the future the

possibility of non-trapping at first contact and the effect on the capture efficiency of

simultaneous multiple particle release.

We finally also showed results of simulations [84] giving insight about the layer of

granular transport in a turbulent flow. The lack of experimental noise allows for a precise

study close to the critical threshold velocity ut that lead us to a parabolic dependence of

the saturated flux. The present model can be extended in many ways including the study

of the dependence of the aeolian transport layer on the grain diameter, the gas viscosity,

and the solid or fluid densities. This would allow to calculate, for instance, the granular

transport on Mars and compare with the expression presented in the literature [78].

∗ ∗ ∗

We thank CNPq, CAPES, FUNCAP and FINEP for financial support.


REFERENCES

[1] Soo S. L., Particles and Continuum: Multiphase Fluid Dynamics (Hemishphere, NewYork) 1989.

[2] Gidaspow D., Multiphase Flow and Fluidization (Academic Press, San Diego) 1994.[3] Pye K. and Tsoar H., Aeolian sand and sand dunes (Unwin Hyman, London) 1990.[4] Dullien F. A. L., Porous Media - Fluid Transport and Pore Structure (Academic, New

York) 1979.[5] Adler P. M., Porous Media: Geometry and Transport (Butterworth-Heinemann,

Stoneham MA) 1992.[6] Sahimi M., Flow and Transport in Porous Media and Fractured Rock (VCH, Boston)

1995.[7] Balzer G., Boell e. A. and Simonin O., Eulerian gassolid flow modelling of dense

fluidised bed, in proc. of ISEF Fluidisation VIII, Tours 1995, pp. 1125–1135.[8] Boelle A., Ph.D. thesis, Universite de Paris 6 (1997).[9] Ferschneider G. and Mege P., Revue de l’IFP, 51, 2 (1996) 301.

[10] Simonin O., 5th workshop on two phase flow predictions (Erlangen RFA) 1990, March19-22.

[11] Soo S., Fluid dynamics of multiphase systems (Baisdell Publishing Co.) 1967.[12] Ergun S., Chemical Engineering Progress, 48, 2 (1952) 89.[13] Hu H., Crochet M. J. and Joseph D., Theoretical and Computational Fluid Dynamics,

3 (1992) 285.[14] Glowinski R., Pan T. and Periaux J., Fictitious domain methods for the simulation of

stokes flow past a moving disk, in proc. of ECCOMAS ’96, Computational Fluid Dynamics1996, pp. 64–70.

[15] Glowinski R., Pan T., Hesla T., Joseph D. and Periaux J., J. Comput. Phys., 162(2001) 363.

[16] Hoomans B. P. B., Kuipers J. A. M., Briels W. J. and Van Swaaij W. P. M.,Chem. Engng. Sci., 51 (1996) 99.

[17] Kalthoff W., Schwarzer S., Ristow G. and Herrmann H., Int. J. Mod. Phys. C,7 (1996) 543.

[18] Schwarzer S., Phys. Rev. E, 52 (1995) 6461.[19] Tsuji Y. and Tanaka T., Int. J. Mod. Phys. B, 7 (1993) 1889.[20] Tsuji Y., Kawaguchi T. and Tanaka T., Powder Technology, 77 (1993) 79.[21] Kawaguchi T., Tanaka T. and Tsuji Y., Powder Technology, 20 (1997) .[22] Tanaka, T.and Kawaguchi T. and Tsuji Y., Int. J. Mod. Phys. B, 7 (1993) 1889.[23] Xu B. H. and Yu A. B., Chem. Engng. Sci., 52, 16 (1997) 2785.[24] Hofler K., Raumliche Simulation von Zweiphasenflussen, Master’s thesis, Universitat

Stuttgart (1997).[25] Fogelson A. and Peskin C., J.Comput. Phys., 79 (1988) 50.[26] Wachmann B., Microsimulation of two-phase system in 3d, Master’s thesis, University

of Stuttgart (1999).[27] Komiwes V., Mege P., Meimon Y. and Herrmann H. J., Granular Matter, 8 (2006)

41.[28] Cundall P. and Strack O., Geotechnique, 29 (1979) 47.[29] Ristow G., Int. J. of Mod. Phys. C, 3 (1992) 1281.[30] Herrmann H., J. Phys. A, 26 (1993) 373.[31] Form W., Ito N. and Kohring G., Int. J. of Mod. Phys. C, 4 (1993) 6.[32] Allen M. and Tildesley D., Computer Simulation of Liquids (Oxford University Press,

Oxford) 1987.


[33] Chorin A., J. of Comput. Phys., 2 (1967) 12.[34] Peyret R. and Taylor T., Computational Methods for Fluid Flow, Springer Series in

Computational Physics (Springer) 1983.[35] Vorst H. A. V. D., SIAM J. Sci. and Stat. Comput., 13, 2 (1992) .[36] Richardson J. and Zaki W., Trans. Instn. Chem. Engrs., 32 (1954) 35.[37] Nicolai H., Herzhaft B., Hinch E., Oger L. and Guazzelli E., Phys. Fluid, 7

(1995) 12.[38] Drazer G., Koplik J., Khusid B. and Acrivos A., J. Fluid Mech., 460 (2002) 307.[39] Abbas M., Climent E., Simonin O. and Maxey M., Phys. Fluids, 18 (2006) 121504.[40] Rouyer F., Martin J. and Salin D., Phys. Rev. Lett., 83 (1999) 1058.[41] Harting J., Herrmann H. J. and Ben-Nain E., Europhys. Lett., 83 (2008) 30001.[42] Ladd A., J. Fluid Mech., 271 (1994) 285.[43] Ladd A., J. Fluid Mech., 271 (1994) 311.[44] Nguyen N. Q. and Ladd A. J. C., Phys. Rev. E, 66 (2002) 046708.[45] Ladd A. J. C. and Verberg R., J. Stat. Phys., 104 (2001) 1191.[46] Nguyen N. Q. and Ladd A. J. C., Phys. Rev. E, 69 (2004) 050401.[47] Komnik A., Harting J. and Herrmann H. J., J. Stat. Mech: Theor. Exp., (2004)

P12003.[48] FLUENT (trademark of FLUENT Inc.) is a commercial package for computational fluid

dynamics.[49] Canceliere A., Chang C., Foti E., Rothman D. H. and Succi S., Phys. Fluids A,

2 (1990) 2085.[50] Kostek S., Schwartz L. M. and Johnson D. L., Phys. Rev. B, 45 (1992) 186.[51] Martys N. S., Torquato S. and P. B. D., Phys. Rev. E, 50 (1994) 403.[52] Andrade J. J. S., Street D. A., Shinohara T., Shibusa Y. and Arai Y., Phys. Rev.

E., 51 (1995) 5725.[53] Koponen A., Kataja M. and Timonen J., Phys. Rev. E, 56 (1997) 3319.[54] Rojas S. and Koplik J., Phys. Rev E, 58 (1998) 4776.[55] Andrade J. J. S., Costa U. M. S., Almeida M. P., Makse H. A. and Stanley

H. E., Phys. Rev. Lett., 82 (1999) 5249.[56] Araujo A. D., Bastos W. B., Andrade J. J. S. and Herrmann H. J., Phys. Rev. E

0511085, 74 (2006) 010401.[57] Marshall H., Sahraoui M. and Kaviany M., Phys. Fluids, 6 (1993) 507.[58] Torquato S., Random Heterogeneous Materials: Microstructure and Macroscopic

Properties (Springer, New York) 2002.[59] Voronoi G. V., J. reine angew. Math., 134 (1908) 198.[60] Watson D. F., The Computer Journal, 24 (1981) 167.[61] Tien C., Granular Filtration of Aerosols and Hydrosols (Butterworths, Boston) 1989.[62] Ghidaglia, C. adn de Arcangelis L., Hinch J. and Guazzelli E., Phys. Rev. E, 53

(1996) R3028.[63] Ghidaglia, C. adn de Arcangelis L., Hinch J. and Guazzelli E., Phys. Fluids, 8

(1996) 6.[64] Lee J. and Koplik J., Phys. Fluids, 13 (2001) 1076.[65] Koch D. L. and Hill R. J., Annu. Rev. Fluid Mech., 33 (2001) 619.[66] Rosner D. E. and Tandon P., Chemical Engineering Science, 50 (1995) 3409.[67] Araujo A. D., Andrade J. J. S. and Herrmann H. J., Phys. Rev. Lett., 97 (2006)

138001.[68] Kubawara S., J. Phys. Soc. Jpn., 14 (1959) 727.[69] Morsi S. A. and Alexander A. J., J. Fluid Mech., 55 (1972) 193.[70] Bagnold R. A., The Physics of Blown Sand and Desert Dunes (Methuen, London) 1941.


[71] Anderson R. S., Sørensen M. and Willetts B. B., Acta Mech. (Suppl.), 1 (1991) 1.[72] Herrmann H. J., The Physics of Granular Media, eds. H. Hinrichsen and D. Wolf

(Wiley VCH, Weinheim) 2004, Ch. 10, pp. 233–252.[73] Bagnold R. A., Proc. R. Soc. Lond A, 167 (1938) 282.[74] Lettau K. and Lettau H. (Eds.), Exploring the World’s Driest Climate (Center for

Climatic Research Univ. of Wisconsin, Madison) 1978.[75] Owen P. R., J. Fluid Mech., 20 (1964) 225.[76] Sørensen M., in proc. of Proc. Int. Wkshp. Physics of Blown Sand, Vol. 1 (Univ. of

Aarhus, Denmark) 1985, p. 141.[77] Sørensen M., Acta Mech. (Suppl.), 1 (1991) 67.[78] White B. R., Geophys. Res., 84 (1979) 4643.[79] Butterfield G. R., Turbulence: Perspectives on Flow and Sediment Transport, eds. N.

J. Cliffod , J. R. French and J. Hardisty (John Wiley) 1993, Ch. 13, p. 305.[80] Nishimura K. and Hunt J. C. R., J. Fluid. Mech., 417 (2000) 77.[81] Ungar J. E. and Haff P. K., Sedimentology, 34 (1987) 289.[82] Andreotti B., J. Fluid Mech., 510 (2004) 47.[83] Nalpanis, P.and Hunt J. C. R. and Barrett C. F., J. Fluid Mech., 251 (1993) 661.[84] Almeida M. P., Andrade Jr. J. S. and Herrman n. H. J., Phys. Rev. Lett., 96 (2006)

018001.[85] Prandtl L., Aerodynamic Theory Vol. III, ed. W. F. Durand (Springer, Berlin) 1935,

p. 34.[86] Anderson R. S. and Haff P. K., Science, 241 (1988) 820.[87] Rioual, F.and Valanc e. A. and Bideau D., Phys. Rev. E, 62 (2000) 2450.[88] Dong Z., Liu X., Wang H., Zhao A. and Wang X., Geomorphology, 49 (2002) 219.[89] Matthes G. H., Scientific American, 188 (1953) 97.[90] Kruszelnicki K., New Scientist, 152 (1996) 26.[91] Bahlmann L., Klaus S., Heringlake M., Baumeier W. and Schmucker P. W.

K. F., Resuscitation, 53 (2002) 101.[92] Anonymous, Science Digest, 92 (1984) 20.[93] Smith E. R., Ohio J. Sci., 46 (1946) 327.[94] Yamasaki S., Sci. Am., 288 (2003) 95.[95] Freundlich H. and Juliusburger F., T. Faraday Soc., 31 (1935) 769.[96] Khaldoun A., Eiser E., Wegdam G. and Bonn D., Nature, 437 (2005) 635.[97] Lambe T. W. and Whitman R. V., Soil Mechanics, Soil Engineering (Wiley, New York)

1969.[98] Lohse D., Rauhe R. and Bergmann R. v. d. M. D., Nature, 432 (2004) 689.[99] El Shamy U. and Zeghal M., J. Engrg. Mech., 131 (2005) 413.

[100] Craig R. F., Soil Mechanics (E & FN Spon, New York) 1997.[101] Huerta D. A., Sosa V., Vargas M. C. and Ruiz-Suarez J. C., Phys. Rev. E, 72

(2005) 031307.[102] Kadau, D.and Herrmann H. and Andrade Jr. J. S., Europ. Phys. J. E, 30 (2009)

275.[103] Schwammle V. and Herrmann H. J., Nature, 426 (2003) 619.[104] Parker W. R., Nature, 210 (1966) 1247.[105] Clayton C. R. I., Matthews M. C. and Simons N. E., Site Investigation (Blackwell

Science, Oxford) 1995.[106] Kadau D., Herrmann H. J., Andrade Jr. J. S., Araujo A., Bezerra L. and Maia

L., Granular Matter, 11 (2009) 67.[107] Danin A., J. Arid Env., 167 (1978) 409.[108] Danin A., J. Arid Env., 21, 2 (1991) 193.


[109] Jean M. and Moreau J. J., Unilaterality and dry friction in the dynamics of rigidbody collections, Contact Mechanics International Symposium (Presses Polytechniques etUniversitaires Romandes, Lausanne) 1992, pp. 31–48.

[110] Moreau J. J., Eur. J. Mech. A-Solid, 13 (1994) 93.

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