# Particles - Tel

Mechanics

Feb 22, 2014 (7 years and 8 months ago)

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Soutenance
de thèse

Transport,

dépôt et relargage

de particules inertielles dans
une fracture à

rugosité périodique

T.
Nizkaya

Directeur
de thèse:

M.
Buès

Co
-
directeur
de thèse:

J.
-
R.
Angilella
,

LAEGO, Université de Lorraine

Ecole doctorale
RP2E

1er
Octobre
2012

Nancy, Lorraine

Laboratoire Environnement, Géomécanique & Ouvrages

1

Particle
-

Photo: NASA's
Goddard Space Flight
Center

Particles
: air and water
pollutants, dust, sprays
and aerosols, etc…

2

Particle
-

Hydrogeology:

Flows through
fractures

often carry particles
(sediments, organic debris etc.).

3

How to model particle
-

Two models of particles

Tracer particles:

point particles

by the fluid

(+
brownian

motion)

Example: dye in water

Inertial particles:

finite size, density

different from fluid.

Example: sand in the air

4

Two

models

of
particles

Tracer
particles
:

point
particles

by the
fluid

(+
brownian

motion
)

Inertial

particles
:

finite

size,
density

different

from

fluid
.

Example
:
sand

in the air

5

-
diffusion

equations for particle

concentration.

Two

models

of
particles

Tracer
particles
:

point
particles

by the
fluid

(+
brownian

motion
)

-
diffusion

equations

for
particle

concentration.

Inertial

particles
:

finite

size,
density

different

from

fluid
.

Particle

inertia

is

important.

Even

weakly
-
inertial

particles

are

6

very

different

from

tracers
!

Clustering

of
inertial

particles

Inertial

particles

tend to cluster in certain zones
of the flow.

Particles

in fractures:
clustering

can

of
particles

across

the fracture?

rain

initiation

Wilkinson &
Mehlig

(2006)

planet

formation

Barge &
Sommeria

(1995)

aerosol

engineering

Fernandez de la Mora (1996)

7

Clustering

of
inertial

particles

Inertial

particles

tend to cluster in certain zones
of the flow.

In

periodic

flows

particle

focus to a single
trajectory
:

Robinson (1955),
Maxey&Corrsin

(1986), etc.

rain

initiation

Wilkinson &
Mehlig

(2006)

planet

formation

Barge &
Sommeria

(1995)

aerosol

engineering

Fernandez de la Mora (1996)

8

Theoretical

study

of
focusing

effect

on
particle

transport in a fracture
with

periodic

corrugations.

Water

+

particles

Goal of the
thesis

«
focusing
»

9

0

h
omogeneos

distribution

I.
Single
-
phase
flow in
a
model
fracture

II.
Focusing of inertial particles in the fracture

III.

Influence of lift force on particle focusing

IV.
Conclusion and perspectives

Outline

of the talk

10

11

I. Single
-
phase flow in a
thin

fracture.

I. Single
-
phase flow in a
thin

fracture.

12

Goal:

Obtain an explicit fluid velocity field for
arbitrary fracture

shapes

Method:

Asymptotic expansions

Simplified

model of a fracture

Model fracture: a
thin

2D
channel

with

«slow» corrugation.

Typical

corrugation
length

L
0
>>
typical

aperture H
0
.

𝜺
=
𝑯

𝑳



Small
parameter
:

𝑳

𝑯

=
𝚽

(
𝑿
)

𝐙
=
𝚽

(
𝑿
)

X

Z

13

Single
-
phase flow in fracture

Single
-
phase flow in fracture:

2D, incompressible,
stationary

𝝆
,
𝝂

𝒖
(
𝒙
,
𝒛
)

𝒖
=

𝒖
=

Streamfunction
:

𝑿
,

=
𝛁
×
𝚿

𝐳
=
𝝓

(
𝒙
)

𝐳
=
𝝓

(
𝒙
)

𝐳

𝐱

𝒙
=
𝑿
𝑳

,

𝒛
=

𝑯

;

𝒖
𝒙
=

𝒙

,

𝒖
𝒛
=
𝜺

𝒛

;

𝝍
=
𝜺
𝚿

;

Non
-
dimensional

variables:


𝑯
=

𝑯

𝝂
=

(

)

Reynolds
number
:

Navier
-
Stokes
equations
:

14

𝜺

𝑯



Equations of inertial lubrication theory

Navier
-
Stokes equations in non
-
dimensional variables:

Boundary conditions:

Hasegawa

and
Izuchi

(1983)

Borisov
(
1982), etc.

No slip at

the walls

15

Navier
-
Stokes
equations

in non
-
dimensional

variables

Boundary

conditions:

Small
parameter

ε

perturbative

method

No slip

at

t
he
walls

16

Equations of
inertial

lubrication

theory

𝜺

𝑯



𝜺



Generalization

of
previous

works

Hasegawa

and
Izuchi

(1983)

Borisov (1982)

Crosnier

(2002)

Present

thesis
:

full
parametrization

of the fracture
geometry
.

}

17

The cross
-
channel

variable:
𝜼

Cross
-
channel variable
:

𝜼
=


𝜼
=



𝜼
=

𝜼

𝒙

h(x)

h(x)

𝐳
=
𝝓
(
𝒙
)

𝒛

𝒙

𝒛
=
𝝓

(
𝒙
)

𝒛
=
𝝓

(
𝒙
)

𝜂
=
𝑧

𝜙
(
𝑥
)

(
𝑥
)

(
𝒙
,
𝒛
)

(
𝒙
,
𝜼
)

h
alf
-
aperture of the channel

middle
-
line profile

18

𝜙
(
𝑥
)
=
𝜙
1
𝑥
+
𝜙
2
(
𝑥
)
2

(
𝑥
)
=
𝜙
2
𝑥

𝜙
1
(
𝑥
)
2

Asymptotic

solution of 2
nd

order

0
th
:

1
st
:

2
nd
:

19

Asymptotic

solution of 2
nd

order

0
th
:

1
st
:

2
nd
:

3
rd
… etc.

v
iscous

correction

inertial

corrections

«local
cubic

law
»

20

Numerical

verification
:
mirror
-
symmetric

---

LCL flow,
2
nd

order

asymptotic
s
,

numerical

simulation

𝜺
=

.




21

Numerical

verification
: flat top
wall

𝜺
=

.




22

---

LCL flow,
2
nd

order

asymptotic
s
,

numerical

simulation

Application: corrections
to
Darcy’s

law







𝑳

Q

𝚫

=






𝛥



Darcy’s

law

𝛥

Inertial

corrections:

analytical

expression?

𝛥

Small flow rates

Larger

flow rates





-

curve

Flow rate
depends

on pressure drop:

23

Corrections to
Darcy’s

law

Pressure drop (
from

2
nd

order

asymptotic

solution):

2
2
3
2
1
3
1
12

Q
K
K
H
Q
L
P
h
24

No

term
!

I
n
accordance
with

Lo
Jacono

et al.

(
2005) and
many

others
.

Corrections to
Darcy’s

law

Pressure drop (
from

2
nd

order

asymptotic

solution):

2
2
3
2
1
3
1
12

Q
K
K
H
Q
L
P
h

(
𝒙
)


(
𝒙
)

𝒛
=
𝝓
(
𝒙
)

𝒙

𝒛

Geometrical

factors
:

25

Corrections to
Darcy’s

law

Pressure drop (
from

2
nd

order

asymptotic

solution):

2
2
3
2
1
3
1
12

Q
K
K
H
Q
L
P
h

(
𝒙
)


(
𝒙
)

𝒛
=
𝝓
(
𝒙
)

𝒙

𝒛

Geometrical

factors
:

Slope

of the
linear

law

depends

on

both

aperture and
shape

of the middle line.

26

Corrections to
Darcy’s

law

Pressure drop (
from

2
nd

order

asymptotic

solution):

2
2
3
2
1
3
1
12

Q
K
K
H
Q
L
P
h
Geometrical

factors
:

Cubic

correction
only

depends

on aperture variation.

27


(
𝒙
)


(
𝒙
)

𝒛
=
𝝓
(
𝒙
)

𝒙

𝒛

Numerical

verification

Darcy’s

law

o
ur

asymptotic

solution

n
umerics

(
mirror
-
symmetric

channel
)

n
umerics

(
channel

with

flat top
wall
)

Pressure drop

v
s

Reynolds
number

28

II. Transport of
particles

in

the
periodic

fracture

29

Periodic

channel

Particles
:
small
, non
-
brownian
, non
-
interacting
,
passive.

𝑳

𝑯

c
orrugation
period

Flow
:
asymptotic

solution (

order
)

«
focusing
»

30

Particle

motion
equations


(
𝑿
𝒑
)




𝒑

1
Re


s
p
aV

𝑠
=

𝑝

𝑓

Particle

dynamics
:

from

Stokes
equations

around

the
particle

Maxey
-
Riley
equations

g
m
F
dt
V
d
m
p
H
p
p




𝒔

Maxey

and Riley (1983)

Gatignol

(1983)

31

Maxey
-
Riley
equations
:

6
1

10
2

6
6

)
(
0
2
2
2
ds
U
a
V
U
ds
d
s
t
U
a
U
Dt
D
dt
V
d
m
U
a
V
U
a
g
m
m
Dt
U
D
m
dt
V
d
m
t
f
p
f
f
f
p
f
f
p
f
f
p
f
f
p
p

d
rag force

f
luid

gravity

mass

Basset’s

m
emory
term

Particle

motion
equations

32

Typical

long
-
time
behaviors

(
numerics

-

LCL flow, no
gravity
)

Heavy

particles

Light

particles

Heavy
particles

can

focus

to a single
trajectory

(or not!)

depending

on
channel

geometry
.

Q

Q

Q

33



Focusing

persists

in
presence

of
gravity
,

if the flow rate
Q

is

high
enough

(permanent suspension)



Low

Q

High Q

Typical

long
-
time
behaviors

(
numerics

-

LCL flow,
with

gravity
)

34

Heavy

particles

Light

particles

Goal:

Find

conditions for
particle

focusing

d
epending

on
channel

geometry

and flow rate
.

Method
:

Poincaré
map

+

a
symptotic

motion
equations

for

weakly
-
inertial

particles

35

Simplified

Maxey
-
Riley
equations

f
p
f

2
2
R
Density

contrast
:

Particle

response

time:

2
0
9
Re
2

H
a
R
H

36

Simplified

Maxey
-
Riley
equations

For
weakly
-
inertial

particles
:

2
/
3
1
)
(
)
(

O
x
v
x
u
x
p
p
f
p

particle

inertia

+
weight

f
luid

velocity

f
p
f

2
2
R
Density

contrast
:

Particle

response

time:

2
0
9
Re
2

H
a
R
H

37

1


f
rom

Maxey
-
Riley
equations

Maxey

(1987)

Poincaré
map

for
weakly
-
inertial

particles

𝜼
𝒌

𝜼
𝒌
+


𝜼


𝜼

𝜂

= rescaled cross
-
channel variable z

𝜂
𝑘
=
𝜂
(
𝑡
𝑘
)

a
fter

k
periods

𝜂
𝑘
+
1
=
𝜂
𝑘
+
𝑓
(
𝜂
𝑘
)

f
rom

simplified

Maxey
-
Riley

equations

38

1


Poincaré
map

for
weakly
-
inertial

particles

𝜼
𝒌

𝜼
𝒌
+


𝜼


𝜼

𝜂
𝑘
+
1
=
𝜂
𝑘
+
𝑓
(
𝜂
𝑘
)

Stable
fixed

point:

𝑓
𝜂

=
0
;

1
<
𝑓′
𝜂

<
0
.

Focusing
!

39

Poincaré
map
:

Particles converge

to the
streamline

𝜂
𝑥
,
𝑧
=
𝜂

z
h
h
G
J
J
R
f
2
2
1
8
9
)
(
'
~
1
2
3
)
(

Analytical

expression for the Poincaré
map

Poincaré
map

for the LCL flow
:

Gravity

number

Channel
geometry

Fluid
/
particle

d
ensity

ratio

1
2
3
1
2
3

R
R
l
ighter

than

fluid

h
eavier

than

fluid

2
1
2
2
'
'
h
h
h
J

2
'
h
h
h
J

Fr
U
L
g
G
z
z
1
2
0
0

40

z
h
h
G
J
J
R
f
2
2
1
8
9
)
(
'
~
1
2
3
)
(

Analytical

expression for the Poincaré
map

Poincaré
map

for the LCL flow
:

Gravity

number

Channel
geometry

Fluid
/
particle

d
ensity

ratio

1
2
3
1
2
3

R
R
l
ighter

than

fluid

h
eavier

than

fluid

2
1
2
2
'
'
h
h
h
J

2
'
h
h
h
J

Fr
U
L
g
G
z
z
1
2
0
0

41

𝑓
𝜂

=
0
;

1
<
𝑓′
𝜂

<
0
.

Attractor

p
osition
𝜂

Focusing
/
sedimentation

diagram

)
(

cr
z
z

Rescaled

gravity
:

Corrugation

asymmetry

factor:

2
2
1
2
2
'
'
'
h
h
h
h

(
analytical

expression)

h
h
J
J
/

2
'
9
8
h
Z
z
h
G

42

Focusing
/
sedimentation

diagram

)
(

cr
z
Case A:

=
0
,


𝑧
=

0
.
217

A

h
h
J
J
/

43

Heavy
particles

Light
particles

Focusing
/
sedimentation

diagram

)
(

cr
z
z

Case B:

=
0
,


𝑧
=

0
.
5

B

h
h
J
J
/

44

Focusing
/
sedimentation

diagram

)
(

cr
z
z

Case C:

=

3
,


𝑧
=
0

C

h
h
J
J
/

45

Focusing
/
sedimentation

diagram

)
(

cr
z
z

Case D:

=

3
,


𝑧
=

0
.
5

D

h
h
J
J
/

46

𝜺

Percentage

of
deposited

particles

Maximal
deposition

length

Focusing

rate

47

Using

the Poincaré
map

we

can

calculate
:

Other

applications of Poincaré
map

Verified

numerically

Ok

Influence of channel geometry

on transport properties

48


(
𝒙
)


(
𝒙
)

𝒛
=
𝝓
(
𝒙
)

𝒙

𝒛



𝒛
=
𝝓


(
𝒙
)

2
1
2
2
'
'
h
h
h
J

2
'
h
h
h
J

2
'
h
J

2
3
1
h
h

Shape
factors

of the
channel

Shape
factors
:

49

𝒛
=
𝝓


(
𝒙
)

«apparent»

aperture

a
perture

variation

m
iddle line

corrugation

difference

between

wall

corrugations

l
h
x
h
dx
x
a
l
a
0
3
2
2
)
(
)
(
1
Aperture
-
weighted

norm
:

Pressure drop
curve
:

Single phase flow:
geometry

influence

2
2
3
2
1
3
1
12

Q
K
K
H
Q
L
P
h
3
3
0
3

h
H
H
h
Slope

of the
linear

law
:

Inertial

correction:

50

2
1
2
2
'
'
h
h
h
J

2
'
h
h
h
J

2
'
h
J

2
3
1
h
h

Shape

factors
:

Weak

dependence

on
channel

shape
!

Particle

Poincaré
map
:

Particle

transport:
geometry

influence

51

)
(
'
~
)
(
1
2
/
3
)
(



P
R
f

z
h
h
G
J
J
P

2
2
1
8
9
)
(

Particle

behavior

depends

strongly

on the
difference

in
wall

corrugations!

2
1
2
2
'
'
h
h
h
J

2
'
h
h
h
J

2
'
h
J

2
3
1
h
h

Shape

factors
:

Particle

transport:
geometry

influence

52

Example
:
channel

with

flat top
wall

and
mirror
-
symmetric

channel
.

Equivalent for single phase flow

but
different

for
particles
.

mirror
h
flat
h
J
J

mirror
h
flat
h
J
J

0

mirror
h
J

flat
h
flat
h
J
J

IV. The
effect

of lift force on

particle

focusing

53

2

)
(
6

)
(

Dt
U
D
dt
V
d
m
V
U
a
g
m
m
Dt
U
D
m
dt
V
d
m
f
p
f
p
f
f
p
f
f
p
p

Particle

motion
equations
:

Motion
equations

with

lift

+ Lift force

)
)
((
)
(
46
.
6
0
2
/
1
2
/
1
2

p
f
f
f
L
V
U
U
a
F

(
Saffman
, 1956)

«
Generalization
» of
Saffman’s

lift:

L
F

P
V

)
(
X
U
f

Lift
appears

when

particle

lags

the
fluid
.

54

Lift in simple

shear

flow

No formula for lift in a
general

flow…

Lift force
induced

by
gravity

55

Heavy
particles

pushed

to the
walls

Light
particles

lag

pushed

to the
center

Effect

opposite to
focusing
!

Gravity

in the direction of the flow (vertical
channel
):

Poincaré
map

with

lift
calculated

analytically

Lift force
induced

by
gravity

56

Effect

opposite to
focusing
!

Two

attracting

streamlines

(
theory
)

Poincaré
map

with

lift shows
splitting

of the
attractor
.

G

Lift
-
induced

chaos
at

finite

response

times

lag

Lift force
induced

by
particle

inertia

Particles

or

lag

because

of
their

proper

inertia
.
The direction of lift changes
many

times.

57

No
gravity

L
F

1

k

L
F

Lift
-
induced

chaos
at

finite

response

times

Lift force
induced

by
particle

inertia

Effect

on
focusing
?

Poincaré
map

does

not
work

here

Particles

or

lag

because

of
their

proper

inertia
.
The direction of lift changes
many

times.

58

No
gravity

Lift
-
induced

chaos
at

finite

response

times

lag

1

k

L
F

L
F

Lift
-
induced

chaos
at

finite

response

times

k

(
response

time)

Chaos!

Period

doubling

Feigenbaum

constants:

59

IV. Conclusion

60

A
new

asymptotic

solution of Navier
-
Stokes
equations

is

obtained

for
thin

channels
.

This solution
generalizes

previous

results

to
arbitrary

wall

shapes
.

Inertial

corrections to
Darcy’s

law

are
calculated

analytically

as
functions

of
channel

geometrical

parameters
.

Conclusions
: single
-
phase flow

61

Particles

transported

in a
periodic

channel

can

focus to
an
attracting

streamline

which

depends

on
channel

geometry
.

This
attractor

persists

in
presence

of
gravity
, if the flow
rate
is

high
enough
.

The
full
focusing
/
sedimentation

diagram

for
particles

in
periodic

channels

has been
obtained

analytically
,
using

Poincaré
map

technique.

Conclusions:
particle

transport

62

Lift has been
taken

into

account

in
form

of a
classical

generalization

of
Saffman

(1965).

In
presence

of
gravity

(vertical
channel
), lift causes
attractor

splitting
:
two

attracting

streamlines

are visible.

In the absence of
gravity
, lift causes a
period
-
doubling

to
chaotic

particle

dynamics
.

Conclusions:
lift
effect

63

Particles

in a non
-
periodic

(
disordered
) fracture

Do
particles

still

cluster? How to
quantify

the
clustering
?

Collisions

Does

focusing

increases

collision rates?

Brownian

particles

with

inertia

Maxey
-
Riley
equations

with

noise?

Experimental

verification

Experimental

setup
is

under

construction
at

LAEGO

Perspectives

64

Thank

you

for attention!

Particles

in a non
-
periodic

(
disordered
) fracture

Do
particles

still

cluster? How to
quantify

the
clustering
?

Collisions

Does

focusing

increases

collision rates?

Brownian

particles

with

inertia

Maxey
-
Riley
equations

with

noise?

Experimental

verification

Experimental

setup
is

under

construction
at

LAEGO.

Perspectives

65