Cem

ef

Octo

ber,

11th

2018,G

DR

TherM

atH

T,Lyon

Tw

o-p

ha

se

mo

de

lin

gu

sin

gth

ep

ha

se

fie

ldth

eo

ry

Fra

nck

Pig

eonneau

and

Pie

rre

Sara

mit

o(L

JK

,G

renoble

)

1/3

3

1.

Scie

ntific

issu

es

intw

o-p

ha

se

flow

s

2.

Ba

sic

so

fth

ep

ha

se

fie

ldth

eo

ry

3.

Nu

me

rica

lre

su

lts

of

two

-ph

ase

flow

s

3.1

Dro

ple

tsh

rin

ka

ge

3.2

Ca

pill

ary

risin

g

3.3

Dro

psp

rea

din

go

na

nh

ori

zo

nta

lw

all

4.

Co

nclu

sio

na

nd

pe

rsp

ective

s

2/3

3

1.

Sc

ien

tifi

cis

su

es

intw

o-p

ha

se

flo

ws

x

z

h(x)

Uθ

liquide

air

Fig

ure

1:

Flu

iddynam

ics

clo

se

toa

conta

ctlin

e.

T

he

velo

city

gra

die

nt

is:

∂u

∂z≈

U

h(x)=

U θx(1

)

T

he

vis

co

us

dis

sip

atio

nis

giv

en

by

1:

Φη=

η

∫R

ǫ

(∂

u

∂z

)2

hdx=

η

∫R

ǫ

(U h

)2

hdx=

ηU

2 θln

(R ǫ

).

(2)

1P.

G.D

eG

ennes:

Wettin

g:

Sta

tics

and

dynam

ics,

in:

Rev.

Mod.

Phys.

57.3

(1985),

pp.827–863.

3/3

3

1.

Sc

ien

tifi

cis

su

es

intw

o-p

ha

se

flo

ws

To

rem

ove

the

sin

gu

lari

ty:

Fig

ure

2:

Pre

curs

or

film

2.

2P.-

G.D

eG

ennes/F

.B

rochard

-Wyart

/D.Q

uéré

:G

outtes,bulle

s,perl

es

et

ondes,

Pari

s2005.

4/3

3

1.

Sc

ien

tifi

cis

su

es

intw

o-p

ha

se

flo

ws

To

rem

ove

the

sin

gu

lari

ty:

u

ls

Fig

ure

3:S

lippage

offluid

son

wall3

.

3L.

M.H

ockin

g:

Am

ovin

gfluid

inte

rface.

Part

2.

The

rem

ova

lofth

efo

rce

sin

gula

rity

by

aslip

flow

,in

:J.

Flu

idM

ech.

79.0

2(1

977),

pp.209–229.

5/3

3

1.

Sc

ien

tifi

cis

su

es

intw

o-p

ha

se

flo

ws

To

rem

ove

the

sin

gu

lari

ty:

Fig

ure

4:C

onta

ctlin

eofdiffu

se

inte

rface

4.

4P.S

eppecher:

Movin

gconta

ctlin

es

inth

eC

ahn-H

illia

rdth

eory

,in

:In

t.J.

Engng

Sci.

34.9

(1996),

pp.977–992.

6/3

3

2.

Ba

sic

so

fth

ep

ha

se

fie

ldth

eo

ryI

E

ach

ph

ase

ism

ark

ed

by

a“p

ha

se

fie

ld”

or

“o

rde

r

pa

ram

ete

r”:ϕ

.

ϕ=

1in

ph

ase

1(ρ

1,η 1

)a

ndϕ=

−1

inp

ha

se

2(ρ

2,η 2

).

Fig

ure

5:S

hear

flow

with

two

phases.

ρ=

ρ1+

ρ2

2+

ρ1−ρ

2

2ϕ.

(3)

T

he

fre

ee

ne

rgy

isw

ritt

en

as

follo

ws

5

7/3

3

2.

Ba

sic

so

fth

ep

ha

se

fie

ldth

eo

ryII

J.D

.va

nder

Waals

(1837-1

923).

F[ϕ]=

∫ Ω

[ Ψ(ϕ

)+

k 2||∇

ϕ||2

] dV.

(4)

Ψ(ϕ

)is

ad

ou

ble

-we

llp

ote

ntia

l.

-2-1.5

-1-0.5

00.5

11.5

20

0.25

0.5

0.751

1.25

1.5

1.752

2.25

φ

ψ

Fig

ure

6:E

xam

ple

ofa

double

-well

pote

ntial:

Ψ=

k(1

−ϕ

2)2/(4ζ

2).

5J.

D.va

nder

Waals

:T

he

therm

odynam

icth

eory

ofcapill

ari

tyunder

the

hypoth

esis

ofa

continuous

vari

ation

ofdensity,

in:

Verh

andel.

Konin

k.

Akad.

Wete

n.

1(1

893),

pp.1–56.

8/3

3

2.

Ba

sic

so

fth

ep

ha

se

fie

ldth

eo

ry

A

te

qu

ilib

riu

m,F[ϕ]

ha

sto

be

min

ima

l.

δF[ϕ]=

∫ Ω

(dΨ

dϕ

−k∇

2ϕ

)δϕ

dV

+

∫ δΩ

∂ϕ

∂nδϕ

dS.

(5)

C

on

se

qu

en

tly:

µ(ϕ

)=

dΨ

dϕ

−k∇

2ϕ=

0,∀x

∈Ω,

(6)

k∂ϕ

∂n

=0,

su

rδΩ

.(7

)

µ(ϕ

)is

the

ch

em

ica

lp

ote

nti

al.

9/3

3

2.

Ba

sic

so

fth

ep

ha

se

fie

ldth

eo

ry

In

1-d

ime

nsio

na

nd

with

the

pre

vio

us

do

uble

-we

llp

ote

ntia

l,

the

ph

ase

fie

ldis

giv

en

by:

ϕ(x)=

tan

h

(x

√2

Cn

),

(8)

x=

x L,

(9)

Cn=

ζ L,

Ca

hn

nu

mb

er.

(10

)

T

he

su

rfa

ce

ten

sio

nis

the

nd

efin

ed

by

σ=

k L

∫∞ −∞

(dϕ

dx

)2

dx=

2√

2k

3ζ

.(1

1)

10

/33

2.

Ba

sic

so

fth

ep

ha

se

fie

ldth

eo

ry

O

uts

ide

the

eq

uili

bri

um

,C

ah

na

nd

Hill

iard

6p

rop

ose

d

∂ϕ ∂t=

−∇

·J,

(12

)

J=

−M∇µ,

(13

)

µ(ϕ

)=

dΨ

dϕ

−k∇

2ϕ.

(14

)

T

ime

be

havio

ro

fth

ep

ha

se

fie

ldd

ue

toth

ed

iffu

sio

no

fth

e

ch

em

ica

lp

ote

ntia

l.

In

vestig

atin

gsp

ino

da

ld

eco

mp

ositio

n.

6J.

W.C

ahn/J

.E

.H

illia

rd:

Fre

eE

nerg

yofa

Nonunifo

rmS

yste

m.

I.

Inte

rfacia

lF

ree

Energ

y,

in:

J.C

hem

.P

hys.

28.2

(1958),

pp.258–267.

11

/33

2.

Ba

sic

so

fth

ep

ha

se

fie

ldth

eo

ry

Nu

me

rica

lsim

ula

tio

no

fa

sp

ino

da

ld

eco

mp

ositio

nin

2D

,

Cn=

10−

2.

12

/33

2.

Ba

sic

so

fth

ep

ha

se

fie

ldth

eo

ryI

!!!!!!!!!!!!!!

!!!!!!!!!!!!!!

Fluid

1

Fluid

2

Wall

θs

Fig

ure

7:C

onta

ctlin

ebetw

een

two

fluid

sand

aw

all.

The

sta

tic

conta

ctangle

isθ s

.

F[ϕ]=

∫ Ω

[ Ψ(ϕ

)+

k 2||∇

ϕ||2

] dV

+

∫ ∂Ω

w

f w(ϕ

)dS.

(15

)

13

/33

2.

Ba

sic

so

fth

ep

ha

se

fie

ldth

eo

ryII

T

he

min

imiz

atio

no

fF[ϕ]7

µ(ϕ

)=

dΨ

dϕ

−k∇

2ϕ=

0,∀x

∈Ω,

(16

)

L(ϕ

)=

k∂ϕ

∂n+

df w dϕ

=0,

on∂Ω

w.

(17

)

J.W

.C

ahn

(1928-2

016).

f w(ϕ

)=

−σ

co

sθ s

ϕ(3

−ϕ

2)

4,(

18

)

k∂ϕ

∂n

=3(1

−ϕ

2)σ

4co

sθ s,

on∂Ω

w.(

19

)

7J.

W.C

ahn:

Cri

ticalpoin

tw

ettin

g,

in:

J.C

hem

.P

hys.

66.8

(1977),

pp.3667–3672.

14

/33

2.

Ba

sic

so

fth

ep

ha

se

fie

ldth

eo

ry

O

uts

ide

the

eq

uili

bri

um

,cre

atio

no

ffo

rce

pro

po

rtio

na

lto

µ∇ϕ

.

S

toke

se

qu

atio

ns:

∇·u

=0,

(20

)

−∇

P+∇

·[2η(ϕ

)D(u

)]+

ρ(ϕ

)g+

µ∇ϕ=

0,

(21

)

C

ah

n-H

illia

rde

qu

atio

n:

∂ϕ ∂t+∇ϕ·u

=∇

·[M(ϕ

)∇µ(ϕ

)],

(22

)

µ(ϕ

)=

λ ζ2

[ ϕ(ϕ

2−

1)−

ζ2∇

2ϕ] .

(23

)

15

/33

2.

Ba

sic

so

fth

ep

ha

se

fie

ldth

eo

ry

U

nd

er

dim

en

sio

nle

ss

form

:

∇·u

=0,(

24

)

−∇

P+

∇·[

2η(ϕ

)D(u

)]+

Bo

Caρ(ϕ

)g+

3

2√

2C

aC

nµ∇ϕ=

0,(

25

)

∂ϕ ∂t+

∇ϕ·u

=1 Pe∇

2µ(ϕ

),(2

6)

µ(ϕ

)=

ϕ(ϕ

2−

1)−

Cn

2∇

2ϕ,(

27

)

D

ime

nsio

nle

ss

nu

mb

ers

:

Bo=

ρ1g

L2

σ,

(28

)C

a=

η 1U σ,

(29

)P

e=

Uζ

2L

Mλ,

(30

)

Cn=

ζ L,

(31

)ρ=

ρ2

ρ1,

(32

)η=

η 2 η 1.

(33

)

16

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.1

Dro

ple

tshri

nkage

S

tud

yth

ise

ffe

ct

of

Ca

hn

nu

mb

er

on

dro

ple

tsh

rin

ka

ge

for

flu

ids

at

rest.

Fig

ure

8:A

sta

tic

dro

ple

tin

aliq

uid

atre

st.

17

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.1

Dro

ple

tshri

nkage

Fig

ure

9:a/a

0vs.

tfo

rth

ree

Cahn

num

bers

.

18

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.1

Dro

ple

tshri

nkage

S

ma

llva

lue

of

Ca

hn

nu

mb

er

pre

ven

tsth

esh

rin

ka

ge.

A

cco

rdin

gto

Yu

ee

ta

l.8,

the

cri

tica

lra

diu

sb

elo

ww

hic

hth

e

sh

rin

ka

ge

occu

rsis

r c=

4√2

1/6

3π

VC

n,

(34

)

r c

=0.7

for

Cn=

10−

1,

r c

=0.6

for

Cn=

5·1

0−

2,

r c

=0.4

for

Cn=

10−

2.

8P.Y

ue/C

.Z

hou/J

.J.

Feng:

Sponta

neous

shri

nkage

ofdro

ps

and

mass

conserv

ation

inphase-fi

eld

sim

ula

tions,

in:

J.C

om

put.

Phys.

223.1

(2007),

pp.1–9

.1

9/3

3

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.2

Capill

ary

risin

g

r

z

00.5

−0.5

1∂ΩN,top

∂ΩN,bottom∂ΩD

Fig

ure

10:

Geom

etr

yofth

e

circula

rtu

be.

T

he

dia

me

ter

of

the

tub

eis

use

da

s

ch

ara

cte

ristic

len

gth

.

Uch

ose

nby

ba

lan

cin

ggra

vity∼

vis

co

us

forc

es⇒

U=

ρg

D2/σ

.

u=

0,∂ϕ

∂n

=(1

−ϕ

2)√

2co

sθ s

2C

n,∂µ

∂n

=0,∀x

∈∂Ω

D,

(35

)

σ·n

=0,∂ϕ

∂n

=∂µ

∂n

=0,∀x

∈∂Ω

N,

top,

(36

)

σ·n

=−(ρ

+1 2)n

,∂ϕ

∂n

=∂µ

∂n

=0,∀x

∈∂Ω

N,

bo

tto

m.

(37

)

20

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.2

Capill

ary

risin

g

A

nu

me

rica

lexa

mp

leh

as

be

en

do

ne

with

:

θ s=

80 ,

Bo=

1,

Cn=

10−

2,

Pe=

10

2.

(38

)

In

itia

lly,

the

he

avy

flu

id(1

)is

be

low

z=

0:

ϕ0(z,r)=

−ta

nh

(z

√2

Cn

).

(39

)

21

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.2

Capill

ary

risin

g

Ca

pill

ary

risin

go

fw

ate

rin

atu

be

with

asta

tic

co

nta

ct

an

gle

eq

ua

lto

4π/9

.

22

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.2

Capill

ary

risin

g

Fig

ure

11:C

onta

ctlin

epositio

nas

afu

nction

oftim

efo

rθ s

=80

and

Bo=

1.

23

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.2

Capill

ary

risin

g

Fig

ure

12:G

eom

etr

yofth

efr

ee

surf

ace,z

vs.

r,fo

rθ s

=80

and

Bo=

1.

24

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.2

Capill

ary

risin

g

Fig

ure

13:z

-axis

velo

city

com

ponentv

vs.

r,ove

ran

hori

zonta

llin

e

localiz

ed

righton

the

conta

ctlin

e.

25

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.2

Capill

ary

risin

g

Fig

ure

14:v/

max(v)

vs.

x∼

4√P

eri

ghton

the

conta

ctlin

efo

rt=

2.5

for

Bo=

1and

Pe=

50,10

2and

10

3.

T

he

diffu

sio

nla

ye

ro

fµ∼

1/

4√P

ea

ssh

ow

nin

9.

9A

.J.

Bri

ant/J.

M.Yeom

ans:

Lattic

eB

oltzm

ann

sim

ula

tions

ofconta

ctlin

e

motion.

II.

Bin

ary

fluid

s,

in:

Phys.

Rev.

E69

(32004),

p.031603.

26

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.3

Dro

pspre

adin

gon

an

hori

zonta

lw

all

00.5

11.5

0

0.51

fluid

1,η 1

fluid

2,η 2

=ηη 1

σ·n=

0,∂ϕ

∂n=

0,∂µ

∂n=

0

u=

0,∂ϕ

∂n=

(1−

ϕ2)√

2cosθ s/(2Cn),

∂µ

∂n=

0

σ·n=0,∂ϕ∂n=0,

∂µ∂n=0

Fig

ure

15:G

eom

etr

yofa

spre

adin

gdro

poffluid

1in

the

initia

l

configura

tion.

27

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.3

Dro

pspre

adin

gon

an

hori

zonta

lw

all

Dro

psp

rea

din

go

fa

dro

po

na

wa

llw

ithθ s

=π/6

,C

n=

10−

2

an

dP

e=

10

2.

28

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.3

Dro

pspre

adin

gon

an

hori

zonta

lw

all

(a)

t=

0.1

(b)

t=

0.5

(c)

t=

1(d

)t=

5

Fig

ure

16:V

elo

city

field

inth

eneig

hbourh

ood

ofth

etr

iple

line.

29

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.3

Dro

pspre

adin

gon

an

hori

zonta

lw

all

Fig

ure

17:D

rop

shape

duri

ng

the

spre

adin

gatdiffe

renttim

es.

30

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.3

Dro

pspre

adin

gon

an

hori

zonta

lw

all

T

he

the

ory

of

the

dyn

am

ics

of

trip

lelin

eh

as

be

en

de

scri

be

dby

Cox

10:

g(θ

d,η)−

g(θ

s,η)=

−C

a∗

lnǫ,

(40

)

inw

hic

hth

efu

nctio

ng(θ,η)

isg

ive

nby

g(θ,η)=

∫θ

0

dα

f(α,η),

(41

)

with

f(α,η

)=

2sinα

η2(

α2−

sin

2α)

+2η[

α(π

−α)+

sin

2α]

+(π

−α)2

−sin

2α

η(α

2−

sin

2α)[π

−α+

sinα

co

sα]+

[

(π−

α)2

−sin

2α]

(α−

sinα

co

sα).

(42

)

10R

.G

.C

ox:

The

dynam

ics

ofth

espre

adin

gofliq

uid

son

asolid

surf

ace.

Part

1.

Vis

cous

flow

,in

:J.

Flu

idM

ech.

168

(1986),

pp.169–194.

31

/33

3.

Nu

me

ric

al

res

ult

so

ftw

o-p

ha

se

flo

ws

3.3

Dro

pspre

adin

gon

an

hori

zonta

lw

all

Fig

ure

18:T

he

dynam

icconta

ctangle

as

afu

nction

of

Ca∗

.

Com

pari

son

betw

een

the

num

eri

calre

sult

and

the

Cox’s

theory

with

ǫ=

10−

1.

32

/33

4.

Co

nclu

sio

na

nd

pe

rsp

ec

tiv

es

N

um

eri

ca

lso

lve

rto

stu

dy

two

-ph

ase

flow

su

sin

gth

ep

ha

se

fie

ldth

eo

ry1

1:

W

ettin

gpro

pert

ies

ofw

all

easy

tota

ke

into

account.

D

yn

am

ics

an

dte

mp

ora

lb

eh

avio

rsw

ell

de

scri

be

d:

C

apill

ary

risin

gin

atu

be;

D

rop

spre

adin

gon

asolid

substr

ate

.

T

he

physic

sis

ma

inly

co

ntr

olle

dby

two

nu

mb

ers

:

Cn≤

10−

2;

Pe≥

10

2.

In

tro

du

ce

a“r

ea

l”th

erm

od

yn

am

ics.

A

pp

lica

tio

ns:

P

hase

separa

tion

ofoxid

egla

sses;

B

ubble

nucle

ation

ingla

ss

form

er

liquid

s.

11F.

Pig

eonneau/E

.H

achem

/P.S

ara

mito:

Dis

continuous

Gale

rkin

finite

ele

mentm

eth

od

applie

dto

the

couple

dS

tokes/C

ahn-H

illia

rdequations,

in:

Int.

J.N

um

.M

eth

.F

luid

sunder

revis

ion

(2018),

pp.1–28.

33

/33