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