Cemef October, 11th 2018, GDR TherMatHT, Lyon n n€¦ · Figure 1: Fluid dynamics close to a...

Cemef October, 11th 2018, GDR TherMatHT, Lyon n n

n Two-phase modeling using the phase fieldtheoryFranck Pigeonneau and Pierre Saramito (LJK, Grenoble)

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1. Scientific issues in two-phase flows

2. Basics of the phase field theory

3. Numerical results of two-phase flows3.1 Droplet shrinkage3.2 Capillary rising3.3 Drop spreading on an horizontal wall

4. Conclusion and perspectives

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x

z

h(x)

U✓

liquide

air

Figure 1: Fluid dynamics close to a contact line.

I The velocity gradient is:

@u@z

⇡ Uh(x)

=U✓x

(1)

I The viscous dissipation is given by1:

�⌘ = ⌘

Z R

✏

✓@u@z

◆2

hdx = ⌘

Z R

✏

✓Uh

◆2

hdx = ⌘U2

✓ln✓

R✏

◆. (2)

1P. G. De Gennes: Wetting: Statics and dynamics, in: Rev. Mod. Phys.57.3 (1985), pp. 827–863.

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I To remove the singularity:

Figure 2: Precursor film2.

2P.-G. De Gennes/F. Brochard-Wyart/D. Quéré: Gouttes, bulles, perles etondes, Paris 2005.

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1. Scientific issues in two-phase flowsI To remove the singularity:

u

l s

Figure 3: Slippage of fluids on wall3.

3L. M. Hocking: A moving fluid interface. Part 2. The removal of the forcesingularity by a slip flow, in: J. Fluid Mech. 79.02 (1977), pp. 209–229.

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I To remove the singularity:

Figure 4: Contact line of diffuse interface4.

4P. Seppecher: Moving contact lines in the Cahn-Hilliard theory, in: Int. J.Engng Sci. 34.9 (1996), pp. 977–992.

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2. Basics of the phase field theory I

I Each phase is marked by a “phase field” or “orderparameter”: '.

I ' = 1 in phase 1 (⇢1, ⌘1) and ' = �1 in phase 2 (⇢2, ⌘2).

Figure 5: Shear flow with two phases.

⇢ =⇢1 + ⇢2

2+

⇢1 � ⇢22

'. (3)

I The free energy is written as follows5

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2. Basics of the phase field theory II

J. D. van derWaals

(1837-1923).

F ['] =

Z

⌦

(') +

k2||r'||2

�dV . (4)

I (') is a double-well potential.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

�

Figure 6: Example of a double-well potential: = k(1 � '2)2/(4⇣2).

5J. D. van der Waals: The thermodynamic theory of capillarity under thehypothesis of a continuous variation of density, in: Verhandel. Konink. Akad.Weten. 1 (1893), pp. 1–56.

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I At equilibrium, F ['] has to be minimal.

�F ['] =

Z

⌦

✓d d'

� kr2'

◆�'dV +

Z

�⌦

@'

@n�'dS. (5)

I Consequently:

µ(') =d d'

� kr2' = 0, 8x 2 ⌦, (6)

k@'

@n= 0, sur �⌦. (7)

I µ(') is the chemical potential.

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I In 1-dimension and with the previous double-well potential,the phase field is given by:

'(x) = tanh✓

xp2 Cn

◆, (8)

x =xL, (9)

Cn = ⇣L , Cahn number. (10)

I The surface tension is then defined by

� =kL

Z 1

�1

✓d'dx

◆2dx =

2p

2k3⇣

. (11)

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I Outside the equilibrium, Cahn and Hilliard6 proposed

@'

@t= �r · J, (12)

J = �Mrµ, (13)

µ(') =d d'

� kr2'. (14)

I Time behavior of the phase field due to the diffusion of thechemical potential.

I Investigating spinodal decomposition.

6J. W. Cahn/J. E. Hilliard: Free Energy of a Nonuniform System. I.Interfacial Free Energy, in: J. Chem. Phys. 28.2 (1958), pp. 258–267.

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Numerical simulation of a spinodal decomposition in 2D,Cn = 10�2.

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spinodalCn1e-2.avi

Media File (video/avi)

2. Basics of the phase field theory I

Fluid 1

Fluid 2

Wall

✓s

Figure 7: Contact line between two fluids and a wall. The staticcontact angle is ✓s.

F ['] =

Z

⌦

(') +

k2||r'||2

�dV +

Z

@⌦w

fw (')dS. (15)

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2. Basics of the phase field theory II

I The minimization of F [']7

µ(') =d d'

� kr2' = 0, 8x 2 ⌦, (16)

L(') = k@'

@n+

dfwd'

= 0, on @⌦w . (17)

J. W. Cahn(1928-2016).

fw (') = �� cos ✓s'(3 � '2)

4, (18)

k@'

@n=

3(1 � '2)�

4cos ✓s, on @⌦w .(19)

7J. W. Cahn: Critical point wetting, in: J. Chem. Phys. 66.8 (1977),pp. 3667–3672.

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I Outside the equilibrium, creation of force proportional toµr'.

I Stokes equations:

r · u = 0, (20)�rP +r · [2⌘(')D(u)] + ⇢(')g + µr' = 0, (21)

I Cahn-Hilliard equation:

@'

@t+r' · u = r · [M(')rµ(')], (22)

µ(') =�

⇣2

h'('2 � 1)� ⇣2r2'

i. (23)

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I Under dimensionless form:

r · u = 0, (24)

�rP +r · [2⌘(')D(u)] +BoCa

⇢(')g +3

2p

2 Ca Cnµr' = 0, (25)

@'

@t+r' · u =

1Pe

r2µ('), (26)

µ(') = '('2 � 1)� Cn2 r2',(27)

I Dimensionless numbers:

Bo =⇢1gL2

�, (28) Ca =

⌘1U�

, (29) Pe = U⇣2LM� , (30)

Cn =⇣

L, (31) ⇢̂ =

⇢2⇢1

, (32) ⌘̂ =⌘2⌘1

. (33)

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3. Numerical results of two-phase flowsh3.1 Droplet shrinkage

I Study this effect of Cahn number on droplet shrinkage forfluids at rest.

Figure 8: A static droplet in a liquid at rest.

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Figure 9: a/a0 vs. t for three Cahn numbers.18 / 33


I Small value of Cahn number prevents the shrinkage.I According to Yue et al.8, the critical radius below which the

shrinkage occurs is

rc =4

s21/6

3⇡V Cn, (34)

I rc = 0.7 for Cn = 10�1,I rc = 0.6 for Cn = 5 · 10�2,I rc = 0.4 for Cn = 10�2.

8P. Yue/C. Zhou/J. J. Feng: Spontaneous shrinkage of drops and massconservation in phase-field simulations, in: J. Comput. Phys. 223.1 (2007),pp. 1–9.

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3. Numerical results of two-phase flowsh3.2 Capillary rising

r

z

0 0.5

�0.5

1@⌦N, top

@⌦N, bottom

@⌦D

Figure 10:Geometry of thecircular tube.

I The diameter of the tube is used ascharacteristic length.

I U chosen by balancing gravity ⇠ viscousforces ) U = ⇢gD2/�.

u = 0,@'

@n=

(1 � '2)p

2 cos ✓s

2 Cn,@µ

@n= 0, 8x 2 @⌦D, (35)

� · n = 0,@'

@n=

@µ

@n= 0, 8x 2 @⌦N, top, (36)

� · n = �(⇢̂+12)n,

@'

@n=

@µ

@n= 0, 8x 2 @⌦N, bottom. (37)

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I A numerical example has been done with:

✓s = 80�, Bo = 1, Cn = 10�2, Pe = 102. (38)

I Initially, the heavy fluid (1) is below z = 0:

'0(z, r) = � tanh✓

zp2 Cn

◆. (39)

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Capillary rising of water in a tube with a static contact angleequal to 4⇡/9.

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capillaryrisingBo1Cn1e-2Pe50.avi



Figure 11: Contact line position as a function of time for ✓s = 80� andBo = 1.

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Figure 12: Geometry of the free surface, z vs. r , for ✓s = 80� andBo = 1.

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Figure 13: z-axis velocity component v vs. r , over an horizontal linelocalized right on the contact line.

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Figure 14: v/max(v) vs. ex ⇠ 4p

Pe right on the contact line for t = 2.5for Bo = 1 and Pe = 50, 102 and 103.

I The diffusion layer of µ ⇠ 1/ 4p

Pe as shown in9.9A. J. Briant/J. M. Yeomans: Lattice Boltzmann simulations of contact line

motion. II. Binary fluids, in: Phys. Rev. E 69 (3 2004), p. 031603.26 / 33

3. Numerical results of two-phase flowsh3.3 Drop spreading on an horizontal wall

0 0.5 1 1.50

0.5

1

fluid 1, ⌘1

fluid 2, ⌘2 = ⌘̂⌘1

� · n = 0, @'@n = 0, @µ

@n = 0

u = 0, @'@n = (1� '2)

p2 cos ✓s/(2Cn),

@µ@n = 0

�·n

=0,

@'

@n=

0,@µ

@n=

0

Figure 15: Geometry of a spreading drop of fluid 1 in the initialconfiguration.

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Drop spreading of a drop on a wall with ✓s = ⇡/6, Cn = 10�2

and Pe = 102.

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dropspreading9030Cn1e-2.avi



(a) t = 0.1 (b) t = 0.5

(c) t = 1 (d) t = 5

Figure 16: Velocity field in the neighbourhood of the triple line.

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Figure 17: Drop shape during the spreading at different times.

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I The theory of the dynamics of triple line has beendescribed by Cox10:

g(✓d , ⌘̂)� g(✓s, ⌘̂) = �Ca⇤ ln ✏, (40)

in which the function g(✓, ⌘̂) is given by

g(✓, ⌘̂) =Z ✓

0

d↵f (↵, ⌘̂)

, (41)

with

f (↵, ⌘̂) =2 sin↵

n⌘̂2

⇣↵2 � sin2 ↵

⌘+ 2⌘̂

h↵(⇡ � ↵) + sin2 ↵

i+ (⇡ � ↵)2 � sin2 ↵

o

⌘̂(↵2 � sin2 ↵) [⇡ � ↵+ sin↵ cos↵] +h(⇡ � ↵)2 � sin2 ↵

i(↵� sin↵ cos↵)

.

(42)

10R. G. Cox: The dynamics of the spreading of liquids on a solid surface.Part 1. Viscous flow, in: J. Fluid Mech. 168 (1986), pp. 169–194.

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Figure 18: The dynamic contact angle as a function of Ca⇤.Comparison between the numerical result and the Cox’s theory with✏ = 10�1.

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4. Conclusion and perspectives

I Numerical solver to study two-phase flows using the phasefield theory11:

I Wetting properties of wall easy to take into account.I Dynamics and temporal behaviors well described:

I Capillary rising in a tube;I Drop spreading on a solid substrate.

I The physics is mainly controlled by two numbers:I Cn 10�2; Pe � 102.

I Introduce a “real” thermodynamics.I Applications:

I Phase separation of oxide glasses;I Bubble nucleation in glass former liquids.

11F. Pigeonneau/E. Hachem/P. Saramito: Discontinuous Galerkin finiteelement method applied to the coupled Stokes/Cahn-Hilliard equations, in:Int. J. Num. Meth. Fluids under revision (2018), pp. 1–28.

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