two-phase hydrodynamic model for air entrainment at moving contact line
DESCRIPTION
Two-phase hydrodynamic model for air entrainment at moving contact line. Tak Shing Chan and Jacco Snoeijer Physics of Fluids Group Faculty of Science and Technology University of Twente. Part one: Introduction. I ntroduction:. air. Static contact angle θ o. liquid. I ntroduction:. - PowerPoint PPT PresentationTRANSCRIPT
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Two-phase hydrodynamic model for air entrainment at moving
contact line
Tak Shing Chan and Jacco Snoeijer
Physics of Fluids GroupFaculty of Science and Technology
University of Twente
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Part one: Introduction
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air
Introduction:
liquid
Static contact angle θo
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Dewetting
(receding contact line): air
U
Ca
Introduction:
liquid
Constant U
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Dewetting
(receding contact line): air
U
Ca
Introduction:
liquid
U > Uc
Bonn et al. (Rev. Mod. Phys. 2009)
e.g. Landau-Levich-Derjaguin film
Lubrication theory
Cac~10-2
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Wetting
(advancing contact line):
air U
Ca
Introduction:
liquid
Constant U
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Wetting
(advancing contact line):
air U
Ca
Introduction:
liquid
U > Uc
Air entrainment ?
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A splash is observed when the speed of the bead is larger than a threshold value.
(Duez, C. et al Nature Phys. 3, 2007)
A fiber is pulled into a liquid bath.
Pressurized liquid, Cac ~ 50
(P.G. Simpkins & V.J. Kuck, J. Colloid & Interface Sci. 263, 2003)
Instability of advancing contact line (experimental motivation)
Dip coating: air bubbles are
observed. Cac ~1
(H. Benkreira & M.I. Khan, Chem. Engineering Sci. 63, 2008)
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Wetting
(advancing contact line):
air U
Ca
Introduction:
liquid
U > Uc
Questions:
What is the mechanism for air entrainment? Can we compute the critical Cac theoretically?
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Wetting
(advancing contact line):
air U
Ca
Introduction:
liquid
U > Uc
Questions:
What is the mechanism for air entrainment? Can we compute the critical Cac theoretically?
Lubrication theory still valid ???
Air flow important ???
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Lorenceau, Restagno, Quere, PRL 2003Eggers PRL 2001
critical Ca depends on viscosity ratio !!
air
liquidIncreasing speed
Analogy with free surface cusp: role of air flow
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Lorenceau, Restagno, Quere, PRL 2003Eggers PRL 2001
critical Ca depends on viscosity ratio !!
air
liquidIncreasing speed
Analogy with free surface cusp: role of air flow
What happens for flow with a contact line?
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Part two: 2-phase hydrodynamic model
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We consider very small Re number (Re << 1)and stationary state ( ) only: 0t
h
Fluid B (e.g. water)
interface
Constant speed U
h
Fluid A (e.g. air)
2-phase model: Assume straight contact line (2D problem)
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We consider very small Re number (Re << 1)and stationary state ( ) only: 0t
h
Young-Laplace equation
BA PP
Fluid B (e.g. water)
interface
Constant speed U
h
Fluid A (e.g. air)
2-phase model: Assume straight contact line (2D problem)
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We consider very small Re number (Re << 1)and stationary state ( ) only: 0t
h
Young-Laplace equation
BA PP
Fluid B (e.g. water)
interface
Constant speed U
h
Fluid A (e.g. air)
2-phase model:
Stokes equation (Re<< 1)
gravityUP
2
Assume straight contact line (2D problem)
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For standard lubrication theory (1 phase, small slope), we use Poiseuille flow to approximate the velocity field.
dx
dh
hh
Ca
dx
hd
)3(
33
3
hx
2-phase model:
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For standard lubrication theory (1 phase, small slope), we use Poiseuille flow to approximate the velocity field.
dx
dh
hh
Ca
dx
hd
)3(
33
3
hx
For two phase flow ??? Huh & Scriven’s solution in straight wedge problem
(C. Huh & L.E. Scriven, Journal of Colloid and Interface Science, 1971).
U
air
liquid
Stream lines
θ
2-phase model:
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With the assumption that the curvature of interface is small, we approximate the flow in our wetting problem by the flow in straight wedge problem.
Our idea is…
……
1 1
22
33
2-phase model:
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cos),(
)3(
32
2
Rfhh
Ca
ds
d B
)]cossin}()({sin}cossin)){((sin[3
}]sin){(}sin)({2)sin([sin2),(
2222
2222223
R
RRRf
hθ
U
Fluid B (e.g. water)
Fluid A (e.g. air)
interface
2-phase model:
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cos),(
)3(
32
2
Rfhh
Ca
ds
d B
)]cossin}()({sin}cossin)){((sin[3
}]sin){(}sin)({2)sin([sin2),(
2222
2222223
R
RRRf
B
AR
B
B
UCa
2-phase model:
o :static contact angle(wettability)
Control parameters:
hθ
U
Fluid B (e.g. water)
Fluid A (e.g. air)
interface
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cos),(
)3(
32
2
Rfhh
Ca
ds
d B
B
AR
B
B
UCa
2-phase model:
o :static contact angle(wettability)
Control parameters:
Boundary conditions: 1. h (at the contact line) = 0
2. θ (at the contact line) = θo
3. θ (at the bath) = π/2
We use shooting method to find the solutions
hθ
U
Fluid B (e.g. water)
Fluid A (e.g. air)
interface
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cos),(
)3(
32
2
Rfhh
Ca
ds
d B
B
AR
B
B
UCa
2-phase model:
o
Control parameters:
Question: How CaBc depends on R and θo ?
:static contact angle(wettability)
hθ
U
Fluid B (e.g. water)
Fluid A (e.g. air)
interface
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Part three: Results
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0 0.05 0.1 0.15 0.2 0.25-2.5
-2
-1.5
-1
-0.5
0
0.5
1
CaB
e.g. fixed θo =50o , fixed R =0.1
Δ
How is critical CaBc found?
air
liquid
Static profile
θo =50o
B
AR
BB
UCa
o :static contact angle (wettability)
Control parameters:
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Δ
How is critical CaBc found?
air
liquid
B
AR
BB
UCa
o :static contact angle (wettability)
Control parameters:
Uniform speed U
e.g. fixed θo =50o , fixed R =0.1
0 0.05 0.1 0.15 0.2 0.25-2.5
-2
-1.5
-1
-0.5
0
0.5
1
CaB
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Δ
How is critical CaBc found?
air
liquid
B
AR
BB
UCa
o :static contact angle (wettability)
Control parameters:
e.g. fixed θo =50o , fixed R =0.1
0 0.05 0.1 0.15 0.2 0.25-2.5
-2
-1.5
-1
-0.5
0
0.5
1
CaB
Uniform speed U
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Δ
How is critical CaBc found?
air
liquid
B
AR
BB
UCa
o :static contact angle (wettability)
Control parameters:
e.g. fixed θo =50o , fixed R =0.1
0 0.05 0.1 0.15 0.2 0.25-2.5
-2
-1.5
-1
-0.5
0
0.5
1
CaB
Uniform speed U
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Δ
How is critical CaBc found?
air
liquid
B
AR
BB
UCa
o :static contact angle (wettability)
Control parameters:
e.g. fixed θo =50o , fixed R =0.1
0 0.05 0.1 0.15 0.2 0.25-2.5
-2
-1.5
-1
-0.5
0
0.5
1
CaB
Uniform speed U
Cac
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0 0.5 1 1.5 2 2.5 3-5
-4
-3
-2
-1
0
1
Ca
R=1R=0.1R=0.01R=0.001R=0
Critical capillary no. (Cac)
fixed θo =50o
B
AR
BB
UCa
o :static contact angle (wettability)
Control parameters: How does CaBc depend on R ?
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-4 -3 -2 -1 0 1 2 3-5
-4
-3
-2
-1
0
1
Log(R)
Lo
g(C
a Bc)
B
AR
B
B
UCa
How does CaBc depend on R ?
U
Fluid A
Fluid B
(fixed θo =50o)
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-4 -3 -2 -1 0 1 2 3-5
-4
-3
-2
-1
0
1
Log(R)
Lo
g(C
a Bc)
B
AR
B
B
UCa
How does CaBc depend on R ?
U
Fluid A
Fluid B
(fixed θo =50o)
Dewetting regime
(-1 scaling)
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-4 -3 -2 -1 0 1 2 3-5
-4
-3
-2
-1
0
1
Log(R)
Lo
g(C
a Bc)
B
AR
B
B
UCa
How does CaBc depend on R ?
U
Fluid A
Fluid B
(fixed θo =50o)
CaBc changes significantly with R, even for small air viscosity !
Wetting regime
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-4 -3 -2 -1 0 1 2 3-5
-4
-3
-2
-1
0
1
Log(R)
Lo
g(C
a Bc)
B
AR
B
B
UCa
How does CaBc depend on R ?
U
Fluid A
Fluid B
(fixed θo =50o)
CaBc changes significantly with R, even for small air viscosity !
Wetting regime
What is the scaling ?
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-4 -3 -2 -1 0 1 2 3-5
-4
-3
-2
-1
0
1
Log(R)
Lo
g(C
a Bc)
B
AR
B
B
UCa
How does CaBc depend on R ?
U
Fluid A
Fluid B
(fixed θo =50o)Wetting regime
Special case : R = 0 (i.e. log(R) → -infinity)
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Special case : R = 0 (i.e. log(R) → -infinity)
How does CaBc depend on R ?
cos)0,(
322
2
Rfh
Ca
ds
d B
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Special case : R = 0 (i.e. log(R) → -infinity)
How does CaBc depend on R ?
Outer region (balance between gravity and viscous force)
)0,(3
cos2
fh
CaB
cos)0,(
322
2
Rfh
Ca
ds
d B
Asymptotic solution when CaB very large
2as
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Special case : R = 0 (i.e. log(R) → -infinity)
How does CaBc depend on R ?
Outer region (balance between gravity and viscous force)
)0,(3
cos2
fh
CaB
cos)0,(
322
2
Rfh
Ca
ds
d B
2as
)0,(3
22
2
f
h
Ca
ds
d B
Inner region (balance between surface tension and viscous force)
innersb /
Asymptotic solution when CaB very large
Asymptotic solution when CaB very large
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Special case : R = 0 (i.e. log(R) → -infinity)
How does CaBc depend on R ?
Outer region (balance between gravity and viscous force)
)0,(3
cos2
fh
CaB
cos)0,(
322
2
Rfh
Ca
ds
d B
)0,(3
22
2
f
h
Ca
ds
d B
Inner region (balance between surface tension and viscous force)
innersb /
innerinner
Asymptotic solution when CaB very large
Asymptotic solution when CaB very large
Matching between inner region and outer region is always possible!
2as
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How does CaBc depend on θo (wettability)?(fixed R = 0.01)
Critical speed decreases significantly for hydrophobic surface !
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
o
Ca cC
aB
c
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How does CaBc depend on θo (wettability)?(fixed R = 0.01)
Critical speed decreases significantly for hydrophobic surface !
(consistent with Duez et al. Nature Physics)
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
o
Ca cC
aB
c
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Conclusion:1. We developed a “lubrication-like” model for two-
phase flow.2. Air dynamics is crucial to find entrainment threshold.
If air flow is neglected (i.e. R=0), there is no air entrainment no matter how large Ca is.
3. Asymptotic scaling of CaBc for small R?
-4 -3 -2 -1 0 1 2 3-5
-4
-3
-2
-1
0
1
Log(R)
Lo
g(C
a Bc)
Dewetting regime
(-1 scaling)
?
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Conclusion:1. We developed a “lubrication-like” model for two-
phase flow.2. Air dynamics is crucial to find entrainment threshold.
If air flow is neglected (i.e. R=0), there is no air entrainment no matter how large Ca is.
3. Asymptotic scaling of CaBc for small R?
-4 -3 -2 -1 0 1 2 3-5
-4
-3
-2
-1
0
1
Log(R)
Lo
g(C
a Bc)
Dewetting regime
(-1 scaling)
?
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