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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 Presentation
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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
Part one: Introduction
air
Introduction:
liquid
Static contact angle θo
Dewetting
(receding contact line): air
U
Ca
Introduction:
liquid
Constant U
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
Wetting
(advancing contact line):
air U
Ca
Introduction:
liquid
Constant U
Wetting
(advancing contact line):
air U
Ca
Introduction:
liquid
U > Uc
Air entrainment ?
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)
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?
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 ???
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
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?
Part two: 2-phase hydrodynamic model
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)
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)
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)
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:
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:
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:
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:
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
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
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
Part three: Results
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:
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
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
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
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
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 ?
-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)
-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)
-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
-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 ?
-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)
Special case : R = 0 (i.e. log(R) → -infinity)
How does CaBc depend on R ?
cos)0,(
322
2
Rfh
Ca
ds
d B
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
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
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
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
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
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)
?
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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