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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