Two-phase hydrodynamic model for air entrainment at moving contact line Tak Shing Chan and Jacco...

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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

description

air Introduction: liquid Static contact angle θ o

Transcript of Two-phase hydrodynamic model for air entrainment at moving contact line Tak Shing Chan and Jacco...

Page 1: 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.

Two-phase hydrodynamic model for air entrainment at moving

contact lineTak Shing Chan and Jacco Snoeijer

Physics of Fluids GroupFaculty of Science and Technology

University of Twente

Page 2: 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.

Part one: Introduction

Page 3: 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.

air

Introduction:

liquid

Static contact angle θo

Page 4: 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.

Dewetting

(receding contact line): air

UCa

Introduction:

liquid

Constant U

Page 5: 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.

Dewetting

(receding contact line): air

UCa

Introduction:

liquid

U > Uc

Bonn et al. (Rev. Mod. Phys. 2009)

e.g. Landau-Levich-Derjaguin film

Lubrication theory

Cac~10-2

Page 6: 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.

Wetting

(advancing contact line):

air UCa

Introduction:

liquid

Constant U

Page 7: 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.

Wetting

(advancing contact line):

air UCa

Introduction:

liquid

U > Uc

Air entrainment ?

Page 8: 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.

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)

Page 9: 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.

Wetting

(advancing contact line):

air UCa

Introduction:

liquid

U > Uc

Questions:

What is the mechanism for air entrainment? Can we compute the critical Cac theoretically?

Page 10: 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.

Wetting

(advancing contact line):

air UCa

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 ???

Page 11: 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.

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

Page 12: 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.

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?

Page 13: 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.

Part two: 2-phase hydrodynamic model

Page 14: 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.

We consider very small Re number (Re << 1)and stationary state ( ) only: 0th

Fluid B (e.g. water)

interface

Constant speed U

h

Fluid A (e.g. air)

2-phase model: Assume straight contact line (2D problem)

Page 15: 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.

We consider very small Re number (Re << 1)and stationary state ( ) only: 0th

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)

Page 16: 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.

We consider very small Re number (Re << 1)and stationary state ( ) only: 0th

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)

Page 17: 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.

For standard lubrication theory (1 phase, small slope), we use Poiseuille flow to approximate the velocity field.

dxdh

hhCa

dxhd

)3(

33

3

hx

2-phase model:

Page 18: 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.

For standard lubrication theory (1 phase, small slope), we use Poiseuille flow to approximate the velocity field.

dxdh

hhCa

dxhd

)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:

Page 19: 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.

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:

Page 20: 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.

cos),()3(

32

2

RfhhCa

dsd B

)]cossin}()({sin}cossin)){((sin[3}]sin){(}sin)({2)sin([sin2),( 2222

2222223

R

RRRf

U

Fluid B (e.g. water)

Fluid A (e.g. air)

interface

2-phase model:

Page 21: 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.

cos),()3(

32

2

RfhhCa

dsd B

)]cossin}()({sin}cossin)){((sin[3}]sin){(}sin)({2)sin([sin2),( 2222

2222223

R

RRRf

B

AR

B

BUCa

2-phase model:

o :static contact angle(wettability)

Control parameters:

U

Fluid B (e.g. water)

Fluid A (e.g. air)

interface

Page 22: 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.

cos),()3(

32

2

RfhhCa

dsd B

B

AR

B

BUCa

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/2We use shooting method to find the solutions

U

Fluid B (e.g. water)

Fluid A (e.g. air)

interface

Page 23: 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.

cos),()3(

32

2

RfhhCa

dsd B

B

AR

B

BUCa

2-phase model:

o

Control parameters:

Question: How CaBc depends on R and θo ?

:static contact angle(wettability)

U

Fluid B (e.g. water)

Fluid A (e.g. air)

interface

Page 24: 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.

Part three: Results

Page 25: 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.

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

BBUCa

o :static contact angle (wettability)

Control parameters:

Page 26: 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.

Δ

How is critical CaBc found?

air

liquid

B

AR

BBUCa

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

Page 27: 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.

Δ

How is critical CaBc found?

air

liquid

B

AR

BBUCa

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

Page 28: 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.

Δ

How is critical CaBc found?

air

liquid

B

AR

BBUCa

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

Page 29: 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.

Δ

How is critical CaBc found?

air

liquid

B

AR

BBUCa

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

Page 30: 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.

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

BBUCa

o :static contact angle (wettability)

Control parameters: How does CaBc depend on R ?

Page 31: 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.

-4 -3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

Log(R)

Log(

Ca B

c)

B

AR

B

BUCa

How does CaBc depend on R ?

U

Fluid A

Fluid B

(fixed θo =50o)

Page 32: 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.

-4 -3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

Log(R)

Log(

Ca B

c)

B

AR

B

BUCa

How does CaBc depend on R ?

U

Fluid A

Fluid B

(fixed θo =50o)

Dewetting regime

(-1 scaling)

Page 33: 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.

-4 -3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

Log(R)

Log(

Ca B

c)

B

AR

B

BUCa

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

Page 34: 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.

-4 -3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

Log(R)

Log(

Ca B

c)

B

AR

B

BUCa

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 ?

Page 35: 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.

-4 -3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

Log(R)

Log(

Ca B

c)

B

AR

B

BUCa

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)

Page 36: 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.

Special case : R = 0 (i.e. log(R) → -infinity)

How does CaBc depend on R ?

cos)0,(322

2

RfhCa

dsd B

Page 37: 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.

Special case : R = 0 (i.e. log(R) → -infinity)

How does CaBc depend on R ?

Outer region (balance between gravity and viscous force)

)0,(3cos 2 fhCaB

cos)0,(322

2

RfhCa

dsd B

Asymptotic solution when CaB very large

2as

Page 38: 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.

Special case : R = 0 (i.e. log(R) → -infinity)

How does CaBc depend on R ?

Outer region (balance between gravity and viscous force)

)0,(3cos 2 fhCaB

cos)0,(322

2

RfhCa

dsd B

2as

)0,(322

2

fhCa

dsd B

Inner region (balance between surface tension and viscous force)

innersb /

Asymptotic solution when CaB very large

Asymptotic solution when CaB very large

Page 39: 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.

Special case : R = 0 (i.e. log(R) → -infinity)

How does CaBc depend on R ?

Outer region (balance between gravity and viscous force)

)0,(3cos 2 fhCaB

cos)0,(322

2

RfhCa

dsd B

)0,(322

2

fhCa

dsd 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

Page 40: 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.

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

a Bc

Page 41: 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.

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

a Bc

Page 42: 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.

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)

Log(

Ca B

c)

Dewetting regime

(-1 scaling)

?

Page 43: 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.

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)

Log(

Ca B

c)

Dewetting regime

(-1 scaling)

?

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