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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
Introduction:




                         air
      Static contact
      angle θo


                       liquid
 Introduction:
                    Constant U




Dewetting                                  U
                                    Ca 
(receding contact                          
                             air
line):



                           liquid
 Introduction:
                               U > Uc


                                      e.g. Landau-Levich-
Dewetting                             Derjaguin film               U
                                                            Ca 
(receding contact                                                  
                                          air
line):

                                                               Cac~10-2
                                       liquid



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

 Lubrication theory
 Introduction:



Wetting                                     U
                                     Ca 
(advancing contact
                            air
                                            
line):

                            liquid




               Constant U
 Introduction:



Wetting                                          U
                                          Ca 
(advancing contact
                          air
                                                    
line):

                          liquid




                                Air entrainment ?
                 U > Uc
Instability of advancing contact line (experimental
motivation)




                                A fiber is pulled into a liquid bath.
                                Pressurized liquid, Cac ~ 50
                                (P.G. Simpkins & V.J. Kuck, J.
                                Colloid & Interface Sci. 263, 2003)
A splash is observed when the                                           Dip coating: air bubbles are
speed of the bead is larger                                             observed. Cac ~1
than a threshold value.
                                                                        (H. Benkreira & M.I. Khan,
(Duez, C. et al Nature Phys.                                            Chem. Engineering Sci. 63,
3, 2007)                                                                2008)
                                           Questions:
 Introduction:
                          What is the mechanism for air entrainment?
                          Can we compute the critical Cac theoretically?



Wetting                                                   U
                                                  Ca 
(advancing contact
                               air
                                                           
line):

                               liquid




                 U > Uc
                                           Questions:
 Introduction:
                          What is the mechanism for air entrainment?
                          Can we compute the critical Cac theoretically?



Wetting                                                    U
                                                  Ca 
(advancing contact
                               air
                                                            
line):

                               liquid




                                        Lubrication theory still valid ???
                 U > Uc                 Air flow important ???
Analogy with free surface cusp: role of air flow
 Lorenceau, Restagno, Quere, PRL 2003
 Eggers PRL 2001
      air




                Increasing speed
  liquid


   critical Ca depends on viscosity ratio !!
Analogy with free surface cusp: role of air flow
 Lorenceau, Restagno, Quere, PRL 2003
 Eggers PRL 2001
      air




                Increasing speed
  liquid


   critical Ca depends on viscosity ratio !!


            What happens for flow with a contact line?
 Part two: 2-phase
hydrodynamic model
2-phase model:               Assume straight contact line (2D problem)
                                                                 h
We consider very small Re number (Re << 1)and stationary state (     0) only:
                                                                 t
              Fluid A (e.g. air)
                                   interface
              h




                    Fluid B (e.g. water)




           Constant speed U
2-phase model:               Assume straight contact line (2D problem)
                                                                 h
We consider very small Re number (Re << 1)and stationary state (     0) only:
                                                                 t
              Fluid A (e.g. air)
                                   interface
              h
                                               Young-Laplace equation
                                                 PA  PB
                    Fluid B (e.g. water)




           Constant speed U
2-phase model:               Assume straight contact line (2D problem)
                                                                 h
We consider very small Re number (Re << 1)and stationary state (     0) only:
                                                                 t
              Fluid A (e.g. air)
                                   interface
              h
                                               Young-Laplace equation
                                                 PA  PB
                    Fluid B (e.g. water)       Stokes equation (Re<< 1)
                                                       
                                               P   U  gravity
                                                      2




           Constant speed U
2-phase model:
For standard lubrication theory (1 phase, small slope), we use Poiseuille
flow to approximate the velocity field.

                                                                 h
                                                                      x

                                        d 3h      3Ca       dh
                                                         
                                        dx 3
                                               h( h  3 ) dx
  2-phase model:
   For standard lubrication theory (1 phase, small slope), we use Poiseuille
   flow to approximate the velocity field.

                                                                                   h
                                                                                       x

                                                   d 3h      3Ca       dh
                                                                    
                                                   dx 3
                                                          h( h  3 ) dx
For two phase flow ??? Huh & Scriven’s solution in straight wedge problem

                                                 Stream lines


                               air
                                                          liquid
                                                     θ
                U

            (C. Huh & L.E. Scriven, Journal of Colloid and Interface Science, 1971).
  2-phase model:
Our idea is…

                                                                   1
                  1




                2                                                           2




                             3            ……                           3




   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.
 2-phase model:
                                                                                          Fluid A (e.g. air)

   d 2     3Ca B
                    f ( , R )  cos                                              h                interface
   ds 2
          h(h  3 )                                                                      θ

                                                                                        Fluid B (e.g. water)




                                                                           U



                 2 sin 3  [ R 2 ( 2  sin 2  )  2 R{ (   )  sin 2  }  {(   ) 2  sin 2  }]
f ( , R ) 
             3[ R (sin 2    2 ){(    )  sin  cos }  {sin 2   (   ) 2 }(  sin  cos )]
 2-phase model:
                                                                                          Fluid A (e.g. air)

   d 2     3Ca B
                    f ( , R )  cos                                              h                interface
   ds 2
          h(h  3 )                                                                      θ

               Control parameters:                                                      Fluid B (e.g. water)

         Ca B 
                U B
                              R  A
                                                B
                o       :static contact angle
                         (wettability)                                     U



                 2 sin 3  [ R 2 ( 2  sin 2  )  2 R{ (   )  sin 2  }  {(   ) 2  sin 2  }]
f ( , R ) 
             3[ R (sin 2    2 ){(    )  sin  cos }  {sin 2   (   ) 2 }(  sin  cos )]
 2-phase model:
                                                                       Fluid A (e.g. air)

  d 2     3Ca B
                   f ( , R )  cos                            h                interface
  ds 2
         h(h  3 )                                                    θ

             Control parameters:                                     Fluid B (e.g. water)

       Ca B 
              U B
                            R  A
                                        B
            o       :static contact angle
                     (wettability)                           U



Boundary conditions: 1. h (at the contact line) = 0
                           2. θ (at the contact line) = θo
                                                             We use shooting method
                           3. θ (at the bath) = π/2          to find the solutions
2-phase model:
                                                               Fluid A (e.g. air)

d 2     3Ca B
                 f ( , R )  cos                      h                interface
ds 2
       h(h  3 )                                              θ

         Control parameters:                                 Fluid B (e.g. water)

   Ca B 
          U B
                        R  A
                                     B
         o       :static contact angle
                  (wettability)                     U




                  Question: How CaBc depends on R and θo ?
Part three: Results
                                                    Control parameters:
How is critical CaBc found?                                  U B
                                                    Ca B              R  A
                                                                               B
e.g.   fixed θo =50o , fixed R =0.1                o :static contact angle (wettability)

                               1
        Static profile
        θo =50o              0.5


              air              0
Δ
                             -0.5
                         




         liquid
                              -1

                             -1.5

                              -2

                             -2.5
                                 0   0.05   0.1           0.15           0.2           0.25
                                                  Ca
                                                      B
                                                   Control parameters:
How is critical CaBc found?                                 U B
                                                   Ca B              R  A
                                                                              B
e.g.      fixed θo =50o , fixed R =0.1            o :static contact angle (wettability)

                              1

                            0.5


                  air         0
Δ
                            -0.5
                        




              liquid
                             -1

                            -1.5
    Uniform speed U
                             -2

                            -2.5
                                0   0.05   0.1           0.15           0.2           0.25
                                                 Ca
                                                     B
                                                   Control parameters:
How is critical CaBc found?                                 U B
                                                   Ca B              R  A
                                                                              B
e.g.      fixed θo =50o , fixed R =0.1            o :static contact angle (wettability)

                              1

                            0.5


                  air         0

                            -0.5
Δ
                        




              liquid
                             -1

                            -1.5

    Uniform speed U
                             -2

                            -2.5
                                0   0.05   0.1           0.15           0.2           0.25
                                                 Ca
                                                     B
                                                   Control parameters:
How is critical CaBc found?                                 U B
                                                   Ca B              R  A
                                                                              B
e.g.      fixed θo =50o , fixed R =0.1            o :static contact angle (wettability)

                              1

                            0.5


                  air         0

                            -0.5
                        




Δ             liquid
                             -1

                            -1.5

                             -2
    Uniform speed U

                            -2.5
                                0   0.05   0.1           0.15           0.2           0.25
                                                 Ca
                                                     B
                                                   Control parameters:
How is critical CaBc found?                                 U B
                                                   Ca B              R  A
                                                                              B
e.g.      fixed θo =50o , fixed R =0.1            o :static contact angle (wettability)

                              1

                            0.5


                  air         0

                            -0.5
                        




Δ             liquid
                             -1

                            -1.5

                             -2
    Uniform speed U

                            -2.5
                                0   0.05   0.1           0.15           0.2           0.25
                                                 Ca
                                                     B                              Cac
How does CaBc depend on R ?       Control parameters:
                                           U B
                                  Ca B                R  A
 Critical capillary no. (Cac)                                  B
                                 o :static contact angle (wettability)

    1
                                R=1
    0                           R=0.1
                                R=0.01                fixed θo =50o
    -1                          R=0.001
                                R=0
    -2





    -3

    -4

    -5
      0   0.5   1    1.5    2   2.5               3
                     Ca
                                                              Fluid A
How does CaBc depend on R ?
                                                           Fluid B
           1

           0                                           U



           -1
                                                   (fixed θo =50o)
Log(Ca )
     Bc




           -2
                                                     R  A
           -3
                                                                B
                                                              U B
                                                     Ca B 
           -4                                                   

           -5
            -4   -3   -2   -1      0   1   2   3
                             Log(R)
                                                                         Fluid A
How does CaBc depend on R ?
                                                                      Fluid B
           1

           0                                                      U



           -1
                                                              (fixed θo =50o)
Log(Ca )
     Bc




           -2
                                                                R  A
                                           Dewetting regime

                                                                           B
                                           (-1 scaling)
           -3
                                                                         U B
                                                                Ca B 
           -4                                                              

           -5
            -4   -3   -2   -1      0   1      2           3
                             Log(R)
                                                                                 Fluid A
How does CaBc depend on R ?
                                                                              Fluid B
           1

           0                                                              U



           -1
                 Wetting regime
                                                                  (fixed θo =50o)
Log(Ca )
     Bc




           -2
                                                                        R  A
           -3
                                                                                   B
                                                                                 U B
                                                                        Ca B 
           -4                                                                      

           -5
            -4       -3       -2   -1      0   1      2      3
                                     Log(R)

    CaBc changes significantly with R, even for small air viscosity !
                                                                                 Fluid A
How does CaBc depend on R ?
                                                                              Fluid B
           1
                                   What is the scaling ?
           0                                                              U



           -1
                 Wetting regime
                                                                   (fixed θo =50o)
Log(Ca )
     Bc




           -2
                                                                        R  A
           -3
                                                                                   B
                                                                                 U B
                                                                        Ca B 
           -4                                                                      

           -5
            -4       -3       -2   -1      0           1   2   3
                                     Log(R)

    CaBc changes significantly with R, even for small air viscosity !
                                                                               Fluid A
How does CaBc depend on R ?
                                                                            Fluid B
           1

           0                                                            U



           -1
                 Wetting regime
                                                                    (fixed θo =50o)
Log(Ca )
     Bc




           -2
                                                                      R  A
           -3
                                                                                 B
                                                                               U B
                                                                      Ca B 
           -4                                                                    

           -5
            -4       -3       -2      -1      0        1    2   3
                                        Log(R)

Special case : R            = 0 (i.e. log(R) → -infinity)
  How does CaBc depend on R ?
Special case : R   = 0 (i.e. log(R) → -infinity)
    d 2 3Ca B
       2
          2 f ( , R  0)  cos
    ds     h
  How does CaBc depend on R ?
Special case : R   = 0 (i.e. log(R) → -infinity)
    d 2 3Ca B
       2
          2 f ( , R  0)  cos
    ds     h
Outer region (balance between gravity and viscous force)

                             Asymptotic solution when CaB very large
         3CaB
   cos  2 f ( ,0)                            as2
          h
  How does CaBc depend on R ?
Special case : R   = 0 (i.e. log(R) → -infinity)
    d 2 3Ca B
       2
          2 f ( , R  0)  cos
    ds     h
Outer region (balance between gravity and viscous force)

                             Asymptotic solution when CaB very large
         3CaB
   cos  2 f ( ,0)                            as2
          h

Inner region (balance between surface tension and viscous force)
                             Asymptotic solution when CaB very large
   d 2 3Ca B
      2
         2 f ( ,0)                          b / sinner
   ds     h
  How does CaBc depend on R ?
                                                                   inner
                                                                   inner

Special case : R   = 0 (i.e. log(R) → -infinity)
    d 2 3Ca B
       2
          2 f ( , R  0)  cos
    ds     h
Outer region (balance between gravity and viscous force)

                             Asymptotic solution when CaB very large
         3CaB
   cos  2 f ( ,0)                            as2
          h

Inner region (balance between surface tension and viscous force)
                             Asymptotic solution when CaB very large
   d 2 3Ca B
      2
         2 f ( ,0)                          b / sinner
   ds     h
                                Matching between inner region and outer region
                                is always possible!
How does CaBc depend on θo (wettability)?
(fixed R = 0.01)
         0.7

         0.6

         0.5
Cac Bc




         0.4
 Ca




         0.3

         0.2

         0.1

          0
           0       0.5         1         1.5          2         2.5    3
                                         
                                           o
    Critical speed decreases significantly for hydrophobic surface !
How does CaBc depend on θo (wettability)?
(fixed R = 0.01)
         0.7

         0.6

         0.5
Cac Bc




         0.4
 Ca




         0.3

         0.2

         0.1

          0
           0         0.5           1           1.5     2        2.5    3
                                               
                                                   o
    Critical speed decreases significantly for hydrophobic surface !
    (consistent with Duez et al. Nature Physics)
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?

                          1

              ?           0

                                                          Dewetting
                          -1
                                                          regime
               Log(Ca )
                    Bc




                          -2                              (-1 scaling)

                          -3

                          -4

                          -5
                           -4   -3   -2   -1      0   1       2          3
                                            Log(R)
  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?
                                                                               Funded by:
                            1

                ?           0

                                                            Dewetting
                            -1
                                                            regime
                 Log(Ca )
                      Bc




Thank you!                  -2                              (-1 scaling)

                            -3

                            -4

                            -5
                             -4   -3   -2   -1      0   1       2          3
                                              Log(R)
(R)

				
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