Sp Spre reading, re retrac action an and sustained oscillations - - PowerPoint PPT Presentation

www.imperial.ac.uk/events

Sp Spre reading, re retrac action an and sustained

• scillations of surfa

factant-laden lenses

• G. Karapetsas1, R. V. Craster2 & O. K. Matar1

1Department of Chemical Engineering 2Department of Mathematics Imperial College London

Workshop on Surfactant Driven Thin Films Flow Fields Institute, Toronto, 24 February, 2012

Motivation

Stocker & Bush JFM 2007 Van Nierop et al. PoF 2006 Daniels et al. 2007

Formulation I

1

2 2

< < =

L V

ε

Surfactant transport and chemical kinetics Approximations

• Lubrication theory
• Rapid vertical diffusion

* 13 12 * 12 13

c S c S

+ ↔ +

* 13 23 * 23 13

c S c S

+ ↔ +

* 23 * 2 23

c c S

↔ +

* 12 * 2 12

c c S

↔ +

* 2 * 2

m Nc

↔

* 23 12 * 12 23

c S c S

+ ↔ +

i Si interface at space empty

=

Formulation II

Governing Equations

x h t

dz u h

      − =

∫

12

1 , 12 x h t

dz u h

      − =

∫

13

1 , 13 x h h h t

dz u dz u h

      + − =

∫ ∫

23 12 12

2 1 , 23

) , ( where

i i i

h f dz u

σ =

∫

( )

[ ]

( )

2 2 12 23 , 2 12 23 2 12 23 , 2 , 2

23 12

J Pe h h m h h dz u h h m m

m x x h h x t

+ − − = − +

∫

( )

[ ]

( )

2 23 2 12 23 23 2 12 2 12 23 12 2 2 12 23 , 2 12 23 2 12 23 , 2 , 2

23 12

J J h h J h h Pe h h c h h dz u h h c c

c c c c c c c c c x x h h x t

− − − − − − − = − +

∫

β β

fluxes sorption

=

i

J

Formulation III

Equation of state

Sheludko 1967

( )

23 23 2 23 , 23 23 23 , , 23 ev c c xx x s t

J J Pe c c u c

+ + = +

( )

12 2 12 , 12 12 12 , , 12 c c xx x s t

J Pe c c u c

+ = +

( )

13 13 , 13 13 13 , , 13 ev xx x s t

J Pe c c u c

+ = +

( ) ( )

( ) ( )

3 3 / 1

1 1 1 1 1

−

− Σ + + Σ + =

i i i i

c

σ

23 , 13 , 12

=

i

( )

* * * im im io i

σ σ σ − = Σ

* 23 * m im i

σ σ δ =

fluxes sorption

=

i

J

Contact line ( )

Formulation IV

Boundary Conditions

• Continuity of pressure
• Mass conservation
• Force balance

c

x x =

( )

2 12 23

2 V dx h h

c

x

= −

∫

1 13 12

2 2 V dx h dx h

x x x

c c

= +

∫ ∫

∞

) , (

,

=

i xx i

h F

σ

) (

, 23 , 12 i x x

f h h

σ + =

) (

, 23 , 13 i x x

g h h

σ + =

23 13 12

h h h

= =

Contact line ( )

Formulation V

Boundary Conditions

23 13 23 13 23 12 23 12 23 , 23 c c c c c c c c x x x

J J Pe c

c

β β + =

=

23 12 12 13 12 13 12 , 12 c c c c c c x x x

J J Pe c

c

+ =

=

β

23 13 12 13 13 , 13 c c c c x x x

J J Pe c

c

+ =

=

c

x x =

Results I

Clean fluid

σ12=1, ρ=1, μ=1 1

12 13

− − = σ σ

S

* * * * im io im i i

σ σ σ σ σ − − =

* 1 * 2

ρ ρ ρ =

* 1 * 2

µ µ µ =

parameter spreading : S

Joanny 1987 Fraaije and Cazabat 1989

7 / 1

~ t xc

  

Results II

Surfactant-laden drop

M=8, δ23=1.9, δ12=1, Σi=0.1, ρ=μ=1

( )

* * * im im io i

σ σ σ − = Σ

* 23 * m im i

σ σ δ =

( )

* * 2 * cmc

c V M M =

Results IIΙ

Effect of M

δ23=1.9, δ12=1, Σi=0.1 Long time drop shapes, t=105

( )

* * * im im io i

σ σ σ − = Σ

* 23 * m im i

σ σ δ =

( )

* * 2 * cmc

c V M M =

Results IV

( ) ( ) [ ]

c

x x c c c c c c

c c c c R k J

=

− − − =

13 12 12 13 12 13 12 13 12 13

1 1

Adsorption at the contact line

M=8, δ23=1.9, δ12=1, Σi=0.1

* 13 12 * 12 13

c S c S

+ ↔ +

Results IV

( ) ( ) [ ]

c

x x c c c c c c

c c c c R k J

=

− − − =

13 12 12 13 12 13 12 13 12 13

1 1

Adsorption at the contact line

M=8, δ23=1.9, δ12=1, Σi=0.1

* 13 12 * 12 13

c S c S

+ ↔ +

Results V

Van Nierop et al. PoF 2006 Oil water interface: Oleic acid + NaOH Na-oleate

Results VI

Stocker & Bush JFM 2007

Results VI

Stocker & Bush JFM 2007

Results VI

Stocker & Bush JFM 2007

Results VI

Stocker & Bush JFM 2007

n evaporatio for parameter kinetic

13 = ev

k

Results VII

Effect of density ratio, ρ

M=8, δ23=1.9, δ12=1, Σi=0.1 Long time drop shapes, t=105

( )

* * * im im io i

σ σ σ − = Σ

* 23 * m im i

σ σ δ =

( )

* * 2 * cmc

c V M M =

* 1 * 2

ρ ρ ρ =

Conclusions

We have studied the spreading of surfactant-laden drops on thin layers of another liquid. The presence of Marangoni stresses gives rise to very rich dynamics which may include:

• Spreading until the drop reaches equilibrium (S < 0).
• Continuous spreading (S > 0)
• Spreading followed by retraction.
• Self-sustained oscillations.

Thank you for your attention!