Jacco Snoeijer

Physics of Fluids - University of Twente

(slides: thanks to Jens Eggers)

matched asymptotics:Landau-Levich-Derjaguin film

dip coating

dip coating

liquid

air

withdraw solid plate at speed U

entrained film,thickness h

liquid

air

dip coating

withdraw solid plate at speed Ca=ηU/γ

Landau and Levich ’42Derjaguin ’43

h ~ lγ Ca2/3entrained film,thickness h

liquid

air

generic scalingdip coating

h ~ lγ Ca2/3

bubble in channel

h ~ r Ca2/3 Bretherton, JFM 1961

outline

• explain method of matched asymptotics(a simple example)

• Landau-Levich-Derjaguin film

literature:

Book by Sam Howison, Chapter 16

A simple example'' (1 ) ' 0y y y! !+ + + = (0) 0

(1) 1

y

y

=

=

0 1

1

?

small ε'' (1 ) ' 0y y y! !+ + + =

(0) 0

(1) 1

y

y

=

=

for 0! " ' 0y y+ =x

y Ae!

=

problem: solution cannot obey both boundary conditions

what happens: ‘boundary layer’ close to x=0

Inner solution'' (1 ) ' 0y y y! !+ + + =

2

1 1/ , '' , 'x y y y y!! !! "

" "= = =

for 0! " 0, y(0)=0y y!! !+ =

( )1y B e!"= " inner solution

Two regimes

1 xy e

!=

( )/1x

y B e!"= " x !" inner solution

(boundary layer)

outer solution(large scale)

we need to match these solutions!(formal procedure: see Howison Chapter 16)

Matching

1 xy e

!=

( )/1x

y B e!"= " x !"

y e! y B! B e=

Exact solution'' (1 ) ' 0y y y! !+ + + =

(0) 0

(1) 1

y

y

=

=

outer solutioninner solution

exactagreement becomes ‘exact’ for ε 0

dip coating

withdraw solid plate at speed Ca=ηU/γ

entrained film,thickness hf

outer

inner

dip coating

withdraw solid plate at speed Ca=ηU/γ

entrained film,thickness hf

The Landau Levich Derjaguin film

fhCa

( )h x

2

31 1

qh

h h

Ca ! "### = $ $% &' (

for a (nearly) flat film

bath

The bath

( )x y! ( )

3/ 22

''0

1 '

xx

x

! =+

cg

!

"=l

l

2 1 sin!= "l (curvature)! = l

outer solution

Inner solution

2

31Q

HH H

! "### = $ $% &' (

LL-equation

2

31 1

qh

h h

Ca ! "### = $ $% &' (

1/3

xh HCa

Ca

!

!"

# $= % &

' (q aQC

!=

0Ca!

1/3

x

Ca!

"#

=

!

0

fH

2, for H a b! ! != + "#

Matching condition

fhCa match!

II

1/3 2 /3 2h a x bC a xa C

!"# +

I: bath

1/ 2 2tan 2 1 sinh x x! !"

# + "

0Ca! 0! =2/3 1/ 2

2Cab!" "=

2 / 3! =

2/30.95f Cah =

experiments?

experiments

Ca

h

perfectly wettingplate

Snoeijer, Ziegler, Andreotti, Fermigier, Eggers, Phys. Rev. Lett. 2008

2/30.95f Cah =

experiments

Ca

h

partial wetting

perfectly wettingplate

Snoeijer, Ziegler, Andreotti, Fermigier, Eggers, Phys. Rev. Lett. 2008

hf ~ Ca1/2

film profilespartial wettingcomplete wetting

Landau-Levich-Derjaguin

Snoeijer, Ziegler, Andreotti, Fermigier, Eggers, Phys. Rev. Lett. 2008

film profilespartial wettingcomplete wetting

Landau-Levich-Derjaguin

thick solutionscontaining ’dimple’

Snoeijer, Ziegler, Andreotti, Fermigier, Eggers, Phys. Rev. Lett. 2008

The dimple!

fh

Ca

I: bath

2! =II

conclusion

- matched asymptotics: powerful tool to solve ’real’ problems

- Landau-Levich-Derjaguin film: nice illustration of matching very important in capillary problems

(see Michiel Kreutzer + Frieder Mugele)

Phase diagram

Ca

fh0Q =

max1.376Q Q= = K

2

31 1

q

h

Ca

h!

" #$ = % %& '( )

( )1/ 2 2 /3

32

f

QCh a Ca= !

new thick filmsolutions!

2/30.95f Cah =

partial wettingcomplete wetting

Experiment:

The Landau Levich Derjaguin film

fhCa

I: bath

2! =

2

31Q

HH H

! "### = $ $% &' (

LL-equation

II

2/3

1/3Ca

Ca

xh H

! "= # $

% &

2/3q QCa=

2/30.95f Cah =

( ) 2H !! " =( ) 0H ! "# =

matching the flux

gravitational draining:

Qfilm

small flux: Qdimple

stationary solution: Qdimple = Qfilm