matched asymptotics: landau-levich-derjaguin film · jacco snoeijer physics of fluids - university...
TRANSCRIPT
Jacco Snoeijer
Physics of Fluids - University of Twente
(slides: thanks to Jens Eggers)
matched asymptotics:Landau-Levich-Derjaguin film
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
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)
Exact solution'' (1 ) ' 0y y y! !+ + + =
(0) 0
(1) 1
y
y
=
=
outer solutioninner solution
exactagreement becomes ‘exact’ for ε 0
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
!
!"
# $= % &
' (q aQC
!=
0Ca!
1/3
x
Ca!
"#
=
!
0
fH
2, for H a b! ! != + "#
Matching condition
fhCa match!
II
1/3 2 /3 2h a x bC a xa C
!"# +
I: bath
1/ 2 2tan 2 1 sinh x x! !"
# + "
0Ca! 0! =2/3 1/ 2
2Cab!" "=
2 / 3! =
2/30.95f Cah =
experiments
Ca
h
perfectly wettingplate
Snoeijer, Ziegler, Andreotti, Fermigier, Eggers, Phys. Rev. Lett. 2008
2/30.95f Cah =
experiments
Ca
h
partial wetting
perfectly wettingplate
Snoeijer, Ziegler, Andreotti, Fermigier, Eggers, Phys. Rev. Lett. 2008
hf ~ Ca1/2
film profilespartial wettingcomplete wetting
Landau-Levich-Derjaguin
Snoeijer, Ziegler, Andreotti, Fermigier, Eggers, Phys. Rev. Lett. 2008
film profilespartial wettingcomplete wetting
Landau-Levich-Derjaguin
thick solutionscontaining ’dimple’
Snoeijer, Ziegler, Andreotti, Fermigier, Eggers, Phys. Rev. Lett. 2008
conclusion
- matched asymptotics: powerful tool to solve ’real’ problems
- Landau-Levich-Derjaguin film: nice illustration of matching very important in capillary problems
(see Michiel Kreutzer + Frieder Mugele)
Phase diagram
Ca
fh0Q =
max1.376Q Q= = K
2
31 1
q
h
Ca
h!
" #$ = % %& '( )
( )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 ! "# =