Modeling of apparent contact lines in evaporating liquid films
Vladimir Ajaev Southern Methodist University, Dallas, TX
joint work with T. Gambaryan-Roisman, J. Klentzman,
and P. Stephan
Leiden, January 2010
Motivating applications
Spray cooling
Sodtke & Stephan (2005)
Motivating applications
Spray cooling Thin film cooling
Sodtke & Stephan (2005) Kabov et al. (2000, 2002)
Disjoining pressure (Derjaguin 1955)
Disjoining pressure (Derjaguin 1955)
Macroscopic equations + extra terms )(d
Apparent contact lines
• Used for both steady and moving contact lines (as reviewed by Craster & Matar, 2009)
• Based on the assumption 3~)( dd
Apparent contact lines
• Used for both steady and moving contact lines (as reviewed by Craster & Matar, 2009)
• Based on the assumption 3~)( dd
Can we use it for partially wetting liquids?
Disjoining pressure curves
0
H
H
H0
adsorbed film thickness, isothermal system
Perfect wetting Partial wetting
Model problem: flow down an incline
Film in contact with saturated vapor
Nondimensional parameters
capillary number
evaporation number
modified Marangoni number
- from interfacial B.C.
10
U
C
,K0
*
STM
Ub
kTCE S
*3/2
xxWxxxxxt JhThM
hhhhEJh )(2
sin)(3
1 223
hK
hTJ xxw
)(
Evolution of the interface
Equation for thickness:
Evaporative flux:
Disjoining pressure models
• Exponential
• Model of Wong et al. (1992)
• Integrated Lennard-Jones
2sech)(
2
213 d
hd
hh
93)(
h
a
hh
2/13
)( dhedh
h
Model problem: scaled apparent contact angle
3/1C
Static contact angle
L.-J.
exponential
Wong et al.
TH
Static contact angle
Isothermal film
0xxh
h
h
xh0
d22
0
d2tantan 3222
h
C
Apparent contact angle:
Adsorbed film: 00 h
0h
Evaporating film
Adsorbed film:
00
hK
hTJ xx 00 Th
Modified Frumkin-Derjaguin eqn.
)( xxi hJKT
Modified Frumkin-Derjaguin eqn.
)( xxi hJKT
dKJTdh
i
h
00
))((2
)(220
Integrate and change variables:
Dynamic contact angle
002.0,5.0 ETH002.0,1 ETH
uCL
001.0,1 ETH
Fingering instability
Huppert (1982)
Mathematical modeling
• Linear stability: Troian et al. (1989), Spaid & Homsy (1996)
Mathematical modeling
• Linear stability: Troian et al. (1989), Spaid & Homsy (1996)
Mathematical modeling
• Linear stability: Troian et al. (1989), Spaid & Homsy (1996)
• Nonlinear simulations: Eres et al. (2000), Kondic and Diez (2001)
)(2
ˆsin)(3
1 223 JhThM
xhhEJt
hW
hK
hTJ w
)( 2
Evolution Equation in 3D
Equation for thickness:
zy
g
h(x,y,t)
Evaporative flux:
0
5
10
15
20
05
1015
20
0
0.01
0.02
0.03
0
Lxx
y
0),,0( tyhx
0),,( tyLh xx
adsx htyLh ),,(
Periodic
Periodic
Initial and Boundary Conditions
constant flux
Weak Evaporation (E = 10-5)
t = 200
t = 40
t = 1
0
20
40
60
0 10 20 30 40 50 60
0
0.01
0.02
0.03
0.04
h0(x,t)
x
y
0
20
40
60
0 10 20 30 40 50 60
0
0.01
0.02
0.03
0.04
h(x,y,t)
x
y
h1(x,y,t) = h(x,y,t) – h0(x,t)
dA)(h2
1||h|| 2
12
1
Integral measure of the instability
0
20
40
60
0 10 20 30 40 50 60
0
0.01
0.02
0.03
0.04
h0(x,t)
x
y
0
20
40
60
0 10 20 30 40 50 60
0
0.01
0.02
0.03
0.04
h(x,y,t)
x
y
Fingering instability development
0,~1 teh
Critical evaporation number (d1=0)
)0(
*
Effects of partial wetting exp. model, d1=20 , perfect wetting
Summary
Apparent contact angle• Defined by maximum absolute value of the slope
of the interface• Not sensitive to details of • Follows Tanner’s law even for strong evaporation
Fingering instability with evaporation:• Growth rate increases with contact angle • Critical wavelength is reduced
)(h
Acknowledgements
This work was supported by the National Science Foundation and the Alexander von Humboldt Foundation