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

UC

,K0

*

STM

UbkTCE S

*3/2

xxWxxxxxt JhThMhhhhEJh )(2

sin)(31 223

hKhTJ 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

hdh

h

93)(ha

hh

2/13)( dhed

hh

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

hxh

0

d22

0

d2tantan 3222

h

C

Apparent contact angle:

Adsorbed film: 00 h0h

Evaporating film

Adsorbed film: 00

hKhTJ 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)(31 223 JhThMxhhEJ

th

W

hKhTJ 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)(h

21||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