Modeling of apparent contact lines in evaporating liquid films Vladimir Ajaev Southern Methodist...

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