1. 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

2. Motivating applications Spray cooling Sodtke & Stephan (2005)

3. Motivating applications Spray cooling Thin film cooling Sodtke & Stephan (2005) Kabov et al. (2000, 2002)

4. Disjoining pressure (Derjaguin 1955)

5. Macroscopic equations + extra terms

6. Apparent contact lines Used for both steady and moving contact lines (as reviewed by Craster & Matar, 2009) Based on the assumption

7. Apparent contact lines Used for both steady and moving contact lines (as reviewed by Craster & Matar, 2009) Based on the assumption Can we use it for partially wetting liquids?

8. Disjoining pressure curves u0u0 H  H  H0H0 adsorbed film thickness, isothermal system Perfect wetting Partial wetting

9. Model problem: flow down an incline Film in contact with saturated vapor

10. Nondimensional parameters capillary number evaporation number modified Marangoni number - from interfacial B.C.

11. Evolution of the interface Equation for thickness: Evaporative flux:

12. Disjoining pressure models Exponential Model of Wong et al. (1992) Integrated Lennard-Jones

13. Model problem: scaled apparent contact angle

14. Static contact angle L.-J. exponential Wong et al. THTH

15. Static contact angle Isothermal film Apparent contact angle: Adsorbed film: Evaporating film Adsorbed film:

16. Modified Frumkin-Derjaguin eqn.

17. Integrate and change variables:

18. Dynamic contact angle u CL

19. Fingering instability Huppert (1982)

20. Mathematical modeling Linear stability: Troian et al. (1989), Spaid & Homsy (1996)

21. Mathematical modeling Linear stability: Troian et al. (1989), Spaid & Homsy (1996)

22. Mathematical modeling Linear stability: Troian et al. (1989), Spaid & Homsy (1996) Nonlinear simulations: Eres et al. (2000), Kondic and Diez (2001)

23. Evolution Equation in 3D Equation for thickness: z y h(x,y,t) Evaporative flux:

24. 0 LxLx x y Periodic Initial and Boundary Conditions constant flux

25. Weak Evaporation (E = 10 -5 ) t = 200 t = 40 t = 1

26. h 0 (x,t) x y h(x,y,t) x y h 1 (x,y,t) = h(x,y,t) – h 0 (x,t) Integral measure of the instability

27. h 0 (x,t) x y h(x,y,t) x y Fingering instability development

28. Critical evaporation number (d 1 =0) *

29. Effects of partial wetting exp. model, d 1 =20, perfect wetting

30. 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

31. Acknowledgements This work was supported by the National Science Foundation and the Alexander von Humboldt Foundation