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Modeling of apparent contact lines in evaporating liquid films

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).

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Modeling of apparent contact lines in evaporating liquid films

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

  4. Disjoiningpressure (Derjaguin 1955)

  5. Disjoiningpressure (Derjaguin 1955) 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. P P H0 H H adsorbed film thickness, isothermal system Disjoining pressure curves Perfect wetting Partial wetting • 0

  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. Wonget al. exponential TH

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

  16. Modified Frumkin-Derjaguin eqn.

  17. Modified Frumkin-Derjaguin eqn. Integrate and change variables:

  18. Dynamic contact angle uCL

  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 z Equation for thickness: y Evaporative flux: h(x,y,t)

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

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

  26. h(x,y,t) h0(x,t) y y x x h1(x,y,t) = h(x,y,t) – h0(x,t) Integral measure of the instability

  27. h(x,y,t) h0(x,t) y y x x Fingering instability development

  28. Critical evaporation number (d1=0) *

  29. Effects of partial wetting exp. model, d1=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

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