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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 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 Thin film cooling Spray cooling Kabov et al. (2000, 2002) Sodtke & Stephan (2005)
Disjoiningpressure (Derjaguin 1955) Macroscopic equations + extra terms
Apparent contact lines • Used for both steady and moving contact lines (as reviewed by Craster & Matar, 2009) • Based on the assumption
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?
P P H0 H H adsorbed film thickness, isothermal system Disjoining pressure curves Perfect wetting Partial wetting • 0
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.
Evolution of the interface Equation for thickness: Evaporative flux:
Disjoining pressure models • Exponential • Model of Wong et al. (1992) • Integrated Lennard-Jones
Static contact angle L.-J. Wonget al. exponential TH
Static contact angle Isothermal film Adsorbed film: Apparent contact angle: Evaporating film Adsorbed film:
Modified Frumkin-Derjaguin eqn. Integrate and change variables:
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)
Evolution Equation in 3D z Equation for thickness: y Evaporative flux: h(x,y,t)
Periodic Periodic y 0 x Lx Initial and Boundary Conditions constant flux
Weak Evaporation (E = 10-5) t = 1 t = 40 t = 200
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
h(x,y,t) h0(x,t) y y x x Fingering instability development
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
Acknowledgements This work was supported by the National Science Foundation and the Alexander von Humboldt Foundation