Nonlinear effect of surface disturbances on mass flux and its modeling in Marangoni dropwise condensation

We study Marangoni dropwise condensation of a binary vapor mixture in the framework of the long-wave model. The model equations for the film thickness of a condensate and the condensation mass flux are derived and numerically solved as an initial value problem. In the dropwise state, rupture of the liquid film is prevented by the effect of mass gain due to condensation, which is enhanced at the troughs of the liquid. The average mass and heat fluxes increase with respect to those in the flat-film state. By simplifying the model equation, we reveal that an increase in the average mass flux arises mainly from nonlinear coupling of the disturbances of the mass flux and the film thickness. Furthermore, we propose a simple expression for the average mass flux. The ratio of the average mass flux to that in the flat-film state becomes larger at smaller ambient vapor concentrations of the lower-boiling-point component, around the bubble point temperature, for thicker condensates or for thinner boundary layers, consistent with experimental results in literature. Our model of the average mass flux is a good approximation for that obtained from the original equation unless the concentration becomes sufficiently small or the temperature approaches that of the bubble point.