Stability of a condensing liquid film in a binary vapor mixture system

We study stability of a condensing liquid film of a binary vapor mixture. When a binary vapor mixture of some kind is cooled on a substrate, a condensing liquid film emerges to take an inhomogeneous form such as a droplet one due to the solutal Marangoni effect. In order to analyze this phenomenon, we apply the long-wave approximation to the condensing liquid film and derive a nonlinear partial differential equation describing the spatio-temporal evolution of the film thickness. An interfacial boundary condition taking account of an effect of mass gain of the liquid film is adopted. Based on this model, we perform a linear stability analysis around a flat-film solution. We obtain an evolution equation of the amplitude of a disturbance, from which the cutoff and fastest growth wavenumbers are deduced. The maximum value of the cutoff wavenumber relative to the film thickness and its film thickness are estimated for water–ethanol mixture at atmospheric pressure. We numerically verify the long-wave nature of the instability of the condensate liquid film in this system. A significant difference in their values is found for low-ethanol fractions of the ambient vapor whether or not the temperature dependence of the mass transfer coefficient is considered. The wavenumber of a pattern of the liquid film observed in the experiment has the same parameter dependence as that of the fastest growth wavenumber.