Time-dependent Marangoni-Bénard instability of an evaporating binary-liquid layer including gas transients

Abstract

We are here concerned with Bénard instabilities in a horizontal layer of a binary liquid, considering as a working example the case of an aqueous solution of ethanol with a mass fraction of 0.1. Both the solvent and the solute evaporate into air (the latter being insoluble in the liquid). The system is externally constrained by imposing fixed “ambient” pressure, humidity, and temperature values at a certain effective transfer distance above the liquid-gas interface, while the ambient temperature is also imposed at the impermeable rigid bottom of the liquid layer. Fully transient and horizontally homogeneous solutions for the reference state, resulting from an instantaneous exposure of the liquid layer to ambient air, are first calculated. Then, the linear stability of these solutions is studied using the frozen-time approach, leading to critical (monotonic marginal stability) curves in the parameter plane spanned by the liquid layer thickness and the elapsed time after initial contact. This is achieved for different ratios of the liquid and gas thicknesses, and in particular yields critical times after which instability sets in (for given thicknesses of both phases). Conversely, the analysis also predicts a critical thickness of the liquid layer below which no instability ever occurs. The nature of such critical thickness is explained in detail in terms of mass fraction profiles in both phases, as it indeed appears that the most important mechanism for instability onset is the solutal Marangoni one. Importantly, as compared to the result obtained previously under the quasi-steady assumption in the gas phase [H. Machrafi, A. Rednikov, P. Colinet, and P. C. Dauby, Eur. Phys. J. Spec. Top. 192, 71 (2011)]10.1140/epjst/e2011-01361-y, it is shown that relaxing this assumption may yield essentially lower values of the critical liquid thickness, especially for large gas-to-liquid thickness ratios. A good-working analytical model is developed for the description of such delicate transient effects in the gas. The analysis reveals that the system considered in this paper is generally highly unstable, the instability setting in even for very small times and liquid thicknesses.