Bénard instabilities in a binary-liquid layer evaporating into an inert gas: Stability of quasi-stationary and time-dependent reference profiles

This study treats an evaporating horizontal binary-liquid layer in contact with the air with an imposed transfer distance. The liquid is an aqueous solution of ethanol (10% wt). Due to evaporation, the ethanol mass fraction can change and a cooling occurs at the liquid-gas interface. This can trigger solutal and thermal Rayleigh-Bénard-Marangoni instabilities in the system, the modes of which corresponding to an undeformable interface form the subject of the present work. The decrease of the liquid-layer thickness is assumed to be slow on the diffusive time scales (quasi-stationarity). First we analyse the stability of quasi-stationary reference profiles for a model case within which the mass fraction of ethanol is assumed to be fixed at the bottom of the liquid. Then this consideration is generalized by letting the diffusive reference profile for the mass fraction in the liquid be transient (starting from a uniform state), while following the frozen-time approach for perturbations. The critical liquid thickness below which the system is stable at all times quite expectedly corresponds to the one obtained for the quasi-stationary profile. As a next step, a more realistic, zero-flux condition is used at the bottom in lieu of the fixed-concentration one. The critical thickness is found not to change much between these two cases. At larger thicknesses, the critical time at which the instability first appears proves, as can be expected, to be independent of the type of the concentration condition at the bottom. It is shown that solvent (water) evaporation plays a stabilizing role as compared to the case of a non-volatile solvent. At last, an effective approximate Pearson-like model is invoked making use in particular of the fact that the solutal Marangoni is by far the strongest as an instability mechanism here.