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-B´enard-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 quasistationary
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.