Improved 1.5-sided model for the weakly nonlinear study of Bénard–Marangoni instabilities in an evaporating liquid layer

Impact of an insoluble surfactant on the thresholds of evaporative Bénard-Marangoni instability under air

It is known that the addition of an insoluble surfactant to a Bénard-Marangoni (BM) layer heated from below or cooled from above can give rise to a supplementary, oscillatory mode of instability. Here the objective is to see how exactly this plays out in the framework of a recently studied and experimentally tested case of a non-long-wavelength BM instability driven by diffusion-limited evaporation into air in isothermal surroundings. Linear stability analysis is accomplished within a now standard reduction to a one-sided model. In the absence of surfactant, we just recover the classical Pearson problem, albeit with an evaporation-specific wavenumber-dependent Biot number potentially attaining large values for strongly volatile liquids. Adding a surfactant not only sharply stabilizes the monotonic Pearson-like mode, but also leads to a more dangerous oscillatory mode, a parametric study of which is here undertaken. Although slanted towards the evaporative case, the present study is also of interest from the general viewpoint of the Pearson problem + an insoluble surfactant, the results for which are scarce in the literature without being obscured by further effects. In particular, an asymptotic analysis based on small interfacial Lewis numbers (diffusion coefficients) is undertaken near the codimension-2 point.

Importance of wave-number dependence of Biot numbers in one-sided models of evaporative Marangoni instability: Horizontal layer and spherical droplet

A one-sided model of the thermal Marangoni instability owing to evaporation into an inert gas is developed. Two configurations are studied in parallel: a horizontal liquid layer and a spherical droplet. With the dynamic gas properties being admittedly negligible, one-sided approaches typically hinge upon quantifying heat and mass transfer through the gas phase by means of transfer coefficients (like in the Newton's cooling law), which in dimensionless terms eventually corresponds to using Biot numbers. Quite a typical arrangement encountered in the literature is a constant Biot number, the same for perturbations of different wavelengths and maybe even the same as for the reference state. In the present work, we underscore the relevance of accounting for its wave-number dependence, which is especially the case in the evaporative context with relatively large values of the resulting effective Biot number. We illustrate the effect in the framework of the Marangoni instability thresholds. As a concrete example, we consider HFE-7100 (a standard refrigerant) for the liquid and air for the inert gas.

Bénard instabilities in a binary-liquid layer evaporating into an inert gas

A linear stability analysis is performed for a horizontal layer of a binary liquid of which solely the solute evaporates into an inert gas, the latter being assumed to be insoluble in the liquid. In particular, a water-ethanol system in contact with air is considered, with the evaporation of water being neglected (which can be justified for a certain humidity of the air). External constraints on the system are introduced by imposing fixed "ambient" mass fraction and temperature values at a certain effective distance above the free liquid-gas interface. The temperature is the same as at the bottom of the liquid layer, where, besides, a fixed mass fraction of the solute is presumed to be maintained. Proceeding from a (quasi-)stationary reference solution, neutral (monotonic) stability curves are calculated in terms of solutal/thermal Marangoni/Rayleigh numbers as functions of the wavenumber for different values of the ratio of the gas and liquid layer thicknesses. The results are also presented in terms of the critical values of the liquid layer thickness as a function of the thickness of the gas layer. The solutal and thermal Rayleigh and Marangoni effects are compared to one another. For a water-ethanol mixture of 10wt.% ethanol, it appears that the solutal Marangoni effect is by far the most important instability mechanism. Furthermore, its global action can be described within a Pearson-like model, with an appropriately defined Biot number depending on the wavenumber. On the other hand, it is also shown that, if taken into account, water evaporation has only minor quantitative consequences upon the results for this predominant, solutal Marangoni mechanism.

Modeling evaporation of sessile drops with moving contact lines

We consider evaporation of pure liquid drops on a thermally conductive substrate. Two commonly used evaporative models are considered: one that concentrates on the liquid phase in determining the evaporative flux and the other one that centers on the gas-vapor phase. A single governing equation for the evolution of drop thickness, including both models, is developed. We show how the derived governing equation can be used to predict which evaporation model is appropriate for different considered experimental conditions.