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.