Onset of Marangoni instability of a two-component evaporating droplet

The temperature and solute concentration reductions across a thin boundary layer near the free surface of an evaporating droplet may induce cellular flow motion in the droplet because of Marangoni instability. The present study is aimed at investigating theoretically the onset of Marangoni instability due to the evaporation of a two-component evaporating droplet.

With the quasi-steady approximation which means that the surrounding gas motion is asymptotically steady, the size change of the droplet is negligible, and the temperature and concentration distributions of the droplet are temporarily frozen at each specified instant of interest, the onset condition for Marangoni instability is obtained through the linear stability analysis.

By assuming the surface tension is a monotonically decreasing function of both temperature and concentration of the higher-volatility substance, the thermocapillary and diffuso-capillary effects augment each other. Therefore, the theoretical analysis predicts a linear relation, with a negative slope, between the onset thermal Marangoni number, Ma_T, and the onset solute Marangoni number, Ma_S. Moreover, when liquid Lewis number Le_l>1, the critical wave number, l_c, may possess different values depending on the variation of the thermocapillary effect and diffuso-capillary effect. In addition, Le_l has a stronger effect on the critical solute Marangoni number Ma_S,C, than on the critical thermal Marangoni number Ma_T,C. That is, as Le_l decreases, Ma_T,C decreases mildly while Ma_S,C increases drastically.