Stability analysis of evaporating falling liquid films

In order to improve the understanding of the wavy dynamics of evaporating falling liquid films, we have performed a linear stability analysis by solving the classical Orr-Sommerfeld eigenvalue problem. The key parameters of the analysis are: (i) Marangoni number $(Ma)$, describing the relation between the thermocapillary stress to the viscous stress, and (ii) the evaporation parameter $(E)$, representing the ratio between the viscous and evaporative time scales.

For small $Ma$, a new unstable mode (S-mode) is observed, resulting from the change of surface tension due to the temperature gradient along the free surface. More interestingly, for a specific Ma, the S-mode and H-mode (hydrodynamic  mode, due to surface inclination) combine into one unstable region, thus reinforcing each other, leading to a possible formation of dry spots. When the liquid is volatile, a new instability is detected due to vapor recoil which destabilizes the flow in a similar fashion as thermocapillarity. Vapour recoil is also accompanied by a significant thinning of the film, accelerating the film's rupture.

Finally, direct numerical simulations are performed to verify the effects of $Ma$ and $E$ on the stability of the liquid films and the results are found to be in agreement with the theory.