Thermal fluctuations in capillary thinning of thin liquid films

Thermal fluctuations have been shown to influence the thinning dynamics of planar thin liquid films, bringing predicted rupture times closer to experiments. Most liquid films in nature and industry are, however, non-planar. Thinning of such films not just results from the interplay between stabilizing surface tension forces and destabilizing van der Waals forces, but also from drainage due to curvature differences. This work explores the influence of thermal fluctuations on the dynamics of thin non-planar films subjected to drainage, with their dynamics governed by two parameters: the strength of thermal fluctuations, $\unicode[STIX]{x1D703}$, and the strength of drainage, $\unicode[STIX]{x1D705}$. For strong drainage ($\unicode[STIX]{x1D705}\gg \unicode[STIX]{x1D705}_{tr}$), we find that the film ruptures due to the formation of a local depression called a dimple that appears at the connection between the curved and flat parts of the film. For this dimple-dominated regime, the rupture time, $t_{r}$, solely depends on $\unicode[STIX]{x1D705}$, according to the earlier reported scaling, $t_{r}\sim \unicode[STIX]{x1D705}^{-10/7}$. By contrast, for weak drainage ($\unicode[STIX]{x1D705}\ll \unicode[STIX]{x1D705}_{tr}$), the film ruptures at a random location due to the spontaneous growth of fluctuations originating from thermal fluctuations. In this fluctuations-dominated regime, the rupture time solely depends on $\unicode[STIX]{x1D703}$ as $t_{r}\sim -(1/\unicode[STIX]{x1D714}_{max})\ln (\sqrt{2\unicode[STIX]{x1D703}})^{\unicode[STIX]{x1D6FC}}$, with $\unicode[STIX]{x1D6FC}=1.15$. This scaling is rationalized using linear stability theory, which yields $\unicode[STIX]{x1D714}_{max}$ as the growth rate of the fastest-growing wave and $\unicode[STIX]{x1D6FC}=1$. These insights on if, when and how thermal fluctuations play a role are instrumental in predicting the dynamics and rupture time of non-flat draining thin films.