Vibration impact on Marangoni instability in a thin film*

We study the influence of a vertical vibration on Marangoni instability in a thin film heated from below. Using a multi-scale expansion the film dynamics is considered in a wide range of the vibration frequency $\omega$: from $\omega t_v\gg 1$ to $\omega t_g=O(1)$, where $t_v$ is the time of viscous relaxation across the layer and $t_g$ is the typical time of the longwave surface dynamics. We have shown that for $\omega t_g \gg 1$ there is no interaction between the Faraday instability and the Marangoni convection because of the large differences in the characteristic time- and length scales (see also [Thiele et al. , JFM (2006)]). Therefore, the averaging technique is applied to derive the equation governing the film dynamics in slow time (in comparison with $1/\omega$). We show that the vibration suppresses the Marangoni instability in a confined cavity; however, the branching remains subcritical. This amplitude equation becomes invalid for the ultra-low frequency, $\omega t_g=O(1)$. In this case the standard amplitude equation [Oron et al., Rev. Mod. Phys. (1997)] is obtained, but with the modulated gravity. The vibration does not change the stability threshold; the subcritical excitation leads to the emergence of a limit cycle instead of a film rupture.