Marangoni convection in a thin film on a vertically oscillating plate

Thermocapillary (Marangoni) convection in a thin film on a plate oscillating with a frequency ranging from ultralow to high is considered. By adjusting the vibration amplitude, the impact of the vibration is kept non-negligible. Using the long-wave approximation framework, the amplitude equations are derived for each frequency interval, and linear and weakly nonlinear stability analyses are performed, supplemented by computations where necessary. In the case of a high vibration frequency, the surface tension effectively increases due to vibration, but the film still ruptures. When the frequency is ultralow, the vibration provides gravity modulation, and the surface deformation emerges subcritically, grows fast, and then decays, all during less than half of the vibration period. In the intermediate regime, the vibration either results in a short-wavelength instability or it does not affect the Marangoni convection.