Linear Stability of Marangoni Convection in a Thin Film under Vertical Vibrations

We investigate the linear stability of an incompressible, viscous liquid thin film placed on a solid substrate subjected to vertical harmonic vibrations in the presence of gravity and a negative temperature gradient. The substrate oscillates with a finite frequency, compared to the viscous time and large amplitude, compared to the film thickness. By separating the governing equations into oscillatory (fast) and time-averaged (slow) components, we obtain an analytical solution for the oscillatory fields and represent their velocity structure through isolines of stream function. Averaging over the fast time scale yields a set of amplitude equations that describe the slow evolution of the free deformable surface. The stability analysis reveals that gravity and surface tension stabilise the interface, while van der Waals attraction and the imposed thermal gradient destabilise. Vertical vibrations may stabilise the surface: at low frequencies even large amplitudes fail to suppress the long-wave instability for moderate and high Marangoni numbers, whereas at moderate to high frequencies sufficiently strong vibrations stabilise the film across the entire wavenumber spectrum. For a huge values of Marangoni number small vibrations are ineffective, but when Marangoni number is small complete stabilisation is achieved at moderate frequencies for all amplitudes considered. Results obtained in limiting cases are consistent with the previous studies for isothermal and non-vibrated cases.