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| We consider a thin liquid film atop a heated substrate. The substrate is subject to vertical oscillations of amplitude $b$ and frequency $\omega$. Two particular cases are analyzed: (i) the isothermal substrate and (ii) the substrate thermally insulated for perturbations. In both cases we consider a wide interval of frequencies $\omega$, from ultra-high (the Stokes layer is thin with respect to mean film thickness) to ultra-low (the vibration period is compared to the typical time of the layer thickness relaxation $\tau$). Two frequency regimes are analyzed in detail: a high frequency ($\omega\tau \gg 1$), when the averaging approach is applied; and an ultra-low frequency ($\omega\tau=O(1)$), when the vibration results in the gravity modulation only. At high frequencies, in case (i) the conventional Kopbosynov--Pukhnachev equation [Oron et al., Rev. Mod. Phys. (1997)] holds with additional vibro-generated averaged terms. In case (ii) the analysis is more involved because the temperature oscillations cannot be neglected; thus additional terms enter the set of amplitude equations derived in [Shklyaev et al., PRE (2012)]. Intermediate asymptotics is developed to match the amplitude equations at high-frequency and ultra-low frequency.
S.Sh. is supported by RFBR (grant N13-01-96010). |
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