Longitudinal Modulation of Marangoni Wave Patterns in Thin Film Heated From Below: Instabilities and Control - Samoilova, Anna; Nepomnyashchy, Alexander (Frontiers in Applied Mathematics and Statistics, 2021)
Nonlinear Marangoni waves, which are generated by the longwave oscillatory instability of the conductive state in a thin liquid film heated from below in the case of the deformable free surface and a substrate of very low conductivity, are considered. Previously, the investigation of traveling Marangoni waves was restricted to the analysis of the bifurcation and stability with respect to disturbances with strongly different wavevectors. In the present paper, for the first time, the modulational instability of traveling waves is investigated. We derive the amplitude equation for modulated traveling wave, which describes nonlinear interaction of the main convective pattern with the perturbations with slightly different wavenumbers. The amplitude equation differs from the conventional complex Ginzburg-Landau equation, as it contains additional term of the local liquid level rise. Linear stability analysis reveals two modulational instability modes: the amplitude modulational and the phase modulational (Benjamin-Feir) ones. It is shown that traveling rolls are stable against the longitudinal modulation for the uncontrolled convection. We also investigate the influence of the nonlinear feedback control, which was applied previously to eliminate subcritical excitation of traveling rolls. Computations reveal both the modulational modes under the nonlinear feedback control. The obtained results show that the modulational instabilities influence significantly the region of parameters where the nonlinear feedback control is efficient for stabilization of waves.