Abstract
Non-linear Marangoni waves, which are generated by the long-wave oscillatory instability of the conductive state in a thin liquid film heated from below in the case of a 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 wave vectors. In the present article, for the first time, the modulational instability of traveling waves is investigated. We derive the amplitude equation for the modulated traveling wave, which describes non-linear 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 an 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 non-linear feedback control, which was applied previously to eliminate subcritical excitation of traveling rolls. Computations reveal both the modulational modes under the non-linear feedback control. The obtained results show that the modulational instabilities significantly influence the region of parameters where the non-linear feedback control is efficient for stabilization of waves.
Highlights
Marangoni instability arises at the interface due to the dependence of the surface tension on temperature
The pattern selection analysis is based on the investigation of the non-linear interaction of perturbations with a specific wavelength that forms the stable spatially periodic patterns
The function 8(f) 8l + 8qf describes non-linear feedback control based on the measurement of the temperature perturbation on the free surface and imposing a corresponding local change of heat flux on the solid substrate [7, 8]
Summary
Marangoni instability arises at the interface due to the dependence of the surface tension on temperature (the Marangoni effect). Long-wave Marangoni instability in a thin liquid film heated from below is governed by the following system of partial differential equations (see [6, 7], a derivation can be found in Supplementary Appendix SA): zh zt The function 8(f) 8l + 8qf describes non-linear feedback control based on the measurement of the temperature perturbation on the free surface and imposing a corresponding local change of heat flux on the solid substrate [7, 8].
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