Nonlinear dynamics of a sustained thermocapillary flow of a thin liquid film in a confined two-layer system under a hydrophobic surface

In this work, we study the nonlinear dynamics of a horizontal two-layer gas-liquid system enclosed between a flat hydrophobic wall from above maintained at a constant temperature and a heat-conducting thick periodically corrugated wall of left-right asymmetric topography uniformly heated at its underside from below. The cases of a lower substrate either more or less (such as some polymers, e.g., PDMS) thermally conducting than the liquid are considered. Hydrophobicity of the upper wall is demonstrated to enable a continuous flow in narrow systems preventing the liquid-film rupture. We find that for moderately large values of the Marangoni number, the liquid-gas interface deforms toward rupture at the upper wall until the onset of its interaction with the latter through the disjoining pressure. The interface finally reaches a steady state allowing for a non-zero average flow rate through the system, which is driven and sustained by thermocapillarity and the asymmetry of the lower substrate. We investigate the variation of the flow rate with respect to the system parameters. The direction of the average flow depends also on the system parameters. It is found that for the lower substrate with a lesser thermal conductivity than that of the liquid, the steady flow rate may be much higher than in the opposite case. In the case of a bottom substrate less conducting than the liquid, there exists a transition zone in terms of Marangoni numbers where the system exhibits time-periodic dynamics. This zone becomes wider with an increase of the wall waviness.