Nonlinear three-dimensional patterns of the Marangoni convection in a thin film on a poorly conducting substrate

We investigate the dynamics of a thin liquid film that is placed atop a heated substrate of very low thermal conductivity. The direct numerical simulation of the stationary long-wave Marangoni instability is performed with the system of coupled partial differential equations. These equations were previously derived within the lubrication approximation; they describe the evolution of film thickness and fluid temperature. We compare our results with the early reported results of the weakly nonlinear analysis. A good qualitative agreement is observed for values of the Marangoni number near the convective threshold. In the case of supercritical excitation, our results for the amplitudes are described by the square root dependence on the supercriticality. In the case of subcritical excitation, we report the hysteresis. For relatively high supercriticality, the convective regimes evolve into film rupture via the emergence of secondary humps. For the three-dimensional patterns, we observe rolls or squares, depending on the problem parameters. We also confirm the prediction of the asymptotic results concerning the nonlinear feedback control for the pattern selection. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.