Nonlinear regimes of the steady Marangoni convection in a thin film heated from below

Abstract

We investigate the dynamics of a thin liquid film that is placed atop a heated substrate of a very low thermal conductivity. The direct numerical simulation of the stationary long-wave Marangoni instability is performed within 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 previous results of the weakly-nonlinear analysis. A good qualitative agreement is observed for values of the Marangoni number near the instability 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 found the hysteresis. In the vicinity of the instability threshold the film interface is nearly sinusoidal, with two vortices under elevated region. As the supercriticality increases, the nonlinear stationary regimes arise, with a local elevation at the global minimum and with two additional vortices, correspondingly. In a case of subcritical excitation, relatively high supercriticality results in that these regimes evolve into film rupture via the emergence of secondary humps. For the supercritical excitation, stationary regime with doubled wavenumber occurs. We also revealed the transition through the traveling wave between two nonlinear regimes with different high of local humps.