Fast-moving pattern interfaces close to a Turing instability in an asymptotic model for the three-dimensional Bénard–Marangoni problem

Together with Bastian Hilder (TU Munich), I have written a new paper. We study an asymptotic model of the Bénard–Marangoni problem, which describes the dynamics of thin liquid films on planes heated from below. Experiments have demonstrated that these liquid films exhibit the formation of regular polygonal patterns if the temperature is increased beyond a certain threshold. The reason for this is the thermocapillary effect, which also explains the convection inside the cells.

We now study an asymptotic long-wave model of the full Bénard–Marangoni problem, describing the dynamics purely in terms of the film height and temperature, but exhibiting the same crucial features for pattern formation. Most importantly, the pure conduction state of constant film height and temperature destabilises via a Turing instability as the Marangoni number \(M\) increases beyond a critical threshold \(M^*\). We show the formation of regular planar patterns such as roll waves, square and hexagonal patterns.

The next natural question is to obtain a dynamical description of the formation of patterns. Experiments have suggested many times that these patterns form in the wake of an invading front. In our paper, we construct fast-moving travelling front solutions that the transition between two states by a planar front.

The preprint can be found on arXiv https://arxiv.org/abs/2410.02708. Further supplementary material including videos of the modulating fronts can be found on GitHub https://github.com/Bastian-Hilder/TuringUnstableThinFilmFronts.