Pattern formation and film rupture in a two-dimensional thermocapillary thin-film model of the Bénard-Marangoni problem

Together with Stefano Böhmer (Lund University) and Bastian Hilder (TU Munich), I have written a new paper. It is a continuation of our mathematical investigations into the Bénard–Marangoni problem. Henri Bénard was the first to describe the emergence of regular polygonal patterns in liquid films heated from below. Experiments in the 1990s have suggested that in certain regimes even dewetting phenomena occur in such liquids. In our new paper, we study an asymptotic model of the Bénard–Marangoni problem that focusses on describing the deformations in the film height.

In a previous work of Bastian Hilder, Gabriele Brüll and myself, we have studied the same equation on one spatial dimension and proved the emergence of stationary periodic film-rupture solutions by means of global analytic bifurcation theory and Hamiltonian dynamics. Now, we have investigated the physically relevant two dimensional model. We can prove that there are global bifurcation branches consisting of hexagonal and square patterns. At least numerically, they tend to film-rupture solutions. This is also substantiated by numerical analyses. In the paper, we use numerical continuation methods to obtain the bifurcation analysis also numerically. But also numerical experiments on the time-dependent problem has given ample evidence for the emergence of ordered film rupture.

The preprint can be found on arXiv https://arxiv.org/abs/2506.19795. Further supplementary material including videos of the modulating fronts can be found on GitHub https://github.com/steboelu/2D_thermoeq.