Bénard–Marangoni convection of a binary mixture as an example of an oscillatory bifurcation under strong symmetry-breaking effects

Surface-tension-driven convection in a binary mixture is examined in the oscillatory regime, where the first instability at threshold leads to patterns built up by plane waves with a finite critical wave vector. Direct numerical treatment of the 3D hydrodynamic basic equations shows a variety of patterns close to onset. We found steady hexagons, traveling hexagons, traveling and standing waves as well as superpositions of these structures. A Karhunen–Loève analysis is used to study these patterns in more detail. We show further that qualitatively the same patterns can be obtained from a reduced and simplified model, the complex Swift–Hohenberg equation. We show by stability analysis that symmetry-breaking terms, here from the non-symmetric boundary conditions at the top and the bottom of the layer caused by the Marangoni effect, play a crucial role in pattern selection. We expect the obtained bifurcation sequences being generic for all instabilities with finite wavelength that occur via a Hopf bifurcation when strong symmetry-breaking effects are present.