Thermocapillary Thin Film Flows: Periodic Steady States and Film Rupture

Thin fluid films on heated planes exhibit the formation of spatially periodic structures. These can take the form of regular polygonal pattern, which was experimentally observed by Henri Bénard in 1900, or film rupture. Later, it was understood that the emergence of these patterns is caused by thermocapillary effects and the mathematical problem is known as the Bénard--Marangoni problem.

In this talk, I will derive a deformational asymptotic model for the Bénard--Marangoni problem in the thin-film limit. In this model, the flat state destabilises via a (conserved) long-wave instability and periodic solutions bifurcate via a subcritical pitchfork bifurcation. I will demonstrate that the bifurcation curve can be extended to a global bifurcation branch. Moreover, I will construct periodic stationary film-rupture solutions as limit points of the bifurcation branch.

This work is a collaboration with Gabriele Brüll and Bastian Hilder.