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

It is experimentally known that thin films of viscous fluids on heated plates develop polygonal, spatially periodic patterns. This is due to a self-sustaining thermocapillary effect causing an instability of the trivial constant state.

Building upon a previous work on the one-dimensional case, we consider a two-dimensional thin-film equation. It can be formally derived from the Bénard-Marangoni problem via a long-wave approximation. We consider the stationary problem, which we are able to reduce to a second-order equation amenable to analytic bifurcation theory. The constant solution destabilizes via a (conserved) long-wave instability and we prove existence of a global bifurcation branch of stationary solutions of fixed mass, which are symmetric and periodic with respect to a fixed square or hexagonal lattice.

We finally analyse qualitative aspects of the solutions on the branch both analytically and numerically. Most importantly, we prove that solutions exhibit film rupture, that is, their minimal height tends to zero, under the condition that the Marangoni number is uniformly bounded on the bifurcation branch. This conditional result is substantiated by numerical experiments.

This is joint work with Bastian Hilder and Jonas Jansen.