Pattern Formation in a 2D Thermocapillary Thin-Film Equation

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. 

We consider a two-dimensional thin-film equation. It can be retrieved as an asymptotic limit of the Boussinesq–Navier–Stokes model for small fluid height. The equation is of fourth order, quasilinear, and degenerate parabolic but 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.