Thermocapillary thin films: periodic steady states and film rupture

Abstract

Abstract We study stationary, periodic solutions to the thermocapillary thin-film model ∂ t h + ∂ x h 3 ∂ x 3 h − g ∂ x h + M h 2 1 + h 2 ∂ x h = 0 , t > 0 ,   x ∈ R , which can be derived from the Bénard–Marangoni problem via a lubrication approximation. When the Marangoni number M increases beyond a critical value M ∗ , the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability.