Thermocapillary Thin Films: Periodic Steady States and Film Rupture

• DOI: 10.48550/arxiv.2308.11279
• Publication Date: 2023-01-01
• Authors: Gabriele Brüll; Bastian Hilder; Jonas Jansen
• Keywords: Stationary state

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