Thermocapillary Thin Films: Periodic Steady States and Film Rupture

• Published in: arXiv.org
• Main Authors: Brüll, Gabriele, Hilder, Bastian, Jansen, Jonas
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 22.05.2024
• Subjects: Bifurcation theory; Dynamic stability; Mathematics - Analysis of PDEs; Physics - Fluid Dynamics; Physics - Pattern Formation and Solitons; Stability analysis; Thin films
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2308.11279

We study stationary, periodic solutions to the thermocapillary thin-film model \begin{equation*} \partial_t h + \partial_x \Bigl(h^3(\partial_x^3 h - 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 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.