{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/thermocapillary-thin-films-periodic-steady","title":"Thermocapillary Thin Films: Periodic Steady States and Film Rupture","arxiv_id":"2308.11279","date":"2023-08-22","proceeding":null,"authors":["Gabriele Brüll","Bastian Hilder","Jonas Jansen"],"abstract":"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 B\\'enard-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.","url_abs":"https://arxiv.org/abs/2308.11279v2","url_pdf":"https://arxiv.org/pdf/2308.11279v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"thermocapillary-thin-films-periodic-steady","repo_url":"https://github.com/bastian-hilder/global-bif-thermocapillary-thin-film-equation","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}