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Steady pattern formation and film rupture in a two-dimensional thermocapillary thin-film model of the Bénard–Marangoni problem

Böhmer, Stefano LU orcid ; Hilder, Bastian LU and Jansen, Jonas LU (2025) In Physica D: Nonlinear Phenomena 484.
Abstract

We study two-dimensional, stationary square and hexagonal patterns in the thermocapillary deformational thin-film model for the fluid height h [Formula presented] that can be formally derived from the Bénard–Marangoni problem via a long-wave approximation. Using a linear stability analysis, we show that the flat surface profile corresponding to the pure conduction state destabilises at a critical Marangoni number M∗ via a conserved long-wave instability. For any fixed absolute wave number k0, we find that square and hexagonal patterns bifurcate from the flat surface profile at M=M∗+4k02. Using analytic global bifurcation theory, we show that the local bifurcation curves can be... (More)

We study two-dimensional, stationary square and hexagonal patterns in the thermocapillary deformational thin-film model for the fluid height h [Formula presented] that can be formally derived from the Bénard–Marangoni problem via a long-wave approximation. Using a linear stability analysis, we show that the flat surface profile corresponding to the pure conduction state destabilises at a critical Marangoni number M∗ via a conserved long-wave instability. For any fixed absolute wave number k0, we find that square and hexagonal patterns bifurcate from the flat surface profile at M=M∗+4k02. Using analytic global bifurcation theory, we show that the local bifurcation curves can be extended to global curves of square and hexagonal patterns with constant absolute wave number and mass. We exclude that the global bifurcation curves are closed loops through a global bifurcation in cones argument, which also establishes nodal properties for the solutions. Furthermore, assuming that the Marangoni number is uniformly bounded on the bifurcation branch, we prove that solutions exhibit film rupture, that is, their minimal height tends to zero. This assumption is substantiated by numerical experiments.

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publication status
published
subject
keywords
Film rupture, Global bifurcation theory, Pattern formation, Planar patterns, Quasilinear degenerate-parabolic equation, Stationary solutions, Thermocapillary instability, Thin-film model
in
Physica D: Nonlinear Phenomena
volume
484
article number
135000
publisher
Elsevier
external identifiers
  • scopus:105021231087
ISSN
0167-2789
DOI
10.1016/j.physd.2025.135000
language
English
LU publication?
yes
id
ee5711f2-c895-4acf-a97f-b5d05f5a6334
date added to LUP
2025-12-08 14:50:05
date last changed
2025-12-08 14:51:18
@article{ee5711f2-c895-4acf-a97f-b5d05f5a6334,
  abstract     = {{<p>We study two-dimensional, stationary square and hexagonal patterns in the thermocapillary deformational thin-film model for the fluid height h [Formula presented] that can be formally derived from the Bénard–Marangoni problem via a long-wave approximation. Using a linear stability analysis, we show that the flat surface profile corresponding to the pure conduction state destabilises at a critical Marangoni number M<sup>∗</sup> via a conserved long-wave instability. For any fixed absolute wave number k<sub>0</sub>, we find that square and hexagonal patterns bifurcate from the flat surface profile at M=M<sup>∗</sup>+4k<sub>0</sub><sup>2</sup>. Using analytic global bifurcation theory, we show that the local bifurcation curves can be extended to global curves of square and hexagonal patterns with constant absolute wave number and mass. We exclude that the global bifurcation curves are closed loops through a global bifurcation in cones argument, which also establishes nodal properties for the solutions. Furthermore, assuming that the Marangoni number is uniformly bounded on the bifurcation branch, we prove that solutions exhibit film rupture, that is, their minimal height tends to zero. This assumption is substantiated by numerical experiments.</p>}},
  author       = {{Böhmer, Stefano and Hilder, Bastian and Jansen, Jonas}},
  issn         = {{0167-2789}},
  keywords     = {{Film rupture; Global bifurcation theory; Pattern formation; Planar patterns; Quasilinear degenerate-parabolic equation; Stationary solutions; Thermocapillary instability; Thin-film model}},
  language     = {{eng}},
  publisher    = {{Elsevier}},
  series       = {{Physica D: Nonlinear Phenomena}},
  title        = {{Steady pattern formation and film rupture in a two-dimensional thermocapillary thin-film model of the Bénard–Marangoni problem}},
  url          = {{http://dx.doi.org/10.1016/j.physd.2025.135000}},
  doi          = {{10.1016/j.physd.2025.135000}},
  volume       = {{484}},
  year         = {{2025}},
}