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Thermocapillary thin films : periodic steady states and film rupture

Bruell, Gabriele ; Hilder, Bastian LU and Jansen, Jonas LU (2024) In Nonlinearity 37(4).
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... (More)

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.

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organization
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type
Contribution to journal
publication status
published
subject
keywords
film rupture, global bifurcation theory, quasilinear degenerate-parabolic equation, stationary solutions, thermocapillary instability, thin-film model
in
Nonlinearity
volume
37
issue
4
article number
045016
publisher
London Mathematical Society / IOP Science
external identifiers
  • scopus:85187699495
ISSN
0951-7715
DOI
10.1088/1361-6544/ad2a8a
language
English
LU publication?
yes
id
92c672eb-c27a-490d-89f6-e6ce724e070b
date added to LUP
2024-03-27 13:44:47
date last changed
2024-03-27 13:46:26
@article{92c672eb-c27a-490d-89f6-e6ce724e070b,
  abstract     = {{<p>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 &gt; 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.</p>}},
  author       = {{Bruell, Gabriele and Hilder, Bastian and Jansen, Jonas}},
  issn         = {{0951-7715}},
  keywords     = {{film rupture; global bifurcation theory; quasilinear degenerate-parabolic equation; stationary solutions; thermocapillary instability; thin-film model}},
  language     = {{eng}},
  month        = {{04}},
  number       = {{4}},
  publisher    = {{London Mathematical Society / IOP Science}},
  series       = {{Nonlinearity}},
  title        = {{Thermocapillary thin films : periodic steady states and film rupture}},
  url          = {{http://dx.doi.org/10.1088/1361-6544/ad2a8a}},
  doi          = {{10.1088/1361-6544/ad2a8a}},
  volume       = {{37}},
  year         = {{2024}},
}