Thermocapillary thin films : periodic steady states and film rupture
(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.
(Less)
- author
- Bruell, Gabriele ; Hilder, Bastian LU and Jansen, Jonas LU
- organization
- publishing date
- 2024-04-01
- 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 > 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}}, }