Thermocapillary Thin Films: Periodic Steady States and Film Rupture
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| author | Brüll, Gabriele Hilder, Bastian Jansen, Jonas |
| author_facet | Brüll, Gabriele Hilder, Bastian Jansen, Jonas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11279 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Thermocapillary Thin Films: Periodic Steady States and Film Rupture Brüll, Gabriele Hilder, Bastian Jansen, Jonas Analysis of PDEs Pattern Formation and Solitons Fluid Dynamics 35B36, 70K50, 35B32, 37G15, 35Q35, 35K59, 35K65, 35D30, 35Q79, 76A20, 35B10, 35B35 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. |
| title | Thermocapillary Thin Films: Periodic Steady States and Film Rupture |
| topic | Analysis of PDEs Pattern Formation and Solitons Fluid Dynamics 35B36, 70K50, 35B32, 37G15, 35Q35, 35K59, 35K65, 35D30, 35Q79, 76A20, 35B10, 35B35 |
| url | https://arxiv.org/abs/2308.11279 |