Report
Thermocapillary Thin Films: Periodic Steady States and Film Rupture
| العنوان: | Thermocapillary Thin Films: Periodic Steady States and Film Rupture |
|---|---|
| المؤلفون: | Brüll, Gabriele, Hilder, Bastian, Jansen, Jonas |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics Nonlinear Sciences Physics (Other) |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, Nonlinear Sciences - Pattern Formation and Solitons, Physics - Fluid Dynamics, 35B36, 70K50, 35B32, 37G15, 35Q35, 35K59, 35K65, 35D30, 35Q79, 76A20, 35B10, 35B35 |
| الوصف: | 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. Comment: 31 pages, 8 figures; we added a remark regarding the instability of positive solutions with additional references |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1088/1361-6544/ad2a8a |
| URL الوصول: | http://arxiv.org/abs/2308.11279 |
| رقم الانضمام: | edsarx.2308.11279 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.1088/1361-6544/ad2a8a |
|---|