Fast-moving pattern interfaces close to a Turing instability in an asymptotic model for the three-dimensional Bénard-Marangoni problem
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912056925487104 |
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| author | Hilder, Bastian Jansen, Jonas |
| author_facet | Hilder, Bastian Jansen, Jonas |
| contents | We study the bifurcation of planar patterns and fast-moving pattern interfaces in an asymptotic long-wave model for the three-dimensional Bénard-Marangoni problem, which is close to a Turing instability. We derive the model from the full free-boundary Bénard-Marangoni problem for a thin liquid film on a heated substrate of low thermal conductivity via a lubrication approximation. This yields a quasilinear, fully coupled, mixed-order degenerate-parabolic system for the film height and temperature. As the Marangoni number $M$ increases beyond a critical value $M^*$, the pure conduction state destabilises via a Turing(-Hopf) instability. Close to this critical value, we formally derive a system of amplitude equations which govern the slow modulation dynamics of square or hexagonal patterns. Using center manifold theory, we then study the bifurcation of square and hexagonal planar patterns. Finally, we construct planar fast-moving modulating travelling front solutions that model the transition between two planar patterns. The proof uses a spatial dynamics formulation and a center manifold reduction to a finite-dimensional invariant manifold, where modulating fronts appear as heteroclinic orbits. These modulating fronts facilitate a possible mechanism for pattern formation, as previously observed in experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_02708 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fast-moving pattern interfaces close to a Turing instability in an asymptotic model for the three-dimensional Bénard-Marangoni problem Hilder, Bastian Jansen, Jonas Analysis of PDEs Dynamical Systems Pattern Formation and Solitons 35B36, 70K50, 35B32, 35Q35, 35K41, 35K59, 35K65, 35Q79, 76A20, 35B06, 34C37, 37L10 We study the bifurcation of planar patterns and fast-moving pattern interfaces in an asymptotic long-wave model for the three-dimensional Bénard-Marangoni problem, which is close to a Turing instability. We derive the model from the full free-boundary Bénard-Marangoni problem for a thin liquid film on a heated substrate of low thermal conductivity via a lubrication approximation. This yields a quasilinear, fully coupled, mixed-order degenerate-parabolic system for the film height and temperature. As the Marangoni number $M$ increases beyond a critical value $M^*$, the pure conduction state destabilises via a Turing(-Hopf) instability. Close to this critical value, we formally derive a system of amplitude equations which govern the slow modulation dynamics of square or hexagonal patterns. Using center manifold theory, we then study the bifurcation of square and hexagonal planar patterns. Finally, we construct planar fast-moving modulating travelling front solutions that model the transition between two planar patterns. The proof uses a spatial dynamics formulation and a center manifold reduction to a finite-dimensional invariant manifold, where modulating fronts appear as heteroclinic orbits. These modulating fronts facilitate a possible mechanism for pattern formation, as previously observed in experiments. |
| title | Fast-moving pattern interfaces close to a Turing instability in an asymptotic model for the three-dimensional Bénard-Marangoni problem |
| topic | Analysis of PDEs Dynamical Systems Pattern Formation and Solitons 35B36, 70K50, 35B32, 35Q35, 35K41, 35K59, 35K65, 35Q79, 76A20, 35B06, 34C37, 37L10 |
| url | https://arxiv.org/abs/2410.02708 |