Flow-induced Oscillations via Hopf Bifurcation in a Fluid-Solid Interaction Problem
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Bonheure, Denis Galdi, Giovanni P. Gazzola, Filippo |
| author_facet | Bonheure, Denis Galdi, Giovanni P. Gazzola, Filippo |
| contents | We furnish necessary and sufficient conditions for the occurrence of a Hopf bifurcation in a particularly significant fluid-structure problem, where a Navier-Stokes liquid interacts with a rigid body that is subject to an undamped elastic restoring force. The motion of the coupled system is driven by a uniform flow at spatial infinity, with constant dimensionless velocity $λ>0$. In particular, if the relevant linearized operator meets suitable spectral properties, there exists a threshold $λ_o>0$ above which a bifurcating time-periodic branch stems out of the branch of steady-state solutions. The most remarkable feature of our result is that no restriction is imposed on the frequency $ω$ of the bifurcating solution, which may thus coincide with one of the natural structural frequencies $ω_{\sf n}$ of the body. Therefore, resonance cannot occur as a result of this bifurcation. However, when $ω\toω_{\sf n}$, the amplitude of oscillations may become very large when the fluid density is negligible compared to the mass of the body. To our knowledge, our result is the first {\it rigorous} investigation of the existence of a Hopf bifurcation in a fluid-structure interaction problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04198 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Flow-induced Oscillations via Hopf Bifurcation in a Fluid-Solid Interaction Problem Bonheure, Denis Galdi, Giovanni P. Gazzola, Filippo Analysis of PDEs 76D05, 74F10, 35B32, 35Q35, 76D03, 35B10 We furnish necessary and sufficient conditions for the occurrence of a Hopf bifurcation in a particularly significant fluid-structure problem, where a Navier-Stokes liquid interacts with a rigid body that is subject to an undamped elastic restoring force. The motion of the coupled system is driven by a uniform flow at spatial infinity, with constant dimensionless velocity $λ>0$. In particular, if the relevant linearized operator meets suitable spectral properties, there exists a threshold $λ_o>0$ above which a bifurcating time-periodic branch stems out of the branch of steady-state solutions. The most remarkable feature of our result is that no restriction is imposed on the frequency $ω$ of the bifurcating solution, which may thus coincide with one of the natural structural frequencies $ω_{\sf n}$ of the body. Therefore, resonance cannot occur as a result of this bifurcation. However, when $ω\toω_{\sf n}$, the amplitude of oscillations may become very large when the fluid density is negligible compared to the mass of the body. To our knowledge, our result is the first {\it rigorous} investigation of the existence of a Hopf bifurcation in a fluid-structure interaction problem. |
| title | Flow-induced Oscillations via Hopf Bifurcation in a Fluid-Solid Interaction Problem |
| topic | Analysis of PDEs 76D05, 74F10, 35B32, 35Q35, 76D03, 35B10 |
| url | https://arxiv.org/abs/2406.04198 |