On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid
We provide sufficient conditions for the occurrence of time-periodic Hopf bifurcation for the coupled system constituted by a rigid sphere, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="scri...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/7/715 |
| _version_ | 1797540039540867072 |
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| author | Giovanni P. Galdi |
| author_facet | Giovanni P. Galdi |
| author_sort | Giovanni P. Galdi |
| collection | DOAJ |
| description | We provide sufficient conditions for the occurrence of time-periodic Hopf bifurcation for the coupled system constituted by a rigid sphere, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula>, freely moving under gravity in a Navier-Stokes liquid. Since the region of flow is unbounded (namely, the whole space outside <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula>), the main difficulty consists in finding the appropriate functional setting where general theory may apply. In this regard, we are able to show that the problem can be formulated as a suitable system of coupled operator equations in Banach spaces, where the relevant operators are Fredholm of index 0. In such a way, we can use the theory recently introduced by the author and give sufficient conditions for time-periodic bifurcation to take place. |
| first_indexed | 2024-03-10T12:54:24Z |
| format | Article |
| id | doaj.art-9cb9f5560f6a4d48adccd6c755eed11d |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T12:54:24Z |
| publishDate | 2021-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-9cb9f5560f6a4d48adccd6c755eed11d2023-11-21T12:03:00ZengMDPI AGMathematics2227-73902021-03-019771510.3390/math9070715On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes LiquidGiovanni P. Galdi0Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15260, USAWe provide sufficient conditions for the occurrence of time-periodic Hopf bifurcation for the coupled system constituted by a rigid sphere, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula>, freely moving under gravity in a Navier-Stokes liquid. Since the region of flow is unbounded (namely, the whole space outside <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula>), the main difficulty consists in finding the appropriate functional setting where general theory may apply. In this regard, we are able to show that the problem can be formulated as a suitable system of coupled operator equations in Banach spaces, where the relevant operators are Fredholm of index 0. In such a way, we can use the theory recently introduced by the author and give sufficient conditions for time-periodic bifurcation to take place.https://www.mdpi.com/2227-7390/9/7/715fluid-structure interactionNavier-Stokes equationsHopf bifurcationfalling sphere |
| spellingShingle | Giovanni P. Galdi On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid Mathematics fluid-structure interaction Navier-Stokes equations Hopf bifurcation falling sphere |
| title | On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid |
| title_full | On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid |
| title_fullStr | On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid |
| title_full_unstemmed | On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid |
| title_short | On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid |
| title_sort | on time periodic bifurcation of a sphere moving under gravity in a navier stokes liquid |
| topic | fluid-structure interaction Navier-Stokes equations Hopf bifurcation falling sphere |
| url | https://www.mdpi.com/2227-7390/9/7/715 |
| work_keys_str_mv | AT giovannipgaldi ontimeperiodicbifurcationofaspheremovingundergravityinanavierstokesliquid |