On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid

• Main Author: Giovanni P. Galdi
• Format: Article
• Language: English
• Published: MDPI AG 2021-03-01
• Series: Mathematics
• Subjects: fluid-structure interaction; Navier-Stokes equations; Hopf bifurcation; falling sphere
• Online Access: https://www.mdpi.com/2227-7390/9/7/715

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.