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

Author

Listed:
  • Giovanni P. Galdi

    (Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15260, USA)

Abstract

We provide sufficient conditions for the occurrence of time-periodic Hopf bifurcation for the coupled system constituted by a rigid sphere, S , freely moving under gravity in a Navier-Stokes liquid. Since the region of flow is unbounded (namely, the whole space outside S ), 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.

Suggested Citation

  • Giovanni P. Galdi, 2021. "On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid," Mathematics, MDPI, vol. 9(7), pages 1-29, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:7:p:715-:d:524213
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