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

TL;DR

This paper establishes conditions under which a sphere moving in a Navier-Stokes fluid undergoes time-periodic bifurcation, addressing challenges posed by the unbounded flow domain and formulating the problem in a suitable functional analytic setting.

Contribution

It introduces a novel functional framework for analyzing bifurcation in unbounded fluid domains and applies recent operator theory to identify conditions for time-periodic bifurcation.

Findings

Sufficient conditions for Hopf bifurcation are derived.

The problem is formulated as coupled operator equations in Banach spaces.

The operators involved are shown to be Fredholm of index 0.

Abstract

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