Flow-induced Oscillations via Hopf Bifurcation in a Fluid-Solid Interaction Problem

Abstract

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 $\lambda>0$. In particular, if the relevant linearized operator meets suitable spectral properties, there exists a threshold $\lambda_o>0$ 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 $\omega$ of the bifurcating solution, which may thus coincide with one of the natural structural frequencies $\omega_{\sf n}$ of the body. Therefore, resonance cannot occur as a result of this bifurcation. However, when $\omega\to\omega_{\sf n}$, 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.