Stability of equilibria and bifurcations for a fluid-solid interaction problem

We study certain significant properties of the equilibrium configurations of a rigid body subject to an undamped elastic restoring force, in the stream of a viscous liquid in an unbounded 3D domain. The motion of the coupled system is driven by a uniform flow at spatial infinity, with constant dimensionless velocity $\lambda$. We show that if $\lambda$ is below a critical value, $\lambda_c$ (say), there is a unique and stable time-independent configuration, where the body is in equilibrium and the flow is steady. We also prove that, if $\lambda<\lambda_c$, no oscillatory flow may occur. Successively, we investigate possible loss of uniqueness by providing necessary and sufficient conditions for the occurrence of a steady bifurcation at some $\lambda_s\ge \lambda_c$.