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

TL;DR

This paper analyzes the stability and bifurcation behavior of a rigid body interacting with a viscous fluid flow, identifying conditions for stable equilibrium and the onset of bifurcations as flow velocity varies.

Contribution

It provides a rigorous analysis of equilibrium stability and bifurcation conditions for a fluid-solid interaction problem in an unbounded domain.

Findings

Existence of a unique stable equilibrium below a critical flow velocity

No oscillatory flow occurs below the critical velocity

Conditions for steady bifurcation at or above the critical velocity

Abstract

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 $\lambda_s\ge \lambda_c$.