Flow-plate interactions: Well-posedness and long-time behavior

Abstract

We consider flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are treated with Neumann type flow conditions, and a novel treatment of the so called Kutta-Joukowsky flow conditions are given in the subsonic case. The goal of the paper is threefold: (i) to provide an accurate review of recent results on existence, uniqueness, and stability of weak solutions, (ii) to present a construction of finite dimensional, attracting sets corresponding to the structural dynamics and discuss convergence of trajectories, and (iii) to state several open questions associated with the topic. This second task is based on a decoupling technique which reduces the analysis of the full flow-structure system to a PDE system with delay.

A thorough look at the (in)stability of piston-theoretic beams

“…On the other hand, such a reduction is a dramatic simplification of complex, multi-physics phenomena; however, focusing on the simple model allows us use robust mathematical theory and perform a thorough numerical study that can be exposited straight-forwardly. (More sophisticated flow-structure models are certainly explored in the rigorous mathematical literature [13,43,12,14]).…”

“…Flutter is fluid-structure (feedback) instability that occurs between elastic displacements of a structure and responsive aerodynamic pressure changes at the flow-structure interface [7,19,10]. The onset of flutter, for particular flow parameters, represents a bifurcation in the linearization of the system [23], and the qualitative properties of the post-flutter dynamics can be analyzed from an infinite dimensional/control-theoretic point of view [24,10,14].…”

“…Indeed, if one can model the dynamic load across the structure, the issue is to determine under what conditions the non-conservative flow loading destabilizes structural (eigen)modes. However, to study the qualitative properties of the dynamics in the post-flutter regime, analysis will require some physical nonlinear restoring force to keep trajectories bounded in time [16,14,10]. Indeed, for the post-flutter dynamics of a given system, the theory of nonlinear dynamical systems is often used, e.g., compact global attractors [12,10] as well as exponential attractors [25,9,12].…”

Dynamics of Quasi-Stable Dissipative Systems

“…In the delay case this method was applied earlier in CHUESHOV/LASIECKA/WEBSTER [63,64] and CHUESHOV/REZOUNENKO [66] for second order models and in CHUESHOV/REZOUNENKO [67] for parabolic equations.…”

Stabilization of a nonlinear flow-plate interaction via component-wise decomposition

“…The classical model [8] is given by a clamped nonlinear plate strongly coupled to a convected wave equation on the half space. In the absence of energy dissipation the plate dynamics converge to a compact and finite dimensional set [6,7]. With a sufficiently large velocity feedback control on the structure we show that the full flow-plate system exhibits strong convergence to the set of stationary states in the natural energy topology.…”