On the Occurrence of Resonance in Elastic-Dissipative Coupled Systems

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GaldiDMS-1614011 Over the past two decades, the study of the motion of a viscous liquid in the presence of rigid or deformable bodies has become one of the main focuses of applied research. The aim of this project is to address significant aspects of the above type of phenomenon and is directed toward the accomplishment of the following objectives. The first one regards occurrence of resonance phenomena in the interaction of a viscous liquid with an elastic structure. For example, in the context of modeling of arterial blood flow it is of the utmost importance to find out whether, for a given model, the pulsatile action of the heart pumping blood would produce an unrealistic high-amplitude oscillation of the arterial wall. The second one deals with the so-called vortex-induced oscillations of a structure in the uniform stream of a viscous liquid. This phenomenon is considered to play a major role in the collapse of chimneys and bridges under wind load. The most famous event in this sense is probably the failure of the Tacoma Narrows bridge. From a technical viewpoint, the above questions are addressed by studying the existence of time-periodic solutions and the occurrence of Hopf bifurcation in coupled systems of nonlinear hyperbolic-parabolic equations that model fluid-structure interactions. One of the main focuses of this project is to understand whether the dissipative mechanism (the parabolic component of the system) can prevent the occurrence of resonance in the elastic structure (the hyperbolic component). Some of these problems need to be formulated in unbounded spatial domains, like infinite pipeline systems or regions exterior to a finite number of bodies. The latter may add an even further complication, in that the linearized relevant operators may possess, in such cases, a non-empty essential spectrum.