Parametric resonance in immersed elastic structures, with application to the cochlea

Examples of fluid motion driven by immersed flexible structures abound in nature. In many biological settings, for instance a beating heart, an active material generates a time-dependent internal forcing on the surrounding fluid. Motivated by such active biological structures, this thesis investigates parametric resonance in fluid-structure systems induced by an internal forcing via periodic modulation of the material stiffness. One particular application that we study is the cochlea which is the primary component for pitch selectivity in the mammalian hearing system. We present a 2D model of the cochlea in which a periodic internal forcing gives rise to amplification of basilar membrane (BM) oscillations. This forcing is inspired by experiments showing that outer hair cells within the cochlear partition change their lengths when stimulated, which can in turn distort the partition and modulate tension across the BM. We demonstrate the existence of resonant (unstable) solutions through a Floquet stability analysis of the linearized governing equations. Moreover, we show that an internal forcing is sufficient to produce travelling waves along the BM in the absence of any external stimulus. We next examine parametric instabilities in a 3D system by considering a closed spherical elastic membrane (or shell) immersed in a viscous, incompressible fluid. A Floquet analysis for both inviscid and viscous systems shows that parametric resonance is possible even in the presence of fluid viscosity. Numerical simulations are presented to verify the analysis and an application to cardiac fluid dynamics is discussed. Finally, we deviate from the topic of parametric instabilities to consider the natural oscillations of unforced spherical elastic membranes. We present a linear stability analysis to obtain a dispersion relation for immersed membrane oscillations for both inviscid and viscous fluids, as well as a nonlinear analysis of immersed membrane oscillations in an inviscid fluid. We then present an experiment where we measure oscillation frequencies of immersed water balloons in an attempt to corroborate the analytical results.