Parametric Resonance in Spherical Immersed Elastic Shells

We perform a stability analysis for a fluid-structure interaction (FSI) problem in which a spherical elastic shell or membrane is immersed in a 3D viscous, incompressible fluid. The shell is an idealized structure having zero thickness, and has the same fluid lying both inside and outside. The problem is formulated mathematically using the immersed boundary framework in which Dirac delta functions are employed to capture the two-way interaction between fluid and immersed structure. The elastic structure is driven parametrically via a time-periodic modulation of the elastic membrane stiffness. We perform a Floquet stability analysis in the case of both a viscous and inviscid fluid, and demonstrate that the forced fluid-membrane system gives rise to parametric resonances in which the solution becomes unbounded even in the presence of viscosity. The analytical results are validated using numerical simulations with a 3D immersed boundary code for a range of wavenumbers and physical parameter values. Moreover, we propose a benchmark computation that is supported by our analytical results and which other FSI software developers can use to validate their simulations. Finally, potential applications to biological systems are discussed, with a particular focus on the human heart and investigating whether or not FSI-mediated instabilities could play a role in cardiac fluid dynamics.