Parametric resonance in spherical immersed elastic shells

TL;DR

This paper analyzes the stability of a spherical elastic shell immersed in a viscous fluid under parametric excitation, revealing resonances that could influence biological systems like the human heart.

Contribution

It introduces a Floquet stability analysis for fluid-structure interactions with parametric forcing in spherical shells, combining analytical and numerical methods.

Findings

Parametric resonance causes unbounded solutions despite viscosity.

Validation of analytical results with 3D numerical simulations.

Potential implications for cardiac fluid dynamics.

Abstract

We perform a stability analysis for a fluid-structure interaction problem in which a spherical elastic shell or membrane is immersed in a 3D viscous, incompressible fluid. The shell is an idealised 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, considering 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. Finally, potential applications to biological systems are discussed, with a particular focus on the human heart and investigating whether or not fluid-structure interaction-mediated instabilities could play a role in cardiac fluid dynamics.