An analytical and numerical study of parametric resonance in immersed elastic boundaries

We investigate a class of resonant instabilities for fluid-structure interaction problems in which an elastic fiber is immersed in a viscous incompressible fluid. The instabilities are excited via internal forcing driven by periodic variations in the material stiffness. The underlying mathematical model is based on the immersed boundary formalism, which we linearize by assuming small perturbations of the fiber around a flat equilibrium state. The stability analysis makes use of Floquet theory to derive an eigenvalue problem relating the key physical parameters. The resulting solution is much simpler than a similar analysis performed in [8] for a circular fiber geometry, and we show that the results are consistent with this previous analysis. We also uncover an interesting behavior for odd modes, in which forcing by a single odd mode generate instabilities that exhibit a combination of wave numbers, something that is not observed for even wavenumbers. Numerical simulations are presented to verify the analytical results.