Parametric Resonance in Immersed Elastic Boundaries

In this paper, we investigate the stability of a fluid-structure interaction problem in which a flexible elastic membrane immersed in a fluid is excited via periodic variations in the elastic stiffness parameter. This model can be viewed as a prototype for active biological tissues such as the basilar membrane in the inner ear, or heart muscle fibers immersed in blood. Problems such as this, in which the system is subjected to internal forcing through a parameter, can give rise to “parametric resonance.” We formulate the equations of motion in two dimensions using the immersed boundary formulation. Assuming small amplitude motions, we can apply Floquet theory to the linearized equations and derive an eigenvalue problem whose solution defines the marginal stability boundaries in parameter space. The eigenvalue equation is solved numerically to determine values of fiber stiffness and fluid viscosity for which the problem is linearly unstable. We present direct numerical simulations of the fluid-structure interaction problem (using the immersed boundary method) that verify the existence of the parametric resonances suggested by our analysis.