Floquet stability analysis of a two-layer oscillatory flow near a flexible wall as a model to study the flow induced in syringomyelia cavities

Syringomyelia is a disorder characterized by the accumulation of fluid in the spinal cord, forming macroscopic fluid-filled cavities, called syrinxes, the growth of which can lead to progressive neurological damage. It is widely accepted that both the hydrodynamics of the cerebrospinal fluid in the spinal subarachnoid space and the motion of the fluid inside the syringomyelia cavities play an important role in their formation and growth. In the present work we are concerned with the coupling between the motion in these two fluid layers, enabled by the flexible nature of the spinal-cord nervous tissue that separates them. In particular, we have studied a simplified model problem involving the Floquet stability analysis of the oscillatory flow of two layers of fluid separated by an initially undeformed flexible wall. The results of the analysis show that, for a given amount of (linear) coupling between the pressure difference across the wall and the wall deformation, there exists a critical value of the Reynolds number above which the flow becomes unstable to perturbations that are synchronous with the basic oscillatory state.