  We investigate the linear Floquet stability of two fluid layers undergoing
oscillations in the direction parallel to the flexible wall that separates
them. This canonical configuration is inspired by the cerebrospinal fluid flow
in the spinal canal of subjects with hydro-/syringomyelia.The analysis focuses
on the marginal conditions for the onset of instability, and how these depend
on the spatial wavelength of the perturbation, and on the values of the control
parameters, which are the two channel widths, the Reynolds number, and the wall
stiffness. Unstable perturbations are found to oscillate synchronous with the
base flow. The wavelength of the most unstable perturbation, of the order of
the stroke length of the basic oscillatory motion, depends strongly on the wall
stiffness, but is only weakly influenced by the channel widths and the Reynolds
number. In general, around criticality, it was found that increasing the
Reynolds number has a destabilizing effect, and that decreasing the canal
widths stabilizes the instability. The wall stiffness on the other hand has a
non-monotonic effect, exhibiting an intermediate value for which the
instability is maximally amplified. The present analysis is a first step
towards a better understanding of the physical mechanisms that govern many
(bio)fluid mechanical problems that involve oscillatory flows near compliant
walls.
