Asymmetric Phonons and the Stability of Hagan-Poiseuille Flow

The compliance of structures bounding a fluid has a significant impact on the transition from laminar to turbulent flow. For example, increasing the compliance of spring-backed plates delays the Tollmien-Schlichting wave instability yet advances the traveling wave flutter instability due to distinct interactions between the unstable modes of the fluid and the phonon modes of the structure. However, mechanical metamaterials offer novel platforms to control these instabilities via tuning of the phonon dispersion. Of particular interest are passive and active metamaterials that yield an asymmetry between forwards- and backwards-traveling waves. This work explores the fluid-structure interactions between Hagan-Poiseuille flow and compliant boundaries for which the in-vacuo dynamics asymmetrically decay in the upstream or downstream direction. A linear stability analysis is performed using the Chebyshev-Tau spectral decomposition of the Orr-Sommerfield equation to construct the neutral curves of the Tollmien-Schlichting wave and traveling wave flutter instabilities for different values of the asymmetry parameter. The respective critical Reynold’s numbers are computed over a broader parameter space of the wall compliance.