Non-monotonicity in the cooperative relaxation of immersed filament pairs

The morphology and evolution of a buckled filament in viscous fluid is a canonical problem in elastohydrodynamics, providing key insights into a range of natural and engineered systems. These insights are critical for understanding the emergent patterns of ciliary carpets, designing bio-mimetic metasurfaces, and in the development of novel flow manipulation and control systems. Despite recent interest in the collective dynamics of densely packed filament arrays, detailed studies on the behavior of a hydrodynamically coupled filament pair remain limited. In this work, we present a combined experimental and numerical study of the evolution of a pair of clamped filaments in a viscous medium. When compressed at a specified rate, the pair exhibits wrinkling, followed by relaxation into a stable buckled configuration. We found that the dynamics of this relaxation are closely linked to the gap between the filaments, with relaxation times displaying a non-monotonic dependence on filament spacing. Interestingly, the peak in relaxation time at intermediate gap sizes corresponds to anti-phase equilibrium configurations. A detailed analysis of the flow field between the filaments highlights how phase mismatch at these intermediate gaps retards viscous dissipation, revealing that partial synchronization can significantly slow down relaxation.