An Elastic filament in a time-periodic linear shear flow

We numerically study the dynamics of a free elastic filament in a highly viscous linear shear flow in the absence of inertia and brownian motion. We use a bead-spring model with Rotne-Pragor viscosity [1]. In time-independent shear flow the filament shows tumbling, C-buckling, and snake-turn, before becoming straight at late times [2,3,4], for different elasto-viscous numbers, $\Gamma$ -- Dimensionless ratio of viscous and elastic stress. In time-periodic flow (Period T), for $\Gamma < \Gamma_1$, as expected the filament comes back to it's initial position after one period. Surprisingly, for $\Gamma_1 < \Gamma < \Gamma_2$, we find two-cycle -- the filament comes back to it's initial shape not after one but two periods. For $\Gamma > \Gamma_2$, we observe complex dynamical behaviors. Our results are independent of choice of initial conditions. \\ (1) H. Wada and R. R. Netz, EPL (Europhysics Letters)75, 645 (2006). (2) L. E. Becker and M. J. Shelley, Physical Review Letters 87, 198301 (2001).\\ (3) L. Guglielmini, A. Kushwaha, E. S. Shaqfeh, and H. A. Stone, Physics of Fluids 24, 123601 (2012). \\ (4) Y. Liu, B. Chakrabarti, D. Saintillan, A. Lindner, and O. du Roure, Proceedings of the National Academy of Sciences 115, 9438 (2018).