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We study a single, freely--floating, inextensible, elastic filament in alinear shear flow: $\mathbf{U}_{0}(x,y) = \dot{\gamma} y \hat{x}$. In ourmodel: the elastic energy depends only on bending; the rate-of-strain,$\dot{\gamma} = S \sin(\omega t)$ is a periodic function of time, $t$; and theinteraction between the filament and the flow is approximated by a localisotropic drag force. Based on the shape of the filament we find five differentdynamical phases: straight, buckled, periodic (with period two, period three,period four, etc), chaotic, and one with chaotic transients. In the chaoticphase, we show that the iterative map for the angle, which the end-to-endvector of the filament makes with the tangent its one end, has period threesolutions; hence it is chaotic. Furthermore, in the chaotic phase the flow isan efficient mixer.

Flexible filament in time-periodic viscous flow: shape chaos and period three