Purcell's elastic swimmer: Drift of elastic hinges in oscillatory shear flows

In low-Reynolds-number flows, time reversibility makes it challenging for particles and organisms to sustain locomotion without the presence of an external force. For example, Purcell demonstrated that simply opening and closing a hinge, which is a successful form of locomotion in fluids at high Reynolds numbers, leads to no net motion of the object under the Stokes flow approximation. Nature has found ways to avoid this constraint such as by using wave-like motions of elastic filaments that are driven by the organism itself. In terms of passive particles, the authors have recently shown that it is possible to enable passive motion across streamlines, i.e., a steady drift, of a rigid slender particle in a steady shear flow if the particle is suitably asymmetric. In this work, we extend these ideas to consider the motions of a symmetrically bent slender filament when it is allowed to operate like a linearly elastic hinge. We consider the case where such a particle is placed into a flow with a sinusoidally varying shear rate, resulting in no net displacement of the fluid packets. We show that by simply adding one degree of elastic freedom to the symmetric hinge-shaped body that the particle will experience a net drift even as the fluid remains stationary on average.