Drift of elastic hinges in quasi-two-dimensional oscillating shear flows

In low-Reynolds-number flows, time reversibility makes it impossible for rigid particles to self-propel using reciprocal motions. When passive particles are freely suspended in a background flow that sinusoidally oscillates and is on average stationary, rigid particles are likewise stationary on average. However, as we demonstrate, an elastic particle in such a flow may experience net translation over each period of the flow oscillation. In this paper we analytically and numerically explore the dynamics of particles in steady and oscillatory shear flows. In particular, we study hinges, which we define as a particle consisting of two slender rigid rods joined at one of their ends at a hinge point. We focus on two classes of hinges: rigid hinges, where the two arms of the hinge are fixed relative to each other, and elastic hinges, where the two arms are allowed to rotate relative to each other with a restoring torque provided by a linear torsional spring. Hinges serve as qualitative analogs for curved fibers, which are an important class of particles in many biological and industrial flow applications. We first analyze the motion of elastic hinges in flow-free environments, providing an asymptotic theory to explain the emergence of symmetry breaking in the net translation between a hinge that is initially open versus one that is initially closed. We next turn to studying hinges in flows. In steady shear, both rigid and elastic hinges undergo periodic motions similar to Jeffery orbits with no steady cross-streamline motion. When the hinges are placed in a shear flow whose shear rate oscillates with time, the rigid hinge undergoes no net motion while an elastic hinge's motion is characterized by attracting cycles in the phase space, which undergo bifurcations as the geometric and flow parameters vary. These cycles lead to nonreciprocal translational motions of the hinge, related to the symmetry-breaking motion presented in the flow-free case, which vary in both magnitude and direction as a function of the controlling parameters. This raises the possibility of designing hinges with particular geometric parameters and then tuning the macroscopic flow properties to control and manipulate the particles while also aiding in particle separation, or, conversely, mixing.