In (Roggeveen & Stone, 2022), we studied the motion of rigid hinges in steady shear, which either oscillate vertically while tumbling or drift at a fixed orientation depending on the degree of asymmetry.
When the rigid hinge is replaced with an elastic hinge, as in (Roggeveen & Stone, 2025), with a torsional spring allowing relative rotation of the two arms, the shape of the trajectory is alerted but not fundamentally changed.
However, when the steady shear flow is replaced with an oscillating shear flow, elastic hinges experience non-reciprocal motion and drift over each period of the flow’s oscillation. This is in contrast to rigid hinges, which experience no net displacement over one period of the flow’s oscillation.
In the paper, we show that the non-reciprocal motion corresponds to one of two limit cycles in the angular coordiantes of the hinge, $\theta$ and $\alpha$. No matter the initial conditions the dynamics eventually settle to one of these two cycles, which correspond to limit cycles in the translational velocity phase plane.
We can use a Poincaré map to study the dynamics of these limit cycles. We find two fixed points and two saddle points, corresponding to the phase boudnary, present in the map. Two examples of hinges started near one of the saddle points are given below, along with their $\theta$ - $\alpha$ dynamics and corresponding Poincaré map.
Hinge example one
Hinge example two
References
2025
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.
2022
Motion of asymmetric bodies in two-dimensional shear flow
At low Reynolds numbers, axisymmetric ellipsoidal particles immersed in a shear flow undergo periodic tumbling motions known as Jeffery orbits, with the orbit determined by the initial orientation. Understanding this motion is important for predicting the overall dynamics of a suspension. While slender fibres may follow Jeffery orbits, many such particles in nature are neither straight nor rigid. Recent work exploring the dynamics of curved or elastic fibres have found Jeffery-like behaviour along with chaotic orbits, decaying orbital constants and cross-streamline drift. Most work focuses on particles with reflectional symmetry; we instead consider the behaviour of a composite asymmetric slender body made of two straight rods, suspended in a two-dimensional shear flow, to understand the effects of the shape on the dynamics. We find that for certain geometries the particle does not rotate and undergoes persistent drift across streamlines, the magnitude of which is consistent with other previously identified forms of cross-streamline drift. For this class of particles, such geometry-driven cross-streamline motion may be important in giving rise to dispersion in channel flows, thereby potentially enhancing mixing.