Geometry-driven dynamics in viscous flows

This project studies the dynamics of passive bodies in low-Reynolds-number shear flows. The classical study of these dynamics dates back to Jeffery, who found that rigid ellipsoids tumble in repeating orbits whose dynamics are governed solely by the particle’s aspect ratio and initial orientation relative to the flow.

Path traced out by major axes of ellipsoids undergoing Jeffery orbits. The trajectory is set by the particle's aspect ratio and the initial orientation of the particle relative to the axis of vorticity.

In (Roggeveen & Stone, 2022), we studied how hinged particles, consisting of two slender bodies joined at a point, moved in similar kinds of flows. In particular, we were interested in how particle geometry, with a particular focus on asymmetry, drives long-time particle motion.

Schematic of the hinged particle system. Note the asymmetry parameter q and the hinge angle α.

A symmetric rigid hinge in steady shear flow simply tumbles, much like an ellipsoid, but with some periodic vertical motion.

An asymmetric rigid hinge adopts a fixed orientation and drifts with a persistent vertical motion (horizontal scales are shrunk by a factor of 10).

We found that some asymmetric shapes adopt fixed orientations and drift persistently across streamlines. This gives a minimal model for understanding how particle geometry can produce cross-streamline migration and influence dispersion in suspensions.

Our later work, (Roggeveen & Stone, 2025), asked what changes when the hinge is allowed to open and close under a torsional spring. In steady shear, the elastic body still undergoes Jeffery-like periodic motion, but elasticity adds an internal degree of freedom that changes the trajectory.

The most interesting behavior appears in oscillating shear. A rigid body in a time-periodic shear flow has no net displacement over a cycle, but an elastic hinge can execute nonreciprocal shape changes and drift. The direction and magnitude of this drift depend on geometry, spring stiffness, and flow parameters.

Effect of changing elasticity on drift speed (left) and the effect of changing equilibrium angle (center) and oscillation frequency (right) on the velocity vector.

The dynamics organize around attracting limit cycles in the angular coordinates of the hinge, θ and α. Initial conditions eventually settle to one of two cycles, which correspond to distinct translational drift behaviors. This gives a compact dynamical-systems description of how a passive elastic particle can move in a time-periodic, force-free environment.

We used a Poincaré map to study the basin structure of these cycles. The map contains attracting fixed points and saddle points that define the phase boundary between the two long-time behaviors.

Poincaré-map view of the angular dynamics. Periodic orbits of the elastic hinge correspond to fixed points of the map, and saddle points organize the phase boundary between long-time drift states.

As the control parameters vary, fixed points and saddles in the Poincaré map collide and reorganize, producing bifurcations and new long-time drift states. The drifting states appear in mirror-symmetric pairs through subcritical pitchfork bifurcations.

Bifurcation map of the Poincaré fixed points as the elasticity is varied, showing both the movement of the points on the map plane and in a more traditional bifurcation diagram.

Long-time drift states

The examples below show hinges initialized near different parts of the phase boundary, together with their θ–α dynamics.

References

2025

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

J. V. Roggeveen, and H. A. Stone

Physical Review Fluids, Mar 2025

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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

James V. Roggeveen , and Howard A. Stone

Journal of Fluid Mechanics, Mar 2022

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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.