Spirographic motion in a vortex

TL;DR

This paper investigates the complex, spirographic trajectories of inertialess dumbbells in two-dimensional vortices, revealing how flow properties influence particle motion and the existence of transport barriers.

Contribution

It introduces a dynamical systems approach to analyze dumbbell motion in vortices, extending understanding beyond point particles to rigid bodies in nonlinear flow fields.

Findings

Dumbbell center of mass follows spirographic trajectories in decreasing angular velocity vortices.

Trajectory shape depends on initial conditions but is qualitatively similar across vortex types.

Transport barriers form in non-monotonic vortices, affecting particle transport.

Abstract

Studies of particle motion in vortical flows have mainly focused on point-like particles, either inertial or self-propelled. This approximation assumes that the velocity field that surrounds the particle is linear. We consider an inertialess rigid dumbbell in a two-dimensional steady vortex. While the system remains analytically tractable, the particle experiences the nonlinearity of the surrounding velocity field. By exploiting the rotational symmetry of the flow, we reduce the problem to that of a two-dimensional dynamical system, whose fixed points and periodic orbits can be used to explain the motion of the dumbbell. For all vortices in which the fluid angular velocity decreases with radial distance, the center of mass of the dumbbell follows a spirographic trajectory around the vortex center. This results from a periodic oscillation in the radial direction combined with revolution around the center. The shape of the trajectory depends strongly on the initial position and orientation of the dumbbell, but the dynamics is qualitatively the same irrespective of the form of the vortex. If the fluid angular velocity is not monotonic, the spirographic motion is altered by the existence of transport barriers, whose shape is now sensitive to the details of the vortex.