A ring with a cross-section that has a blunt inner and sharper outer edge can attain an equilibrium orientation in a Newtonian fluid subject to a low Reynolds number simple shear flow. This may be contrasted with the continuous rotation exhibited by most rigid bodies. Such rings align along an orientation when the rotation due to fluid vorticity balances the counter-rotation due to the extensional component of the simple shear flow. While the viscous stress on the particle tries to rotate it, the pressure can generate a counter-vorticity torque that aligns the particle. Using boundary integral computations, we demonstrate ways to effectively control this pressure by altering the geometry of the ring cross-section, thus leading to alignment at moderate particle aspect ratios. Aligning rings that lack fore–aft symmetry can migrate indefinitely along the gradient direction. This differs from the periodic spatial trajectories of fore–aft asymmetric axisymmetric particles that rotate in periodic orbits. The mechanism for migration of aligned rings along the gradient direction is elucidated in this work. The migration speed can be controlled by varying the cross-sectional shape and size of the ring. Our results provide new insights into controlling motion of individual particles and thereby open new pathways towards manipulating macroscopic properties of a suspension.
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Local Stochastic Algorithms for Alignment in Self-Organizing Particle Systems
We present local distributed, stochastic algorithms for alignment in self-organizing particle systems (SOPS) on two-dimensional lattices, where particles occupy unique sites on the lattice, and particles can make spatial moves to neighboring sites if they are unoccupied. Such models are abstractions of programmable matter, composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational capabilities. We consider oriented particle systems, where particles are assigned a vector pointing in one of q directions, and each particle can compute the angle between its direction and the direction of any neighboring particle, although without knowledge of global orientation with respect to a fixed underlying coordinate system. Particles move stochastically, with each particle able to either modify its direction or make a local spatial move along a lattice edge during a move. We consider two settings: (a) where particle configurations must remain simply connected at all times and (b) where spatial moves are unconstrained and configurations can disconnect.
Our algorithms are inspired by the Potts model and its planar oriented variant known as the planar Potts model or clock model from statistical physics. We prove that for any q ≥ 2, by adjusting a single parameter, these self-organizing particle systems can be made to collectively align along a single dominant direction (analogous to a solid or ordered state) or remain non-aligned, in which case the fraction of particles oriented along any direction is nearly equal (analogous to a gaseous or disordered state). In the connected SOPS setting, we allow for two distinct parameters, one controlling the ferromagnetic attraction between neighboring particles (regardless of orientation) and the other controlling the preference of neighboring particles to align. We show that with appropriate settings of the input parameters, we can achieve compression and expansion, controlling how tightly gathered the particles are, as well as alignment or nonalignment, producing a single dominant orientation or not. While alignment is known for the Potts and clock models at sufficiently low temperatures, our proof in the SOPS setting are significantly more challenging because the particles make spatial moves, not all sites are occupied, and the total number of particles is fixed.
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- Award ID(s):
- 2106687
- NSF-PAR ID:
- 10425738
- Date Published:
- Journal Name:
- Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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