Geometric control of particle manipulation in a two-dimensional fluid

Abstract

Abstract-Manipulation of particles suspended in fluids is crucial for many applications, such as precision machining, chemical processes, bio-engineering, and self-feeding of microorganisms. In this paper, we study the problem of particle manipulation by cyclic fluid boundary excitations from a geometric-control viewpoint. We focus on the simplified problem of manipulating a single particle by generating controlled cyclic motion of a circular rigid body in a two-dimensional perfect fluid. We show that the drift in the particle location after one cyclic motion of the body can be interpreted as the geometric phase of a connection induced by the system's hydrodynamics. We then formulate the problem as a control system, and derive a geometric criterion for its nonlinear controllability. Moreover, by exploiting the geometric structure of the system, we explicitly construct a feedback-based gait that results in attraction of the particle towards the rigid body. We argue that our gait is robust and model-independent, and demonstrate it in both perfect fluid and Stokes fluid.

Quasisteady Particle Transport in Slowly Varying Periodic Streaming Flows

“…Stokes drift provides another example of geometric phase. The deliberate displacement of fluid particles as a result of cyclic displacements in the position of a cylindrical probe was treated as a control problem in Or et al (2009). The objective therein was the design of trajectories in the manifold of probe positions to generate desired fiberwise displacements in a symplectic bundle over this manifold.…”

Hamiltonian mechanics and nonlinear dynamics of a body subject to time-varying gyroscopic and potential forces

“…A geometric theory of controllability for symplectic control systems (i.e., two-dimensional Liouville systems) has been developed in [7], while the extension to higher-dimensional Liouville systems will be the subject of future work.…”

The control of particles in the Stokes limit

“…With this richness of emergent behaviours, the consideration of multi-particle controllability thereby warrants the treatment of non-local interactions, of particular relevance given recent developments in experimental control via optical tweezers (Zou et al 2020). One such exploration, though notably in the case of inviscid flows, considered the control of a passive particle in a two-dimensional fluid, with control effected by the movement of a cylinder and the induced flow (Or et al 2009), though similar such evaluations in Stokes flow are currently lacking, to the best of the authors' knowledge.…”