Jeffery's orbits and microswimmers in flows: A theoretical review

Abstract

In this review, we provide a theoretical introduction to Jeffery's equations for the orientation dynamics of an axisymmetric object in a flow at low Reynolds number, and review recent theoretical extensions and applications to the motions of self-propelled particles, so-called microswimmers, in external flows. Bacteria colonize human organs and medical devices even with flowing fluid, microalgae occasionally cause huge harmful toxic blooms in lakes and oceans, and recent artificial microrobots can migrate in flows generated in well-designed microfluidic chambers. The Jeffery equations, a simple set of ordinary differential equation, provide a useful building block in modeling, analyzing, and understanding these microswimmer dynamics in a flow current, in particular when incorporating the impact of the swimmer shape since the equations contain a shape parameter as a single scalar, known as the Bretherton parameter. The particle orientation forms a closed orbit when situated in a simple shear, and this non-uniform periodic rotational motion, referred to as Jeffery's orbits, is due to a constant of motion in the non-linear equation. After providing a theoretical introduction to microswimmer hydrodynamics and a derivation of the Jeffery equations, we discuss possible extensions to more general shapes, including those with rapid deformation. In the latter part of this review, simple mathematical models of microswimmers in different types of flow fields are described, with a focus on constants of motion and their relation to periodic motions in phase space, together with a breakdown of degenerate orbits, to discuss the stable, unstable, and chaotic dynamics of the system. The discussion in this paper will provide a comprehensive theoretical foundation for Jeffery's orbits and will be useful to understand the motions of microswimmers under various flows.

arXiv comment26 pages, 13 figures. To appear in the Journal of the Physical Society of Japan

References in corpus (72)

• Active Particles in Complex and Crowded Environments
• Machine Learning for Fluid Mechanics
• The hydrodynamics of swimming microorganisms
• Physics of Microswimmers - Single Particle Motion and Collective Behavior
• Swimming in circles: Motion of bacteria near solid boundaries
• Hydrodynamic attraction of swimming microorganisms by surfaces
• Fluid dynamics and noise in bacterial cell-cell and cell-surface scattering
• Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations
• The 2019 Motile Active Matter Roadmap
• Direct measurement of the flow field around swimming microorganisms
• Physical Models of Collective Cell Migration
• Green Algae as Model Organisms for Biological Fluid Dynamics
• Nonlinear dynamics of a microswimmer in Poiseuille flow
• Rheotaxis facilitates upstream navigation of mammalian sperm cells
• Active Turbulence
• Self-Propelled Rods: Insights and Perspectives for Active Matter
• Oscillatory Flows Induced by Microorganisms Swimming in Two-dimensions
• Flow Navigation by Smart Microswimmers via Reinforcement Learning
• Artificial Rheotaxis
• Periodic and Quasiperiodic Motion of an Elongated Microswimmer in Poiseuille Flow
• Efficiency optimization and symmetry-breaking in a model of ciliary locomotion
• Squirmers with swirl -- a model for $Volvox$ swimming
• Oscillatory surface rheotaxis of swimming E. coli bacteria
• Separation of microscale chiral objects by shear flow
• Rheotaxis of spherical active particles near a planar wall
• Upstream swimming in microbiological flows
• Cross-stream migration of active particles
• Machine learning strategies for path-planning microswimmers in turbulent flows
• Optimal steering of a smart active particle
• Rotation of a spheroid in a simple shear at small Reynolds number
• Active turbulence in microswimmer suspensions -- the role of active hydrodynamic stress and volume exclusion
• Derivation of a hydrodynamic theory for mesoscale dynamics in microswimmer suspensions
• Locomotion in complex fluids: Integral theorems
• Finding Efficient Swimming Strategies in a Three Dimensional Chaotic Flow by Reinforcement Learning
• Chirality-induced bacterial rheotaxis in bulk shear flows
• Accumulation of motile elongated micro-organisms in turbulence
• Self-learning how to swim at low Reynolds number
• Effect of weak fluid inertia upon Jeffery orbits
• Flexible filaments buckle into helicoidal shapes in strong compressional flows
• Relating microswimmer synthesis to rheotactic tunability
• The Bank Of Swimming Organisms at the Micron Scale (BOSO-Micro)
• Swimming bacteria in Poiseuille flow: the quest for active Bretherton-Jeffery trajectories
• Oscillatory rheotaxis of active droplets in microchannels
• Motility and Phototaxis of $Gonium$, the Simplest Differentiated Colonial Alga
• Gyrotactic trapping in laminar and turbulent Kolmogorov flow
• Reinforcement learning of optimal active particle navigation
• Reinforcement learning for pursuit and evasion of microswimmers at low Reynolds number
• Encounter rates between bacteria and small sinking particles
• Surfing on turbulence: a strategy for planktonic navigation
• Self-organized swimming with odd elasticity
• Optimising low-Reynolds-number predation via optimal control and reinforcement learning
• Gyrotactic phytoplankton in laminar and turbulent flows: a dynamical systems approach
• Tuning upstream swimming of micro-robots by shape and cargo size
• Preferential rotation of chiral dipoles in isotropic turbulence
• Rheotaxis of Active Droplets
• Flexible fibers in shear flow approach attracting periodic solutions
• Getting drowned in a swirl: deformable bead-spring model microswimmers in external flow fields
• Transport barriers to self-propelled particles in fluid flows
• Angular dynamics of small crystals in viscous flow
• The N-flagella problem: Elastohydrodynamic motility transition of multi-flagellated bacteria
• A deformable microswimmer in a swirl: capturing and scattering dynamics
• Chiral Separation by Flows: The Role of Flow Symmetry and Dimensionality
• Microswimmers in an axisymmetric vortex flow
• Trapping of swimmers in a vortex lattice
• Focusing of Active Particles in a Converging Flow
• Reaching for the surface: Spheroidal microswimmers in surface gravity waves
• Lord Kelvin's isotropic helicoid
• Elastohydrodynamics of swimming helices: effects of flexibility and confinement
• Swinging Motion of Active Deformable Particles in Poiseuille Flow
• Alignment of elongated swimmers in a laminar and turbulent Kolmogorov flow
• Rototaxis: localization of active motion under rotation
• Effective aspect ratio of helices in shear flow

Cited by in corpus (7)

• A review of active matter reviews
• Active particle motion in Poiseuille flow through rectangular channels
• Exploring noisy Jeffery orbits: A combined Fokker-Planck and Langevin analysis in 2D and 3D
• A Hamiltonian formulation for the motion of an active spheroidal particle suspended in laminar straight duct flow
• Rapidly yawing spheroids in viscous shear flow: Emergent loss of symmetry
• Motility and rotation of multi-timescale microswimmers in linear background flows
• Simulations of Three-dimensional Nematic Guidance of Microswimmers