Periodic and Chaotic Orbits of Plane-Confined Micro-rotors in Creeping Flows

We explore theoretically the complex dynamics and emergent behaviors of spinning spheres immersed in viscous fluid. The particles are coupled to each other via the fluid in which they are suspended: Each particle disturbs the surrounding fluid with a rotlet field and that fluid flow affects the motion of the other particles. We notice the emergence of intricate periodic or chaotic trajectories that depend on the rotors initial position and separation. The point-rotor motions confined to a plane bear similarities to the classic 2D point-vortex dynamics. Our analyses highlight the complexity of the interaction between just a few rotors and suggest richer behavior in denser populations. We discuss how the model gives insight into more complex systems and suggest possible extensions for future theoretical studies.

[1]  S. Grill,et al.  Active chiral processes in thin films. , 2013, Physical review letters.

[2]  S. Ramaswamy,et al.  Hydrodynamics of soft active matter , 2013 .

[3]  Salima Rafaï,et al.  Suspensions with a tunable effective viscosity: a numerical study , 2012, Journal of Fluid Mechanics.

[4]  G M Whitesides,et al.  Dynamic, self-assembled aggregates of magnetized, millimeter-sized objects rotating at the liquid-air interface: macroscopic, two-dimensional classical artificial atoms and molecules. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  J. F. Brady,et al.  Chaos and threshold for irreversibility in sheared suspensions , 2005, Nature.

[6]  M. J. Kim,et al.  Enhanced diffusion due to motile bacteria , 2004 .

[7]  Chjan C. Lim,et al.  Periodic solutions for three sedimenting spheres , 1988 .

[8]  John F. Brady,et al.  Accelerated Stokesian Dynamics simulations , 2001, Journal of Fluid Mechanics.

[9]  C. Peskin,et al.  Modeling and simulation of active suspensions containing large numbers of interacting micro-swimmers , 2013 .

[10]  M. Leoni,et al.  Dynamics and interactions of active rotors , 2010, 1012.2768.

[11]  H. Aref.  INTEGRABLE, CHAOTIC, AND TURBULENT VORTEX MOTION IN TWO-DIMENSIONAL FLOWS , 1983 .

[12]  Kyongmin Yeo,et al.  Rheology and ordering transitions of non-Brownian suspensions in a confined shear flow: effects of external torques. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  David Saintillan,et al.  Electrohydrodynamic interaction of spherical particles under Quincke rotation. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Ramin Golestanian,et al.  Synchronization in a carpet of hydrodynamically coupled rotors with random intrinsic frequency , 2010 .

[15]  H. Rubinsztein-Dunlop,et al.  Optical alignment and spinning of laser-trapped microscopic particles , 1998, Nature.

[16]  George M. Whitesides,et al.  Dynamic self-assembly of magnetized, millimetre-sized objects rotating at a liquid–air interface , 2000, Nature.

[17]  Paul K. Newton,et al.  Vortex Lattice Theory: A Particle Interaction Perspective , 2009, SIAM Rev..

[18]  David G. Grier,et al.  Optical tweezers in colloid and interface science , 1997 .

[19]  Yang Wang,et al.  Dynamic interactions between fast microscale rotors. , 2009, Journal of the American Chemical Society.

[20]  I Pagonabarraga,et al.  Hydrodynamic regimes of active rotators at fluid interfaces , 2008, The European physical journal. E, Soft matter.

[21]  Hassan Aref,et al.  Gröbli's Solution of the Three-Vortex Problem , 1992 .

[22]  Neil Pomphrey,et al.  Integrable and chaotic motions of four vortices. I. The case of identical vortices , 1982, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[23]  Martin R. Maxey,et al.  Simulation of concentrated suspensions using the force-coupling method , 2010, J. Comput. Phys..

[24]  George M. Whitesides,et al.  Dynamic Aggregation of Chiral Spinners , 2002, Science.

[25]  Aparna Baskaran,et al.  Cooperative self-propulsion of active and passive rotors , 2011, 1107.0347.

[26]  Tamás Tél,et al.  THE VORTEX DYNAMICS ANALOGUE OF THE RESTRICTED THREE-BODY PROBLEM : ADVECTION IN THE FIELD OF THREE IDENTICAL POINT VORTICES , 1997 .

[27]  George Em Karniadakis,et al.  Dynamic Self-Assembly of Spinning Particles , 2007 .

[28]  D. Wolf,et al.  Chaotic particle dynamics in viscous flows: the three-particle stokeslet problem , 1997 .

[29]  Enkeleida Lushi,et al.  Collective dynamics in a binary mixture of hydrodynamically coupled microrotors. , 2015, Physical review letters.

[30]  Petia M Vlahovska,et al.  Electrohydrodynamic rotations of a viscous droplet. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  A. Menzel.  Tuned, driven, and active soft matter , 2015, 1501.07266.

[32]  Sangtae Kim,et al.  Microhydrodynamics: Principles and Selected Applications , 1991 .

[33]  Neil Pomphrey,et al.  Integrable and chaotic motions of four vortices , 1980 .

[34]  Jean-Baptiste Caussin,et al.  Emergence of macroscopic directed motion in populations of motile colloids , 2013, Nature.

[35]  M. Shelley,et al.  Emergence of coherent structures and large-scale flows in motile suspensions , 2012, Journal of The Royal Society Interface.