Synchronization and Collective Dynamics in a Carpet of Microfluidic Rotors

Abstract

We study synchronization of an array of rotors on a substrate that are coupled by hydrodynamic interaction. Each rotor, which is modeled by an effective rigid body, is driven by an internal torque and exerts an active force on the surrounding fluid. The long-ranged nature of the hydrodynamic interaction between the rotors causes a rich pattern of dynamical behaviors including phase ordering and self-proliferating spiral waves. Our results suggest strategies for designing controllable microfluidic mixers using the emergent behavior of hydrodynamically coupled active components. PACS numbers: 87.19.rh,07.10.Cm,47.61.Ne,87.80.Fe,87.85.Qr Introduction. Microorganisms and the mechanical components of the cell motility machinery such as cilia and flagella operate in low Reynolds number conditions where hydrodynamics is dominated by viscous forces [1]. The medium thus induces a long-ranged hydrodynamic interaction between these active objects, which could lead to emergent many-body behaviors. Examples of such cooperative dynamical effects include sperms beating in harmony [2], metachronal waves in cilia [3][4][5], formation of bound states between rotating microorganisms [6], and flocking behavior of red blood cells moving in a capillary [7]. For a collection of free swimmers, such as microorganisms [8], hydrodynamic interactions have been shown to lead to instabilities [9,10] that can result in complex dynamical behaviors [10,11]. In the context of simple microswimmer models where hydrodynamic interactions coupled to internal degrees of freedom can be studied with minimal complexity, it has been shown that the coupling could result in complex dynamical behaviors such as oscillatory bound states between two swimmers [12], and collective many-body swimming phases [13,14].A particularly interesting aspect of such hydrodynamic coupling is the possibility of synchronization between different objects with cyclic motions [4,5,[15][16][17][18][19][20][21]. This effect has mostly been studied in simple systems such as two interacting objects or linear arrays and very little is known about possible many-body emergent behaviors of a large number of active objects with hydrodynamic coupling. For example, in a recent experiment [22], Darnton et al. observed chaotic flow patterns with complex vortices above a carpet of bacteria with their heads attached to a substrate and their flagella free to interact with the fluid (see also [23]). On the other hand, recent advances from micron-scale magnetically actuated tails [24] to synthetic molecular rotors [25] now allow fabrication of arrays of active tails that can stir up the fluid. It is therefore very important to explore the possible complexity of the phase behavior of such an actively stirred microfluidic system.Here, we consider a simple generic model of rotors [26] positioned on a regular 2D array on a substrate and study

Emergence of metachronal waves in cilia arrays

“…The first class of models consists either of rotors-spheres orbiting on quasi-elliptical trajectories near a wall (19)(20)(21)(22)-or rowers-spheres that oscillate on a line with different hydrodynamic radii in the two directions of motion (18, 23)-in both cases under a constant driving force. Two rotors have been found to show asynchronous dynamics-taken as an indication for metachronal coordination-if their distance is close enough or their relative orientation is perpendicular to the beat direction (19).…”

Two-Fluid Equilibrium Considerations of <i>T</i>_e/<i>T</i>_i≫ 1, Collisionless ST Plasmas Sustained by RF Electron Heating

“…Such plasmas are similar in nature to those also observed in LATE [5], QUEST [6], and MAST [7]. Calculation of such equilibrium will help understand the electron and ion fluid force balance properties in these plasmas, and provide an equilibrium basis for further studies of particle orbits, plasma stability, transport, current drive, boundary, and the plasma-wall interface.…”

“…The properties characterizing these plasmas can be described with the help of the MHD equilibrium modeled for an RF-only LATE plasma [5] shown in Fig. 1.…”

Energetics of synchronization in coupled oscillators rotating on circular trajectories

“…The relationship between synchronization and energy dissipation has been attracting much interest, e.g., in the context of low Reynolds-number hydrodynamics [4][5][6][7][8][9] since Taylor's classical work on hydrodynamic synchronization of active objects with periodic motions [10]. Recent extensive theoretical and experimental studies on beating eukaryotic flagella and cilia have elucidated the underlying physical mechanism of hydrodynamic synchronization based on a simplified phase-description without losing its essence [5,[11][12][13][14][15][16][17][18][19][20]. In this phase-description, they are simply modeled as coupled oscillators whose periodic motions are described by phase equations.…”