Hydrodynamic synchronization of nonlinear oscillators at low Reynolds number

Department of Mathematics, University of Bristol, Clifton, Bristol BS8 1TW, United Kingdom.
Physical Review E (Impact Factor: 2.29). 04/2012; 85(4 Pt 1):040901. DOI: 10.1103/PhysRevE.85.040901
Source: PubMed


We introduce a generic model of a weakly nonlinear self-sustained oscillator as a simplified tool to study synchronization in a fluid at low Reynolds number. By averaging over the fast degrees of freedom, we examine the effect of hydrodynamic interactions on the slow dynamics of two oscillators and show that they can lead to synchronization. Furthermore, we find that synchronization is strongly enhanced when the oscillators are nonisochronous, which on the limit cycle means the oscillations have an amplitude-dependent frequency. Nonisochronity is determined by a nonlinear coupling α being nonzero. We find that its (α) sign determines if they synchronize in phase or antiphase. We then study an infinite array of oscillators in the long-wavelength limit, in the presence of noise. For α>0, hydrodynamic interactions can lead to a homogeneous synchronized state. Numerical simulations for a finite number of oscillators confirm this and, when α<0, show the propagation of waves, reminiscent of metachronal coordination.

Download full-text


Available from: Tanniemola B. Liverpool, Oct 06, 2015
22 Reads
  • Source
    • "The biological significance of this property for the proper functioning of pollen tube growth is unknown. It can be speculated that, in the presence of many self-sustained and coupled oscillators as may be the case in pollen tube growth, the ability of the different oscillators to adjust their frequency allows them to better synchronize [22], and avoid destructive interference, as would be the case between different non-synchronous oscillators. The synchronicity of different coupled oscillators may, by preventing destructive interference, increase the robustness of the pollen tube oscillation in the repeatedly changing environment to which the elongating pollen tube is exposed while traversing the different pistillar tissues. "
    [Show abstract] [Hide abstract]
    ABSTRACT: Pollen tubes are tip growing plant cells that display oscillatory growth behavior. It has been demonstrated experimentally that the reduction of the average pollen tube growth rate through elevated extracellular calcium or borate concentrations coincides with a greater amplitude of the growth rate oscillation and a lower oscillation frequency. We present a simple numerical model of pollen tube growth that reproduces these results, as well as analytical calculations that suggest an underlying mechanism. These data show that the pollen tube oscillator is non-isochronous, and is different from harmonic oscillation.
    Mathematical Modelling of Natural Phenomena 07/2013; 8(4):25-34. DOI:10.1051/mmnp/20138403 · 0.81 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: Two-state switching rowers, or simply rowers, are model self-sustained oscillators in a fluid at a low Reynolds number, introduced in Cosentino Lagomarsino et al (2003 Phys. Rev. E 68 021908) and realized experimentally in Kotar et al (2010 Proc. Natl Acad. Sci. USA 107 7669–73). Here we present a new approach for investigating the hydrodynamic synchronization of a pair of rowers with the same and different frequencies. Our analysis is geometrical, in the spirit of the qualitative theory of dynamical systems. By taking advantage of the separation of timescales in the model, the dynamics can be decomposed into a sequence of fast changes followed by slow relaxations. In this framework we discuss how synchronization is determined by the dominant mode of the relaxation dynamics. For rowers with the same frequencies, our analysis recovers naturally the anti-phase synchronized motion, its stability and the basin of attraction (Cosentino Lagomarsino et al 2003 Phys. Rev. E 68 021908; Kotar et al 2010 Proc. Natl Acad. Sci. USA 107 7669–73); in the case of rowers with different frequencies we are able to provide upper bounds for the phase-locked solution and to determine a critical value of the frequency mismatch after which the coordination is lost. Our estimates are in good agreement with numerical simulations. In this way we provide a simple and robust explanation for the transition from the phase-locked state to the loss of synchrony.
    Nonlinearity 09/2012; 25(10):2953. DOI:10.1088/0951-7715/25/10/2953 · 1.21 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: This review summarizes theoretical progress in the field of active matter, placing it in the context of recent experiments. This approach offers a unified framework for the mechanical and statistical properties of living matter: biofilaments and molecular motors in vitro or in vivo, collections of motile microorganisms, animal flocks, and chemical or mechanical imitations. A major goal of this review is to integrate several approaches proposed in the literature, from semimicroscopic to phenomenological. In particular, first considered are “dry” systems, defined as those where momentum is not conserved due to friction with a substrate or an embedding porous medium. The differences and similarities between two types of orientationally ordered states, the nematic and the polar, are clarified. Next, the active hydrodynamics of suspensions or “wet” systems is discussed and the relation with and difference from the dry case, as well as various large-scale instabilities of these nonequilibrium states of matter, are highlighted. Further highlighted are various large-scale instabilities of these nonequilibrium states of matter. Various semimicroscopic derivations of the continuum theory are discussed and connected, highlighting the unifying and generic nature of the continuum model. Throughout the review, the experimental relevance of these theories for describing bacterial swarms and suspensions, the cytoskeleton of living cells, and vibrated granular material is discussed. Promising extensions toward greater realism in specific contexts from cell biology to animal behavior are suggested, and remarks are given on some exotic active-matter analogs. Last, the outlook for a quantitative understanding of active matter, through the interplay of detailed theory with controlled experiments on simplified systems, with living or artificial constituents, is summarized.
    Review of Modern Physics 07/2013; 85(3). DOI:10.1103/RevModPhys.85.1143 · 29.60 Impact Factor
Show more