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"Hydrodynamic Synchronization at Low Reynolds Number," Soft Matter 7

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View Online / Journal Homepage / Table of Contents for this issue Soft Matter Dynamic Article LinksC< Cite this: Soft Matter, 2011, 7, 3074 www.rsc.org/softmatter REVIEW Hydrodynamic synchronization at low Reynolds number† Ramin Golestanian,*a Julia M. Yeomans*a and Nariya Uchida*b Received 7th October 2010, Accepted 6th December 2010 DOI: 10.1039/c0sm01121e After a long gap following the classic work of Taylor, there have recently been several studies dealing with hydrodynamic synchronization. It is now apparent that synchronization driven by hydrodynamic interactions is not only possible, but relevant to the efficiency of pumping by arrays of cilia and to bacterial swimming. Recent work has included experiments demonstrating synchronization, both in model systems and between bacterial flagella. The effect has been demonstrated in model swimmers and pumps, and large scale simulations have been used to investigate synchronization of cilia and of sperm cells. In this review article, we summarize the various experimental and theoretical studies of hydrodynamic synchronization, and put them in a framework which draws parallels between the different systems and suggests useful directions for further research. I. Introduction typical velocity v in a region controlled by boundaries of typical length scale l, we have Re ¼ rvl/h. In the low Re regime, which Reynolds number (Re) is defined as the ratio between typical could be due to small size and/or high viscosity, hydrodynamics scales of the inertial forces and the viscous forces in the Navier– is governed by viscous forces. For microorganisms in water, with Stokes equation, which describes the time evolution of fluid flow. 1 4 typical values l 10 mm and v 10 mmsÀ , Re 10À . Therefore, For a fluid with mass density r and viscosity h that flows with microorganisms live the ‘‘life at low Reynolds number’’.1 Towards the end of his distinguished career, which resulted in aThe Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 numerous ground-breaking contributions to solid mechanics and Keble Road, Oxford, OX1 3NP, United Kingdom. E-mail: ramin. fluid dynamics,2 G. I. Taylor turned his attention to swimming [email protected]; [email protected] 3 Downloaded by University of Cambridge on 02 September 2012 and hydrodynamic activities of microorganisms. He realized bDepartment of Physics, Tohoku University, Sendai, 980-8578, Japan. Published on 04 January 2011 http://pubs.rsc.org | doi:10.1039/C0SM01121E that the absence of inertia means that the conventional knowl- E-mail: [email protected] † This paper is part of a Soft Matter themed issue on Active Soft Matter. edge about hydrodynamics of swimming cannot be used to Guest editors: Fred MacKintosh and Mike Cates. describe how microorganisms propel themselves, and showed, Ramin Golestanian is Professor Julia Yeomans is Professor of at the Rudolf Peierls Centre for Physics at the Rudolf Peierls Theoretical Physics, and is Centre for Theoretical Physics at affiliated with the Centre for the University of Oxford. She is Soft and Biological Matter, at a member of the Oxford Centre the University of Oxford. He for Soft and Biological Matter, received his PhD from the and is Pauline Chan Fellow at St Institute for Advanced Studies in Hilda’s College. She obtained her Basic Sciences (IASBS) in D. Phil. from Oxford and has Iran, and has been a Visiting worked at Cornell University, Scholar at MIT, Postdoctoral USA, and the University of Fellow at the Kavli Institute for Southampton. Her research Theoretical Physics at UCSB, interests include swimming at low Ramin Golestanian Joliot Chair and CNRS Visiting Julia M: Yeomans Reynolds number, wetting Professor at ESPCI, and dynamics and microfluids, and the Visiting Professor at College de hydrodynamics of complex fluids. France. Before joining Oxford, he has held academic positions at IASBS and the University of Sheffield. His research interests are in statistical and mechanical properties of soft and biological matter. 3074 | Soft Matter, 2011, 7, 3074–3082 This journal is ª The Royal Society of Chemistry 2011 View Online using extensive arguments and explicit calculations, how force- which defines the Green’s function for viscous hydrodynamics, free swimming in a viscous medium is possible. In the same eqn (2), called the Oseen tensor.5 In analogy to electrostatics, the paper, he then went on to propose an original idea, which forms Green’s function for point forces near planar6 or spherical5 the subject of this review article. Through interactions with the boundaries can also be calculated and understood via the method Cambridge zoologist James Gray, Taylor was fascinated to learn of images. of the observations made by several people that when two or The long-ranged nature of hydrodynamic interaction apparent more spermatozoa get close to each other their tails tend to beat from eqn (3) suggests that a collection of active components that synchronously.4 He proposed that hydrodynamic interaction dynamically exert forces on the fluid medium they are immersed could explain this dynamical phenomenon, and made a calcula- in could influence each other very strongly, leading to the tion that suggested having in-phase synchronized beating mini- possibility of novel collective behaviors. This suggests a strong mizes the rate of energy dissipation.3 analogy to chemotaxis, where the slowly decaying concentration Viscous hydrodynamics is governed by Stokes equation for the field near a particle source, C(r) 1/r, is used for chemical sig- fluid velocity field vi(r)(i ¼ x,y,z), which reads nalling. It would be interesting to discover what capacities exist for hydrodynamic signalling. 2 Àhv vi ¼vip + fi, (1) Since the original work of Taylor in 1951, active hydrody- namics at low Reynolds number has been the subject of intense v 2 investigation in a variety of different contexts. Here, we aim to where vih , v ^ vivi with summation over repeated indices vri review a collection of these studies, with an emphasis on the implicit, p is the hydrostatic pressure, and fi is the density of the possibility of achieving synchronization, solely via hydrody- body force exerted on the fluid. The incompressibility of the fluid namic interactions, between active components that undergo puts a constraint on the velocity field, vjvj ¼ 0, which can be used independent cyclic motion or deformation. 1 The rest of the paper is organized as follows. Section II gives to nominally solve for pressure in eqn (1) as p ¼ 2 vj fj. v a brief overview of the general subject of low Reynolds number Putting the expression for pressure back in eqn (1), we find the hydrodynamics of active particles, such as swimming microor- governing equation for viscous hydrodynamics ganisms and beating cilia, with emphasis on elements that can lead to hydrodynamic synchronization. Section III is devoted to 2 vivj Àhv vi ¼ dij À fj: (2) v2 a comprehensive discussion of the many studies of hydrodynamic synchronization, ranging from detailed simulations of beating Eqn (2) shows that the velocity profile in the medium is elastic filaments to synchronization of swimmers and minimal determined by the distribution of body forces as well as boundary model systems represented by simple beads, through to experi- conditions, through a tensorial Poisson equation that suggests an ments. This is followed by discussion of a number of generic electrostatic analogy. Eqn (2) is reversible, in the sense that for features of hydrodynamic synchronization in Sec. IV, and any solution v corresponding to f, the reverse flow with velocity concluding remarks in Sec. V. Àv will be a solution when the force changes to Àf. This poses Downloaded by University of Cambridge on 02 September 2012 a fundamental difficulty in creating a net directed flow from Published on 04 January 2011 http://pubs.rsc.org | doi:10.1039/C0SM01121E II. Active hydrodynamics at low Reynolds number a periodic activity that involves equal half cycles of f and Àf, such as swimming or pumping strokes. For a point force F located at Before focusing on the specific dynamical phenomenon of the origin, eqn (2) yields a fluid velocity at point r that reads synchronization, we first give a brief overview of the subject of 7 1 r r active viscous hydrodynamics. v ðrÞ¼ d þ i j F ; (3) i 8phr ij r2 j A. Observations, discoveries, and experimental developments Nature has developed a number of mechanically active compo- Nariya Uchida received his Ph.D. nents such as cilia and flagella that are capable of moving the from Kyoto University in 2000 neighboring fluid for the purpose of motility, transport (of under the supervision of Professor mucus, for example), feeding (e.g. for sponges), or pumping. Akira Onuki, and was a JSPS Bacterial flagella are relatively rigid helical proto-filaments that postdoctral fellow in 2001–2002 at are attached to highly efficient rotary motors in bacteria such as Yukawa Institute of Theoretical E. coli.8–10 An E. coli could have 6–10 such flagella around its Physics, Kyoto University and body, and their coordinated action could lead to two distinct Max Planck Institute for Polymer types of motion, termed as ‘‘run’’ (when the flagella bundle up Research. Since 2002, he is an and corkscrew together) and ‘‘tumble’’ (when the flagella disor- Assistant Professor at the ganize and net motility is lost).11 Cilia (and eukaryotic flagella) Department of Physics, Tohoku are elastic filaments that can cyclically actuate into different University in Sendai. His current conformations.12 This ability comes from the so-called axonemal research is focused on hydrody- structure, which is an assembly of nine microtubule doublet Nariya Uchida namic description of hybrid soft filaments that can slide along one another via dynein motor materials and non-linear dynamics proteins.13,14 The oscillating beating patterns are believed to be of active microfluidic systems.
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