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Geometría y Física del Movimiento de Microorganismos Jair Koiller, FGV/RJ, GMC y AGIMB UniAndes, Deciembre El bobo, Doctor Universalis

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Outline I.Rowers and Squirmers (Taylor, Lighthill, Purcell, 1950 …) Aristotelian mechanics, gauge theory of micro-swimming Purcell a short course: Geometry: the hydro-dynamical connection Optimizing Efficiency: the sub-riemannian control problem II. Singers/shakers (joint work with Kurt Ehlers, 2008, unpublished) Acoustic streaming AS in micro-engineering devices AS in locomotion strategies: diatoms and cyanobacteria Collaboration with Kurt Ehlers

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I. Rowers and SquirmersRowers and Squirmers Aristotelian mechanics, gauge theory of micro-swimming PurcellPurcell Geometry: the hydro-dynamical connection Optimizing Efficiency: sub-riemannian control problem a short course (JK): seminar A survey in bacterial motility: Howard Berg (Harvard)Howard Berg (Harvard

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II. Singers/shakers (joint work with Kurt Ehlers, 2008, unpublished)unpublished Acoustic streaming AS in micro-engineering devices AS in locomotion strategies: diatoms and cyanobacteria

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Acoustic streaming Acoustics timelineAcoustics timeline Lord Rayleigh (Theory of Sound, 1896)Lord Rayleigh Theory of Sound Nyborg, Westervelt (RNW streaming, 1953) Lighthill (1978):Lighthill “not only can a jet generate sound, but also sound can generate a jet” AS holds for all Reynolds numbers focus here in low Reynolds regime

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MEMS devicesMEMS devices NEMS devicesNEMS devices MITMIT GalleryGallery Berkeley Nanophysics Roukes New stuff AS in MEMS MonashMonash Research TanetalResearchTanetal Near future: NEMS ReviewNEMSReview Nanotech Old stuff: MoroneyMoroney HashimotoHashimoto

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Two main types of acoustic streaming: 1.The quartz wind effect. Here the attenuation takes place in the bulk of the fluid. Streaming is normal to the source. (When piezoelectrically excited, the faces of a quartz crystal vibrate, creating an ultrasonic beam. AS generates a turbulent jet with velocities reaching 10’s of cm/s.) 2.Boundary induced streaming. Here the attenuation takes place near a solid surface. The induced streaming is tangential to the surface.

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Quartz wind effect Attenuation of the sound wave occurs in the bulk of the fluid 800kHz ultrasonic wave in glycerol W. Dridi, V. Button, X. Escriva, H. BenHadid, D. Henry Video1 Video2

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Boundary induced streaming Let U be an irrotational oscillatory vector field in a fluid representing an acoustic wave. Owing to the no slip boundary condition, U must vanish at a solid boundary. There is a thin layer (the Stokes boundary layer) where U is rotational. The thickness of the Stokes boundary layer is 5√( where is the kinematic viscosity and is the frequency of U. Within the Stokes boundary layer shear stresses cause strong attenuation of U leading to streaming.

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Rayleigh’s Law In the late 19 th century Lord Rayleigh showed that the streaming velocity at the edge of the boundary layer due to an oscillatory vector field U=U(x) is -3/(4 U(x)U’(x) and that the streaming is in the direction of the nodes:

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Kundt’s Tube A resonant standing acoustic wave is established using a sound transducer at one end of the tube. (Quartz wind) Boundary induced streaming blows dust into piles at the nodes.

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We propose two possible mechanisms for self-propulsion via acoustic streaming: 1.The Quartz Wind (QW) model. 2.The Surface Acoustic Wave (SAW) model based on boundary induced streaming

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The Quartz Wind model In this model the spicules in a small region vibrate at a high frequency buckling the crystalline shell in a manner similar to an electric door buzzer. Our inspiration for this mechanism came from the cuica: aBrazilian samba instrument A flow, normal to the cell, is generated by attenuation in the bulk of the fluid. Problem: Low efficiency. (Quartz wind swimmers are Hummers!)

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Power/Efficiency estimate for Synechococcus employing the QWmechanism: Lighthill defines the efficiency to be the ratio of the power required to push the cell through the water to the power required by the mechanism (P): η = viscosity, for water η = 0.01 g / cm sec a = radius, for Synechococcus a = 10^(-4) cm V = velocity, for Synechococcus V = 2.5×10^(-3) cm / sec The force (F) exerted on the fluid by the QW effect and acoustic power (P) are related by P=Fc where c is the speed of sound. The force required to drive the cell with velocity V is F = 6πμaV making the power output for Synechococcus P=7×10^(-10) watts.

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The efficiency of the QW mechanism for Synechococus is then η=1.7×10^(-6) % The squirming and boundary induced streaming mechanisms have efficiencies between 0.1-1%. We have not ruled out the QW mechanism completely. There are possible power enhancement mechanisms. Example: Bubble induced streaming. Here submicro-bubbles adhere to the CS. Being of characteristic size, the bubbles resonate enhancing the local streaming.

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Boundary induced streaming: the SAW mechanism In this model, the cell propagates a high frequency traveling wave along the CS. Attenuation of the wave within the Stokes boundary layer generates a mean flow just outside this layer creating an effective ‘slip’ velocity. Longuet-Higgins (1953) derived a generalization of Rayleigh’s law for streaming due to a traveling wave. The limiting streaming velocity at the edge of the Stokes boundary layer is the real part of (* = complex conjugate) where is the tangential velocity at the CS and is the solution to the linearized NS equations outside the Stokes boundary layer (=0 for us).

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We model the cell as a sphere of radius a with spherical coordinates where is the azimuthal coordinate. The traveling wave is where ϕ m is a material point on the CS. The slip velocity due to streaming leads to a swimming velocity of which is 2.5 times that predicted by the squirming mechanism.

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Efficiencies for the boundary induced streaming mechanism The efficiency compares well with other known strategies. But … are the required frequencies biologically feasible?

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Question: Is singing biologically feasible? Bacterial flagellar motors are large membrane embedded structures and have been observed to rotate at 300Hz when unloaded. From E-Coli in Motion HC Berg (2004, Springer) The required frequency for acoustic streaming is biologically feasible.

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More details for people who know some fluid mechanics Streaming flow = what survives after averaging out the fluctuating part due to some external source or to internal waves this idea is also used in statistical turbulence Averaging already present in the very formulation of Navier Stokes equations Equations of motion in AS: time-averaged Navier Stokes equations. Reynolds stress tensor = gives the mean momentum flux. Its gradient is a force, non-zero when an attenuation mechanism is present. Attenuation is necessary for streaming can occur in the body of the fluid or in a thin Stokes boundary layer surrounding a surface.

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What is Reynolds stress? (following Lighthill, Waves in fluids pg 338) acceleration frames produce inertial forces (general fact in mechanics) Eulerian velocities are different at each point in the fluid. typically, these inertial forces appear when one averages the turbulent variations in fluid velocity about a mean flow, or when waves produce fluid motions. In short: Acoustic streaming is the result of a gradient in the Reynolds stress associated with high frequency (acoustic) oscillations in the fluid. Reynolds stress = mean value of u i u j But … attenuation is needed for a net nonzero forcing

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Why is attenuation necessary? Formal argument in Lighthill pag For unattenuated internal waves: fluid velocity is parallel to surfaces of constant phase, the gradient is perpendicular to them. For unattenuated sound waves: force is gradient of a scalar, will be cancelled by the gradient of a mean pressure Will see: attenuation works as an asymmetry leading to directed motion analogy with other phenomena

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Next : analogies for AS / attenuation as an asymmetry Summary:

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Analogies for non-fluidmechanicists (like me) AS is one among many phenomena in which: small vibrations are rectified to organized macroscopic motion (linear or rotational) this always involves an averaging process, usually second order in the amplitude EscaladoEscalado An asymetry is required

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Example: holonomy in principal bundles (requires: nonintegrable distributions) Berry phases (+ topological interpretations) In Acoustic Streaming: assymetry = attenuation Example: holonomy due to a time periodic potential

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A digression, specially for biologists: molecular motorsmolecular motors (hot topic!) Feynman’s Ratchet Eindhoven AdelaideAdelaide this is wrong! Relation with game theory Parrondo games this is correct (need T2 > T1 By second law of thermodynamics )

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MEMS devicesMEMS devices NEMS devicesNEMS devices MITMIT GalleryGallery Berkeley Nanophysics Roukes AS in MEMS MonashMonash Research TanetalResearchTanetal Old stuff: MoroneyMoroney HashimotoHashimoto

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Biological applications Artificial cilia WixforthWixforth dropletsdroplets RenaudinRenaudin nanoquakesnanoquakes Advalytix Microchip

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AS in biological MEMS devices? Diatoms Diatoms Diatoms Diatoms Diatoms DiatomsDiatoms Nanotechnology Startreck Life in glass houses

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Bengtsson, Martin; Laurell, Thomas, Analytical and Bioanalytical Chemistry, Volume 378, Number 7, April 2004, pp Analytical and Bioanalytical Chemistry Experiment on a diatom? Sandra Azevedo’s lab, UFRJSandra Azevedo’s lab, UFRJ

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Experiment on Synechococcus? From: Sent: Mon 12/8/ :46 AM To: Jair Koiller Hi Jair, For the experiment we will need to work out the details of what to expect. The wave in the fluid would not cause much bulk flow but boundary induced streaming would move a 'dead' cell. Maybe Berg's tracking microscope could be used to figure out what the wave does to a single bacterium on average over a time period. I'll try to think things out once classes end (this week). Enjoy Colombia! (Berg's wife is from Bogota!) Cheers, Kurt Nanotech meeting

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GRACIAS!

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