T he development of a general theoretical framework for describing the behaviour of a crystal driven far from equilibrium has proved difficult 1 . Microfluidic crystals, formed by the introduction of droplets of immiscible fluid into a liquid-filled channel, provide a convenient means to explore and develop models to describe non-equilibrium dynamics 2-11 . Owing to the fact that these systems operate at low Reynolds number (Re), in which viscous dissipation of energy dominates inertial effects, vibrations are expected to be over-damped and contribute little to their dynamics 12-14 . Against such expectations, we report the emergence of collective normal vibrational modes (equivalent to acoustic 'phonons') in a one-dimensional microfluidic crystal of water-in-oil droplets at Re ∼ 10 −4 . These phonons propagate at an ultra-low sound velocity of ∼100 μm s and nucleoprotein filaments 20 . To investigate many-body effects of one-dimensional (1D) hydrodynamic crystals, we built a microfluidic water-in-oil droplet generator 2 (Fig. 1a, Methods section and Supplementary Information, Movie S1). Water droplets formed at a T-junction between water and oil channels under continuous flow, emanating at a constant rate with uniform radii R (10-15 μm) and fixed interdroplet distances a (10-200 μm). The thin channel (h = 10 μm) deformed the droplets into discs, confining their motion to 2D and exerting friction with the floor and ceiling (Fig. 1b). Owing to friction, the droplets were dragged by the oil at a velocity u d (150−800 μm s −1 ) that was slower than that of the oil (u oil ∼ 5u d ). Symmetry was broken by the relative motion of the oil with respect to the droplet crystal. Thus, we obtained a flowing 1D crystal of droplets that can move in 2D. The crystal exhibited visible longitudinal and transversal fluctuations, which were reminiscent of solid-state phonons (we henceforth term these normal modes 'phonons') (Fig. 1c,d and Supplementary Information, Movies S2-S4). We explored these modes by measuring their wave dispersion relations (Fig. 1e-h). This was done by tracking the positions of droplets in time and applying a Fourier transform to obtain the power spectrum of vibrations in terms of the wavevector k and frequency ω (see the Methods section). We then extracted the dispersion relations of waves in the crystal, ω (k). Surprisingly, the dispersion relations reveal the existence of acoustic phonons that propagate in the crystal at ultra-low frequencies of a few hertz. Manifestly, at Re ∼ 10 −4 , collective modes at such low frequencies cannot arise from inertial effects and are likely to be due to hydrodynamic interactions within the crystal. The main feature of the dispersion is an unusual sine-like curve that spans the Brillouin zone (0 ≤ k ≤ π/a) and has unique properties (Fig. 1). The linear behaviour of the curve ω(k) = C s k around k = 0 shows that these waves are acoustic and propagate at a sound velocity of C s = (∂ω/∂k) k→0 ≈ 250 μm s −1 . This velocity is some six orders of magnitude slower than sound in ...

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Owing to aerodynamic instabilities, stable flapping flight requires ever-present fast corrective actions. Here, we investigate how flies control perturbations along their body roll angle, which is unstable and their most sensitive degree of freedom. We glue a magnet to each fly and apply a short magnetic pulse that rolls it in mid-air. Fast video shows flies correct perturbations up to 1008 within 30 + 7 ms by applying a stroke-amplitude asymmetry that is well described by a linear proportional-integral controller. For more aggressive perturbations, we show evidence for nonlinear and hierarchical control mechanisms. Flies respond to roll perturbations within 5 ms, making this correction reflex one of the fastest in the animal kingdom.

We investigate the acoustic normal modes (''phonons'') of a 1D microfluidic droplet crystal at the crossover between 2D flow and confined 1D plug flow. The unusual phonon spectra of the crystal, which arise from long-range hydrodynamic interactions, change anomalously under confinement. The boundaries induce weakening and screening of the interactions, but when approaching the 1D limit we measure a marked increase in the crystal sound velocity, a sign of interaction strengthening. This nonmonotonous behavior of the phonon spectra is explained theoretically by the interplay of screening and plug flow.Microfluidic two-phase flow offers experimental tools to investigate dissipative nonequilibrium dynamics [1][2][3][4][5][6][7][8][9][10][11][12]. Microfluidic crystals-ordered arrays of water-in-oil droplets driven by flow-are governed by long-range dipolar interactions and exhibit acoustic normal modes, akin to solid-state phonons [11,12]. These interactions share common themes with other systems driven by a symmetry-breaking field, such as dusty-plasma crystals [13,14], vortices in superconductors [15,16], active membranes [17], and nucleoprotein filaments [18,19]. Thus, microfluidic crystals offer a vista, in the linear flow regime, into many-body physics far from equilibrium. Long-range forces, such as the hydrodynamic dipolar force, are known to be radically affected by boundaries and dimensionality. This has been recently shown for two disordered systems: Brownian particles confined to 1D and 2D [20 -22] and sedimenting particles [23,24].In this Letter we examine the direct influence of boundaries on the normal modes of an ordered many-body system at low Reynolds number (Re 5 10 ÿ4 ). We investigated 1D microfluidic droplet crystals under different degrees of confinement ranging from unconfined 2D flow to 1D flow, where the channel is nearly blocked by droplets (plug flow). The interdroplet forces that fall off as r ÿ2 in 2D cross over under confinement to decay as exp ÿ2 r=W [20,25,26], where the screening length W is the width of the channel. However, close to plug flow, and despite the weakening of interactions due to screening, the magnitude of interdroplet forces increases as tan R=W due to the crystal's incompressibility, R being the droplet radius. This interplay between hydrodynamic screening and incompressibility is reflected in a nonmonotonous behavior of the phonon spectra. Confinement breaks the translational invariance, which is manifested in the breaking of the x-y antisymmetry of unconfined spectra. Additionally, the approach to incompressibility in the 1D limit implies a divergence of sound velocity and, indeed, we observed its marked increase.Experimental setup.-The microfluidic device (Fig. 1) was fabricated using standard soft lithography and made of poly-dimethyl-siloxane (PDMS) [1,11]. Water droplets formed at a T junction between water and oil channels under continuous flow, emanating with uniform radii R and fixed interdroplet distance a. Channel height h was 10 m and droplets had a disk...

Flapping insect flight is a complex and beautiful phenomenon that relies on fast, active control mechanisms to counter aerodynamic instability. To directly investigate how freely flying Drosophila melanogaster control their body pitch angle against such instability, we perturbed them using impulsive mechanical torques and filmed their corrective maneuvers with high-speed video. Combining experimental observations and numerical simulation, we found that flies correct for pitch deflections of up to 40 deg in 29±8 ms by bilaterally modulating their wings' front-most stroke angle in a manner well described by a linear proportional-integral (PI) controller. Flies initiate this corrective process only 10±2 ms after the perturbation onset, indicating that pitch stabilization involves a fast reflex response. Remarkably, flies can also correct for very largeamplitude pitch perturbations -greater than 150 deg -providing a regime in which to probe the limits of the linear-response framework. Together with previous studies regarding yaw and roll control, our results on pitch show that flies' stabilization of each of these body angles is consistent with PI control.