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If one opens today a textbook on fluid mechanics, it seems that whereas French scientists of the eighteenth and nineteenth centuries are frequently mentioned, those from more recent times occur rarely; in particular, French contributions to the major breakthroughs of the first half of the twentieth century (boundary layers and turbulence) would appear quite modest. However, study of contemporary documents (PhD theses, journals, correspondence, etc.) reveals remarkable work undertaken by outstanding personalities. A key instigator of these achievements was the French Air Ministry, which, starting in 1929, and with great open-mindedness, created and generously financed four Institutes of Fluid Mechanics and five teaching centers in faculties of science. This reveals the 1930s in their true light as a fruitful decade, with achievements that explain the prominent role played by France in the creation of the International Union of Theoretical and Applied Mechanics (IUTAM) in 1946. This review recounts the story of fluid mechanics in France, with emphasis on the connection between scientific questions and social and cultural issues, in a period marked by two world wars and great strengthening of international relationships.

International audience; The motion of isolated heavy particles on the bottom wall of a turbulent shear flow is investigated experimentally (using high-frequency particle tracking) and correlated to that of the fluid (from high-resolution LDV measurements). Numerous particle diameters, particle densities, and flow velocities are considered, where the particles remain within the viscous sublayer (particle Reynolds number lower than 10). It is shown that fluid flow fluctuations within the viscous sublayer (varying Couette flow) induce particle velocity fluctuations of the same magnitude, slightly smaller. The mean particle velocity, standard deviation, and Lagrangian correlation time are found to scale with the viscous scales. Particles roll and slide without takeoff, unlike what happens in viscous laminar flow where the lift force may overcome the immersed particle weight. Arguments are given for this difference. Probability density functions for the streamwise and spanwise velocity are found to follow Gamma and Gaussian laws, respectively, which also scale with the viscous scales.

L’institut de mécanique des fluides de Toulouse (IMFT), issu d’un laboratoire d’hydraulique créé en 1913, a célébré son centenaire en 2016 -- année marquant ses 50 ans d’association au CNRS. The Institute of Fluid Mechanics of Toulouse (IMFT), created in 1913 from a former hydraulic laboratory, celebrated its centenary in 2016. This article presents the milestones in the history of this century-old laboratory, in the initial context of the regional development of hydropower, then, in the 1930s, in the context of a national aeronautics policy.

Flow over an uneven terrain is a complex phenomenon that requires a chain of approximations in order to be studied. In addition to modelling the intricacies of turbulence if present, the problem is classically first linearized about a flat bottom and a locally parallel flow, and then asymptotically approximated into an interactive representation that couples a boundary layer and an irrotational region through an intermediate inviscid but rotational layer. The first of these steps produces a stationary Orr–Sommerfeld equation; since this is a one-dimensional problem comparatively easy for any computer, it would seem appropriate today to forgo the second sweep of approximation and solve the Orr–Sommerfeld problem numerically. However, the results are inconsistent! It appears that the asymptotic approximation tacitly restores some of the original problem’s non-parallelism. In order to provide consistent results, Benjamin’s version of the Orr–Sommerfeld equation needs to be modified into Hanratty’s. The large difference between Benjamin’s and Hanratty’s formulations, which arises in some wavenumber ranges but not in others, is here explained through an asymptotic analysis based on the concept of admittance and on the symmetry transformations of the boundary layer. A compact and accurate analytical formula is provided for the wavenumber range of maximum laminar shear-stress response. We highlight that the maximum turbulent shear-stress response occurs in the quasi-laminar regime at a Reynolds-independent wavenumber, contrary to the maximum laminar shear-stress response whose wavenumber scales with a power of the boundary-layer thickness. A numerical computation involving an eddy-viscosity model provides a warning against the inaccuracy of such a model. We emphasize that the range $k\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}} of the spectrum remains essentially unexplored, and that the question is still open whether a fully developed turbulent regime, similar to the one predicted by an eddy-viscosity model, ever exists for open flow even in the limit of infinite wavelength.

International audience; The question of the place where a fluid flow separates on a curved obstacle is among the most difficult in fluid mechanics. A criterion based on the cancellation, at that place, of the inviscid flow singularity was proposed by Brillouin [Annal. Chimie Phys. 23, 145 (1911)] and Villat [J. Math. Pures Appl. 10, 231 (1914)], which is at the starting point of the triple-deck theory. A similar question arises for the flow over an erodible sand dune with sharp crest, where the a priori unknown dune profile and location of the crest result from the coupling of the fluid flow with the sand motion at the dune surface. We show, by computing the potential flow with Levi-Civita’s conformal transformation and using a standard closure law for the shear stress driving the sand flux, that the Brillouin-Villat condition here provides the appropriate criterion. We emphasize that within the present model where the bed shear stress is in phase with the flow velocity, so that the flat bed is linearly stable, it is the separation which allows the existence of a self-preserving dune shape. For a dune traveling without deformation on a nonerodible ground, the Brillouin-Villat condition eventually selects its velocity. A parallel is drawn with the Kutta-Joukovski condition for the flow around an airfoil.

International audience; En 1913, Charles Camichel crée le laboratoire d’hydraulique de l’institut électrotechniquede l’université de Toulouse, institut qu’il avait lui-même fondé six ans plus tôt. Celaboratoire acquiert rapidement une grande notoriété pour la qualité de sa rechercheexpérimentale et pour ses succès dans la transposition à l’hydraulique des concepts etméthodes de la physique et de la mécanique. En 1930, le ministère de l’Air lui associeun institut de mécanique des fluides, et finance une grande soufflerie. Un siècle plus tard,l’IMFT a diversifié son activité dans de nombreux domaines autour d’enjeux scientifiqueset d’applications très variés. Il est l’un des plus grands laboratoires de sa disciplineà l’échelle internationale. Cet article en retrace l’histoire jusqu’un 1970, en insistantplus particulièrement sur les remarquables contributions scientifiques des trois premièresdécennies.

International audience; The linear response of the wall-shear stress of a turbulent flow to wall waviness is analyzed in the context of a comparison between existing experiments, direct numerical simulations, and analytical approximations. The spectral region where the response is largest is found to be amenable to a simplified quasilaminar analysis. The end result is a parameterless description of this phenomenon that completely captures its physics in a single analytical formula, a Padé approximation of the response function.

International audience; After a quick overview of recent research on sediment transport by shear flow and ripple and dune formation, original numerical results are presented from two-phase flow modelling of the interaction between a viscous flow and a bed of particles. Good agreement is found with previous experiments or numerical simulation, notably for the particle flux and velocity profiles within the moving layer. Bed instability is also found, giving rise to ripples whose characteristics are discussed.

International audience; The long wave-length dynamics and stability of a bed of sand occupying the lower segment of a circular pipe are studied analytically up to first-order in the small parameter characterizing the slope of the bed. The bed is assumed to be at rest, with at most a thin sand layer (the bedload) moving at the sheared interface. When the sand bed is plane, with depth independent of position z along the axis of the pipe, the velocity of the liquid is known from previous studies of stratified laminar flow of two Newtonian liquids (the lower one with infinite viscosity representing the sand bed). When the depth of the sand bed varies with z, secondary flows develop in the cross-sectional (x, y) plane, and these are computed numerically, assuming that the sand bed remains a straight horizontal line in the cross-sectional plane. The mean shear stress acting on the perturbed sand bed is then determined both from the computed secondary flows and by means of the averaged equations of Luchini and Charru. The latter approach requires knowledge only of the flow over the unperturbed, flat sand bed, combined with an accurate approximation of the distribution of the perturbed stresses between the pipe wall and the sand bed. The perturbed stresses determined by the two methods agree well with each other. Using these stresses, it is then possible to apply standard theories of bed stability to determine the balance between the destabilizing effect of inertial (out-of-phase) stresses and the stabilizing effects of gravity and relaxation of the particle flux, and various examples are considered.

We report an experimental study of the turbulent flow above a barchan dune in a channel, from particle image velocimetry measurements, for Reynolds numbers ranging from 9000, just below the threshold for particle motion, up to 24 000, where the dune moves. Two calculations of the speed-up over the dune are compared, the usual ‘same-elevation’ and the more relevant ‘Lagrangian’, showing that the latter is smaller by a factor of two. The two-layer structure of the flow disturbance – an essentially inviscid outer layer and a turbulent inner layer of thickness ${\delta }_{i} $ – is assessed. In the outer layer, streamline curvature is shown to be responsible for half of the Lagrangian speed-up, from the comparison of the velocity measurements with two Bernoulli calculations. In the inner layer, detailed measurements of the velocity and stresses are provided, down to ${y}^{+ } \approx 1$, and the momentum budget is discussed. The Reynolds shear stress decreases monotonically towards the dune surface, according to the standard mixing-length closure, whereas the total shear stress increases strongly in the viscous sublayer. Along the dune surface, the shear stress increases up to the crest where it reaches twice its unperturbed value. A good estimate of the surface stress is provided by a parabolic fit of the inner velocity profile matching the outer flow at ${y}_{d} \approx {\delta }_{i} $. Doubling the Reynolds number, the surface shear stress and the speed-up decrease by ∼30 %. The implications of these results on the dune motion, presented in Part 1 of this study (Franklin & Charru, J. Fluid Mech., vol. 675, 2011, pp. 199–222), are finally discussed.

International audience; The non-linear dynamics of thin liquid films sheared by a laminar gas flow in a channel is investigated. Such a two-layer flow is driven by pressure gradient and possibly by the gravity force. We describe the liquid phase with a long-wave integral model, with the aim to save computational cost with respect to the full Direct Numerical Simulation (DNS) of the Navier–Stokes equations. We derive this long-wave model by the integration of the Navier–Stokes equations over the film thickness, and by an asymptotic expansion up to the first order in terms of a long-wave parameter. These depth-integrated (or shallow water) equations are discretized by means of an augmented system, which holds an evolution equation for the surface tension in order to avoid numerical instabilities of classical upwind and centered schemes. On the other side, we study the gas phase with compressible Navier–Stokes equations, and we discretize them by means of a lowMach scheme, accounting also for moving meshes (ALE). In order to analyze liquid–gas interactions, we introduce then a coupling methodology between depth-integrated equations and Navier–Stokes equations. This approach represents a compromise between the two existing methods: the full DNS, and the full long-wave model applied to both phases. In order to validate this approach, we present comparisons with DNS, showing a good agreement of spatio-temporal evolutions of the film thickness and the stress field. Furthermore, interfacial shear stress and pressure gradient evolutions are shown to be in accordance with those provided by two-layer second-order low-dimensional models.

Experiments are reported on the formation and migration of isolated dunes in a turbulent channel flow. These dunes have a very robust crescentic shape with horns pointing downstream, very similar to that of the barchan dunes observed in deserts at a much larger scale. Their main geometrical and dynamical properties are studied in detail, for four types of grains: the conditions for their formation, their morphology, the threshold shear stress for their motion, their velocity, erosion rate, minimum size and the longitudinal stripes of grains hollowed by fluid streaks in the boundary layer. In particular, the law for the dune velocity is found to involve two dimensionless parameters, the Shields number and the sedimentation Reynolds number, in contrast with predictions based on classical laws for particle transport. As the dune migrates, its size slowly decreases because of a small leakage of particles at the horn tips, and the erosion law is given. A minimum size is evidenced, which is shown to increase with the friction velocity and scale with a settling length.

The formation of sand ripples in oscillating flows is thought to be due to a steady streaming current which near the bottom is towards the crests. We present quantitative observations of this mean flow over self-formed and artificial ripples, by observing the displacement of a coloured filament after a number of oscillations in the simple situation of viscous Couette flow. Confusingly, the filament moves in the ‘wrong’ direction, because it follows the Lagrangian mean flow. We calculate the Lagrangian mean flow. A complication is that the amplitudes of the oscillations in the experiments are not small. We compare the predictions with the experimental observations of displacements of the filament, showing good agreement.

The moving layer at the surface of a granular bed sheared by a viscous flow has been investigated experimentally. The fluid and particle velocities have been measured using particle imaging velocimetry (PIV) and particle tracking, respectively, with a technique of matched index of refraction. The mean velocity profiles are found to be parabolic. The models of Bagnold (Phil. Trans. R. Soc. Lond. A, vol. 249, 1956, pp. 235–297) and Leighton & Acrivos (Chem. Engng Sci., vol. 41, 1986, pp. 1377–1384) fail to account for the observations. A simplified model assuming uniform particle concentration provides good agreement close to the threshold.

The transport of solid particles entrained by a fluid flow is frequently found in industrial applications. A better knowledge of it, is of importance to improve particle related industrial processes. When shear stresses exerted by the fluid on the bed of particles are bounded to some limits, a mobile layer of particles known as bed-load takes place in which the particles stay in contact with the fixed bed. If it takes place over a non-erodible ground, and if the particle flow rate is small enough, an initial thin continuous layer of particles becomes discontinuous and composed of isolated dunes. We present here an experimental study to understand some features of the dynamics of isolated dunes under a fluid flow using a closed-conduit experimental loop made of transparent material. Acquired data concerns mainly dune morphology and displacement velocity under different conditions: different types of beads (diameters and densities) and different water flow conditions. We observed that an initial pile of beads placed in the conduit is rapidly deformed by the water flow, adopting a "croissant" shape, like barchan dunes found in deserts at a much larger scale. We observed also self-similarity in dunes dimensions and that dune displacement velocity scales with the inverse of their dimensions. The variation of the dune displacement velocity with the fluid shear velocity is discussed here., This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

Symposium on Powder Science and Technology - Powders and Sintered Material, Albi, FRANCE, MAY 23-25, 2007Extrait de : STP 2007 - Science et technologie des poudres et Poudres et matériaux frittés / sous la dir. de M. BARON, C. CARRY, J. DODDS et al.; International audience; The dust generation occurring during the handling of bulk materials in free falls or at the impact on a stockpile can be a source of danger for the operators health. Proper design of control systems of fugitive dust requires knowledge of the behavior of the free falling powder, the air it entrains, and the concentration of dust liberated. This paper presents first a simple model for a free falling column of bulk solids and compares it with relevant previous research. This two phase model predicts the particle and air velocities, and especially the volumetric flow of induced air in the column without dependence on any empirical constant like the entrainment constant used in the plume model. For small drop heights, the predictions of the theory appear to be in qualitative agreement with the available data for the quantity of air entrained, but the theory needs to be extended in the case of large drop heights, when the expansion of the jet of particles is large. in a second part, the description of an expanded jet of particles is experimentally studied with PIV measurements. The data obtained are well fitted by the model by Liu, when the entrainment constant is taken as the angle of expansion of the jet obtained from the velocity field.

The motion of a spherical particle along a rough bed in a simple shear viscous flow is studied experimentally for a wide range of parameters, varying the particle size and density, the fluid viscosity and the shear rate γ. The instantaneous particle velocity is calculated in the stream, transverse and vertical directions, using a high-speed video imaging system. It is found that the normalized streamwise mean particle velocity U/US, where US is the Stokes settling velocity, depends only on the dimensionless shear rate μ γ/(Δ ρ g d), this relationship being independent of the particle Reynolds number Rep. This result holds for small Rep, which was the case in our experiments (Rep < 10). The characteristic amplitude and frequency of the velocity fluctuations are also given and discussed. A model is then proposed for the mean streamwise velocity, based on ideas of Bagnold (Proc. R. Soc. Lond. A, vol. 332, 1973, p. 473) and calculations of Goldman et al. (Chem. Engng Sci., vol. 22, 1967b, p. 653) for the velocity of a particle close to a smooth plane. From this model an equivalent bed roughness and an effective friction coefficient are deduced.

A linear stability analysis of a shear flow in the presence of a continuous but steep variation of viscosity between two layers of nearly uniform viscosity is presented. This instability is investigated in relation to the known interfacial instability for the parallel flow of two superposed fluids of different viscosity. With respect to this configuration, the stability of our problem depends on two new parameters: the interface thickness $\delta$ and the Peclet number $\hbox{\it Pe}$ , which accounts for diffusion effects when viscosity perturbations, coupled to the velocity perturbations, are allowed. We show that instability still exists for the continuous viscosity profile, provided the thickness of the interface is small enough and $\hbox{\it Pe}$ sufficiently large. Small and large wavenumbers are found to be stable, at variance with the discontinuous configuration. Of particular interest is also the possibility of obtaining higher growth rates than in the discontinuous case for suitable $\hbox{\it Pe}$ and $\delta$ ranges.

The instability of a bed of particles sheared by a viscous fluid is investigated theoretically. The viscous flow over the wavy bed is first calculated, and the bed shear stress is derived. The particle transport rate induced by this bed shear stress is calculated from the viscous resuspension theory of Leighton & Acrivos (1986). Mass conservation of the particles then gives explicit expressions for the wave velocity and growth rate, which depend on four dimensionless parameters: the wavenumber, the fluid thickness, a viscous length and the shear stress. The mechanism of the instability is given. It appears that for high enough fluid-layer thickness, long-wave instability arises as soon as grains move, while short waves are stabilized by gravity. For smaller fluid thickness, the destabilizing effect of fluid inertia is reduced, so that the moving at bed is stable for small shear stress, and unstable for high shear stress. The most amplified wavelength scales with the viscous length, in agreement with the few available experiments for small particle Reynolds numbers. The results are also compared with related studies for turbulent flow.

Phys. Rev. E ISI Document Delivery No.: AE1BR Times Cited: 0 Cited Reference Count: 33 Capron, M. Tordjeman, Ph. Charru, F. Badel, E. Cochard, H. Amer physical soc College pk; The xylem vessels of trees constitute a model natural microfluidic system. In this work, we have studied the mechanism of air flow in the Populus xylem. The vessel microstructure was characterized by optical microscopy, transmission electronic microscopy (TEM), and atomic force microscopy (AFM) at different length scales. The xylem vessels have length approximate to 15 cm and diameter approximate to 20 mu m. Flow from one vessel to the next occurs through similar to 10(2) pits, which are grouped together at the ends of the vessels. The pits contain a thin, porous pit membrane with a thickness of 310 nm. We have measured the Young's moduli of the vessel wall and of the pits (both water-saturated and after drying) by specific nanoindentation and nanoflexion experiments with AFM. We found that both the dried and water-saturated pit membranes have Young's modulus around 0.4 MPa, in agreement with values obtained by micromolding of pits deformed by an applied pressure difference. Air injection experiments reveal that air flows through the xylem vessels when the differential pressure across a sample is larger than a critical value Delta P-c = 1.8 MPa. In order to model the air flow rate for Delta P >= Delta P-c, we assumed the pit membrane to be a porous medium that is strained by the applied pressure difference. Water menisci in the pit pores play the role of capillary valves, which open at Delta P = Delta P-c. From the point of view of the plant physiology, this work presents a basic understanding of the physics of bordered pits.

A unified view is given of the instabilities that may develop in two-layer Couette flows, as a ‘phase diagram’ in the parameter space. This view is obtained from a preliminary study of the single-fluid Couette flow over a wavy bottom, which reveals three flow regimes for the disturbances created at the bottom, each regime being characterized by a typical penetration depth of the vorticity disturbances and an effective Reynolds number. It appears that the two-layer flow exhibits the same flow regimes for the disturbances induced by the perturbed interface, and that each type of instability can be associated with a flow regime. Typical curves giving the growth rate versus wavenumber are deduced from this analysis, and favourably compared with the existing literature. In the second part of this study, we propose a mechanism for the long wavelength instability, and provide simple estimates of the wave velocity and growth rate, for channel flows and for semi-bounded flows. In particular, an explanation is given for the ‘thin-layer effect’, which is typical of multi-layer flows such as pressure driven flows or gravity driven flows, and according to which the flow is stable if the thinner layer is the less viscous, and unstable otherwise.

Instabilities in two-layer Couette flow are investigated from a small Reynolds number expansion of the eigenvalue problem governing linear stability. The wave velocity and growth rate are given explicitly, and previous results for long waves and short waves are retrieved as special cases. In addition to the inertial instability due to viscous stratification, the flow may be subject to the Rayleigh–Taylor instability. As a result of the competition of these two instabilities, inertia may completely stabilise a gravity-unstable flow above some finite critical Froude number, or conversely, for a gravity-stable flow, inertia may give rise to finite wavenumber instability above some finite critical Weber number. General conditions for these phenomena are given, as well as exact or approximate values of the critical numbers. The validity domain of the many asymptotic expansions is then investigated from comparison with the numerical solution. It appears that the small-Re expansion gives good results beyond Re = 1, with an error less that 1%. For Reynolds numbers of a few hundred, we show from the eigenfunctions and the energy equation that the nature of the instability changes: instability still arises from the interfacial mode (there is no mode crossing), but this mode takes all the features of a shear mode. The other modes correspond to the stable eigenmodes of the single-layer Couette flow, which are recovered when one fluid is rigidified by increasing its viscosity or surface tension.

We report experiments on the stability of interfacial waves in a two-layer Couette flow. As the shear rate is increased, the periodic wave train arising from the primary instability undergoes a secondary instability which results in wave coalescence or nucleation, after a long transient. This secondary instability crucially involves the coupling with a long wave mode, which corresponds to variations of the mean interface level. These observations are favourably compared to stability results on travelling wave solutions for a set of two coupled equations, one for the envelope of a weakly unstable wave packet, and the other for the marginal long wave mode with zero wave number. A physical mechanism for this instability is proposed, as well as an interpretation for the onset of chaos.

We consider the stability of interfacial waves in a two-layer viscous shear flow. The weakly nonlinear dynamics are governed by a set of two coupled amplitude equations (Renardy and Renardy, 1993): a complex Ginzburg-Landau equation for the travelling wave with finite wavenumber kc, and a Burgers-type equation for the neutral mode with wavenumber k = 0. We study here the linear stability, against long wavelength disturbances, of the travelling wave solutions of these equations. For the travelling wave with k = kc, the Lange & Newell criterion for the Benjamin-Feir instability is modified by the coupling, and a new kind of instability may arise from the neutral mode k = 0. For travelling waves with k ≠ kc, several wavenumber bands may be Eckhaus-unstable, with growth rate much larger than for the classical Eckhaus instability. A physical mechanism for this instability is proposed. Beyond the case of interfacial waves in viscous shear flows, the results apply for physical systems with translational and galilean invariances, and no-space-reflection symmetry.

We study a simple mechanical oscillator the bifurcations of which illustrate first- and second-order phase transitions. The phase diagram of the oscillator exhibits a coexistence curve. This curve ends at a critical point, where three critical exponents can be defined. A metronome may be used to illustrate the main results. We then consider a linear array of such oscillators with elastic coupling, which is governed by the damped Klein - Gordon equation. The classical solutions of this equation, such as fronts propagating in an unstable or in a metastable state, can be guessed at and discussed from the point of view of a mechanical model. Nous etudions un oscillateur mecanique simple dont les bifurcations illustrent les transitions de phase du premier et du second ordre. Le diagramme des phases de l'oscillateur met en evidence une courbe de coexistence des phases. Cette courbe se termine sur un point critique, ou peuvent etre definis trois exposants critiques. Un metronome permet d'illustrer les resultats principaux. Nous considerons ensuite une ligne d'oscillateurs couples elastiquement, dont le mouvement est regi par une equation de Klein - Gordon avec dissipation. Les solutions classiques de cette equation, telles que les fronts se propageant dans une phase instable ou metastable, peuvent etre pressenties et discutees a partir du modele mecanique.

International audience; An erodible bed sheared by a fluid flow, gas or liquid, is generally unstable, and bed forms grow. This review discusses the following issues, in light of the recent literature: What are the relevant dynamical mechanisms controlling the emergence of bed forms? Do they form by linear instability or nonlinear processes such as pattern coarsening? What determines their timescales and length scales, so different in air and water? What are the similarities and differences between aeolian and subaqueous patterns? What is the influence of the mode of transport: bed load, saltation, or suspension? Can bed forms emerge under any hydrodynamical regime, laminar and turbulent? Guided by these questions, we propose a unified description of bed-form growth and saturation, emphasizing the hydrodynamical regime in the inner layer and the relaxation phenomena associated with particle transport

Interfacial stability of two-layer Couette flow was investigated experimentally in a channel bent into an annular ring. This paper is focused on the supercritical long-wave instability which arises for a broad range of flow parameters. Above the critical upper plate velocity, a slowly growing long wave appears with wavelength equal to the perimeter of the channel. Transients of this wave were studied within the theoretical frame of amplitude equations obtained from the long-wave interface equation. Near the onset of instability, the unstable fundamental harmonic is described by the Landau–Stuart equation, and the nonlinear dynamics of the harmonics closely follows the central and slaved modes analysis. For the higher upper plate velocity, harmonics gain some autonomy but they eventually are enslaved by the fundamental, through remarkable collapses of amplitudes and phase jumps leading to wave velocity and frequency locking. Dispersive effects play a crucial role in the nonlinear dynamics. Far from the threshold, the second harmonic becomes unstable and bistability appears: the saturated wave is dominated either by the fundamental harmonic, or by the even harmonics, after periodic energy exchange.

International audience; Sap hydrodynamics in vascular cells of trees seems to be controlled by small membranes called pits. Understanding how the pit junctions regulate the sap flow and stop embolism by cavitation is today a challenging issue. The hypothesis that the pit porosity adjusts the flow under negative pressure and stops the air bubble diffusion need to be validated. In this talk, we will present the experimental results on Populus trees that support the idea that pits operate 'passively' in a biological point of view. This work is based on atomic force microscope (AFM) experiments, which have been realised to measure quantitatively the mechanical properties of pits at the nanoscale.

Using a perturbation method up to the second order, the equation for long waves at the interface between two viscous fluids is derived for plane Couette–Poiseuille flow, and for moderate surface tension. The leading order equation is the Burgers equation. Higher order terms take into account linear dispersive effects and stabilizing effect of surface tension, and involve two more nonlinear terms. The exactness of the coefficients of this equation has been checked by using symmetry properties. For zero gravity, Poiseuille flow is shown to be stable if and only if the velocity profile is convex. Stable Couette flow can become unstable when the Reynolds number of one fluid is decreased, while keeping the other dimensionless parameters unchanged. Introducing characteristic length scales, the interface equation is put into ‘‘canonical’’ form.