How vortices mix

International audienceThe advection of a passive scalar blob in the deformation field of an axisymmetric vortex is a simple mixing protocol for which the advection-diffusion problem is amenable to a near-exact description. The blob rolls-up in a spiral which ultimately fades away in the diluting medium. The complete transient concentration field in the spiral is accessible from the Fourier equations in a properly chosen frame. The concentration histogram of the scalar wrapped in the spiral presents unexpected singular transient features and its long time properties are discussed in connection with mixtures from the real world

Van Hove singularities in Probability Density Functions of scalars

A general theory for the Probability Density Function (PDF) of a scalar stirred in an axisymmetric time-dependent flow is derived. This theory reveals singularities, discontinuities and cusps occurring as soon as the spatial gradient of the scalar concentration vanishes somewhere in the field. These singularities are similar to the Van Hove singularities obtained in the density of vibration modes of a crystal. This feature, ubiquitous in convection–diffusion problems, is documented experimentally for the mixing of a dye in a Lamb–Oseen vortex

The destabilization of an initially thick liquid sheet edge

International audienceBy forcing the sudden dewetting of a free soap film attached on one edge to a straight solid wire, we study the recession and subsequent destabilization of its free edge. The newly formed rim bordering the sheet is initially thicker than the film to which it is attached, because of the Plateau border preexisting on the wire. The initial condition is thus that of an immobile massive toroidal rim connected to a thin liquid film of thickness h. The terminal Taylor-Culick receding velocity V = sqrt(2 sigma/rho h), where sigma and rho are the liquid surface tension and density, respectively, is only reached after a transient acceleration period which promotes the rim destabilization. The selected wavelength and associated growth time coincide with those of an inertial instability driven by surface tension

Fragmentation de liquides et de solides

On s'intéresse à la forme et à la dynamique d'objets déformables lors d'un impact avec une source de quantité de mouvement dirigée. On envisagera le cas d'un objet dur (élastique) choqué par un autre objet dur pour des géométries simples (tiges, feuilles), le cas d'un objet mou (liquide) impactant un solide, et le cas d'un objet mou se déformant dans un milieu encore plus mou (gaz). On soulignera l'intérêt de ces observations pour le problème de la fragmentation en général, et pour la compréhension de certains phénomènes naturels comme la pluie

Impacts on thin elastic sheets

International audienceWe study transverse impacts of rigid objects on a free elastic membrane, using thin circular sheets of natural rubber as experimental models. After impact, two distinct axisymmetric waves propagate in and on the sheet. First a tensile wave travels at sound speed leaving behind the wave front a stretched domain. Then, a transverse wave propagates on the stretched area at a lower speed. In the stretched area, geometrical confinement induces compressive circumferential stresses leading to a buckling instability, giving rise to radial wrinkles. We report on a set of experiments and theoretical remarks on the conditions of occurrence of these wrinkles, their dynamics and wavelength

Experimental measurement of the Melnikov function

International audienceWe study the transport properties of a genuine two-dimensional flow with a large mean velocity perturbed periodically in time by means of an original experimental technique. The flow, generated by the co-rotation of two cylinders is both stratified with a linear density gradient using salted water, and viscous in order to prevent Ekman pumping and centrifugal instabilities. Thus, the mean flow contains a hy-perbolic point with a homoclinic streamline, which we perturb periodically by an extra oscillation. A blob of scalar injected close to the stagnation point contracts on the stable manifold, and stretches in the unstable direction. The distance between the stable and the unstable manifolds is measured as the distance between the maximum and the minimum of the dye undulating pattern, and is recorded as a function of the perturbation frequency. This distance, also called the Melnikov function, presents a maximum when the residence time of a fluid particle in the mean flow is about half a perturbation period. This resonance criterion is recovered with good quantitative agreement by the theoretical prediction of the Melnikov function computed for this flow