Acoustic Resonance Frequencies of Underwater Toroidal Bubbles.

Underwater bubbles display an acoustic resonance frequency close to spherical ones. In order to obtain a resonance significantly deviating from the spherical case, we stabilize bubbles in toroidal frames, resulting in bubbles which can be slender while still compact. For thin tori the resonance frequency increases greatly. Between a pair of bubble rings, we can achieve a flat acoustic pressure field for a critical distance between rings, a condition reminiscent of Helmholtz coils in magnetostatics. This opens the possibility to shape the acoustic field using long tunnels of rings.

Acoustic tokamak with strongly coupled toroidal bubbles.

Gas bubbles stabilized in toroidal 3D-printed cages are good acoustic resonators with an unusual topology. We arrange them in a circular array to obtain what we call an "acoustic tokamak" because of the torus shape of the whole array. We demonstrate experimentally and theoretically that the system features several acoustic modes resulting from the acoustic interaction between tori. The fundamental acoustic mode has a much lower frequency than that of the individual bubbles. The acoustic field along the circle inside the acoustic tokamak is remarkably homogeneous, as shown by our 3D simulations.

Understanding and predicting viscous, elastic, plastic flows.

Foams, gels, emulsions, polymer solutions, pastes and even cell assemblies display both liquid and solid mechanical properties. On a local scale, such "soft glassy" systems are disordered assemblies of deformable rearranging units, the complexity of which gives rise to their striking flow behaviour. On a global scale, experiments show that their mechanical behaviour depends on the orientation of their elastic deformation with respect to the flow direction, thus requiring a description by tensorial equations for continuous materials. However, due to their strong non-linearities, the numerous candidate models have not yet been solved in a general multi-dimensional geometry to provide stringent tests of their validity. We compute the first solutions of a continuous model for a discriminant benchmark, namely the flow around an obstacle. We compare it with experiments of a foam flow and find an excellent agreement with the spatial distribution of all important features: we accurately predict the experimental fields of velocity, elastic deformation, and plastic deformation rate in terms of magnitude, direction, and anisotropy. We analyse the role of each parameter, and demonstrate that the yield strain is the main dimensionless parameter required to characterize the materials. We evidence the dominant effect of elasticity, which explains why the stress does not depend simply on the shear rate. Our results demonstrate that the behaviour of soft glassy materials cannot be reduced to an intermediate between that of a solid and that of a liquid: the viscous, the elastic and the plastic contributions to the flow, as well as their couplings, must be treated simultaneously. Our approach opens the way to the realistic multi-dimensional prediction of complex flows encountered in geophysical, industrial and biological applications, and to the understanding of the link between structure and rheology of soft glassy systems.

Nonspherical vibrations of microbubbles in contact with a wall: a pilot study at low mechanical index.

Radially oscillating microbubbles can deform when in contact with a wall. These nonspherical shapes have a preferential orientation perpendicular to the wall. Conventional microscope setups for microbubble studies have their optical axis perpendicular to the wall (top view); consequently they have a limited view of the deformation of the bubble. We developed a method to image the bubble in a side view by integrating a mirror in the microscope setup. The image was recorded at 14.5 million frames per second by a high-speed camera. When insonified by a 1-MHz, 140-kPa ultrasound pulse, a 9-microm diameter coated bubble appeared spherical in the top view, but strongly nonspherical in the side view. Its shape was alternatively oblate and prolate, with maximum second order spherical harmonic amplitude equal to the radius.

Investigating acoustic-induced deformations in a foam using multiple light scattering.

We have studied the effect of an external acoustic wave on bubble displacements inside an aqueous foam. The signature of the acoustic-induced bubble displacements is found using a multiple light scattering technique, and occurs as a modulation on the photon correlation curve. Measurements for various sound frequencies and amplitudes are compared to analytical predictions and numerical simulations. These comparisons finally allow us to elucidate the nontrivial acoustic displacement profile inside the foam; in particular, we find that the acoustic wave creates a localized shear in the vicinity of the solid walls holding the foam, as a consequence of inertial contributions. This study of how bubbles "dance" inside a foam as a response to sound turns out to provide new insights on foam acoustics and sound transmission into a foam, foam deformation at high frequencies, and analysis of light scattering data in samples undergoing nonhomogeneous deformations.

Discrete rearranging disordered patterns, part I: robust statistical tools in two or three dimensions.

Discrete rearranging patterns include cellular patterns, for instance liquid foams, biological tissues, grains in polycrystals; assemblies of particles such as beads, granular materials, colloids, molecules, atoms; and interconnected networks. Such a pattern can be described as a list of links between neighbouring sites. Performing statistics on the links between neighbouring sites yields average quantities (hereafter "tools") as the result of direct measurements on images. These descriptive tools are flexible and suitable for various problems where quantitative measurements are required, whether in two or in three dimensions. Here, we present a coherent set of robust tools, in three steps. First, we revisit the definitions of three existing tools based on the texture matrix. Second, thanks to their more general definition, we embed these three tools in a self-consistent formalism, which includes three additional ones. Third, we show that the six tools together provide a direct correspondence between a small scale, where they quantify the discrete pattern's local distortion and rearrangements, and a large scale, where they help describe a material as a continuous medium. This enables to formulate elastic, plastic, fluid behaviours in a common, self-consistent modelling using continuous mechanics. Experiments, simulations and models can be expressed in the same language and directly compared. As an example, a companion paper (P. Marmottant, C. Raufaste, and F. Graner, this issue, 25 (2008) DOI 10.1140/epje/i2007-10300-7) provides an application to foam plasticity.

Yield drag in a two-dimensional foam flow around a circular obstacle: effect of liquid fraction.

We study the two-dimensional flow of foams around a circular obstacle within a long channel. In experiments, we confine the foam between liquid and glass surfaces. In simulations, we use a deterministic software, the Surface Evolver, for bubble details and a stochastic one, the extended Potts model, for statistics. We adopt a coherent definition of liquid fraction for all studied systems. We vary it in both experiments and simulations, and determine the yield drag of the foam, that is, the force exerted on the obstacle by the foam flowing at very low velocity. We find that the yield drag is linear over a large range of the ratio of obstacle to bubble size, and is independent of the channel width over a large range. Decreasing the liquid fraction, however, strongly increases the yield drag; we discuss and interpret this dependence.

Elasticity and plasticity of two-dimensional amorphous solid layers of beta-lactoglobulin.

We investigate the mechanical properties of a two-dimensional amorphous solid. It is formed spontaneously by the adsorption of a protein (the beta-lactoglobulin) at the surface of water. We measure its mechanical response in both elastic and plastic regimes by applying a point-like force (using a glass fiber). We compare our results with previous measurements of shear moduli using a floating torsion device.