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.

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.

Effect of Gaussian curvature modulus on the shape of deformed hollow spherical objects.

A popular description of soft membranes uses the surface curvature energy introduced by Helfrich, which includes a spontaneous curvature parameter. In this paper we show how the Helfrich formula can also be of interest for a wider class of spherical elastic surfaces, namely with shear elasticity, and likely to model other deformable hollow objects. The key point is that when a stress-free state with spherical symmetry exists before subsequent deformation, its straightforwardly determined curvature ("geometrical spontaneous curvature") differs most of the time from the Helfrich spontaneous curvature parameter that should be considered in order to have the model being correctly used. Using the geometrical curvature in a set of independent parameters unveils the role of the Gaussian curvature modulus, which appears to play on the shape of an elastic surface even though this latter is closed, contrary to what happens for surfaces without spontaneous curvature. In appendices, clues are given to apply this alternative and convenient formulation of the elastic surface model to the particular case of thin spherical shells of isotropic material (TSSIMs).

Lamb-type waves generated by a cylindrical bubble oscillating between two planar elastic walls.

The volume oscillation of a cylindrical bubble in a microfluidic channel with planar elastic walls is studied. Analytical solutions are found for the bulk scattered wave propagating in the fluid gap and the surface waves of Lamb-type propagating at the fluid-solid interfaces. This type of surface wave has not yet been described theoretically. A dispersion equation for the Lamb-type waves is derived, which allows one to evaluate the wave speed for different values of the channel height h. It is shown that for h<λt, where λt is the wavelength of the transverse wave in the walls, the speed of the Lamb-type waves decreases with decreasing h, while for h on the order of or greater than λt, their speed tends to the Scholte wave speed. The solutions for the wave fields in the elastic walls and in the fluid are derived using the Hankel transforms. Numerical simulations are carried out to study the effect of the surface waves on the dynamics of a bubble confined between two elastic walls. It is shown that its resonance frequency can be up to 50% higher than the resonance frequency of a similar bubble confined between two rigid walls.

Ultrasonic emissions reveal individual cavitation bubbles in water-stressed wood.

Under drought conditions, the xylem of trees that conducts ascending sap produces ultrasonic emissions whose exact origin is not clear. We introduce a new method to record simultaneously both acoustic events and optical observation of the xylem conduits within slices of wood that were embedded in a transparent material setting a hydric stress. In this article, we resolved the rapid development of all cavitation bubbles and demonstrated that each ultrasound emission was linked to the nucleation of one single bubble, whose acoustic energy is an increasing function of the size of the conduit where nucleation occurred and also of the hydric stress. We modelled these observations by the fact that water columns in conduits store elastic energy and release it into acoustic waves when they are broken by cavitation bubbles. Water columns are thus elastic, and not rigid, 'wires of water' set under tension by hydric stresses. Cavitation bubbles are at the origin of an embolism, whose development was followed in our experiments. Such an embolism of sap circulation can result in a fatal condition for living trees. These findings provide new insights for the non-destructive monitoring of embolisms within trees, and suggest a new approach to study porous media under hydric stress.

Plastic and viscous dissipations in foams: cross-over from low to high shear rates.

Soft glassy materials made of deformable cells, such as liquid foams, simultaneously display elastic, plastic and viscous behaviours. Bubble deformation is elastic until the material plastically yields and bubbles swap neighbours, then bubbles relax dissipatively towards a new energy minimum. This relaxation occurs in a finite time, and shearing a foam at a fast strain rate compared to that time leads to a viscous flow. To describe such an elastic, plastic and viscous behaviour we introduce a simplified scalar model of foam deformation and flow with a periodic pinning potential. The continuum mechanics behaviour of the foam emerges as an ensemble average over disordered units without requiring that they are coupled. Our model captures surprisingly well various features of the viscous dissipation during plastic deformation. At low shear rates, the time averaged stress is smaller than the static yield stress. A critical shear rate exists: any flow at fixed stress has a shear rate above this critical value. Moreover, the model only involves measurable parameters, which enables us to compare it with existing experiments and simulations.

Discrete rearranging disordered patterns: prediction of elastic and plastic behavior, and application to two-dimensional foams.

We study the elasto-plastic behavior of materials made of individual (discrete) objects such as a liquid foam made of bubbles. The evolution of positions and mutual arrangements of individual objects is taken into account through statistical quantities such as the elastic strain of the structure, the yield strain, and the yield function. The past history of the sample plays no explicit role except through its effect on these statistical quantities. They suffice to relate the discrete scale with the collective global scale. At this global scale, the material behaves as a continuous medium; it is described with tensors such as elastic strain, stress, and velocity gradient. We write the differential equations which predict their elastic and plastic behavior in both the general case and the case of simple shear. An overshoot in the shear strain or shear stress is interpreted as a rotation of the deformed structure, which is a purely tensorial effect that exists only if the yield strain is at least of order 0.3. We suggest practical applications including the following: when to choose a scalar formalism rather than a tensorial one; how to relax trapped stresses; and how to model materials with a low, or a high, yield strain.

Numerical modelling of foam Couette flows.

A numerical computation based on a tensorial visco-elasto-plastic model based on continuous mechanics is compared to experimental measurements on liquid foams for a bidimensional Couette flow between two glass plates, both in stationary and transient cases. The main features of the model are elasticity up to a plastic yield stress, and viscoelasticity above it. The effect of the friction of the plates is taken into account. The numerical modelling is based on a small set of standard material parameters that are fully characterised. Shear localisation as well as acute transient observations are reproduced and agree with experimental measurements. The plasticity appears to be the fundamental mechanism of the localisation of the flow. Finally, the present approach could be extended from liquid foams to similar materials such as emulsions, colloids or wet granular materials, that exhibit localisation.

Discrete rearranging disordered patterns, part II: 2D plasticity, elasticity and flow of a foam.

The plastic flow of a foam results from bubble rearrangements. We study their occurrence in experiments where a foam is forced to flow in 2D: around an obstacle; through a narrow hole; or sheared between rotating disks. We describe their orientation and frequency using a topological matrix defined in the companion paper (F. Graner, B. Dollet, C. Raufaste, and P. Marmottant, this issue, 25 (2008) DOI 10.1140/epje/i2007-10298-8), which links them with continuous plasticity at large scale. We then suggest a phenomenological equation to predict the plastic strain rate: its orientation is determined from the foam's local elastic strain; and its rate is determined from the foam's local elongation rate. We obtain a good agreement with statistical measurements. This enables us to describe the foam as a continuous medium with fluid, elastic and plastic properties. We derive its constitutive equation, then test several of its terms and predictions.

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.

An elastic, plastic, viscous model for slow shear of a liquid foam.

We suggest a scalar model for deformation and flow of an amorphous material such as a foam or an emulsion. To describe elastic, plastic and viscous behaviours, we use three scalar variables: elastic deformation, plastic deformation rate and total deformation rate; and three material-specific parameters: shear modulus, yield deformation and viscosity. We obtain equations valid for different types of deformations and flows slower than the relaxation rate towards mechanical equilibrium. In particular, they are valid both in transient or steady flow regimes, even at large elastic deformation. We discuss why viscosity can be relevant even in this slow shear (often called "quasi-static") limit. Predictions of the storage and loss moduli agree with the experimental literature, and explain with simple arguments the non-linear large amplitude trends.

Transient Surface Tension of an Expanding Liquid Sheet.

We address the problem of dynamic surface tension using measurements of sheet diameters that results from the impact of a liquid jet of diameter d(0) on a small disk of diameter d(i) (d(i)/d(0) approximately 4). At low velocities, the sheet diameter D is related to d(0) by the Weber number We, constructed with the liquid density rho, the jet velocity u(0), and the surface tension sigma at the rim: D/d(0)=18 We=18 [rho u(0)(2)/(sigma/d(0))]. This relation expresses the equilibrium between inertial forces and surface tension forces at the sheet rim. When a surfactant has been dissolved in the bulk of the liquid prior to the formation of the initial jet, the rim surface tension, and therefore the sheet diameter, depends on the amount of surfactant adsorbed at the rim. This amount is fixed by a competition between surface formation induced by radial extension and repopulation of the liquid interface in surfactant. The experimental setup proposed here provides a method to measure dynamic surface tension from sheet diameter measurements and symmetrically to monitor the adsorption of a surfactant on a liquid surface. The available adsorption time ranges from 10 to 100 ms. Experimental data obtained with two surfactants are in agreement with a model of a diffusion-controlled adsorption at the interface. Copyright 2000 Academic Press.