Two-dimensional flow of foam around a circular obstacle: local measurements of elasticity, plasticity and flow

We investigate the two-dimensional flow of a liquid foam around circular obstacles by measuring all the local fields necessary to describe this flow: velocity, pressure, bubble deformations and rearrangements. We show how our experimental setup, a quasi-2D "liquid pool" system, is adapted to the determination of these fields: the velocity and bubble deformations are easy to measure from 2D movies, and the pressure can be measured by exploiting a specific feature of this system, a 2D effective compressibility. To describe accurately bubble rearrangements, we propose a new, tensorial descriptor. All these quantities are evaluated via an averaging procedure that we justify showing that the fluctuations of the fields are essentially random. The flow is extensively studied in a reference experimental case; the velocity presents an overshoot in the wake of the obstacle, the pressure is maximum at the leading side and minimal at the trailing side. The study of the elastic deformations and of the velocity gradients shows that the transition between plug flow and yielded regions is smooth. Our tensorial description of T1s highlight their correlation both with the bubble deformations and the velocity gradients. A salient feature of the flow, notably on the velocity and T1 repartition, is a marked asymmetry upstream/downstream, signature of the elastic behaviour of the foam. We show that the results do not change qualitatively when various control parameters vary, identifying a robust quasistatic regime. These results are discussed in the frame of the actual foam rheology literature, and we argue that they constitute a severe test for existing rheological models, since they capture both the elastic, plastic and fluid behaviour of the foam.Comment: 41 pages, 25 figures, submitted to Journal of Fluid Mechanics (but not in JFM style), short version of the abstrac

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 (Marmottant, Raufaste and Graner, joint paper) provides an application to foam plasticity