Geometric mixing.

Mixing fluids often involves a periodic action, like stirring one's tea. But reciprocating motions in fluids at low Reynolds number, in Stokes flows where inertia is negligible, lead to periodic cycles of mixing and unmixing, because the physics, molecular diffusion excepted, is time reversible. So how can fluid be mixed in such circumstances? The answer involves a geometric phase. Geometric phases are found everywhere in physics as anholonomies, where after a closed circuit in the parameters, some system variables do not return to their original values. We discuss the geometric phase in fluid mixing: geometric mixing. This article is part of the theme issue 'Stokes at 200 (part 2)'.

Structural and dynamic properties of soda-lime-silica in the liquid phase.

Soda-lime-silica is a glassy system of strong industrial interest. In order to characterize its liquid state properties, we performed molecular dynamics simulations employing an aspherical ion model that includes atomic polarization and deformation effects. They allowed us to study the structure and diffusion properties of the system at temperatures ranging from 1400 K to 3000 K. We show that Na+ and Ca2+ ions adopt a different structural organization within the silica network, with Ca2+ ions having a greater affinity for non-bridging oxygens than Na+. We further link this structural behavior to their different diffusivities, suggesting that escaping from the first oxygen coordination shell is the limiting step for the diffusion. Na+ diffuses faster than Ca2+ because it is bonded to a smaller number of non-bridging oxygens. The formed ionic bonds are also less strong in the case of Na+.

Topological Symmetry Breaking in Viscous Coarsening.

The crucial role of hydrodynamic pinch-off instabilities is evidenced in the coarsening stage of viscous liquids. The phase separation of a barium borosilicate glass melt is studied by in situ synchrotron tomography at high temperature. The high viscosity contrast between the less viscous phase and the more viscous phase induces a topological symmetry breaking: capillary breakups occur preferentially in the less viscous phase. As a result, contrasting morphologies are obtained in the two phases. This symmetry breaking is illustrated on three different glass compositions, corresponding to different volume fractions of the two phases. In particular, a fragmentation phenomenon, reminiscent of the end-pinching mechanism proposed by Stone and co-workers is evidenced in the less viscous phase.

Fragmentation and limits to dynamical scaling in viscous coarsening: an interrupted in situ x-ray tomographic study.

X-ray microtomography was used to follow the coarsening of the structure of a ternary silicate glass experiencing phase separation in the liquid state. The volumes, surfaces, mean, and Gaussian curvatures of the domains of minority phase were measured after reconstruction of the 3D images and segmentation. A linear growth law of the characteristic length scale ℓ∼t was observed. A detailed morphological study was performed. While dynamical scaling holds for most of the geometrical observables under study, a progressive departure from scaling invariance of the distributions of local curvatures was evidenced. The latter results from a gradual fragmentation of the structure in the less viscous phase that also leads to a power-law size distribution of isolated domains.

Topology of chaotic mixing patterns.

A stirring device consisting of a periodic motion of rods induces a mapping of the fluid domain to itself, which can be regarded as a homeomorphism of a punctured surface. Having the rods undergo a topologically complex motion guarantees at least a minimal amount of stretching of material lines, which is important for chaotic mixing. We use topological considerations to describe the nature of the injection of unmixed material into a central mixing region, which takes place at injection cusps. A topological index formula allow us to predict the possible types of unstable foliations that can arise for a fixed number of rods.

Analyzing microtomography data with Python and the scikit-image library.

The exploration and processing of images is a vital aspect of the scientific workflows of many X-ray imaging modalities. Users require tools that combine interactivity, versatility, and performance. scikit-image is an open-source image processing toolkit for the Python language that supports a large variety of file formats and is compatible with 2D and 3D images. The toolkit exposes a simple programming interface, with thematic modules grouping functions according to their purpose, such as image restoration, segmentation, and measurements. scikit-image users benefit from a rich scientific Python ecosystem that contains many powerful libraries for tasks such as visualization or machine learning. scikit-image combines a gentle learning curve, versatile image processing capabilities, and the scalable performance required for the high-throughput analysis of X-ray imaging data.

Topological mixing with ghost rods.

Topological chaos relies on the periodic motion of obstacles in a two-dimensional flow in order to form nontrivial braids. This motion generates exponential stretching of material lines, and hence efficient mixing. Boyland, Aref, and Stremler [J. Fluid Mech. 403, 277 (2000)] have studied a specific periodic motion of rods that exhibits topological chaos in a viscous fluid. We show that it is possible to extend their work to cases where the motion of the stirring rods is topologically trivial by considering the dynamics of special periodic points that we call "ghost rods", because they play a similar role to stirring rods. The ghost rods framework provides a new technique for quantifying chaos and gives insight into the mechanisms that produce chaos and mixing. Numerical simulations for Stokes flow support our results.

Geometric Mixing, Peristalsis, and the Geometric Phase of the Stomach.

Mixing fluid in a container at low Reynolds number--in an inertialess environment--is not a trivial task. Reciprocating motions merely lead to cycles of mixing and unmixing, so continuous rotation, as used in many technological applications, would appear to be necessary. However, there is another solution: movement of the walls in a cyclical fashion to introduce a geometric phase. We show using journal-bearing flow as a model that such geometric mixing is a general tool for using deformable boundaries that return to the same position to mix fluid at low Reynolds number. We then simulate a biological example: we show that mixing in the stomach functions because of the "belly phase," peristaltic movement of the walls in a cyclical fashion introduces a geometric phase that avoids unmixing.

scikit-image: image processing in Python.

scikit-image is an image processing library that implements algorithms and utilities for use in research, education and industry applications. It is released under the liberal Modified BSD open source license, provides a well-documented API in the Python programming language, and is developed by an active, international team of collaborators. In this paper we highlight the advantages of open source to achieve the goals of the scikit-image library, and we showcase several real-world image processing applications that use scikit-image. More information can be found on the project homepage, http://scikit-image.org.

Moving walls accelerate mixing.

Mixing in viscous fluids is challenging, but chaotic advection in principle allows efficient mixing. In the best possible scenario, the decay rate of the concentration profile of a passive scalar should be exponential in time. In practice, several authors have found that the no-slip boundary condition at the walls of a vessel can slow down mixing considerably, turning an exponential decay into a power law. This slowdown affects the whole mixing region, and not just the vicinity of the wall. The reason is that when the chaotic mixing region extends to the wall, a separatrix connects to it. The approach to the wall along that separatrix is polynomial in time and dominates the long-time decay. However, if the walls are moved or rotated, closed orbits appear, separated from the central mixing region by a hyperbolic fixed point with a homoclinic orbit. The long-time approach to the fixed point is exponential, so an overall exponential decay is recovered, albeit with a thin unmixed region near the wall.