No abstract

Fracture or cracking essentially involves the formation of new interfaces. These patterns are usually studied as twodimensional mosaics. The new surface that opens up is in the third dimension, along the thickness of the sample. The thickness is usually very small compared to the lateral dimensions of the pattern. A spectacular and distinctive departure from these everyday examples of cracks are columnar joints. Here, molten volcanic lava, by the sea, cools and cracks under appropriate thermal and elastic conditions, causing the crack system to grow downward, creating long, vertical columns with polygonal cross-section. The focus of this paper is the study of the elongated interfaces of these columns: how the crosssection of their outlines gradually undergoes a metamorphosis from a disordered-looking Gilbert tessellation to a well-ordered hexagonal Voronoi pattern. As the columns grow downward to lengths of several meters (in natural systems), their outline continuously changes, the center may shift, causing the column to twist. For the first time, the evolution of these crack mosaics has been simulated and mapped as a trajectory of a 4-vector tuple in a geometry-topology domain. The trajectory of the columnar joint systems is found to depend on the crack seed distribution and crack orientation. An empirical relationship between the system energy and the crack mosaic shape parameter λ has been proposed on the basis of principles of fracture mechanics. The total system energy shows a power-law dependence on λ with the exponent β ∼ 0.3 and λ ≈ 0.75 at crack maturation. The parameter values are validated by matching the proposed relation with energy estimates existing in the literature. The relation not only matches the visible changes in geometry but also provides a feasible measure of the energy of the system. The geometric energy for the polygonal mosaics in the transverse section has also been estimated as a function of time. The geometric energy moves toward a minimum as the mosaic becomes more Voronoi-like at maturation.

In this paper, we propose a mathematical picture of flow in a drying multiphase droplet. The system studied consists of a suspension of microscopic polystyrene beads in water. The time development of the drying process is described by defining the “Euler characteristic surface,” which provides a multiscale topological map of this dynamical system. A novel method is adopted to analyze the images extracted from experimental video sequences. Experimental image data are converted to binary data through appropriate Gaussian filters and optimal thresholding and analyzed using the Euler characteristic determined on a hexagonal lattice. In order to do a multiscale analysis of the extracted image, we introduce the concept of Euler characteristic at a specific scale r > 0. This multiscale time evolution of the connectivity information on aggregates of polysterene beads in water is summarized in a Euler characteristic surface and, subsequently, in a Euler characteristic level curve plot. We introduce a metric between Euler characteristic surfaces as a possible similarity measure between two flow situations. The constructions proposed by us are used to interpret flow patterns (and their stability) generated on the upper surface of the drying droplet interface. The philosophy behind the topological tools developed in this work is to produce low-dimensional signatures of dynamical systems, which may be used to efficiently summarize and distinguish topological information in various types of flow situations.