K-gamma Distribution Research Articles

Vortex line entanglement in active Beltrami flows

Over the last decade, substantial progress has been made in understanding the topology of quasi-two-dimensional (2-D) non-equilibrium fluid flows driven by ATP-powered microtubules and microorganisms. By contrast, the topology of three-dimensional (3-D) active fluid flows still poses interesting open questions. Here, we study the topology of a spherically confined active flow using 3-D direct numerical simulations of generalized Navier–Stokes (GNS) equations at the scale of typical microfluidic experiments. Consistent with earlier results for unbounded periodic domains, our simulations confirm the formation of Beltrami-like bulk flows with spontaneously broken chiral symmetry in this model. Furthermore, by leveraging fast methods to compute linking numbers, we explicitly connect this chiral symmetry breaking to the entanglement statistics of vortex lines. We observe that the mean of linking number distribution converges to the global helicity, consistent with the asymptotic result by Arnold [In Vladimir I. Arnold – Collected Works (ed. A.B. Givental, B.A. Khesin, A.N. Varchenko, V.A. Vassiliev & O.Y. Viro), pp. 357–375. Springer]. Additionally, we characterize the rate of convergence of this measure with respect to the number and length of observed vortex lines, and examine higher moments of the distribution. We find that the full distribution is well described by a k-Gamma distribution, in agreement with an entropic argument. Beyond active suspensions, the tools for the topological characterization of 3-D vector fields developed here are applicable to any solenoidal field whose curl is tangent to or cancels at the boundaries of a simply connected domain.

DISTRIBUTION AND EVOLUTION OF FREE VOLUME OF ELLIPSOIDAL PARTICLE SYSTEMS DURING SHEARING

Granular material is a complex multi-body interaction system which is composed of a large number of discrete particles and their surrounding free volume. Although the correlation between free volume and the mechanical properties as well as the deformation characteristics of granular materials has been proved, the local free volume of non-spherical particles is not fully understood at present due to the difficulties in characterizing. In this paper, the combined finite and discrete element method (FDEM) is used to simulate the triaxial tests of ellipsoidal particles with different principal axis lengths, and the Set Voronoi tessellation method is applied to construct the Voronoi cells of the particles during shearing. The statistical distribution and evolution of the local free volume of the granular systems during shearing are analyzed, and the influence of particle shape on the evolution of free volume is studied. Our results show the anisotropy of Voronoi cells gradually increases during shearing, and the degree of anisotropy increase will be intensified with the increase of particle shape asphericity, which means the granular assembly with a larger asphericity will experience more intense rearrangement during shearing. The local void ratio of ellipsoidal particle systems with different asphericity statistically complies with a k-gamma distribution, which is controlled by the global void ratio of granular assembly and not affected by particle shape and shear state. The local void ratio fluctuations follow an Asymmetric Laplace Distribution (ALD), and its asymmetric parameter which has a linear relationship with the global void ratio of granular assembly describes the competition between contraction and dilatation of local free volume.

Local responses in 2D assemblies of elliptical rods when subjected to biaxial shearing

This study explored the characteristics of local responses in 2D assemblies of elliptical particles when subjected to shearing. A biaxial shearing system was designed for this investigation, and two assemblies of elliptical rods with different eccentricities were used. To quantify the local responses, the assemblies were partitioned into individual Set Voronoi cells, and in this partition, the scaling method was proposed to tackle the interparticle overlapping problem. Then, in the Voronoi cells, the local strain was measured, accounting for the effect of interparticle rotation, and the local structures in terms of the local volume and anisotropy were estimated using Minkowski functionals. In the two samples, discontinuity by shearing was observed for these local quantities. Before the peak stress, the local strain developed randomly and the local volume followed a k-gamma distribution. After shearing, shear band and a mixed k-gamma distribution of the local volume emerged. For the local anisotropy, the variation inside shear band was significantly increased by shearing. Finally, the responses inside and outside the shear band were taken out separately for analysis, and the free-volume model was used to explain the differences between these two regions.

Contact forces distribution for a granular material from a Monte Carlo study on a single grain

The force network ensemble is one of the most promising statistical descriptions of granular media, with an entropy accounting for all force configurations at mechanical equilibrium consistent with some external stress. It is possible to define a temperature-like parameter, the angoricity $$\alpha ^{-1}$$ , which under isotropic compression is a scalar variable. This ensemble is frequently studied on whole packings of grains; ho wever, previous works have shown that spatial correlations can be neglected in many cases, opening the door to studies on a single grain. Our work develops a Monte Carlo method to sample the force ensemble on a single grain at constant angoricity on two and three-dimensional mono-disperse granular systems, both with or without static friction. The results show that, despite the steric exclusions and the constrictions of Coulomb’s limit and repulsive normal forces, the pressure per grain always show a k-gamma distribution with scale parameter $$\nu =\alpha ^{-1}$$ and shape parameter k close to $$k'$$ , the number of degrees of freedom in the system. Moreover, the average pressure per grain fulfills an equipartition theorem $$\langle p \rangle = k' {\alpha }^{-1} $$ in all cases (in close parallelism with the one for an ideal gas). Those results are in good agreement with the analysis of previous experimental and numerical results on many-grain systems, and with our own molecular dynamics simulations. These results suggest the existence of $$k'$$ independent random variables (i.e. elementary forces) with identical exponential distributions as the basic elements for describing the force network ensemble at low angoricities under isotropic compression, in analogy with the volume ensemble of granular materials.

Pore Characterisation in Monodisperse Granular Assemblies

The micro-scale geometric arrangement of pores was quantitatively characterised for monodisperse granular assemblies, particularly in relation to pore volume distribution and pore orientation characteristics. Using physical experiments and numerical simulations, the pore volume distribution was uniquely described by the analytical k-gamma distribution function [1-2]. A pore orientation tensor was defined to determine the preferred orientation of individual pores. This was subsequently used to define a global orientation tensor that revealed an isotropic pore network for the monodisperse granular assemblies considered in this study. The global orientation tensor was analytically linked to the parameters defining the pore volume distribution.

Pore shapes, volume distribution and orientations in monodisperse granular assemblies

The complex mechanical behaviour of granular materials is commonly studied by considering the evolving particle contact network. An often overlooked feature is the influence of micro-scale geometric configuration of pores on the macroscopic response. This paper presents a series of tools to quantify the shape, volume distribution and orientation characteristics of the pore space. The proposed approach is compared against data extracted from physical and numerical experiments with monodisperse assemblies of spheres. Individual pores are represented by polyhedral pore unit cells obtained by combining the Delaunay tessellation with an algorithm for merging Delaunay cells based on the concept of maximal inscribed sphere, after Al-Raoush et al. (Soil Sci Soc Am J 67(6):1687–1700, 2003). A pore shape parameter is proposed that considers pore volume and surface, and is analytically related to the void ratio and the number of edges forming the polyhedral pore unit cell. The pore volume distribution is shown to be uniquely described by the analytical k-gamma distribution proposed by Aste and Di Matteo (Phys Rev E 77(2):021309, 2008). A pore orientation tensor is introduced to define the principal orientation of individual pore units. This is subsequently used to define a global orientation tensor that reveals an isotropic pore network for the reference monodisperse assemblies. The global orientation tensor is analytically expressed in terms of the parameters defining the pore volume distribution.

Physical reconstruction of packed beds and their morphological analysis: Core–shell packings as an example

We report a fast, nondestructive, and quantitative approach to characterize the morphology of packed beds of fine particles by their three-dimensional reconstruction from confocal laser scanning microscopy images, exemplarily shown for a 100 μm i.d. fused-silica capillary packed with 2.6 μm-sized core–shell particles. The presented method is generally applicable to silica-based capillary columns, monolithic or particulate, and comprises column pretreatment, image acquisition, image processing, and statistical analysis of the image data. It defines a unique platform for fundamental comparisons of particulate and monolithic supports using the statistical measures derived from their reconstructions. Received morphological data are column cross-sectional porosity profiles and chord length distributions from the interparticle macropore space, which are a descriptor of local density and can be characterized by a simplified k-gamma distribution. This distribution function provides a parameter of location and a parameter of dispersion which can be correlated to individual chromatographic band broadening processes (i.e., to transchannel and short-range interchannel contributions to eddy dispersion, respectively). Together with the transcolumn porosity profile the presented approach allows to analyze and quantify the packing microstructure from pore to column scale and therefore holds great promise in a comparative study of packing conditions and particle properties, particularly for characterizing and minimizing the packing process-specific heterogeneities in the final bed structure.

On the k-gamma q-distribution

Abstract We provide combinatorial as well as probabilistic interpretations for the q-analogue of the Pochhammer k-symbol introduced by Díaz and Teruel. We introduce q-analogues of the Mellin transform in order to study the q-analogue of the k-gamma distribution.

Emergence of Gamma distributions in granular materials and packing models

We study the distribution of volume fluctuations in experiments and numerical simulations concerning equal-sized sphere packings prepared with different techniques. We show that the distribution of the local volumes (Voronoï cells) and also the distributions of the global volumes (whole samples) follow remarkably well a shifted and rescaled Gamma distribution that we name a k-Gamma distribution. Such agreement is robust over a broad range of packing fractions and it is observed for several distinct systems. This distribution is characterized by the average packing fraction and a shape parameter "k" which is very sensitive to changes in the structural organization. A statistical mechanics approach predicts such k-Gamma distribution at statistical equilibrium and it links the parameter k with the number of elementary cells which are exchanging volume during the system preparation. The thermodynamical equivalent of k and its relation with the "granular temperature" are also discussed.