Erratum: Particle capture in a model chaotic flow [Phys. Rev. E 104, 064203 (2021)].

This corrects the article DOI: 10.1103/PhysRevE.104.

Percolation of a fine particle in static granular beds.

We study the percolation of a fine spherical particle under gravity in static randomly packed large-particle beds with different packing densities ϕ and large to fine particle size ratios R ranging from 4 to 7.5 using discrete element method simulations. The particle size ratio at the geometrical trapping threshold, defined by three touching large particles, R_{t}=sqrt[3]/(2-sqrt[3])=6.

Segregation patterns in three-dimensional granular flows.

Flow of size-bidisperse particle mixtures in a spherical tumbler rotating alternately about two perpendicular axes produces segregation patterns that track the location of nonmixing islands predicted by a dynamical systems approach. To better understand the paradoxical accumulation of large particles in regions defined by barriers to transport, we perform discrete element method (DEM) simulations to visualize the three-dimensional structure of the segregation patterns and track individual particles. Our DEM simulations and modeling results indicate that segregation pattern formation in the biaxial spherical tumbler is due to the interaction of size-driven radial segregation with the weak spanwise component of the advective surface flow.

Particle capture in a model chaotic flow.

To better understand and optimize the capture of passive scalars (particles, pollutants, greenhouse gases, etc.) in complex geophysical flows, we study capture in the simpler, but still chaotic, time-dependent double-gyre flow model. For a range of model parameters, the domain of the double-gyre flow consists of a chaotic region, characterized by rapid mixing, interspersed with nonmixing islands in which particle trajectories are regular.

Identifying invariant ergodic subsets and barriers to mixing by cutting and shuffling: Study in a birotated hemisphere.

Mixing by cutting and shuffling can be mathematically described by the dynamics of piecewise isometries (PWIs), higher dimensional analogs of one-dimensional interval exchange transformations. In a two-dimensional domain under a PWI, the exceptional set, E[over ¯], which is created by the accumulation of cutting lines (the union of all iterates of cutting lines and all points that pass arbitrarily close to a cutting line), defines where mixing is possible but not guaranteed. There is structure within E[over ¯] that directly influences the mixing potential of the PWI.

Granular segregation induced by a moving subsurface blade.

Size-driven particle segregation can occur when an object such as a blade moves through an otherwise static bed of granular material. Here we use discrete element method (DEM) simulations to study segregation resulting from a subsurface blade moving through a bed of size-bidisperse spherical particles. Segregation increases with each pass of the blade until a surface layer of mostly large particles forms above a small-particle layer adjacent to the bottom wall.

Pattern formation in a fully three-dimensional segregating granular flow.

Segregation patterns of size-bidisperse particle mixtures in a fully three-dimensional flow produced by alternately rotating a spherical tumbler about two perpendicular axes are studied over a range of particle sizes and volume ratios using both experiments and a continuum model. Pattern formation results from the interaction of size segregation with chaotic regions and nonmixing islands of the flow. Specifically, large particles in the flowing surface layer are preferentially deposited in nonmixing islands despite the effects of collisional diffusion and chaotic transport.

Cutting and shuffling a hemisphere: Nonorthogonal axes.

We examine the dynamics of cutting-and-shuffling a hemispherical shell driven by alternate rotation about two horizontal axes using the framework of piecewise isometry (PWI) theory. Previous restrictions on how the domain is cut-and-shuffled are relaxed to allow for nonorthogonal rotation axes, adding a new degree of freedom to the PWI. A new computational method for efficiently executing the cutting-and-shuffling using parallel processing allows for extensive parameter sweeps and investigations of mixing protocols that produce a low degree of mixing.

Modeling Segregation in Granular Flows.

Accurate continuum models of flow and segregation of dense granular flows are now possible. This is the result of extensive comparisons, over the last several years, of computer simulations of increasing accuracy and scale, experiments, and continuum models, in a variety of flows and for a variety of mixtures. Computer simulations-discrete element methods (DEM)-yield remarkably detailed views of granular flow and segregation.

Persistent structures in a three-dimensional dynamical system with flowing and non-flowing regions.

Mixing of fluids and mixing of solids are both relatively mature fields. In contrast, mixing in systems where flowing and non-flowing regions coexist remains largely unexplored and little understood. Here we report remarkably persistent mixing and non-mixing regions in a three-dimensional dynamical system where randomness is expected.

Effect of pressure on segregation in granular shear flows.

The effect of confining pressure (overburden) on segregation of granular material is studied in discrete element method (DEM) simulations of horizontal planar shear flow. To mitigate changes to the shear rate due to the changing overburden, a linear with depth variation in the streamwise velocity component is imposed using a simple feedback scheme. Under these conditions, both the rate of segregation and the ultimate degree of segregation in size bidisperse and density bidisperse granular flows decrease with increasing overburden pressure and scale with the overburden pressure normalized by the lithostatic pressure of the particle bed.

Continuum modelling of segregating tridisperse granular chute flow.

Segregation and mixing of size multidisperse granular materials remain challenging problems in many industrial applications. In this paper, we apply a continuum-based model that captures the effects of segregation, diffusion and advection for size tridisperse granular flow in quasi-two-dimensional chute flow. The model uses the kinematics of the flow and other physical parameters such as the diffusion coefficient and the percolation length scale, quantities that can be determined directly from experiment, simulation or theory and that are not arbitrarily adjustable.

Transient response in granular quasi-two-dimensional bounded heap flow.

We study the transition between steady flows of noncohesive granular materials in quasi-two-dimensional bounded heaps by suddenly changing the feed rate. In both experiments and simulations, the primary feature of the transition is a wedge of flowing particles that propagates downstream over the rising free surface with a wedge front velocity inversely proportional to the square root of time. An additional longer duration transient process continues after the wedge front reaches the downstream wall.

Mixing and transport from combined stretching-and-folding and cutting-and-shuffling.

While structures and bifurcations controlling tracer particle transport and mixing have been studied extensively for systems with only stretching-and-folding, and to a lesser extent for systems with only cutting-and-shuffling, few studies have considered systems with a combination of both. We demonstrate two bifurcations for nonmixing islands associated with elliptic periodic points that only occur in systems with combined cutting-and-shuffling and stretching-and-folding, using as an example a map approximating biaxial rotation of a less-than-half-full spherical granular tumbler. First, we characterize a bifurcation of elliptic island containment, from containment by manifolds associated with hyperbolic periodic points to containment by cutting line tangency.

Predicting mixing via resonances: Application to spherical piecewise isometries.

We present an analytic method to find the areas of nonmixing regions in orientation-preserving spherical piecewise isometries (PWIs), and apply it to determine the mixing efficacy of a class of spherical PWIs derived from granular flow in a biaxial tumbler. We show that mixing efficacy has a complex distribution across the protocol space, with local minima in mixing efficacy, termed resonances, that can be determined analytically. These resonances are caused by the interaction of two mode-locking-like phenomena.

Mixing and the fractal geometry of piecewise isometries.

Mathematical concepts often have applicability in areas that may have surprised their original developers. This is the case with piecewise isometries (PWIs), which transform an object by cutting it into pieces that are then rearranged to reconstruct the original object, and which also provide a paradigm to study mixing via cutting and shuffling in physical sciences and engineering. Every PWI is characterized by a geometric structure called the exceptional set, E, whose complement comprises nonmixing regions in the domain.

A review on locomotion robophysics: the study of movement at the intersection of robotics, soft matter and dynamical systems.

Discovery of fundamental principles which govern and limit effective locomotion (self-propulsion) is of intellectual interest and practical importance. Human technology has created robotic moving systems that excel in movement on and within environments of societal interest: paved roads, open air and water. However, such devices cannot yet robustly and efficiently navigate (as animals do) the enormous diversity of natural environments which might be of future interest for autonomous robots; examples include vertical surfaces like trees and cliffs, heterogeneous ground like desert rubble and brush, turbulent flows found near seashores, and deformable/flowable substrates like sand, mud and soil.

Mixing with piecewise isometries on a hemispherical shell.

We introduce mixing with piecewise isometries (PWIs) on a hemispherical shell, which mimics features of mixing by cutting and shuffling in spherical shells half-filled with granular media. For each PWI, there is an inherent structure on the hemispherical shell known as the exceptional set E, and a particular subset of E, E+, provides insight into how the structure affects mixing. Computer simulations of PWIs are used to visualize mixing and approximations of E+ to demonstrate their connection.

Principles of appendage design in robots and animals determining terradynamic performance on flowable ground.

Natural substrates like sand, soil, leaf litter and snow vary widely in penetration resistance. To search for principles of appendage design in robots and animals that permit high performance on such flowable ground, we developed a ground control technique by which the penetration resistance of a dry granular substrate could be widely and rapidly varied. The approach was embodied in a device consisting of an air fluidized bed trackway in which a gentle upward flow of air through the granular material resulted in a decreased penetration resistance.

Shear-Rate-Independent Diffusion in Granular Flows.

We computationally study the behavior of the diffusion coefficient D in granular flows of monodisperse and bidisperse particles spanning regions of relatively high and low shear rate in open and closed laterally confined heaps. Measurements of D at various flow rates, streamwise positions, and depths collapse onto a single curve when plotted as a function of γd2, where d is the local mean particle diameter and γ is the local shear rate. When γ is large, D is proportional to γd2, as in previous studies.

Force and flow at the onset of drag in plowed granular media.

We study the transient drag force FD on a localized intruder in a granular medium composed of spherical glass particles. A flat plate is translated horizontally from rest through the granular medium to observe how FD varies as a function of the medium's initial volume fraction, ϕ. The force response of the granular material differs above and below the granular critical state, ϕc, the volume fraction which corresponds to the onset of grain dilatancy.

Competitive autocatalytic reactions in chaotic flows with diffusion: prediction using finite-time Lyapunov exponents.

We investigate chaotic advection and diffusion in autocatalytic reactions for time-periodic sine flow computationally using a mapping method with operator splitting. We specifically consider three different autocatalytic reaction schemes: a single autocatalytic reaction, competitive autocatalytic reactions, which can provide insight into problems of chiral symmetry breaking and homochirality, and competitive autocatalytic reactions with recycling. In competitive autocatalytic reactions, species B and C both undergo an autocatalytic reaction with species A such that [Formula: see text] and [Formula: see text].

Slow axial drift in three-dimensional granular tumbler flow.

Models of monodisperse particle flow in partially filled three-dimensional tumblers often assume that flow along the axis of rotation is negligible. We test this assumption, for spherical and double cone tumblers, using experiments and discrete element method simulations. Cross sections through the particle bed of a spherical tumbler show that, after a few rotations, a colored band of particles initially perpendicular to the axis of rotation deforms: particles near the surface drift toward the pole, while particles deeper in the flowing layer drift toward the equator.

Flipper-driven terrestrial locomotion of a sea turtle-inspired robot.

To discover principles of flipper-based terrestrial locomotion we study the mechanics of a hatchling sea turtle-inspired robot, FlipperBot (FBot), during quasi-static movement on granular media. FBot implements a symmetric gait using two servo-motor-driven front limbs with flat-plate flippers and either freely rotating or fixed wrist joints. For a range of gaits, FBot moves with a constant step length.

Stratification, segregation, and mixing of granular materials in quasi-two-dimensional bounded heaps.

Segregation and mixing of granular mixtures during heap formation has important consequences in industry and agriculture. This research investigates three different final particle configurations of bidisperse granular mixtures--stratified, segregated and mixed--during filling of quasi-two-dimensional silos. We consider a large number and wide range of control parameters, including particle size ratio, flow rate, system size, and heap rise velocity.