The role of mentorship in protégé performance.

The role of mentorship in protégé performance is a matter of importance to academic, business and governmental organizations. Although the benefits of mentorship for protégés, mentors and their organizations are apparent, the extent to which protégés mimic their mentors' career choices and acquire their mentorship skills is unclear. The importance of a science, technology, engineering and mathematics workforce to economic growth and the role of effective mentorship in maintaining a 'healthy' such workforce demand the study of the role of mentorship in academia. Here we investigate one aspect of mentor emulation by studying mentorship fecundity-the number of protégés a mentor trains-using data from the Mathematics Genealogy Project, which tracks the mentorship record of thousands of mathematicians over several centuries. We demonstrate that fecundity among academic mathematicians is correlated with other measures of academic success. We also find that the average fecundity of mentors remains stable over 60 years of recorded mentorship. We further discover three significant correlations in mentorship fecundity. First, mentors with low mentorship fecundities train protégés that go on to have mentorship fecundities 37% higher than expected. Second, in the first third of their careers, mentors with high fecundities train protégés that go on to have fecundities 29% higher than expected. Finally, in the last third of their careers, mentors with high fecundities train protégés that go on to have fecundities 31% lower than expected.

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. While initial conditions of unmixed materials add a layer of complexity, the inherent structure of E+ defines fundamental aspects of mixing by cutting and shuffling.

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. However, for γd2 below a critical value, D is independent of γd2. The acceleration due to gravity g and particle stiffness (or, equivalently, the binary collision time t(c)) together determine the transition in D between regimes. This suggests that while shear rate and particle size determine diffusion at relatively high shear rates in surface-driven flows, diffusion at low shear rates is an elastic phenomenon with time and length scales dependent on gravity (sqrt d/g) and particle stiffness (t(c)sqrt(dg), respectively.

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]. Small amounts of initially spatially localized B and C and a large amount of spatially homogeneous A are advected by the velocity field, diffuse, and react until A is completely consumed and only B and C remain. We find that local finite-time Lyapunov exponents (FTLEs) can accurately predict the final average concentrations of B and C after the reaction completes. The species that starts in the region with the larger FTLE has, with high probability, the larger average concentration at the end of the reaction. If B and C start in regions with similar FTLEs, their average concentrations at the end of the reaction will also be similar. When a recycling reaction is added, the system evolves towards a single species state, with the FTLE often being useful in predicting which species fills the entire domain and which is depleted. The FTLE approach is also demonstrated for competitive autocatalytic reactions in journal bearing flow, an experimentally realizable flow that generates chaotic dynamics.

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])=6.464, divides percolation behavior into passing and trapping regimes. However, the mean percolation velocity and diffusion of untrapped fine particles, which depend on both R and ϕ, are similar in both regimes and can be collapsed over a range of R and ϕ with the appropriate scaling. An empirical relationship for the local percolation velocity based on the local pore throat to fine particle size ratio and packing density is obtained, which is valid for the full range of size ratio and packing density we study. Similarly, in the trapping regime, the probability for a fine particle to reach a given depth is well described by a simple statistical model. Finally, the percolation velocity and fine particle diffusion are found to decrease with increasing restitution coefficient.

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. Specifically, we find that after large particles segregate to the surface, slow axial drift in the flowing layer, which is inherent to spherical tumblers, is sufficient to drive large particles across nominal transport barriers and into nonmixing islands predicted by an advective flow model in the absence of axial drift. Axial drift alters the periodic dynamics of nonmixing islands, turning them into "sinks" where large particles accumulate even in the presence of collisional diffusion. Overall, our results indicate that weak perturbation of chaotic flow has the potential to alter key dynamical system features (e.g., transport barriers), which ultimately can result in unexpected physical phenomena.

Cascading failure and robustness in metabolic networks.

We investigate the relationship between structure and robustness in the metabolic networks of Escherichia coli, Methanosarcina barkeri, Staphylococcus aureus, and Saccharomyces cerevisiae, using a cascading failure model based on a topological flux balance criterion. We find that, compared to appropriate null models, the metabolic networks are exceptionally robust. Furthermore, by decomposing each network into rigid clusters and branched metabolites, we demonstrate that the enhanced robustness is related to the organization of branched metabolites, as rigid cluster formations in the metabolic networks appear to be consistent with null model behavior. Finally, we show that cascading in the metabolic networks can be described as a percolation process.

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. Capture units placed within the domain remove all particles that cross their perimeters without altering the velocity field. To predict the capture capability of a unit at an arbitrary location, we characterize the trajectories of a uniformly seeded ensemble of particles as chaotic or nonchaotic, and then use them to determine the spatially resolved fraction of time that the flow is chaotic. With this information, we can predict where to best place units for maximum capture. We also examine the time dependence of the capture process, and demonstrate that there can be a trade-off between the amount of material captured and the capture rate.

Onset mechanism for granular axial band formation in rotating tumblers.

The mechanism for band formation of a granular mixture in long rotating tumblers is unresolved 70 years after the phenomenon was first observed. We explore the onset mechanism for axial segregation of a bidisperse mixture of particles of different sizes using the discrete element method. End walls initiate axial band formation via an axial flow due to friction at the end walls. The nonuniform distribution of axial velocity in the flow together with simultaneous radial segregation due to percolation result in small particles being driven further from the end walls, while larger particles accumulate at the end walls. Once this occurs, a cascading effect likely causes other bands to form due to the axial gradient in particle concentrations.

Chaotic mixing via streamline jumping in quasi-two-dimensional tumbled granular flows.

We study, numerically and analytically, the singular limit of a vanishing flowing layer in tumbled granular flows in quasi-two-dimensional rotating containers. The limiting behavior is found to be identical under the two versions of the kinematic continuum model of such flows, and the transition to the limiting dynamics is analyzed in detail. In particular, we formulate the no-shear-layer dynamical system as a piecewise isometry. It is shown how such a discontinuous map, through the concordant mechanism of streamline jumping, leads to the physical mixing of granular matter. The dependence of the dynamics of Lagrangian particle trajectories on the tumbler fill fraction is also established through Poincaré sections, and, in the special case of a half-full tumbler, chaotic behavior is shown to disappear completely in the singular limit. At other fill levels, stretching in the sense of shear strain is replaced by spreading due to streamline jumping. Finally, we use finite-time Lyapunov exponents to establish the manifold structure and understand "how chaotic" the limiting piecewise isometry is.

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. Here we provide computational and analytical formalisms for examining this structure by way of measuring the density and connectivity of ɛ-fattened cutting lines that form an approximation of E[over ¯]. For the example of a PWI on a hemispherical shell studied here, this approach reveals the subtle mixing behaviors and barriers to mixing formed by invariant ergodic subsets (confined orbits) within the fractal structure of the exceptional set. Some PWIs on the shell have provably nonergodic exceptional sets, which prevent mixing, while others have potentially ergodic exceptional sets where mixing is possible since ergodic exceptional sets have uniform cutting line density. For these latter exceptional sets, we show the connectivity of orbits in the PWI map through direct examination of orbit position and shape and through a two-dimensional return plot to explain the necessity of orbit connectivity for 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. Conti-nuum models, however, offer the best possibility for parametric studies of outcomes in what could be a prohibitively large space resulting from the competition between three distinct driving mechanisms: advection, diffusion, and segregation. We present a continuum transport equation-based framework, informed by phenomenological constitutive equations, that accurately predicts segregation in many settings, both industrial and natural. Three-way comparisons among experiments, DEM, and theory are offered wherever possible to validate the approach. In addition to the flows and mixtures described here, many straightforward extensions of the framework appear possible.

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. The protocol-dependent structure of the unstable manifolds of the flow surrounding the nonmixing islands provides further insight into why certain segregation patterns are more robust than others.

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. The rate of segregation decreases with each pass so that the degree of segregation asymptotically approaches its maximum value, and the number of passes to reach a steady segregation state increases as the bed depth is increased or the blade height decreased. In shallow beds, the characteristic number of passes for segregation, τ, scales with the inverse of the granular inertial number, I. In deep beds with small blade heights, the effect of the blade is more localized to its immediate vicinity, resulting in many more passes of the blade to reach a steady segregation state, and a corresponding deviation from the shallow bed scaling of τ with I^{-1}.

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. Nonorthogonal rotation axes break some of the symmetries that produce poor mixing with orthogonal axes and increase the overall degree of mixing on average. Arnold tongues arising from rational ratios of rotation angles and their intersections, as in the orthogonal axes case, are responsible for many protocols with low degrees of mixing in the nonorthogonal-axes parameter space. Arnold tongue intersections along a fundamental symmetry plane of the system reveal a new and unexpected class of protocols whose dynamics are periodic, with exceptional sets forming polygonal tilings of the hemispherical shell.

Evolving loop structure in gradually tilted two-dimensional granular packings.

Granular packings, especially near the jamming transition, form fragile networks where small perturbations can lead to destabilization and large scale rearrangements. A key stabilizing element in two dimensions is the contact loop, yet surprisingly little is known about contact loop statistics in realistic granular networks. In this paper, we use particle dynamics to study the evolution of contact loop structure in a gradually tilted two-dimensional granular bed. We find that the resulting contact loop distributions (1) are sensitive to material properties, (2) deviate from the expected structure of a randomly wired lattice, and (3) are uniquely dependent on tilting angle. Also, we introduce a quantitative measure of loop stability xi and show that increased tilting results in a gradual destabilization of individual loops. We briefly discuss the considerations for extending our approach to three dimensions.

Subsurface granular flow in rotating tumblers: a detailed computational study.

To better understand the subsurface velocity field and flowing layer structure, we have performed a detailed numerical study using the discrete element method for the flow of monodisperse particles in half-full three-dimensional (3D) and quasi-2D rotating tumblers. Consistent with prior measurements at the surface, a region of high speed flow with axial components of velocity occurs near each endwall in long tumblers. This region can be eliminated by computationally omitting the friction at the endwalls, confirming that a mass balance argument based on the slowing of particles immediately adjacent to the frictional endwalls explains this phenomenon. The high speed region with the associated axial flow near frictional endwalls persists through the depth of the flowing layer, though the regions of high velocity shift in position and the velocity is lower compared to the surface. The axial flow near the endwalls is localized and independent with the length of the tumbler for tumblers longer than one tumbler diameter, but these regions interact for shorter tumblers. In quasi-2D tumblers, the high speed regions near the endwalls merge resulting in a higher velocity than occurs in a long tumbler, but with a flowing layer that is not as deep. Velocity fluctuations are altered near the endwalls. Particle velocity fluctuations are greatest just below the surface and diminish through the depth of the flowing layer.

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. At overburdens greater than approximately 20 times the lithostatic pressure at the bottom of the bed, the density segregation rate is zero while the size segregation rate is small but nonzero, suggesting that different physical mechanisms drive the two types of segregation. The segregation rate scales close to linearly with the inertial number for both size bidisperse and density bidisperse mixtures under various flow conditions, leading to a proposed pressure dependence term for existing segregation velocity correlations. Surprisingly, particle stiffness has only a minor effect on segregation, although it significantly affects the packing density.

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. The predictions from the model are consistent with experimentally validated discrete element method (DEM) simulations over a wide range of flow conditions and particle sizes. The degree of segregation depends on the Péclet number, Pe, defined as the ratio of the segregation rate to the diffusion rate, the relative segregation strength κij between particle species i and j, and a characteristic length L, which is determined by the strength of segregation between smallest and largest particles. A parametric study of particle size, κij , Pe and L demonstrates how particle segregation patterns depend on the interplay of advection, segregation and diffusion. Finally, the segregation pattern is also affected by the velocity profile and the degree of basal slip at the chute surface. The model is applicable to different flow geometries, and should be easily adapted to segregation driven by other particle properties such as density and shape.

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. A spherical shell half-filled with dry non-cohesive particles and periodically rotated about two horizontal axes generates complex structures that vary non-trivially with the rotation angles. They result from the interplay between fluid-like mixing by stretching-and-folding, and solids mixing by cutting-and-shuffling. In the experiments, larger non-mixing regions predicted by a cutting-and-shuffling model alone can persist for a range of protocols despite the presence of stretching-and-folding flows and particle-collision-driven diffusion. By uncovering the synergy of simultaneous fluid and solid mixing, we point the way to a more fundamental understanding of advection driven mixing in materials with coexisting flowing and non-flowing regions.