Richard M. Lueptow

Festschrifts can provide valuable vantage points for reflecting on a discipline’s evolution and future possibilities. This article honoring Devang Vipin Khakhar (DVK) offers such an opportunity─not through a comprehensive career review, but by examining one dimension of scholarly achievements through the lens of collaborative work spanning 40 years and three generations of researchers. Specifically, we review DVK’s contributions to our understanding of flowing granular materials across three domains: surface flows, mixing and segregation, and computational modeling. This retrospective traces the field’s evolution from early explorations of chaos and mixing in rotating tumblers to sophisticated analyses of flow kinematics and segregation in heaps. We use this opportunity to chart the emergence of granular mechanics as a predictive science. In addition, we look forward to identifying persistent challenges and emerging opportunities: modeling unconfined geometries with evolving boundaries, extending segregation theory across wide size-ratio regimes, addressing cohesive effects and particle breakage, and integrating multiscale computational approaches with machine learning.

We numerically investigate percolation of independent frictionless glued-sphere rod-like particles under gravity through a disordered static bed of larger spheres. We identify two distinct regimes: a \emph{trapping} regime, where rods stop after percolating a limited distance in the bed and a \emph{passing} regime, where rods percolate continuously with constant mean velocity. The transition between these regimes is governed by the length of the rod and the geometrical trapping threshold for spherical particles based on the rod diameter and the minimum pore throat diameter defined by three touching large spheres. The percolation velocity for all rod geometries, including the single sphere limit, collapses onto a single curve when scaled with the gravitational acceleration and the bed sphere diameter. The results also demonstrate that short rods percolate nearly twice as fast as long rods due to the geometric constraints associated with the disordered pore structure of the static bed. Consequently, long rods are more susceptible to trapping via specific contact configurations with the bed spheres, which differ from those for short rods. These results reveal how shape anisotropy introduces dynamical constraints and thresholds in granular percolation, with implications for predicting segregation in mixtures of non-spherical particles.

We numerically investigate percolation of independent frictionless glued-sphere rod-like particles under gravity through a disordered static bed of larger spheres. We identify two distinct regimes: a \emph{trapping} regime, where rods stop after percolating a limited distance in the bed and a \emph{passing} regime, where rods percolate continuously with constant mean velocity. The transition between these regimes is governed by the length of the rod and the geometrical trapping threshold for spherical particles based on the rod diameter and the minimum pore throat diameter defined by three touching large spheres. The percolation velocity for all rod geometries, including the single sphere limit, collapses onto a single curve when scaled with the gravitational acceleration and the bed sphere diameter. The results also demonstrate that short rods percolate nearly twice as fast as long rods due to the geometric constraints associated with the disordered pore structure of the static bed. Consequently, long rods are more susceptible to trapping via specific contact configurations with the bed spheres, which differ from those for short rods. These results reveal how shape anisotropy introduces dynamical constraints and thresholds in granular percolation, with implications for predicting segregation in mixtures of non-spherical particles.

Percolation of fine particles (fines) in a static bed of larger particles is central to many industrial and natural processes. Non-cohesive fines either pass through the bed or become trapped depending on multiple factors including particle sizes, friction and restitution coefficients, and size-polydispersity. Here we consider the additional factor of cohesion. We use the discrete element method to simulate gravity-driven percolation of cohesive fine particles through a static bed of randomly packed large particles; fines interact with bed particles but not with each other. A large-to-fine particle diameter ratio of 7 geometrically permits non-cohesive fines to pass the narrowest pore throats formed by the large particles so they can freely percolate. However, sufficiently large cohesion and friction lead to non-geometric trapping. Fines are trapped when they fail to rebound after a collision, due to large cohesion, low restitution, and low collision velocity, and any subsequent rolling or sliding is insufficient to cause detachment. This establishes a sequence of local interactions -- collision, adhesion, and post-contact motion -- that governs the ultimate fate of a fine particle. A collisional model that incorporates a trapping probability per collision and a collision frequency predicts the trapping distance in the regime dominated by collision-induced trapping. For non-rebounding collisions, frictional effects are enhanced by cohesion and, when large enough, prevent the fine particle from subsequently detaching. A static equilibrium condition based on force balance predicts whether a fine particle remains stationary after contact. These results show that percolation of cohesive fine particles is not determined by geometric accessibility alone, but also by particle-scale interaction dynamics that can override geometric expectations.

Percolation of fine particles (fines) in a static bed of larger particles is central to many industrial and natural processes. Non-cohesive fines either pass through the bed or become trapped depending on multiple factors including particle sizes, friction and restitution coefficients, and size-polydispersity. Here we consider the additional factor of cohesion. We use the discrete element method to simulate gravity-driven percolation of cohesive fine particles through a static bed of randomly packed large particles; fines interact with bed particles but not with each other. A large-to-fine particle diameter ratio of 7 geometrically permits non-cohesive fines to pass the narrowest pore throats formed by the large particles so they can freely percolate. However, sufficiently large cohesion and friction lead to non-geometric trapping. Fines are trapped when they fail to rebound after a collision, due to large cohesion, low restitution, and low collision velocity, and any subsequent rolling or sliding is insufficient to cause detachment. This establishes a sequence of local interactions -- collision, adhesion, and post-contact motion -- that governs the ultimate fate of a fine particle. A collisional model that incorporates a trapping probability per collision and a collision frequency predicts the trapping distance in the regime dominated by collision-induced trapping. For non-rebounding collisions, frictional effects are enhanced by cohesion and, when large enough, prevent the fine particle from subsequently detaching. A static equilibrium condition based on force balance predicts whether a fine particle remains stationary after contact. These results show that percolation of cohesive fine particles is not determined by geometric accessibility alone, but also by particle-scale interaction dynamics that can override geometric expectations.

The influences of restitution coefficient, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msub> <a:mi>e</a:mi> <a:mi>n</a:mi> </a:msub> </a:math> , interparticle friction, <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>μ</b:mi> </b:math> , and size ratio, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>R</c:mi> </c:math> , on gravity-driven percolation of fine particles through static beds of larger particles in the free-sifting regime ( <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>R</d:mi> <d:mo>≳</d:mo> <d:mn>6.5</d:mn> </d:mrow> </d:math> ) remain largely unexplored. Here, we use discrete element method simulations to study the fine particle percolation velocity, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msub> <e:mi>v</e:mi> <e:mi>p</e:mi> </e:msub> </e:math> , and velocity fluctuations, <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:msub> <f:mi>v</f:mi> <f:mi>rms</f:mi> </f:msub> </f:math> , for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mn>7</g:mn> <g:mo>≤</g:mo> <g:mi>R</g:mi> <g:mo>≤</g:mo> <g:mn>50</g:mn> </g:mrow> </g:math> and a range of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:msub> <h:mi>e</h:mi> <h:mi>n</h:mi> </h:msub> </h:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>μ</i:mi> </i:math> . Varying <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:msub> <j:mi>e</j:mi> <j:mi>n</j:mi> </j:msub> </j:math> modulates the degree of particle excitation and thereby alters the nature of particle trajectories: at low <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:msub> <k:mi>e</k:mi> <k:mi>n</k:mi> </k:msub> </k:math> , percolation is dominated by gravity, whereas increasing <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:msub> <l:mi>e</l:mi> <l:mi>n</l:mi> </l:msub> </l:math> amplifies velocity fluctuations thereby reducing the mean percolation velocity. Increasing <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> </m:math> decreases <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:msub> <n:mi>v</n:mi> <n:mi>rms</n:mi> </n:msub> </n:math> but its influence on <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:msub> <o:mi>v</o:mi> <o:mi>p</o:mi> </o:msub> </o:math> varies with <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:msub> <p:mi>v</p:mi> <p:mi>rms</p:mi> </p:msub> </p:math> , decreasing <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"> <q:msub> <q:mi>v</q:mi> <q:mi>p</q:mi> </q:msub> </q:math> for low <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"> <r:msub> <r:mi>v</r:mi> <r:mi>rms</r:mi> </r:msub> </r:math> and increasing <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"> <s:msub> <s:mi>v</s:mi> <s:mi>p</s:mi> </s:msub> </s:math> for high <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"> <t:msub> <t:mi>v</t:mi> <t:mi>rms</t:mi> </t:msub> </t:math> . Although the influence of size ratio is weaker, larger values of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML"> <u:mi>R</u:mi> </u:math> increase both <v:math xmlns:v="http://www.w3.org/1998/Math/MathML"> <v:msub> <v:mi>v</v:mi> <v:mi>p</v:mi> </v:msub> </v:math> and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML"> <w:msub> <w:mi>v</w:mi> <w:mi>rms</w:mi> </w:msub> </w:math> . We also assess the influence of different excitation mechanisms, specifically using static, randomly excited, and sheared beds, finding that an inverse correlation between <x:math xmlns:x="http://www.w3.org/1998/Math/MathML"> <x:msub> <x:mi>v</x:mi> <x:mi>p</x:mi> </x:msub> </x:math> and <y:math xmlns:y="http://www.w3.org/1998/Math/MathML"> <y:msub> <y:mi>v</y:mi> <y:mi>rms</y:mi> </y:msub> </y:math> persists across all cases and is well-described by the Drude model, where increased scattering reduces mobility, when <z:math xmlns:z="http://www.w3.org/1998/Math/MathML"> <z:msub> <z:mi>v</z:mi> <z:mi>rms</z:mi> </z:msub> </z:math> is large. However, for weakly excited particles with low <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML"> <ab:msub> <ab:mi>v</ab:mi> <ab:mi>rms</ab:mi> </ab:msub> </ab:math> , the Drude analogy breaks down. In this regime, we introduce a staircase-inspired model that accounts for the gravitationally dominated percolation behavior. These findings provide fundamental insight into the mechanisms governing percolation dynamics in porous media and granular systems.

Size-bidisperse granular material flowing down a rough incline, or chute, may develop a Rayleigh-Taylor instability even when the two particle species have the same density. Unlike fluid instability, which occurs under quiescent conditions, the granular material must be flowing for the instability to occur. Initially, size segregation results in a monodisperse layer of large particles above a monodisperse layer of small particles with an interfacial layer of mixed particles that has a higher volume fraction and, hence, is denser than the monodisperse layers above and below it. The interfacial layer destabilizes via a Rayleigh-Taylor instability and forms dense descending plumes of mixed particles and ascending plumes of less dense pure small particles. As a result of these plumes, the upper layer of large particles breaks and accumulates above the descending plumes; the small-particle plumes reach the free surface. The instability evolves into persistent longitudinal rolls corresponding to streamwise bands, or stripes, of small and large particles at the free surface. The appearance of the instability depends on the particle and mixture properties as well as the flow conditions. In all cases, the propensity for the appearance of the instability and subsequent band formation can be traced back to the total particle volume fraction, or packing density, in the mixed-particle layer, which depends on the particle size ratio, fraction of each particle species, and thickness of the flow.

Granular materials, composed of discrete macroscopic particles such as sand, are ubiquitous in both natural and industrial contexts.These materials exhibit unique mechanical and transport properties due to their discrete nature and interparticle interactions through contact forces.Transport of fine particles within granular media plays a central role in processes ranging from chute flows and silos to geophysical flows and powder handling 1 .In such systems, the interplay of segregation 2 , confinement 3 , and diffusion 4 leads to complex dynamics that are not fully captured by existing models.A particular challenge arises at large particle size ratios, where fines can navigate void networks within the granular bed, resulting in transport mechanisms distinct from those in monodisperse or low size ratio systems 5 .We investigate fine particle diffusion across varying fine-particle concentrations using large-scale Discrete Element Method (DEM) simulations and find that the diffusion coefficient decreases with increasing concentration and size ratio.Drawing inspiration from kinetic theory, we develop a scaling framework that links particle concentration, size ratio, and bed geometry to diffusion behavior in dense granular beds.The framework has broad relevance for both industrial applications, such as mixers, separators, and hoppers, and fundamental studies of diffusion in heterogeneous media.

The influences of restitution coefficient, $e_n$, inter-particle friction, $μ$, and size ratio, $R$, on gravity-driven percolation of fine particles through static beds of larger particles in the free-sifting regime ($R \gtrsim 6.5$) remain largely unexplored. Here we use discrete element method simulations to study the fine particle percolation velocity, $v_p$, and velocity fluctuations, $v_{rms}$, for $7 \le R \le 50$ and a range of $e_n$ and $μ$. Increasing $e_n$ increases velocity fluctuations and reduces percolation velocity. Increasing $μ$ decreases $v_{rms}$ but its influence on $v_p$ varies with $v_{rms}$, decreasing $v_p$ for low $v_{rms}$ and increasing $v_p$ for high $v_{rms}$. Although the influence of size ratio is weaker, larger values of $R$ increase both $v_p$ and $v_{rms}$. We also assess the influence of different excitation mechanisms, specifically using static, randomly excited, and sheared beds, finding that an inverse correlation between $v_p$ and $v_{rms}$ persists across all cases and is well-described by the Drude model, where increased scattering reduces mobility, when $v_{rms}$ is large. However, for weakly excited particles with low $v_{rms}$, the Drude analogy breaks down. In this regime, we introduce a staircase-inspired model that accounts for the gravitationally dominated percolation behavior. These findings provide fundamental insight into the mechanisms governing percolation dynamics in porous media and granular systems.

Removing dispersed material, such as pollutants, from dynamic fluid environments like the ocean or the atmosphere is challenging when the flow is chaotic. Here the capture of passive tracer particles by a mobile collector (MC) is studied in a model two-dimensional chaotic flow with vortices. Four simple capture strategies for determining the MC direction are considered, all of which rely on periodic measurement of the local particle distribution. The ultimate success of a strategy depends on its associated motion and detection parameters as well as the underlying fluid flow. When the flow is fully chaotic or the relative velocity of the MC is large, the four strategies exhibit nearly equal effectiveness. However, when the flow is less chaotic and the relative MC velocity is small, the collector can become trapped in or outside of a vortex. Changing the particle detection parameters can prevent trapping, which improves capture. In the absence of trapping and for both high and low relative velocities of the MC, a scaling analysis explains the dependence of the capture rate on the relevant dimensionless variables based on timescales for the mobile collector and the underlying flow. For a wide range of parameters and all four capture strategies, the capture timescale depends linearly on a combination of the characteristic kinematic timescale related to the relative motion of the collector and the gradient timescale related to the underlying flow field, confirming that the capture process is properly characterized.

Predicting particle segregation has remained challenging due to the lack of a general model for the segregation velocity that is applicable across a range of granular flow geometries. Here, a segregation-velocity model for dense granular flows is developed by exploiting force balance and recent advances in particle-scale modelling of the segregation driving and drag forces over the entire particle concentration range, size ratios up to 3 and inertial numbers as large as 0.4. This model is shown to correctly predict particle segregation velocity in a diverse set of idealised and natural granular flow geometries simulated using the discrete element method. When incorporated in the well-established advection–diffusion–segregation formulation, the model has the potential to accurately capture segregation phenomena in many relevant industrial applications and geophysical settings.

Polyamide membranes, such as nanofiltration (NF) membranes, are widely used for water purification. However, the mechanisms of solute transport and solute rejection due to solute charge interactions with the membrane remain unclear at the molecular level. Here, we use molecular dynamics simulations to examine the transport of single-solute feeds through charged nanofiltration membranes with different membrane charge concentrations of COO- and NH+2 resulting from the deprotonation or protonation of polymeric end groups according to the pH level that the membrane experiences. The results show that Na+ and Cl- solute ions are better rejected when the membrane has a higher concentration of negatively charged groups, corresponding to a higher pH, whereas CaCl2 is well rejected at all pH levels studied. These results are consistent with those of experiments performed at the same pH conditions as the simulation setup. Moreover, solute transport behavior depends on the membrane functional group distribution. When COO- functional groups are concentrated at membrane feed surface, ion permeation into the membrane is reduced. Counter-ions tend to associate with charged functional groups while co-ions seem to pass by the charged groups more easily. In addition, steric effects play a role when ions of opposite charge cluster in pores of the membrane. This study reveals solute transport and rejection mechanisms related to membrane charge and provides insights into how membranes might be designed to achieve specific desired solute rejection.

Lift and drag forces on moving intruders in flowing granular materials are of fundamental interest but have not yet been fully characterized. Drag on an intruder in granular shear flow has been studied almost exclusively for the intruder moving across flow streamlines, and the few studies of the lift explore a relatively limited range of parameters. Here, we use discrete element method simulations to measure the lift force, $F_{{L}}$ , and the drag force on a spherical intruder in a uniformly sheared bed of smaller spheres for a range of streamwise intruder slip velocities, $u_{{s}}$ . The streamwise drag matches the previously characterized Stokes-like cross-flow drag. However, $F_{{L}}$ in granular shear flow acts in the opposite direction to the Saffman lift in a sheared fluid at low $u_{{s}}$ , reaches a maximum value and then decreases with increasing $u_{{s}}$ , eventually reversing direction. This non-monotonic response holds over a range of flow conditions, and the $F_{{L}}$ versus $u_{{s}}$ data collapse when both quantities are scaled using the particle size, shear rate and overburden pressure. Analogous fluid simulations demonstrate that the flow around the intruder particle is similar in the granular and fluid cases. However, the shear stress on the granular intruder is notably less than that in a fluid shear flow. This difference, combined with a void behind the intruder in granular flow in which the stresses are zero, significantly changes the lift-force-inducing stresses acting on the intruder between the granular and fluid cases.

Polyamide membranes, such as nanofiltration (NF) and reverse osmosis (RO) membranes, are widely used for water desalination and purification. However, the mechanisms of solute transport and solute rejection due to charge interactions remain unclear at the molecular level. Here we use molecular dynamics (MD) simulations to examine the transport of single-solute feeds through charged nanofiltration membranes with different membrane charge concentrations of COO$^{\text{-}}$ and NH$_2\!^+$ corresponding to different pH levels. Results show that Na$^+$ and Cl$^{\text{-}}$ solute ions are better rejected when the membrane has a higher concentration of negatively charged groups, corresponding to a higher pH, whereas CaCl$_2$ is well-rejected at all pH levels studied. These results are consistent with experimental findings which are performed at the same pH conditions as simulation setup. Moreover, solute transport behavior depends on the membrane functional group distribution. When COO$^{\text{-}}$ functional groups are concentrated at membrane feed surface, ion permeation into the membrane is reduced. Counter-ions tend to associate with charged functional groups while co-ions seem to pass by the charged groups more easily. In addition, steric effects play a role when ions of opposite charge cluster in pores of the membrane. This study reveals solute transport and rejection mechanisms related to membrane charge and provides insights into how membranes might be designed to achieve specific desired solute rejection.

The Discrete Element Method is widely employed for simulating granular flows, but conventional integration techniques may produce unphysical results for simulations with static friction when particle size ratios exceed $R \approx 3$. These inaccuracies arise because some variables in the velocity-Verlet algorithm are calculated at the half-timestep, while others are computed at the full timestep. To correct this, we develop an improved velocity-Verlet integration algorithm to ensure physically accurate outcomes up to the largest size ratios examined ($R=100$). The implementation of this improved integration method within the LAMMPS framework is detailed, and its effectiveness is validated through a simple three-particle test case and a more general example of granular flow in mixtures with large size-ratios, for which we provide general guidelines for selecting simulation parameters and accurately modeling inelasticity in large particle size-ratio simulations.

Lift and drag forces on moving intruders in granular materials are of fundamental interest. While the drag force on an intruder in granular flow has been studied, the few studies characterizing the lift force explore a relatively limited range of parameters. Here we use discrete element method (DEM) simulations to measure the lift force, $F_\mathrm{L}$, on a spherical intruder in a uniformly sheared bed of smaller spheres for a range of intruder slip velocities, $u_\mathrm{s}$, relative to the unperturbed flow. In what at first appears as a puzzling result, $F_\mathrm{L}$ in granular shear flow acts in the opposite direction to the Saffman lift force on a sphere in a sheared fluid at low $u_\mathrm{s}$, reaches a maximum value, and then decreases, eventually reversing direction and becoming comparable to $F_\mathrm{L}$ for a fluid. This non-monotonic response holds over a range of flow conditions, and the $F_\mathrm{L}$ versus $u_\mathrm{s}$ data can be collapsed by scaling both quantities using the particle sizes, shear rate, and overburden pressure. Analogous fluid simulations demonstrate that the flow field around the intruder particle is similar in the granular and fluid cases. However, the shear stress acting on the intruder in a granular shear flow is much less than that in a fluid shear flow. This difference, combined with a void region behind the intruder in granular flow, which alters the pressure and shear stress on the trailing side of the intruder, significantly changes the lift-force inducing stresses acting on the intruder between the granular and fluid cases.

This corrects the article DOI: 10.1103/PhysRevE.104.064203.

The origin of large and small particle axial bands in long rotating tumblers is a long-standing question. Using DEM simulations, we show that this axial segregation is due to a Rayleigh-Taylor instability which is characterized by the fact that the density of a granular medium increases with mixing and decreases with segregation. For initially mixed particles, segregation and collisional diffusion in the flowing layer balance and lead to a three-layer system, with a layer of large particles over a layer of small particles, and, interposed between these layers, a layer of more densely packed mixed particles. The higher density mixed particle layer over the lower density small particle layer induces a Rayleigh-Taylor instability, evident as waviness in the interface between the layers. The waviness destabilizes into ascending plumes of small particles and descending plumes of mixed particles with large particles enriched near the surface, which become evident as small and large particle bands visible at the free surface. Rolls driven by segregation at the tilted interface between plumes maintain the pattern of frozen plumes. Published by the American Physical Society 2024

Abstract Membranes facilitate scalable and continuous lithium concentration from hypersaline salt lakes and battery leachates. Conventional nanofiltration (NF) membranes, however, exhibit poor monovalent selectivity in high‐salinity environments due to weakened exclusion mechanisms. This study examines polyamide NF membranes coated with polyelectrolytes enriched with ammonium groups to maintain high monovalent cation selectivity in hypersaline conditions. Over 8000 ion rejection measurements are recorded using salt lake brines and battery leachates. The experiments exemplify the coated membrane's ability to reduce magnesium concentrations to 0.14% from salt lakes and elevate lithium purity to 98% from battery leachates, in a single filtration stage. The membrane's selectivity is retained after 12 weeks in acidic conditions. Molecular dynamics analyses reveal that the ammonium groups create an electrostatic barrier at low pH, selectively hindering multivalent cation transport. This is corroborated by the Coulombic attraction between cations and carboxylate groups, along with a repulsive barrier from ammonium groups. Despite a 14.7% increase in specific energy, a two‐stage NF system using the coated membranes for lithium recovery significantly reduces permeate magnesium composition to 0.031% from Chilean salt lake brines. For NMC leachates, the coated membranes achieve permeate lithium purity exceeding 99.5%, yielding enhanced permeate quality with minor increases in energy demands.

Particle segregation in dense flowing size-disperse granular mixtures is driven by gravity and shear, but predicting the associated segregation force due to both effects has remained an unresolved challenge. Here, a model of the combined gravity- and kinematics-induced segregation force on a single intruder particle is integrated with a model of the concentration dependence of the gravity-induced segregation force. The result is a general model of the net particle segregation force in flowing size-bidisperse granular mixtures. Using discrete element method simulations for comparison, the model correctly predicts the segregation force for a variety of mixture concentrations and flow conditions in both idealized and natural shear flows.

Abstract Static granular packings play a central role in numerous industrial applications and natural settings. In these situations, fluid or fine particle flow through a bed of static particles is heavily influenced by the narrowest passage connecting the pores of the packing, commonly referred to as pore throats, or constrictions. Existing studies predominantly assume monodisperse rigid particles, but this is an oversimplification of the problem. In this work, we illustrate the connection between pore throat size, polydispersity, and particle deformation in a packed bed of spherical particles. Simple analytical expressions are provided to link these properties of the packing, followed by examples from Discrete Element Method (DEM) simulations of fine particle percolation demonstrating the impact of polydispersity and particle deformation. Our intent is to emphasize the substantial impact of polydispersity and particle deformation on constriction size, underscoring the importance of accounting for these effects in particle transport in granular packings.

Small particles fall through sheared beds of larger particles in settings ranging from geophysics to industry, but the study of large-to-small size ratios <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>R</a:mi></a:math>, spanning the trapping threshold <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>R</b:mi><b:mi>t</b:mi></b:msub><b:mo>,</b:mo></b:mrow></b:math> has been neglected. In simulations of noncohesive spheres for <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>R</c:mi><c:mo>&lt;</c:mo><c:msub><c:mi>R</c:mi><c:mi>t</c:mi></c:msub><c:mo>,</c:mo></c:mrow></c:math> the small-sphere vertical velocity <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi>v</d:mi><d:mi>p</d:mi></d:msub></d:math> first increases with shear rate <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mover accent="true"><e:mi>γ</e:mi><e:mo>̇</e:mo></e:mover></e:math> as trapping time decreases, but <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mi>v</g:mi><g:mi>p</g:mi></g:msub></g:math> then decreases as velocity fluctuations frustrate downward mobility. For <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>R</h:mi><h:mo>&gt;</h:mo><h:msub><h:mi>R</h:mi><h:mi>t</h:mi></h:msub></h:mrow><h:mo>,</h:mo><h:mo> </h:mo><h:msub><h:mi>v</h:mi><h:mi>p</h:mi></h:msub></h:math> is constant at low <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mover accent="true"><i:mi>γ</i:mi><i:mo>̇</i:mo></i:mover><i:mo>,</i:mo></i:mrow></i:math> but again decreases at high <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mover accent="true"><k:mi>γ</k:mi><k:mo>̇</k:mo></k:mover></k:math>. We model these behaviors and discuss analogies with electron transport in solids. Published by the American Physical Society 2024

Static granular packings play a central role in numerous industrial applications and natural settings. In these situations, fluid or fine particle flow through a bed of static particles is heavily influenced by the narrowest passage connecting the pores of the packing, commonly referred to as pore throats or constrictions. Existing studies predominantly assume monodisperse rigid particles, but this is an oversimplification of the problem. In this work, we illustrate the connection between pore throat size, polydispersity, and particle deformation. Simple analytical expressions are provided to link these properties of the packing, followed by examples from Discrete Element Method (DEM) simulations of fine particle percolation demonstrating the impact of polydispersity and particle deformation. Our intent is to emphasize the substantial impact of polydispersity and particle deformation on constriction size, underscoring the importance of accounting for these effects in particle transport in granular packings.

We study the percolation velocity, $v_p,$ of a fine spherical particle in a sheared large-particle bed under gravity using discrete element method simulations for large-to-fine particle diameter ratios, $R=d/d_f,$ below and above the free-sifting threshold, $R_t\approx6.5.$ For $RR_t$, $v_p$ is constant at low $\dotγ$ but decreases toward zero at higher shear rates due to fine-particle excitation.

Particle segregation in dense flowing size-disperse granular mixtures is driven by gravity and shear, but predicting the associated segregation force due to both effects has remained an unresolved challenge. Here, a model of the combined gravity- and kinematics-induced segregation force on a single intruder particle is integrated with a model of the concentration dependence of the gravity-induced segregation force. The result is a general model of the net particle segregation force in flowing size-bidisperse granular mixtures. Using discrete element method simulations for comparison, the model correctly predicts the segregation force for a variety of mixture concentrations and flow conditions in both idealized and natural shear flows.

In 1923, the Philosophical Transactions published G. I. Taylor’s seminal paper on the stability of what we now call Taylor–Couette flow. In the century since the paper was published, Taylor’s ground-breaking linear stability analysis of fluid flow between two rotating cylinders has had an enormous impact on the field of fluid mechanics. The paper’s influence has extended to general rotating flows, geophysical flows and astrophysical flows, not to mention its significance in firmly establishing several foundational concepts in fluid mechanics that are now broadly accepted. This two-part issue includes review articles and research articles spanning a broad range of contemporary research areas, all rooted in Taylor’s landmark paper. This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical Transactions paper (part 2)’.

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.

In 1923, the Philosophical Transactions published G. I. Taylor’s seminal paper on the stability of what we now call Taylor–Couette flow. In the century since the paper was published, Taylor’s ground-breaking linear stability analysis of fluid flow between two rotating cylinders has had an enormous impact on the field of fluid mechanics. The paper’s influence has extended to general rotating flows, geophysical flows and astrophysical flows, not to mention its significance in firmly establishing several foundational concepts in fluid mechanics that are now broadly accepted. This two-part issue includes review articles and research articles spanning a broad range of contemporary research areas, all rooted in Taylor’s landmark paper. This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical Transactions paper (part 1)’.

Abstract In dense flowing bidisperse particle mixtures varying in size or density alone, smaller particles sink (percolation‐driven) and lighter particles rise (buoyancy‐driven). But when particle species differ from each other in both size and density, percolation and buoyancy can either enhance (large/light and small/heavy) or oppose (large/heavy and small/light) each other. In the latter case, a local equilibrium can exist in which the two mechanisms balance and particles remain mixed: this allows the design of minimally segregating mixtures by specifying particle size ratio, density ratio, and mixture concentration. Using DEM simulations, we show that mixtures specified by the design methodology remain relatively well‐mixed in heap and tumbler flows. Furthermore, minimally segregating mixtures prepared in a fully segregated state in a tumbler mix over time and eventually reach a nearly uniform concentration. Tumbler experiments with large steel and small glass particles validate the DEM simulations and the potential for designing minimally segregating mixtures.