Fine particle percolation in a sheared granular bed

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 $R<R_t,$ $v_p$ initially increases with increasing shear rate, $\dot\gamma,$ as shear-driven bed rearrangement reduces fine-particle trapping but then decreases toward zero due to fine-particle excitation for $\dot\gamma\sqrt{d/g}\gtrsim 0.1$. For $R>R_t$, $v_p$ is constant at low $\dot\gamma$ but decreases toward zero at higher shear rates due to fine-particle excitation.

Mixtures of granular materials often segregate due to differences in size, density, or other physical properties, and understanding this phenomenon is often critical for predicting and controlling various natural [1-3] and industrial processes [4,5].Recent studies have advanced the modeling of granular segregation [6,7], with size segregation in dense granular flows receiving the majority of attention [3,[8][9][10][11][12][13][14][15].Nearly all of these studies consider mixtures with large-to-small particle diameter ratios, R = d/d f ≲ 2, where interparticle contacts are enduring [11][12][13].In these cases, segregation can be characterized by a concentration-dependent percolation velocity, v p , which is typically predicted and observed to increase monotonically with both R and the shear rate, γ.
For larger R, where small particles are referred to as "fine," the v p -dependence on γ and R in sheared flows changes significantly.In particular, for R ≳ 2 and low fine-particle concentration, v p increases dramatically with increasing R [15,16], but v p is nearly Rindependent at larger fine-particle concentration (above 10%) for 2 ≲ R ≲ 4 [14].Here we focus on fineparticle segregation in uniform shear flow in the zeroconcentration limit where an increasing tendency toward free sifting (or spontaneous percolation), with increasing R, leads to qualitative changes in the dependence of v p on γ and other parameters.
Free sifting has been investigated primarily in static beds for R > R t [17][18][19][20][21], where the free-sifting threshold, R t , is R t0 = (2/ √ 3 − 1) −1 ≈ 6.46 for rigid monodisperse spheres [22] but is larger in polydisperse mixtures of "soft" particles.Free sifting can also occur for R < R t in randomly packed static beds when a sub-population of pore throats-the minimum opening between neighboring bed spheres-exceeds the fine-particle diameter.In this case, percolation is necessarily transient since a fine particle will inevitably encounter an impassible pore throat [23].Despite the ubiquity of fine particles in industrial solids processing [24,25], their potential for increasing the mobility of various geophysical flows [26][27][28], and their importance in sediment infiltration that shapes river dynamics, channel morphology and ecological habi-tats [29,30], few studies have focused on fine-particle percolation velocity in granular shear flows [15,16,31].
We show here that free sifting is pronounced and unavoidably coupled with shear for R < R t , because fines that would be trapped in a static bed are repeatedly re-mobilized by shear-induced particle rearrangements.In past work, the complexity of this problem and the limited parameter-space explored, produced puzzling inconsistencies regarding the dependence of v p on γ [31] and R [14][15][16].In this Letter we resolve these issues by characterizing the fine-particle percolation velocity in large-particle beds for size ratios spanning the freesifting threshold (2 ≤ R ≤ 10) and spatially-uniform shear rates covering the quasi-static and rapid dense flow regimes [32].Our results reveal a non-monotonic dependence of v p on γ, provide relations for predicting v p in the low-and high-shear rate regimes, and add insight into the dominant physics in each regime.
Methods-LIGGGHTS [33], a discrete element method code, is used to simulate single fine-particle percolation in a confined dense granular flow with a prescribed linear velocity profile [34].The flow domain is periodic in the streamwise and spanwise directions and confined in the depthwise (y) direction by two horizontal planar walls roughened by randomly attached bed particles.A constant downward force on the top wall, which is otherwise free to move vertically and in the spanwise direction, sets the bed overburden pressure, P 0 , which is increased with increasing γ to maintain a constant volume fraction ϕ ≈ 0.58 of bed particles.The bottom wall is stationary while the top wall is translated in the streamwise direction with velocity γh(t), where the timedependent bed height, h(t), accommodates dilation due to shear.Parameters are set as follows: bed-particle diameter d = 5 mm with 10% uniform polydispersity, gravitational acceleration g = 9.81 m s −2 (in the −y direction), restitution coefficient e = 0.8, friction coefficient µ = 0.5, and bed-and fine-particle densities of ρ = ρ f = 2500 kg m −3 .We also vary d, g, e, and ρ f to explore their effects on v p .Depending on R, between ∼ 10 3 (R = 2) and ∼ 10 4 (R = 10) single fine particles with initial streamwise velocity matching the porous upper moving wall are dropped into the sheared bed.Fine particles interact with bed particles but not with each other to examine the zero-concentration limit.
Percolation velocity-Figure 1 plots the scaled fineparticle percolation velocity, v * p = −v p / √ gd, versus scaled shear rate, γ * = γ d/g, for 2 ≤ R ≤ 10.As in static beds, single fine particles always percolate downward on average even at the largest γ * , and v * p increases monotonically with increasing R for all γ * .However, the dependence of v * p on γ * is strongly R-dependent.First, for size ratios corresponding to the static-bed passing regime (R > 6.5 > R t0 here due to overburden-pressuredriven deformation and polydispersity of bed particles that decreases the minimum pore throat diameter relative to rigid monodisperse bed particles), v * p initially remains constant at its static-bed value as γ * is increased from zero.Hence, v p ∝ √ gd for R ≳ R t .However, for γ * ≳ 0.03, v * p decreases with increasing γ * .Second, for size ratios in the static-bed trapping regime (R ≤ 6.5), v * p increases from zero with increasing γ * , similar to segregation with R ≲ 2 [11][12][13]15].However, v * p reaches a maximum near γ * ∼ 0.1 and then decreases toward zero with further increase in γ * .Note that the previously observed γ-independence of v p for R ≈ 2.5 [31] results from that study's limited shear rate range, 0.04 < γ * < 0.14, which brackets the peak in v p and about which v p is nearly constant (e.g., see R = 2 in Fig. 1).To test the nondimensionalization of v p and γ, Fig. 1 also includes data where d and g differ from the values used in the other simulations.This additional data (magenta symbols) overlays the data for d = 5 mm and g = 9.81 m/s 2 at the corresponding R values, indicating that the scaling is correct in both low and high γ regimes.
The value of γ * where v * p begins to drop decreases with increasing R (dashed curve), e.g., γ * ≈ 0.14 for R = 2, while γ * ≈ 0.03 for R = 6.5.This sensitivity to R along with the decrease in v p with increasing γ for γ * ≳ 0.1 is due to increasing fine-particle velocity fluctuations (here characterized by fluctuations in the vertical velocity component, v rms ), which frustrate percolation and increase with increasing R or γ * , as described later.In static beds, a similar decrease in v p is observed with increasing e due to velocity fluctuations for both 4 ≤ R ≤ R t [23] and for R > R t [17,19,20,23], as fine particles rebound more energetically after colliding with bed particles.
Figure 1 and previous work in static beds [23] suggest that fine-particle percolation in sheared beds depends on three mechanisms: geometric trapping, which is possible when R < R t ; bed particle rearrangement due to shear; and fine-particle velocity fluctuations, which frustrate percolation.Details of their contributions to fineparticle transport are developed below, but the basics are as follows: First, when fine particles with R < R t are trapped, they re-mobilize due to shear-driven bed rearrangements at a rate that is proportional to γ and increases with R, indicating that passable voids are generated at a higher rate for smaller fine particles.Second, the average time to pass through a void increases with increasing excitation of the fine particles, measured in terms of v rms .Consequently, at high shear rates, where trapping times are short for R < R t and v rms is large (v rms ∝ γ) for all R, v p decreases with increasing γ.
Low-shear-rate regime-To better understand the dominant physics and develop a model for v p in this regime, we start with the percolation depth model for R < R t in static beds, p(∆y) ∝ P ∆y d p , where p(∆y) is the probability that a fine particle falls a distance ∆y or more from its starting height, and P p represents the probability of a fine particle passing through a randomly selected pore throat, i.e., the fraction of constrictions with diameters larger than d f [23].In static beds, p(∆y) is the proportion of trapped fine particles that exceed a depth ≥ ∆y, assuming that the passage of fine particles through consecutive pore throats is independent [23,35].Since untrapped fine particles percolate with mean velocity v p,s = −c 1 √ gd [23], p(∆y) can be reformulated as a function of time using ∆y = −v p,s t as p(t) ∝ P −vp,s t d p , where p(t) is the probability that a fine particle is untrapped after time t.The average percolation velocity over t is then For sheared systems, we assume that the time interval between significant bed rearrangements scales as t b = c 2 γ−1 , where c 2 depends on R. Substituting t b for t in Eq. 1 gives v p as a function of shear rate, bed structure (via P p ) and its variation (via c 2 ), bed particle diameter, and gravitational acceleration: This relation is alternatively expressed as where γ′ = −C γ d/g with C −1 = c 1 c 2 ln P p as the single model parameter.Eq. 2 exhibits the appropriate limiting behaviors under its assumption that velocity fluctuations are small: i) as γ → ∞ (t b → 0), v p → v p,s ∝ √ gd for all R; ii) as γ → 0, v p ∝ d γ for the trapping regime (R < R t ) as in most shear-driven percolation models for small R [11][12][13]15] and is independent of g; iii) in the passing regime (R > R t , P p = 1) v p ∝ √ gd independent of γ as in i).
To compare Eq. 2 to our data, we determine P p by characterizing the pore throat size distribution using Delaunay triangulation [23,36,37].For ϕ ≈ 0.58, P p is nearly independent of shear rate for γ ≲ 0.1, and increases from 0.17 to 0.93 as R is increased from 2 to 6. From [23], c 1 = 0.09 √ R for ϕ ≈ 0.58 and e = 0.8.Fits of Eq. 2 to simulation results for three R values obtained by adjusting the one free parameter, c 2 , match the simulation data at low γ * , as shown in Fig. 1 (solid curves).The inset in Fig. 2 shows that P p increases with R and c 2 decreases with R, as would be expected.
All data in Fig. 1 is compared to the universal form of the model (Eq.3) in Fig. 2, which plots the percolation velocity scaled by the untrapped percolation velocity from the static bed, v ′ p = v p /v p,s , versus the rescaled shear rate γ′ = −C γ * .Data for all R as well as varying g and d (magenta) collapse onto the model prediction (red curve) in the low-shear-rate region.Note that for freesifting cases (R > 6.5), γ′ = ∞ since P p = 1, and the corresponding symbols fall on the far right of Fig. 2 and go to v ′ p = 1 (yellow star) in the low-shear-rate regime.High-shear-rate regime-When v * p for γ * ≳ 0.1 is plotted versus γ * on a log-scale in Fig. 3(a), it is clear that v * p ∝ 1/ γ * for γ * ≳ 0.4 and different R, g and d.This behavior is related to increasing fine-particle velocity fluctuations, which frustrate percolation.To demonstrate the relation between γ * and v rms , we first plot the scaled vertical root-mean-square velocity fluctuations of  fine particles, v * rms = v rms / √ gd, versus γ * for various R in Fig. 3(b).For context, the vertical velocity fluctuations of bed particles (×) increase linearly with γ * , as would be expected from the corresponding increase in inter-particle collisions.Similarly, for γ * ≳ 0.4, v * rms ∝ γ * for all R, indicating that fine-particle velocity fluctuations are linked to the bed-particle velocity fluctuations in the high γ * regime.In comparison, for γ * ≲ 0.1, gravity-driven fluctuations dominate, so that v * rms is either constant (free-sifting regime, R > 6.5) or decreases slower than γ * (trapping regime, R ≤ 6.5).Simulations with different g and d values at R = 7 [magenta triangles in Fig. 3(b)] confirm the scaling between v rms and γ, which indicates that v rms is gravity independent where v * rms ∝ γ * but proportional to √ gd where v * rms is constant.
For all γ * , v * rms is always larger for larger R and appears to approach a limiting curve for large R, as Fig. 3(b) shows.Additional simulations with fine-particle density varied by two orders of magnitude (250 kg m −3 to 2.5 × 10 4 kg m −3 ) at constant R change v rms by < 7% (v p is also minimally affected), thereby indicating that the increase in v rms with increased R is due to decreased fine-particle diameter (i.e., smaller fine particles are less constrained by bed particles than larger fine particles) rather than decreased fine-particle mass.
Having demonstrated the linear dependence of v rms on shear rate at high γ, Fig. 3(c) tests our hypothesis that the percolation velocity decreases with increasing v rms .Indeed, the data show that v * p ∝ 1/v * rms when v * rms ≳ 0.4 for all R. In dimensional form, v p ∝ gd/v rms ∝ g/ γ, where the linear dependence of v p versus γ on g alone at high γ contrasts with the low-shear-rate scaling of v p with √ gd in the free-sifting regime and with d alone in the trapping regime.Equally significant, the figure also includes data for varying restitution coefficient between bed and fine particles, 0.2 ≤ e ≤ 1 for R = 5 and 7, indicating that different combinations of e and γ producing the same v rms yield the same v p .Hence, it is v rms that determines v p in this regime.
The v * p ∝ 1/v * rms relationship can be understood in terms of the different rates at which fine particles exit bed particle voids traveling down versus up.This process mimics gas molecules escaping through a hole smaller than their mean free path, which is described by Graham's law of effusion for the flux, Φ = P g A/ √ 2πmk B T , where m is the molecular mass, P g is the gas pressure, A is hole area, and T is the temperature.The analogy follows by replacing P g A with the gravity induced force differential in a trapping void, mg, and k B T with mv 2 rms .Multiplying Φ by the characteristic length d to form a velocity gives v p ∝ gd/v rms , which is the dimensional form of the scaling shown in Fig. 3(c).A second analogy is the Drude model for electron transport in metals due to an electric field, E, in which the mean electron momentum is p = m e v = qEτ, where m e and q are the electron mass and charge, and τ is the characteristic time interval between collisions with heavier ions.Replacing v with v p , qE/m e with g, and τ with d/v rms (since the fine-particle mean free path is proportional to d) yields v p ∝ gd/v rms .
Discussion-Our simulations of gravity-driven percolation of single fine particles in sheared granular beds display different dominant physics at low and high shear rates.For low shear rates, γ d/g ≲ 0.1, as the shear rate increases from zero, bed-particle rearrangements due to shear reduce fine-particle trapping to increase the perco-lation velocity, v p .A statistical model of this mechanism (Eqs. 2 or 3) accurately predicts v p for a wide range of conditions.In the high-shear-rate regime, γ d/g ≳ 0.1, increasing γ results in increasing fine-particle velocity fluctuations which frustrate percolation such that v p is inversely proportional to the velocity fluctuations and thus inversely proportional to γ, as is evident from Fig. 3.
These results are far from complete, and suggest that many interesting questions and challenges remain.For instance, our model and scalings accurately capture the dependence of v p on γ, d, and g, but require additional inputs to describe the effects of restitution coefficient, size ratio, and volume fraction.Understanding how to incorporate these parameters in expressions for v * p and v rms is likely to be non-trivial.For example, a scaling based on the mean free path of a fine particle, c 3 d − d f = d(c 3 − 1/R), collapses the data in Figs.3(a) and (c), but cannot be rigorously justified.
Finally, our results apply to the single-fine-particle limit, but extending the key conclusions to binary mixtures with finite fine-particle concentrations, c f , would also be valuable.Preliminary heap flow simulations with R > 4 and global c f up to 30%, exhibit high-shear regions with local c f < 5% where insights from the single-fineparticle limit are likely applicable.However, in low-shear regions, fine particles pack densely around bed particles, forming a continuous fine-particle phase that greatly reduces their vertical mobility.
t e x i t s h a 1 _ b a s e 6 4 = " R d M p u I 3 r S v x y w o r k N
Error bars for selected cases indicate standard error.Red curves are predictions of the low-shear-rate regime model (Eq.2) for R = 3, 6, and 7.