Granular segregation across flow geometries: a closure model for the particle segregation velocity

TL;DR

This work delivers a general, first-principles closure for the segregation velocity $w_i$ in dense bidisperse granular flows by balancing the particle weight, a force of segregation $F^S_i$, and granular drag $F^D_i$, with diffusion corrections. By combining a gravity- and kinematics-enabled segregation force model with a Stokes-like drag law informed by $ ext{μ(I)}$ rheology and a volume-flux conservation constraint, the authors derive closed-form expressions for the species-specific segregation velocities in mixtures and demonstrate their validity against extensive DEM simulations across controlled and natural flow geometries. Incorporating diffusion through $D = A \\dot\gamma ar d^2$ yields net velocities that accurately capture segregation fluxes in inhomogeneous concentration fields. The framework, which can be embedded in the advection-diffusion-segregation equation, advances predictive capability for industrial and geophysical granular flows, while acknowledging current limitations in large-size-ratio, cohesive, and flow-coupled regimes and outlining clear paths for extension.

Abstract

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 momentum balance and recent advances in particle-scale modelling of the segregation driving and drag forces over a wide range of particle concentrations, size and density ratios, and flow conditions. This model is shown to correctly predict particle segregation velocity in a diverse set of idealized 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 application and geophysical settings.

Figures (9)

• Figure 1: (left) DEM simulation example of large (4 mm, blue) and small (2 mm, red) spheres in a uniform shear flow with downward gravity (negative $z$-direction), partitioned into 2.5$d_l$ high layers (shading) for characterizing depth-varying segregation velocity. Here, large particles rise while small particle sink. The segregation direction varies in the different flow configurations analyzed later. (right) Force balances on a large particle and a small particle corresponding to equation (\ref{['force_balance']}) and species-specific vertical segregation velocities, $w_i$.
• Figure 2: (a) Large particle drag coefficient, $C^D_l$, vs. large particle species concentration, $c_l$, in a uniformly sheared flow for size ratios of $R_d=1.5$ at $I\approx 0.08$ (blue crosses) and $R_d=2$ at $I\approx 0.12$ (black circles) for $g=0$. Error bars show the standard deviation of $C^D_{l}$ over a 1 s window for $R_d=2$; error bars for $R_d=1.5$ are similar but omitted for clarity. Horizontal solid black line corresponds to $C^D_{i,0}$ for $R_d=2$; horizontal dashed blue line corresponds to $C^D_{i,0}$ for $R_d=1.5$. (b) Comparison of $C^D_{i,0}$ with $C^D_i$ for varying size ratio. The single intruder drag coefficient, $C^D_{i,0}$ is calculated from (\ref{['eq:drag_coefficient']}) for large ($i=l$ for $R_d\ge1$) (solid black curve) and small ($i=s$ for $R_d<1$) (dashed black curve) intruder particles. The mixture drag coefficient, $C^D_i$, (red curve) is calculated from (\ref{['eq:drag_coefficient']}) for $R_d\ge1$ and (\ref{['eq:conservation']}) for $R_d<1$. Both curves represent predictions for $I=0.2$. Predictions of the mixture model for $I$ values ranging from 0 (lower bound) to 0.4 (upper bound), which are typical of dense granular flows, are indicated by the shaded band.
• Figure 3: Depth profiles (rows) of time averaged simulation results (symbols) and predictions (dashed black curves) for the four controlled shear flows (columns) in steady state at $R_d=2$. (a) Streamwise mean velocity $u$, (b) normalized segregation force on a large particle $\hat{F}^S_l=F^S_l/m_l g_0$, (c) bulk viscosity $\eta$, and (d) segregation velocity, $w_i$ for small (red) and large (blue) particles measured from the simulation (symbols) and predicted via (\ref{['eq:w']}) (curves). Dotted vertical lines in (b) indicate segregation force equal to particle weight. In all cases, $U=20\,$m s$^{-1}$, $c_l=c_s=0.5$, and $H\approx 0.2\,$m.
• Figure 4: Profiles of the segregation velocity $w_i$ for large (blue) and small (red) particles with $R_d=2$ for the exponential velocity profile with $g=g_0$ and bulk large particle concentrations of (a) $c_l=0.2$, (b) $c_l=0.5$, and (c) $c_l=0.8$, based on the prescribed velocity profiles (solid curves) compared to DEM measurements (symbols) averaged over 1 s after the flow reaches steady state.
• Figure 5: Effect of (a) three different spatially varying concentration profiles (columns) on the segregation velocity $w_i$ for (b) linear ($u=Uz/H$) and (c) exponential ($Ue^{k(z/H-1)}$) velocity profiles with $g=g_0$ and $R_d=2$. In rows (b) and (c), dashed curves represent model predictions using equation (\ref{['eq:w']}) for $w_i$, solid curves represent predictions corrected by the diffusion flux, i.e. $w^{net}_i$ from (\ref{['eq:w_net']}), and symbols indicate measurements from DEM simulations.