2.1. Particle force balance in a mixture

Consider an incompressible flow of a size-disperse mixture of two species, such as the simple shear flow shown schematically in figure 1(a). The two particle species have volume concentrations $c_i$ , where $i=l,s$ for large and small particles, respectively, and $c_s+c_l=1$ . We neglect vertical acceleration terms, which is reasonable for the relatively slow segregation observed in many common granular flows including heap, chute and rotating-tumbler flows (Gray Reference Gray2018; Umbanhowar et al. Reference Umbanhowar, Lueptow and Ottino2019), though not necessarily all flows (such as some high-speed geophysical flows).

Figure 1. (a) A DEM simulation example of large (4 mm, blue) and small (2 mm, red) spheres in a uniform shear flow with streamwise velocity $u(z)$ , top wall velocity $U=u(H)$ where $H$ is the height of the top wall above the stationary bottom wall, overburden pressure $P_0$ and downward gravity (negative $z$ -direction), partitioned into 2.5 $d_l$ high layers (shading) for characterising depth-varying segregation velocity. Here, large particles rise while small particles sink. The segregation direction varies in the different flow configurations analysed later. (b) Force balances on a large particle and a small particle corresponding to (2.1) and species-specific vertical segregation velocities, $w_i$ .

A force balance at the particle level in the vertical (gravitational, $g$ ) direction for an individual non-accelerating particle with mass $m_i$ in the flowing mixture, shown in figure 1(b), includes the segregation force, $F^S_i$ , the weight, $m_i g$ , and the drag force, $F^D_i$ such that (Jing et al. Reference Jing, Ottino, Umbanhowar and Lueptow2022)

(2.1) \begin{equation} F^S_i-m_i g+ F^D_i =0. \end{equation}

Much like in the analysis of terminal velocity in a fluid, the segregation velocity in granular flows appears in the drag force, $F^D_i$ , as we will show shortly. Thus, with an appropriate model for the segregation force, $F^S_i$ , and a model for the dependence of the drag force, $F^D_i$ , on the segregation velocity, we can use (2.1) to calculate the segregation velocity. The key is having appropriate models for $F^S_i$ and $F^D_i$ on an individual particle in the flowing mixture, which are described shortly.

Using force balance at the particle level, i.e. (2.1), differs somewhat from the continuum description of segregation using the mixture theory framework. Within this framework, the momentum balance for each species along the segregation direction (negative $z$ -direction) in a simple shear flow scenario can be expressed as (Gray & Thornton Reference Gray and Thornton2005; Tunuguntla, Weinhart & Thornton Reference Tunuguntla, Weinhart and Thornton2017)

(2.2) \begin{equation} -\frac {\partial p_i}{\partial z}-\rho _i g+ {\beta }_i=0. \end{equation}

Here, $ -\partial p_i / \partial z= n_i F^S_i$ is the partial pressure gradient, where $p_i$ is the partial pressure of species $i$ , and $z$ is in the direction of gravity, and $\rho _i$ is the density of species $i$ . The interspecies momentum exchange is $\beta _i=n_i F^D_i$ , where $n_i = c_i \phi /V_i$ represents the particle number density, $\phi$ is the bulk solid volume fraction, and $V_i$ denotes the individual particle volume of species $i$ . Combined with the bulk pressure gradient $\partial p/\partial z = -\phi \rho g$ , where it is assumed that both species have the same density $\rho$ , the ratio of the pressure contribution of species $i$ to the bulk pressure $p$ , or normal stress fraction, is $f_i=p_i/p= n_i F^S_i/\phi \rho g=c_i F^S_i/m_i g$ in the simplified case where $F^S_i$ remains constant with depth. Prior studies adopting the momentum-based approach have often assumed a quadratic-dependence or a particle-size or volume-weighted dependence of $f_i$ on $c_i$ (Marks et al. Reference Marks, Rognon and Einav2012; Tunuguntla, Bokhove & Thornton Reference Tunuguntla, Bokhove and Thornton2014; Gray & Ancey Reference Gray and Ancey2015; van der Vaart et al. Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015; Rousseau et al. Reference Rousseau, Chassagne, Chauchat, Maurin and Frey2021; Trewhela et al. Reference Trewhela, Ancey and Gray2021), along with a linear drag model (Gray & Thornton Reference Gray and Thornton2005). While these approaches have been proposed for species concentration profiles in specific scenarios, the approach we use here matches direct DEM measurements of $f_i$ (Duan et al. Reference Duan, Jing, Umbanhowar, Ottino and Lueptow2022), and using (2.1) allows us to consider the segregation velocity of not only mixtures but also intruders for a wide range of granular flow conditions and geometries.

2.2. Segregation force, $F^S_i$

Determining the segregation force $F^S_i$ at finite concentration starts with the segregation force on a single intruder, $F^S_{i,0}$ (subscript $0$ indicates the single-intruder limit of species $i$ , $c_i \to 0$ ). Here, $F^S_{i,0}$ can be modelled with two additive terms, one related to gravity and the other to flow kinematics. Inspired by the observations of Fan & Hill (Reference Fan and Hill2011a ,Reference Fan and Hill b ), the flow kinematics term was initially scaled with the shear stress gradient (Guillard et al. Reference Guillard, Forterre and Pouliquen2016) but was later linked to the shear rate gradient (Jing et al. Reference Jing, Ottino, Lueptow and Umbanhowar2021; Singh et al. Reference Singh, Liu and Henann2024). The segregation force can be expressed as (Jing et al. Reference Jing, Ottino, Lueptow and Umbanhowar2021)

(2.3) \begin{equation} F^S_{i,0}=-f^g(R_d)\frac {\partial {p}}{\partial {z}}V_i+f^k(R_d)\frac {p}{\dot \gamma }\frac {\partial \dot \gamma }{\partial {z}}V_i, \end{equation}

where superscripts $g$ and $k$ indicate gravity- and kinematics-related mechanisms, respectively, $V_i$ is the intruder particle volume, $\dot \gamma$ is the local shear rate, and $\rho$ is the density of both the intruder and the bed particles. The gravity term is buoyancy-like, and the kinematic term depends on the shear rate gradient in the flow. The empirical dimensionless functions $f^g(R_d)$ and $f^k(R_d)$ depend on the intruder-to-bed-particle-diameter ratio $R_d=d_i/d_j$ (Jing et al. Reference Jing, Ottino, Lueptow and Umbanhowar2021),

(2.4a) \begin{equation} f^g(R_d)=\left [ 1-c^g_1 \exp \left(-\frac {R_d}{R^g_1}\right) \right] \left [ 1+c^g_2\exp \left(-\frac {R_d}{R^g_2}\right) \right], \end{equation}

(2.4b) \begin{equation} f^k(R_d)=f^k_\infty \left [ \tanh {\left(\frac {R_d-1}{R^k_1}\right)} \right] \left [ 1+c^k_2\exp \left(-\frac {R_d}{R^k_2}\right) \right], \end{equation}

where $R^g_1=0.92$ , $R^g_2=2.94$ , $c^g_1=1.43$ , $c^g_2=3.55$ , $f^k_\infty =0.19$ , $R^k_1=0.59$ , $R^k_2=5.48$ and $c^k_2=3.63$ are fitting parameters appropriate for a range of flow conditions. In applying these functions over a range of concentrations, we need to consider both a large intruder particle surrounded by a bed of small particles and a small intruder particle surrounded by a bed of large particles corresponding to intruder-to-bed particle-size ratios of $d_l/d_s$ and $d_s/d_l$ , respectively.

To predict the segregation force $F^S_i$ in mixtures at arbitrary non-zero concentrations, the intruder-segregation-force model in (2.3) is extended using semi-empirical relations for mixtures (rather than an intruder particle) based on DEM simulations (Duan et al. Reference Duan, Jing, Umbanhowar, Ottino and Lueptow2022). For large particles in a bidisperse mixture of concentration $c_l$ (Duan et al. Reference Duan, Jing, Umbanhowar, Ottino and Lueptow2024),

(2.5a) \begin{equation} F^S_l=m_l g\cos {\theta }+(F_{l,0}^S-m_l g\cos {\theta })\textrm {tanh}\left ( \frac {m_l g\cos {\theta }- F_{s,0}^S }{F_{l,0}^S-m_l g\cos {\theta }}\frac {c_s}{c_l} \right), \end{equation}

and for small particles,

(2.5b) \begin{equation} F_{s}^S=m_s g\cos {\theta }-(F_{l,0}^S-m_s g\cos {\theta }){\frac {c_l}{c_s}}\textrm {tanh}\left ( \frac {m_s g\cos {\theta }-F_{s,0}^S}{{F}_{l,0}-m_s g\cos {\theta }}\frac {c_s}{c_l} \right), \end{equation}

where $\theta$ is the angle between gravity and the segregation direction, which is generally normal to the flow.

Equations (2.5a ) and (2.5b ) can be rewritten in terms of the net forces ( $T_i=F^S_i-m_ig\cos \theta$ ) that balance the interspecies drag (2.1) such that

(2.6a) \begin{equation} T_l=T_{l,0}\textrm {tanh}\left ( -\frac {{T}_{s,0}}{{T}_{l,0} }\frac {m_lc_s}{m_sc_l} \right), \end{equation}

(2.6b) \begin{equation} T_{s}=-{T}_{l,0}{\frac {m_sc_l}{m_lc_s}}\textrm {tanh}\left ( -\frac {{T}_{s,0}}{{T}_{l,0} }\frac {m_lc_s}{m_sc_l} \right), \end{equation}

where $T_{i,0}=F_{i,0}-m_ig\cos \theta$ , and $F_{i,0}$ and $F_{i}$ denote the segregation forces for a single intruder and for mixtures, respectively. The force balance in (2.1) can be rewritten for a mixture as

(2.7) \begin{equation} T_i+ F^D_i =0. \end{equation}

What remains to be specified is an expression for the drag force in a mixture, $F^D_i$ , including its dependence on the segregation velocity, which is described next.

2.3. Drag force, $F^D_i$

As with the approach for the segregation force, we start with the drag force on a single intruder particle within a monodisperse flow of bed particles, $F^D_{i,0}$ . A Stokes drag formulation can be employed to express the drag force (Tripathi & Khakhar Reference Tripathi and Khakhar2013; Jing et al. Reference Jing, Ottino, Umbanhowar and Lueptow2022; He et al. Reference He, Zhang, Ottino, Umbanhowar and Lueptow2025) in terms of the intruder segregation velocity relative to the local bulk flow velocity in the segregation direction, $w_{i,0}$ , as

(2.8) \begin{equation} F^D_{i,0}=-C^D_{i,0} \pi \eta d_i w_{i,0}, \end{equation}

where $C^D_{i,0}$ is the drag coefficient for a single intruder, and $\eta$ is the effective bulk granular viscosity calculated from the $\mu (I)$ rheology as described shortly. For Stokes drag on a spherical particle at low Reynolds number in a viscous fluid, $C^D_{i,0}=3$ . However, the value of $C^D_{i,0}$ for an intruder in a granular flow is not as simply specified. For a single spherical intruder particle, $C^D_{i,0} \approx 2.1$ for $1\leqslant R_d \leqslant 5$ , but the precise value depends on the flow conditions (Jing et al. Reference Jing, Ottino, Umbanhowar and Lueptow2022), specifically, the size-bidisperse mixture inertial number (Rognon et al. Reference Rognon, Roux, Naaïm and Chevoir2007; Tripathi & Khakhar Reference Tripathi and Khakhar2011),

(2.9) \begin{equation} { I =\dot \gamma \sqrt {\frac {\rho }{p}}\sum _{i=s,l}d_ic_i.} \end{equation}

For large intruders with $R_d\geqslant 1$ ,

(2.10) \begin{equation} C^D_{i,0} =[k_1-k_2\exp (-k_3 R_d)]+s_1 I R_d +s_2 I (R_\rho -1), \end{equation}

where $k_1=2$ , $k_2=7$ , $k_3=2.6$ , $s_1=0.57$ and $s_2=0.1$ are fitting parameters determined across a wide range of flow conditions ( $0.6\leqslant R_d \leqslant 5$ , $1\leqslant R_{\rho } \leqslant 20$ and $I \lesssim 1$ ), and $R_\rho$ is the intruder-to-bed-particle density ratio (Jing et al. Reference Jing, Ottino, Umbanhowar and Lueptow2022). The granular viscosity $\eta$ is estimated from the $\mu (I)$ rheology (GDR-MiDi 2004) as

(2.11a) \begin{equation} \eta =\mu _{\it{eff}} \frac {p}{\dot \gamma }, \end{equation}

where the effective friction coefficient is

(2.11b) \begin{equation} \mu _{\it{eff}}=\tau /p=\mu _s+\frac {\mu _2-\mu _s}{(I_c/I)+1}, \end{equation}

where $\tau$ is the shear stress and $\mu _s$ , $\mu _2$ and $I_c$ are granular material specific parameters. The flows simulated in this study generally follow the $\mu (I)$ rheology for dense flows with the usual caveats for collisional flow at larger $I$ than considered here and for quasi-static flow as $I\rightarrow 0$ , see Appendix A.

The drag coefficient, $C^D_{i,0}$ , in (2.10) is for a single intruder particle in an otherwise homogeneous bed of the other species. Since we are interested in the segregation velocity of particles in a mixture, it is necessary to determine the dependence of the drag force in a mixture, $F^D_i,$ on species concentration. Previous approaches to determine the mixture $C^D_i$ have focused predominantly on density-bidisperse mixtures where $R_d=1$ (Tripathi & Khakhar Reference Tripathi and Khakhar2013; Duan et al. Reference Duan, Umbanhowar, Ottino and Lueptow2020; Bancroft & Johnson Reference Bancroft and Johnson2021) due to the simplicity of estimating the segregation force in terms of buoyancy. Reported values of $C^D_i$ at $R_d=1$ based on this approach range from 1.7 to 3.7 depending on the volume fraction, and $C^D_i$ is independent of $c_i$ for density-bidisperse mixtures. However, the concentration dependence of $C^D_i$ in size-bidisperse mixtures ( $R_d\ne 1$ and density ratio $R_\rho =1$ ) has not been considered explicitly, although Bancroft & Johnson (Reference Bancroft and Johnson2021) mention it in passing. We use simulations of controlled shear flow later in this paper (§ 4) to demonstrate that the drag coefficient is nearly independent of the mixture concentration for the conditions we consider here. For now, in order to proceed with the analysis, we simply assume that $C^D_i \approx C^D_{i,0}$ . Hence,

(2.12) \begin{equation} F^D_i \approx F^D_{i,0}=-C^D_i \pi \eta d_i w_i, \end{equation}

where the mixture drag coefficient, $C^D_i$ , has been substituted for the intruder drag coefficient, $C^D_{i,0}$ , and the species-specific segregation velocity in the mixture, $w_i$ , for the intruder segregation velocity, $w_{i,0}$ , in (2.8).

2.4. Segregation velocity

The species-specific segregation velocity, $w_i$ , relative to the local bulk flow velocity in the segregation direction is now easily calculated by substituting the mixture drag force (2.12) and the segregation-force model (2.6) into the force balance (2.7) and solving for the segregation velocities of the large-particle species, $w_l$ , and small-particle species, $w_s$ :

(2.13a) \begin{equation} w_l=\frac { T_{l,0}\textrm {tanh}\left ( -\dfrac{{T}_{s,0}}{{T}_{l,0} }\dfrac{m_lc_s}{m_sc_l} \right)}{C^D_l \pi \eta d_l} \end{equation}

and

(2.13b) \begin{equation} w_s=-\frac { {T}_{l,0}{\dfrac{m_sc_l}{m_lc_s}}\textrm {tanh}\left ( -\dfrac{{T}_{s,0}}{{T}_{l,0} }\dfrac{m_lc_s}{m_sc_l} \right) }{C^D_s \pi \eta d_s}. \end{equation}

Recall that we assume $C^D_i \approx C^D_{i,0}$ , independent of species concentration $c_i$ , which we confirm in § 4.

2.5. Effect of diffusion on segregation velocity

A final consideration is the diffusive flux of species driven by collisional diffusion and its effect on the measured segregation velocity. The diffusion contribution is most clearly framed in terms of the advection–diffusion–segregation transport equation based on mass balance that has been successfully used to model segregation in flowing granular media (Bridgwater et al. Reference Bridgwater, Foo and Stephens1985; Dolgunin & Ukolov Reference Dolgunin and Ukolov1995; Gray Reference Gray2018; Umbanhowar et al. Reference Umbanhowar, Lueptow and Ottino2019). Within this continuum framework, the concentration of species $i$ can be expressed as

(2.14) \begin{equation} \frac {\partial c_i}{\partial t} + {\boldsymbol{\nabla } \boldsymbol{\cdot } (\boldsymbol{u}_i c_i)}={\boldsymbol{\nabla } \boldsymbol{\cdot } (D\nabla c_i)}. \end{equation}

Here, $\boldsymbol{u}_i$ is the diffusionless velocity, and the local collisional diffusion coefficient $D$ is a scalar, although in general it is a tensor. This approximation is accurate for flows with a single dominant shear direction (Umbanhowar et al. Reference Umbanhowar, Lueptow and Ottino2019). Note that the diffusionless velocity, $\boldsymbol{u}_i$ , differs slightly from the overall velocity, which is a combined effect of both advection and diffusion. With the usual assumptions of two-dimensional flow and gradual development in the streamwise direction, (2.14) in the $z$ -direction can be written as

(2.15) \begin{equation} \frac {\partial c_i}{\partial t} +\frac {\partial (w_i+w)c_i}{\partial z}=\frac {\partial }{\partial z} \left ( D\frac {\partial c_i}{\partial z} \right), \end{equation}

or, rearranging, as

(2.16) \begin{equation} \frac {\partial c_i}{\partial t} +\frac {\partial \left [(w_i+w)c_i-D(\partial c_i/\partial z) \right] }{\partial z}=0, \end{equation}

where $w$ is the local vertical velocity of the bulk. Note that $w=0$ in the reference frames associated with the example flows used in this study.

When the normal component of flux for species $i$ is measured from DEM simulation, it is the entire quantity within the square brackets of (2.16) that is measured. In other words, the measured normal flux $(w_i+w)c_i-D(\partial c_i/\partial z)$ is a combination of both the total segregation flux, $(w_i+w)c_i$ , and the diffusion flux, $-D(\partial c_i/\partial z)$ . To compare the segregation-velocity model predictions developed in this paper with DEM measurements of the segregation velocity, the segregation flux needs to be combined with the diffusion flux, such that the net species velocity is

(2.17) \begin{equation} w^{net}_i=w_i-\frac {D}{c_i}\frac {\partial c_i}{\partial z}, \end{equation}

which can be measured directly in situations where there is a concentration gradient.

Both experimental (Bridgwater Reference Bridgwater1980; Utter & Behringer Reference Utter and Behringer2004) and computational (Fan et al. Reference Fan, Umbanhowar, Ottino and Lueptow2015; Cai et al. Reference Cai, Xiao, Zheng and Zhao2019; Fry et al. Reference Fry, Umbanhowar, Ottino and Lueptow2019) studies of dense granular flows suggest that the diffusion coefficient, $D$ , is proportional to the product of the local shear rate and the square of the local mean particle diameter,

(2.18) \begin{equation} D=A \dot \gamma \bar d ^2, \end{equation}

where $\bar d = \sum c_i d_i$ and $A$ is a constant with reported values in the range 0.01–0.1 (Savage & Dai Reference Savage and Dai1993; Hsiau & Shieh Reference Hsiau and Shieh1999; Utter & Behringer Reference Utter and Behringer2004; Fan et al. Reference Fan, Schlick, Umbanhowar, Ottino and Lueptow2014, Reference Fan, Schlick, Umbanhowar, Ottino and Lueptow2015; Cai et al. Reference Cai, Xiao, Zheng and Zhao2019; Fry et al. Reference Fry, Umbanhowar, Ottino and Lueptow2019). In this study, $A=0.046$ based on diffusion coefficient data measured from heap-flow simulations (Duan et al. Reference Duan, Jing, Umbanhowar, Ottino and Lueptow2022).

With the exception of demonstrating that the drag coefficient for a mixture is similar to that for an intruder particle, as noted in § 2.3, we now have all of the relationships necessary to calculate the segregation velocity. Before addressing drag in mixtures, it is first necessary to describe the simulation approach that we use.

5.1. Controlled shear flows

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) normalised 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 (2.13) (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.

The simulation results and predicted values for the segregation velocity for the four controlled shear flows at $R_d=2$ are presented in figure 3. Figure 3(a) shows the prescribed and measured streamwise velocity profiles along depth, $z$ . The close agreement between the DEM data points and the curves representing the target velocity profile demonstrates the effectiveness of the control scheme in achieving the desired velocity profiles.

The prescribed velocity functions in figure 3(a) are used to calculate the $z$ -profiles of $\dot \gamma$ and $\partial \dot \gamma /\partial z$ , while the pressure is estimated based on an idealised hydrostatic pressure, $p=P_0+\rho \phi g (H-z)$ . These results are then used to calculate the predicted mixture segregation force, shown in figure 3(b), which we normalise with the particle weight, $\hat F^S_i=F^S_i/m_i g_0$ . The dashed curves represent $\hat F^S_l$ calculated using (2.3)–(2.5) based on the prescribed pressure and shear states. The symbols indicate $\hat F^S_l$ measured directly from the simulation using the extended virtual spring approach, where forces proportional to the offset of the centres of mass of the two species are applied uniformly to particles of each species to suppress segregation (Duan et al. Reference Duan, Jing, Umbanhowar, Ottino and Lueptow2024). Error bars indicate the standard deviation of DEM data averaged over the 1 s measurement window. The agreement between the DEM data and the model predictions for $\hat F^S_l$ across all four controlled shear flows confirms the general applicability of the segregation-force model (2.5) (see also Duan et al. Reference Duan, Jing, Umbanhowar, Ottino and Lueptow2024). Results for $\hat F^S_s$ are similar (not shown). Note that the total segregation force equals the depth-wise component of the particle weight for steady flows, specifically, $c_l \hat F^S_l + c_s \hat F^S_s=\cos \theta$ , where the value on the right-hand side of this equation is shown as a vertical dotted line in figure 3(b). The large difference between measured and predicted values of $\hat F^S_l$ in the vicinity of $z/H=0.5$ for the parabolic velocity profile occurs because $\dot \gamma (H=0.5)=0$ , leading to a singularity in the second term in (2.3), see Duan et al. (Reference Duan, Jing, Umbanhowar, Ottino and Lueptow2024). However, this does not affect the segregation velocity because of a corresponding singularity in the viscosity, as explained next.

The effective granular viscosity, $\eta$ , measured as the ratio of shear stress $\tau$ to shear rate $\dot \gamma$ , (2.11a ), averaged over the 1 s window from DEM simulation data, is plotted in figure 3(c). The dashed curve represents $\eta$ estimated from the $\mu (I)$ rheology (2.11) for the corresponding prescribed pressure-shear state. The predicted and measured values of viscosity match well except near the flow boundaries (see Appendix A). Similar to the segregation force, the viscosity exhibits a singularity at $\dot \gamma \approx 0$ for the parabolic velocity profile. This singularity, however, is eliminated as $\dot \gamma \to 0$ when the segregation force, along with the drag force, are incorporated into the force and momentum balance, because both forces scale as $1/\dot \gamma$ .

With the knowledge of segregation force (figure 3 b), viscosity (figure 3 c) and drag coefficient ((2.10) for large particles and (4.4) for small particles), (2.13) allows the calculation of segregation velocity. Moreover, the depth-dependent segregation velocity can be measured directly for each horizontal layer by quantifying the rate of centre of mass offset between the two species (3.1) in a 1 s window after the flow reaches steady state. Figure 3(d) shows both the model predictions and DEM measurements of the segregation velocity for the four controlled shear flows.

For the linear velocity profile (first column of figure 3), the segregation velocity decreases from top to bottom, despite a constant depth-independent segregation force. This decrease is due to the increase of pressure with depth, which increases the viscosity term in the drag force, thereby slowing the segregation. A similar pattern is observed for the exponential velocity profile without gravity in the second column of figure 3. Despite the constant segregation force and pressure, the shear rate decreases with depth, which increases the viscosity and, hence, the drag. For the parabolic velocity profile (third column of figure 3), the segregation force changes sign at $\dot {\gamma }=0$ . Consequently, the segregation velocity also changes sign at this point, consistent with large particles segregating to regions of higher shear (Fan & Hill Reference Fan and Hill2011a ). While the model accurately predicts this sign change, the flow in the $z/H\approx 0.5$ region has negligible shear and, hence, is quasi-static. The continuum framework upon which the model is based is difficult to apply in this regime, resulting in significant discrepancies between the model predictions and the simulation measurements. Finally, in the case of the exponential velocity profile with gravity (fourth column of figure 3), the segregation force increases linearly with depth. However, the viscosity experiences a more pronounced increase, resulting in a decrease in segregation velocity with depth.

Overall, the consistent agreement between the model predictions and the simulation results across all four cases in both magnitude and sign demonstrates the effectiveness of (2.13) in predicting segregation velocity based on the local pressure-shear state. These results further indicate that the model effectively captures the underlying physical mechanisms driving segregation and the associated segregation velocity. The match between the measured segregation velocity and its predicted values in figure 3(d) is not perfect. Nevertheless, given the relatively straightforward model, the simplifying assumptions used in the model, the difficulty in isolating the various forces acting on individual particles, the complication of considering mixtures (rather than an intruder), and the inherent stochastic nature of granular flows, the match between the measured and predicted segregation velocities is surprisingly good.

5.2. Varying species concentration

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 with DEM measurements (symbols) averaged over 1 s after the flow reaches steady state.

The analysis in § 5.1 focuses on uniformly mixed systems with equal volume fractions of small and large particles, $c_l=c_s=0.5$ . However, the concentration dependence of the segregation velocity as described by (2.13) should hold for any species concentration within the range $0\leqslant c_l\leqslant 1$ , where $c_s=1-c_l$ . To verify this, we consider controlled shear flows with the exponential velocity profile and $g=g_0$ , because for these conditions both the segregation force and the viscosity vary with depth (see the last column of figure 3). Figure 4 shows the model predictions for $w_i$ at $R_d = 2$ with uniform species concentrations $c_l = 0.2$ and 0.8, along with the results for $c_l = 0.5$ (repeated from figure 3 with a different horizontal scale). While a wider scatter of data points is observed with $c_l=0.2$ , the predicted segregation velocity matches the measured segregation velocity reasonably well. Furthermore, a general trend of decreasing segregation velocity with depth is evident for both particle species, similar to $c_l=0.5$ . However, the segregation velocity for large particles increases for $c_l=0.2$ , and the segregation velocity for small particles decreases, compared with $c_l=0.5$ . That is, for $c_l=0.2$ , a large particle among mostly small particles segregates faster than a small particle among few large particles. In contrast, for $c_l=0.8$ , the model underpredicts the segregation velocity, possibly due to the difficulty in accurately determining $\hat F^S_{s}$ for small intruders in the sea of large particles, which is highly sensitive to $R_d$ and $c_l$ near the monodisperse limit of $c_l=1$ (Jing et al. Reference Jing, Ottino, Lueptow and Umbanhowar2021). Nevertheless, the decreased segregation velocity with depth and the increased (decreased) segregation velocity for small (large) particles compared with $c_l=0.5$ is clear. Here, a small particle among many large particles segregates faster than a large particle among many small particles, consistent with previously observed asymmetry in the segregation velocity (van der Vaart et al. Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015; Jing, Kwok & Leung Reference Jing, Kwok and Leung2017; Jones et al. Reference Jones, Isner, Xiao, Ottino, Umbanhowar and Lueptow2018). This asymmetry in the segregation velocity originates as an asymmetry in the segregation force (2.5), which in turn can be traced back to segregation being intruder-like for small particles at large $c_l$ and intruder-like for large particles at small $c_l$ . The species concentration range that the segregation force is intruder-like is narrower for small particles and wider for large particles (see figures 2 and 3 in Duan et al. Reference Duan, Jing, Umbanhowar, Ottino and Lueptow2022), which is reflected in the hyperbolic tangent dependence of the segregation force on concentration (2.5) and, consequently, in the form of the segregation velocity (2.13). It is also consistent with the kinetic sieving model of Savage & Lun (Reference Savage and Lun1988), as demonstrated by Jones et al. (Reference Jones, Isner, Xiao, Ottino, Umbanhowar and Lueptow2018) by comparing DEM segregation-velocity results to the solution of the Savage and Lun model over a range of size ratios and species concentrations (see figures 5 and 7 in Jones et al. Reference Jones, Isner, Xiao, Ottino, Umbanhowar and Lueptow2018).

Figure 5. Effect of three different spatially varying concentration profiles (columns and plotted in (a) using the large particle concentration, $c_l$ ) on the segregation velocity $w_i$ versus depth for (b) linear ( $u=Uz/H$ ) and (c) exponential ( $U\textrm {e}^{k((z/H)-1)}$ ) velocity profiles with $g=g_0$ and $R_d=2$ . In the graphs in (b) and (c), dashed curves represent model predictions using (2.13) for $w_i$ , solid curves represent predictions corrected by the diffusion flux, i.e. $w^{net}_i$ from (2.17), and symbols indicate measurements from DEM simulations. Note that the volume fraction, $\phi$ , in (a) varies only weakly with $c_l$ .

To this point, the segregation-velocity model (2.13) has been validated against different flow configurations with uniform mixture concentrations throughout the flow domain. However, the model can be extended to scenarios where particle species concentration varies with depth. In such cases, concentration gradients induce a diffusion flux that can enhance or counteract the segregation flux, as described in § 2.5. Because differentiating between these two fluxes is impractical from a measurement standpoint, the observed segregation velocity represents the net effect of both fluxes (see (2.17)).

To consider varying species concentrations, three distinct large-particle concentration profiles, shown in figure 5(a), are investigated: $c_l$ increasing with depth, $c_l$ decreasing with depth, and $c_l$ decreasing in the upper half of the flow and increasing in the lower half of the flow. Although these concentration profiles are initialised to vary linearly with depth, variations in concentration occur as the flow is established, resulting in slightly nonlinear $c_l$ profiles at the start of the 1 s measurement window. The particle volume fraction, $\phi$ , remains approximately constant with depth in all three cases. Figure 5(b) shows the segregation velocity for an imposed linear velocity profile with $g=g_0$ in the $z$ -direction. Neglecting the diffusion flux leads to an underestimation of the segregation velocity (dashed curves) for both small and large particles, as both the diffusion flux and the segregation flux act in the same direction. Specifically, both segregation and diffusion contribute to the upward movement of large particles. The improved agreement between the diffusion-corrected velocity ( $w_i^{net}$ , solid curves) and the DEM measurements shows the importance of the diffusion effect in inhomogeneous-concentration flows across all three initial conditions for $c_l$ . Results for an exponential velocity profile with $g=g_0$ in the $z$ -direction in figure 5(c) indicate a generally good agreement between the predicted segregation velocity and the DEM results even without accounting for the diffusion flux, except near the top boundary where the correction for diffusion flux is clearly necessary. This is because the diffusion flux scales with shear rate, which is maximal at the top boundary and decreases rapidly with increasing depth.

For all cases in figure 5, the predicted segregation velocity accounting for diffusion matches the measured segregation velocity well. It is also worth noting that the force-based segregation-velocity model accurately captures the change in segregation direction in scenarios where $c_l$ reaches a minimum value at a mid-depth position for both linear and exponential velocity profiles. This not only highlights the model’s ability to predict segregation velocities, even with substantial variations in the concentration gradient, but also reveals that diffusion can, under certain conditions, become the dominant mechanism, overpowering the effects of size-induced segregation.

5.3. Natural flows

Figure 6. Depth profiles (rows) of time-averaged simulation results (symbols) for the four natural shear flows (columns) in steady state at $R_d = 1.5$ . (a) Streamwise mean velocity $u$ , (b) normalised 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 by (2.13) (dashed curves) and considering diffusion (2.17) (solid curves). Dotted vertical lines in (b) indicate segregation force equal to particle weight. In all cases, $c_l=c_s=0.5$ and $H\approx 0.2\,$ m.

As demonstrated earlier, the segregation-velocity model (2.13) works well for arbitrary concentration fields in a variety of flows where the velocity field is artificially prescribed. We now examine four uncontrolled wall- or gravity-driven flows, illustrated in figure 6(a), in which the velocity field develops naturally via the boundary conditions and gravity-induced body forces. Note that particles in each layer of the flow are constrained vertically using the restoring force approach so that they cannot segregate until they are released when the velocity profile reaches steady state. In wall-driven plane shear flow, which results from upper and lower walls moving in opposite directions with velocity $U=10$ m s $^{-1}$ , there is no gravity, so the fully developed velocity profile is linear with a nearly constant inertial number of 0.2 (Duan et al. Reference Duan, Jing, Umbanhowar, Ottino and Lueptow2024). In the three gravity-driven cases, the orientation of gravity relative to the flow direction is parameterised by $\theta$ . For wall-driven plane shear flow, gravity acts perpendicular to the flow direction (aligned with the $z$ -direction such that $\theta =0$ ), resulting in a nonlinear velocity profile due to the increasing pressure with depth in the flow. In vertical chute flow, gravity aligns with the $z$ -direction ( $\theta =\pi /2$ ), such that there is no pressure gradient in the $z$ -direction, $\partial p/\partial z=0$ , resulting in a blunt velocity profile with zero velocity at the two confining walls. Finally, for inclined chute flow with $\theta =28^\circ$ , which exceeds the critical angle required for flow, the resulting velocity profile is nonlinear with a maximum velocity at the free surface.

Following a similar methodology to the analysis of controlled-velocity flow fields, we plot dimensionless depth profiles of $u$ , $\hat F^S_i$ , $\eta$ and $w_i$ for the natural flows with $c_l=c_s=0.5$ , as shown in figure 6. Here we consider $R_d=1.5$ to demonstrate results for a different size ratio than used earlier. Unlike the controlled flows where $\dot \gamma$ , $\partial \dot \gamma /\partial z$ , $p$ and $\eta$ can be determined from prescribed functions (streamwise velocity, pressure) or assumed constant (concentration), these same variables for natural flow are measured directly from DEM simulations.

In the absence of an intrinsic velocity scale for vertical and inclined chute flows, the kinematic terms are non-dimensionalised using the acceleration due to gravity at the earth’s surface ( $g_0$ ) and the flow depth ( $H$ ). Unlike the controlled flows in the previous sections, here there are no predicted values for $\hat F^S_i$ and $\eta$ because $u$ and, hence, $\dot \gamma$ , $\partial \dot \gamma /\partial z$ , $p$ and $\eta$ depend on the naturally developed flow conditions, rather than being prescribed. Nevertheless, the predicted value for the segregation velocity, $w_i$ , based on measured $z$ -dependent values of $\dot \gamma$ , $\partial \dot \gamma /\partial z$ , $p$ and $\eta$ , can be compared with the measured value of $w_i$ .

Consider first the wall-driven plane shear flow without gravity (first column of figure 6) where the streamwise velocity decreases linearly with increasing depth, resulting in a constant shear rate. This leads to a negligible segregation force across the depth, with only minor deviations observed near the walls. Furthermore, the viscosity remains relatively constant throughout the flow domain. Consequently, the segregation velocity in the first column of figure 6(c) is approximately zero over most of the depth, except close to the walls. The minor deviation from perfect symmetry about $z/H=0.5$ near the top and bottom walls is likely a consequence of two factors. First, the initial packing procedure, which involves particles falling freely under gravity in the negative $z$ -direction to fill the domain, may lead to a slight initial segregation of particles. Second, random variations in roughness between the top and bottom boundaries could also contribute to the asymmetry.

With gravity (second column in figure 6), the wall-driven plane shear flow velocity profile is steep near the upper moving wall but flattens in the bottom half of the flow where the pressure is higher. As a result, the segregation force increases with depth except within the region $z/H\lt 0.3$ . In this region, a rapid decrease in segregation force is observed, accompanied by a reversal in its sign at approximately $z/H\approx 0.2$ . This is due to the interaction of particles with the moving bottom wall, leading to complex variations in the shear rate gradient for small $z/H$ . A similar trend also occurs for the viscosity. Consequently, the segregation velocity in the second column of figure 6(d) demonstrates a nonlinear variation with depth, with a reversal in segregation direction at $z/H\approx 0.2$ . Nevertheless, the predicted segregation velocity matches the measured values well except where it is affected by the walls.

The vertical chute flow (third column of figure 6) has a plug-like velocity profile, resulting in a segregation force that is antisymmetric about $z/H=0.5$ . Within the central region of the flow, specifically $0.2\leqslant z/H\leqslant 0.8$ , the shear rate is negligible, leading to difficulty in accurately estimating the local viscosity, which is typical of the quasi-static flow regime. Despite this problem, the model accurately predicts the measured segregation velocity to be approximately zero within this quasi-static central region, as well as the reversal of segregation direction above and below this region.

The curvature of the velocity profile for the inclined chute with $\theta =28^\circ$ (fourth column of figure 6) is opposite that of the wall-driven flow with gravity (second column of figure 6). The combined effects of shear rate and shear rate gradient result in an almost uniform segregation force except near the static bottom boundary. However, the increase in viscosity with depth leads to a gradual decrease in the segregation velocity, except near the bottom wall, as shown in the fourth column of figure 6(c). Additionally, the model overpredicts the segregation velocity near the free surface ( $z/H \geqslant 0.9$ ). This discrepancy is attributed to the inherent difficulty in accurately resolving the free surface using bin-averaging, compounded by the dilute flow regime ( $I\gt 5$ ) that prevails near the free surface. Such conditions deviate from the dense flow regime for which the model was developed, resulting in the observed overprediction of $w_i$ . Nevertheless, through most of the depth of the chute flow, the predicted segregation velocity is consistent with the measured values.

Figure 7. Segregation velocity $w_i$ for small (red) and large (blue) particles measured from the simulation (symbols) and predicted via (2.13) (dashed curve) and after considering diffusion via (2.17) (solid curve) for chute flow inclined at $28^\circ$ with different size ratios. In all cases, $c_l = c_s = 0.5$ and $H\approx 0.2\,$ m.

To further confirm the validity of our approach for calculating the segregation velocity, model predictions are compared with chute flow simulations for size ratios other than $R_d=1.5$ . Figure 7 shows reasonably close agreement between the model predictions and DEM measurements for $R_d =2$ , $2.5$ and $3$ , although the model underpredicts the segregation velocity. The overall segregation velocity is largest for $R_d=2$ and $2.5$ , but is smaller for $R_d=1.5$ and $3$ , consistent with previous findings (Alonso, Satoh & Miyanami Reference Alonso, Satoh and Miyanami1991; Félix & Thomas Reference Félix and Thomas2004; Thornton et al. Reference Thornton, Weinhart, Luding and Bokhove2012; Jones et al. Reference Jones, Isner, Xiao, Ottino, Umbanhowar and Lueptow2018). Similar to the results for $R_d=1.5$ in figure 6, the discrepancy for $z/H\geqslant 0.9$ is a result of dilute flow near the free surface. Nevertheless, the agreement observed between the model predictions and the simulation results, not only for various natural flow configurations but also across different size ratios, shows the broad applicability of (2.13) to a wide range of size-bidisperse granular flows at inertial numbers typical of dense flows.