# Comparison of empirical and particle force-based density segregation models

Soniya Kumawat  Vishnu Kumar Sahu  Anurag Tripathi<EMAIL_ADDRESS>
###### Abstract

The empirical and particle force-based models of granular segregation due to
density differences among the species are compared in this work. Dependency of
the empirical segregation parameters on the initial configuration, the
observation time duration, inclination angle, and mixture composition are
discussed in detail. The parameters obtained from empirical models are used to
predict the steady-state concentration profiles for different density ratios
and compositions. In addition, we utilize the predictions from the particle
force-based segregation model and compare them with the predictions of the
empirical segregation models. Our results show that the linear empirical
segregation model predictions agree well with the simulation results for
mixtures rich in light species where as quadratic empirical segregation model
works better for mixtures rich in heavy species. Particle force-based
segregation model, on the other hand, seems to be in very good agreement with
the DEM simulation data across all mixture compositions.

###### keywords:

Granular mixtures, Discrete Element Method, Density segregation, Empirical
segregation model, Particle-force based segregation model

††journal: Chemical Engineering Science

[inst1]organization=Department of Chemical Engineering,addressline=Indian
Institute of Technology Kanpur, city=Uttar Pradesh, postcode=208016,
country=India [inst2]organization=Department of Chemical Engineering and
Materials Science,addressline=University of Minnesota, city=Minneapolis,
postcode=MN 55455, country=USA

Measurement duration and initial configuration influence empirical parameter
values

Linear model parameters are more sensitive compared to quadratic model
parameters

Empirical models show larger deviations compared to particle force based
models

##  1 Introduction

Granular mixtures often undergo segregation during their flow due to
differences in the sizes and$/$or densities of the particles. Variety of
geometries such as chute (Tripathi and Khakhar, 2013; Deng et al., 2018;
Tripathi et al., 2021), vibrated bed (Yang., 2006; Tai et al., 2010; Jain et
al., 2013), hopper (Xiao et al., 2019), heap (Baxter et al., 1998; Schlick et
al., 2016; Fan et al., 2017; Deng et al., 2020), and rotating tumbler (Pereira
et al., 2014; Schlick et al., 2015b; Chen et al., 2016; Deng et al., 2019)
have been used to understand and predict the segregation of granular mixtures.
Discrete element method simulations are commonly utilized to obtain various
properties of the flowing mixture which are then used to propose segregation
models, often in the form of correlations. These segregation models are
typically used in the continuum framework by employing the convection-
diffusion equation to predict the concentration of different species in
flowing granular mixtures.

One of the most popular approaches for proposing these segregation models
involves empirically correlating the species segregation velocity to the flow
properties of the mixture. Early researchers investigating segregation of
different size mixtures (Savage and Lun (1988); Gray and Chugunov (2006); Gray
and Ancey (2011)) assumed the segregation velocity of the species to be
linearly dependent on the concentration of other species. Using the
segregation model of Gray and Chugunov (2006), however, showed significant
difference from the experimental concentration profiles for a binary mixture
of different size grains in the study of Wiederseiner et al. (2011). The
authors attributed these differences in the concentration profiles to possible
dependence of the segregation velocity on the local shear rate. Later studies
(Fan et al. (2014); Schlick et al. (2015a)) confirmed this local shear rate
dependency of the segregation velocity and utilized the combined shear rate
and concentration dependent segregation velocity to predict size segregation
in heap flow. Both linear (Fan et al. (2014); Schlick et al. (2015a, 2016);
Deng et al. (2018, 2019, 2020); Gao et al. (2021)) as well as nonlinear
(Gajjar and Gray (2014); Van der Vaart et al. (2015); Jones et al. (2018))
forms have been proposed to account for the concentration dependency. A linear
dependency of the segregation velocity has been found on the shear rate in all
these studies.

This dependency on the shear rate as well as the species concentration has
been also confirmed for mixtures with same size but different density
particles (Xiao et al. (2016); Fry et al. (2018); Jones et al. (2018); Fry et
al. (2019)). By performing DEM simulations of quasi 2D heap flows, Xiao et al.
(2016) obtained the linear dependence of the segregation velocity $v_{seg,i}$
on the local concentration of other species $f_{j}$ and shear rate for binary
mixtures differing in density. The authors proposed a correlation for the
segregation velocity $v_{seg,i}=S_{D}\dot{\gamma}f_{j}$ as proposed by Fan et
al. (2014) and Schlick et al. (2015a) in the case of size segregation. The
segregation length scale $S_{D}$ in different density mixtures was found to be
proportional to the diameter $d$ of the particles and logarithm of the density
ratio $\rho$ and was given as $S_{D}=dC_{D}\ln{\rho}$. The value of the
empirical parameter ($C_{D}$) was estimated by linearly fitting the $S_{d}$
vs. $\rho$ data on a semi-log plot. Fry et al. (2018) modified this linear
empirical segregation model to account for the effect of the confining
pressure in addition to the local concentration and shear rate in case of
plane shear flows. In a followup study (Fry et al., 2019), the authors showed
that the model is able to predict segregation in plane shear flow geometry for
different densities as well as different size mixtures.

Jones et al. (2018) studied the segregation for a binary mixture due to
density difference using DEM simulations in $2D$ bounded heap flow. The
authors also investigated the size segregation of binary mixtures in the same
geometry. Based on their findings, the authors revised the empirical model for
the segregation velocity and proposed a quadratic dependence on the
concentration along with the linear dependency on the shear rate. Duan et al.
(2021) showed that the combined effect of size and density segregation in $2D$
quasi-heap flow can also be described well using the quadratic empirical
model. The empirical parameters are found to be dependent on the size ratio as
well as the density ratio. Duan et al. (2022) showed that this approach is
also able to capture the segregation in a rotating tumbler.

Another promising approach to predict segregation is referred to as the
particle force-based approach. By considering the forces on a high density
intruder settling in a flowing granular medium, Tripathi and Khakhar (2013)
proposed the particle force-based theory for predicting density segregation of
binary mixtures in chute flow. In a recent study (Sahu et al., 2023), the
particle force-based theory is extended for multi-component granular mixtures
of different densities. This model is successfully able to capture the
segregation in binary and multi-density mixtures. A similar approach is
utilized by Tripathi et al. (2021) to obtain the theoretical expression for
the segregation velocity of large size particles and predict the concentration
profiles in binary size mixtures for periodic chute flows.

Both empirical as well as particle force-based approaches account for the
effect of the species concentration and shear rate. However, the dependence on
the species densities appears to be very different in these two approaches.
While the particle force-based model suggests segregation velocity to be
proportional to the density difference between the same size particles, the
linear empirical model suggests that the density ratio of the species
determines the segregation. A quick comparison of the particle force-based
segregation velocity expression given by Duan et al. (2020) indeed suggests
that the segregation parameter for the linear empirical model is proportional
to the density difference and inversely proportional to the product of
pressure and effective friction coefficient (i.e., the shear stress). Given
the fact that the shear stress in the case of chute flows depends on the
species concentration and inclination angle, the dependence of the empirical
segregation parameter on these two can’t be ruled out. Further, the particle
force-based theory suggests strong inter-coupling of rheology and segregation
with each other during the flow. The flow kinematics for systems starting from
different initial configurations may be significantly different due to such
inter-coupling of rheology and segregation. Since the empirical segregation
parameters are obtained from the kinematics data during the flow evolution,
the dependency of these parameters on flow configuration also remains to be
explored. In this study, we compare the empirical and particle force-based
segregation models in an identical geometry. Specifically, we chose periodic
chute flow of different density particles over a bumpy base and explore the
effect of mixture composition, measurement time duration, density ratios at
different inclination angles, and different initial configurations on the
empirical segregation parameters.

The organization of the paper is as follows. Section §2 describes the
simulation methodology for DEM simulations. Details of the calculation of the
empirical segregation parameters estimation along with the effect of the
measurement time and initial configuration are discussed in §3. This detailed
analysis helps us identify the limitations of the empirical models and
determine the best way to estimate the appropriate values of the empirical
parameters. The theory utilizing the two empirical models to predict the
concentration profile in the periodic chute is presented in §4. In addition,
the predicted profiles are compared with the DEM simulation and particle
force-based theory predictions in this section. The conclusions are presented
in section §5.

##  2 Simulation methodology

(a)(b)

Figure 1: DEM simulation snapshots for binary granular mixture having (a)
$20\%$ and (b) $80\%$ light particles with density ratio $\rho=3.0$ flowing
over an inclined plane at an inclination angle of $\theta$. Green and blue
particles represent the heavy and light particles, respectively. Black
particles represent the rough bumpy base of thickness $1.2d$.

We simulate granular mixtures of different density particles, modelled as
slightly inelastic and frictional spheres, flowing over a bumpy inclined
surface. At the beginning of the simulation, the particles are positioned in
such a way that their centers form a cubic lattice, ensuring that there is no
contact between any pair of particles. Additionally, these particles are
assigned random initial velocities. First, we settle the particles at a zero
inclination angle and then increase the inclination to the desired angle.
Figure 1 shows the uniformly mixed arrangement of high-density (green)
particles of mass $m_{H}$ and the light density (blue) particles of mass
$m_{L}$ for two different mixture compositions. The bumpy chute base, shown as
black spheres in figure 1, is formed by taking a slice of $1.2d$ thickness
from a randomly arranged, densely packed arrangement of spheres of the
diameter of $d$. The simulation box has equal length and width of $20d$ in the
flow ($x$) and the neutral ($z$) directions. The height of the simulation box
in the $y$-direction is chosen to be sufficiently large in order to make sure
that the particles never reach the top surface to effectively mimic free
surface flow. A total of $10000$ flowing particles are simulated within the
system so that the typical flowing layer height $H\approx 25d$. The
simulations are performed for binary as well as ternary density mixtures over
a wide range of compositions, and inclination angles. We consider three
different initial configurations for density segregation: light near base,
heavy near base, and well-mixed configuration.

The flow and segregation behavior of the mixtures is simulated using the
Discrete Element Method (DEM) simulation. Following Tripathi and Khakhar
(2011a), we use linear spring dashpot contact force model, and the Coulomb’s
friction law is used to limit the maximum permissible tangential force. The
equations of motion are numerically integrated using a sufficiently small time
step. Both gravity as well as inter-particle contact forces acting on the
particles are accounted for to obtain the positions and velocities of every
particle. All the particle properties used in the DEM simulations are
identical to those used in Tripathi and Khakhar (2011a) and Sahu et al.
(2023). Using the positions and velocities of the particles, all the relevant
quantities of interest are calculated. Due to the periodic boundaries in the
$x$ and $z$ direction, properties are obtained across different strips in the
$y$ direction. The sampling volume is taken as a strip of thickness $d$
spanning the entire area of the simulation box in the $x-z$ plane. In order to
get a good statistical average, the properties are averaged over a total of
$20$ time units obtained at regular intervals of $0.1$ time unit.

Packing fraction of $j^{th}$ species in the mixture is calculated as
$\phi_{j}=(1/V)\sum_{i}\phi_{j,i}$, where $\phi_{j,i}$ is the volume fraction
of $i^{th}$ particle of the $j^{th}$ species in the sampling volume $V$. The
total packing fraction of the mixture is obtained as $\phi=\sum_{j}\phi_{j}$.
The concentration of $j^{th}$ species at any $y$ location is obtained as

$f_{j}=\frac{\phi_{j}}{\phi}.$ (1)

Species velocity is calculated as,

$\vec{v}_{j}=\frac{\sum\vec{c}_{j,i}\phi_{j,i}}{\sum\phi_{j,i}},$ (2)

where $\vec{c}_{j,i}$ is the instantaneous velocity of particle $i$ belonging
to species $j$. The average velocity of all the $N$ particles partially or
completely situated in the sampling volume $V$ is calculated as
$\vec{v}=\sum_{k=1}^{N}\phi_{k}\vec{c}_{k}$ where $\phi_{k}$ is the volume
fraction of particle $k$ in the sampling volume and $\vec{c}_{k}$ is the
instantaneous velocity of particle. The shear rate at each $y$ location is
calculated by numerically differentiating the velocity profile using forward
difference method. Following Fan et al. (2014), Xiao et al. (2016), and Jones
et al. (2018), the segregation velocity of $j^{th}$ species $\vec{v}_{seg,j}$
is defined as the local velocity of species $j$ with respect to mean flow
$\vec{v}_{mix}$ in the cases where contribution due to diffusional velocity is
neglected, i.e.,

$\vec{v}_{seg,j}=\vec{v}_{j}-\vec{v}_{mix}.$ (3)

Data for the segregation velocity is obtained for each strip of thickness $1d$
across the entire flowing layer and is averaged over an interval of $100$ time
units. Steady-state is characterized by a constant kinetic energy of the
system and a constant height of the centre of masses of all the species in the
mixture.

##  3 Calculation of empirical segregation parameter

A binary granular mixture flowing over an inclined surface is known to
segregate in regions rich in one component. Previous studies (see Fan et al.
(2014); Schlick et al. (2015a); Xiao et al. (2016); Jones et al. (2018); Duan
et al. (2021)) use the empirical approach to predict the segregation due to
size or density differences in free surface flows such as heap and rotating
tumbler. All of these approaches utilize the flow and kinematics data starting
from an initially well-mixed configuration for obtaining segregation
parameters. Since this parameter determination utilizes the data during the
flow evolution, the possibility of empirical parameters being dependent on the
duration of observation can’t be ignored. Hence, we performed DEM simulations
and explored the effect of the observation time duration on the empirical
parameter calculation. We consider a periodic chute flow simulation of a
binary mixture having two different density particles. The usage of this setup
provides a much better control over the inclination angle (and the shear rate)
compared to other setups and facilitates investigation of the effect of the
measurement time duration on empirical segregation parameters. Using this
simple setup, we estimate the segregation time scale of the species using the
data of centre of mass variation with the time. We first, discuss the duration
of segregation for different compositions of binary density mixtures. Figure 2
shows the variation of the centre of mass ($y_{com}$) with time for three
different mixtures. Figure 2a shows the data for a mixture having $20\%$ light
species ($f^{T}_{L}=0.20$) whereas figures 2b and 2c show the data for
$f^{T}_{L}=0.50$ and $f^{T}_{L}=0.80$ respectively. The data presented
corresponds to a binary mixture with a density ratio of $\rho=3.0$ flowing
over an inclined plane at inclination $\theta=25\degree$. Following the
studies mentioned in the beginning of this section, we start the simulations
from an initial condition where the light and heavy particles are uniformly
mixed. Hence, the centre of mass position of both the species at time $t=0$ is
nearly identical for all three cases. The particles are settled under the
influence of gravity at zero inclination angle, resulting in a packing
fraction close to a randomly closed-packed configuration. Shearing of such a
closed-packed layer undergoes dilation as it starts to flow. Such a dilation
at early times is observable in the inset of figures 2a, 2b, and 2c. Note that
both heavy and light species exhibit a rise in the centre of mass position in
the beginning due to the dilation of the layer. After an initial period of
dilation, the centre of mass position of heavy species (green squares)
decreases while that of light species (blue circles) increases with time. The
$y_{com}$ position of both the species becomes constant at larger times,
indicating that steady-state segregation is achieved.

(a)(b)(c)

Figure 2: Centre of mass variation of light (blue circles) and heavy (green
squares) species with time for a binary mixture having density ratio
$\rho=3.0$ starting from an initially well-mixed configuration. The
composition of the light species in the mixture is (a) $f^{T}_{L}=0.20$ , (b)
$f^{T}_{L}=0.50$, and (c) $f^{T}_{L}=0.80$. Symbols represent the DEM data and
the lines are the exponential fits to the data.

Ignoring the dilation for the initial times, the subsequent centre of mass
variation can be captured very well using an exponential function. The
following mathematical expression is used to fit the data shown in figures
2a-2c,

$y_{com}(t)=y^{0}_{com}+(y^{\infty}_{com}-y^{0}_{com})\left(1-e^{-(t-t_{0})/\tau}\right).$
(4)

In this expression, $y^{0}_{com}$ and $y^{\infty}_{com}$ are the initial and
final centre of mass positions of the species. $\tau$ is the time scale that
characterizes the segregation phenomena and $t_{0}$ is the dilation time
beyond which $y_{com}$ of heavy species starts to decrease. A similar
expression has been utilized by Staron and Phillips (2014) to calculate the
segregation time scale. The exponential curve (shown using solid lines in
figure 2) describes the data reasonably well for both the species for all
three mixture compositions. The segregation time scale $\tau$ obtained by
fitting equation 4 to the species centre of mass profiles is reported in table
1. Note that the segregation time scale is found to be different for the two
species and varies with the composition of the mixture.

Segregation time scale | $f^{T}_{L}=0.2$ | $f^{T}_{L}=0.5$ | $f^{T}_{L}=0.8$
---|---|---|---
Heavy species ($\tau_{H}$) | 1157 | 799 | 478
Light species ($\tau_{L}$) | 1237 | 916 | 599
Table 1: Segregation time scale for heavy and light species for three
different mixture compositions shown in fig 2

Table 1 also shows that the time scale for both the species decreases with an
increase in the composition of the light particles. Additionally, the time
scale of the light species is found to be approximately $80$ to $120$ time
units larger than that of heavy species. The average value of the time scales
of the two species i.e., $\tau=(\tau_{H}+\tau_{L})/2$ is taken as the mixture
segregation time scale for further analysis.

###  3.1 Linear empirical models

(a)(b)

Figure 3: (a) DEM simulation data of shear rate scaled segregation velocity
for light (circles) and heavy (squares) species of a binary mixture having
density ratio $\rho=3.0$ flowing over chute at inclination angle
$\theta=25\degree$. Different colours correspond to different mixture
compositions of light species in the range $f^{T}_{L}=0.2$ to $f^{T}_{L}=0.8$.
(b) DEM simulation data for all the compositions averaged over every $\Delta
f_{i}=0.02$. Solid lines represent the linear fit to the averaged data.

(a)(b)(c)(d)

Figure 4: Dependency of the segregation parameter on the observation time
duration (a) $0.5\tau\leq t\leq 1.0\tau$, (b) $1.5\tau\leq t\leq 2.0\tau$, (c)
$2.5\tau\leq t\leq 3.0\tau$ and (d) $3.5\tau\leq t\leq 4.0\tau$. Blue symbols
represent the light species, and green symbol represents the heavy species.
Solid lines represent the linear fit $v_{seg,i}/\dot{\gamma}=S_{D}f_{j}$. Data
reported for inclination angle $\theta=25^{\circ}$ and density ratio
$\rho=3.0$.

Following Fan et al. (2014), Schlick et al. (2015a) & Xiao et al. (2016), we
define the segregation velocity as the species velocity relative to the
mixture velocity using equation 3. Using this definition, these studies have
shown that the species segregation velocity scaled by the local shear rate
i.e, $v_{seg,i}/\dot{\gamma}$, varies linearly with other species
concentration $f_{j}$ for bounded heap flows and rotating cylinders. Here, we
report the data for the segregation velocity in periodic chute flow obtained
from DEM simulations in figure 3. These data correspond to a binary mixture
density ratio $\rho=3.0$ flowing at an inclination angle of
$\theta=25\degree$. We consider the data for an observation duration of
approximately two-time scales ($t_{0}\leq t\leq 2\tau$). The data during the
initial dilation period $t<t_{0}$ is not included. Following Xiao et al.
(2016), the flow is started from an initially mixed configuration. The light
species tends to rise towards the free surface, resulting in a positive
segregation velocity. Conversely, heavy species move towards the base, leading
to a negative segregation velocity.

Figure 3a shows the variation of the segregation velocity scaled by the shear
rate $v_{seg,i}/\dot{\gamma}$ with the concentration of other species $f_{j}$.
Different coloured symbols represent the local segregation velocity for
different mixture compositions. Total five different mixtures compositions
with $f^{T}_{L}=0.2,0.4,0.5,0.6$, and $0.8$ have been considered to ensure the
full range of $f_{j}$ from $0$ to $1$ in figure 3. The large scatter in the
data points of figure 3a is substantially reduced by averaging it over bins of
width $\Delta f_{i}=0.02$ spanning over the entire concentration range. This
data, shown in figure 3b, can be described very well using a straight line.
The slope of the fitted line for the heavy species is $S_{D,H}=-0.023$ while
that for the light species is $S_{D,L}=0.023$. Note that the magnitude of
slopes (termed as the segregation or percolation length scale in previous
studies) for both species is identical. These results indicate that the
segregation velocity scaling proposed by Xiao et al. (2016) for the density
segregation holds true in the case of the chute flow as well. Thus, the
segregation velocity of species $i$ can be described using the following
relation,

$\centering v_{seg,i}=S_{D}(i,j)\dot{\gamma}f_{j}.\@add@centering$ (5)

Recall that the data reported in figure 3 correspond to an observation
duration of approximately two-time scales. We now investigate the effect of
the observation time on the empirical parameters. We first report the
variation of the scaled segregation velocity with time. To explore this
effect, we collect the scaled segregation velocity data for different time
intervals. The data for the initial duration of layer dilation are discarded.
Figure 4 illustrates the shear rate scaled segregation velocity for both the
species for four distinct time intervals, each with a duration of $0.5\tau$.
As before, the data are presented for a binary mixture with $\rho=3.0$ at
$25^{\circ}$ starting from the mixed configuration. The segregation velocity
scaled by the shear rate as a function of concentration is shown for
$0.5\tau\leq t\leq 1.0\tau$ in figure 4a, for $1.5\tau\leq t\leq 2.0\tau$ in
figure 4b, for $2.5\tau\leq t\leq 3.0\tau$ in figure 4c, and for $3.5\tau\leq
t\leq 4.0\tau$ in figure 4d. Symbols represent the DEM data, and lines
represent the linear fit to data. The slope of the fitted lines is mentioned
for each time interval. The shear rate-scaled segregation velocities for both
the species exhibit higher values within the time range of $0.5\tau\leq t\leq
1.0\tau$. However, these scaled velocities decrease as time progresses.
Notably, the values of $v_{seg,i}/\dot{\gamma}$ for both the species are
significantly low during the time interval of $2.5\tau\leq t\leq 3.0\tau$ &
$3.5\tau\leq t\leq 4.0\tau$. We find that the linear fit (equation 5)
accurately describes the data for all the time intervals. However, the
empirical parameters estimated from the linear fit to DEM data differ for
different time intervals. The decreasing slope of fitted lines in figure 4
confirms that the species segregation velocity keeps decreasing with time and
becomes nearly zero at steady state. This is in contradiction with the
empirical models for segregation velocity. The linear empirical relation of
equation 5 states that segregation velocity depends only on the shear rate
$\dot{\gamma}$ and concentration of other species $f_{j}$. At steady state,
the shear rate has a finite value, and except for the limiting cases of
$f_{i}=0$ & $1$, $f_{j}$ is also non-zero. Therefore, according to equation 5
the segregation velocity should have a finite non-zero value at steady state.
We consider this issue again later while discussing the role of diffusional
velocity in estimation of segregation velocity using DEM. The time-dependent
slope of scaled percolation velocity suggests that a time-averaged slope may
be more appropriate for usage in equation 5. Hence we consider data from
$t_{0}\leq t\leq t_{f}$, where $t_{f}$ ranges from $1\tau$ to $2.5\tau$. For a
time duration larger than $2.5\tau$, the segregation velocity becomes very
small and remains practically indistinguishable from zero as the segregation
process is around $95\%$ complete. We find that even after averaging the data
over various observation time frames, substantial differences remain
noticeable and the value of $S_{D}$ varies substantially for different values
of $t_{f}$. Table 2 reports the absolute value of the segregation length scale
empirically obtained from the DEM simulations for four different observation
times. The segregation parameter has a negative value for heavy species and a
positive value for light species.

Segregation parameter | $\tau$ | $1.5\tau$ | $2\tau$ | $2.5\tau$
---|---|---|---|---
Heavy $|S_{D,H}|$ | 0.031 | 0.027 | 0.023 | 0.020
Light $|S_{D,L}|$ | 0.027 | 0.025 | 0.023 | 0.022
Average ($S_{D}$) | 0.029 | 0.026 | 0.023 | 0.021
Table 2: Segregation length scale for heavy and light species for different
observation time duration $t_{0}\leq t\leq t_{f}$ for different values of
$t_{f}$ in the range $\tau$ to 2.5$\tau$. Average value of the segregation
parameter is calculated as $S_{D}=[|S_{D,L}|+|S_{D,H}|]/2$.

While the magnitude of segregation parameters of the two species are found to
be reasonably close to each other, the values of the parameter seem to
decrease as observation time increases. This decrease in the slopes is
attributed to the fact that more data near the steady state are considered in
the case of the higher value of $t_{f}$. It appears that a time duration of
twice the segregation time scale is appropriate as it provides a long enough
time window to capture most of the finite segregation velocity data without
capturing the near-zero segregation velocity close to the steady state. Hence,
we use the segregation parameter obtained over twice of segregation time scale
($2\tau$) in the rest of the paper for the linear empirical model. Using the
data over $2\tau$, we report the segregation length scale for a range of
density ratios $1.5\leq\rho\leq 10$.

Figure 5: Variation of non-dimensional segregation parameter $S_{D}/d$ with
density ratio $\rho=\rho_{H}/\rho_{L}$ on a semi-log plot for binary mixture
at $\theta=25\degree$ for different $t_{f}$.

Figure 5 shows that $S_{D}/d$ varies linearly with the density ratio on the
semi-logarithm scale for all five different time durations considered in this
study. The data presented in this figure corresponds to a binary mixture
flowing at an inclination angle of $25^{\circ}$ starting from the mixed
configuration. The data can be described using the relation
$S_{D}(i,j)/d=C_{D}\ln(\rho_{j}/\rho_{i})$ where $C_{D}$ is an empirical
parameter. This relation is identical to that reported by Xiao et al. (2016).
However, the time duration for segregation velocity measurement has not been
mentioned by the authors. Our results show that the value of empirical
parameter $C_{D}$ (i.e the slope of fitted lines in figure 5) decreases by
almost $23\%$ by increasing the time duration from $1\tau$ to $3\tau$. Figure
5 inset shows the data for $S_{D}/d$ variation at different inclination
$\theta$ for a binary mixture of density ratio $\rho=3.0$. Evidently, the
$S_{D}/d$ value increases with decreasing the inclination angle $\theta$. The
dashed line shows the extrapolated line for lower inclination angles. Xiao et
al. (2016) reported that the segregation parameter $S_{D}$ is $0.15$ mm for
the binary mixture having identical size particles of diameter $d=3$ mm and
density ratio $\rho=3.0$ flowing at inclination angle $\theta=16\degree$
(Figure 7 of their study). Notably, the dimensionless value of $S_{D}=0.05$ is
found to be close to our extrapolated value of $0.043$. Based on their
results, we use the value of $C_{D}$ corresponding to a measurement time of
$2\tau$ while using the linear empirical segregation model and predict the
theoretical concentration profile for binary as well as multi-component
mixture flowing over a chute in section §4.

###  3.2 Quadratic empirical model

Linear empirical models (Fan et al. (2014), Schlick et al. (2015a), and Xiao
et al. (2016)) describing the segregation due to either size differences or
density differences utilize equation 3 to measure the segregation velocity
from DEM simulations. A recent study by Duan et al. (2021) dealing with the
combined size and density segregation suggests that this method of measurement
of the segregation velocity using DEM simulations is not appropriate. The
authors revise the method to calculate segregation velocity and incorporate
the presence of the diffusional velocity using the following expression

$v_{seg,i}=v_{i}-v_{mix}+\frac{1}{f_{i}}D\dfrac{\partial f_{i}}{\partial y}.$
(6)

(a)$t_{0}\leq t\leq 2\tau$(b)$2\tau\leq t\leq 4\tau$

(c)(d)(e)(f)

Figure 6: Shear rate scaled segregation velocity (empty symbols) and
diffusional velocity (filled symbols) with the concentration of other species
in a mixture for two different time durations $t_{0}\leq t\leq 2\tau$ (left
column) and $2\tau\leq t\leq 4\tau$ (right column). Results for the same size
and same density particle mixture are shown in (a) and (b). Results for a
binary mixture of density ratio $\rho=3.0$ starting from a mixed initial
configuration are shown in (c) and (d). (e) and (f) show data for heavy near
base initial configuration. Blue circles, and green squares represent the
light and heavy species respectively.

(a)(b)(c)

Figure 7: Shear rate scaled segregation velocity variation with concentration
for binary mixture having for density ratio (a) $\rho=1.5$; (b) $\rho=3.0$ at
inclination angle $\theta=25\degree$, (c) $\rho=3.0$ at inclination angle
$\theta=29\degree$. Blue symbols represent the light species, and green symbol
represents the heavy species. Blue and green solid lines represent the
quadratic fit to the data $v_{seg,i}/\dot{\gamma}=df_{j}(A_{ij}+B_{ij}f_{j}$).

The rationale for the addition of the last term can be understood by
multiplying the above equation with $f_{i}$. The equation then states that the
flux of the species $i$ in the $y$-direction $f_{i}v_{seg,i}$ is due to the
combined effect of convective flux $f_{i}(v_{i}-v_{mix})$ and the diffusion
flux $D\partial f_{i}/\partial y$. In the case of significant size or density
differences between the species, the contribution of the convective flux is
much larger than the diffusive flux, and hence ignoring the latter leads to
little error. It is important to note that the above discussion pertains to
the empirical measurement of the segregation velocity using the DEM
simulations. The previous studies used only the relative velocity of the
species with respect to the mixture velocity to measure the segregation
velocity at any location. The revised method accounts for the correction due
to the presence of diffusion. Given the focus of the present study, we
investigate the role of convective and diffusional contributions to the
segregation velocity calculations in detail.

To better understand this, we consider a system in which only diffusion is
present, and the segregation effect is absent. We consider mono-disperse
particles, identical in all aspects but tagged as two different species
flowing over an inclined surface. The particles in the lower portion are
tagged as species $1$ and the upper portion particles are tagged as species
$2$. In this case, the sizes and densities of both species are equal, and
hence we expect the segregation velocity to be zero and only diffusion
velocity to be present. The figure 6a and 6b show the shear rate scaled
segregation velocity $\frac{v_{seg,i}}{\dot{\gamma}}$ where $v_{seg,i}$ is
computed using equation 6, as well as shear rate scaled diffusional velocity
$\frac{v_{diff,i}}{\dot{\gamma}}$ (where
$v_{diff,i}=\frac{1}{f_{i}}D\frac{\partial f_{i}}{\partial y}$). Filled
symbols represent the diffusional velocity and empty symbols represent the
segregation velocity. Evidently, we find that the segregation velocity using
equation 6 is close to zero at all times. This indeed appears to be the case
for not only the first two time scales (figure 6a) but also the next two time
scales (figure 6b). Diffusional velocity, represented with filled symbols, is
noticeable only for initial times $t_{0}\leq t\leq 2\tau$ (figure 6a). It
becomes close to zero for time $2\tau\leq t\leq 4\tau$ as both identical
species mix together, and the concentration gradient becomes zero (figure 6b).
As in figure 3, the data shown in figure 6 are obtained by considering five
different mixtures of composition $20\%,40\%,50\%,60\%$ and $80\%$ of one of
the species. In figures 6a and 6b, we also use the data for $10\%$ and $90\%$
compositions to obtain the entire range of concentration from $0$ to $1$.

Figure 6c & 6d show the diffusional velocity along with the segregation
velocity (given by equation 6), both scaled by the shear rate. The data are
shown for a binary mixture of density ratio $\rho=3.0$ flowing at
$\theta=25\degree$. During the $t_{0}\leq t\leq 2\tau$ time interval, the
diffusional velocity is found to be close to zero (figure 6c) since we start
from a uniformly mixed configuration. Hence the segregation velocity arises
primarily due to the convective motion of the species relative to the mixture.
Figure 6d shows the same data for a time interval ranging from two to four
time scales, i.e., $2\tau\leq t\leq 4\tau$. In contrast to figure 6c, the
diffusional velocities of both species appear to be almost equal to the
segregation velocities confirming that the convective contribution
$v_{i}-v_{mix}$ (not shown here) to the segregation velocity is negligible.
Figures 6e & 6f, show the same data for a mixture starting from a completely
segregated initial configuration with heavy particles near the base. As
before, figure 6e shows the data for time up to two-time scale, while figure
6f shows the data for a time interval ranging from two to four time scales. In
contrast to figure 6c & 6d for mixed configuration, it is evident that
convective flux is negligible not only for time $2\tau\leq t\leq 4\tau$ but
also for time $t_{0}\leq t\leq 2\tau$.

The observations made in figure 6c-6f can be understood as follows. Starting
from a well-mixed configuration, the concentration gradient $\partial
f_{i}/\partial y$ and hence diffusional contribution remains close to zero
(figure 6c) initially, and the segregation occurs entirely due to the relative
motion of the species with respect to each other. This relative motion leads
to the build-up of concentration gradient $\partial f_{i}/\partial y$ across
the layer, which leads to increasing importance of the diffusional
contribution at later times (figure 6d). In the case of starting from an
initial configuration of segregated case, the high concentration gradients
exist since beginning and hence the behaviour in figure 6e is similar to
figure 6d. As the final steady state configuration also has heavy particles
concentrated near the base, no difference in the figure 6e and figure 6f
noticeable. Results shown in figure 6c-6f confirm that the assumption of
diffusional contribution being small in case of density segregation is not
correct and the relative importance of the diffusion flux and convective flux
depends upon the initial configuration and may also change as the flow
evolves.

Figure 7 shows the variation of the shear rate scaled segregation velocity
with the concentration of other species for the initially mixed configuration.
The segregation velocity $v_{seg,i}$ is computed using equation 6 by utilizing
the data for time duration $2\tau\leq t\leq 4\tau$. Green square symbols
represent the DEM data for heavy species, and blue circles represent the light
species. Figures 7a - 7b show the effect of density ratio on shear rate scaled
velocity variations at inclination angle $\theta=25\degree$. Figure 7a
corresponds to density ratio $\rho=1.5$ and 7b corresponds to $\rho=3.0$.
Figure 7c shows data for density ratio $\rho=3.0$ at $\theta=29\degree$.
Comparing the magnitude of $v_{seg,i}/\dot{\gamma}$ in figures 7a and 7b, it
is clear that it increases with increase in the density ratio. Comparison of
figures 7b and 7c shows the effect of inclination angle at a given density
ratio $\rho=3.0$. An increase in the inclination angle from $\theta=25\degree$
to $\theta=29\degree$ leads to a slight reduction in the shear rate scaled
segregation velocity. The solid lines correspond to a quadratic fit confirming
that the shear rate-scaled segregation velocity exhibits a quadratic variation
with concentration. We use the segregation velocity relation proposed by Jones
et al. (2018) and Duan et al. (2021) to fit the DEM data shown in figure 7.
The authors used the following expression relating shear rate segregation
velocity with concentration,

$\centering v_{seg,i}/\dot{\gamma}=df_{j}(A_{ij}+B_{ij}f_{j}),\@add@centering$
(7)

where $A_{ij}$ and $B_{ij}$ are the empirical parameters. We report the values
of these parameters for both species in table 3. We observe that the
magnitudes of $A_{LH}$ and $B_{LH}$ for light species increase with increase
in the density ratio, and decreases with increase in the inclination angle.
This is consistent with the findings of Jones et al. (2018) and Duan et al.
(2021). From these results, we conclude that quadratic variation of shear rate
scaled segregation velocity with concentration is indeed observed in the case
of chute flow as well. However, the dependency on inclination angle $\theta$
is also observable since the parameters $A_{ij}$ & $B_{ij}$ for
$\theta=25\degree$ & $\theta=29\degree$ differ substantially from each other.
While increase in $\theta$ leads to higher shear rates, the model is unable to
account for this dependency on $\theta$ using the shear rate scaling.

Density ratio ($\rho$) |  inclination angle ($\theta$) | $A_{LH}$ | $B_{LH}$ | $A_{HL}$ | $B_{HL}$
---|---|---|---|---|---
$\rho=1.5$ |  $\theta=25\degree$ | 0.022 | -0.013 | -0.001 | -0.026
$\rho=3.0$ |  $\theta=25\degree$ | 0.082 | -0.054 | -0.003 | -0.083
$\rho=3.0$ |  $\theta=29\degree$ | 0.047 | -0.028 | -0.004 | -0.057
Table 3: Empirical parameters for the species obtained using quadratic fitting
(equation 7) of $v_{seg,i}/\dot{\gamma}$ vs $f_{j}$ data for binary mixture,
shown in figure 7.

(a)

HNB

(c)(e)(b)

LNB

(d)(f)

Figure 8: Snapshots showing an initial configuration of (a) Heavy near base
(HNB) case and (b) Light near base (LNB) case for a binary mixture of equal
compositions of each species at an inclination angle $\theta=25^{\circ}$ with
density ratio $\rho_{H}:\rho_{L}=3:1$. Black spheres represent the bumpy base.
Shear rate scaled segregation velocity (neglecting diffusional velocity) used
in linear empirical model for the (c) HNB case and (d) LNB case (error bars
shown only for alternate data points for the sake of clarity). The data are
taken for $t_{0}\leq t_{f}\leq 2.0\tau$. Shear rate scaled segregation
velocity (accounting for diffusional velocity) used in quadratic empirical
model for the (e) HNB case and (f) LNB case. The data are taken for
$2.0\tau\leq t_{f}\leq 4.0\tau$.

We next explore the effect of initial configuration on the empirical
parameters estimated using the two segregation models. We estimate the
empirical parameters for binary granular mixtures starting from two different
initial configurations. We follow the terminology used by Sahu et al. (2023).
The configuration with the heavy species initially concentrated near the base
is referred to as the heavy near base (HNB) case. A snapshot for the HNB case
is shown in figure 8a. The light near base (LNB) case, depicted in figure 8b,
consists of the low-density species concentrated entirely near the base and
the high-density species entirely near the free surface. In figures 8a and 8b,
the green spheres represent the high-density particles, and blue spheres
represent the low-density particles. The inclined surface has an angle of
$25\degree$ and the mixture has a density ratio $\rho=3.0$. Figures 8c and 8d
show the variation of $v_{seg,i}/\dot{\gamma}$ with the concentration of other
species $f_{j}$ for HNB and LNB cases, respectively for the linear empirical
model. $v_{seg,i}$ (computed using equation 3) divided by the shear rate is
plotted with $f_{j}$ by utilizing the data for observation time up to twice
the segregation time scale ($2\tau$). The magnitude of
$v_{seg,i}/\dot{\gamma}$ for both HNB and LNB case are different from that for
the mixed state (shown in figure 3b). Since the HNB configuration is already
close to the final segregated state, very low values of the percolation
velocities are obtained. The shear rate scaled segregation velocity for the
LNB case is approximately $10$ times higher than the HNB case. Using linear
fit to get the empirical parameters for linear model appears to differ by
nearly an order of magnitude for the two configurations. Further, the data for
the LNB case does not seem to be described well using the linear fit for
$f_{j}\geq 0.50$. These results confirm the dependency of initial
configuration on the parameter estimation using the linear empirical model.

Figure 8e shows the $v_{seg,i}/\dot{\gamma}$ calculated using equation 6 and
relate it with $f_{j}$ using the quadratic empirical approach for HNB case.
Figure 8f shows the same data for the LNB case. In both the figures, we employ
data beyond twice the segregation time scale $2\tau$ (i.e., $2\tau\leq t\leq
4\tau$). The magnitude of the $v_{seg,i}/\dot{\gamma}$ in both the
configurations is found to be comparable to each other as well as to the mixed
case discussed in figure 7b. Fitting equation 7 to the data reveals that the
values of the empirical parameters $A_{ij}$ and $B_{ij}$ for both species are
slightly different from each other. However, the predicted segregation
velocity seems to differ only for large values of $f_{j}$. Thus we can
conclude that the initial configuration doesn’t play as significant role in
empirical parameter estimation in the case of the quadratic empirical model as
in the linear case.

##  4 Comparison of empirical and particle force-based segregation models

The studies for heap flow and rotating cylinders suggest that the empirical
models of segregation velocity coupled with the convection-diffusion-
segregation equation can predict the concentration profile in good agreement
with DEM simulations and experiments. In this section, we explore the
applicability of the empirical approach for periodic chute flows. We solve the
steady state $1D$ diffusion-segregation equation (Tripathi and Khakhar, 2013;
Sahu et al., 2023) along with the momentum balance equations and the inertial
number-based rheology following the approach Tripathi and Khakhar (2011b) and
Sahu et al. (2023). Both empirical as well as particle force-based segregation
models have been utilized to predict the steady-state concentration profiles
theoretically.

###  4.1 Theory

We consider a fully developed flow over an inclined surface which is simulated
by employing a periodic chute in the DEM simulations. For this system, the
concentration gradient along the flow direction vanishes, and hence, the
convection term in the convection-diffusion-segregation equation (Xiao et al.
(2016); Duan et al. (2021)) becomes zero. At steady state, the transient term
also vanishes, and the equation reduces to a simple flux balance equation for
the species (Tripathi and Khakhar (2013), Sahu et al. (2023)),

$\Phi_{seg,i}+\Phi_{diff,i}=0.$ (8)

Using the expression for the diffusion flux $\Phi_{diff,i}=-D\frac{\partial
f_{i}}{\partial y}$ and segregation flux $\Phi_{seg,i}=v_{seg,i}f_{i}$, we
obtain an ordinary differential equation

$v_{seg,i}f_{i}-D\frac{df_{i}}{dy}=0,$ (9)

where $v_{seg,i}$ is the segregation velocity and $D$ is the diffusivity.
Several studies, including Fan et al. (2014), Schlick et al. (2016), Deng et
al. (2018), and Deng et al. (2020) utilized a constant value of the
diffusivity in the theoretical predictions. Utilizing a constant diffusivity
in the empirical approach to solve equation 9 would require the knowledge of
$\dot{\gamma}$ due to the dependence of percolation velocity $v_{seg,i}$ on
the local shear rate $\dot{\gamma}$. For this, one needs to incorporate
information about flow kinematics for predicting segregation. DEM simulations
confirm that the average value of diffusivity depends on inclination angle
$\theta$ (see Figure 8 by Tripathi and Khakhar (2013)) and hence separate
diffusivity values need to be used for each inclination angle to get
reasonable predictions.  Table S1 in the appendix shows that a very weak
dependency of the average diffusivity on the mixture composition is observed.
However, the average diffusivity values for $\theta=25^{\circ}$ and
$\theta=29^{\circ}$ differ significantly from each other. Figure S1 confirms
that the predictions using the particle force-based theory also differ
significantly if the diffusivity value corresponding to the particular
inclination angle is not used.

Variation of the diffusivity with the shear rate has been reported by many
researchers (Bridgwater et al. (1985); Utter and Behringer (2004); Tripathi
and Khakhar (2013); Fan and Hill (2015); Fry et al. (2019); Sahu et al.
(2023)). It is observed that the diffusivity ($D$) varies linearly with the
shear rate, $D=b\dot{\gamma}d^{2}_{mix}$, where $d_{mix}$ is the local volume
average diameter. $b$ is a constant parameter obtained using DEM simulation.
We use $b=0.041$, obtained by Tripathi and Khakhar (2013) and Sahu et al.
(2023) in the case of density segregation. For different density and same size
particle mixtures, $d_{mix}=d$. By utilizing the variation of diffusivity on
the local shear rate and concentration using relation
$D=b\dot{\gamma}d^{2}_{mix}$, the need for the detailed information about flow
kinematics is eliminated, and the equation can be solved analytically. The
usage of diffusivity linearly scaling with the shear rate automatically takes
care of diffusivity variation with inclination and hence should be preferred.
In this study, we account for the shear rate dependence of the diffusivity and
use this relation to obtain analytical solutions of equation 9 for both linear
and quadratic empirical relation of $v_{seg,i}$. For the linear segregation
model (given by equation 5) equation 9 simplifies to

$b\dot{\gamma}d^{2}\frac{df_{i}}{dy}=S_{D}(i,j)\dot{\gamma}f_{j}f_{i}.$ (10)

Using $f_{j}=1-f_{i}$ for the binary mixture case and integrating equation 10
with boundary condition $f_{i}(y=0)=f_{i0}$, we get

$f_{i}=\frac{f_{i0}\exp{\left(\frac{S_{D}}{bd^{2}}y\right)}}{1-f_{i0}\left(1-\exp\left(\frac{S_{D}}{bd^{2}}y\right)\right)}.$
(11)

Equation 11 describes the variation of species concentration $f_{i}$ with $y$
across the layer using an explicit function for the linear empirical
segregation model. On the other hand, using quadratic empirical relation
(given by equation 7) in the flux balance equation (equation 9), we get

$b\dot{\gamma}d^{2}\frac{df_{i}}{dy}=\dot{\gamma}df_{j}(A_{ij}+B_{ij}f_{j})f_{i},$
(12)

where $f_{j}=1-f_{i}$ is the concentration of other species. Integrating
equation 12 with boundary condition $f_{i}(y=0)=f_{i0}$, we get an implicit
equation for $f_{i}$ given as

$y=\frac{1}{bd}\left[\frac{B_{ij}}{A_{ij}(A_{ij}+B_{ij})}\log\left(\frac{A_{ij}+B_{ij}-B_{ij}f_{ij}}{A_{ij}+B_{ij}-B_{ij}f_{i0}}\right)\quad\quad\quad\quad\quad\quad+\frac{1}{A_{ij}+B_{ij}}\log\left(\frac{f_{i}}{f_{i0}}\right)-\frac{1}{A_{ij}}\log\left(\frac{1-f_{i}}{1-f_{i0}}\right)\right].$
(13)

Both equation 11 and equation 13 require the knowledge of $f_{i0}$ i.e., the
value of $f_{i}$ at $y=0$ to predict the concentration variation across the
layer. This value, however, is not known apriori. In order to obtain the value
of $f_{i0}$, we make use of the overall species balance across the flowing
layer. For this mass balance, the depth-averaged concentration of the species
in the layer must be equal to the total mixture composition $f^{T}_{i}$. In
other words, the integral of the species concentration $f_{i}(y)$ across the
flowing layer divided by the layer thickness $H$ must be equal to the total
composition of the species $i$, i.e.,

$\frac{1}{H}\int_{0}^{H}f_{i}dy-f_{i}^{T}=0.$ (14)

Equation 14 can be solved analytically for the linear empirical model.
Specifically, we use equation 11 in the equation 14 to get

$\frac{1}{H}\int_{0}^{H}\frac{f_{i0}\exp(\frac{S_{D}}{bd^{2}}y)}{1-f_{i0}+f_{i0}\exp(\frac{S_{D}}{bd^{2}}y)}dy=f_{i}^{T}.$
(15)

Equation 15 can be easily integrated to get the value of the unknown
concentration $f_{i}$ at $y=0$, i.e., $f_{i0}$

$f_{i0}=\frac{1-\exp{(\frac{S_{D}}{bd^{2}}Hf_{i}^{T})}}{1-\exp(\frac{S_{D}}{bd^{2}}H)}.$
(16)

In the case of the quadratic model analytical solution for $f_{i0}$ can’t be
obtained. In this case, we obtain $f_{i0}$ iteratively by following the
approach used by the Tripathi and Khakhar (2013). We start by assuming an
initial guess value of $f_{i0}$ at $y=0$ and solve the equation 13 to compute
the species concentration $f_{i}$ across the layer height $H$. Subsequently,
we numerically calculate the $LHS$ of equation 14, which should be equal to
$0$ to ensure the species mass balance. Numerically this condition is assumed
to be satisfied if the absolute value of the $LHS$ of equation 14 turns out to
be less than the tolerance value. In this work, we use a tolerance value of
$10^{-4}$. Given that $f^{T}_{i}\geq 0.1$, this tolerance value leads to less
than $0.1\%$ error in the mass balance. If this condition is not satisfied, we
update the value of $f_{i0}$ by a small increment (or decrement) of $\Delta
f_{0}=10^{-5}$. We repeat this process until the tolerance criteria is
satisfied to obtain $f_{i0}$.

While the flux balance equations for the binary mixture case (equations 10 and
13) can be solved analytically, the equations for mixtures with more than two
components require a numerical solution. We note that the linear empirical
model has been extended for multi-component mixtures differing in density by
Deng et al. (2018). The particle force-based segregation model has also been
extended for multi-component mixtures by Sahu et al. (2023). However, the
quadratic empirical model has not been extended for multi-component density
segregation. Studies involving multi-component mixtures differing in size
(Gray and Ancey (2011); Deng et al. (2019)) as well density (Deng et al.
(2018); Sahu et al. (2023)) suggest that the segregation flux
$\phi_{seg,i}^{Multi}$ of a species $i$ in a multi-component mixture of $N$
species can be written as the summation of the binary segregation fluxes with
respect to all the other species in the mixture, i.e.,
$\phi_{seg,i}^{Multi}=\sum_{j=1,j\neq i}^{N}\phi_{seg,i}^{Binary}(i,j)$. The
species flux balance (equation 8) for multi-component mixtures can be written
as

$-D\frac{\partial f_{i}}{\partial y}+\sum_{j=1,j\neq
i}^{N}\phi_{seg,i}^{Binary}(i,j)=0.$ (17)

Employing the linear empirical model given by equation 5, i.e.,
$v_{seg,i}=S_{D}(i,j)\dot{\gamma}f_{j}$ and $\phi_{seg,i}=v_{seg,i}f_{i}$, we
have $\phi_{seg,i}^{Binary}(i,j)=S_{D}(i,j)\dot{\gamma}f_{j}f_{i}$. Since
$D=b\dot{\gamma}d_{mix}^{2}$, the dependency on the shear rate is eliminated
for the empirical model and equation 17 simplifies to

$\dfrac{\partial f_{i}}{\partial y}=\frac{1}{bd^{2}}\sum_{j=1,j\neq
i}^{N}S_{D}(i,j)f_{i}f_{j}.$ (18)

In the case of the quadratic empirical model
$\phi_{seg,i}^{Binary}=(A_{ij}+B_{ij}f_{j})f_{j}f_{i}$ and the equation 17
simplifies to

$\dfrac{\partial f_{i}}{\partial y}=\frac{1}{bd}\sum_{j=1,j\neq
i}^{N}(A_{ij}+B_{ij}f_{j})f_{i}f_{j}.$ (19)

for  _$i\leftarrow 1$ to $N-1$_ do

Initialize $f_{i}(y)=f_{i}^{T}$ at all $y$;

Initialize $f_{i0}$;

end for

define $error=1$; $tolerance=10^{-4}$; $\Delta f_{0}=10^{-5}$;

while  _$error >tolerance$ _ do

for use in ODE45 solver;

for  _$i\leftarrow 1$ to $N-1$_ do

Define $f_{i}(y)$ in symbolic form;

end for

for  _$i\leftarrow 1$ to $N-1$_ do

Solve equations 18 (19) using ode45 with boundary condition
$f_{i}(y=0)=f_{i0}$ to compute new $f_{i}(y)$;

Compute difference between the calculated and actual total composition of the
species using 14;

end for

Compute $f_{N}=1-\sum\limits_{i=1}^{N-1}f_{i}$ ;

Define
$error=\sqrt{\sum\limits_{i=1}^{N-1}\left|\dfrac{1}{H}\int_{0}^{H}f_{i}dy-
f_{i}^{T}\right|^{2}}$ ;

for  _$i\leftarrow 1$ to $N-1$_ do

if  _$\dfrac{1}{H}\int_{0}^{H}f_{i}dy <f_{i}^{T}$_ then

$f_{i0}=f_{i0}+\Delta f_{0}$;

else

$f_{i0}=f_{i0}-\Delta f_{0}$;

end if

end for

end while

Algorithm 1 Algorithm used for predicting concentration of $N$ component
mixture using empirical segregation model

Equations 18 and 19 represent the spatial variation of species $i$ in a
mixture of $N$ components. Hence $i$ can take any value from $1$ to $N$. Since
$\sum^{N}_{i=1}f_{i}=1$, we get $N-1$ inter-coupled, non-linear differential
equations that need to be solved simultaneously. We solve equations 18 and 19
numerically using $ode45$ in MATLAB with an initial guess of $f_{i0}$ by
following the Algorithm 1. The iterative method is employed in the quadratic
approach to obtain $f_{i0}$. The empirical parameters are estimated for
different combinations of density ratios. In the linear empirical approach,
the relationship of $S_{D}$ with density ratios
$S_{D}(i,j)=C_{D}d\ln{\rho_{j}/\rho_{i}}$ facilitates solution for any generic
pair of $\rho_{i}$ and $\rho_{j}$. However, for the quadratic model,
parameters $A_{ij}$ and $B_{ij}$ are obtained for a particular density ratio.

Note that it is possible to obtain the relation of quadratic empirical model
parameters $A_{ij}$ and $B_{ij}$ with density ratio as reported by Duan et al.
(2021). However, given the focus of the paper, we use the $A_{ij}$ and
$B_{ij}$ for the specific combinations of density ratios corresponding to the
binary mixture of the same density ratio. Using the equations described above,
we obtain the concentration profiles of species across the layer for empirical
segregation models (linear as well quadratic). The predictions for the
particle force-based theory for binary mixtures are done following the
approach of Tripathi and Khakhar (2013). Specifically, we use the algorithm
given in Table 5 of their work. For ternary mixtures, we follow algorithm
given by Sahu et al. (2023). We utilize the same set of parameters as used in
these references and the interested reader is encouraged to look at these
references for more details.

###  4.2 Comparison of theoretical predictions

Next, we compare the theoretical prediction using the linear empirical model
of Xiao et al. (2016) and quadratic empirical model of Duan et al. (2021) with
the particle force-based theory proposed by Tripathi and Khakhar (2013) and
Sahu et al. (2023). As shown in section §3, the empirical parameters for the
mixed configuration obtained using the linear segregation model are more
reliable in comparison to LNB and HNB configurations. Hence, we utilize the
empirical parameters obtained for the mixed configuration in these theoretical
predictions. While the empirical parameters obtained using the quadratic
approach are found to be relatively less dependent on the initial
configuration, we use the parameters for mixed configuration for quadratic
model as well.

(a)(b)(c)(d)

(e)(f)(g)(h)

Figure 9: (a-d) Comparison of theoretical predictions using linear (dashed
lines) and quadratic (dotted) empirical approach along with (e-h) theoretical
predictions of particle force-based segregation model (solid lines) with DEM
simulation data (symbols). The results are shown for the three different
mixture compositions $f^{T}_{L}=0.20$ (black), $0.50$ (green) and $0.80$
(brown) for binary mixtures having density ratio $\rho=1.5$ in (a,e),
$\rho=2.0$ in (b,f), $\rho=3.0$ in (c,g), and $\rho=5.0$ in (d,h) at
inclination angle $\theta=25^{\circ}$.

Figures 9a - 9d show the comparison of the theoretical predictions using
linear (dashed lines) and quadratic (dotted lines) empirical models. The
steady-state DEM simulation data are shown as symbols for comparison. The
concentration profiles of light species are shown for the three different
mixture compositions of light species $f^{T}_{L}=0.20$ (black), $0.50$ (green)
and $0.80$ (brown). Figure 9a, 9b, 9c, & 9d show the results for density ratio
$\rho=1.5$, $2.0$, $3.0$ and $5.0$ respectively using the empirical model.
Figure 9e, 9f, 9g, and 9h show the predictions using the particle force-based
theory using solid lines for the same density ratios. Thus each column in
figure 9 corresponds to a particular density ratio.

Let us first consider mixtures with the composition of light species
$f^{T}_{L}=0.20$. Figure 9a shows that the predictions using the linear
segregation model (black dashed line) show significant differences with DEM
data (black circles) at density ratios $\rho=1.5$. Similar behaviour is
observed for $\rho=2.0$ (figure 9b), $\rho=3.0$ (figure 9c), and $\rho=5.0$
(figure 9d). However, these differences in the predictions appear to be
decreasing with the increase in the density ratio. On the other hand, the
quadratic empirical model (black dotted line) shows relatively small
differences and seems to be describing the simulation data comparatively
better for the density ratio of $\rho=1.5$ for $f^{T}_{L}=0.20$ (figure 9a).
At higher density ratios of $\rho=2.0$, $\rho=3.0$, and $\rho=5.0$, the
differences between data & theory get substantially reduced, and the black
dotted line follows the black circles very well, indicating that the quadratic
empirical segregation model predicts segregation very well for a dilute
mixture of light particles for $\rho\geq 2.0$. For $f^{T}_{L}=0.50$,
predictions using the quadratic approach match better with the DEM results in
comparison to linear empirical model for higher density ratios. At lower
density ratios, the two models appear to be similar in their agreement with
the DEM simulation data. For the mixtures having high composition of light
species $f^{T}_{L}=0.80$, the predictions using the linear empirical model
(dashed lines) match with DEM data very well. However, the predictions using
the quadratic empirical model show noticeable differences at low density
ratios. The deviation between the data (symbols) and dotted lines decreases
with increase in $\rho$. Theoretical predictions using the particle force-
based theory, shown in figures 9e, 9f, 9g, and 9h, on the other hand are found
to match well with DEM data across the entire composition range and various
density ratios.

$\rho=1.5$(a)$\rho=2.0$(b)$\rho=3.0$(c)$\rho=5.0$(d)

Figure 10: Variation of light species root mean square deviation (RMSD) value
with mixture composition for three different segregation model density ratios
(a) $\rho=1.5$, (b) $\rho=2.0$, (c) $\rho=3.0$, and (d) $\rho=5.0$. Solid
lines correspond to the particle force-based model, whereas the dashed and the
dotted lines correspond to the linear and quadratic empirical models,
respectively.

In order to quantify the deviations of the theoretical predictions with the
simulation data, we compute the root mean square deviation (RMSD) and compare
these three segregation models, namely the particle force-base model, and the
linear and quadratic empirical segregation model. The RMSD value, calculated
as

$RMSD_{j}=\sqrt{\frac{\sum_{i=1}^{N_{bin}}\left[f^{th}_{j}\left(y_{i}\right)-f^{DEM}_{j}\left(y_{i}\right)\right]^{2}}{N_{bin}}},$
(20)

quantifies the deviation of the theoretical predictions of the concentration
profile $f^{th}_{j}(y)$ obtained using the segregation model from that of DEM
simulations data $f^{DEM}_{j}(y)$ for the $j^{th}$ species in the mixture.
Here, $N_{bin}$ denotes the number of DEM data points in the concentration
profile obtained from DEM simulations. Since we use bin sizes of $1d$
thickness to obtain DEM simulations, $N_{bin}$ equals $H/d=24$. Figure 10
shows the RMSD values for different composition mixtures at density ratio
$\rho=1.5$, $\rho=2.0$, $\rho=3.0$, and $\rho=5.0$ shown in figures 10a, 10b,
10c, and 10d respectively. As before, the dashed and dotted lines correspond
to the linear and the quadratic empirical models, respectively. The solid
lines correspond to the particle force-based theory. Figures 10a - 10d show
that for mixtures having $f^{T}_{L}\leq 0.50$, the RMSD values for the linear
model (dashed lines) are larger than that of the quadratic model (dotted
lines). However, for compositions in the range $f^{T}_{L}\geq 0.50$, the
linear model (dashed lines) exhibits lower RMSD values compared to the
quadratic model (dotted lines) except for $\rho=5.0$. In that case, the RMSD
value of the linear model is smaller than the quadratic only for
$f^{T}_{L}=0.80$. The RMSD value decreases with increase in $f^{T}_{L}$ (total
composition of light species) for the linear empirical model for all four
density ratios. The RMSD for the quadratic model, on the other hand, appears
to increase with increasing $f^{T}_{L}$. This quantitative measurement
confirms our qualitative observations discussed above, where we observe that
the linear empirical model works well for mixtures with high composition of
light particles, whereas the quadratic model works better for mixtures with
low composition of light species. The particle force-based theory, however,
shows that RMSD values remain nearly constant and do not vary with the
composition of the mixture. The RMSD value using the particle force-based
model is significantly smaller compared to the other two empirical approaches
across the wide range of compositions and density ratios considered in this
study. The above results for binary mixtures confirm that particle force-based
theory is able to capture the segregation much more accurately compared to the
empirical model for a wide range of composition and density ratios.

(a)(b)(c)

Figure 11: Comparison of theoretical predictions using linear (dashed lines)
and quadratic (dotted) empirical model along with the particle force-based
model (solid lines) with DEM simulation data (symbols) for equal composition
ternary mixture of density ratio $\rho_{H}:\rho_{M}:\rho_{L}=3:2:1$.
Concentration profile of (a) light-density species, (b) medium-density
species, and (c) high-density species for $\theta=25\degree$.

We next report results for ternary mixtures. Figure 11 shows the steady state
concentration profiles for a ternary mixture having an equal composition of
the three species. The data corresponds to a density ratio of
$\rho_{H}:\rho_{M}:\rho_{L}=3:2:1$ and inclination angle of
$\theta=25\degree$. Figure 11a shows the concentration profile of light
species, whereas figures 11b and 11c show that of the medium and the heavy
species, respectively. Theoretical predictions using the linear empirical
segregation model (dashed lines) differ from the DEM data (symbols) for light
and medium species in the ternary mixture. Such differences are not observable
for the heavy species. On the other hand, predictions using the particle
force-based segregation model (solid lines) and quadratic empirical
segregation model (dotted lines) match with the DEM results very well for all
three species of the mixture. These observations can be confirmed by the RMSD
values reported in the legends of figure 11. While the RMSD values for the
heavy species are found to be nearly identical for all three segregation
models, the linear empirical model RMSD values are found to be significantly
higher than the other two models for both light as well as medium-density
species. We also find that RMSD values for the quadratic empirical model are
not very different from the RMSD values of the particle force-based theory for
all three species.

(a)(b)(c)

Figure 12: Comparison of theoretical predictions using linear (dashed lines)
and quadratic (dotted lines) empirical model along with the particle force-
based model (solid lines) with DEM simulation data (symbols) for three
different compositions in a ternary mixture of density ratio
$\rho_{H}:\rho_{M}:\rho_{L}=3:2:1$. Blue circles correspond to light species
while red squares and green diamonds correspond to medium and heavy species.
Concentration profiles for (a) $f^{T}_{L}=70\%,f^{T}_{M}=20\%,f^{T}_{H}=10\%$,
(b) $f^{T}_{L}=10\%,f^{T}_{M}=70\%,f^{T}_{H}=20\%$, and (c)
$f^{T}_{L}=20\%,f^{T}_{M}=10\%,f^{T}_{H}=70\%$.

(a)(b)(c)

Figure 13: The root mean square deviation (RMSD) values for light (blue),
medium (red) and heavy (green) species using three different segregation
models in ternary mixtures of density ratio
$\rho_{H}:\rho_{M}:\rho_{L}=3:2:1$. Three different mixture compositions of
(a) $f^{T}_{L}=70\%,f^{T}_{M}=20\%,f^{T}_{H}=10\%$, (b)
$f^{T}_{L}=10\%,f^{T}_{M}=70\%,f^{T}_{H}=20\%$, and (c)
$f^{T}_{L}=20\%,f^{T}_{M}=10\%,f^{T}_{H}=70\%$ are considered.

These results confirm that both the quadratic empirical model and the particle
force-based model are capable of accurately predicting segregation for equal
composition ternary mixture. In order to explore the applicability of the
empirical model over a wide range of mixture compositions, we select mixtures
with $70\%$ composition of a particular species and $20\%$ and $10\%$ of the
other two species. We obtain the theoretical predictions for each of these
mixtures and compare them with their DEM simulations.

(a)(b)

Figure 14: (a) Shear scaled segregation velocity of heavy species for
different composition mixture. Symbols represent DEM data, and solid lines
represent the quadratic empirical fits of form
$v_{seg,i}/\dot{\gamma}=df_{j}(A_{ij}+B_{ij})f_{j}$. (b) Steady-state
concentration profiles of light species in a binary mixture of density ratio
$\rho=1.5$. Symbols represent DEM data. Lines represent quadratic empirical
model predictions. Dashed lines correspond to the empirical parameters
estimated by averaging data over a wide range of compositions while dotted
lines correspond to empirical parameters estimated for individual mixture
compositions of light species.

Figure 12a shows the results for the mixture with $70\%$ of light species
i.e., $f^{T}_{L}=0.70$ with $f^{T}_{M}=0.20$ & $f^{T}_{H}=0.10$. Theoretical
predictions using the linear empirical segregation model (dashed lines) match
well with the DEM data for all three species. In contrast to the equal
composition mixtures case, the quadratic empirical segregation model
predictions (dotted lines) deviate strongly from the DEM results and particle
force-based segregation model predictions. In this case, the quadratic
approach fails to capture the DEM results throughout the entire layer height.
Figure 12b shows predictions for the mixture with $70\%$ medium species
composition; the heavy and light species composition being $20\%$ and $10\%$,
respectively. For this mixture, the predictions using the quadratic empirical
model as well as the linear empirical model differ substantially from the DEM
data for both light (blue) and medium (red) species. On the other hand, the
predictions based on the particle force model exhibit very good match with the
DEM data. Figure 12c shows the results for a ternary mixture where the
composition of heavy species is $70\%$ and that of the light species is
$20\%$. Predictions using the quadratic approach for the light species are
found to be in good agreement with DEM data. Differences for the medium-
density and high-density species are noticeable. The linear empirical
segregation model (dashed line) shows significant deviation for all three
species. Theoretical predictions obtained using the particle force-based model
exhibit excellent agreement with the DEM data for all three species for this
case as well. These observations can be confirmed by comparing the RMSD values
of the three models for each of the mixtures. Figure 13 shows the RMSD value
for the light (blue), medium (red) and heavy (green) species for each
segregation model. In all the cases, the RMSD value for the particle force-
based theory is much smaller compared to both the empirical models. This
confirms that the particle force-based segregation model is able to capture
the segregation of various kinds of mixtures. In contrast, the linear and
quadratic empirical models seem to work well in some cases and fail in some
other situations.

In order to understand the inability of the quadratic model to predict
concentration in the mixtures rich in either in light or in medium species, we
plot the shear rate scaled segregation velocity of the heavy species for four
different compositions of binary mixtures in figure 14a. Data have been shown
for a binary mixture of density ratio $\rho=1.5$ flowing over an inclination
angle of $\theta=25\degree$. Different symbols correspond to different mixture
compositions. Evidently, the variation of the shear rate scaled segregation
velocity is different for different compositions of the mixture. For example,
the value of $v_{seg,i}/\dot{\gamma}$ for a mixture composition of
$f^{T}_{L}=0.8$ is less than one-fourth of the value for a composition mixture
of $f^{T}_{L}=0.2$. The magnitude of $v_{seg,i}/\dot{\gamma}$ of heavy species
decreases with an increase in the composition of light species in the mixture.
Hence the empirical parameters obtained for different mixture compositions are
expected to show significant differences.

$f^{T}_{L}$ | 0.20 | 0.40 | 0.50 | 0.60 | 0.80
---|---|---|---|---|---
$A_{HL}$ | 0.013 | 0.040 | 0.037 | 0.032 | 0.037
$B_{HL}$ | -0.132 | -0.123 | -0.101 | -0.084 | -0.070
Table 4: Quadratic empirical parameters for heavy species for five different
composition mixtures of density ratio $\rho=1.5$.

Table 4 shows the parameters for each composition obtained by fitting the
quadratic model to the data shown in figure 14a. Solid lines in figure 14a
show the fitted quadratic curve using the values reported in Table 4. Figure
14b shows the predictions obtained using the quadratic empirical parameters
for the respective composition of the mixture using dotted lines. The
predictions match accurately with DEM data for all three compositions.
However, employing the averaged empirical parameters across all the
compositions, the predictions (shown by dashed lines) deviate significantly
from the DEM data. The RMSD values for the dashed lines (using the same
parameters for all compositions) are found to be $0.06$, $0.06$, & $0.10$ for
$f_{L}^{T}=20\%,50\%$, and $80\%$ mixtures, respectively. However, using the
values corresponding to the mixture composition, the RMSD values reduce to
$0.05$, $0.04$, & $0.03$, respectively. Our analysis confirms that the large
RMSD values for $f^{T}_{L}>0.5$ in the quadratic empirical model case shown in
figure 10 can be attributed to this concentration dependence of the
parameters. For example, using the revised parameters for $\rho=3.0$ and
$f_{L}^{T}=80\%$ mixture (figure 10c) reduces the RMSD value to $0.02$ from
$0.08$. Thus we conclude that the quadratic empirical parameters indeed depend
on the total composition of the species in the mixture. This is in contrast to
the observations reported by Jones et al. (2018) who reported that the inlet
composition in heap flow does not affect the empirical segregation velocity
relation for quadratic segregation models. A similar comparison for the linear
empirical model, however, does not show the dependency of the fitted
parameters on mixture composition. This is in agreement with Xiao et al.
(2016) who showed that the RMSD values for the predictions do not change
significantly when using the linear empirical parameters obtained for
individual composition.

##  5 Conclusion

In this study, we have investigated the suitability of empirical segregation
models for predicting segregation in periodic chute flows. This simple system
allows us to explore the effect of the inclination angle, different initial
configurations, and the duration of measurement on the segregation parameters.
We explore the two existing empirical approaches by accounting for or ignoring
diffusional contribution in the computation of segregation velocity
$v_{seg,i}$ from DEM simulations. While the parameters are estimated using
transient data in the case of the linear empirical model, the quadratic model
uses steady-state data for estimating the parameters. We find that the time
period of measurement of $v_{seg,i}$ affects the values of the empirical
parameters estimated in the case of the linear empirical model. In addition,
the initial configuration also seems to have a significant effect on the
estimated values of these parameters. The appropriate values of empirical
parameters in the case of the linear empirical model can be obtained by
measuring percolation velocity for approximately twice the segregation time
scale in a system starting with mixed configuration. The other configurations
should be avoided for determining the linear empirical model parameters. The
quadratic model parameters, however, are found to be less sensitive to the
initial configurations used. In addition, the time scale of observation does
not affect these values as well since steady-state data are used.

Using the appropriate values of empirical parameters for both the empirical
models, we predict the concentration profiles for mixtures of different
compositions and density ratios. In addition, we also predict the
concentration profile from the particle force-based segregation model. We
compute RMSD values to quantify the deviation of the predicted profiles with
the corresponding DEM simulation data. A comparison of the RMSD values shows
that the empirical model RMSD values are significantly larger than those of
the particle force-based model predictions. The linear empirical model has a
larger RMS deviation for mixtures rich in heavy species. The quadratic
empirical model predictions have larger RMS deviation in the case of mixture
rich in light species. The deviation in the quadratic model case arises due to
the dependence of empirical parameters on mixture composition and can be
substantially reduced by using parameters corresponding to the specific
composition.

We also compare the predictions of the empirical segregation models and
particle force-based theory with DEM simulations for ternary mixtures. While
the linear empirical as well as particle force-based models have been used to
predict segregation in multi-component mixtures, the quadratic model
predictions have been restricted to binary mixtures only. We extended the
methodology to predict segregation in the case of ternary mixtures using
quadratic empirical model. A comparison of the predictions from these three
segregation models for ternary mixtures of different compositions reconfirms
that the particle force-based theory is able to predict the simulation data in
much better agreement compared to the empirical models. Owing to the fact that
the segregation model in the particle force-based approach is obtained from
the forces acting on the particles, the species segregation velocity in this
approach accounts for the effect of the inclination angle and viscosity
dependence on local mixture composition. Such effects are not accounted in the
empirical approach.

A similar comparison of empirical and particle force-based segregation models
also needs to be done for mixtures having different sizes. It remains to be
seen whether the dependencies of the segregation parameters on the measurement
time, initial configuration, and inclination angle are observable for
empirical models in the case of size segregation. In addition, mixtures
differing in both size and density also need to be considered. While empirical
models for such mixtures have been proposed, a particle force-based theory for
the combined effect of size and density is still lacking and will be reported
in the near future. Finally, the particle force-based theory also needs to be
explored in few other flow configurations to establish its applicability in
experimentally realizable systems.

[Funding]AT gratefully acknowledges the financial support provided by the
Indian Institute of Technology Kanpur via the initiation grant
IITK/CHE/20130338. AT and SK gratefully acknowledge the funding support
provided to SK by the Prime Minister’s Research Fellowship (Government of
India) grant. [Declaration of Interest]The authors report no conflict of
interest. [Author ORCIDs]

Soniya Kumawat https://orcid.org/0000-0003-3314-9875;

Vishnu Kumar Sahu https://orcid.org/0000-0002-7768-0505;

Anurag Tripathi https://orcid.org/0000-0001-9945-3197.

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##  Appendix A Effect of diffusivity values on segregation predictions

Fan et al. (2014) and Gao et al. (2021) considered both spatially varying
diffusivity values as well as a constant diffusivity value in the prediction
of binary size segregation using linear empirical model. The authors report
that the spatial variation in the diffusivity has a negligible effect on the
predictions of the continuum model in the case of quasi-2D heap flows. Table
S1 shows that while the mean diffusivity values for a given inclination do not
differ a lot from each other due to change in mixture composition, the effect
of inclination on the mean diffusivity is very significant and hence needs to
be accounted for if one wishes to use a constant diffusivity.

$f^{T}_{L}$ | 0.20 | 0.40 | 0.50 | 0.60 | 0.80
---|---|---|---|---|---
$\theta=25\degree$ | 0.0121 | 0.0114 | 0.0113 | 0.0114 | 0.0127
$\theta=29\degree$ | 0.0361 | 0.0325 | 0.0321 | 0.0322 | 0.0345
Table S1: Mean diffusivity ($D_{avg}$) values for five different binary
mixture compositions at two different inclination angles $\theta=25\degree$
and $29\degree$.

(a)(b)

Figure S1: Comparison of theoretical predictions using a constant diffusivity
and a spatially varying diffusivity $D=b\dot{\gamma}d^{2}_{mix}$ for a binary
mixture of density ratio $\rho=3.0$ for (a) $\theta=25\degree$ and (b)
$\theta=29\degree$. Symbols represent DEM data, solid lines represent the
predictions using spatially varying diffusivity $D=b\dot{\gamma}d^{2}_{mix}$.
Dashed lines represent the predictions using constant diffusivity $D$
corresponding to inclination angle $\theta$ for $50\%-50\%$ binary mixture.
Dotted lines represent the predictions using a constant value of diffusivity
$D$ taken for $\theta=29^{\circ}$ in (a) and $\theta=25^{\circ}$ in (b).

Figure S1 shows the comparison of theoretical predictions using particle
force-based theory using both constant as well as spatially varying
diffusivity across the layer. Data are shown for three different binary
mixtures having light species composition $f^{T}_{L}=0.20$ (black),
$f^{T}_{L}=0.50$ (green), and $f^{T}_{L}=0.80$ (brown) with density ratio
$\rho=3.0$. Figure S1a shows the predictions at inclination angle
$\theta=25\degree$ while figure S1b shows the predictions at inclination angle
$\theta=29\degree$. Solid lines represent the theoretical predictions
accounting for the spatial variation of the diffusivity using the relation
$D=b\dot{\gamma}d^{2}_{mix}$. The dashed lines represent the predictions using
a constant value of average diffusivity $D_{avg}$ corresponding to the
inclination angle for $50\%-50\%$ binary mixture. Both solid and dashed lines
accurately match with the DEM data (symbols) for all three composition
mixtures at two different inclination angles.

Dotted lines in figure S1a correspond to predictions obtained using $D_{avg}$
for $\theta=29\degree$ instead of $25\degree$. The discrepancy from the DEM
simulation data (symbols) is evident. This mismatch confirms that using the
average value of diffusivity at one angle for predicting segregation of
mixtures flowing at another angle gives highly inaccurate results. Similarly,
using data for $D_{avg}$ corresponding to $\theta=25\degree$ to predict
segregation for $\theta=29\degree$ case also leads to highly inaccurate
predictions (figure S1b). The difference is attributed to the fact that the
diffusivity values at $\theta=29\degree$ are nearly three times larger than
$\theta=25\degree$ case. This indicates that accounting for the variation of
the diffusivity with the shear rate using the relation
$D=b\dot{\gamma}d^{2}_{mix}$ should be preferred as it captures the influence
of the inclination angle on the diffusivity as well.