3.1. Segregation model

Let us consider a multicomponent mixture consisting of $N$ species differing only in density. The concentration of the $i\mathrm{th}$ species in the mixture is $f_i$ with $\sum _{i=1}^N f_i = 1$. The expression for segregation velocity by considering the buoyancy, gravity and drag forces acting on a single intruder of the $i\mathrm{th}$ species in a multicomponent mixture using the particle force-based theory of Sahu et al. (Reference Sahu, Kumawat, Agrawal and Tripathi2023) is given as

(3.6) \begin{equation} v_{i} = -\dfrac {g\cos \theta }{c\pi \eta d}{\sum _{j=1}^{N} \left(m_{i} - m_{j}\right) f_{j} }, \end{equation}

where $f_j$ is concentration of the $j\mathrm{th}$ species in mixture. Here, $m_i$ and $m_j$ are masses of the $i\mathrm{th}$ and $j\mathrm{th}$ species in the mixture, and $\eta$ is the mixture viscosity which is obtained as the ratio of the magnitude of the shear stress to the shear rate i.e. $|\tau _{yx}|/|\dot \gamma |$. Stokes’ drag coefficient $c$ at different inclination angles is taken from the simulation study by Tripathi & Khakhar (Reference Tripathi and Khakhar2013). The value of $c$ lies in the range of $2.5$–$3.7$ for dense flows with solid fractions $\phi$ ranging between $0.5\ \mathrm{ and }\ 0.59$. Equation (3.6) states that the segregation velocity of any species at any location and time depends on the local concentration of all other species and the mass difference of the species with all other species. The mixture velocity in $y$ direction is calculated as $v_{mix} = \sum _{i=1}^{N} f_i v_i$. Using (3.6) for $v_i$, we find that $\sum _{i = 1}^{N} f_i v_i = 0$. Thus, the mixture velocity in the vertical $y$ direction remains zero for multicomponent mixtures as well.

Substituting (3.6) in the segregation–diffusion equation (3.5), we get $N$ partial differential equations as follows:

(3.7) \begin{equation} \frac {\partial f_i}{\partial t} = \frac {\partial }{\partial y } \left ( \dfrac {g\cos \theta }{c\pi \eta d}{\sum _{j=1}^{N} \left(m_{i} - m_{j}\right) f_{j} } f_i + b \dot \gamma d^2 \frac {\partial f_i}{\partial y}\right ); \,\,\, i\in [1:N]. \end{equation}

To get the concentration profiles of $N$ species in mixture, we need to solve $N-1$ partial differential equations simultaneously. The concentration of the $N\mathrm{th}$ species can be obtained using the identity $\sum _{i=1}^{N}f_i=1$. For example, in the case of ternary mixtures, we need to solve only two partial differential equations for any two (e.g. high- and low-density) species. The concentration of the third species (e.g. medium-density species) can be obtained as $f_M = 1 - f_L - f_H$. The initial and boundary conditions needed to solve (3.7) are given as follows:

(3.8) \begin{align}f_{i}(y,0) = f_{i,ini}(y); \,\,\, i\in [1:N -1],\nonumber\\ J_{iy}^{S}(0,t) + J_{iy}^{D}(0,t) = 0; \,\,\, i\in [1:N - 1],\nonumber\\ J_{iy}^{S}(h,t) + J_{iy}^{D}(h,t) = 0; \,\,\, i\in [1:N - 1].\end{align}

Here, $f_{i,ini}(y)$ is the initial concentration profile of the $i\mathrm{th}$ species. In the case of starting from a well-mixed configuration, $f_{i,ini}$ is equal to the total composition of that species in mixture i.e. $f^T_i$. For the cases when the mixture is nearly segregated, $f_{i,ini}(y)$ is given by a step function. Note that the initial concentration profiles of all the $N-1$ species need to be provided at $t=0$. To enable comparison with our DEM simulation results, the initial concentration profile is approximated by fitting a sigmoid function to the concentration profile obtained from the DEM simulation data. Similarly, the boundary conditions (zero flux at the free surface and base) also need to be provided for all the $N-1$ species. The boundary conditions correspond to no net mass flow in the $y$ direction at the free surface ($y=h$) and the base ($y=0$), and state that the sum of segregation and diffusion fluxes will be zero at extreme values of $y$.

3.2. Momentum balance equations and mixture rheological model

The convection–diffusion–segregation equation (3.7) for mixtures incorporates the influence of the local shear rate and local viscosity of mixtures on the concentration profile. The viscosity of the flowing granular medium depends on the local shear rate and the local pressure. Hence, the knowledge of the flow kinematics and pressure is needed to predict the species concentrations. For this, we solve the momentum balance equations for the unidirectional, fully developed flow (in the $x$ direction) over an inclined surface at an inclination angle $\theta$. The simplified momentum balance equations in the $x$ and $y$ directions for this case reduce to

(3.9) \begin{align} \rho _b \frac {\partial v_x }{\partial t} = \rho _b g \sin \theta - \frac {\partial \tau _{yx}}{\partial y},\end{align}

(3.10) \begin{align} 0 = - \frac {\partial \sigma _{yy}}{\partial y} - \rho _b g \cos \theta .\end{align}

Here, $v_x$ is the velocity in the $x$ direction, $\rho _b$ is the bulk density and $\tau _{yx}$ is the shear stress. The normal stress $\sigma _{yy}$ at any location $y$ is obtained by integrating (3.10) with the boundary condition $\sigma _{yy} = 0$ at $y = h$. We also account for the anisotropy of stress tensor due to non-zero normal stress differences in the case of granular flow (Tripathi & Khakhar Reference Tripathi and Khakhar2011) by means of the parameter $a$ and obtain the expression for the pressure:

(3.11) \begin{equation} P(y,t) = g \cos \theta (1 - a \tan \theta )\int _{y}^{h} \rho _b(y,t) {\rm d}y. \end{equation}

We follow the approach of Sahu et al. (Reference Sahu, Kumawat, Agrawal and Tripathi2023) and account for the dependence of the stress anisotropy parameter upon the inertial number as $a = (0.35 - 0.27 I_{mix})/3$. In agreement with Sahu et al. (Reference Sahu, Kumawat, Agrawal and Tripathi2023), we also find that using this relation leads to better predictions of the mixture velocity. However, the concentration profiles and the centre of mass variation with time are found to be practically the same as that obtained by using a constant value of the stress anisotropy parameter used by Tripathi & Khakhar (Reference Tripathi and Khakhar2011).

The bulk density of the mixture is given by $\rho _b = \phi \rho _{mix}$, where $\rho _{mix}(y,t)=\sum _{i=1}^{N} \rho _i f_i(y,t)$ is the local volume average density of the mixture. Here, $\rho _i$ is the density and $f_i$ is the local concentration of species $i$ in the mixture. The local packing fraction $\phi$ is obtained using the dilatancy law with the generalised inertial number for the granular mixtures (Tripathi & Khakhar Reference Tripathi and Khakhar2011)

(3.12) \begin{equation} \phi (y,t)=\phi _{max}- \beta I_{mix}. \end{equation}

Here, $\phi _{max}$ and $\beta$ are the dilatancy law parameters obtained from DEM simulation data. The expression of inertial number for granular mixture having identical size ($d_{mix} = d$) and different density particles is given by $ I_{mix}=|\dot \gamma | d /\sqrt {P/\rho _{mix}}$. Additionally, $\dot \gamma ={\rm d}v_x/{\rm d}y$ is the local shear rate. The shear stress ($\tau _{yx}$) and pressure ($P$) relate to each other by means of the inertial number dependent effective friction coefficient $\mu (I_{mix}) = |\tau _{yx}|/ P$. The JFP model by Jop, Forterre & Pouliquen (Reference Jop, Forterre and Pouliquen2006) empirically relates the variation of $\mu$ with $I$ as

(3.13) \begin{equation} \mu \left(I_{mix}\right) = \mu _{s}+\frac {\mu _{m}-\mu _{s}}{1+I_{0}/I_{mix}}, \end{equation}

where, $\mu _s$, $\mu _m$ and $I_0$ are the rheological model parameters. The values of these parameters are taken from Tripathi & Khakhar (Reference Tripathi and Khakhar2013); Sahu et al. (Reference Sahu, Kumawat, Agrawal and Tripathi2023) and are reported in table 1. Using the expression for the bulk density $\rho _b$ and shear stress $\tau _{yx}$, the momentum balance (3.9) reduces to

(3.14) \begin{equation} \phi \rho _{mix} \frac {\partial v_x }{\partial t} = \phi \rho _{mix} g \sin \theta + \frac {\partial }{\partial y} \left(\mu \left(I_{mix}\right) P\right). \end{equation}

Table 1.

Values of the rheological model parameters (same as in Tripathi & Khakhar (Reference Tripathi and Khakhar2011)) used in this study.

In our simulations, the flow is started from an initial condition of zero velocity and we incorporate a rough, bumpy base to ensure the no-slip boundary condition at the base. Hence, we use the following initial and boundary conditions to solve the partial differential equation (3.14):

(3.15) \begin{align} IC: v_{x}(y,0) = 0,\nonumber\\ BC1: v_{x}(0,t) = 0,\nonumber\\ BC2: \tau _{yx}(h,t) =0. \end{align}

Further, the species concentration and velocity fields are obtained by solving the segregation–diffusion equation (3.7) and momentum balance equation (3.14) simultaneously using the PDEPE solver along with the initial and boundary conditions given in (3.8) and (3.15), respectively. The PDEPE solver uses numerical discretisation to approximate the derivatives by dividing the domain into finite spatial and temporal grids. A total of $N = 200$ spatial grids along the $y$ direction are used, which corresponds to a grid size of approximately one-eighth of the particle diameter. We choose the time step $\Delta t = 0.1$ dimensionless time units. This time step corresponds to approximately $10^{-3}\ \mathrm{s}$ for a 1 mm particle size for earth’s gravitational acceleration. To facilitate comparison with DEM simulations, the instantaneous properties from the continuum model predictions are also obtained by averaging $20$ snapshots over a time period of $2$ time units. The detailed description for solving these equations using the MATLAB PDEPE solver is given in the supplementary material, SM1.

4.1. Comparison of DEM simulations with continuum model predictions

Figure 2.

(a) Comparison of the instantaneous profiles for an equal composition binary mixture having density ratio $\rho = 2.0$. (a) Concentration $f_L$, (b) pressure $P$, (c) shear stress $\tau _{yx}$, (d) mixture velocity $v_x$, (e) inertial number $I_{mix}$ and (f) solids fraction $\phi$. Symbols represent the DEM data and solid lines represent the continuum predictions with each colour corresponding to a different time instant. The flow starts from the initial well-mixed configuration at inclination angle $\theta = 25^\circ$.

We first report the results for an equal composition binary mixture with density ratio $\rho = 2.0$. The flow is started from an initially well-mixed state at an inclination angle of $\theta = 25^{\circ}$. Figure 2 shows the instantaneous concentration profiles along with other flow properties of the binary mixture at different time instants. Different symbols represent the DEM data at different times, while solid lines represent the continuum model predictions. The DEM data as well as continuum predictions at a particular instant are represented by a specific colour for the ease of comparison. Figure 2(a) shows the concentration profile of light species $f_L$ in the mixture. Since the flow begins from an initially well-mixed state i.e. $f_L(y,0) = 0.5$, the continuum model predictions for the concentration profile of light species at time $t = 2.0$ remains almost constant (black line) with $f_L \approx 0.50$ for all $y$ values following the DEM data (black circles) closely. As time proceeds, the light particles move upwards and heavy particles move downwards leading to increase (decrease) in the $f_L$ values near the free surface (base) which is clearly observable from the DEM data at $t = 500,1000,2000$ and $6000$. The predicted concentration profiles also depict the same trend and follow the DEM data very well. We observe that the concentration profiles become steady after $t=6000$ indicating that steady-state segregation is achieved. Figures 2(b)–2(f) show the pressure, shear stress, velocity, inertial number and packing fraction profiles at different time instants. Figures 2(b) and 2(c) demonstrate that the pressure and shear stress profiles exhibit relatively small change over time due to segregation.

Figure 2(d) shows the variation of velocity in the flow direction ($v_x$) with the distance from the base. The predicted velocity profiles for $50$ and $100$ time units are found to be reasonably higher compared with the DEM data. The deviation in the continuum model predictions at initial times can also be observed in the variation of the inertial number, shown in figure 2(e). The origin of these differences lies in the inertial number based $\mu {-}I$ rheological model. Figure S1 in the supplementary material shows that the fits for $\mu (I)$ and $\phi (I)$ significantly deviate form the DEM data for low inertial numbers. We also note that the validity of the inertial number based rheological description becomes questionable at very low values of the inertial numbers (Andreotti, Forterre & Pouliquen Reference Andreotti, Forterre and Pouliquen2013). After this initial time period, the predicted velocity profiles match very well with the DEM data. As expected, the instantaneous velocity at $t=500$ time units is found to be higher compared with that at $t=200$ as the flow evolves. However, the steady-state velocity at $t=6000$ is found to be lower than the instantaneous velocity at $t=500$ time units. This very interesting feature is accurately predicted by the continuum model as well. We discuss this in more detail later while discussing the results for figure 3. However, the prediction for later times agree reasonably well with DEM simulations. The variation of solids fraction ($\phi$) at different times is shown in figure 2(f) and the predicted profiles match well with the DEM data.

Figure 3.

Comparison of evolution of concentration of light species ($f_L$) with time for an equal composition binary mixture with density ratio $ \rho = 2.0$. (a) DEM data and (b) continuum simulation. Variation of (c) the species centre of mass and (d) average mixture velocity with time. Symbols represent the DEM data, while solid lines represent the continuum model predictions.

Figure 4.

Variation of average velocity with time for monodisperse mixture. Red squares represent DEM data, while the red line represents theory data for monodisperse mixture. Purple circles/lines represent the data for the case where we increased the mass of randomly selected $50\,\%$ particles by a factor of two, starting from the steady-state positions and velocities of the particles corresponding to monodisperse case after $500$ time units. Inset shows the data for the monodisperse case up to $500$ time units and the sudden mass change case for $t \geqslant 500$. Black dashed line represents model predictions for a binary mixture shown by the solid line in figure 3(d).

Variation of the light species concentration across the layer in figure 2(a) could be shown only for a few specific time instants. To get a better understanding of evolution of the concentration of light particles across both spatial as well as temporal coordinates, we plot the light species concentration using a colour map across the space–time plane in figures 3(a) and 3(b). We report the data for the case shown in figure 2, i.e. density ratio $\rho = 2.0$ and inclination angle $\theta = 25^{\circ}$. Figure 3(a) shows the concentration colour map of light species obtained from the DEM simulations, while the predictions from the theory are shown in figure 3(b). In this representation, yellow denotes $f_L = 1$, i.e. pure light species, whereas black corresponds to $f_L = 0$, i.e. pure heavy species. The spatial distribution of light and heavy species across flowing layer height using the DEM data (figure 3a) matches well with the predictions (figure 3b) throughout the space–time plane. The concentration using DEM data in figure 3(a) exhibits more discrete variations compared with the continuum model, which can be attributed to the large grid size (equal to $d$) used in the case of DEM. The rough nature of contour lines for DEM data in figure 3(a) compared with the smooth contour lines of continuum simulations in figure 3(b) can also be attributed to the same reason. The black, dark grey, light grey and white colour contour lines correspond to $f_L = 0.8,0.6,0.4$ and $0.2$, respectively. The white region near the top represents the region beyond the free surface.

For a more quantitative comparison of the theory with DEM simulations, we report the variation of the centre of mass of the species $y_{com}$ with time $t$. Figure 3(c) shows the variation of $y_{com}$ for both light (blue) and heavy (green) species in the mixture. Symbols represent the DEM data, while solid lines represent the continuum model predictions. Initially at time $t = 0$, $y_{com}$ is identical for both species since the flow is started from a well-mixed configuration. The centre of mass of the heavy species decreases while that of light species starts to increase until becoming constant for $t \geqslant 5000$ and the continuum predictions describe the DEM data very well. Figure 3(d) shows the variation of average mixture velocity along the flow direction ($v_{avg}$) with time. While a rapid increase in $v_{avg}$ is observable at early times, the flow velocity reaches a maximum around $500$ time units and then slowly decreases afterwards. This non-monotonic variation of the average flow velocity with time is very well captured by the continuum model as well. This demonstrates the capability of the one-dimensional continuum model which is able to capture the intricate details and subtle nuances of the flow and segregation features observed in DEM simulations. The non-monotonic variation of $v_{avg}$ with time occurs due to the inter-coupling of the rheology and segregation. This peculiar behaviour can be better understood by considering two different flow situations that minimise the effects of the inter-coupling between rheology and segregation. In the first case, we remove the segregation altogether and consider the average velocity variation with time for a monodisperse granular material starting from rest. We plot $v_{avg}$ for monodisperse material in figure 4 using red squares (DEM data) and solid red lines (model). For this case, $v_{avg}$ increases with time and reaches a plateau around $500$ time units. This rising behaviour at early times is similar to the early-time behaviour observed in figure 3(d), confirming that the early rise of $v_{avg}$ with time occurs due to the acceleration of the layer. To simulate a situation where the inter-coupling of segregation with rheology is minimal, we use the following protocol. Starting from the steady-state positions and velocities of the particles corresponding to the monodisperse case, we randomly selected $50\,\%$ particles in the layer and increased their mass by a factor of two. This protocol provides us with a well-mixed mixture with an initial velocity equal to that for the steady flow of monodisperse granular material of the same density. Using this combination of the initial species composition and velocity of the particles, we performed the DEM simulations as well as continuum simulations and observed the variation of the average flow velocity and species centre of mass. The results from these DEM simulations are shown using purple circles and the continuum model predictions are shown using solid lines of the same colour in figure 4. The $v_{avg}$ for this case keeps decreasing with time. The inset in figure 4 shows that figure 3(d) can be approximately reconstructed by combining the monodisperse data (red symbols) of figure 4 up to $500$ time units with the data for the sudden change in the particle density case for the fully developed flow (purple symbols). The black dashed line in the inset corresponds to the model predictions shown in figure 3(d). The excellent agreement of the black dashed line with the data confirms that the initial rise in the average velocity occurs primarily due to acceleration of the layer. Further, the slow decreasing behaviour of $v_{avg}$ in figure 3(d) occurs due to the slow evolution of segregation during the flow. Due to the dependence of the mixture viscosity on local concentration in the inertial number based rheological model, a complex interplay of the segregation and rheology during the flow occurs. Since the segregation occurs over a large time scale, the viscosity of the granular mixture also keeps changing and leads to slow decrease of the average velocity. The proposed continuum model accounts for this inter-coupled nature of rheological and segregation model, and hence, this interesting non-monotonic behaviour of average velocity is captured very accurately by the model. It is important to note that in the absence of the inter-coupling of the rheology with segregation, such flow features cannot be captured.

Figures 2 and 3 confirm that the particle force-based theory is successfully able to predict the time-dependent segregation for equal composition binary mixtures. We next report the results for two different mixture compositions. As before, the flow is started from an initial well-mixed state at an inclination angle $\theta = 25^{\circ}$ and the density ratio of the heavy to light species is $\rho = 2.0$. Figure 5 shows the time variation of the centre of mass position of light and heavy species for two different composition mixtures. Empty symbols correspond to DEM data for a mixture having $20\,\%$ light particles ($f^T_L = 0.20$), while filled symbols correspond to DEM data for mixture having $80\,\%$ light particles ($f^T_L = 0.80$). Starting from a uniform concentration of $f_L = f^T_L$, the concentration of light species keeps decreasing near the base and keeps increasing near the top as time progresses. For the case of $f^T_L = 0.20$, the centre of mass for light species (blue colour) rises from $y_{com}\approx 12$ to $y_{com}\approx 22$, whereas for $f^T_L = 0.80$, it reaches only up to to $y_{com}\approx 15$. Figure 5 shows that the model predictions (dashed lines for $f^T_L = 0.20$ and solid lines for $f^T_L = 0.80$) are in reasonably good agreement with DEM data (symbols). Figure S2 (refer to § SM3 of the supplementary material) also shows that the instantaneous concentration profiles and colour maps of light species using the continuum model agree well with the DEM data.

Figure 5.

Time evolution of centre of mass ($y_{com}$) for two different composition mixture for density ratio $\rho = 2.0$ at inclination angle $\theta = 25^\circ$. Empty symbols along with the dashed lines represent the data for a mixture with $20\,\%$ light species ($f^T_L = 0.2$), while filled symbols along with the solid lines correspond to the $f^T_L = 0.8$ case.

Figure 6.

Time variation of (a) centre of mass ($y_{com}$) and (b) average velocity ($v_{avg}$) for an equal composition binary mixture with density ratio $\rho = 3.0$ flowing at $\theta = 25^\circ$ and $\theta = 29^\circ$.

Next, we explore whether the proposed theory is able to capture the influence of the inclination angle on the mixture flow and segregation. Figure 6 shows the results for an equal composition binary mixture with density ratio $\rho = 3.0$. As before, the data are reported for a mixture starting from well-mixed initial configuration. Figure 6(a) shows time variation of the species centre of mass for both heavy (green) and light (blue) species flowing at inclination angles $\theta = 25^{\circ}$ and $\theta = 29^{\circ}$. Here, the empty symbols along with the dashed lines represent the DEM data and model predictions for inclination angle $\theta = 25^{\circ}$. The filled symbols and solid lines correspond to the DEM data and the prediction of the model for $\theta = 29^{\circ}$. The corresponding average mixture velocity is shown in figure 6(b). As before, the average velocity for both inclinations rises quickly, achieves a maximum value and then starts to slowly decrease due to the inter-coupling of rheology and segregation. The steady-state average mixture velocity at inclination angle $\theta = 29^{\circ}$ (red) is more than twice of the steady-state average velocity at $\theta = 25^{\circ}$ (black). Continuum model predictions (shown using lines) are in very good agreement with DEM simulation data (symbols) for both these inclination angles. We have also performed simulations for different binary mixtures with density ratios $\rho = 1.5, 2.0, 3.0$ and $5.0$ over a wide range of compositions and inclination angles. While these results are not reported here to avoid repetition, we find that the continuum model predictions for all these different cases are found to be in very good agreement with DEM simulations. Hence, we conclude that the particle force-based theory for a binary mixture is successfully able to predict the evolution of segregation over a wide range of concentrations, inclination angles and density ratios. In addition, we find that the continuum model is able to account for the dependence of the DEM simulation parameters (§ SM4 of the supplementary material). The relative importance of various continuum model parameters has also been investigated and is reported in § SM5 of the supplementary material.

4.2. Comparison with kinetic theory (Larcher Jenkins 2015)

Figure 7.

Time evolution of centre of mass ($y_{com}$) for an equal composition binary mixture having density ratio $\rho = 2.0$ at inclination angle $\theta = 25^\circ$ for two different initial configurations: (a) heavy-near-base and (b) well-mixed configuration. Symbols represent the DEM data, solid lines represent the predictions using the particle force-based segregation model, while dashed lines represent predictions using the kinetic theory model of Larcher & Jenkins (Reference Larcher and Jenkins2015). The blue colour corresponds to the light species and the green colour corresponds to the heavy species.

Larcher & Jenkins (Reference Larcher and Jenkins2015) have proposed a kinetic theory-based model to predict the evolution of segregation of binary granular mixtures in the dense flow regime for small size and density differences. In this section, we use their kinetic theory-based model to predict density segregation in a periodic chute and compare the predictions of their model with the present work. We solve the partial differential equation given by Larcher & Jenkins (Reference Larcher and Jenkins2015) (($3.10$) of their work) along with the boundary conditions (($4.3$) of their work) using the MATLAB PDEPE solver. For these predictions, we have used the same model parameters as used by these authors. For a meaningful comparison, the flowing layer thickness has been chosen equal to that used in the present work. Using their model, we predict the evolution of segregation for two different initial configurations of the mixture, namely heavy-near-base configuration and well-mixed configuration. Figure 7 shows the results for an equal composition binary mixture having density ratio $\rho = 2.0$ flowing at inclination angle $\theta = 25^\circ$. Figure 7(a) shows the time variation of the centre of mass of both heavy (green) and light (blue) species starting from the heavy-near-base configuration. Solid lines represent the predictions using the particle force-based model, while dashed lines represent predictions using the kinetic theory model of Larcher & Jenkins (Reference Larcher and Jenkins2015). Both models capture the evolution of segregation very well for this initial segregated state. Figure 7(b) shows the time variation of the centre of mass of both heavy (green) and light (blue) species starting from the well-mixed configuration. Predictions using kinetic theory (dashed lines) significantly deviate from the DEM data for both the species. Specifically, the time scale for segregation using the kinetic theory model for this initial state is much larger compared with that observed in the DEM simulation. While the DEM simulation shows that the centre of mass of the species becomes nearly constant after approximately 4000 time units, kinetic theory predictions for the species centre of mass do not reach a constant value even after 10 000 time units. Similar discrepancies are reported by the authors in the case of size segregation (figure 1 of Larcher & Jenkins Reference Larcher and Jenkins2015) for the well-mixed configuration of different size mixtures. The authors do acknowledge this limitation of their model prediction and report that the rate of evolution of segregation according to their theory is slower as compared with DEM simulations and experimental data available in the literature. While the authors suggest that adjustment of the kinetic theory model parameters may be helpful in predicting this rate of evolution in better agreement by DEM simulations, we were unable to observe good agreement of the time evolution and the steady-state values of the centre of mass of the species with the DEM simulations despite varying the parameters in the kinetic theory over a wide range.

6.1. Heavy-near-base configuration

Figure 10.

(a) Time variation of species centre of mass ($y_{com}$) starting from the initial concentration profile ($t = 0$) shown in the inset. Variation of concentration of light species across the layer height with time for a ternary mixture with $\rho _H: \rho _M: \rho _L = 3:2:1$ starting from the heavy-near-base initial configuration at an inclination angle $\theta = 25^\circ$ from (b) DEM simulations and (c) continuum model predictions.

We first consider mixtures with high-density species located near bottom of the layer. The results for the binary mixture case are shown in figure S6 of the supplementary material and we present the results only for the ternary mixture case here. Figures 10(a)–10(c) show the results for an equal composition ternary mixture having density ratios $\rho _H: \rho _M: \rho _L = 3:2:1$ and flowing at $\theta = 25^\circ$. The flow starts from a heavy-near-base configuration in which high-density particles are kept near the bottom and light-density particles are kept near the surface with the medium-density particles sandwiched in between the two. Time variation of $y_{com}$ for all the three species are shown in figure 10(a). Symbols represent the DEM data and solid lines represent continuum predictions. A very good match between the two is observed. Figures 10(b) and 10(c) show concentration colour maps of the light species obtained from the DEM and continuum simulations, respectively. Since the initial concentration of light species is present only near the upper one third of the layer in this case, the yellow region with $f_L = 1$ is observed near the top portion of the layer at early times along with narrow regions of red and purple (corresponding to the linear part of the concentration data shown in the inset of figure 10a). As time increases, the span of the yellow region across the layer appears to decrease, while that of the red and purple regions increase due to the diffusional mixing. This diffusional mixing observed in the concentration colour maps obtained from the DEM simulation is very well captured by the continuum model as well. The white, light grey, dark grey and black colour contour lines correspond to $f_L =0.2, 0.4, 0.6$ and $0.8$, respectively. Again, the agreement between DEM simulations and continuum predictions in figures 10(b) and 10(c) is found to be pretty good. Similar results are observed for heavy $f_H$ and medium species $f_M$ (not shown here) as well.

Figure 11.

(a) DEM snapshot for an equal composition binary mixture of density ratio ($\rho = 2.0$) for the light-near-base configuration. Green spheres represent the high-density particles, while blue spheres represent the low-density particles. The orange arrow indicates the direction of flow. (b) Time evolution of $y_{com}$ for both light (blue) and heavy (green) species using DEM data (symbol) and model predictions (solid lines). Colour map of light species concentration in the flowing layer over time range using (c) DEM and (d) continuum model at inclination angle $\theta = 25^{\circ}$.

6.2. Light-near-base configuration

We next report the results for the cases where the heavy species is near the free surface and the light species is near the base at the starting of the simulation. Figure 11(a) shows the DEM snapshot of the initial configuration for an equal composition binary mixture having density ratio $\rho = 2.0$. The flow occurs at $\theta = 25^\circ$ and the variation of $y_{com}$ of both light and heavy species with time is shown in figure 11(b). The centre of masses of both the species rise for a brief initial period (up to $250$ time units) due to the layer dilation. After that, a sharp rise in the $y_{com}$ of light species (blue circles) and sharp decline in the $y_{com}$ of heavy species (green squares) is observed from the DEM simulations (symbols) during the next $250$ time units. After $t = 500$ time units, $y_{com}$ of both the species change gradually. In contrast to all other cases considered before, the predicted centre of mass from the continuum model (shown using solid lines) deviates significantly from the DEM data. It is evident that the model fails to capture the centre of mass behaviour observed in the case of DEM simulations.

The concentration colour map of the light species, as obtained from the DEM simulations, is shown in figure 11(c). The predictions from the continuum model are shown in figure 11(d). Lines in the colourmaps correspond to contour lines of different $f_L$ values as before. Evidently, the predicted behaviour from the model is qualitatively different from that observed in the DEM simulations. The concentration colour map from the DEM simulations (figure 11c) show that the yellow region (having pure light species) in the lower part of the layer disappears around $500$ time units and very quickly appears near the top part of the layer. In addition, the concentration of heavy species in the lower part of the layer remains high (purple region) for all times larger than $750$ time units except for a very brief duration, (shown by the red coloured region in the lower part of the layer), where both the species appear to be well-mixed. The continuum model predictions in figure 11(d), however, show a gradual decrease in the light species concentration in the lower part of the layer and high concentration of heavy species (purple region) is observable only after $3000$ time units. Similarly significant qualitative differences are observable in the upper part of the layer as well where the contour line for a given colour corresponding to a particular $f_L$ shows very different trend in DEM simulation as compared with the continuum prediction.

We further investigate this configuration for a ternary mixture, in which particles with low density are concentrated near the base, medium-density particles are positioned in the middle and particles with high density are located at the top. Figure 12(a) shows the DEM snapshot of such an initial configuration where green, red and blue spheres represent the high-, medium- and low-density particles, respectively, and black spheres represent the static particles that form the rough, bumpy base. Figure 12(b) shows the variation of $y_{com}$ for all three species in the mixture. As before, symbols represent the DEM data and lines correspond to predictions obtained from continuum model. As in the case of a binary mixture, DEM data show a very quick change in the centre of mass of the heavy (green) and light (blue) species, while the theoretical lines show a gradual variation. Evidently, the quick segregation observed in the initial $500$ time units is not captured by the model. Visible qualitative differences are observable in the concentration colour maps of the light species obtained from the DEM simulations (figure 12c) and the continuum model (figure 12d). Figures 12(e) and 12(f) show the concentration colour map for the heavy species and notable qualitative differences are observable between the model and the DEM simulations. The contour lines for the DEM and continuum predictions show very different trends for this initial configuration in the case of a ternary mixture as well. Similar differences are also observed in the concentration colour maps of medium-density species as well (not shown here).

Figure 12.

DEM snapshot of a ternary mixture having density ratio $\rho _H: \rho _M: \rho _L = 3:2:1$ for the light-near-base initial configuration at inclination angle $\theta = 25^\circ$. Green, red and blue particles represent the high-, medium- and low-density particles, respectively. Black particles represent the rough bumpy base of thickness $1.2d$. The orange arrow indicates the direction of flow. (b) Variation of $y_{com}$ with time for all species. Colour map for the concentration of light species using (c) DEM and (d) continuum model. (e) and (f) The same for the heavy species.