Comparing and contrasting sizebased particle segregation models
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Abstract
Over the last 12 years, numerous new theoretical continuum models have been formulated to predict particle segregation in the sizebased bidisperse granular flows over inclined channels. Despite their presence, to our knowledge, no attempts have been made to compare and contrast the fundamental basis upon which these continuum models have been formulated. In this paper, firstly, we aim to illustrate the difference in these models including the incompatible nomenclature which impedes direct comparison. Secondly, we utilise (i) our robust and efficient inhouse particle solver MercuryDPM, and (ii) our accurate micro–macro (discrete to continuum) mapping tool called coarsegraining, to compare several proposed models. Through our investigation involving sizebidisperse mixtures, we find that (i) the proposed total partial stress fraction expressions do not match the results obtained from our simulation, and (ii) the kinetic partial stress fraction dominates over the total partial stress fraction and the contact partial stress fraction. However, the proposed theoretical total stress fraction expressions do capture the kinetic partial stress fraction profile, obtained from simulations, very well, thus possibly highlighting the reason why mixture theory segregation models work for inclined channel flows. However, further investigation is required to strengthen the basis upon which the existing mixture theory segregation models are built upon.
Keywords
Micro–Macro mapping Coarsegraining Granular media Particle segregation Mixture theory Discrete particle simulations1 Introduction
Granular materials in nature [10, 14] and industry [14] often comprise highly polydisperse particle mixtures. The constituents of these mixtures can vary in size, density, inelasticity, shape, surface roughness, etc. When such polydisperse mixtures are subjected to external forces such as shaking, stirring or shearing [10], these mixtures often segregate, leading to complex pattern formations such as particle segregation. Several factors have been reported to be responsible for segregation or demixing in polydisperse mixtures, where individual studies confirm the influences of varying the size [48], density [40], inelasticity [4], shape [30] and surface roughness [43]. However, in rapid freesurface flows over inclined channels, the differences in size and density are the dominant factors [2, 5, 6] leading to particle segregation.
Kinetic sieving [9] is the dominant mechanism in dense granular flows. Although it is an easytocomprehend mechanism, its effects can be mindboggling. In order to illustrate the idea of kinetic sieving, let us consider a sizebidisperse granular mixture flowing down an inclined channel. As the flow progresses, fluctuations in the local pore space cause particles to fall under gravity into the space/gaps that are created beneath them. As a result, smallersized particles fall easily into these gaps leading to their gradual percolation towards the base of the flow (i.e. in the direction of gravity). Simultaneously, force imbalances lever/squeeze particles towards the surface; this process was named as squeeze expulsion by Savage and Lun [31]. The combination of kinetic sieving and squeeze expulsion results in a net migration of large particles upwards and small particles towards the base. As a result, this simple mechanism results in stratified layers, which one terms as particle segregation.
In 2011, Fan and Hill [11] proposed an alternative kineticstressdriven mechanism for segregation. The model was originally derived for vertical chutes; however, it was later extended to include the gravitydriven mechanism [20, 21] and then applied to mixture flows over inclined planes and in rotating drums. Their model is very similar to the gravitydriven models and still uses the ideas of kinetic sieving; however, it is driven by gradients in kinetic stress rather than lithostatic pressure as in the case of the gravitydriven mixture theory models. In the kineticstressdriven mechanism, all particles are squeezed away from regions of higher fluctuation energy. During this process, smaller particles filter through the matrix of other particles, analogous to the gravitydriven voidfilling mechanism, resulting in a net migration of small particles towards regions of lower fluctuation kinetic energy. Previously, WindowsYule et al. [50] experimentally investigated the competition between gravitydriven, kineticstressdriven, and other segregation mechanisms in axially nonuniform drums. Here, we consider a simpler scenario of sizebidisperse mixtures rapidly flowing down an inclined plane.
In rapid freesurface flows, opposing kinetic sieving is diffusive remixing, which causes the random motion of particles as they collide and shear over each other [22]. Based on the relative strength of these competing mechanisms, the mixture strongly or weakly segregates; the relative strength is captured by the segregation Peclet number [18]. However, both experiments [48] and particle simulations [38] have shown that, in rapid chute flows, the effect of diffusive remixing is around 10 % of the strength of segregation.
Apart from kinetic sieving, which is a purely sizebased effect, buoyancy effects due to differences in particle density also play a major role in particle segregation [23]. For bidisperse mixtures, varying in size and density, experiments [13] and particle simulations [41] indicate that a balance between the two driving mechanisms is possible, i.e. kinetic sieving and buoyancy effects, which in turn can keep the mixture homogeneously mixed.
Although complete understanding of the dynamics of segregation is beyond the scope of this paper, it becomes increasingly difficult as one allows more particle properties to be varying, i.e. size, density, shape, roughness, etc. In order to carry out the comparison between the existing particle segregation continuum models, we focus on the leadingorder effect of bidisperse mixtures varying in size only. As an alternative to experiments, we employ discrete particle simulations (DPMs) [24] from which macroscopic quantities, appearing in the continuum models, are extracted. To perform this accurately, we use an appropriate micro–macro mapping technique called coarsegraining [1, 16], as this method gives continuum fields that exactly satisfy the continuum equations. The method has previously been extended to include the effects of boundaries or discontinuities [47] and more recently to unsteady polydisperse mixtures [42]. The technique has been successfully applied to investigate monodisperse shallow granular flows [46] and sizebidisperse mixtures [38, 45]. However, Weinhart et al. [45] focussed on steady bidisperse flows alone; here, we consider both steady and transient data. For our simulations, we use our inhouse opensource code MercuryDPM [36, 37, 39] which includes all the coarsegraining tools utilised in this paper.
2 Theoretical models
In this section, we present a brief overview of the existing sizebased segregation continuum models, which have been formulated in the past few decades.
A few years later, Savage and Lun [31] used statistical mechanics and information entropy theory to arrive at a segregation model from the first principles. Their model was formulated in terms of number densities and fluxes. Although the model of Savage and Lun [31] considered various functional forms for the shear rate, i.e. different downslope velocity profiles u(z), it certainly had a downside because their model predicts segregation even in the absence of gravity, which is odd given kinetic sieving is a gravitydriven process.
In 2012, Marks et al. [26] significantly extended the particle segregation continuum theories to polydisperse flows, which also allows for varying density and size effects. Besides this, they also attempt to incorporate the shearrate dependency in a consistent way. By doing so, they include the dependence of percolation velocities on spatially varying shear rate and particle sizeratio, i.e. q in (3) is a function of both shear rate (\(\dot{\gamma }\)) and particle sizeratio (\(\widehat{s}\)). Moreover, they were also the first to attempt to quantify the segregation flux, \(F(\phi )\), in terms of the real particle properties such as the particle sizeratio (\(\widehat{s}\)) itself.
New extensions to the Gray and Thornton [19] continuum framework were added, in the year 2014, by Tunuguntla et al. [41], where they make subtle fundamental changes to the basis upon which Gray and Thornton [19] model is built upon. However, this did not alter the resulting framework presented in (3). Nevertheless, both, Tunuguntla et al. [41] and Gajjar and Gray [15] presented different possible forms for segregation fluxes, \(F(\phi )\), which intended to quantify segregation in more realistic scenarios. Besides these extensions, the Gray and Thornton framework was further developed to accord for the effects due to density differences [17, 26, 41], thus enabling us to predict particle segregation in a wider range of applications.
During the same period, 2014 to present, an alternative transport equation approach is adopted by Leuptow and coworkers [12, 32, 33, 51], who use the same general framework (3) to model particle segregation in twodimensional bounded heaps, circular drums and chute flows. Similar to several extensions of mixture theory segregation models, listed above, they have also extended their models to account for polydisperse size effects [33] and density differences [51]. However, to close the model, they utilise particle simulations to determine the flow kinematics and physical parameters such as the incompressible bulk velocity field and diffusion coefficient. Similar to Marks et al. [26] and Tunuguntla et al. [41], Leuptow and coworkers [12, 32, 33, 51] also consider the percolation velocity to be dependent on the spatially varying shear rates and particle sizeratios. This approach of closing their models with particle simulations does produce relatively good results; however, it is unable to capture flow transitions that lead to stratification patterns [25, 49], which the mixture theory models are able to predict. Moreover, it should be noted that they use a binning method to extract their continuum fields, which are required to close their models. The binning method has extra degrees of freedom compared with the coarsegraining method we utilise here, namely how to split the stress between the large and small particles. Making different choices of this split has a large effect on the results, and crucially incorrect splitting of the stress can even change the directions of segregation; see the discussion in Staron and Phillips [34] for more details. For this reason, we utilise our coarsegraining method for bidiperse mixtures, which is summarised in Sect. 4.
In this paper, we focus on sizebased mixture theory segregation models and utilise particle simulations as a validation tool alone. The key idea behind mixture theory segregation models is that particle segregation is caused by a gradient in lithostatic pressure caused by gravity, whereas in the case of the FanHill model [11] segregation is caused by a gradient in kinetic stress. Thus, through the use of particle simulations and advance micro–macro tools, we scrutinise and quantify each of this mechanism and, more importantly, see how these two mechanisms play a role in the process of sizebased particle segregation. Moreover, as implied from this paper’s title, we also compare and contrast the different proposed forms of these models. The ultimate aim is to develop a theoretical model that can accurately predict particle sizesegregation, thus eliminating the need for particles simulations entirely. However, the work in this paper is only a stepping stone towards this goal.
As a first step towards our analysis, in the following section, we review the basic background theory upon which these mixture theory models are based on and propose an unified notation to make model comparisons easier, both, for us and for future research.
2.1 Mixture theory framework
Mixture theory deals with partial variables that are defined per unit volume of the mixture rather than with the intrinsic variables associated with the material, i.e. the values one would measure experimentally, such as the material density of glass or steel particles.
The basic mixture postulate states that every point in the mixture is simultaneously occupied by all constituents. Hence, at each point in space and time, there exist overlapping fields (displacements, velocities, densities) associated with different constituents.
2.2 Gravitydriven segregation
Notations utilised in the existing mixture theory segregation models
Stress fractions corresponding to small constituents. Note that we assume the model from Bridgwater et al. [3] to be a mixture theory model, for which we back compute the pressure scaling for the smaller constituent of the mixture
Model  \(f^{s*}\)  \(B^s[\phi ]\) 

Gray and Thornton [19]  \( 1  b (1\phi )\)  \(b\) (constant) 
Marks et al. [26]  \( \dfrac{1}{\phi + \hat{s}(1\phi )}\)  \( \dfrac{1  \hat{s}}{\phi + \hat{s}(1\phi )}\) 
Tunuguntla et al. [41]  \( \dfrac{1}{\phi + \hat{s}^{3}(1\phi )}\)  \(\dfrac{1  \hat{s}^3}{\phi + \hat{s}^3(1\phi )}\) 
Gajjar and Gray [15]  \(1 b A_\gamma (1\phi )(1\gamma \phi )\)  \( b A_\gamma (1\gamma \phi )\) 
Bridgwater et al. [3]  \( 1  (1\phi )^2\)  \((1\phi )\) 
Stress fractions corresponding to large constituents
Model  \(f^{l*}\)  \(B^l[\phi ]\) 

Gray and Thornton [19]  \( 1 + b \phi \)  b (constant) 
Marks et al. [26]  \(\dfrac{\hat{s}}{\phi + \hat{s}(1\phi )}\)  \(\dfrac{1  \hat{s}}{\phi + \hat{s}(1\phi )}\) 
Tunuguntla et al. [41]  \(\dfrac{\hat{s}^3}{\phi + \hat{s}^{3}(1\phi )}\)  \(\dfrac{1  \hat{s}^3}{\phi + \hat{s}^3(1\phi )}\) 
Gajjar and Gray [15]  \(1 + b A_\gamma \phi (1\gamma \phi )\)  \(b A_\gamma (1\gamma \phi )\) 
Bridgwater et al. [3]  \( 1 + \phi ^2\)  \(\phi \) 
In the following sections, we verify and compare the existing forms of scaling functions, listed in Tables 2 and 3, which subdivide the bulk pressure among the constituents. This is done by utilising information rich discrete particle simulations, which are set up as described in following section.
3 Simulation setup
Fully threedimensional (3D) discrete particle simulations (DPMs) are used, as an alternative to experiments, to investigate segregation dynamics in a sizebidisperse mixture flowing over inclined channels. The simulations are set up in our inhouse opensource particle solver, MercuryDPM [36, 37, 39].
To begin with, we consider a cuboidal box inclined at \(26^\circ \) to the horizontal and is periodic in x and ydirection. The box has dimensions \(L \times W \times H = 30 d_m \times 10 d_m \times 10 d_m\), where \(d_m\) is the mean particle diameter defined as \(d_m = \phi ^s d_s + \phi ^l d_l\). To create a rough base (bottom), we fill the box with a randomly distributed set of particles with uniform diameter \(d_m\) and simulate them until a static layer of about 12 particles thickness is produced. Then a slice of particles with centres between \(z \in [9.3, 11]d_m\) is taken and translated 11 mean particle diameters downwards, to form the rough base of the box. To ensure no flowing particles fall through the base, a solid wall is placed underneath this static layer. Once the rough base is created, the box is inclined and filled with a homogeneously mixed bidisperse mixture of particle diameters \(d_s\) and \(d_l\) and equal material densities, i.e. \(\rho ^{s*} = \rho ^{l*}\), as illustrated in Fig. 1; see Weinhart et al. [46] for more details.
In our DPM simulations, we nondimensionalise the parameters such that the mean particle diameter \(\widehat{d}_m=1\), the mean particle mass \(\widehat{m}_m=1\), the magnitude of gravity \(\widehat{g}=1\). This implies that the mean particle density \(\widehat{\rho }_m=\widehat{m}_m/\widehat{V}_m=6/\pi \) and the mean particle volume \(\widehat{V}_m=\pi (\widehat{d}_m)^3/6\). The nondimensional quantities are denoted by ‘ \(\widehat{}\) ’ . In this paper, we consider three particular bidisperse mixtures with \(\hat{s}=d_l/d_s = \{1.3,1.5,1.7\}\) without any size distribution around its particle size. Hence, the use of term perfectly in the subtitle of the paper.
Furthermore, a linear springdashpot model is used, where the spring stiffness and dissipation for each collision is chosen such that the collision/contact time \(t_c=0.005\,\sqrt{d_m/g}\) and the coefficient of restitution \(r_c=0.88\) are constant. The microscopic sliding friction coefficient is taken to be 0.5 and no rolling friction is considered. More details about the model can be found in [7, 24, 46]. Besides the contact model, we use the velocityVerlet timestepping algorithm.
Once the particle sizeratio, total number of particles and the contact model parameters are given as an input, the particles are inserted into the box with dimensions \(\widehat{V}_{box}=30 \times 10 \times H\) where H is defined as \((N_s + N_l)/300\). If the inserted particle at any position overlaps with another particle, the insertion is rejected and the insertion domain is enlarged by increasing H to \(H+0.01\) to ensure that there is enough volume for all the particles, thus leading to a loosely packed mixture initially. Once the simulation starts, the mixture compacts enough, see Fig. 1, giving the particles enough energy to initialise flow. For more details, see Weinhart et al. [46].
Given the particle simulations are setup in the above described manner, we still need to extract continuum fields to compare different forms of stress fractions, \(f^\nu \), listed in Tables 2 or 3. This is the focus of the following section, i.e. how to perform the micro–macro step accurately?
4 Micro to Macro: coarsegraining (CG)
Compared to other, simpler methods of averaging such as binning or the method of planes, the coarsegraining method has the following advantages: (i) the resulting macroscopic fields exactly satisfy the equations of continuum mechanics, even near the base of the flow, see Weinhart et al. [47], (ii) the particles are neither assumed to be spherical or rigid, (iii) the resulting fields are even valid for a single particle, as no averaging over an ensemble of particles is required, (iv) the fields are determined at every point in space, not just at the centre of averaging cells as in the case of binning and (v) in a contact between different types of particles i.e. large and small here, the stresspartition is clearly defined. However, the coarsegraining method does assume (i) each particle pair to have a single point of contact, i.e. the particles are convex in shape; (ii) that the contact area can be replaced by a contact point, implying the particles are not too soft; and (iii) that collisions are not instantaneous (i.e., particles cannot be perfectly rigid).
Considering the above advantages and assumptions, in this section, we briefly elucidate the idea behind the coarsegraining technique and, more importantly, the mapping expressions, which will be employed to extract continuum partial densities, velocities, stresses and the interaction force density (interspecies drag force) from the discrete particle simulations setup in Sect. 3. For more information regarding the technique, please see Tunuguntla et al. [42], where they not only derive the coarsegraining expressions systematically but also focus on its application in detail. More importantly, they also present a general mixturetheorybased CG framework that can be easily extended to polydisperse mixtures without any loss of generality.
4.1 Nomenclature
In the following sections, we first present the idea of coarsegraining (CG) and then list the CG expressions for the partial and bulk quantities, using the above nomenclature.
4.2 Idea behind coarsegraining
4.3 Coarsegraining expressions: novel micro–macro map
On utilising the above CG expressions, stated in Sect. 4.3, the following section focusses on extracting the continuum fields from, both, the transient and steady particle data of our sizebidisperse mixtures.
5 Applying coarsegraining to the DPM simulations
In the simulations setup in Sect. 3, the temporal derivatives \(\partial _t (\rho ^\nu )\) and \(\partial _t (\rho ^\nu \mathbf {u}^\nu )\) vanish after a short time interval \(\hat{t}\in [0,\hat{t}_e\approx 50]\), and thereafter the slow process of segregation dominates the transient flow dynamics. For the given particle sizeratio, this transient flow behaviour or the process of segregation approximately happens within the first 2000 DPM time units. For example, see Fig. 3 where for particle sizeratio \(\hat{s} = 1.3\), the vertical centres of mass of, both, large and small particles, are tracked. Therefore, we focus on the time interval before the process of particle segregation has reached a steady state, i.e. when \(\hat{t} \in [50,2000]\), where ‘\(\,\,\hat{}\,\,\)’ denotes nondimensional quantities such as \(\hat{t} = t/\sqrt{d_m/g}\). For the purpose of our investigation, particle data is stored at every 5000 simulation time steps^{5}.
As in Tunuguntla et al. [42], we use the coarsegraining expressions of Sect. 4.3 to spatially coarsegrain the particle data. For our analysis, we need continuum fields which are a function of, both, time (\(\hat{t}\)) and flow depth (\(\hat{z}\)). To do so, for a given spatial coarsegraining scale (\(\widehat{w} = w/d_m\)), we, first, spatially average the extracted continuum fields in x and ydirection, thus resulting in averaged quantities, \(\bar{\zeta }(\hat{t},\hat{z})\), as a function of both time \(\hat{t}\) and flow depth \(\hat{z}\) \(=\) \(z/d_m\). However, to construct macroscopic continuum fields in the temporal direction, we further need to average \(\bar{\zeta }(\hat{t},\hat{z})\) temporally over a time interval \(\left[ \hat{t}  \widehat{w}_t, \hat{t} + \widehat{w}_t\right] \), where \(\widehat{w}_t\) is defined as the temporal averaging scale.
6 Analysis and discussion
With coarsegrained quantities available at the transient stages of particle segregation, we initially begin by looking at the evolution of the local solid volume fraction of, both, types and typel constituents as a function of the flowdepth.
6.1 Local mass fractions
Utilising these expressions, for \(\hat{s} = \{1.3, 1.5, 1.7\}\), Fig. 4 illustrates the evolution of local mass fraction for, both, the bulk and the mixture constituents. As seen, flows segregate faster with an increase in the particle sizeratio. Moreover, as the flow segregates, a pure layer of small particles develops at the base. Right above this layer, \(\varLambda ^l\) shows an oscillating behaviour at \(t_n = [700, 2000, 3000]\) (Fig. 4), indicating that layers of large particles develop on the small particle bed. Once the flow is fully segregated, at \(t_n = 3000\), only a single layer right above the layer of pure small particles remains, see the circle highlighting this in the plots in the rightmost column of Fig. 4a–c. This might be an artefact of using monodisperse constituents, i.e. no size distribution around \(d_s\) or \(d_l\). Apart from these oscillations, \(\varLambda ^l\) increases steadily towards the freesurface, forming a layer of large particles at the freesurface.
6.2 Transient vs. steady state analysis
The stress fractions determine the amount of normal stresses to be distributed among the small and large constituents. In gravitydriven segregation models, segregation is driven by the pressure gradient which in turn is scaled by the difference between the total partial stress fraction \(f^\nu \) and the partial volume fraction \(\phi ^\nu \), i.e. \(f^{\nu }  \phi ^{\nu }\), see (19). On the other hand, in the mechanism proposed by Hill and Tan, (22) implies that it is the difference between the contact and kinetic partial stress fractions (\(f^{con,\nu }  f^{kin,\nu }\)) and (\(f^{con,\nu }  \phi ^\nu \)) which scales the strength of either of the stress gradients, \(\partial \sigma _{zz}^{kin} / \partial z\) and \(\partial \sigma _{zz}/ \partial z\). If (\(f^{con,\nu }  f^{kin,\nu }\))\(\partial \sigma ^{kin}_{zz}/ \partial z\) > (\(f^{con,\nu }  \phi ^\nu \))\(\partial \sigma _{zz}/ \partial z\), segregation is majorly driven by gradient in kinetic stress. If not, then segregation is driven by the pressure gradients, as seen in (22). This is where our dimensionless number (23) comes into picture. Moreover, on reexamining (19) and (22), we also see that the pressure gradients are scaled by two prefactors (\(f^{\nu }  \phi ^{\nu }\)) or (\(f^{con,\nu }  \phi ^{\nu }\)), respectively.
As the process of segregation progresses, \(\hat{t}_n = 700\) to 2000, the relative total and contact partial stress fractions appear to have unfurled from their complicated initial profiles to a much more structured one. The initially strong oscillations observed, near the freesurface (red circles and diamonds), in the profiles of \(f^\nu  \phi ^\nu \) and \(f^{con,\nu }  \phi ^\nu \), disappear over time and the high concentration of data points observed at intermediate volume fractions, \(0.25<\phi <0.75\) initially, also resolve over time. However, the fraction of the total and contact stress, (\(\sigma _{zz}\)) and (\(\sigma ^{con}_{zz}\)), bore by the large and small constituents is still nearly the same as its local volume fraction. Moreover, when closely observed, the profiles of relative kinetic stress fraction (\(f^{kin,\nu }  \phi ^\nu \)) and (\(f^{con,\nu }  f^{kin,\nu }\)) stay identical throughout the transient stages of segregation, i.e. the amount of kinetic stress borne by the small and large particles remains the same during the process of segregation (overlooking the points near the freesurface). Strikingly, this is also true compared with the corresponding profile in steady state at \(\hat{t}_n=3000\). Thus, based on the illustrated profiles in Figs. 5, 6, 7, it implies that in a sizebidisperse mixture with given particle sizeratio, the smaller particles support a fraction of the kinetic stress larger than their volume fraction, as \(f^{con,\nu } \approx \phi ^\nu \), thereby complementing the findings of Weinhart et al. [45] and Hill and Tan [21].
Moreover, Fig. 8 further compares the steady state profiles of relative kinetic stress fraction (\(f^{kin,\nu }  \phi ^\nu \)) and (\(f^{con,\nu }  f^{kin,\nu }\)) for increasing particle sizeratio. As the particle sizeratio increases, the smallersized constituents support larger fraction of the kinetic stress and, interestingly, we also observe that the relative kinetic stress fractions become more asymmetric with the increase in particle sizeratio. However, more detailed study is required to explain this asymmetry.
6.3 Comparison of segregation models
In the previous section, we closely looked at the relative stress fractions, also known as the prefactors in (19) and (22), computed from the simulations of a sizebidisperse mixture flowing over an inclined channel. In this section, we utilise these coarsegrained profiles and compare them with the existing theoretical forms of stress fractions that are listed in Table 2 and Table 3.
Given this, at \(\hat{t}_n = 3000\), we compared the total partial stress fraction profiles corresponding to the large constituents with the scalings listed in Table 3, see Fig. 9. As illustrated, with \(b=1\), \(\gamma =0.9\) and \(\widehat{s}=\{1.3,1.5,1.7\}\), none of the expressions for the stress fractions seem to match the profiles computed from the particle simulations, (\(f^l_{sim}\)). Note that the values of b and \(\gamma \) could be modified such the proposed functional forms are closer to the relative total partial scaling profiles. However, this would not be of much help as the relative total partial scaling profiles obtained from the simulations (\(f^l_{sim}\)), for different bidisperse mixtures, are approximately the same as those of its local volume fraction, \(\phi ^\nu \), thus implying negligible relative percolation velocity which in turn implies very weak segregation; see (19), thereby forcing us to ask a very simple question: How do these gravitydriven models even work, when none of the currently proposed scaling forms match in Fig. 9? Even if they do, which of them is relatively good at quantitatively predicting particle segregation?
6.4 The kineticstress model
Functional forms for partial intrinsic kinetic stress fractions corresponding to small constituents
Model  \(f^{kin,s*}\)  \(B^{kin,s}[\phi ]\) 

Gray and Thornton [19]  \( 1 + b \phi \)  b (constant) 
Marks et al. [26]  \(\dfrac{\hat{s}}{\phi + \hat{s}(1\phi )}\)  \(\dfrac{1  \hat{s}}{\phi + \hat{s}(1\phi )}\) 
Tunuguntla et al. [41]  \(\dfrac{\hat{s}^3}{\phi + \hat{s}^{3}(1\phi )}\)  \(\dfrac{1  \hat{s}^3}{\phi + \hat{s}^3(1\phi )}\) 
Gajjar and Gray [15]  \(1 + b A_\gamma \phi (1\gamma \phi )\)  \(b A_\gamma (1\gamma \phi )\) 
Bridgwater et al. [3]  \( 1 + \phi ^2\)  \(\phi \) 
As a result, one could say that the gravitydriven segregation models work because the suggested functional forms for total partial stress fractions capture the shape of the partial kinetic stress fraction profiles obtained from the simulations. They implicitly support the fact that smaller particles support a fraction of kinetic stress larger than their volume fractions. As seen, three out of the four suggested functional forms are able to capture the segregation dynamics, with the closest being the form suggested by Gajjar and Gray [15]. However, there is still scope for work to be carried out in this direction, where fundamental changes are needed to be made in the basis upon which the mixture models are constructed. Moreover, on a different note, it also raises the interesting question of why did the model of Tununguntla et al. best determine the zero segregation line [41] if this model is incorrectly capturing the details of the stress distribution.
7 Summary and conclusions
In this work, we have reviewed the different segregation models that have recently been developed and have also defined a common unifying notation that allows for different models to be easily compared and contrasted.
In order to do so, we utilise particle simulations combined with an advance micro–macro tool, called coarsegraining, to analyse the different assumptions made in these models. For the first time, we analyse data corresponding to transient stages of a segregating flow rather than simply using data corresponding to steady state.

The kinetic stress bore by the large and small constituents remains the same during the whole process of segregation, except near the freesurface of the flow where we observe some fluctuations in the initial stages of the flow.

Near the freesurface of the flow, initially the stress (contact and kinetic) profiles are different; but, gradually they relax onto the steadyprofile. This process happens from the centre outwards, i.e., the closer your location to the centre the quickly the stress relaxes to the steadystate value.

The contact stress has a much more complicated evolution; however, this seems to be associated with layering effects, that is present in the early stages of segregation. These layers slowly melt as time progress.

We confirm as previously reported by Weinhart et al. [45] and Hill and Tan [21] that, for the given particle sizeratio, the smaller constituents support a larger fraction of the normal kinetic stress.

With rightly chosen set of fitting parameters, the shape of the partial kinetic stress fraction is best captured by the functional form suggested by Gajjar and Gray [15].

The measured relative stress fractions are asymmetric as observed in the experiments of van der Vaart et al. [44]. However, they considered varying fill fractions, whereas here we consider varying particle sizeratio. Moreover, the asymmetry increases with the increase in particle sizeratio.

Firstly the strong layer we observed in the earlier stages of segregation, i.e. earlier time data, may be due to the use of a perfectly bidispersed mixture. Thereby, a small size distribution around two distinct mean values should be used.

Also the flows used here are of intermediate thickness; and deeper flows should be used to see the effect of absolute depth.

None of the partial stress fractions suggested in the literature match the simulation results; however, three of the forms suggested, do closely match the profile of the kineticstress fraction obtained from the simulation. So this opens the question such as what is the correct functional form? More interestingly, why did the form of Tunuguntla et al. [41] capture the zerosegregating line so well?

Moreover, the effect of basal roughness and varying fill concentrations on the stress fractions also needs to be studied and included in the theoretical models.
Footnotes
 1.
The bulk is defined as \(\mathcal {F}^s \cup \mathcal {F}^l\), see Fig. 1, excluding the interstitial porespace.
 2.
Stress fractions can also be called as pressure scaling functions, see [19].
 3.
For details regarding the contact and kinetic stress and its corresponding stress fractions, \(f^{con,\nu }\) and \(f^{kin,\nu }\), please see Sect. 4.
 4.
For more details regarding the coarsegraining functions, see Tunuguntla et al. [42].
 5.
More particle data can be used for coarsegraining, if the coarsegraining is applied while the simulation is running (livestatistics); however, this is time consuming and was not deemed necessary for this study.
Notes
Acknowledgments
The authors would like to thank the Dutch Technology Foundation STW for its financial support of STWVidi Project 13472, Shaping Segregation: Advanced Modelling of Segregation and its Application to Industrial Processes. We also thank Kasper van der Vaart for useful discussions.
Compliance with Ethical Standards
Funding:
This study was funded by STW VIDI Project (13472). The authors declare that they have no conflict of interest.
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