2.1. Bidisperse particle-size segregation equations

We consider non-cohesive granular media of bidisperse, rigid spherical particles – of differing small $d_{s}$ and large $d_{l}>d_{s}$ diameters, but uniform density $\rho _{*}$ – within a fixed volume $V$. The mass of the mixture is $M = M_{l}+M_{s}$, with $M_{s} = \rho _{*}n_{s} ({\rm \pi} /6)d^{3}_{s}$ and $M_{l} = \rho _{*}n_{l} ({\rm \pi} /6)d^{3}_{l}$ the masses of the small $n_{s}$ particles and the large $n_{l}$ particles, respectively. The solids volume fraction $\varPhi$ is then determined by the ratio between $M$ and the reference mass $M_{*}=\rho _{*}V$, so $\varPhi =\varPhi _{l}+\varPhi _{s}$. In dynamic systems, each concentration $\varPhi _{\nu }$ ($\nu \in \{s,l\}$) is, in principle, space- and time-dependent. However, the concentration $\varPhi$ usually varies little in most dense granular flows. By assuming $\varPhi$ to be constant, we can express each particle-size concentration in terms of its corresponding species concentration, $\phi _{s}=\varPhi _{s}/\varPhi$ and $\phi _{l}=\varPhi _{l}/\varPhi$. Thus $\phi _{s}+\phi _{l}=1$ establishes the inherent mass conservation law for the granular system.

The bidisperse granular material can be treated as a continuum medium, for which mixture theory applied to the granular bulk yields the convective–diffusive equations for segregation,

(2.1a)$$\begin{gather} \frac{\partial \phi_{s}}{\partial t}+\boldsymbol{\nabla} \boldsymbol{\cdot}(\phi_{s}\boldsymbol{u})+ \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\,f_{sl}\phi_{s}\phi_{l}\,\frac{\boldsymbol{g}}{|\boldsymbol{g}|}\right)= \boldsymbol{\nabla}\boldsymbol{\cdot}(\mathcal{D}_{sl}\,\boldsymbol{\nabla}\phi_{s}), \end{gather}$$

(2.1b)$$\begin{gather}\frac{\partial \phi_{l}}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}(\phi_{l}\boldsymbol{u})-\boldsymbol{\nabla}\boldsymbol{\cdot} \left(\,f_{sl}\phi_{s}\phi_{l}\,\frac{\boldsymbol{g}}{|\boldsymbol{g}|}\right)= \boldsymbol{\nabla}\boldsymbol{\cdot}(\mathcal{D}_{sl}\,\boldsymbol{\nabla}\phi_{l}), \end{gather}$$

where $\boldsymbol {u}$ is the divergence-free granular bulk velocity, $\mathcal {D}_{sl}$ is the particle diffusivity, and $f_{sl}\phi _{s}\phi _{l}\boldsymbol {g}/|\boldsymbol {g}|$ characterises the segregation flux, with $f_{sl}$ the particle-size segregation velocity magnitude (Trewhela et al. Reference Trewhela, Ancey and Gray2021a). The gravitational norm vector $\boldsymbol {g}/|\boldsymbol {g}|$ is included to confer a direction to size segregation, determined by the gravity-driven kinetic sieving and squeeze expulsion mechanisms (Gray Reference Gray2018). The functional form of $f_{sl}$ has been proposed to be quadratic or cubic, symmetrical or asymmetrical (Gajjar & Gray Reference Gajjar and Gray2014; Umbanhowar et al. Reference Umbanhowar, Lueptow and Ottino2019). Recent scaling laws have aimed to parametrise $f_{sl}$ in terms of physical quantities, such as pressure $p$, shear rate $\dot {\gamma }$, local particle concentrations $\phi _{s}$, and particle diameter $d$. With that goal, Trewhela et al. (Reference Trewhela, Ancey and Gray2021a) proposed that the segregation velocity magnitude $f_{sl}$ could be described as

(2.2)\begin{equation} f_{sl}=\left(\frac{\mathcal{B}\rho_{*}g\dot{\gamma} \bar{d}^{2}}{p}\right)[(R_{d}-1)+\mathcal{E}(1-\phi_{s})(R_{d}-1)^{2}], \end{equation}

where $\mathcal {B}=0.374$ and $\mathcal {E}=2.096$ are empirical constants, $\bar {d}=\phi _{s}d_{s}+\phi _{l}d_{l}$ denotes the particles’ concentration-averaged diameter, and $R_{d}=d_{l}/d_{s}$ defines the particles’ size ratio. Merging the above expressions for $\bar {d}$ and $R_{d}$, the concentration-averaged diameter can be expressed as $\bar {d}=(1-\chi _{d}\phi _{s})R_{d}d_{s}$, which captures the asymmetric behaviour of size segregation by defining $\chi _{d}=(R_{d}-1)/R_{d}$ as the asymmetry coefficient (Gajjar & Gray Reference Gajjar and Gray2014; van der Vaart et al. Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015; Trewhela Reference Trewhela2024). Note that the function (2.2) is not particular or case-dependent; it describes the segregation velocity for small and large particles at variable concentrations $\phi _{s}$ and $\phi _{l}=(1-\phi _{s})$, respectively, and it also compiles a comprehensive set of experimental and numerical results that hints at its dependence with pressure $p$, shear rate $\dot {\gamma }$ and size ratio $R_{d}$ (Golick & Daniels Reference Golick and Daniels2009; Thornton et al. Reference Thornton, Weinhart, Luding and Bokhove2012; Fry et al. Reference Fry, Umbanhowar, Ottino and Lueptow2018; Chassagne et al. Reference Chassagne, Frey, Maurin and Chauchat2020; Trewhela, Gray & Ancey Reference Trewhela, Gray and Ancey2021b; Tripathi et al. Reference Tripathi, Kumar, Nema and Khakhar2021).

From the definition of $\varPhi _{\nu }$, (2.1a) can be expressed directly as

(2.3)\begin{equation} \frac{\partial \varPhi_{\nu}}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}(\varPhi_{\nu}\boldsymbol{u})+ \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{F}_{\varPhi_{\nu}}= \boldsymbol{\nabla}\boldsymbol{\cdot}(\mathcal{D}_{sl}\,\boldsymbol{\nabla}\varPhi_{\nu}), \end{equation}

where the segregation flux $\boldsymbol {F}_{\varPhi _{\nu }}$ for the $\nu$-particle species is defined as

(2.4)\begin{equation} \boldsymbol{F}_{\varPhi_{\nu}} = \begin{cases} +f_{sl}\varPhi_{s}[1-\phi_{s}]\,\dfrac{\boldsymbol{g}}{|\boldsymbol{g}|} & \textrm{for}\ \nu=s,\\ -f_{sl}\varPhi_{l}[1-\phi_{l}]\,\dfrac{\boldsymbol{g}}{|\boldsymbol{g}|} & \textrm{for}\ \nu=l, \end{cases} \end{equation}

whereas the particles’ diffusivity $\mathcal {D}_{sl}=\mathcal {A}\dot {\gamma }\bar {d}^{2}$ (Utter & Behringer Reference Utter and Behringer2004) is responsible for diffusive remixing and controls the final segregation state, with $\mathcal {A}=0.108$ an empirical constant (table 1).

Table 1. Parameters used in this work: frictional parameters $\mu _{1}$, $\mu _{2}$ and $I_{0}$ for the $\mu (I)$-rheology (Jop et al. Reference Jop, Forterre and Pouliquen2006); coefficient $\mu _{\infty }$ with the fitting parameters $\alpha$, $I_{1}$ and $A$ for the partial regularisation of the $\mu (I)$-rheology, measured by Barker & Gray (Reference Barker and Gray2017) for $d_{l}=143\,\mathrm {\mu }$m glass beads; the grains’ intrinsic density $\rho _{*}$; the solids volume fraction $\varPhi$; the specific values for pressure $p_{0}$ and velocity $u_{0}$ at the wall; and the frictional asymmetry coefficient $\chi _{\mu }$ (Trewhela Reference Trewhela2024). Finally, we give the constants for diffusivity $\mathcal {A}$ (Utter & Behringer Reference Utter and Behringer2004) and segregation $\mathcal {B}$, $\mathcal {E}$ (Trewhela et al. Reference Trewhela, Ancey and Gray2021a).

3.1. Energetics for the GPE

Building upon the convective–diffusive equations for $\phi _{\nu }$ in (2.3), the evolution equation for the GPE density of the $\nu$-particle species is given by

(3.3)\begin{equation} \frac{\partial \mathscr{E}^{(\nu)}_{gp}}{\partial t} = \rho_{*}gz[- \boldsymbol{\nabla}\boldsymbol{\cdot}(\varPhi_{\nu}\boldsymbol{u})- \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{F}_{\varPhi_{\nu}}+ \boldsymbol{\nabla}\boldsymbol{\cdot}(\mathcal{D}_{sl}\,\boldsymbol{\nabla}\varPhi_{\nu})]. \end{equation}

Utilising vector identities and the divergence theorem, the evolution equation for the GPE in the volume $V$ unfolds as

(3.4)\begin{align} \frac{{\rm d}E^{(\nu)}_{gp}}{{\rm d}t} &= \underbrace{-\oint_{\partial\varOmega}\rho_{*}gz [\varPhi_{\nu}\boldsymbol{u}+\boldsymbol{F}_{\varPhi_{\nu}}-\mathcal{D}_{sl}\,\boldsymbol{\nabla}\varPhi_{\nu}] \boldsymbol{\cdot}\hat{\boldsymbol{n}}\,{\rm d}S}_{\varSigma^{(\nu)}_{gp}} \nonumber\\ &\quad +\underbrace{\int_{\varOmega}\rho_{*}g\varPhi_{\nu}w\,{\rm d}V}_{\varPsi^{(\nu)}_{c}} + \underbrace{\int_{\varOmega}\rho_{*}g\boldsymbol{F}_{\varPhi_{\nu}}\boldsymbol{\cdot} \hat{\boldsymbol{k}}\,{\rm d}V}_{\varPsi^{(\nu)}_{gps}} -\underbrace{\int_{\varOmega}\rho_{*}g\mathcal{D}_{sl}\, \frac{\partial\varPhi_{\nu}}{\partial z}\,{\rm d}V}_{\varPsi^{(\nu)}_{gpd}}. \end{align}

Therefore, the rate of change of $E^{(\nu )}_{gp}$ is determined by four main energetics that characterise the injection/extraction, transformation and dissipation of GPE due to redistribution and dynamics of the $\nu$-particle species.

The first term on the right-hand side of (3.4) quantifies the energetics associated with the net boundary energy flux $\varSigma ^{(\nu )}_{gp}$, which integrates fluxes resulting from the advection, segregation and diffusion of particles at the boundary of the granular bulk. In the case of granular systems with adiabatic boundaries, $\varSigma ^{(\nu )}_{gp}$ equals 0. Conversely, when granular flows experience mass exchange at the boundaries, there is a resultant gain or loss of GPE in which $\varSigma ^{(\nu )}_{gp}>0$ or $\varSigma ^{(\nu )}_{gp}<0$, respectively. The second term $\varPsi ^{(\nu )}_{c}$ quantifies the rate of change of $E^{(\nu )}_{gp}$ due to the energetics associated with vertical convective flows that redistribute particles within the moving granular mixture. Whereas the third $\varPsi ^{(\nu )}_{gps}$ and fourth $\varPsi ^{(\nu )}_{gpd}$ terms characterise the rate of change of $E^{(\nu )}_{gp}$ owing to segregation and diffusion of particles in the vertical direction, respectively. The nature of the last two energetics reflects competitive dynamics, where both may contribute to increase or decrease the GPE, depending on the $\nu$-particle species involved. For instance, in a gravity-driven granular current, larger particles ($\nu =l$) tend to segregate to the free surface, thereby raising the GPE at a rate $\varPsi ^{(l)}_{gps}$. However, the energetics associated with the diffusion of large particles $\varPsi ^{(l)}_{gpd}$ acts to counterbalance this by hindering complete segregation, hence reducing the maximum attainable GPE. Conversely, in the same granular flow, smaller particles ($\nu =s$) tend to segregate (percolate) to the base due to kinetic sieving, resulting in an energy flux that decreases GPE at a rate $\varPsi ^{(s)}_{gps}$. In this case, the vertical diffusion of small particles works to raise GPE at a rate $\varPsi ^{(s)}_{gpd}$, opposing the system's tendency towards its minimum GPE.

3.2. Energetics for the KE

The evolution equation for the KE density is derived from

(3.5)\begin{equation} \frac{\partial \mathscr{E}^{(\nu)}_{k}}{\partial t} = \rho_{*}\, \frac{1}{2}\,|\boldsymbol{u}|^{2}\,\frac{\partial \varPhi_{\nu}}{\partial t} + \rho_{*}\varPhi_{\nu}\boldsymbol{u}\boldsymbol{\cdot} \frac{\partial \boldsymbol{u}}{\partial t}, \end{equation}

and the continuum linear momentum equation for the particles mixture,

(3.6)\begin{equation} \rho_{*}\varPhi\left[\frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{\nabla}\left(\frac{1}{2}\,|\boldsymbol{u}|^{2}\right) \right] = \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{\mathsf{T}}+\rho_{*}\varPhi\boldsymbol{g}. \end{equation}

The stress tensor can be decomposed into $\boldsymbol{\mathsf{T}} = \boldsymbol {\tau }-p\boldsymbol{\mathsf{1}}$, with $p$ the pressure multiplying the tensor identity, and $\boldsymbol {\tau }$ the deviatoric shear stress. Consequently, from (2.3) and (3.6), the evolution equation of $\mathscr {E}^{(\nu )}_{k}(t,\boldsymbol {x})$ is given by

(3.7)\begin{align} \frac{\partial \mathscr{E}^{(\nu)}_{k}}{\partial t} &= \frac{1}{2}\,\rho_{*}\,|\boldsymbol{u}|^{2}\,\{-\boldsymbol{\nabla}\boldsymbol{\cdot} (\varPhi_{\nu}\boldsymbol{u})-\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{F}_{\varPhi_{sl}}+\boldsymbol{\nabla}\boldsymbol{\cdot} (\mathcal{D}_{sl}\,\boldsymbol{\nabla}\varPhi_{\nu})\} \nonumber\\ &\quad + \boldsymbol{u}\boldsymbol{\cdot}\left\{-\rho_{*}\varPhi_{\nu}\, \boldsymbol{\nabla}\left(\frac{1}{2}\,|\boldsymbol{u}|^{2}\right)+ \phi_{\nu}\,\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{\mathsf{T}} + \rho_{*}\varPhi_{\nu}\boldsymbol{g}\right\}. \end{align}

Integrating (3.7) in the volume $V$, the evolution equation for the KE is given by

(3.8)\begin{align} \frac{{\rm d}E^{(\nu)}_{k}}{{\rm d}t} &= \underbrace{-\int_{\partial\varOmega}\left[\frac{\rho_{*}}{2}\,|\boldsymbol{u}|^{2}\, (\varPhi_{\nu}\boldsymbol{u}+\boldsymbol{F}_{\varPhi_{\nu}}-\mathcal{D}_{sl}\, \boldsymbol{\nabla}\varPhi_{\nu})-(\phi_{\nu}\boldsymbol{u} \boldsymbol{\cdot}\boldsymbol{\mathsf{T}})\right]\boldsymbol{\cdot} \hat{\boldsymbol{n}}\,{\rm d}S}_{\varSigma^{(\nu)}_{k}} \nonumber\\ &\quad +\underbrace{\int_{\varOmega}\boldsymbol{F}_{\varPhi_{\nu}}\boldsymbol{\cdot} \boldsymbol{\nabla}\left(\frac{\rho_{*}}{2}\,|\boldsymbol{u}|^{2}\right){\rm d}V}_{\varPsi^{(\nu)}_{ks}}- \underbrace{\int_{\varOmega}\mathcal{D}_{sl}\,\boldsymbol{\nabla}\varPhi_{\nu}\boldsymbol{\cdot} \boldsymbol{\nabla}\left(\frac{\rho_{*}}{2}\,|\boldsymbol{u}|^{2}\right){\rm d}V}_{\varPsi^{(\nu)}_{kd}} \nonumber\\ &\quad -\varPsi^{(\nu)}_{c} -\underbrace{\int_{\varOmega}\boldsymbol{\nabla} (\phi_{\nu}\boldsymbol{u}):\boldsymbol{\mathsf{T}}\,{\rm d}V}_{\varPsi^{(\nu)}_{\mu}}. \end{align}

Therefore, the rate of change of $E^{(\nu )}_{k}$ is determined by five main energetics, which characterise the injection/extraction, transformation and dissipation of KE due to redistribution and dynamics of the $\nu$-particle species.

The first term $\varSigma ^{(\nu )}_{k}$ on the right-hand side of (3.8) quantifies the energetics associated with the net boundary energy flux, which integrates fluxes resulting from advection, segregation, diffusion and interparticle friction. For granular systems with adiabatic and frictionless boundaries, $\varSigma ^{(\nu )}_{k}$ equals zero. Otherwise, boundaries act as surface areas through which energy is either injected or extracted by friction, being shear-driven granular flows, a canonical example of a system wherein $\varSigma ^{(\nu )}_{k}$ is positive, thereby contributing to the sourcing of KE.

The second term $\varPsi ^{(\nu )}_{ks}$ on the right-hand side of (3.8) quantifies the energetics of $E^{(\nu )}_{k}$ due to $\nu$-particle species segregation within the moving granular bulk. Since $\boldsymbol {F}_{\varPhi _{\nu }}$ is parallel to the gravitational acceleration vector $\boldsymbol {g}/|\boldsymbol {g}|$, commonly characterised by $-\hat {\boldsymbol {z}}$ (see figure 1), spatial gradients of $\rho _{*}\,|\boldsymbol {u}|^{2}/2$ in directions perpendicular to $\boldsymbol {g}/|\boldsymbol {g}|$ do not contribute to $\varPsi ^{(\nu )}_{ks}$. As an example, let us consider a bidisperse granular flow driven by shear stress exerted at the granular bulk's surface, moving at a speed $u_{0}$. The resulting velocity profile increases linearly with height $z$, i.e. $\boldsymbol {u}=u_{0}(z/h)\,\hat {\boldsymbol {x}}$, with $h$ the sheared layer. In this context, the energetics of segregation simplifies to $\varPsi ^{(\nu )}_{ks}=\int _{\varOmega } \boldsymbol {F}_{\varPhi _{\nu }}\boldsymbol {\cdot } (\rho _{*}u\,\partial u/\partial z)\,\hat {\boldsymbol {z}}\,{\rm d} V$. Interestingly, the sign of $\varPsi ^{(\nu )}_{ks}$ depends on the $\nu$-particle species. If $u(\partial u/\partial z)>0$, then the KE flux owing to segregation is positive for large particles ($\nu =l$): $\varPsi ^{(l)}_{ks}=\int _{\varOmega } f_{sl}\varPhi _{l}[1-\phi _{l}] (\rho _{*}u\,\partial u/\partial z)\,{\rm d}V > 0$. In this scenario, large particle species segregate into upper regions characterised by a granular flow with higher inertia and density of KE. Conversely, for the case of small particles ($\nu =s$), the KE flux owing to segregation is negative: $\varPsi ^{(s)}_{ks}=-\int _{\varOmega } f_{sl}\varPhi _{s}[1-\phi _{s}] (\rho _{*}u\,\partial u/\partial z)\,{\rm d}V < 0$. Here, small-particle species segregate into deeper regions characterised by a granular flow with a lower inertia and density of KE, resulting in a global loss of their KE.

Figure 1. (a) Schematic of the case study and an example of the numerical simulations. (b,c) Temporal evolutions of the small-particle species concentration $\phi _{s}(\hat {t},\hat {z})$ and velocity profile $\hat {u}(\hat {t},\hat {z})$ for $R_{d}=2.5$, $\mu _{0}=0.5$.

The third $\varPsi ^{(\nu )}_{kd}$ and fourth $\varPsi ^{(\nu )}_{c}$ terms on the right-hand side of (3.8) correspond to the energetics that quantify the rate of change of $E^{(\nu )}_{k}$ owing to diffusion and convective flows that redistribute $\nu$-particle species vertically within the granular matrix. We remark that the sign of the energy flux due to diffusion $\varPsi ^{(\nu )}_{kd}$ also depends on the $\nu$-particle species. To illustrate this, we consider the same bidisperse granular flow discussed earlier, which allows us to simplify the term to $\varPsi ^{(\nu )}_{kd}=\int _{\varOmega } \mathcal {D}_{sl}(\partial \varPhi _{\nu }/\partial z)(\rho _{*}u\,\partial u/\partial z)\,{\rm d}V$. Assuming that $u(\partial u/\partial z)>0$, the actual sign of $\varPsi ^{(\nu )}_{kd}$ is determined by the gradient $\partial \varPhi _{\nu }/\partial z$, which may also change over time, as shown later in figure 1(b) and evaluated in § 4. Finally, the term $\varPsi ^{(\nu )}_{\mu }$ is general and characterises the energetics associated with the irreversible loss of KE due to interparticle friction, commonly referred to as the KE dissipation rate. Note that a closed-form mathematical expression for $\varPsi ^{(\nu )}_{\mu }$ requires a constitutive relationship between the stress tensor $\boldsymbol{\mathsf{T}}$ and the granular mixture rheology. In this direction, the proposed framework is flexible enough to allow the usage of local or non-local rheological models (Kamrin et al. Reference Kamrin, Hill, Goldman and Andrade2024), without loss of generality.

4.1. Stress tensor and $\mu (I)$-rheology for bidisperse granular mixtures

To calculate $\varSigma ^{(\nu )}_{k}$ and $\varPsi ^{(\nu )}_\mu$ in (3.8), the stress tensor $\boldsymbol{\mathsf{T}}$ needs to be described thoroughly. A widely used relationship to do so is the $\mu (I)$-rheology, that relates the shear $\tau$ and normal $p$ stresses through a Coulombic relation $\tau =\mu (I)\,p$. Embedded within this definition, the frictional coefficient $\mu (I)$ is found to be dependent on the inertial number (e.g. MiDi Reference GDR MiDi2004)

(4.1)\begin{equation} I=\frac{\dot{\gamma}\bar{d}}{\sqrt{p/\rho^{*}}}.\end{equation}

The frictional coefficient is obtained by fitting the empirically determined function $\mu (I) = \mu _{1} +(\mu _{2}-\mu _{1})/(I_{0}/I+1)$, where $\mu _{1}$ and $\mu _{2}$ are the static and dynamic frictional coefficients, respectively (Jop, Forterre & Pouliquen Reference Jop, Forterre and Pouliquen2006). These two coefficients, alongside the material parameter $I_{0}$, correspond to the experimental parameters that define the $\mu (I)$-rheology. However, the above relationship for $\mu (I)$ is ill-posed. Barker & Gray (Reference Barker and Gray2017) partially regularised the $\mu (I)$ definition with the extended relationship

(4.2)\begin{equation} \mu(I)=\left\{\begin{array}{@{}ll@{}} \displaystyle \sqrt{\alpha/\log(A/I)}, & \text{for}\ I\leq I_{1},\\ \displaystyle\dfrac{\mu_{1}I_{0}+\mu_{2}I+\mu_{\infty}I^{2}}{I+I_{0}}, & \text{for}\ I > I_{1},\end{array}\right. \end{equation}

where $\mu _{\infty }$, $\alpha$ and $I_{1}$ are material-dependent coefficients. The two latter parameters with $A$ serve as fitting constants for the partially regularised $\mu (I)$-rheology, which guarantees continuity of (4.2). For this work, the values for all these parameters and constants defining the $\mu (I)$-rheology are presented in table 1.

To include bidispersity effects in the bulk's frictional response, we define a concentration-averaged frictional coefficient $\bar {\mu }=\mu _{s}\phi _{s}+\mu _{l}\phi _{l}=(1-\chi _{\mu }\phi _{s})R_{\mu }\mu _{s}$, where $\mu _{\nu }$ denotes the frictional coefficient for the $\nu$ species, and $R_{\mu }=\mu _{l}/\mu _{s}$ is a ratio that defines the asymmetry parameter $\chi _{\mu }=(R_{\mu }-1)/R_{\mu }$ (Trewhela Reference Trewhela2024) (see table 1). From the above definitions, the stress tensor components are given by

(4.3)\begin{equation} T_{ij}=\left\{\begin{array}{@{}ll@{}} \displaystyle \bar{\mu}(I)\,p-p, & \text{for }i=j,\\ \displaystyle \bar{\mu}(I)p, & \text{for } i\neq j, \end{array}\right.\quad {\rm with}\ i,j=\{x,z\}, \end{equation}

which are naturally averaged over the concentrations of particle species.

4.2. Energetics of a sheared bidisperse granular layer

As a case study, we consider a bidisperse granular bulk of depth $h$ sustaining shear exerted by a top plate imposing a uniform normal pressure distribution $p=p_{0}$ and moving at a constant velocity $u_{0}$ in the streamwise $x$-direction (figure 1a). Based on the rheological model in § 4.1, the pressure $p_0$, velocity $u_0$ and frictional coefficient $\mu _{0}$ set a shear $\tau _{xz} = \tau _{0}=p_{0}\mu _{0}$ (table 1) at the top boundary. In contrast, the granular material in contact with the motionless bottom does not slip or roll. Under these conditions, the granular system develops a zero-gravity shear flow, which, for simplicity, is treated as two-dimensional and periodic in the $x$-direction. As a result, the sheared layer depends on the upper boundary condition (John Soundar Jerome & Di Pierro Reference John Soundar Jerome and Di Pierro2018)

(4.4)\begin{equation} h=\left(\frac{\mu_{0}}{\mu_{1}}-1\right)\frac{p_{0}}{\rho_{*}g\varPhi}. \end{equation}

The top and bottom boundaries are closed, so fluxes across them are null. Without loss of generality, we examine the momentum–segregation equations of the small-particle species, i.e. $\nu =s$, which, in non-dimensional form, are given by

(4.5a,b)\begin{equation} \frac{\partial \hat{u}}{\partial \hat{t}}=\frac{\partial}{\partial \hat{z}} ([1-\chi_{\mu}\phi_{s}]R_{\mu}\mu_{s}\hat{p}),\quad \frac{\partial \phi_{s}}{\partial \hat{t}}=\frac{\partial}{\partial \hat{z}} \left(\hat{f}_{sl}\phi_{s}[1-\phi_{s}]+\hat{\mathcal{D}}_{sl}\, \frac{\partial\phi_{s}}{\partial \hat{z}}\right), \end{equation}

where $\hat {u}=u/u_{0}$, $\hat {p}=p/p_{0}$, $\hat {z}=z/h$, $\hat {t}=t/(h/u_{0})$, $\hat {f}_{sl}=f_{sl}/u_{0}$ and $\hat {\mathcal {D}}_{sl}=\mathcal {D}_{sl}/(u_{0}h)$ denote the non-dimensional variables (Trewhela Reference Trewhela2024). From (4.5a,b) and considering the energy scale per unit area $\varPhi \rho _{*}gh^{2}$, the evolution equation for the non-dimensional GPE, $\hat {E}^{(s)}_{gp}=\int ^{1}_{0}\phi _{s}\hat {z}\,{\rm d}\hat {z}$, reduces to

(4.6) \begin{equation} \frac{{\rm d}\hat{E}^{(s)}_{gp}}{{\rm d}t} = \underbrace{\int^{1}_{0}-\hat{f}_{sl} \phi_{s}[1-\phi_{s}]\,{\rm d}\hat{z}}_{\hat{\varPsi}^{(s)}_{gps}}- \underbrace{\int^{1}_{0}\hat{\mathcal{D}}_{sl}\, \frac{\partial\phi_{s}}{\partial \hat{z}}\,{\rm d}\hat{z}}_{\hat{\varPsi}^{(s)}_{gpd}}. \end{equation}

Likewise, considering the same energy scale, the evolution equation for the non-dimensional KE, $\hat {E}^{(s)}_{k}=\frac {1}{2}Fr^{2}\int ^{1}_{0}\hat {u}^{2}\,{\rm d}\hat {z}$, simplifies to

(4.7)\begin{equation} \frac{{\rm d}\hat{E}^{(s)}_{k}}{{\rm d}\hat{t}}=\underbrace{Fr^{2}\, (\hat{u}[1-\chi_{\mu}\phi_{s}]R_{\mu}\mu_{s}\hat{p})|_{\hat{z}=1}}_{\hat{\varSigma}^{(s)}_{k}} - \underbrace{Fr^{2}\int^{1}_{0}[1-\chi_{\mu}\phi_{s}]R_{\mu}\mu_{s}\hat{p}\, \frac{\partial \hat{u}}{\partial \hat{z}}\,{\rm d}\hat{z}}_{\hat{\varPsi}^{(s)}_{\mu}}, \end{equation}

where $Fr = u_{0}/\sqrt {gh}$ is the Froude number of the bulk granular flow. The energetics $\hat {\varSigma }^{(s)}_{k}$ and $\varPsi ^{(s)}_{\mu }$ are positive definite and determine the rate of kinetic (mechanical) energy production due to surface shear and KE dissipation owing to friction, respectively. In contrast, the energetics $\hat {\varPsi }^{(s)}_{gps}$ and $\hat {\varPsi }^{(s)}_{gpd}$ quantify the transformation rate of potential (mechanical) energy due to vertical segregation and diffusion of particles, respectively; their signs are not defined a priori. The energy pathways controlled by the processes responsible for transforming the mechanical energy – i.e. segregation, particle diffusion and friction – are unknown and investigated next.

4.3. Numerical experiments

Following Trewhela (Reference Trewhela2024), we resolved numerically the coupled, nonlinear partial differential equation system in (4.5a,b) using the method of lines. This numerical scheme is unconditionally stable and considered a constant pressure distribution $p=p_{0}$, so its robustness is not compromised by the ill-posed $\mu (I)$-rheology, even though it is regularised in (4.2). A total of 18 simulations were run, considering first a set of parameters $\mu _{0}\times R_{d}=\{0.5, 0.8\}\times \{1.25,1.5,1.75,2.0,2.25,2.5\}$ yielding 12 simulations to understand mid-range values, and second the set $\{0.4, 1.2, 1.6\}\times \{1.25,2.5\}$ to explore extreme values for $\mu _{0}$.

From the parameters introduced in table 1, the size ratio $R_{d}$ and frictional coefficient $\mu _{0}$ are the most determinants for segregation dynamics. The $\mu (I)$-rheology defining parameters are also relevant. Still, these are material dependent and altogether compile a large set of parameters that are particular for a type of grain. It is then hard to vary each parameter separately while extracting relevant information for segregation, and it is also at risk of exponentially increasing the number of simulations. Next, the empirical constants from the Trewhela et al. (Reference Trewhela, Ancey and Gray2021a) scaling law and Utter & Behringer (Reference Utter and Behringer2004) need further validation in other flow configurations and particles, yet these have delivered consistent results against experimental and numerical data (Maguire et al. Reference Maguire, Barker, Rauter, Johnson and Gray2024; Singh, Liu & Henann Reference Singh, Liu and Henann2024). For completion, Trewhela (Reference Trewhela2024) analysed the relevance of the parameters at the wall imposing shear, namely $p_{0}$, $u_{0}$ and $\mu _{0}$ in this work. These parameters control the time scales at which segregation operates, but only $\mu _{0}$ determines the segregation–diffusion balance or the resulting concentration $\phi _{s}$ profiles. Also, Trewhela (Reference Trewhela2024, figure 5) shows that $R_{\mu }$ does not have a determinant role in the segregation–diffusion balance, hence the resultant small-particle distribution $\phi _{s}$, only on the velocity profiles $\hat {u}$. Thus we propose a brief but comprehensive set of simulations to shed light on the energetics under various frictional regimes and a wide range of segregation–mixing dynamics. Figure 1 illustrates an example of the spatiotemporal evolution of small-particle concentration $\phi _{s}(\hat {t},\hat {z})$ and vertical profiles of the shear-driven granular flow $\hat {u}(\hat {t},\hat {z})$, which are utilised to compute the energetics in (4.6) and (4.7).

Figure 2(a) shows time series of the segregation energy flux $\hat {\varPsi }^{(s)}_{gps}$ and the diffusive energy flux $\hat {\varPsi }^{(s)}_{k}$ for the frictional coefficient $\mu _{0}=0.8$ and a range of size ratio values $1.25\leq R_{d}\leq 2.5$. The energy flux $\hat {\varPsi }^{(s)}_{gps}$ is always negative, i.e. segregation continuously consumes the available GPE that the small particles have. Yet $\hat {\varPsi }^{(s)}_{gps}$ exhibits two striking phases. The first phase is characterised by an exponential-like increase of the segregation flux, supported by the diffusive flux $\hat {\varPsi }^{(s)}_{gpd}$ that also consumes GPE during this phase. The second phase is characterised by a change in the diffusive flux direction, which starts to raise the GPE until the granular flow balances into ${\rm d}\hat {E}^{(s)}_{gp}/\text {d}\hat {t}\rightarrow 0$. This final state equilibrium is shown in figure 2(c), which illustrates ${\rm d}\hat {E}^{(s)}_{gp}/{\rm d}\hat {t}$ versus time. Figure 2(b) shows time series of the KE production due to shear taken by the small particles $\hat {\varSigma }^{(s)}_{k}$ and the KE dissipation rate owing to friction $\hat {\varPsi }^{(s)}_{\mu }$. The magnitude of KE energetics is larger than the GPE energetics, showing that the system is dominated by particle–particle interactions with a short-lived transient phase, in which shear production at the boundary $\hat {\varSigma }_{k}^{(s)}$ is slightly higher than friction dissipation $\hat {\varPsi }_{\mu }^{(s)}$. This results in the KE reaching a shear friction equilibrium significantly faster than the GPE equilibrium, which is hindered due to the slow pace segregation and the counter effect of diffusive remixing, as shown in figure 2(d). At the equilibrium state, the KE injected by shear is utilised to mobilise grains horizontally and maintain the bidisperse granular material fully or partially segregated across the sheared layer $h$. Such a balanced state – and therefore its GPE – should strongly depend on the competition between segregation and diffusion fluxes, which fosters mixing within the granular bulk.

Figure 2. Numerical results for $\mu _{0}\times R_{d}=\{0.8\}\times \{1.25,1.5,1.75,2.0,2.25,2.5\}$. (a) Energetics of GPE, $\hat {\varPsi }^{(s)}_{gps}$ and $\hat {\varPsi }^{(s)}_{gpd}$ versus time $\hat {t}$. (b) Energetics of KE, $\hat {\varSigma }^{(s)}_{k}$ and $\hat {\varPsi }^{(s)}_{\mu }$ versus time $\hat {t}$. (c) Temporal gradient of KE and its (e) budget versus time $\hat {t}$. (d) Temporal gradient of GPE and its (f) budget versus time $\hat {t}$.

The temporal evolutions of the bulk GPE and KE are shown in figures 2(e,f). These two plots confirm and aggregate the observations made from the overall energetics and their rates in figures 2(a–d). Much of the variation in $\hat {E}^{(s)}_{gp}$ comes from segregation that decreases the small particles’ energy as a result of the onset of the kinetic sieving mechanism. Naturally, this decrease comes to a halt when it is balanced with diffusion, as observed in figure 2(a). The behaviour of the KE $\hat {E}^{(s)}_{k}$ is different, showing a strong and immediate increment due to the upper plate transferring momentum to the small particles located at the top. After the momentum is transferred through the whole bulk, the transient velocity profile is achieved, producing a plateau in $\hat {E}^{(s)}_{k}$. Finally, the KE plateau value is then modified by the segregation and diffusion balancing out into a steady state that varies with $\mu _{0}$ and $R_{d}$. Notably, the plateau values for $\hat {E}^{(s)}_{k}$ show higher KE values for larger $R_{d}$ values, which is consistent with the notion that larger particles result in larger inertia. This trend is inverted after segregation and diffusion balance, with smaller $R_{d}$ values having more KE. The latter shows the influence of diffusion and the effect of partially averaged quantities.

4.4. Non-dimensional parameters and scalings

From the numerical solutions (figures 1b,c) and the results shown in figure 2, the roles of segregation, diffusion and friction (or dissipation) are evident in the energetics distribution and balance. Besides the energy injection $\hat {\varSigma }^{(s)}_{k}$, the other three energetics represent well-defined processes that control the bidisperse granular flow dynamics. When combining and balancing these three main energetics, several non-dimensional parameters arise. A first quantity appears when analysing the GPE energetics: the competition between segregation $\varPsi ^{(s)}_{gps}$ and diffusion $\varPsi ^{(s)}_{gpd}$ is controlled by the segregation Péclet number

(4.8)\begin{equation} Pe=\frac{hf_{sl}}{\mathcal{D}_{sl}}, \end{equation}

which directly compares the segregation velocity $f_{sl}$ to the particles’ diffusion velocity scale $\mathcal {D}_{sl}/h$ (Gray & Chugunov Reference Gray and Chugunov2006; Wiederseiner et al. Reference Wiederseiner, Andreini, Épely-Chauvin, Moser, Monnereau, Gray and Ancey2011; Gray Reference Gray2018), with $h$ the thickness of the active sheared layer, which depends on the boundary condition parameters $\mu _{0}$ and $p_{0}$ (see (4.4)). Yet in terms of the physical forcing of our system, it appears naturally that two other non-dimensional quantities balance the energetics. The second non-dimensional parameter, the Schmidt number $Sc=\mathcal {D}_{sl}/\nu _{g}$, balances particle diffusivity $\mathcal {D}_{sl}$ ($\varPsi ^{(s)}_{gpd}$) and momentum diffusion ($\varPsi ^{(s)}_{\mu }$), represented by the granular kinematic viscosity $\nu _{g}$. This viscosity is defined as $\nu _{g}=\mu p/(\rho \dot {\gamma })$ when the Coulombic relation $\tau =\mu p$ combines with the equivalent Newtonian description $\tau =\eta _{g}\dot {\gamma }$ for the granular flow. Thus the remaining combination from the main three energetics is that of a balance between segregation $\varPsi ^{(s)}_{gps}$ and momentum diffusion or frictional dissipation $\varPsi ^{(s)}_{\mu }$. This balance results in the segregation–rheology Péclet number, which can be interpreted alternatively as a segregation Reynolds number and can be obtained as a combination of the other two non-dimensional numbers (Trewhela Reference Trewhela2024):

(4.9)\begin{equation} Pe_{sr}=\frac{Pe}{Sc}=\frac{hf_{sl}}{\nu_{g}}. \end{equation}

The competition between segregation and frictional dissipation is condensed by the segregation–rheology Péclet number $Pe_{sr}$, which in our numerical experiments spans $Pe_{sr}=Pe/Sc\in [0.4,300]$. For the specific case study, the granular kinematic viscosity $\nu _{g}$ is introduced following Trewhela (Reference Trewhela2024, (3.5)), which depends on size ratio $R_{d}$ via the shear rate $\dot {\gamma }$. This results in a segregation–rheology Péclet number that can be defined and computed as

(4.10)\begin{equation} Pe_{sr}=\frac{I^{2}}{\varPhi_{P}\hat{\mu}_{0}}\ \mathcal{F}(R_{d},\phi_{s})= \frac{\rho_{*} g h}{p_{0}}\,\frac{\rho_{*}\dot{\gamma}^{2}\bar{d}^{2}}{\mu_{0}p_{0}}\ \mathcal{B}(R_{d}-1)(1+\mathcal{E}(R_{d}+1)(1-\phi_{s})), \end{equation}

where $\varPhi _{P}=\rho _{*} gh/p_{0}$ is a non-dimensional pressure that can also be interpreted as a solids volume fraction, and $\hat {\mu }_{0}$ is a $\chi _{\mu }$-dependent frictional coefficient that approximates to $\mu _{0}$ when $\chi _{\mu }\rightarrow 0$, in our numerical solutions.

However, as emphasised earlier, Péclet numbers do not provide explicit information about the energy state of granular flows. To understand how segregation, mixing and dissipation dynamics affect the energy partition between GPE and KE, we can borrow the Richardson number $Ri$ definition for stratified flows, weighting the balance between buoyancy and inertia (e.g. Caulfield Reference Caulfield2021). Estimating $Ri$ a priori is challenging, which leads us to explore a potential link between $Ri$ and $Pe_{sr}$ directly from our numerical experiments.

To unravel the relation between $Pe_{sr}$ and $Ri$, we first examine the partition of the mechanical energy components (of the small-particle species) versus the segregation–rheology Péclet number. Figure 3(a) shows $\hat {E}^{(s)}_{gp}/\hat {E}^{(s)}_{me}$ (blue) and $\hat {E}^{(s)}_{k}/\hat {E}^{(s)}_{me}$ (red) versus $Pe_{sr}$, with $\hat {E}^{(s)}_{me}=\hat {E}^{(s)}_{k}+\hat {E}^{(s)}_{gp}$ the mechanical energy. For low $Pe_{sr}$ (${\leq }10$), the KE is substantially smaller than the GPE, whereas for high $Pe_{sr}$ (${\geq }10^{2}$), the KE scales with or surpasses the GPE. From these results, we therefore expect the segregation–mixing state to be sensible to the energy partition, and in particular to the ratio between GPE and KE, here termed as the bulk Richardson number $Ri$ of the granular flow:

(4.11)\begin{equation} Ri \equiv \frac{\hat{E}^{(s)}_{gp}} {\hat{E}^{(s)}_{k}}= \frac{gh}{u^{2}_{0}/2}\,\frac{\displaystyle\int^{1}_{0}\phi_{s}\hat{z}\,{\rm d}\hat{z}}{ \displaystyle\int^{1}_{0}\hat{u}^{2}\,{\rm d}\hat{z}}. \end{equation}

This specific $Ri$ is defined for small particles, but is extensible for other particle species.

Figure 3. Results for all 18 numerical solutions. (a) Mechanical energy partition versus the segregation–rheology Péclet number $Pe_{sr}$. Reddish markers denote $\hat {E}^{(s)}_{k}/\hat {E}^{(s)}_{me}$, blueish markers denote $\hat {E}^{(s)}_{gp}/\hat {E}^{(s)}_{me}$, with $\hat {E}^{(s)}_{me}=\hat {E}^{(s)}_{k}+\hat {E}^{(s)}_{gp}$. (b) Ratio of the mechanical energy components, i.e. the Richardson number $Ri$ versus $Pe_{sr}$. The inset shows experimental data of $^{\dagger}$van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015) and $^\ddagger$Wiederseiner et al. (Reference Wiederseiner, Andreini, Épely-Chauvin, Moser, Monnereau, Gray and Ancey2011) compared to the proposed scaling $Ri\sim Pe_{sr}^{-1/2}$.

Figure 3(b) plots $Ri$ versus $Pe_{sr}$ for our numerical solutions and experimental data from the literature. For the latter, we used the results from a 50 : 50 mixed shear cell experiment of van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015), and the chute flow experiments of Wiederseiner et al. (Reference Wiederseiner, Andreini, Épely-Chauvin, Moser, Monnereau, Gray and Ancey2011). These studies provide measurements of the resulting small-particle distribution $\phi _{s}$ at steady-state conditions, which is key to determining the energy partition, hence the ratio between GPE and KE. We determined the small particle concentration $\phi _{s}$ profiles of Wiederseiner et al. (Reference Wiederseiner, Andreini, Épely-Chauvin, Moser, Monnereau, Gray and Ancey2011) using their reported segregation rates $q$, diffusion coefficients $D$, and flow heights $h$ (for their runs 22–25), and we directly used their flume slope (equivalent to $\mu _{0}$ in steady state), particle density and shear rates to calculate the granular viscosity $\nu _{g}$. Hence with their reported Péclet numbers $Pe$, we could obtain $Pe_{sr}=Pe/(\nu _{g}/D)$. For the van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015) experiment, we computed $Pe$ at the middle of their shear cell $\hat {z}=1/2$ – in similar fashion to Trewhela et al. (Reference Trewhela, Ancey and Gray2021a) – as $Pe$ varies with depth $\hat {z}$ in such an experimental configuration. While a range of $Pe$ values could be used, the balance between segregation and diffusion is most relevant at the small–large particle interface, which is precisely at $\hat {z}=1/2$ in their experiment.

The results of this numerical and experimental dataset exhibit the distinct scaling relationship $Ri \propto Pe_{sr}^{-1/2}$ for six decades of $Pe_{sr}$, a direct result from a linear regression of the plot in logarithmic scale (dash-dotted line in figure 3b). Intriguingly, the empirical scaling law for $Ri$–$Pe_{sr}$ does not seem straightforward to justify theoretically. However, a possible explanation for this scaling law relies upon the relationship between the active shear layer $h$ – where the mechanical energy is stored and transformed – and the segregation–rheology Péclet and bulk Richardson numbers. Following the definitions in (4.10) and (4.4) yields that $Pe_{sr}\propto h^{-2}$. At the same time, the relation $Ri\propto h$ is retrieved directly from the bulk Richardson number definition. Therefore, combining both relationships, one may derive that $Ri\propto Pe^{-1/2}_{sr}$. Remarkably, this scaling shows excellent agreement with our compiled numerical results and experimental data from the literature (see the inset of figure 3b). Despite the fact that these experiments inherently have dispersion in their measurements, which we could not quantify from the reported data, the proposed scaling proves to be consistent as a follow-up to our numerical results. From this scaling, the energy partition of the granular flow can be readily predicted since $Pe_{sr}$ is determined from prescribed system parameters in most cases, as our calculations from previous experiments demonstrate.

4.5. Degree of mixing (and segregation)

Yet how does the energy partition relate to the segregation–mixing state? We investigate this question through the ‘degree of mixing’

(4.12)\begin{equation} \mathscr{M}_{\phi}=1-(\sigma_{\phi}/\sigma_{sg})^{2}, \end{equation}

with $\sigma ^{2}_{sg}=1/2$ the variance of the fully segregated state of the species concentration $\phi _{s}$, and $\sigma ^{2}_{\phi }$ the variance of $\phi _{s}$. Thus $\mathscr {M}_{\phi }=0$ denotes a perfectly segregated state, and $\mathscr {M}_{\phi }=1$ a perfectly mixed state. It is apparent that the ‘degree of segregation’ in the system can be quantified from the reciprocal $\mathscr {S}_{\phi }=1-\mathscr {M}_{\phi }$. To explore first how the degree of mixing evolves in time throughout our numerical solutions, we plot $\mathscr {M}_{\phi }$ for five different cases in figure 4(a). The selected cases cover the whole range of friction coefficients $\mu _{0}$ and size ratios $R_{d}$, and they result in differing mixing degrees $\mathscr {M}_{\phi }$ at steady state. These results show that all solutions pass by an intermediate, yet maximum, apparent mixing state, independently of whether the steady-state solution is mixed or segregated. Interestingly, the moment when this apparent mixing state happens is not the same for all cases. It appears to be controlled by diffusion, based on the results shown in figure 2(a). We plot in figure 4(b) the degree of mixing as a function of time for the experimental data of van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015) against an equivalent numerical solution of their 50 : 50 mix experiment. Despite the evident point-to-point differences, numerical and experimental results show remarkable agreement – in magnitude and trend – both passing by the apparent mixing state, with the difference in $\mathscr {M}_{\phi }$ closing out at steady state. Notice that the evolution of the degree of mixing $\mathscr {M}_{\phi }$ in figure 4(b) differs from that observed in figure 4(a); this is because in the shear cell experiment, the onset of the shear rate is almost immediate compared to the shear flow studied here, which requires momentum to be transferred first.

Figure 4. (a) Degree of mixing $\mathscr {M}_{\phi }$ as a function of time $\hat {t}$ for a subset of our numerical simulations. (b) A comparison between the degree of mixing $\mathscr {M}_{\phi }$ in the 50 : 50 mix experiment of van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015) and its corresponding numerical solution, both as a function of the van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015) non-dimensional time $\hat {t}^{*}$, which depends on their shear cell oscillation period. (c) Degree of mixing $\mathscr {M}_{\phi }$ at steady state as a function of the Péclet number $Pe$ for our numerical simulations. Experimental data from $^{\dagger}$van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015) and $^\ddagger$Wiederseiner et al. (Reference Wiederseiner, Andreini, Épely-Chauvin, Moser, Monnereau, Gray and Ancey2011) are also included and plotted by considering their depth-averaged small-particle concentration. The whole dataset is compared to the here introduced theoretical expression $\mathscr {M}_{\phi }=(4/Pe)\tanh (Pe/4)$.

Finally, to focus on the segregation–mixing state attained by the system, we examine $\mathscr {M}_{\phi }$ in terms of the segregation Péclet number $Pe$ instead of $Pe_{sr}$. This allows us to isolate the mechanisms controlling mass distribution and GPE, excluding the friction needed to account for KE. Figure 4(c) shows $\mathscr {M}_{\phi }$ versus $Pe$. The results show that $Pe$ strongly controls the degree of mixing, characterised by a nonlinear yet monotonic relationship. Within our parameter space, the maximum mixing state is attained at $Pe \approx 1$, with $\mathscr {M}_{\phi } \approx 0.99$. Conversely, the minimum mixing state, $\mathscr {M}_{\phi }\approx 0.05$, is achieved at $Pe \approx 10^{2}$, indicating the onset of a saturation state. Statistically, we expect $\mathscr {M}_{\phi }$ not to be zero at high $Pe$ values as the transition region between small and large particles experiences a persistent remixing–segregation process, as seen in laboratory and numerical experiments (e.g. Golick & Daniels Reference Golick and Daniels2009; van der Vaart et al. Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015; Ferdowsi et al. Reference Ferdowsi, Ortiz, Houssais and Jerolmack2017; Singh & Henann Reference Singh and Henann2024).

The nonlinear relationship between the degree of mixing and the segregation Péclet number can be derived theoretically from the steady-state solution for $\phi _{s}$. From the segregation–diffusion equation in (4.5a,b), $\phi _{s}$ can be integrated to yield

(4.13) \begin{equation} \phi_{s}=\frac{{\rm e}^{{Pe\,(\mathcal{K}-\hat{z})}}}{1+{\rm e}^{{Pe\,(\mathcal{K}-\hat{z})}}}, \end{equation}

which is analogue to the solutions of Trewhela (Reference Trewhela2024, (3.3)), Wiederseiner et al. (Reference Wiederseiner, Andreini, Épely-Chauvin, Moser, Monnereau, Gray and Ancey2011, (11)) and Gray & Chugunov (Reference Gray and Chugunov2006, (3.4)). The integration constant $\mathcal {K}$ corresponds to the depth-averaged concentration, i.e. $\overline {\phi _{s}}=0.5$ for the studied case, and variable for the Wiederseiner et al. (Reference Wiederseiner, Andreini, Épely-Chauvin, Moser, Monnereau, Gray and Ancey2011) experiments. With this analytical solution for $\phi _{s}$, its variance can be expressed as $\sigma ^{2}_{\phi }=\mathbb {E}(\phi _{s}^{2})-\mathbb {E}(\phi _{s})^{2}$, where $\mathbb {E}(\cdot )$ denotes the expected value. The theoretical calculation of $\sigma _{\phi }^{2}$ alongside the value $\sigma _{sg}=1/2$ yields an analytical expression for the degree of mixing:

(4.14)\begin{equation} \mathscr{M}_{\phi}=-\frac{4}{Pe}\left(\frac{1-{\rm e}^{Pe/2}}{1+{\rm e}^{Pe/2}}\right)= \frac{4}{Pe}\tanh\left(\frac{Pe}{4}\right). \end{equation}

Reciprocally, the latter expression provides an explicit function for the degree of segregation as a function of the Péclet number for segregation:

(4.15)\begin{equation} \mathscr{S}_{\phi}=1-\mathscr{M}_{\phi}=1-\frac{4}{Pe}\tanh\left(\frac{Pe}{4}\right), \end{equation}

where $Pe$ for a first-order approximation in $R_{d}-1$ is a function of the empirical constants $\mathcal {A}$ (Utter & Behringer Reference Utter and Behringer2004) and $\mathcal {B}$ (Trewhela et al. Reference Trewhela, Ancey and Gray2021a) corresponding to $Pe=\mathcal {B}(R_{d}-1)\rho gh/(\mathcal {A}p_{0})$.

In figure 4(c), our compiled dataset of numerical solutions and experimental data from van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015) and Wiederseiner et al. (Reference Wiederseiner, Andreini, Épely-Chauvin, Moser, Monnereau, Gray and Ancey2011) shows excellent agreement with the proposed theoretical curve in (4.14) (dash-dotted line), which indicates that mixing (and segregated) states depend solely on $Pe$, hence on the parameters that define it. The latter provides a general relationship that does not depend on flow configuration or scale, a fact inherited from the variable conditions considered in our shear flow case, and the shear cell and chute flow configurations from the previous experimental results included here.