3.1. Effect of size ratio in the segregation and velocity profiles

To demonstrate the crucial role of particle sizes and segregation in the shear flow dynamics, the vertical distributions of small particles $\phi _{s}$ and horizontal velocity $\hat {u}$ are computed as functions of time $\hat {t}$ for three particles size ratios $R_{d}=\{1.25,2,3.33\}$, keeping $p_{w}=100$ Pa, $\mu _{w}=0.5$ and $u_{w}=0.2$ m s$^{-1}$ constant (figure 3). The initially unstable arrangement of small particles over large ones influences the velocity $\hat {u}$ profiles; all reach a transient state at $\hat {t}\approx 10^{2}$, only to be then altered by segregation. From $\hat {t}>10^{2}$, the linear velocity profiles $\hat {u}$ for both species, and featuring a slope change at the small–large particle interface, changes into a nonlinear smooth distribution for the final steady-state condition. At initial times, segregation does not influence the velocity profile $\hat {u}$ or the time scale to attain the transient linear state. Therefore, momentum transfer is uncoupled from segregation, with the latter acting at longer time scales, an observation made in various experiments and flow configurations (Golick & Daniels Reference Golick and Daniels2009; Tripathi & Khakhar Reference Tripathi and Khakhar2011; van der Vaart et al. Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015; Trewhela & Ancey Reference Trewhela and Ancey2021). Nonetheless, size ratio $R_{d}$ influences the transient linear state, determining the velocity vertical gradient, a result also observed by Tripathi & Khakhar (Reference Tripathi and Khakhar2011) and corroborated by Barker et al. (Reference Barker, Rauter, Maguire, Johnson and Gray2017). The asymmetric nature of segregation is captured clearly in figures 3(a–c). As the size ratio $R_{d}$ increases, small particles segregate faster to the bottom compared to large particles arriving at the top. As a result, when asymmetry $\chi _{d}$ is increased, segregation fluxes are fast and dynamic, achieving the nonlinear steady-state swiftly (as observed for $R_{d}=3.33$, $\chi _{d}=0.7$ in figures 3c,f).

Table 2.

Definition of the parameters used to set the presented numerical simulations in this work. The numerical values for the size asymmetry coefficient $\chi _{d}$ (hence particle-size ratio $R_{d}$) and the frictional asymmetry coefficient $\chi _{\mu }$ are used to set the particle parameters. The numerical values for pressure $p_{w}$, velocity $u_{w}$ and frictional coefficient $\mu _{w}$ at the upper plate or wall (see figure 1) resulted in a range of values for $h$ (see (2.4)) and set the granular rheology, together with the values of table 1. Finally, the empirical parameters $\mathcal {A}$, $\mathcal {B}$, $\mathcal {E}$ proposed by Trewhela et al. (Reference Trewhela, Ancey and Gray2021a) and Utter & Behringer (Reference Utter and Behringer2004) control the segregation dynamics in the numerical solutions.

Figure 3.

(a–c) Vertical and temporal distributions of small particle concentrations $\phi _{s}$ for $u_{w}=0.2$ m s$^{-1}$, $p_{w}=100$ Pa, $\mu _{w}=0.5$ and $R_{d}=\{1.25,2,3.33\}$ values. (d–f) Small particle concentration profiles $\phi _{s}$ at different $\hat {t}$ times for the corresponding $R_{d}$ values. (g–i) Velocity profiles $\hat {u}$ at different $\hat {t}$ instants for the corresponding $R_{d}$ values, reflected in its asymmetrical coefficient $\chi _{d}=(R_{d}-1)/R_{d}$ value. The thicker line in each of (d–i) highlights the final solution of each profile. Animated solutions of (d–i) are in supplementary movie 1, available at https://doi.org/10.1017/jfm.2024.168.

Other striking differences appear when comparing the final small particle concentrations $\phi _{s}$. Larger $R_{d}$ values produce sharper profiles with strongly segregated $\phi _{s}$ profiles and a marked small–large interface. Subsequently, this segregated and nonlinear distribution is imprinted into the shape of the steady-state velocity profile $\hat {u}$, thus a direct result of particle-size segregation effectiveness. For size ratios $R_{d}\rightarrow 1$, apparent mixing (i.e. diffusive remixing) is observed. Controlled by diffusion, mild segregation produces quasi-linear $\hat {u}$ profiles, as shown in figure 3(d). In contrast to these quasi-linear profiles, $\hat {u}$ solutions for large $R_{d}$ show a sharp break in the velocity profile, drawing a rhomboid or lens geometry. The rhomboid is defined graphically by the linear responses expected for the normally graded and inversely graded conditions for particle-size distribution.

3.2. Initially mixed conditions and resulting profiles

The normally graded condition shown in figure 1(a) is a widely explored initial condition for segregation in experiments and numerical simulations (e.g. van der Vaart et al. Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015; Gray Reference Gray2018; Chassagne et al. Reference Chassagne, Maurin, Chauchat, Gray and Frey2020). However, in the context of mixed bidisperse granular flows, the initially mixed condition illustrated in figure 1(b) also provides a valuable assessment on how segregation affects the rheology of granular mixtures. Figure 4 shows the resulting small particle concentration $\phi _{s}$ and velocity $\hat {u}$ profiles for three different initially mixed conditions of small and large particles at $\phi _{s}^{0}=25$, 50 and 75 % homogeneously mixed concentrations. To compare with the results shown previously in figures 3(c,f,i), an $R_{d}=3.33$ value is used. The steady state $\phi _{s}$ distribution and the $\hat {u}$ profile observed in figures 4(b,e,h) are identical to those observed in figures 3(c,f,i). This agreement proves that steady-state solutions are independent of the normally graded or mixed initial conditions. Such independence is just about the particles’ initial arrangement, and it is not extended to the average particle concentration $\phi _{s}^{0}$, a condition that results in differing profiles for $\phi _{s}$ and $\hat {u}$ (figure 4).

Figure 4.

(a–c) Vertical and temporal distributions of small particle concentrations $\phi _{s}$ for $u_{w}=0.2$ m s$^{-1}$, $p_{w}=100$ Pa, $\mu _{w}=0.5$ and $R_{d}=3.33$ for initially-mixed conditions of small and large particles, $\phi ^{0}_{s}=\{0.25,0.5,0.75\}$. (d–f) Small particle concentration $\phi _{s}$ and (g–i) velocity $\hat {u}$ profiles at different $\hat {t}$ instants for the corresponding $\phi _{s}^{0}$ and the same wall parameters and $R_{d}$ value. The thicker line in each of (d–i) highlights the final solution of each profile. Animated solutions of (d–i) are available in supplementary movie 2.

As expected, there are important differences in the intermediate solutions for both $\phi _{s}$ and $\hat {u}$ due to the change in the initial conditions. Homogeneously mixed initial $\phi _{s}^{0}$ concentrations produce not only different velocity $\hat {u}$ and skewed – but similar – $\phi _{s}$ profiles, but also delay segregation. The delayed steady-state solutions observed in figures 4(a–c) for larger $\phi _{s}^{0}$ concentrations are tied inherently to the asymmetric nature of size segregation, i.e. the segregation flux $\boldsymbol {F}_{s}=f_{sl}\phi _{s}\phi _{l}\boldsymbol {g}/|\boldsymbol {g}|\sim (1-\chi _{d}\phi _{s})^{2}\phi _{s}(1-\phi _{s})$. Small particles at low $\phi _{s}$ concentrations segregate at a faster rate than large particles at low $\phi _{l}$ concentrations (Gajjar & Gray Reference Gajjar and Gray2014; van der Vaart et al. Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015; Trewhela et al. Reference Trewhela, Ancey and Gray2021a). If segregation fluxes were to be symmetric $\boldsymbol {F}_{s}= \phi _{s}\phi _{l}\boldsymbol {g}/|\boldsymbol {g}|$, then segregation rate would be maximum at $\phi _{s}^{0}=0.5$, and at identical rates for $\phi _{s}^{0}=\{0.25,0.75\}$ (Gray Reference Gray2018). Therefore, figure 4 reproduce the expected asymmetry. For $\phi _{s}^{0}=0.25$, segregation alters the velocity profile solution swiftly from the mixed condition (at $\hat {t}\approx 10^{2}$) into the steady-state condition at almost $\hat {t}\approx 10^{4}$. In the case of $\phi _{s}^{0}=0.75$, segregation also initiates at $\hat {t}\approx 10^{2}$, but steady-state solutions are reached at a slower pace, i.e. at $\hat {t}\approx 10^{5}$. These differences hint at a $\phi _{s}$-dependent time scale for segregation that controls the final duration of the segregation–rheology feedback.

3.3. Influence of differences in the particles’ frictional coefficients

An additional effect in the Stokes’ problem is explored by changing the small particles’ frictional coefficient $\mu _{s}$ while keeping the size ratio $R_{d}=2$ fixed, along with $p_{w}=300$ Pa, $\mu _{w}=0.5$ and $u_w=0.2$ m s$^{-1}$. Differences in $\mu _{s}$, relative to $\mu _{l}$ via $R_{\mu }$, introduce a non-zero $\chi _{\mu }$ value in (2.9a), which is null when small and large particles share the same $\mu (I)$-rheology parameters (table 1). Changes in $\chi _{\mu }$ lead to velocity differences at the small–large particles interface, compromising the numerical scheme due to $\hat {u}$ discontinuity. Numerical solutions are found only for slight changes in $\chi _{\mu }$, within the interval $\chi _{\mu }=[-0.033,0.013]$ for $R_{d}=2$. This interval depends on the size ratio as well; as the size ratio $R_{d}$ increases, the $\chi _{\mu }$ interval for the solutions narrows.

In figure 5, $\phi _{s}$ and $\hat {u}$ solutions are plotted for various $\chi _\mu$ values. While the small particle concentration $\phi _{s}$ solutions do not show significant differences between the different $\chi _\mu$ cases, the velocity profiles $\hat {u}$ display interesting behaviour. For negative $\chi _\mu$ values, $\mu _{s}$ is larger than $\mu _{l}$, resulting in the transient solution having inverted slopes compared to the case with positive $\chi _\mu$ values; see figures 5(f,j) for comparison. Intermediate negative values for $\chi _\mu$ seem to counteract the $\hat {u}$ profile nonlinearity generated by the size difference in $R_{d}$ (figures 5g,h). This finding is consistent with the proposed theoretical framework and could illustrate a way to obtain linear velocity profiles in granular flows with particles of varying sizes and frictional properties. Therefore, to control the particles’ frictional properties is to partially control the segregation–rheology feedback, despite having a segregated bulk with different grain sizes.

Figure 5.

Numerical solutions for (a–e) small particle concentrations $\phi _{s}$, and (f–j) velocity profiles $\hat {u}$ with constant wall parameters $u_{w}=0.2$ m s$^{-1}$, $p_w=300$ Pa, $\mu _{w}=0.5$, $R_{d}=2$, and variable $\chi _{\mu }=\{-0.03,-0.02,-0.01,0,0.01\}$ for (a,f), (b,g), (c,h), (d,i) and (e,j), respectively. The thicker line in each plot highlights the final solution of each variable. Solutions plotted in (d,f–j) are shown as animations in supplementary movie 3.

3.4. Steady-state analytical solutions and comparison with numerical solutions

A solution for the transient velocity profiles $\hat {u}$ can be determined analytically from the steady-state condition in (2.9a), i.e. $\bar {\mu }=\mu _{w}$. This condition is replaced in the quadratic expression for $I$, obtained from the regularized $\mu (I)$-rheology in (2.2):

(3.1)\begin{equation} \mu_{\infty}I^{2}+(\mu_{2}-\mu_{w})I+(\mu_{1}-\mu_{w})I_{0}=0.\end{equation}

The positive solution for $I$ is then balanced with the inertial number $I$ definition to yield a differential equation for the velocity $\hat {u}$. This ODE is solved by separating variables and using the no-slip boundary condition $\hat {u}|_{\hat {z}=1}=0$ to yield

(3.2)\begin{equation} \hat{u}=\frac{h}{u_{w}\bar{d}}\,\sqrt{\frac{p_{w}}{\rho_{*}}}\, \mathcal{I}(\mu_{w})\,(1-\hat{z}),\end{equation}

where $\mathcal {I}(\mu _{w})=((\mu _{w}-\mu _{2})+\sqrt {(\mu _{w}- \mu _{2})^{2}+4\mu _{\infty }(\mu _{w}-\mu _{1})I_{0}})/2\mu _{\infty }$ is the aforementioned positive solution of (3.1), i.e. the inertial number $I$. Equation (3.2) results in the rhomboid formation, already described and discussed in § 3.1. If the concentration-averaged diameter $\bar {d}$ is evaluated with the normally-graded size distribution, $\phi _{s}\big |_{\hat {z}=0}^{\hat {z}=1/2}=1$ and $\phi _{s}\big |_{\hat {z}=1/2}^{\hat {z}=1}=0$, and the inversely-graded size distribution, $\phi _{s}\big |_{\hat {z}=0}^{\hat {z}=1/2}=0$ and $\phi _{s}\big |_{\hat {z}=1/2}^{\hat {z}=1}=1$, then the rhomboid can be diagrammed. The solution captured by (3.2) provides a good approximation of the transient velocity profile without considering segregation in its development. When segregation determines the velocity profile, a steady-state solution for the small particle concentration $\phi _{s}$ can be obtained. For that, the procedure used by Trewhela et al. (Reference Trewhela, Ancey and Gray2021a) is followed, yet their solution is implicit for $\phi _{s}$. Here, a simpler differential equation is developed from $\mathcal {E}=0$ (a first-order approximation in $R_{d}-1$), resulting in the explicit solution

(3.3)\begin{equation} \phi_{s}=\frac{{\rm e}^{(\hat{z}+\mathcal{K})/\lambda}}{1+ {\rm e}^{(\hat{z}+\mathcal{K})/\lambda}},\end{equation}

where $\mathcal {K}=-1/2$ is the constant of integration, which can be calculated using the depth-averaged condition $\bar {\phi }_{s}=\int ^{1}_{0}\phi _{s}\,\text {d}\hat {z}=0.5$, and $\lambda =\mathcal {A}p_{w}/(\mathcal {B}\rho g h(R_{d}-1))$. This small particle concentration $\phi _{s}$ profile serves as input for the generalized ODE:

(3.4)\begin{equation} \frac{\text{d}\hat{u}}{\text{d}\hat{z}}=-\frac{h}{u_{w}\bar{d}}\,\sqrt{\frac{p_{w}}{\rho_{*}}}\,\mathcal{I}(\hat{\mu}_{w}), \end{equation}

where $\hat {\mu }_{w}=\mu _{w}(1-2.7(1-\phi _{s})\chi _{\mu })/(1-\chi _{\mu }\phi _{s})$ is a $\phi _{s}$- and $\chi _{\mu }$-dependent frictional coefficient, with $\bar {d}$ a function of the former as well. Due to this dependence, the ODE's solution is computed using a Runge–Kutta fourth-order method.

Figure 6 summarizes the core behaviour of the segregation–rheology feedback. The transient-state solution of (3.2) and its reciprocal stable state (red and blue toned rhomboids) are shown in figure 6(a). These rhomboids emphasize the slope change in $\hat {u}$ as $R_{d}$ increases, with the fully linear case $R_{d}=1$ as reference. In figure 6(b), the generalized solution in (3.4) is drawn for the $R_{d}=2$ case and various $\chi _{\mu }$ values, with the reference curve $\chi _{\mu }=0$ at the middle. Negative $\chi _{\mu }$ values tend to displace the steady-state velocity profile $\hat {u}$ towards the linear reference, acting as a counterpart for the deviation generated by the size ratio $R_{d}$. Conversely, positive $\chi _{\mu }$ values exacerbate the nonlinear behaviour of the velocity profiles $\hat {u}$, displacing the curves beyond the reciprocal stable-state curve (with large particles over small ones). Moreover, solutions attempting to pass beyond the stable state could explain the difficulties in obtaining solutions when the frictional asymmetric coefficient is $\chi _\mu >0.013$ for $R_{d}=2$. A detailed evolution of the velocity profile $\hat {u}$ solutions is shown in figure 6(c). The velocity profile $\hat {u}$ develops from the initial condition to the transient state $\hat {t}\sim 10^{2}$, adjusting well to the analytical solution of (3.2) (left rhomboid linear curves in figure 6(c), taken from $R_{d}=2$ in figures 6a,b). Then segregation alters this state, rearranging particles according to their size, achieving a steady state at $\hat {t}\approx 10^{5}$ that fits well the ODE solution in (3.4) (dot-dashed line in figure 6c). In figure 6(c) the segregation–rheology feedback ends up being described graphically by the changes in $\hat {u}$ from the transient to the steady state, hence analytically bounded within the $R_{d}$-dependent rhomboid region.

Figure 6.

(a) Velocity profiles $\hat {u}$ for the steady-state condition in (3.2) for various $R_{d}=[1,4]$ values, represented in blue and red tones for $d_{s}$ and $d_{l}$, respectively. (b) Velocity profiles $\hat {u}$ of the numerically solved (3.4) for variable $\chi _{\mu }=[-0.05,0.05]$ and constant $R_{d}=2$, with its corresponding rhomboid from (a). (c) Temporal evolution of the velocity profile $\hat {u}$ for the $R_{d}=2$ and $\chi _{\mu }=0.01$ case compared with the corresponding steady-state solutions from (a) ((3.2), plotted as a rhomboid) and (b) ((3.4), plotted with a dot-dashed line). Note how the velocity profiles $\hat {u}$ first collapse around the rhomboid solution, and then converge to the steady-state solution (red dot-dashed line). Fully animated solutions (for $\chi _{\mu }=\{-0.03,-0.01,0.01\}$ and constant $R_{d}=2$), compared to the steady-state solutions of (3.2) and (3.4), are shown in supplementary movies 4–6.

3.5. Granular kinematic viscosity, time scales and segregation–momentum balances

From the ODE for the non-dimensional shear rate $\hat {\dot {\gamma }}=|\text {d}\hat {u}/\text {d}\hat {z}|$ in (3.4), a general definition for the granular kinematic viscosity $\nu _{g}$ can be formulated. Using the theoretical constitutive relation for a granular dynamic viscosity $\tau =\mu p=\eta _{g}\dot {\gamma }$ (Jing et al. Reference Jing, Ottino, Umbanhowar and Lueptow2022), the dimensional kinematic viscosity can be defined as

(3.5)\begin{equation} \nu_{g}=\frac{\hat{\mu}_{w}\bar{d}}{\mathcal{I}(\hat{\mu}_{w})}\, \sqrt{\frac{p_{w}}{\rho_{*}}}.\end{equation}

Compared to the granular viscosity definition of Jerome & Di Pierro (Reference Jerome and Di Pierro2018), in (3.5) this bidisperse granular viscosity is $R_{d}$-, $\chi _{\mu }$- and $\phi _{s}$-dependent (via $\bar {d}$ and $\hat {\mu }_{w}$), and is still determined by the wall conditions and the $\mu (I)$-rheology parameters (table 1). It is worth recalling that the $\mathcal {I}$ function is in fact the inertial number $I$, but for the Stokes’ problem analysed in this work, it is left as the function defined in (3.2).

The definition of $\nu _{g}$ allows the analysis of the different time scales at which the segregation–rheology feedback develops. Evidently, a first time scale appears from the non-dimensional time $\hat {t}$ definition, where $t_{\dot {\gamma }}\sim h/u_{w}$ can be defined as a convective or shear rate time scale, inferred from $1/\dot {\gamma }=(\text {d}u/\text {d}z)^{-1}\approx h/u_{w}$ (as in the definition of $I$; GDR-MiDi 2004). A second time scale can be associated with momentum diffusing through the granular bulk and derived directly from (3.5), yielding

(3.6)\begin{equation} t_{\nu_{g}}\sim \frac{h^{2}}{\nu_{g}}=\frac{h^{2}\,\mathcal{I}(\hat{\mu}_{w})}{\hat{\mu}_{w} \bar{d}}\,\sqrt{\frac{\rho_{*}}{p_{w}}}.\end{equation}

Then two more time scales appear from the particle-segregation-related processes. A segregation time scale is drawn by the time it takes particles of diameter $\bar {d}$ to segregate through the sheared layer $h$ with velocity $f_{sl}$:

(3.7)\begin{equation} t_{f_{sl}}\sim \frac{h}{f_{sl}}=n_{\bar{d}}P\,\mathcal{F}(R_{d},\phi_{s})^{-1}\,t_{\dot{\gamma}}, \end{equation}

where $n_{\bar {d}}=h/\bar {d}$ is the number of particle layers through the sheared layer depth, $\mathcal {F}(R_{d},\phi _{s})=\mathcal {B}(R_{d}-1)[1+\mathcal {E}(R_{d}-1)(1-\phi _{s})]$ is a function defined from the segregation velocity $f_{sl}$ function, and $P=p_{w}/(\rho _{*}g\bar {d})$ corresponds to a non-dimensional pressure (Trewhela et al. Reference Trewhela, Ancey and Gray2021a). The segregation time scale is proportional to $t_{\dot {\gamma }}$ and $p_{w}$, in agreement with the clear segregation dependence on shear rate and pressure (Golick & Daniels Reference Golick and Daniels2009; Trewhela et al. Reference Trewhela, Gray and Ancey2021b). Thus the segregation time scale $t_{f_{sl}}$ is the shear rate time scale $t_{\dot {\gamma }}$ modulated by pressure conditions at the wall, segregation, and the number of layers that particles must segregate through. Finally, the particle diffusion time scale can be defined as

(3.8)\begin{equation} t_{\mathcal{D}_{sl}}\sim \frac{h^{2}}{\mathcal{D}_{sl}}= \frac{n_{\bar{d}}^{2}}{\mathcal{A}}\,t_{\dot{\gamma}}, \end{equation}

which again is proportional to $t_{\dot {\gamma }}$. Although these time scales do not provide the actual time that the velocity $\hat {u}$ and small particle concentration $\phi _{s}$ profiles take to reach their corresponding steady-state solutions, which is out of the scope of this work, they serve to describe the progression of the segregation–rheology feedback.

For the numerical solutions computed in § 3.1, $t_{\dot {\gamma }}=0.0157$ s and the momentum diffusion time scale share similar values $t_{\nu _{g}}\approx 0.15$ s, yielding a non-dimensional time $\hat {t}=t_{\nu _{g}}/t_{\dot {\gamma }}\approx 10$ for the $\hat {u}$ profiles to develop into the transient solution. This value is close to what is shown in figure 3, with $\hat {t}\approx 10^2$ being the time at which the transient solution is achieved in the computations. In terms of segregation, there are larger differences in $t_{f_{sl}}\approx \{100,20,7\}$ s and $t_{\mathcal {D}_{sl}}=\{80,120,170\}$ s for $R_{d}=\{1.25,2,3.33\}$, respectively. These $t_{f_{sl}}$ values capture well the faster set in of segregation at $\hat {t}=t_{f_{sl}}/t_{\dot {\gamma }} \approx \{6\times 10^{3},10^{3},5\times 10^{2}\}$ for the same $R_{d}$ values. However, these $t_{f_{sl}}$ are balanced out by shorter or larger diffusive time scales $t_{\mathcal {D}_{sl}}$, which finally settle the concentration profiles into their steady-state solution. Consistent results are observed when analysing these time scales in the case of the initially mixed condition solutions at fixed $R_{d}=3.33$. These yield the same $t_{\dot {\gamma }}$ and $t_{\nu _{g}}$ values for all solutions, but $t_{f_{sl}}\approx \{3,7,20\}$ s and $t_{\mathcal {D}_{sl}}\approx \{100,170,310\}$ s increase with $\phi _{s}^{0}=\{0.25,0.5,0.75\}$, resulting in the monotonic increase of $\hat {t}=t_{\mathcal {D}_{sl}}/t_{\dot {\gamma }} \approx \{7\times 10^3,10^4,2\times 10^4\}$. These values are also similar to the times found in the numerical solutions (figure 4). In the case of the varying $\chi _\mu$ solutions, there are only slight differences in $t_{\nu _{g}}$. Values decrease from approximately 0.9 s to 0.7 s as $\chi _\mu$ goes from $-$0.03 to 0.01.

From these results and the analysed time scales, it is thus natural to compare the two diffusion processes acting in the bidisperse granular Stokes’ problem. With the definition of the granular momentum diffusivity time scale, or its diffusivity coefficient in $\nu _{g}$, the Schmidt number or the ratio of the momentum and particle diffusivity time scales is obtained directly as

(3.9)\begin{equation} Sc=\frac{t_{\nu_{g}}}{t_{\mathcal{D}_{sl}}}=\frac{\nu_{g}}{\mathcal{D}_{sl}}= \frac{\hat{\mu}_{w}}{\mathcal{A}I^{2}}, \end{equation}

where the inertial number is reintroduced for simplicity and generalization; for that matter, $I$ can be considered equal to $d\dot {\gamma }/\sqrt {p/\rho _*}$ or equal to the definition of $\mathcal {I}$ made after (3.2). It is worth noting that the $I^{2}$ term corresponds to the Savage or Coulomb number (Savage Reference Savage1984; Ancey, Coussot & Evesque Reference Ancey, Coussot and Evesque2000). Low $Sc$ values indicate an important particle diffusivity (diffusive remixing), dominating over momentum diffusion, which at first glance can be achieved with low $\mu _{w}$, high $\mathcal {A}$ or high inertial numbers $I$. Conversely, a large $Sc$ points towards a more viscous response from the granular material, with a lesser role played by particle diffusion in the overall flow dynamics. Interestingly, this definition of the Schmidt number depends implicitly not only on the particles’ concentration-averaged diameter $\bar {d}$ (via $I$), hence their size ratio $R_{d}$ and size asymmetry coefficient, but also on the frictional asymmetry coefficient $\chi _{\mu }$ and the particles’ frictional parameters.

To illustrate these dependencies, figure 7(a) shows the Schmidt number $Sc$ as a function of both $\mu _{w}$ and an overall small particle concentration, which can be interpreted as the initial small particle concentration $\phi _{s}^{0}$ or the depth-averaged small particle concentration $\overline {\phi _{s}}=\int _{0}^{1}\phi _{s}\,\text {d}\hat {z}$. In figure 7(a), the contours’ slope is determined by $\chi _{\mu }$: positive $\chi _{\mu }$ yields a negative slope (dashed contours), and vice versa (filled contours). For the computed values in the domain $\mu _{w}=[0.4,1.2]$ and $\phi _{s}=[0,1]$, $Sc$ varies from $10^{3}$ to $10^{-1}$, decreasing linearly in two regions marked approximately by $\mu _{w}\lessgtr \mu _{2}=0.557$. This is a direct result of the partially regularized $\mu (I)$-rheology, which allows $\mu _{w}>\mu _{2}$, therefore extending the range for $Sc$. Overall, $Sc$ decreases with $\mu _{w}$, and increases slightly with higher $\phi _{s}^{0}$, when $\chi _{\mu }<0$. The former observation contradicts the linear dependence of $Sc$ on $\mu _{w}$, but is consistent with the fact that high $\mu _{w}$ values result in high $I$ values as well, since $Sc\sim I^{-2}$. (For more results shedding light on the role of $\mu _{w}$, see Appendix A.) While particle diffusivity scales with the shear rate $\mathcal {D}_{sl}\sim \dot {\gamma }$ (Utter & Behringer Reference Utter and Behringer2004), the granular viscosity scales inversely with the latter $\nu _{g}\sim \dot {\gamma }^{-1}$; so larger friction at the wall produce high shear rates and will result in low $Sc$. But as stated before, the segregation–rheology feedback is not a purely momentum- or particle-diffusion-dominated process; segregation also controls the flow dynamics.

Figure 7.

(a) Filled contours for the Schmidt number $Sc=t_{\nu _{g}}/t_{\mathcal {D}_{sl}}=\nu _{g}/\mathcal {D}_{sl}$ as a function of both $\mu _{w}$ and the overall small particle concentration $\phi _{s}^{0}$, in the case $\chi _{\mu }=-0.03$. Dashed contours show the change of slope in the case $\chi _{\mu }=0.03$ for the same $Sc$ values of the filled contours. (b) Filled contours for the segregation–rheology Péclet number $Pe_{sr}=t_{f_{sl}}/t_{\nu _{g}}=hf_{sl}/\nu _{g}$ as a function of $\mu _{w}$ and the size ratio $R_{d}$, in the case $\chi _{\mu }=0.01$. (c) Variation of the segregation–rheology Péclet number $Pe_{sr}$ as a function of the wall friction $\mu _{w}$ and the frictional asymmetry parameter $\chi _{\mu }$. For comparison, the constant frictional parameter $\mu _{2}=0.557$ is plotted alongside the same $Pe_{sr}$, but using the non-regularized inertial number definition $\mathcal {I}_{nr}=I_{0}(\mu _{w}-\mu _{1})/(\mu _{2}-\mu _{w})$, which is independent of $\chi _{\mu }$ (dot-dashed thick line).

To understand better at which extent segregation controls the segregation–rheology feedback, a particle segregation–diffusion balance is borrowed from previous work. Following the definitions made by Gray (Reference Gray2018) and Trewhela et al. (Reference Trewhela, Ancey and Gray2021a), a Péclet number for segregation $Pe$ can be defined as the ratio between the segregation velocity $f_{sl}$ and particles’ diffusivity $\mathcal {D}_{sl}$, i.e. $Pe=hf_{sl}/\mathcal {D}_{sl}$, or alternatively, $t_{f_{sl}}/t_{\mathcal {D}_{sl}}$. With this definition, a specific Péclet number for the segregation–rheology feedback can be introduced as

(3.10)\begin{equation} Pe_{sr}=\frac{Pe}{Sc}=\frac{hf_{sl}}{\nu_{g}}= \frac{t_{f_{sl}}}{t_{\nu_{g}}}=\frac{I^{2}}{\varPhi_{p} \hat{\mu}_{w}}\,\mathcal{F}(R_{d},\phi_{s}), \end{equation}

where $\varPhi _{p}=p_{w}/(\rho g h)$ is a non-dimensional pressure, which can be interpreted as an equivalent solids volume fraction. Again, this second non-dimensional quantity depends on $I^{2}$ (or the Savage number). Compared to the previous $Sc$ definition, the segregation–rheology Péclet number $Pe_{sr}$ depends on the size ratio $R_{d}$ and the empirical constants determined by Trewhela et al. (Reference Trewhela, Ancey and Gray2021a). The proposed $Pe_{sr}$ somehow confirms the results discussed previously in §§ 3.1 and 3.3, where segregation profiles were sharper for large $R_{d}$ (figure 3) and positive $\chi _{\mu }$ values (figure 5).

Figures 7(b,c) show the variation of the segregation–rheology Péclet number $Pe_{sr}$ as a function of $\hat {\mu }_{w}$, $\mu _{w}$ and $R_{d}$. Broadly, the $Pe_{sr}$ dependence on $R_{d}$ is well captured in figure 7(b). As expected, $Pe_{sr}$ is minimum for $R_{d}\rightarrow 1$, and interestingly, larger size differences do not necessarily result in predominant segregation effects over momentum diffusion, since $Pe_{sr}=0.01$ for $R_{d}=4$ and $\mu _{w}=0.4$. Therefore, the results presented in the previous subsections are summarized conceptually in the $Pe_{sr}$ definition; the segregation–rheology feedback is relevant when a combination of both high friction and large size differences is in place. Nonetheless, high $\mu _{w}$ is still enough to promote segregation, even accompanied by low $R_{d}$. All this, as long as there is sufficient pressure $p_{w}$ or the bulk is compressed ($\varPhi _{p}$) enough at the wall to transfer shear without restraining both momentum and segregation. The latter opens the possibility of including compressibility effects in the proposed system and solutions. In figure 7(c), the $Pe_{sr}$ curves are plotted for various $\chi _{\mu }$ values, all of which show similar behaviour. However, when $Pe_{sr}$ is calculated using the non-regularized $\mu (I)$-rheology, i.e. $\mathcal {I}_{nr}=I_{0}(\mu _{w}-\mu _{1})/(\mu _{1}-\mu _{w})$, the $Pe_{sr}$ value diverges as $\mu _{w}\rightarrow \mu _{2}$ (thick dashed line in figure 7c). This shows the importance of having a well-posed extended formulation for the $\mu (I)$-rheology. When using the partially regularized $\mu (I)$-rheology, $Pe_{sr}$ continues to increase monotonically with $\mu _{w}>\mu _{2}$, but at a lower rate than for $\mu _{w}<\mu _{2}$, and without a clear cutoff value.