Transport analogy for segregation and granular rheology

Here, we show a direct connection between density-based segregation and granular rheology that can lead to insight into both problems. Our results exhibit a transition in the rate of segregation during simple shear that occurs at $I\sim 0.5$ and mimics a coincident regime change in flow rheology. We propose scaling arguments that support a packing fraction criterion for this transition that can both explain our segregation results as well as unify existing literature studies of granular rheology. By recasting a segregation model in terms of rheological parameters, we establish an approach that not only collapses results for a wide range of conditions, but also yields a direct relationship between the coordination number $z$ and the segregation velocity. Moreover, our approach predicts the precise location of the observed regime change or saturation. This suggests that it is possible to rationally design process operating conditions that lead to significantly lower segregation extents. These observations can have a profound impact on both the study of granular flow or mixing as well as industrial practice.

Figure 1

Schematic of the simulated plane shear geometry. The 3D flow is periodic in both the streamwise ($x$) and transverse ($z$) directions. Blue (dark) particles are heavy intruders while yellow (light) particles are lower density particles. We employ either constant pressure or constant volume boundary conditions.

Figure 2

Segregation velocity under varying conditions of shear rate, density ratio, particle diameter, and boundary conditions. Differing colors represent boundary conditions [constant pressures: 78 Pa, red (dark gray); 117 Pa, blue (solid light gray); 156 Pa, green (open light gray); constant volume, solid circles; full gravity effects, dotted and crossed circles] while shape represents the density ratio (circle, $\overline{\rho }=2$; triangle, $\overline{\rho }=3$; square, $\overline{\rho }=6$). While most particles are 9.0 mm in diameter, the thick-walled open circles represent a range from 6.0 to 18.0 mm. (a) The dimensionless segregation velocities are plotted vs the shear rate made dimensionless with $\sqrt{g/d_p}$. The inset shows packing fraction as a function of $I$. (b) In this panel we have replotted the ${\overline{v}}_s$ as a function of inertia number ($I$). Note that the varying boundary conditions all collapse onto individual curves corresponding to different density ratios. In all figures, error bars on the data are smaller than the symbol sizes chosen.

Figure 3

Variation of dimensionless segregation velocity with varying density at fixed values of the inertia number (upright triangle, $I=0.1193$; diamond, $I=0.2350$; square, $I=0.4563$; inverted triangle, $I=0.8627$). The inset shows the traditional scaling of the segregation rate with the dimensionless density difference. Note that, in contrast to previous studies, we find a power-law relationship with exponents that range from 0.6 to 0.75. In contrast, when we plot the segregation velocity vs our proposed density scaling, we obtain straight lines.

Figure 4

Traditional scaled segregation velocity under varying conditions of shear rate, density ratio, particle diameter, and boundary conditions. Differing colors represent boundary conditions [constant pressures: 78 Pa, red (dark gray); 117 Pa, blue (solid light gray); 156 Pa, green (open light gray); constant volume, solid circles; full gravity effects, dotted and crossed circles] while shape represents the density ratio (circle, $\overline{\rho }=2$; triangle, $\overline{\rho }=3$; square, $\overline{\rho }=6$). While most particles are 9.0 mm in diameter, the thick-walled open circles represent a range from 6.0 to 18.0 mm. The dimensionless segregation velocities are plotted vs the shear rate made dimensionless with $\sqrt{g/d_p}$. The magnitude of the segregation velocity is scaled by the traditional density scaling [that is, $({\rho }_h/{\rho }_l-1)]$. Note that, particularly in the saturated rate region, it is clear that this scaling does not collapse the data.

Figure 5

Rheology and segregation in a sheared cell system under varying conditions of shear rate, density ratio, particle diameter, and boundary conditions (symbols explained in Fig. 2). (a) shows how the effective friction coefficient changes with the inertia number. The inset shows the variation of the coordination number with $I$. Note that both rheological quantities display a regime change near a value of $I=0.5$. (b) shows the dimensionless segregation velocity rescaled with our proposed density scaling [Eq. (7)] and plotted against $I$. Note that all results fall on a master curve regardless of gravitational condition, boundary condition, or other process parameters. The included line represents the model proposed in Eq. (9). The inset shows the packing fraction as a function of $I$.