Subdiffusive Axial Transport of Granular Materials in a Long Drum Mixer

Granular mixtures segregate radially by size when tumbled in a partially filled horizontal drum. The smaller component moves toward the axis of rotation and forms a buried core, which then splits into axial bands. Models have generally assumed that the axial segregation is opposed by diffusion. Using narrow pulses of the smaller component as initial conditions, we have characterized axial transport in the core. We find that the axial advance of the segregated core is well described by a self-similar concentration profile whose width scales as $t^{\alpha }$, with $\alpha \sim 0.3<1/2$. Thus, the process is subdiffusive rather than diffusive as previously assumed. We compare our results to two one-dimensional model equations which contain self-similarity and subdiffusion: a linear fractional diffusion model and the nonlinear porous medium equation.

Figure 1

(a) An image of the shadow of the radial core formed by $177–212\mu \mathrm{m}$ sand grains within $300–420\mu \mathrm{m}$ salt grains. (b) The detected edge used to determine the vertical extent $h(x,t)$ of the core. (c) The $x$ integral of $h$ is not constant in time and is thus not proportional to the volume of small grains contained within the radial core. (d) The integral of $h^2$ is constant in time and is proportional to the desired concentration.

Figure 2

(a) Concentration profiles of a spreading radial core pulse of sand grains within salt grains at various times. (b) Power-law scaling of the FWHM of the radial core pulse. From the linear fit (solid line) we find that the width $\propto t^{0.38}$. (c) Collapsed concentration profiles of the radial core pulse corresponding to (a). The collapse parameter is $\alpha =0.37$.

Figure 3

(a) Concentration profiles of a mixing pulse of dyed black salt grains surrounded by white salt grains. (b) Collapsed concentration profiles corresponding to (a). The collapse parameter is $\alpha =0.3$.

Figure 4

Collapsed concentration profiles of a black sand radial core in salt grains fit to (a) the fractional diffusion equation (solid line) and (b) the porous medium equation (solid line). Collapsed concentration profiles of mixing dyed black salt grains fit to (c) the fractional diffusion equation (solid line) and (d) the porous medium equation (solid line).