Steadily oscillating axial bands of binary granules in a nearly filled coaxial cylinder

Granular materials often segregate under mechanical agitation such as flowing, shaking, or rotating, in contrast to an expectation of mixing. It is well known that bidisperse mixtures of granular materials in a partially filled rotating cylinder exhibit monotonic coarsening dynamics of segregation. Here we report the steady oscillation of segregated axial bands under the stationary rotation of a nearly filled coaxial cylinder for $O\left({10}^3\right)$ revolutions. The axial bands demonstrate steady back-and-forth motion along the axis of rotation. Experimental findings indicated that these axial band dynamics are driven by global convection throughout the system. The essential features of the spatiotemporal dynamics are reproduced with a simple phenomenological equation that incorporates the effect of global convection.

Figure 1

(a) Photo of the experimental setup. (b) Sketch of a lateral image of the coaxial cylinder.

Figure 2

Series of snapshots (top) and spatiotemporal plots (bottom) capturing the exterior segregation patterns at a rotation rate of 27 rpm: (a) traveling waves moving from the center toward both ends, observed at $\varphi =0.932$ and (b) bidirectional steady oscillation of the bands at $\varphi =0.977$. The scale bar indicates 10 cm.

Figure 3

Phase diagram of the spatiotemporal segregation dynamics. The spatiotemporal patterns of segregation can be classified into the following four steady states: monotonic coarsening (open triangle), traveling wave (squares), oscillation (open circles), and ceasing (cross). There are also nonsteady states: intermittent oscillation (closed circles) and monotonic coarsening accompanied by a traveling wave (closed triangles).

Figure 4

(a) Band velocity profile given as a color representation, for a steady oscillation pattern at $\varphi =0.977$ and $\Omega =27$ rpm. The phase is determined with a period $\tau =48.7$ min. (b) Schematic drawing of the profile of the velocity field in the coaxial cylinder. The upper part of the cross section (drawn as a rectangle) corresponds to the outer cylinder and the lower part corresponds to the inner cylinder.

Figure 5

(a1)–(d1) Space-time plots of the segregation dynamics. (a2)–(d2) Band velocity profiles as a function of position. Shown is the traveling wave from (a) experiment $(\varphi =0.913$ and $\Omega =27$ rpm) and (b) the numerical model and the oscillatory pattern from (c) experiment $(\varphi =0.977$ and $\Omega =27$ rpm) and (d) the numerical model. The time periods of the spatiotemporal diagrams are (a1) and (c1) 340 min, while those of numerical models are (b1) 140 time units and (d1) 340 time units. The model is numerically solved with the following parameters in the case of a traveling wave (oscillatory pattern, if not identical): time intervals $\Delta T=5.0\times {10}^{-4}$ $\left(3.125\times {10}^{-4}\right)$, cylinder length $L=8\pi$, space interval $\Delta X=L/400,\lambda =0.225$ $\left(0.18\right)$, $\epsilon =0.0225,\alpha =0.15L/2\pi$ $\left(0.089L/2\pi \right)$, $u_0=0.2$ $\left(0.07\right)$, ${\beta }_0=3,{\beta }_1=0$ $\left(0.75\right)$, $c_1=-0.15,c_2=0.3$, and $c_3=-0.7$.