Pattern scaling in the axial segregation of granular materials in a rotating tube

Granular mixtures frequently segregate by grain size along the axis of partially-filled, horizontal, rotating tubes. When segregation approaches saturation at the surface, a well-defined pattern of bands with wavelength $\lambda$ emerges. The long-term dynamics of the pattern involves a much slower coarsening process. We characterized the initial saturated wavelength $\lambda$ as a function of the diameter of the tube $D$ for a filling fraction of 30 %, for $1.9 {\rm cm} \leq D \leq 11.5 {\rm cm}$. We also studied the initial growth-rate of the bands as $D$ varies. We find that $\lambda/D$ is not constant, but rather increases rapidly for small $D$. The growth-rate of bands decreases with smaller $D$ and segregation is suppressed completely for sufficiently small $D$. These relatively simple features are not captured by any of the existing models of axial segregation.