Transition to density waves of granular particles flowing through a vertical pipe

We show experimental results regarding how density waves of granular particles flowing through a vertical glass pipe emerge by controlling two kinds of parameters; air flow out of a flask attached to the bottom-end of the pipe and inflow of granular materials from a hopper attached to the top-end. It was found, by controlling the former parameter, that the most important factor for the onset of density waves is the strong interaction between granular particles and the air. It was also found, by changing the latter parameter, that there is a sharp transition from the uniform flow to the density waves at the threshold value of the inflow rate of granules. Power spectra of density waves show a clear power-law form S(f) similar to f(-alpha), where alpha similar or equal to 4/3. The value of alpha is robust in that, once density waves occurred, it was independent of the two parameters described above.