Dynamics and structure of granular flow through a vertical pipe

Mitsugu Matsushita

Department of Physics, Chuo University,
Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan

Static, dynamic and statistical properties of granular materials are one of the most important topics in current science and its application to technology. In some situations granules behave like ordinary fluid or ordinary solid. On the other hand, a variety of unusual motions peculiar to granules can be observed, such as size segregation, bubbling, standing waves and localized excitations under vertical vibrations, avalanche and other unusual motions in a rotating mill, chute flow down a slope and a fluidized bed due to air injected inside a box containing granules. Emergence of density waves of granules flowing through a vertical pipe is also a typical and the simplest example of unusual features of granular motion.

We have shown how density waves of granular particles (ordinary sand) emerge, while they flow through a vertical glass pipe, by controlling air flow out of a flask attached to the bottom-end of the pipe. When the cock attached to the flask is fully open, air is dragged by falling granules and flows together with them. No density waves are observed for this situation. As the cock is gradually closed, however, the pressure gradient of air inside the pipe becomes gradually large, inducing the velocity difference between granules and air. As a result, density waves emerge from the lower part of the pipe. The smaller the rate of air flow, i.e., the more the cock is closed, the higher the onset point (along the pipe) of density waves. The onset of density waves is characterized by the growth of the lower frequency part of the power spectra of time-series signals of density waves. The power spectra of density waves display a clear power-law form $P(f) \sim f^{-\alpha}$ with the value of the exponent $\alpha = 1.33 \pm 0.06$, which is very close to 4/3. The value of $\alpha$ is robust even under the medium flow or variation of the pipe diameter, as far as density waves can be seen.

Very recently we have also controlled the flow rate of granules under the condition that the cock is completely closed, i.e., medium air does not flow with granules. When the flow rate of granules is small, they flow homogeneously with no density waves. The power spectra of the flow exhibit white-noise-like behavior. This is an expected result, just as raindrops fall homogeneously. What we observed rather unexpectedly is the following. If you gradually increase the flow rate of granules, density waves emerge suddenly at some threshold value of the flow rate. Above this threshold the power spectra exhibit clear power-law form with the same exponent $\alpha = 4/3$ robustly.

References
Jaeger H M, Nagel S R and Behringer R P 1996 Rev. Mod. Phys. 68 1259
Kadanoff L P 1999 Rev. Mod. Phys. 71 435
Peng G and Herrmann H J 1995 Phys. Rev. E 51 1745
Horikawa S, Nakahara A, Nakayama T and Matsushita M 1995 J. Phys. Soc. Jpn. 64 1870
Moriyama O, Kuroiwa N, Matsushita M and Hayakawa H 1998 Phys. Rev. Lett. 80 2833
Moriyama O, Kuroiwa N, Kanda M and Matsushita M 1998a J. Phys. Soc. Jpn. 67 1603
Moriyama O, Kuroiwa N, Kanda M and Matsushita M 1998b J. Phys. Soc. Jpn. 67 1616

Pattern formation in bacterial colonies

Mitsugu Matsushita

Department of Physics, Chuo University,
Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan

We have studied the growth mechanism and morphological change in colony formation of bacteria from the viewpoint of physics of pattern formation. Even very small number of bacterial cells, once they are inoculated on the surface of an appropriate medium such as semi-solid and nutrient-rich agar plate and incubated for a while, repeat the growth and cell division many times. Eventually the cell number of the progeny bacteria becomes huge, and they swarm on the medium to form a visible colony. The colony changes its form sensitively with the variation of environmental conditions. This implies that although usual bacteria such as Escherichia coli are regarded as single cell organisms, they never make their colony independently and randomly but somehow collaborate multicellularly. We have thus tried to extract some simple and universal behavior in growth from such complex bacterial systems.

Here we varied only two parameters to investigate the colony growth; concentrations of nutrient $C_n$ and agar $C_a$ in a thin agar plate as the incubation medium. Other parameters specifying experimental conditions such as temperature were kept constant. We mainly used a typical bacterial species Bacillus subtilis. Otherwise the experimental procedures are standard. It was found that colonies show characteristic patterns in the specific regions of values of $C_n$ and $C_a$ in the morphological diagram and the patterns change drastically from one region to another. They were classified into five types; fractal DLA-like, compact Eden-like, concentric ring-like, simple disk-like and densely branched DBM-like. We have experimentally elaborated characteristic properties for each of these colony patterns.

We have also examined colony formation of a species Proteus mirabilis, which forms concentric-ring-like colonies that look much more regular than those produced by Bacillus subtilis. The colony grows cyclically with the interface repeating an advance (migration) and a stop (consolidation) alternately. Our experimental results suggest that macroscopically the most important factor for its repetitive growth is the cell population density, i.e., that there seem to be higher threshold of the cell population density to start migrating and lower one to stop migrating.

We have tried to construct a phenomenological model which produces characteristic colony patterns observed in our experiments. The basic idea is that the main features of individual biological organisms are, if focusing on their population behavior, reproduction and active motion. Our modeling is, therefore, based on the reaction-diffusion-type approach for the population density of bacterial cells and the concentration of nutrient.

References
Matsushita M 1997 in Bacteria as Multicellular Organisms eds. J A Shapiro and M Dworkin (New York: Oxford UP) pp366-393
Wakita J, Ràfols I, Itoh H, Matsuyama T and Matsushita M 1998 J. Phys. Soc. Jpn. 67 3630
Itoh H, Wakita J, Matsuyama T and Matsushita M 1999 J. Phys. Soc. Jpn. 68 1436
Mimura M, Sakaguchi H and Matsushita M 2000 Physica A 282 283
Wakita J et al. 2001 J. Phys. Soc. Jpn. 70 911
Czirók A, Matsushita M and Vicsek T 2001 Phys. Rev. E 63 No.031915, 1
Watanabe K et al. 2002 J. Phys. Soc. Jpn. 71 650