PATTERN FORMATION IN GRANULAR FLOWS

Moreover, we recently showed the phenomenon of granular self-stratification: a spontaneous periodic pattern arising as a consequence of flow instabilities of granular mixtures poured in a vertical cell. This striking behavior and the importance of mixing problems from a technological point of view have led to a broad interest in granular materials in the physics and engineering community.

Strat. experiment Strat. model

Seg., experiment Seg., model

Spontaneous stratification in granular mixtures---i.e. the formation of alternating layers of small-rounded and large-faceted grains when one pours a random mixture of the two types of grains into a quasi-two dimensional vertical Hele-Shaw cell---has been recently reported by H. A. Makse, S. Havlin, P. R. King, and H. E. Stanley, "Spontaneous stratification in Granular Mixtures", [ Nature 386, 379 (1997)].

The typical experimental set consists of a vertical ``quasi-two-dimensional'' Hele-Shaw cell with a narrow gap of 5 mm separating two transparent plates of 300 mm by 200 mm. We close the left edge of the cell leaving the right edge free, and we pour continuously, near the left edge, an equal-volume mixture of grains differing in size and shape.

Spontaneous stratification, arises when the grains composing the mixture differ not only in size but also in shape (or friction properties). When a mixture of large grains that are more faceted and small grains that are less faceted is poured in a ``granular Hele-Shaw cell'' (two vertical slabs separated by a gap of typically 5--10 mm), the mixture spontaneously stratifies into alternating layers of larger faceted grains and smaller rounded grains. The Figure above shows an example of such stratification. A mixture of large cubic sugar grains (typical diameter 0.8 mm) and smaller spherical glass beads (diameter 0.19 mm) is poured in the cell. We notice the striped pattern with approximately constant wavelength.

In contrast, when the mixture is composed of larger less faceted grains and smaller more faceted grains, the mixture only segregates---i.e., the small more-faceted grains are found preferentially at the top of the cell, while the large less-faceted grains are found near the bottom. The Figure above shows an example of such segregation, when a mixture of small faceted sand grains (typical size 0.3 mm) and large spherical glass beads (typical size 0.8 mm) is poured in the cell.

Theoretical studies

We use a formalism developed by Bouchaud and colaborators, as well as cellular automata models. In collaboration with Pierre Gilles de Gennes and Thomas Boutreux from College de France, Paris, we have developed a Canonical model of segregation in granular flows [ T. Boutreux, H. A. Makse and P. G. de Gennes, Surface Flows of Granular Mixtures: III. Canonical Model, Eur. Phys. J.-B 9, 105-115 (1999).]

The results of our simulations can be seen in the figures above. In the theoretical formalism, the grains are considered to belong to one of two phases: a static or bulk phase if the grain is part of the solid sandpile, and a rolling or liquid phase if the grain is not part of the sandpile but rolls downward on top of the static phase. Using this formalism, we have reproduced and studied the dynamics of spontaneous stratification which is governed by the existence of a kink at which the grains are stopped during an avalanche.

Stratification is an instability developed due to a competition between size segregation and shape segregation as we studied in H. A. Makse, Stratification Instability in Granular Flows, Phys. Rev. E, 56, 7008-7016 (1997). We show that the stable solution of the system is a segregation solution due to size (large grains tend to segregate downhill near the substrate and small grains tend to segregate uphill) and shape (rounded grains tend to segregate downhill and more faceted grains tend to segregate uphill). As a result, the segregation solution of the system is realized for mixtures of large-rounded grains and small-cubic grains with the large-rounded grains segregating near the bottom of the pile. Stability analysis reveals the instability mechanism driving the system to stratification as a competition between size-segregation and shape-segregation taking place for mixtures of large-cubic grains and small-rounded grains. The large-cubic grains tend to size-segregate at the bottom of the pile, while at the same time, they tend to shape-segregate near the pouring point. Thus, the segregation solution becomes unstable, and the system evolves spontaneously to stratification.

Experimental studies

We studied experimentally the dynamical processes leading to spontaneous stratification [H. A. Makse, R. C. Ball, H. E. Stanley, and S. Warr, Dynamics of Granular Stratification, Phys. Rev. E 58, 3357 (1998)]. I used a high speed video camera at the Cavendish Laboratory at Cambridge University, UK. We divide the dynamical process of stratification into three stages to study the dynamics: (a) The avalanche of grains down the slope, and size segregation of grains in the rolling phase due to ``percolation''. (b) The formation of the ``kink''---an uphill wave at which grains are stopped. (c) The uphill motion of the kink and formation of a pair of layers.

We also studied the well-developed flow regime where the rolling grains segregate during the flow [ H. A. Makse, Kinematic Segregation of Flowing Grains in Sandpiles, Eur. Phys. J. B, 7, 271-276 (1999).] In this regime the thickness of the layer of rolling grains is larger than the typical size of a grain d (typically 5d), and the smaller rolling grains are found to percolate downward in the rolling phase to form a sublayer of smaller rolling grains underneath the sublayer of larger rolling grains. This dynamical size segregation process, known as ``percolation'' or ``kinematic sieving'' contributes to the stratification of grains.

Why Do We Care about this Phenomenon?

There are always two reasons to care about a research project, one is practical and the other scientific. One practical feature rest on the fact that understanding the basic properties of segregation is of immense importance to many industries, such as pharmaceutical, chemical and agricultural industries. For example, over a trillion kilograms of granular materials are produced per year in the U.S., and 61 billion are linked to granular material technology in the chemical industry alone. Clearly, establishing the basic principles of granular segregation remains a very important issue.

A second practical reason is to try to understand periodic pattern formation in sedimentary structures such as sandstone. This fact is of vital importance to the oil industry, since much of the earth's oil is trapped in sandstone. The phenomenon of sandstone stratification is familiar to specialist and layman alike. However, the longstanding question of how such periodic patterns are generated during sand ripple migration has not been properly addressed. When there is a windstorm, the wind blows very strongly and picks up grains off the ground. These grains fly through the air and ultimately come back to the ground again by falling relatively freely out of the sky. And as they fall, apparently, the layers can be formed of alternating sizes of grains. We propose to develop a possible unifying mechanism for the origin of the most common sedimentary structures observed in Nature.

The understanding of layering in sandstone is not the only practical reason. In a recent ``News & Views'' article in Nature, J. Fineberg described the relevance of segregation in avalanches to a phenomenon which is quite dramatic. Namely that in an avalanche of rock from a height of a thousand meters, the flow after the avalanche can be as much as ten times bigger. The flow can be dramatic--- e.g., the flow from one particular such rock slide that took place in Frank, Alberta, Canada in 1903 actually wiped out the town which was quite a distance away---over 4 km--- from the mountain that initiated the avalanche. The mechanism of this long-runout rock slides has been the subject of much speculation. Possible explanations include a segregation of small particle at the bottom of the flow which acts as a ball-bearing mechanism, or an acoustic fluidization of a narrow zone beneath the flow. However, so far, there is no conclusive evidence to understand this phenomenon, which we believe is closely related to the avalanche dynamics we see in our experiments and models.

Links in the press

Our research has been featured in the press:

• Nature Editorial, News & Views editorial by J. Fineberg, Nature 386, 323 (1997).
• Discovery Magazine, ``Top 100 discoveries of the year'', January 1998.
• Science News 151, 206 (5 April 1997) ``Grains sort themselves into layers'';
• Pour la Science, France, "Les couches de sable", No. 237 July 1997
• Daily Telegraph (London), March 29, 1997, ``Rocks that roll across the plain'';
• MRS Bulletin, March 1997, p. 72;
• The Ottawa Citizen (Canada), April 6, 1997, ``Deadly rock slide explained at last''
• Physics World, May 1997, p. 29
• Super Interessante, Brazil, N. Worcman, August 1997, p. 18
• Bostonia, T. McNeil,``Angles of repose'', p. 44, June 1997
• Geo Magazine, ``Schutten schafft Ordnung'', June 1997, p. 134
• Frankfurter Allgemeine Zeitung, R. Scharf, June 1997;
• BBC World News, ``World in Action'', August 1997, Radio 4 article;
• New Scientist, 7 December 1996; H. E. Stanley review of "The Web of Life" by F. Capra.

Collaborators

P. Cizeau and H. E. Stanley (BU), P.-G. de Gennes and T. Boutreux (College de France), H. Herrman (ESPCI, Paris), S. Warr and R. C. Ball (Cambridge University, UK), S. Havlin (Bar-Ilan), and P. R. King (BP).