Granular materials are ubiquitous in our daily lives. While they have been the subject of intensive engineering research for centuries, in the last two decades granular matter has attracted significant attention from physicists. Yet despite major efforts by many groups, the theoretical description of granular systems remains largely a plethora of different, often contradictory concepts and approaches. Various theoretical models have emerged for describing the onset of collective behavior and pattern formation in granular matter. This review surveys a number of situations in which nontrivial patterns emerge in granular systems, elucidates important distinctions between these phenomena and similar ones occurring in continuum fluids, and describes general principles and models of pattern formation in complex systems that have been successfully applied to granular systems.

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Patterns and collective behavior in granular media: Theoretical concepts

Yield stress fluids are encountered in a wide range of applications: toothpastes, cements, mortars, foams, muds, mayonnaise, etc. The fundamental character of these fluids is that they are able to flow (i.e., deform indefinitely) only if they are submitted to a stress above some critical value. Otherwise they deform in a finite way like solids. The flow characteristics of such materials are difficult to predict as they involve permanent or transient solid and liquid regions that are generally hard to locate a priori. Here we review the present state of the art as it appears from experimental data for flows of simple (non-thixotropic) yield stress fluids under various conditions, viz., uniform flows in straight channels or rheometrical geometries, complex stationary flows in channels of varying cross-section such as extrusion, expansion, flow through a porous medium, transient flows such as flows around obstacles, spreading, spin-coating, squeeze flow, and elongation. The effects of surface tension, confinement, and secondary flows are also reviewed. We focus especially on experimental work identifying internal flow characteristics that can be compared with numerical predictions. It is shown in particular that: (i) deformations in the solid regime can play a critical role in transient flows; (ii) the yield character is not apparent in the flow field when the boundary conditions impose large deformations; (iii) the yield character is lost in secondary flows.

Yield stress fluid flows: A review of experimental data

Experiments on the low-speed impact of solid objects into granular media have been used both to mimic geophysical events1,2,3,4,5 and to probe the unusual nature of the granular state of matter6,7,8,9,10. Observations have been interpreted in terms of conflicting stopping forces: product of powers of projectile depth and speed6; linear in speed7; constant, proportional to the initial impact speed8; and proportional to depth9,10. This is reminiscent of high-speed ballistics impact in the nineteenth and twentieth centuries, when a plethora of empirical rules were proposed11,12. To make progress, we developed a means to measure projectile dynamics with 100 nm and 20 μs precision. For a 1-inch-diameter steel sphere dropped from a wide range of heights into non-cohesive glass beads, we reproduce previous observations6,7,8,9,10 either as reasonable approximations or as limiting behaviours. Furthermore, we demonstrate that the interaction between the projectile and the medium can be decomposed into the sum of velocity-dependent inertial drag plus depth-dependent friction. Thus, we achieve a unified description of low-speed impact phenomena and show that the complex response of granular materials to impact, although fundamentally different from that of liquids and solids, can be simply understood.

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Unified force law for granular impact cratering

Direct measurements of the acceleration of spheres and disks impacting granular media reveal simple power law scalings along with complex dynamics which bear the signatures of both fluid and solid behavior. The penetration depth scales linearly with impact velocity while the collision duration is constant for sufficiently large impact velocity. Both quantities exhibit power law dependence on sphere diameter and density, and gravitational acceleration. The acceleration during impact is characterized by two jumps: a rapid, velocity-dependent increase upon initial contact and a similarly sharp depth-dependent decrease as the impacting object comes to rest. Examination of the measured forces on the sphere in the vicinity of these features leads to an experimentally based granular force model for collision. We discuss our findings in the context of recently proposed phenomenological models that capture qualitative dynamical features of impact but fail both quantitatively and in their inability to capture significant acceleration fluctuations that occur during penetration and which depend on the impacted material.

Scaling and dynamics of sphere and disk impact into granular media.

Very fine sand is prepared in a well-defined and fully decompactified state by letting gas bubble through it. After turning off the gas stream, a steel ball is dropped on the sand. On impact of the ball, sand is blown away in all directions ("splash") and an impact crater forms. When this cavity collapses, a granular jet emerges and is driven straight into the air. A second jet goes downwards into the air bubble entrained during the process, thus pushing surface material deep into the ground. The air bubble rises slowly towards the surface, causing a granular eruption. In addition to the experiments and the discrete particle simulations we present a simple continuum theory to account for the void collapse leading to the formation of the upward and downward jets.

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Impact on soft sand: Void collapse and jet formation

We present the results of experiments on impact craters formed by dropping a steel ball vertically into a container of small glass beads. As the energy of impact increases, we observe a progression of crater morphologies analogous to that seen in craters on the moon. We find that both the diameter and the depth of the craters are proportional to the 1/4 power of the energy. The ratio of crater diameter to rim-to-floor depth is constant for low-energy impacts, but increases at higher energy, similar to what is observed for lunar craters.

http://fluid.ippt.gov.pl/ictam04/CD_ICTAM04/FM17/12338/FM17_12338.pdf

Morphology and scaling of impact craters in granular media.

We study fingering that occurs when hot glycerine displaces cooler, more viscous glycerine in a radial Hele-Shaw cell. We find that fingering occurs for a sufficiently large initial viscosity contr...

Viscous fingering with a single fluid

We study the spreading and fingering of drops of silicone oil on a rotating substrate for a range of rotation speeds and drop volumes. The spreading of the drop prior to the onset of fingering is found to follow the theoretically predicted time dependence, but with a large shift in time scale. For the full range of experimental parameters studied, the contact line becomes unstable and fingers develop when the radius of the drop becomes sufficiently large. We study the growth of perturbations around the perimeter of the drop and find the growth rate of the most unstable mode to agree well with the predictions of lubrication theory. The number of fingers which form around the perimeter of the drop is found to be a function of both rotation speed and drop volume, and is also in excellent agreement with theoretical predictions.

Spreading and fingering in spin coating.

We have performed numerical simulations of the flow of hot glycerine as it displaces colder, more viscous glycerine in a radial Hele–Shaw cell. We find that fingering occurs for sufficiently high i...

Numerical simulations of a viscous-fingering instability in a fluid with a temperature-dependent viscosity

We study the deformation, spreading, and fingering of small droplets of a yield-stress fluid subjected to a centrifugal force on a rotating substrate. At low rotation rates and for small enough droplets, the droplets deform elastically but retain their essentially circular contact line. For large enough droplet volumes and rotation speeds, however, one or more fingers eventually form and grow at the edge of the drop. This fingering is qualitatively different from the contact line instability observed in other fluids, and appears to be a localized phenomenon that occurs when the stress at some point on the perimeter of the drop exceeds the yield stress.

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Kristi E. Holloway

Papers

Morphology and scaling of impact craters in granular media.

Viscous fingering with a single fluid

Spreading and fingering in spin coating.

Numerical simulations of a viscous-fingering instability in a fluid with a temperature-dependent viscosity

Spreading and fingering in a yield-stress fluid during spin coating