Flow under gravity acceleration

Another way to probe hydrodynamic descriptions of rapid granular dynamics is the study of flows along inclined channels. In this kind of experiments the whole material is accelerated by gravity, but the friction with the plane induce shearing, so that measurements similar to the ones performed in Couette cells can be performed. The first experiments in this configuration were performed by Ridgway and Rupp [188], and reviews can be found in the works of Savage [192] and Drake [76]. Interest has focused on constitutive relations, as before, but also on the profiles of the hydrodynamic fields, mainly flow velocity and solid fraction: computer simulations (see for example Campbell and Brennen [53] and for an exhaustive review the classical work of Campbell [52]) have allowed the measurement of the temperature field: this has confirmed the picture of a gas-like behavior, explaining the reduction of density (solid fraction) near the bottom by means of an increase of granular temperature, due to the shear work. In this framework the scheme representing the ``mechanical energy path'' sketched by Campbell in his review on rapid granular flow [52] is enlightening. The external driving force (i.e. gravity) induces mean motion (kinetic energy) which consequently generates friction with boundaries, that is shear work (granular temperature). The randomization represented by the granular temperature induces collisions among the grains, which are dissipative and therefore produce heat. Moreover, granular temperature generates internal (transversal as well as normal) stresses.

Figure 1.8: The draining from the bottom of a silo: it is clear the separation between a region where grain move downward and a region where grain do not move at all

Another configuration of granular flow under the force of gravity is the simple hopper geometry (a hopper is a funnel-shaped container in which materials, such as grain or coal, are stored in readiness for dispensation). The bottom of hopper is opened and the grains start to pour out. As already discussed the pressure (and therefore the flow rate) does not depend upon the height of the column of material. However the flux of grains leaving the container produces complex flow regions inside the container. Brown and Hawksley [47] identified four regions of density and velocity, most notably a tongue of dense motion just above the aperture and an area of no grain motion below a cone extending upwards from the opening (a similar can be observed in a silo, see Fig. fig_draining). Baxter et al. [19] have showed that for large opening angles, density waves propagate upward from above the aperture against the direction of particle flow, but downwards for small angles. The flow can even stop due to ``clogging'', i.e. the grains can form big arches above the aperture and sustain the entire weight of the column.

More recent experiments have been performed on granular flows along inclined planes or chutes, evidencing other interesting phenomena:

• Validations of kinetic theory Azanza et al. [7] have recently repeated the experiment of grain flow along an inclined channel, studying the stationary profiles of velocity, solid fraction and granular temperature. They have verified that there is a limited range of inclinations of the channel that allow for a stationary flow. Moreover they have probed the validity of the kinetic theories developed in the previous years [194,121,152,120,150], based on the assumption of slight perturbation to the Maxwellian equilibrium. The profiles of hydrodynamic fields show two different regions: a collisional region (higher density) where the transport is mainly due to collisions, and a ballistic region (on the upper free surface) where the grains fly almost ballistically.

• Clustering: Kudrolli and Gollub [131,133] have studied the formation of clusters measuring the density distribution in an experiment consisting of steel balls rolling on a smooth surface which could or could not be inclined with a vibrating side. The experiment takes into account a monolayer (not completely covered) of grains, in order to study a true $2d$ setup. In both cases (inclined or horizontal), at high enough global densities, the distribution of density (going from Poissonian to exponential) indicates strong clustering.

• Non-Gaussian velocity distributions: Kudrolli and Henry [132] have studied the distributions of velocities in the same setup of the previous experiment, with varying angles of inclination, obtaining non-Gaussian statistics with enhanced high energy tails; moreover they have seen that increasing the angle of inclination the distributions tends toward the Maxwellian (see Fig. fig_kudrolli).

  Figure 1.9: The experiment of Kudrolli and Henry [132]: distributions of horizontal velocities of grains rolling on an inclined plane, with the inferior wall vibrating.

  

• Velocity correlations: A very recent experiment of Blair and Kudrolli [71] with the same experimental setup of the previous ones has revealed strong correlations between velocity particles.

• Size segregation in silo filling or emptying: Samadani et al. [191] have recently studied the phenomena of size segregation in a quasi-two dimensional silo emptying out of an orifice. They [190] have also studied the effects of interstitial fluids.

• Size segregation in rotating drums: another typical experiment, inspired to many industrial situations, is the tumbling mixer, or rotating drum, i.e. a container with some shape that rotate around a fixed axis, usually used to mix different kind of granular materials (typically powders, in the pharmaceutical, chemical, ceramic, metallurgical and construction industry). Depending on the geometry of the mixer, the shapes of the grains, the parameters of the dynamics and so on, the grains can mix as well as separate. A very large literature exists on this phenomena (see a recent review of Ottino and Khakhar [177]). Usually segregation is strictly tied to convection: there is a shallow flowing layer on the surface of the material inside the rotating drum, the grains at the end of it are transported into the bulk and follow a convective path so that they emerge again in another point of the surface. Segregation happens in many different ways: segregated bands appear and slowly enlarge (like in a coarsening model), segregation can emerge in different directions, e.g. parallel to the rotation axis as well as transversal to it.