Couette cylinders

The first tentatives of studying granular media under the point of view of rheology, i.e. transport properties (discussed in more detail in section 2.4) have been performed using typical shear experiments used to probe ordinary fluids. In particular the Couette geometry has been largely used and is still now an important tool of investigation.

Figure 1.7: The experiment of Mueth and coworkers on a Couette cylinder: the paths of the internal forces are evidenced by means of non-invasive X-ray imaging

Even if there were earlier important experimental studies on the flow properties of granular materials (mainly initiated by Hagen [104] and Reynolds [187]), the modern pioneering work on the constitutive behavior of rapid granular flows was Bagnold's experimental study [8] of wax spheres, suspended in a glycerin-water-alcohol mixture and sheared in a coaxial cylinder rheometer (Couette experiment). His main finding was a constitutive relation between internal stresses and shear rate:

$$\mathcal{T}_{ij}=\rho_p\sigma^2\gamma^2 \mathcal{G}_{ij}(n)$$ (1.13)

with $\rho_p$ the particle density, $\sigma$ the particle radius, $\gamma$ the shear rate and $\mathcal{G}_{ij}$ a tensor-valued function of the solid fraction $\phi$. This relation has been confirmed in shear-cell experiments with both wet or dry mixtures by Craig et al. [70], Hanes et al. [105], Savage et al. [195], and in many computer simulations [53,54,221].

Bagnold measured not only shear stresses (i.e. transversal components, say $i \neq j$ in $\mathcal{T}_{ij}$), but also normal stress ($i=j$), that is the analogous of pressure in gas kinetics: he referred to them as ``dispersive stresses'' as they tend to cause dilation of the material.

More recent experiments have focused on different phenomena observed in the Couette rheometer:

• Fluctuations of stresses: already in the experiments of Savage and Sayed [195] large fluctuations of internal (normal) stresses were observed; more recently Howell and Behringer [113] have seen that in a 2D Couette experiment the mean internal stress follows a continuous transition when the packing fraction of the granular material changes and passes through a critical value $\phi_c=0.776$: when the packing fraction is above the critical threshold the material shows strong fluctuations of internal stress, while under the threshold the stresses are averagely zero and the system is highly compressible.

• 3d experiments: Mueth et al. [167] have studied the formation of microstructures in the dense shearing regime in a 3D Couette rheometer, using non-invasive imaging by X-Ray microtomography (see Fig. fig_couette); they have found that the velocity parallel to the shear direction decays more rapidly than linear (from exponential to Gaussian-like decay, depending upon the regularity of the grains). A similar strong decay of the flow with the distance from the moving wall was observed in many experiments, for example recently by Losert et al. [145]

• Diluted (air-fluidized) shear: Losert et al. [143] have performed a Couette experiment with a flow of air coming from the bottom of the cylinder, in order to fluidize the material and obtaining smoother profiles. They have put in relation the RMS fluctuations of velocity and the shear forces, observing that $T^{1/2} (y) \sim \gamma(y)^{\alpha}$ with $\alpha= \simeq 0.4$, and suggesting a phenomenological model that explains the shear velocity profiles.

• Size segregation: Khosropour et al. [123] have observed convection patterns and size segregation in a Couette flow with spherical glass beads; they also checked the effect of interstitial fluids finding it irrelevant.

• Planetary rings: planetary rings (those of Saturn for example) have been sometime studied in the framework of granular rheology, whereas the ``geometry'' of the planetary experiment is similar to a Couette cell (grains are circularly sheared because the angular velocity depends upon the distance from the planet). A review of these study can be found in the work of Brahic [37].