Granular flows

. Our understanding of granular flows has progressed considerably over the past two decades. In this talk, I will first review the advances made in the rheology of dense granular media based on the concept of pressure-imposed rheology, discussing the different regimes, from viscous to inertial. I will also illustrate how these rheological descriptions can be used in a two-phase flow approach to model more complex flow configurations, at least in simple small scaled experiments mimicking granular avalanche flows. Can these advances help describe and predict debris flows, is not an easy question, which I will try to discuss by pointing out the limitations and robustness of the rheological models.


Introduction
Mixture of grains and fluids are ubiquitous in many geophysical situations and understanding their flow behaviors is of crucial importance to better predict and describe natural hazards. To progress in this quest one can consider much simpler system consisting of model granular materials. However, even for these systems, the description of the flow properties remains a challenge, mainly because this multi-body system involves complex particle interactions [1]. The search for a continuum description of particulate flows has therefore attracted a lot of effort at the frontier between physics, rheology, geophysics. In this talk, we present a review of the recent advances in the rheology of granular flows and illustrate how the study of simple small-scale systems can help to understand the underlying physics and develop relevant two-phase flow modelling.

Pressure imposed versus volume imposed rheology
To study the rheology of complex fluids (i.e. the flow behavior in response to an applied force), the traditional approach is to consider the plane shear configuration. The fluid of interest is confined between two plates separated by a fixed distance and sheared at a constant shear rate. Conventional rheology consists in measuring how the shear stress τ varies with the shear rate . In the case of granular materials, another crucial parameter must be considered, namely the granular pressure Pp. During shear, the grains push on the wall and exert a normal stress Pp on the wall. Consequently, there are two ways to study the rheology of granular systems. In the classical method, the material is sheared at a constant volume, keeping the distance between the two plates constant. In this case, it is necessary to measure how both the shear stress τ and the normal granular stress Pp vary with shear rate for a given grain volume fraction φ, where φ is the ratio of the volume occupied by the grains over the total volume of the sample. However, a second method has proven to be very relevant for granular media and consists of shearing the material by imposing the granular pressure Pp. In this latter case, the volume occupied by the grains φ is free to adjust. The latter configuration is reminiscent of avalanche flow where gravity prescribes the confining stress and the material is free to dilate. In our group, we have developed a custom rheometer able to investigate this pressure imposed rheology as well as volume imposed rheology of particulate systems. The pressure imposed approach provides a unique way to study the very dense flow regime and has been used to study many different systems such as dry granular material [2], viscous suspensions [3], suspensions in non newtonian fluids [3].
For dry granular media made of rigid particles of size d and density ! , dimensional analysis dictates that when the material is sheared under a prescribed pressure Pp the system is controlled by a single dimensionless parameter called the inertial number: =̇/)Pp/ ! [3]. Therefore, the shear stress τ must be proportional to the pressure Pp (the only stress scale in the problem), with a coefficient of proportionality (a macroscopic coefficient of friction) that is a function of I. The volume fraction φ must also be a function of the inertial number I, such that: These scaling laws have been measured experimentally both in experiments and DEM numerical simulations [2,3], providing empirical expressions for the friction coefficient and the dilatancy law. When approaching the quasi-static flows regime (I->0), both the friction coefficient and the volume fraction tend to constants " and " ( . 1 ) defining what the physicist called the jamming transition.
The case of a suspension whose particles are immersed in a liquid of the same density and of viscosity # can also be studied under pressure-imposed conditions (Fig. 1b). Experimentally, the suspension is sheared by a porous top plate that is free to move vertically [2,4]. The mesh size of the plate is smaller than the particle size and the liquid can flow through it. In the limit of the viscous regime, when inertia plays no role, the rheology is controlled by a single dimensionless number called the viscous number J= #̇/ Pp. The friction coefficient and the volume fraction are then given by the following relations: Experiments were conducted showing the relevance of the approach and empirical formulations for the constitutive laws have been proposed [4]. It is interesting to compare this approach with the conventional rheology carried out with a constant volume fraction . The shear stress τ and the granular pressure Pp are then functions of the shear rate ̇ and of the volume fraction φ. Dimensional analysis implies in this case that the shear and the normal stress τ and Pp for a dry granular material are given by the following expressions with Bagnoldian scaling : where % (ϕ), and & (ϕ), are two dimensionless increasing functions of φ, which appear to diverge close to the maximum volume fraction " (Fig. 1c). For viscous suspensions, the volume-imposed rheology is given by a Newtonian scaling: where % (ϕ) and & (ϕ) are called the shear and normal relative viscosity respectively, and are two dimensionless increasing functions of φ, which also diverge close the maximum volume fraction φc (Fig.1d). Empirical and theoretical expression have been developed in the literature for the constitutive laws [5].
The existence of two different descriptions may be confusing for the neophyte interested in granular rheology. The material is described as a frictional material in the pressure-imposed approach, with a yield stress and a shear stress proportional to the pressure, whereas it is described as a viscous or Bagnold liquid in the volume-imposed approach, with no yield stress. This illustrates the fact that the behavior of granular materials strongly depends on the way they are manipulated, and on the control parameters. The two approaches are of course fully equivalent and one can easily switch from one description to another. In geophysical applications and gravity flows, pressure-imposed expressions are often the most relevant.

Viscous/ Inertial transition
While the above two extreme regimes are relatively well documented, the transition from the Newtonian rheology in the viscous limit to a Bagnoldian rheology when inertia is increased is not well deciphered. It is considered to take place when the viscous and inertial stresses have the same order of magnitude, i.e., to be described by the ratio between the inertial and viscous stress scales, namely the Stokes number = 2̇/ . However, since the seminal work of Bagnold [6] who identified the two regimes, the studies are scarce and still inconclusive. Numerical studies [7][8][9] finds that it happens at a transitional Stokes number around 1-2, independent of the volume fraction, while experiments [10] and theoretical work for frictionless particles suggest that the transitional Stokes number vanishes when approaching the jamming transition. We have used our custom rheometer which can be run in a pressure-or a volume-imposed mode to examine this transition in the dense regime close to jamming [11]. By systematically varying the interstitial fluid, shear rate, and packing fraction in volume-imposed measurements, we have shown that the transition takes place at a Stokes number of 10 independent of the packing fraction (Fig. 2). Using pressure-imposed rheometry, we also investigated whether the inertial and viscous regimes can be unified as a function of a single dimensionless number combining I and J, based on an assumption of stress additivity [11].

Towards a hydrodynamic description of granular flows
The knowledge of the response of a granular medium to a plane shear is a starting point to develop full tensorial rheological models, able to describe complex flow configurations with shear in different directions.

Continuum modeling of dry granular media
For dry granular media, the μ(I) friction law has been generalized to a tensorial form, assuming that the material is incompressible and that the shear stress tensor is collinear with the strain rate tensor [12]. This approach is equivalent to a visco-plastic description, in which both the yield stress and the viscosity are pressure-dependent quantities. Such a simple description has been implemented in codes, and quantitative predictions have been made for flows on inclined plane, roll waves instability, flows in silo, collapses of columns. Limits exist for slow flows where non-local effects start to play a role, and for which more sophisticated models are developed [13].

Continuum modeling of immersed granular media
The case of grains immersed in a liquid relevant to debris flows is more complex than the case of dry granular media due to the coupling with the interstitial fluid. In many situations, a relative motion takes place between the liquid and the grains, and the dynamics of both phases need to be properly described. Two-phase flow models have been developed, which consist in considering the fluid and the grains as two intricated continuum phases, and in writing the mass and momentum equations for each of them. The difficulty lies in the choice of the stresses for each phase and in the choice of the interphase force [5]. The granular rheology discussed in the previous section provides expressions for the granular stresses, which can be injected in two-phase flow models to give predictions in different configurations, especially for avalanche flow down an inclined plane.
However, the constitutive laws are based on the generalization of the properties of steady uniform plane shear flows. Although real successes have been obtained in predicting complex configurations, this simple approach fails to properly capture the details of unsteady and non uniform flows.
An important missing ingredient concerns transient flows. It is well known that a granular material can be prepared at rest at different volume fractions, above or below the maximum volume fraction measured under steady shear φc. This means that under shear, a medium prepared at a volume fraction denser than φc should dilate (the well-known Reynolds dilatancy), and that a packing looser than φc should compact. As a result, a pile initially prepared in a dense state does not behave the same as a pile prepared in a loose state [14,15]. This variation of volume fraction implies a coupling through the pore pressure that is known to play a crucial role and that needs to be taken into account if one want to correctly describe for example the triggering of avalanches.
We have studied this problem experimentally by investigating the initiation of the flow of a uniform granular layer immersed in a viscous fluid [14] and during the collapse of an immersed column [15]. In such small-scale experiments where all parameters can be controlled, we have been able to study precisely how the initial volume fraction influences the dynamics.    conditions but a smaller µ sf = 0.21 ± 0.04 with UCON mixtures.
The rheological measurements were undertaken with a custom rheometer [11] depicted in Fig. 1(c). This shearing device is composed of an annular cylinder (of inner radius 43.95 mm and outer radius 90.28 mm) that is attached to a bottom plate and covered by a top permeable plate. Both plates are made rough by using wire meshes with openings of 6.3 mm corresponding to 1.3 d. The top plate can be moved vertically with a linear positioning stage and enables the fluid to flow through it but not the particles. The bottom plate can be rotated at a constant angular velocity to produce a linear shear. The range of shear rate that can be achieved is 1 ˙ 50 s −1 . The thickness of the cell, h, can be varied between 24.8 and 27.5 mm, i. e. 5.3 d h 6 d. The shear stress, ⌧ , is deduced from measurements of the torque exerted on the top plate after calibration with the pure fluid. The component of the normal stress perpendicular to the top plate, simply referred as the particle pressure P , is given by a precision scale attached to the translation stage after correction for buoyancy. The volume fraction, , can be deduced from the position of the top plate recorded by a sensor. We estimated the accuracy based on the fluctuations around the stationary values to be 7%, 3%, and ±0.002 on the measurements of ⌧ , P , and , respectively.
The rheometer could be run in a pressure-imposed or a volume-imposed mode using a feedback control loop involving the scale measurement or the position of the top data collected in volume-imposed rheome pensions having suspending fluids of varia ent . A Newtonian regime in which the mal viscosities, ⌘ s = ⌧ ⌘ f˙ and ⌘ n = P ⌘ f are independent of St (i.e. independent of with increasing is observed for St < 10. A the rheology transitions smoothly from continuously shear thickening. For St > 10 regime in which ⌘ s and ⌘ n scale linearly w ˙ ) is reached. We have also plotted in F fective friction coe cient µ = ⌧ P = ⌘ s ⌘ n recover the Stokes regime for St < 10. µ decreases with increasing St as ⌘ s pre transition to the Bagnoldian regime than We have also reported in these graphs ing from Fig.7 of Bagnold's paper [3] obta transition region for a single = 0.555 ( cyan). Strikingly, although they were obt ferent particles (spheres of nearly 50% mix wax and lead stearate with d = 1.32 mm that of water), di↵erent fluids (water wit and glycerin mixture with ⌘ f = 7 mPa s), experimental device, these data show a sim around St ≈ 10 and are located in betwe data at = 0.545 and 0.567.
The fact the transition takes place at a number independent of is evidenced i Fig. 2(a,b)  To theoretically capture these effects, an additional law describing the dilatation/compaction process has been proposed and coupled to the rheological description discussed in the previous section, leading to predictions for the initiation of submarine granular avalanches [14].

Conclusions
Our understanding of granular flows has improved during the past decade and there exists rheological description capable of predicting non trivial behavior, at least for simple materials. However, debris flows are far from being simple granular media. Several major difficulties make it difficult to transpose the granular rheology to debris flows without care. The first is the polydispersity, and the very wide range of particle sizes present in debris flow. Although segregation models are rapidly developing in the community [16], the role of fine grains and of the continuous gradation present in natural events is unclear. Another difficulty lies in the boundaries at the bottom of the flow, where erosion occurs, which is not yet fully understood. the flow may erode an a priori cohesive soil, a problem that to our knowledge has not been studies in detail. Last but not least, the description and prediction of the onset of catastrophic events remains a challenge, which goes beyond the rheological approach.