Nonlocal granular rheology: Role of pressure and anisotropy

We probe the secondary rheology of granular media, by imposing a main flow and immersing a vane-shaped probe into the slowly flowing granulate. The secondary rheology is then the relation between the exerted torque T and rotation rate ω of our probe. In the absence of any main flow, the probe experiences a clear yield-stress, whereas for any finite flow rate, the yield stress disappears and the secondary rheology takes on the form of a double exponential relation between ω and T . This secondary rheology does not only depend on the magnitude of T , but is anisotropic — which we show by varying the relative orientation of the probe and main flow. By studying the depth dependence of the three characteristic torques that characterize the secondary rheology, we show that for counter flow, the dominant contribution is frictional like — i.e., T and pressure are proportional for given ω — whereas for co flow, the situation is more complex. Our experiments thus reveal the crucial role of anisotropy for the rheology of granular media. We still lack a full description of slow, dense flows of granular media — given boundary conditions, what determines the flow rate? Granular flows differ in two important ways from Newtonian flows. First, friction plays a central role — when grains have persistent contacts, as is the case in slow granular flows, inter-particle friction provides the main channel by which energy can be dissipated. Experiments in which the resistance to granular flow is probed by controlling the stress on a single moving boundary, such as in Couette, split-bottom or vane geometries [1–3], find that the shear stresses are proportional to the confining pressure, and that the ratio of shear to compressive stress, which can be seen as an effective friction coefficient, does not vary strongly with rate [1–5]. The crucial consequence from this rate independence is that the stress plateaus for strain rates going to zero. In other words, the stresses in slow granular flows are rate independent, and therefore the stress is not sufficient to set the strain rate. The second ingredient is non-locality. Several recent experiments [6–8] and theoretical works [9–12] indicate that for matter with granularity, the ”fluidity” in location A, i.e., the local relation between stress and strain rate, can be strongly influenced by the flow in location B. Such non-local behavior was first observed in the flow of emulsions [8], has also been observed in foams [13], and has been modeled by a diffusive model for the fluidity, leading to the introduction of a length scale which characterizes the nonlocality [8,10,12]. Kamrin and coworkers have adapted these ideas for frictional rheologies, capturing granular Couette flows [9], and more recently, the full flow profiles and height dependence of split bottom granular flows [11,14–18]. In all these studies, there is a single source driving the flow, and the nonlocality manifests itself via the spatial flow profiles. However, as flow in location B influences the fluidity in A, it is natural to study situations where the driving of the main flow and the probing of the rheology are independent. Two examples of such granular experiments which probe the “secondary rheology” concern the sinking of a passive probe into a granular bed that is stirred far away from the probe (in a split-bottom cell) [7], and the rheology probed by rods submersed in and dragged through a granular medium that is itself stirred in a Couette geometry [6]. Recently, numerical studies in which a plate is dragged through a 2D granular simple shear flow have been performed [10]. In all cases, the yield stress experienced by the probes was found to vanish as soon as there is any external flow imposed, and the rate of the probe was proportional to the stirring rate, sup-