Erratum: Viscous to Inertial Transition in Dense Granular Suspension [Phys. Rev. Lett. 129, 078001 (2022)].

This corrects the article DOI: 10.1103/PhysRevLett.129.078001.

Finite size effects during the penetration of objects in a granular medium.

In many industrial or geotechnical applications, objects move through a granular medium and an important issue is the prediction of the force that develops during the motion of the intruder. In this paper, we experimentally study the vertical penetration of intruders into granular media and analyze both the average force and the fluctuations during motion. We investigate configurations where the size of the intruder becomes close to a few grain sizes, a regime that has not been studied before. Finite size effects are observed, showing that both the mean force and the fluctuations significantly increase when decreasing the ratio of the intruder size to the particle size, and scaling laws are identified to characterize this effect. The role of a conical tip in front of the cylinder to facilitate the penetration is also studied, showing that it is more efficient when the aspect ratio between the intruder size and the grain size is low.

Insights into the rheology of cohesive granular media.

Characterization and prediction of the "flowability" of powders are of paramount importance in many industries. However, our understanding of the flow of powders like cement or flour is sparse compared to the flow of coarse, granular media like sand. The main difficulty arises because of the presence of adhesive forces between the grains, preventing smooth and continuous flows. Several tests are used in industrial contexts to probe and quantify the "flowability" of powders. However, they remain empirical and would benefit from a detailed study of the physics controlling flow dynamics. Here, we attempt to fill the gap by performing intensive discrete numerical simulations of cohesive grains flowing down an inclined plane. We show that, contrary to what is commonly perceived, the cohesive nature of the flow is not entirely controlled by the interparticle adhesion, but that stiffness and inelasticity of the grains also play a significant role. For the same adhesion, stiffer and less dissipative grains yield a less cohesive flow. This observation is rationalized by introducing the concept of a dynamic, "effective" adhesive force, a single parameter, which combines the effects of adhesion, elasticity, and dissipation. Based on this concept, a rheological description of the flow is proposed for the cohesive grains. Our results elucidate the physics controlling the flow of cohesive granular materials, which may help in designing new approaches to characterize the "flowability" of powders.

Viscous to Inertial Transition in Dense Granular Suspension.

Granular suspensions present a transition from a Newtonian rheology in the Stokes limit to a Bagnoldian rheology when inertia is increased. A custom rheometer that can be run in a pressure- or a volume-imposed mode is used to examine this transition in the dense regime close to jamming. By varying systematically the interstitial fluid, shear rate, and packing fraction in volume-imposed measurements, we show that the transition takes place at a Stokes number of 10 independent of the packing fraction. Using pressure-imposed rheometry, we investigate whether the inertial and viscous regimes can be unified as a function of a single dimensionless number based on stress additivity.

Gravisensors in plant cells behave like an active granular liquid.

Plants are able to sense and respond to minute tilt from the vertical direction of the gravity, which is key to maintain their upright posture during development. However, gravisensing in plants relies on a peculiar sensor made of microsize starch-filled grains (statoliths) that sediment and form tiny granular piles at the bottom of the cell. How such a sensor can detect inclination is unclear, as granular materials like sand are known to display flow threshold and finite avalanche angle due to friction and interparticle jamming. Here, we address this issue by combining direct visualization of statolith avalanches in plant cells and experiments in biomimetic cells made of microfluidic cavities filled with a suspension of heavy Brownian particles. We show that, despite their granular nature, statoliths move and respond to the weakest angle, as a liquid clinometer would do. Comparison between the biological and biomimetic systems reveals that this liquid-like behavior comes from the cell activity, which agitates statoliths with an apparent temperature one order of magnitude larger than actual temperature. Our results shed light on the key role of active fluctuations of statoliths for explaining the remarkable sensitivity of plants to inclination. Our study also provides support to a recent scenario of gravity perception in plants, by bridging the active granular rheology of statoliths at the microscopic level to the macroscopic gravitropic response of the plant.

Revealing the hierarchy of processes and time-scales that control the tropic response of shoots to gravi-stimulations.

Gravity is a major abiotic cue for plant growth. However, little is known about the responses of plants to various patterns of gravi-stimulation, with apparent contradictions being observed between the dose-like responses recorded under transient stimuli in microgravity environments and the responses under steady-state inclinations recorded on earth. Of particular importance is how the gravitropic response of an organ is affected by the temporal dynamics of downstream processes in the signalling pathway, such as statolith motion in statocytes or the redistribution of auxin transporters. Here, we used a combination of experiments on the whole-plant scale and live-cell imaging techniques on wheat coleoptiles in centrifuge devices to investigate both the kinematics of shoot-bending induced by transient inclination, and the motion of the statoliths in response to cell inclination. Unlike previous observations in microgravity, the response of shoots to transient inclinations appears to be independent of the level of gravity, with a response time much longer than the duration of statolith sedimentation. This reveals the existence of a memory process in the gravitropic signalling pathway, independent of statolith dynamics. By combining this memory process with statolith motion, a mathematical model is built that unifies the different laws found in the literature and that predicts the early bending response of shoots to arbitrary gravi-stimulations.

Brownian Granular Flows Down Heaps.

We study the avalanche dynamics of a pile of micrometer-sized silica grains in water-filled microfluidic drums. Contrary to what is expected for classical granular materials, avalanches do not stop at a finite angle of repose. After a first rapid phase during which the angle of the pile relaxes to an angle θ_{c}, a creep regime is observed where the pile slowly flows until the free surface reaches the horizontal. This relaxation is logarithmic in time and strongly depends on the ratio between the weight of the grains and the thermal agitation (gravitational Péclet number). We propose a simple one-dimensional model based on Kramers' escape rate to describe these Brownian granular avalanches, which reproduces the main observations.

Depth-independent drag force induced by stirring in granular media.

The drag force experienced by a horizontal cylinder rotating around the vertical axis in a granular medium under gravity is experimentally investigated. We show that, for deeply buried objects, the drag force dramatically drops after half a turn, as soon as the cylinder crosses its own wake. Whereas the drag during the first half turn increases linearly with the depth, the drag after several rotations appears to be independent of depth, in contradiction with the classical frictional picture stipulating that the drag is proportional to the hydrostatic pressure. We systematically study how the saturated drag force scales with the control parameters and show that this effect may be used to drill deeply in a granular medium without developing high torques.

A constitutive law for dense granular flows.

A continuum description of granular flows would be of considerable help in predicting natural geophysical hazards or in designing industrial processes. However, the constitutive equations for dry granular flows, which govern how the material moves under shear, are still a matter of debate. One difficulty is that grains can behave like a solid (in a sand pile), a liquid (when poured from a silo) or a gas (when strongly agitated). For the two extreme regimes, constitutive equations have been proposed based on kinetic theory for collisional rapid flows, and soil mechanics for slow plastic flows. However, the intermediate dense regime, where the granular material flows like a liquid, still lacks a unified view and has motivated many studies over the past decade. The main characteristics of granular liquids are: a yield criterion (a critical shear stress below which flow is not possible) and a complex dependence on shear rate when flowing. In this sense, granular matter shares similarities with classical visco-plastic fluids such as Bingham fluids. Here we propose a new constitutive relation for dense granular flows, inspired by this analogy and recent numerical and experimental work. We then test our three-dimensional (3D) model through experiments on granular flows on a pile between rough sidewalls, in which a complex 3D flow pattern develops. We show that, without any fitting parameter, the model gives quantitative predictions for the flow shape and velocity profiles. Our results support the idea that a simple visco-plastic approach can quantitatively capture granular flow properties, and could serve as a basic tool for modelling more complex flows in geophysical or industrial applications.

Unifying suspension and granular rheology.

Using an original pressure-imposed shear cell, we study the rheology of dense suspensions. We show that they exhibit a viscoplastic behavior similarly to granular media successfully described by a frictional rheology and fully characterized by the evolution of the friction coefficient µ and the volume fraction ϕ with a dimensionless viscous number I(v). Dense suspension and granular media are thus unified under a common framework. These results are shown to be compatible with classical empirical models of suspension rheology and provide a clear determination of constitutive laws close to the jamming transition.

An Integrative Model of Plant Gravitropism Linking Statoliths Position and Auxin Transport.

Gravity is a major cue for the proper growth and development of plants. The response of plants to gravity implies starch-filled plastids, the statoliths, which sediments at the bottom of the gravisensing cells, the statocytes. Statoliths are assumed to modify the transport of the growth hormone, auxin, by acting on specific auxin transporters, PIN proteins. However, the complete gravitropic signaling pathway from the intracellular signal associated to statoliths to the plant bending is still not well-understood. In this article, we build on recent experimental results showing that statoliths do not act as gravitational force sensor, but as position sensor, to develop a bottom-up theory of plant gravitropism. The main hypothesis of the model is that the presence of statoliths modifies PIN trafficking close to the cell membrane. This basic assumption, coupled with auxin transport and growth in an idealized tissue made of a one-dimensional array of cells, recovers several major features of the gravitropic response of plants. First, the model provides a new interpretation for the response of a plant to a steady stimulus, the so-called sine-law of plant gravitropism. Second, it predicts the existence of a gravity-independent memory process as observed recently in experiments studying the response to transient stimulus. The model suggests that the timescale of this process is associated to PIN turnover, calling for new experimental studies.

Cohesion-controlled granular material.

We present a simple method to prepare a granular material with a controlled cohesion between particles. The granular material is made of spherical glass beads coated with a polyborosiloxane polymer. This material is proved to be stable in time and nonsensitive to temperature and humidity. The interparticle force is measured and related to the size of the grain and the polymer coating thickness. Classical measurements (packing fraction, repose angle, macroscopic cohesion) are performed with this cohesion-controlled granular material. This model material opens many perspectives to study in a controlled manner the flow of cohesive grains.

Inclination not force is sensed by plants during shoot gravitropism.

Gravity perception plays a key role in how plants develop and adapt to environmental changes. However, more than a century after the pioneering work of Darwin, little is known on the sensing mechanism. Using a centrifugal device combined with growth kinematics imaging, we show that shoot gravitropic responses to steady levels of gravity in four representative angiosperm species is independent of gravity intensity. All gravitropic responses tested are dependent only on the angle of inclination from the direction of gravity. We thus demonstrate that shoot gravitropism is stimulated by sensing inclination not gravitational force or acceleration as previously believed. This contrasts with the otolith system in the internal ear of vertebrates and explains the robustness of the control of growth direction by plants despite perturbations like wind shaking. Our results will help retarget the search for the molecular mechanism linking shifting statoliths to signal transduction.

Origin of a depth-independent drag force induced by stirring in granular media.

Experiments have shown that when a horizontal cylinder rotates around the vertical axis in a granular medium, the drag force in the stationary regime becomes independent of the depth, in contradiction with the frictional picture stipulating that the drag should be proportional to the hydrostatic pressure. The goal of this study is to understand the origin of this depth independence of the granular drag. Intensive numerical simulations using the discrete element method are performed giving access to the stress distribution in the packing during the rotation of the cylinder. It is shown that the rotation induces a strong anisotropy in the stress distribution, leading to the formation of arches that screen the hydrostatic pressure in the vicinity of the cylinder and create a bubble of low pressure.

A non-local rheology for dense granular flows.

A non-local theory is proposed to model dense granular flows. The idea is to describe the rearrangements occurring when a granular material is sheared as a self-activated process. A rearrangement at one position is triggered by the stress fluctuations induced by rearrangements elsewhere in the material. Within this framework, the constitutive law, which gives the relation between the shear rate and the stress distribution, is written as an integral over the entire flow. Taking into account the finite time of local rearrangements, the model is applicable from the quasi-static regime up to the inertial regime. We have checked the prediction of the model in two different configurations, namely granular flows down inclined planes and plane shear under gravity, and we show that many of the experimental observations are predicted within the self-activated model.

Aeolian sand ripples: experimental study of fully developed states.

We report an experimental investigation of aeolian sand ripples, performed both in a wind tunnel and on stoss slopes of dunes. Starting from a flat bed, we can identify three regimes: appearance of an initial wavelength, coarsening of the pattern, and finally saturation of the ripples. We show that both initial and final wavelengths, as well as the propagative speed of the ripples, are linear functions of the wind velocity. Investigating the evolution of an initially corrugated bed, we exhibit nonlinear stable solutions for a finite range of wavelengths, which demonstrates the existence of a saturation in amplitude. These results contradict most of the models.