We investigate the effects of lateral walls on the angle of movement and on the angle of repose of a granular pile. Our experimental results for beads immersed in water are similar to previous results obtained in air and to recent numerical simulations. All of these results, showing an increase of pile angles with a decreasing gap width, are explained by a model based on the redirection of stresses through the granular media. Two regimes are observed depending on the bead diameter. For large beads, the range of wall effects corresponds to a constant number of beads whereas it corresponds to a constant characteristic length for small beads as they aggregate via van der Waals forces.
We study experimentally the shape of a granular heap formed pouring a granular material into a vertical Hele-Shaw cell and analyze the effect of the grain impact energy. We propose a continuous model for the steady profile of the heap that explicitly considers energy dissipation of flowing grains through inelastic collisions. We solve the model analytically and analyze the resulting height profile as a function of several parameters, such as the restitution coefficient of the grains and their impact energies. We find good agreement between theory and experiments.
In this paper, we consider the effect of cohesion on the stability of a granular heap. We first briefly review literature results on the cohesion force between two rough granular beads and specifically consider the dependence of the adhesion force on the normal load. We then compute the dependence of the maximum angle of stability of the heap as a function of the cohesion. We point out that the dependence of the cohesive forces on the external normal load between grains is a key point in determining the localization of the failure plane. While for a constant cohesive force, slip occurs deep in…
We experimentally study the transition from steady flow to unsteady flow in a quasi-2D granular heap when small amounts of water are added to monodisperse glass spheres. Particles flow uniformly down both sides of the heap for low water content, but unsteady flow occurs as the water content increases. The unsteady flow mode consists of a non-depositing downslope avalanche and an upslope propagating granular jump. The transition from steady to unsteady flow occurs when the slope exceeds a critical angle as a result of water-induced cohesion. Under unsteady flow conditions, the deposited heap co…
Gravity-driven thick granular flows are relevant to many industrial and geophysical processes. In particular, it is important to know and understand the particle velocity distributions as we get deeper into the flow from the free surface. In this paper, we use dimensional analysis as a tool to reproduce the velocity profile experimentally reported for granular flows down a confined heap for the so-called flowing layer and the creep layer underneath: the grains velocity first decrease linearly from the free surface, and then exponentially.
In this paper we report experiments where we shear granular rods in split-bottom geometries, and find that a significant heap of height of least 40% of the filling height can form at the particle surface. We show that heaping is caused by a significant secondary flow, absent for spherical particles. Flow reversal transiently reverses the secondary flow, leading to a quick collapse and slower regeneration of the heap. We present a symmetry argument and experimental data that show that the generation of the secondary flow is driven by a misalignment of the mean particle orientation with the stre…
Segregation and mixing of granular mixtures during heap formation have important consequences in industry and agriculture. This research investigates three different final particle configurations of bi-disperse granular mixtures -- stratified, segregated and mixed -- during filling of quasi-two dimensional silos. We consider a larger number and relatively wider range of control parameters than previous studies, including particle size ratio, flow rate, system size and heap rise velocity. The boundary between stratified and unstratified states is primarily controlled by the two-dimensional flow…
We present a continuum approach to model segregation of size-bidisperse granular materials in unsteady bounded heap flow as a prototype for modeling segregation in other time varying flows. In experiments, a periodically modulated feed rate produces stratified segregation like that which occurs due to intermittent avalanching, except with greater layer-uniformity and higher average feed rates. Using an advection-diffusion-segregation equation and characterizing transient changes in deposition and erosion after a feed rate change, we model stratification for varying feed rates and periods. Feed…
If a granular material is poured from above on a horizontal surface between two parallel, vertical plates, a sand heap grows in time. For small piles, the grains flow smoothly downhill, but after a critical pile size $X_c$, the flow becomes intermittent: sudden avalanches slide downhill from the apex to the base, followed by an "uphill front" that slowly climbs up, until a new downhill avalanche interrupts the process. By means of experiments controlling the distance between the apex of the sandpile and the container feeding it from above, we show that $X_c$ grows linearly with the input flux,…
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…
In two-dimensional rotating drum experiments, we find two separate influences of the packing fraction of a granular heap on its stability. For a fixed grain shape, the stability increases with packing fraction. However, in determining the relative stability of different grain shapes, those with the lowest average packing fractions tend to form the most stable heaps. We also show that only the configuration close to the surface of the pile figures prominently.
We study the transition between steady flows of non-cohesive granular materials in quasi-2D bounded heaps by suddenly changing the feed rate. In both experiments and simulations, the primary feature of the transition is a wedge of flowing particles that propagates downstream over the rising free surface with a wedge front velocity inversely proportional to the square root of time. An additional longer duration transient process continues after the wedge front reaches the downstream wall. The entire transition is well modeled as a moving boundary problem with a diffusion-like equation derived f…
We report on grain dynamics versus depth for steady-state gravity-driven flow of grains along a heap formed between two parallel sidewalls. Near the surface the flow is steady and fast, while far below there is no flow whatsoever; therefore, a jamming transition occurs where depth is an effective control parameter for setting the distance from the transition. As seen previously, the time-averaged velocity profile along the sidewall exhibits a nearly exponential decay vs depth. Using speckle-visiblility spectroscopy (SVS), we find that velocity fluctuations grow relative to the average on appro…
We have experimentally observed a new phenomenon in the surface flow of a granular material. A heap is constructed by injecting sand between two vertical glass plates separated by a distance much larger than the average grain size, with an open boundary. As the heap reaches the open boundary, "soliton-like" fluctuations appear on the flowing layer, and move "up the hill" (i.e., against the direction of the flow). We explain the phenomenon in the context of stop-and-go traffic models, and show that soliton-like behavior is allowed within a Saint-Venant description for the granular flow.
Granular flow dynamics on a vertically vibrated pile is studied by means of both laboratory experiments and numerical simulations. As already revealed, the depth-averaged velocity of a fully-fluidized granular pile under strong vibration, which is measured by a high-speed laser profiler in the experiment, can be explained by the nonlinear diffusion transport model proposed by our previous paper (Tsuji et al, Phys. Rev. Lett. 120, 128001 (2018)). In this paper, we report that a similar transport model can be applied to the relation between the surface velocity and slope in the experiment. These…
We measure stability of two-dimensional granular mixtures in a rotating drum and relate grain configurations to stability. For our system, the smaller but smoother grains cluster near the center of the drum, while the larger, rougher grains remain near the outer edge. One consequence of the size segregation is that the smaller grains heavily influence the stability of the heap. We find that the maximum angle of stability is a non-linear function of composition, changing particularly rapidly when small grains are first added to a homogeneous pile of large grains. We conclude that the grain conf…
We have investigated by rotating drum experiments the influence of a small amount of fine particles on the stability of a granular heap. The fine particles are shown to have a strong and ambivalent influence. For low fine particle content, the heap destabilizes by avalanches, and the fine particles tend to fluidize the heap. In contrast, for high fine particle content, they increase the cohesion of the heap, which destabilizes through stick-slip at the drum wall. We interpret and model the fluidification in the avalanching regime, which we show is independent of humidity, by granular lubricati…
As if it has become a consensus that the slope of the granular heap is straight, and from that, an angle of repose is defined. However, closer inspection shows that the slopes are not truly straight, instead, more often they have convex shapes, although with small convexity. Similar curvature is observed on the sand dunes' surfaces. We derive equations together with experiments on 10 types of granular materials to show that, in general, the slope of the granular heap is convex while the straight slope is a critical state that rarely occurs. The angle of repose is allowed to vary within a narro…
We present a method to estimate the segregation parameter, $S,$ a key input in a continuum transport model of particulate flows. $S$ is determined by minimizing the difference between measured and model-predicted concentration profiles. To validate the approach, we conduct discrete element method simulations of size-bidisperse mixtures in quasi-2D bounded heap flow; the resulting data show that $S$ calculated from concentration profiles is consistent with the directly measured value. The method's accuracy depends critically on the velocity profile during filling, but only weakly on the diffusi…
Vibrating walls, used to maintain the temperature in a granular gas, modify the system strongly. Most conspicuously, the usual one-peak velocity distribution splits into two, asymmetrically positioned. A surgical repair of the usual hydrodynamic description is presented that provides an account for, and an understanding of, the situation.
DEM simulations were used to study the sensitivity of a granular system to the change of the mechanical properties of the grains, applied to the penetration of a cylindrical intruder near a vertical wall. The simulations reproduce a real experimental setup, changing the friction, the Young modulus and the restitution coefficient of the particles.
When an object penetrates into granular matter near a boundary, it experiences a horizontal repulsion due to the intruder-grain-wall interaction. Here we show experimentally that a square cuboid intruder, released from rest with no initial velocity, neara vertical wall to its left, goes through three distinct kinds of motion: it first tilts clockwise, then 'slides' away from the wall,and finally tilts counterclockwise. This dynamic highly favors both repulsion and penetration of the cuboid intruder as compared to that observed from its release farther away from the wall
We investigate the onset of the wall-induced convection in vertically vibrated granular matter by means of experiments and two-dimensional computer simulations. In both simulations and experiments we find that the wall-induced convection occurs inside the bouncing bed region of the parameter space in which the granular bed behaves like a bouncing ball. A good agreement between experiments and simulations is found for the peak vibration acceleration at which convection starts. By comparing the results of simulations initialised with and without defects, we find that the onset of convection occu…
Boundary shape, particularly roughness, strongly controls the amount of wall slip in dense granular flows. In this paper, we aim to quantify and understand which aspects of a dense granular flow are controlled by the boundary condition, and to incorporate these observations into a cooperative nonlocal model characterizing slow granular flows. To examine the influence of boundary properties, we perform experiments on a quasi-2D annular shear cell with a rotating inner wall and a fixed outer wall; the later is selected from among 6 walls with various roughness, local concavity, and compliance. W…
We consider a granular gas under the action of gravity, fluidized by a vibrating base. We show that a horizontal temperature gradient, here induced by limiting dissipative lateral walls (DLW), leads always to a granular thermal convection (DLW-TC) that is essentially different from ordinary buoyancy-driven convection (BD-TC). In an experiment where BD-TC is inhibited, by reducing gravity with an inclined plane, we always observe a DLW-TC cell next to each lateral wall. Such a cell squeezes towards the nearest wall as the gravity and/or the number of grains increase. Molecular dynamics simulati…
The side-wall pressure distribution in static silos is an important and open problem related to storage safety of granular materials. Referring to Lvin's theory [Powder Technology 4, 280 (1971)] and Rodolfo's 2D numerical result [Phys. Rev. E 97.1(2018)], a new theoretical model of side wall pressure is established by assuming the friction coefficient as well as the ratio of horizontal to vertical stress changing with depth position in silo. Furthermore, the transition of stress state between hydrostatic-like and Janssen-like is systematically investigated by our model and experiment, and fina…
In this paper we study the steady uniform flows that develop when granular material is released from a hopper on top of a static pile in a channel. We more specifically focus on the role of side walls by carrying out experiments in setup of different widths, from narrow channels 20 particle diameters wide to channels 600 particle diameters wide. Results show that steady flows on pile are entirely controlled by side wall effects. A theoretical model, taking into account the wall friction and based on a simple local constitutive law recently proposed for other granular flow configurations (GDR M…
The effects of movement of the side walls of a confined granular packing are studied by discrete element, molecular dynamics simulations. The dynamical evolution of the stress is studied as a function of wall movement both in the direction of gravity as well as opposite to it. For all wall velocities explored, the stress in the final state of the system after wall movement is fundamentally different from the original state obtained by pouring particles into the container and letting them settle under the influence of gravity. The original packing possesses a hydrostatic-like region at the top …
When a cylindrical object penetrates granular matter near a vertical boundary, it experiences two effects: its center of mass moves horizontally away from the wall, and it rotates around its symmetry axis. Here we show experimentally that, if two identical intruders instead of one are released side-by-side near the wall, both effects are also detected. However, unexpected phenomena appear due to a cooperative dynamics between the intruders. The net horizontal distance traveled by the common center of mass of the twin intruders is much larger than that traveled by one intruder released at the s…
We study experimentally the influence of confinement on the penetration depth of impacting spheres into a granular medium contained in a finite cylindrical vessel. The presence of close lateral walls reduces the penetration depth, and the characteristic distance for these lateral wall effects is found to be of the order of one sphere diameter. The influence of the bottom wall is found to have a much shorter range.
Buoyancy-driven (thermal) convection in dilute granular media, fluidized by a vibrating base, is known to appear without the need of lateral boundaries in a restricted region of parameters (inelasticity, gravity, intensity of energy injection). We have recently discovered a second buoyancy-driven convection effect which occurs at any value of the parameters, provided that the impact of particles with the lateral walls is inelastic (Pontuale et al., Phys. Rev. Lett. 117, 098006 (2016)). It is understood that this novel convection effect is strictly correlated to the existence of perpendicular e…
We investigate a granular gas in a shaken quasi-two-dimensional box in molecular dynamics computer simulations. After a sudden change (quench) of the shaking amplitude, transient density correlations are observed orders of magnitude beyond the steady-state correlation length scale. Propagation of the correlations is ballistic, in contrast to recently investigated quenches of Brownian particles that show diffusive propagation [Rohwer et al., Phys. Rev. Lett., 118, 015702 (2017), Rohwer et al., Phys. Rev. E, 97, 032125 (2018)]. At sufficiently strong cooling of the fluid the effect is overlaid b…
We discuss the advantages and results of using a mixing-length, compressible model to account for shear banding behaviour in granular flow. We formulate a general approach based on two function of the solid fraction to be determined. Studying the vertical chute flow, we show that shear band thickness is always independent from flowrate in the quasistatic limit, for Coulomb wall boundary conditions. The effect of bin width is addressed using the functions developed by Pouliquen and coworkers, predicting a linear dependence of shear band thickness by channel width, while literature reports contr…
Here we show that variations of temperature, even of a few degrees in amplitude, induce the ageing of a granular pile. In particular, we report measurements of physical properties of a granular heap submitted to thermal cycles. Namely, we focus on the evolution of the thermal linear-expansion coefficient and of the thermal conductivity of the pile with the number of cycles. The present contribution nicely supplements a recent article we published elsewhere [Phys. Rev. Lett. \textbf{101}, 148303 (2008)] and introduces a different and promising method to impose temperature cycles to a granular p…
Size bidisperse granular materials in a spherical tumbler segregate into two different patterns of three bands with either small particles at the equator and large particles at the poles or vice versa, depending upon the fill level in the tumbler. Here we use discrete element method (DEM) simulations with supporting qualitative experiments to explore the effect of the tumbler wall roughness on the segregation pattern, modeling the tumbler walls as either a closely packed monolayer of fixed particles resulting in a rough wall, or as a geometrically smooth wall. Even though the tumbler wall is i…
Vertically vibrated rod-shaped granular materials confined to quasi-2D containers self organize into distinct patterns. We find, consistent with theory and simulation, a density dependent isotropic-nematic transition. Along the walls, rods interact sterically to form a wetting layer. For high rod densities, complex patterns emerge as a result of competition between bulk and boundary alignment. A continuum elastic energy accounting for nematic distortion and local wall anchoring reproduces the structures seen experimentally.
A significant dependence on the relative humidity H for the apparent mass (Mapp) measured at the bottom of a granular packing inside a vertical tube in relative motion is demonstrated experimentally. While the predictions of Janssen's model are verified for all values of H investigated (25%< H <80%), Mapp increases with time towards a limiting value at high relative humidities (H>60%) but remains constant at lower ones (H=25%). The corresponding Janssen length is nearly independent of the tube velocity for H>60% but decreases markedly for H=25%. Other differences are observed on the motion of …
We report numerical simulations on granular shear flows confined between two flat but frictional sidewalls. Novel regimes differing by their strain localization features are observed. They originate from the competition between dissipation at the sidewalls and dissipation in the bulk of the flow. The effective friction at sidewalls is characterized (effective friction coefficient and orientation of the friction force) for each regime, and its interdependence with slip and force fluctuations is pointed out. We propose a simple scaling law linking the slip velocity to the granular temperature in…
This study presents numerical simulations of the resonance of a finite-length granular chain of dissipative grains driven by a harmonically vibrated tube. Multiple gradual resonant modes, namely, non-resonance mode, partial-resonance mode, and complete-resonance mode, are identified. With a fixed vibration frequency, increased vibration acceleration leads to a one-by-one increase in the number of grains participating in resonance, which is equal to the number of grain-wall collisions in a vibration period. Compared with the characteristic time of the grain-grain and the grain-wall collisions, …
Granular material exposed to shear shows a variety of unique phenomena: Reynolds dilatancy, positional order and orientational order effects may compete in the shear zone. We study granular packings consisting of macroscopic prolate, oblate and spherical grains and compare their behaviour. X-ray tomography is used to determine the particle positions and orientations in a cylindrical split bottom shear cell. Packing densities and the arrangements of individual particles in the shear zone are evaluated. For anisometric particles, we observe the competition of two opposite effects. One the one ha…
A two-dimensional granular packing under horizontally circular shaking exhibits various collective motion modes depending on the strength of the oscillation and the global packing density. For intermediate packing density and oscillation amplitude, a high density phase travels along the container's side wall in clockwise direction, while the oscillation itself is anti-clockwise. Further increasing packing density towards the hexagonal packing, the whole packing rotates collectively in clockwise direction. The core of the packing rotates as a solid and is separated from the boundary by a fluid-…
Granular crystallisation is an important phenomenon whereby ordered packing structures form in granular matter under vibration. However, compared with the well-developed principles of crystallisation at the atomic scale, crystallisation in granular matter remains relatively poorly understood. To investigate this behaviour further and bridge the fields of granular matter and materials science, we simulated mono-disperse spheres confined in cylindrical containers to study their structural dynamics during vibration. By applying adequate vibration, disorder-to-order transitions were induced. Such …
From event-driven simulations of a gravity-driven channel flow of inelastic hard-disks, we show that the velocity distribution function remains close to a Gaussian for a wide range densities (even when the Knudsen number is of order one) if the walls are smooth and the particle collisions are nearly elastic. For dense flows, a transition from a Gaussian to a power-law distribution for the high velocity tails occurs with increasing dissipation in the center of the channel, irrespective of wall-roughness. For a rough wall, the near-wall distribution functions are distinctly different from those …
Two-dimensional Particle Tracking Velocimetry (PTV) is a promising technique to study the behaviour of granular flows. The aim is to experimentally determine the free surface width and position of the shear band from the velocity profile to validate simulations in a split-bottom shear cell geometry. The position and velocities of scattered tracer particles are tracked as they move with the bulk flow by analyzing images. We then use a new technique to extract the continuum velocity field, applying coarse-graining with the postprocessing toolbox MercuryCG on the discrete experimental PTV data. F…
The structure and stresses of static granular packs in cylindrical containers are studied using large-scale discrete element molecular dynamics simulations in three dimensions. We generate packings by both pouring and sedimentation and examine how the final state depends on the method of construction. The vertical stress becomes depth-independent for deep piles and we compare these stress depth-profiles to the classical Janssen theory. The majority of the tangential forces for particle-wall contacts are found to be close to the Coulomb failure criterion, in agreement with the theory of Janssen…
An ensemble of inelastically colliding grains driven by a vibrating wall in 2D exhibits density clustering. Working in the limit of nearly elastic collisions and employing granular hydrodynamics, we predict, by a marginal stability analysis, a spontaneous symmetry breaking of the extended clustering state (ECS). 2D steady-state solutions found numerically describe localized clustering state (LCSs). Time-dependent granular hydrodynamic simulations show that LCSs can develop from natural initial conditions. The predicted instability should be observable in experiment.
Granular material in a swirled container exhibits a curious transition as the number of particles is increased: at low densities the particle cluster rotates in the same direction as the swirling motion of the container, while at high densities it rotates in the opposite direction. We investigate this phenomenon experimentally and numerically using a co-rotating reference frame in which the system reaches a statistical steady-state. In this steady-state the particles form a cluster whose translational degrees of freedom are stationary, while the individual particles constantly circulate around…
Thermal wall is a convenient idealization of a rapidly vibrating plate used for vibrofluidization of granular materials. The objective of this work is to incorporate the Knudsen temperature jump at thermal wall in the Navier-Stokes hydrodynamic modeling of dilute granular gases of monodisperse particles that collide nearly elastically. The Knudsen temperature jump manifests itself as an additional term, proportional to the temperature gradient, in the boundary condition for the temperature. Up to a numerical pre-factor of order unity, this term is known from kinetic theory of elastic gases. We…
The behavior of an ultrasonic shot peening process is observed and analyzed by using a model of inelastic hard spheres in a gravitational field that are fluidized by a vibrating bottom wall (sonotrode) in a cylindrical chamber. A marked heterogeneous distribution of impacts appears when the collision between the shot and the side wall becomes inelastic with constant dissipation. This effect is one order of magnitude larger than the simple heterogeneity arising from boundary collision on the cylinder. Variable restitution coefficients bring the simulation closer to the general observation and a…
Nonlocal rheologies allow for the modeling of granular flows from the creeping to intermediate flow regimes, using a small number of parameters. In this paper, we report on experiments testing how particle properties affect model parameters, using particles of three different shapes (circles, ellipses, and pentagons) and three different materials, including one which allows for measurements of stresses via photoelasticity. Our experiments are performed on a quasi-2D annular shear cell with a rotating inner wall and a fixed outer wall. Each type of particle is found to exhibit flows which are w…
When grains are added to a cylinder, the weight at the bottom is smaller than the total weight of the column, which is partially supported by the lateral walls through wall/grain frictional forces. This is known as the Janssen effect. Via a combined experimental and numerical investigation, here we demonstrate a reverse Jansen effect whereby the fraction of the weight supported by the base overcomes one. We characterize the dependence of this phenomenon on the various control parameters involved, rationalize the physical process responsible for the emergence of the compressional frictional for…
We study fluidized granular gases in a stationary state determined by the balance between an external driving and the bulk dissipation. The two considered situations are inspired by recent experiments, where the gravity plays a major role as a driving mechanism: in the first case gravity acts only in one direction and the bottom wall is vibrated, in the second case gravity acts in both directions and no vibrating walls are present. Simulations performed under the molecular chaos assumption show averaged profiles of density, velocity and granular temperature which are in good agreement with the…
We show that the van der Waals-like phase separation instability of a driven granular gas at zero gravity, previously investigated in two-dimensional settings, persists in three dimensions. We consider a monodisperse granular gas driven by a thermal wall of a three-dimensional rectangular container at zero gravity. The basic steady state of this system, as described by granular hydrodynamic equations, involves a denser and colder layer of granulate located at the wall opposite to the driving wall. When the inelastic energy loss is sufficiently high, the driven granular gas exhibits, in some ra…
We have studied the two-dimensional flow of balls in a small angle funnel, when either the side walls are rough or the balls are polydisperse. As in earlier work on monodisperse flows in smooth funnels, we observe the formation of kinematic shock waves/density waves. We find that for rough walls the flows are more disordered than for smooth walls and that shock waves generally propagate more slowly. For rough wall funnel flow, we show that the shock velocity and frequency obey simple scaling laws. These scaling laws are consistent with those found for smooth wall flow, but here they are cleane…
Experimental results are presented for a vertically shaken granular layer. In the range of accelerations explored, the layer develops a convective motion in the form of one or more rolls. The velocity of the grains near the wall has been measured. It grows linearly with the acceleration, then the growing rate slows down. A rescaling with the amplitude of the wall velocity and the height of the granular layer makes all data collapse in a single curve. This can provide insights on the mechanism driving the motion.
In this paper, we have designed and employed a suspended-wall silo to remove Janssen effect in order to explore directly the local pressure dependence of Granular Orifice Flow (GOF) systematically. We find that as Janssen effect is removed, the flow rate Q changes linearly with the external pressure. The slope α of the linear change decays exponentially with the ratio of the silo size and the size of the orifice Φ/D, which suggests the existence of a characteristic ratio λ (~2.4). When Φ/D > λ, α gradually decays to zero, and the effect of external pressure on the GOF becomes negligible, where…
Symmetry-breaking instability of a laterally uniform granular cluster (strip state) in a prototypical driven granular gas is investigated. The system consists of smooth hard disks in a two-dimensional box, colliding inelastically with each other and driven, at zero gravity, by a "thermal" wall. The limit of nearly elastic particle collisions is considered, and granular hydrodynamics with the Jenkins-Richman constitutive relations is employed. The hydrodynamic problem is completely described by two scaled parameters and the aspect ratio of the box. Marginal stability analysis predicts a spontan…
