Reorganization of a granular medium around a localized transformation Aymeric Merceron, Alban Sauret, and Pierre Jop * Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, 39, quai Lucien Lefranc, F-93303 Aubervilliers, Cedex, France (Dated: June 24, 2016)

Physical and chemical transformation processes in reactive granular media involve the reorganization of the structure. In this paper, we study experimentally the rearrangements of a two-dimensional (2D) granular packing undergoing a localized transformation. We track the position and evolution of all the disks that constitute the granular packing when either a large intruder shrinks in size or is pulled out of the granular structure. In the two situations the displacements at long time are similar to 2D quasistatic silo flows whereas the short-time dynamic is heterogeneous in both space and time. We observe an avalanchelike behavior with power-law distributed events uncorrelated in time. In addition, the instantaneous evolutions of the local solid fraction exhibit self-similar distributions. The averages and the standard deviations of the solid fraction variations can be rescaled, suggesting a single mechanism of rearrangement.

I. INTRODUCTION

Sintering, glass melting and other industrially relevant processes turn large assemblies of grains into homogeneous final products. Such processes are often complex as they can involve both chemical and physical transformations of the granular packing leading to the rearrangement of the structure of the material. For instance, dilatations of the material and phase transitions affect the volume of the particles [1,2]. Consequently, the shapes of particles are modified [3] as well as their flowing properties through a lubrication film [4] or the formation of capillary bridges [5]. In addition, chemical reactions can also modify the surface and nature of each individual grain [6]. These different phenomena occur at the same time and lead to the final homogeneous product through complex paths. Modeling the behavior of such systems involves different fields of research. Typically, mechanics and statistical physics are used to predict the behavior of granular assemblies under flow, rearrangements or perturbations [START_REF] Andreotti | Granular media: between fluid and solid[END_REF]. The rich behavior of granular media has led to extensive studies considering their rheology [START_REF] Jop | [END_REF], the criteria of mechanical stability [9], and the properties of wet granular assemblies [10]. Other fields of research have been considered to characterize a granular assembly in more complex situations. For instance, the behavior of reactive powders is modeled in chemical engineering [11] and rock transformations have been characterized in geology by focusing on thermodynamics of reactions [12].

However, despite these different approaches, the reorganization of a granular assembly at short time scales remains poorly understood. In particular, few studies have considered a granular medium whose particles are undergoing reactive transformations. Some recent studies have characterized the rearrangement of a reactive granular packing, often icy particles than can melt [1,13], but experimental investigations remain rare, likely because of the difficulties in monitoring the location of the individual grains as well as their properties. Therefore, most studies have focused on granular flows around a unique intruder using mechanical [14][15][16], rheological [17], or purely kinematic approaches [18,19]. In addition, these different studies were often performed using an intruder penetrating the granular packing. Flows of granular materials induced by gravity in silos and hoppers were also largely considered [20] and mainly deal with mean field flows of the granular material over a long time. But such flows induce significant reorganizations of the granular packing leading to jamming problems [21,22]. Similar silo problems have been considered in mining applications where grains are also extracted from a bottom orifice under gravity at very low flow rate [23,24]. However, all these studies only characterized and modeled the mean displacement fields of the granular assembly but did not characterize the instantaneous response of the packing and the distribution of the size of events during the flowing process. The rearrangements at small time scales and length scales remain poorly explored although they can govern the evolution of a reactive granular packing.

In this paper, we experimentally investigate the quasistatic response of a two-dimensional (2D) granular assembly undergoing a localized transformation. We model the transformation using two methods: (i) with a single large grain that shrinks in size and (ii) with the slow extraction of a large intruder in the granular assembly. The position of each grain during the rearrangement process is accurately characterized. This method also allows us to observe the rearrangements of the granular assembly over large spatial and time scales. We thus quantify the typical amplitude of a rearrangement -the fraction of the granular assembly that is involved in the localized transformation. We also determine how the granular structure is modified during the reorganization. We present our experimental methods in Sec. II. We qualitatively describe the long-term displacements using a melting grain and a mechanical intruder in Sec. III. We then rely on the experiments performed with a mechanical intruder to characterize the statistics of reorganization events. These quantitative results are reported in Sec. IV where we describe the reorganization process in term of both the dynamics of displacements and instantaneous structural evolutions. Finally, the results are discussed in Sec. V.

II. EXPERIMENTS

The reorganization of a granular medium around a localized transformation is studied using a two-dimensional system, shown in Figs. 1(a)-(c), which allows us to track the time evolution of the position of each individual grain. We consider two types of localized transformation: the shrinkage of a large grain and the extraction of a large intruder. The first method is called the melting block experiment and the second one the mechanical-intruder experiment. In both cases, we use a two-dimensional cell made of two parallel glass plates (30 cm high and 30 cm wide for the melting block experiment or 60 cm wide for the mechanical-intruder experiment) separated by a gap of 3.1 mm. The cell is filled with 5000 or 3000 (depending on the width of the cell) bidisperse rigid disks 4 and 5 mm in diameter (proportion 10:7) to avoid crystallization. In the following all lengths are expressed in terms of the diameter of smallest grains, d g = 4 mm.

In a melting block experiment, a pure dimethylsulfoxide (DMSO) disk 40 mm in diameter (10 d g ) and 3 mm thick is placed at the bottom of the granular assembly and surrounded by inert metal disks [Fig. 1(c)]. To prevent the formation of capillary bridges during the melting process and to provide a better thermal homogeneity, the entire granular packing is immersed into a mixture of DMSO and water (20% v/v) initially thermalized at -5 o C. The melting point of the DMSO is 18.5 o C; therefore, the reactive grain slowly melts while the system warms up to room temperature, leading to the reorganization process. Hydrodynamic interactions are present but they remain negligible compared to the weight of the disks since the density of the metal disk is ρ g = 8 g cm -3 and the dynamic viscosity of the water/DMSO mixture is η mixture ≈ 8 mPa s. The typical time scale of the melting of the DMSO disk is about 40 min, which ensures a quasistatic evolution of the surrounding granular assembly. However, the control of these experiments is very sensitive as some heterogeneities can be present in the DMSO disk leading to variations in the melting process. In addition, this method, although the closest to a reactive transformation, makes it difficult to conduct a large quantity of experiments. To overcome the challenges, we use an alternative method, the mechanical-intruder experiment, that also produces a quasistatic reorganization of the granular assembly.

In a mechanical-intruder experiment, a semicircular metallic intruder 40 mm in diameter (10 d g ) and 3 mm thick is inserted at the bottom of the cell and is surrounded by inert disks [Figs. 1(a) and 1(b)]. To mimic a decrease of volume similar to the melting grain, the intruder is slowly pulled down out of the cell at a constant speed of 0.05 mm s -1 using a linear translation stage. Similarly to the melting block experiment, the granular assembly follows a quasistatic evolution. In this situation, the granular packing remains perfectly dry in contrast with the melting block experiment so that grains only interact through friction.

In both situations, the experimental protocol is the following. First, the inert grains are mixed and horizontally inserted in the 2D cell at the same moment as the intruder to ensure that no preloading forces are exerted close to the intruder which could lead to an inhomogeneity in the packing. The method ensures that the granular packing forms a dense and disordered assembly at the beginning of the experiment. Metallic weights are also set horizontally. Then, the cell is tilted into a vertical position so that the system is driven by gravity. Metallic weights at the top ensure a uniform load all along the top of the granular assembly throughout the experiment. Because of the low friction forces compared to the weight of the disks, the granular packing instantaneously reaches an initial stable state. Therefore, the rearrangements that may appear at the beginning of an experiment are not due to initial fragile states. Finally, depending on the experiment, the intruder is attached to the linear translation stage and then pulled down or the packing is immersed in the mixture mentioned above and the disk starts to melt, which defines the time t = 0 in our experiments. A light-emitting diode panel is set behind the granular assembly and a high-resolution camera (Nikon D7000) records the time evolution of the packing at time intervals δt = 10 s for the melting block experiment and δt = 5 s for the mechanical-intruder experiment with a resolution of 4928 × 3264 pixels. For the mechanical intruder, the vertical displacement of the intruder between two snapshots is equal to 0.25 mm = d g /16. The pictures recorded during the experiments are processed with a segmentation protocol [START_REF] Zhao | Length scales in granular matter[END_REF] to obtain a subpixel accuracy (inferior to 33 µm) of the position of the disks. Then, tracking techniques are used to measure the short-term and long-term displacements. We have also performed tessellation methods to analyze the structural evolutions at different spatial scales. It is worth noting that the length scale of both the translation of the mechanical intruder and the shrinking intruder between two snapshots is far above the resolution of the image processing. To obtain reliable statistics and compare the two situations, we have performed 20 times the melting block experiment and 100 times the mechanical-intruder experiment.

We first present the phenomenological behavior observed in the experiments. As the intruder is melting or as it is being pulled down, we observe that the response of the granular packing is heterogeneous in space and time. For instance, although some reorganization events involve only few grains, large events mobilize almost the entire granular packing. In addition, the trajectories of some grains can be relatively steady in time whereas strongly intermittent behavior can be found elsewhere in the packing. Moreover, local structural variations, i.e., grain density, can be observed through compacting and dilating zones that are not necessarily close to the intruder.

III. LONG TERM DISPLACEMENTS

We first analyze the long-term displacements and compare the two methodologies. The long-term vertical displacements inside the packing (predominant to horizontal ones due to gravity) are shown in Fig. 2(a) for the melting block and Fig. 2(b) for the mechanical intruder. These results are obtained by averaging the total vertical displacement fields over each set of experiments. As the main reorganization effects are induced by gravity, static zones are observed below the intruder and we thus choose to set the intruders at the bottom of the cell. Both vertical displacement fields share strong qualitative similarities even if the amplitudes are different. Indeed, in a melting block experiment, because of the quasiradial shrinking of the intruder, the maximum vertical displacement is limited to 6.6 d g instead of 10 d g that is the intruder's stroke in a mechanical-intruder experiment. However, normalizing the amplitude of the displacements by the maximum amplitude leads to a collapse of both experiments as depicted in Fig. 2(c) along the vertical axis at x = 0. As a consequence long range displacements are qualitatively equivalent for both experiments and the mechanical intruder is a good method to characterize the displacement due to a shrinking grain although the amplitudes of some radial components are reduced.

In addition, the two displacement fields present similarities with velocity profiles obtained in bidimensional silos when a hole replaces the intruder. However in this study, the amplitude of displacements of the grains is limited by the shape and the geometrical evolution of the intruder and not by free fall dynamics as observed in silos. Our setups are indeed closer to experiments performed for block caving applications [24]. In that study, the authors suggest an analytical formulation for the ver-tical displacement field in 2D quasistatic silolike systems:

δ(x, y) = - δ 0 2 erf x + R 4D p y -erf x -R 4D p y , (1)
where x and y are the horizontal and vertical coordinates, respectively; δ 0 is the vertical displacement at the aperture, R is the half width of the aperture; and finally D p is the diffusion coefficient that links the vertical to horizontal displacements. For the calculation, R is set equal to 20 mm, the radius of the intruder, and δ 0 to the maximum displacement at the edge of the intruder. The only fitting parameter is D p . In our case, this parameter is found to be close to 6 d g for both systems. The diffusion coefficient D p shows large discrepancies in the literature and is usually closer to d g [20], and eventual dependence on the system parameters is not described. For instance, the geometry of the intruder could play a role. Vertical displacements experimentally found along several horizontal axes are compared to the Eq. ( 1) in Fig. 2(d) for the mechanical intruder. The agreement is again very good for the shrinking intruder as soon as δ 0 is properly set. In summary, these similarities imply a silolike behavior of our experiments for long-term displacements but give no information on the short-term dynamic. In the next section, we focus on the instantaneous responses of the granular packing.

IV. SHORT-TERM DISPLACEMENTS

Because of several experimental limitations mentioned in Sec. II, melting block experiments are not conducive to reliable statistics. Besides, as the size of the intruder is evolving in time, the response of the granular medium is time-dependent and the averaged displacements decrease with time (Fig. 3). Indeed, reducing the block size below a few d g increases the probability for the surrounding structure to jam. Therefore, instantaneous responses at different moments cannot be compared quantitatively. Therefore, we mostly analyzed the short-term displacements using the mechanical-intruder experiment and some qualitative comparisons are established with the melting block experiment.

A. Avalanches

The instantaneous response of the granular packing is analyzed for the mechanical-intruder experiment. An avalanche of size N A is defined as the number of disks that have an absolute displacement larger than the displacement of the intruder between two consecutive snapshots (d g /16). No phenomenological differences on displacements have been found between the beginning and the end of a mechanical-intruder experiment as shown by the dimensionless mean displacement h averaged over all disks between two consecutive snapshots with respect to the normalized time in Fig. 3. Consequently, the avalanche sizes for the 100 mechanical-intruder experiments have been gathered on a single distribution in Fig. 4. The distribution spans over three decades, up to large ones involving about one third of the total number of disks. This distribution is characteristic of the length scales and the granular nature of the system. First, there is a nonzero probability to have N A = 0 that can be assimilated to quasijamming events even for an intruder as wide as 10 d g . Quasijamming events cannot be assimilated to purely jamming events because the amplitudes of FIG. 4. Probability density function of the avalanche sizes for 16000 events with the mechanical intruder. The frequency distribution has a slope of -2.5 and the dotted vertical line corresponds to the number of disks in contact with the top of the intruder. The blue region shows the events used for the analysis performed in Fig. 5.

displacements are never strictly null all over the packing. Second, the most frequent avalanche size corresponds to the mean number of disks in contact with the intruder as shown by the dotted vertical line in Fig. 4. Most of the time, only disks touching the intruder are involved in the avalanche and the perturbation induced by the intruder is damped by frictions or geometrical frustrations over grains. Third, largest events are distributed following a power law. This behavior is characteristic of processes ruled by scale invariance like snow avalanches [27] and earthquakes [28]. Note that this property is independent of the threshold used to define avalanches. For the system size and the chosen threshold, a slope of the power-law equal to -2.5 captures the behavior of the large events. Although slightly different, this result is qualitatively consistent with other avalanche like systems, in which the slope of the probability of occurrence of largest events is usually found between -1 and -2 [29][30][31].

In avalanche like systems, the predictability of largest events is a crucial point [29]. The distribution of the waiting times between two large events is shown in Fig. 5. Here, the largest avalanches have been arbitrarily defined as N A ≥ 30. This threshold has been chosen for being in the power-law distribution of avalanche sizes and keeping a good probability of occurrence (Fig. 4). Another cutoff in the power law would provide the same qualitative results. For correlated events, a power-law distribution of waiting times is expected [32]. However, the largest avalanches exhibit an exponential decay as shown in Fig. 5, with a characteristic time of occurrence τ of five time steps for the threshold chosen (τ ∼ 5 δt):

p(t w ) ∝ exp(-t w / τ ) (2)
This exponential decay can be related to uncorrelated events in time. For short waiting times, a steeper slope is observed and can be attributed to time clustering of avalanches. This corresponds to quasiconsecutive large events assimilated to foreshocks and aftershocks even if the sequence of events seems unpredictable [29].

To conclude, the short-term displacements share similarities with avalanche like systems, i.e., power-law distributed events uncorrelated in time. In the following part, the structural response is analyzed with respect to the event amplitudes.

B. Structural evolution

Dynamical rearrangements entail microstructural reorganizations at different spatial scales. In this part, the evolution of the granular structure during an event is discussed. To sort the events in order of increasing amplitudes in the rearrangement process, we measure the mean displacement h averaged over all disks between two consecutive snapshots, which is less arbitrary than the avalanche size. Solid fractions at disk scales are measured using Voronoi tessellations [START_REF] Okabe | Spatial tessellations: concepts and applications of Voronoi diagrams[END_REF]. Such a construction defines a unique spatial division that allocates a local volume to each disk, thus defining the solid fraction at the scale of disks. The local densities are tracked over time and the corresponding local solid fraction variations are then measured between two consecutive snapshots with an accuracy of 6 × 10 -3 . The forthcoming analysis has been performed on a subdomain, in restricted populations of grains, with different geometries to focus on relevant data: the rectangular frame depicted in Fig. 2(a) or the radial geometry in Fig. 6, for example.

A spatial view of the local solid fraction variations is plotted in Fig. 6 for an event corresponding to h = 0.018. Significant amplitudes of compaction (δφ > 0) and decompaction (δφ < 0) occur all over the system and not only close to the intruder. This observation indicates that the granular medium as a whole is involved in the reorganization process even for a localized perturbation. Moreover, the regions undergoing compaction and decompaction are not isolated, but rather interconnected.

The probability density function of the local solid fraction variations p(δφ) is displayed in Fig. 7 for three events with different amplitudes ( h 1 = 0.014, h 2 = 0.031, and h 3 = 0.052). These distributions are almost centered on δφ = 0 implying that most of the disks do not undergo strong structural variations. Then, large local compactions always go along with large local decompactions, which is consistent with Fig. 6. An explanation could be that the packing needs to create voids somewhere to unjam elsewhere. Finally as the amplitude of events decreases, the local solid fraction variation distribution gets narrower. This result suggests that there is a relation between the structural evolution and the amplitude of events.

The same conclusions can be obtained in the melting block experiment, as shown in Fig. 8. We note that the amplitudes of displacements are different because the populations considered are not the same. Indeed, we select a smaller population of grains for the melting block experiment due to the smaller size of cell and boundary effects. However, the probability density functions of the local solid fraction variations are quantitatively close for similar amplitudes of displacements in both experiments. We then conclude that the structural evolutions observed in the mechanical-intruder experiment are very similar to the ones observed in the melting block experiment. In the following, we expand on the results obtained with the mechanical intruder rather than on the ones of the too sensitive experiment of the melting block.

In the following, statistical informations on the distribution of the solid fraction variations [average δφ and standard deviation σ(δφ)] is plotted with respect to the mean displacements: the average δφ in Fig. 9(a) and the standard deviation σ(δφ) in Fig. 9(b) respectively. All events associated to the mechanical-intruder experiments are plotted in this figure and so correspond to a reliable statistics of about 16 000 points. In Fig. 9, the points are dense around small values of h , in agreement with Fig. 4, where small avalanches are very frequent. The selected population of grains (the same as in Fig. 6) shows higher mean displacements than in the overall packing (Fig. 3). Looking at moving averages in Fig. 9, several qualitative features can be deduced. First, the average solid fraction variations close to zero correspond to short-range mean displacements. Low values of solid fraction variations are associated with small displacements in the packing. Second, a high level of compactions is associated with high average displacements. To get denser, the medium starts from a looser state that allows large amplitude displacements. Third, the moving average of standard deviations monotonically increases with h . Since each distribution is mainly centered around zero, higher standard deviations necessarily correspond to higher amplitudes of local solid fraction variations and so wider distributions. Knowing the link between event amplitudes and parameters of local solid fraction variations distributions, we center and normalize the previous distributions in the inset of Fig. 7 by their average δφ and standard deviation σ(δφ). The three curves collapse on the same distribution. Therefore events with different amplitudes share a common rearrangement mechanism that leads to self-similar statistics of solid fraction variations. This common rearrangement mechanism is described by two parameters: δφ , the mean value of local solid fraction variations basically being the outcome response, and σ(φ), the standard deviation of local solid fraction variations.

We ensured that this result does not depend on the position of disks. Indeed, the same analysis is performed for three populations of grains at different distances from the intruder. Given the number of disks in each population, the distribution of the solid fraction variation is measured over three experiments to obtain sufficient statistics. These distributions are shown in Fig. 10 and colored according to their vertical positions in the packing. Normalized distributions also collapse (Fig. 11). Choosing a different geometry or another location does not change the conclusions supporting the universality of structural rearrangements in a granular medium undergoing a localized transformation.

V. DISCUSSION

In this part, the underlying mechanism leading to these distributions of local solid fraction variations is discussed. The forms of the distributions can give clues about the mechanism involved (e.g., [START_REF] Radjai | [END_REF]). In our study, the shapes of the distributions are conserved independently of the event amplitude. Those distributions are then fitted with Gaussian and exponential curves on Fig. 11. A Gaussian fit does not capture the tails on both sides of the solid fraction variation distributions whereas exponential fit does. Other studies on two-dimensional silo discharges [35,36] show similar probability density functions with fat tails on relative displacement fluctuations as long as mean displacements are lower than a particle diameter. Interestingly, this threshold is found to set the transition from a superdiffusive to a diffusive regime. Even if the fluctuations of displacements and the variations of the solid fraction at the grain scale are not strictly equivalent, they still depict the same phenomenon: indeed a small fluctuation of relative displacements induces a local density variation. Such shapes are found in our study for mean displacements h between 0.03 and 0.2 that are consistent with the superdiffusive regime found in both references. In addition, it is worth noting that the FIG. 12.

Averaged local solid fraction variations in mechanical-intruder experiments as a function of the corresponding standard deviations using the data of Fig. 9.

previous references deal with flowing grains whereas we analyze quasistatic experiments. Looking at the exponential tails, we propose a scenario that can give some explanations about the evolution of solid fraction variations. During an event, each disk has a given probability to reach a characteristic value of compaction or decompaction. This value sets the range of solid fraction variations and certainly depends on disk properties. But during this process of reorganization, all the disks evolve until they suddenly stop at a random time which does not depend on either the elapsed time or the amplitude of solid fraction variations reached. Such a scenario would explain the tendencies observed in Fig 11 . Indeed, exponential tails on both sides indicate a memoryless process for which a uniform and random process is involved.

In addition, structural variations of the packing are found to be correlated to the amplitudes of displacements as depicted in Fig. 9. Interestingly, in compaction experiments [37], similar conclusions can be found linking the amplitude of the shear perturbation to the amplitude of compaction at the scale of the packing. Our results show that only two parameters are needed to entirely describe the probability density functions of local density variations during an event: the mean and the standard deviation of the solid fraction variations. Moreover, this description could be done with only one parameter, since σ(δφ) and δφ appear to be strongly correlated as illustrated in Fig. 12.

The circular geometries have been chosen to induce radial solicitations and may promote the formation of arches. It is worth noting that the mechanical-intruder experiment does not fully reproduce the lateral influence induced by a pure radial melting like in a melting block experiment. However, because of the gravity, the vertical motions remain predominant in the rearrangement process, reducing the phenomenological differences between both experiments. Moreover, as depicted in Figs. 7 and8, quantitative similarities are found between the structural evolutions of the packing between both experiments. However, some differences still remain. As depicted in Fig. 3, the characteristic size of the melting block plays an important role in the reorganization process. The probability to get large mean displacements decreases with the intruder size, as observed in silos, since the number of disks in contact with the block decreases. In a mechanical-intruder experiment, the mean number of disks in contact with the top disk-shaped region of the intruder remains constant, so this effect does not appear. In addition, the melting of a block within the granular structure remains intrinsically closer to real processes than a mechanical intruder. But such experiments are still too sensitive to get reliable statistics, that is why we decided to expand our results on mechanical-intruder experiments that are more accurate. Therefore, numerical simulations could benefit this study to explore the process of rearrangements with different dynamics of melting or with higher numbers of melting grains with a better repeatability.

In conclusion, we have studied experimentally the rearrangements of a granular medium undergoing a localized transformation. Mainly driven by gravity, the granular systems exhibit long-term displacements similar to those from 2D quasistatic silos. In the mechanical-intruder experiment, local instantaneous displacements are analyzed statistically through the concept of avalanches whose sizes range over three decades. The size distribution of the largest events follows a power law, which is characteristic of processes ruled by scale invariance. Moreover, these large events are uncorrelated in time. Finally, all events participating in the structural evolution of the system share a common mechanism of rearrangements independently of their amplitudes. It shows that only two parameters are necessary to fully describe these structural reorganizations: the mean value that sets the outcome response and the standard deviation. We propose that the origin of the underlying mechanism is governed by a homogeneous and random process.

Most industrial processes involve fully reactive packings. In this study, we only considered a localized transformation due to a single intruder. Even for such a localized perturbation, the granular packing exhibits a longrange response that can involve almost all the packing. Therefore, we expect that multiple and localized transformations, such as a second intruder, may have a significant influence on the packing and exhibit cooperative effects [38].

FIG. 1 .

 1 FIG. 1. (a) Picture of the experimental setup showing the mechanical intruder and the 2D packing. (b) Zoom on the intruder used in the mechanical-intruder experiment and on (c) the initial DMSO disk used in the melting block experiment. Pictures in both (b) and (c) are 6.5 cm wide.

FIG. 2 .

 2 FIG. 2. Vertical displacement field between the initial and the final state for (a) the melting block setup (averaged over 20 experiments) and (b) the mechanical-intruder setup (averaged over 100 experiments). (c) Vertical normalized displacement field obtained at x=0 for the melting block (brown crosses) and mechanical-intruder (green circles) experiments. (d) Horizontal normalized displacement field extracted at y= 2.7, 13.6, 27.1 for the mechanical-intruder experiments. In (d) the dotted line corresponds to the equation for 2D quasistatic silos [26].

FIG. 5 .

 5 FIG. 5. Probability density function associated with the waiting durations between two events (only events with NA ≥ 30 are considered) normalized by the time step ∆t = 5 s for the mechanical-intruder experiments.

FIG. 6 .

 6 FIG.6. Solid fraction variations at the grain scale following a rearrangement event of h = 0.018 in a mechanical-intruder experiment. The range of solid fraction variations has been limited to -2×10 -2 and 2×10 -2 . Being under the accuracy of the measurement, grains with solid fraction variations smaller than 6 × 10 -3 appear in white. Darkest regions correspond to largest amplitudes of solid fraction variations.

FIG. 7 .

 7 FIG. 7. Probability density function of local solid fraction variations for three events of different amplitudes in mechanical-intruder experiments: h1 = 0.014 (yellow squares), h2 = 0.031 (orange circles) and h3 = 0.052 (brown triangles). Inset: Probability density function of rescaled local solid fraction variations.

FIG. 8 .

 8 FIG. 8. Probability density functions associated to the variations of the local solid fractions for three different events showing different mean displacements in melting block experiments. The average variations of solid fractions are δφ1 = -2 × 10 -4 (yellow squares), δφ2 = -10 -4 (orange triangles) and δφ3 = 6 × 10 -4 (red circles).

FIG. 9 .

 9 FIG. 9. (a) Mean value δφ and (b) standard deviations σ(δφ) of local solid fraction variations as a function of the mean displacements h in mechanical-intruder experiments. White circles correspond to averaged values. In (b) the hashed area corresponds to the limit of detection.

FIG. 10 .

 10 FIG. 10. Probability density functions of local solid fraction variations in different regions on top of the intruder (see inset) in mechanical-intruder experiments.

ACKNOWLEDGMENTS

This work has benefited from interesting discussions with P. Levitz and O. Ramos.


H Boutayeb

K Mahdjoubi

A C Tarot

Analysis of Radius-Periodic Cylindrical Structures

The purpose of this communication is to present an analysis method for cylindrical structure witch are periodic along their radii and are excited by a line source placed in the center. At first, we derive analytical formulas for the transmitted wave and the wave inside the structure of a single layer cylindrical structure, which allow to extract the characteristics of the cylindrical layer. Then we obtain the dispersion equation and the transmission coefficient of multiple layer periodic structures.

I -Introduction

There is an important recent interest in periodic structures associated with microwave antennas and devices. Cylindrical periodic structures are however not enough used, probably because their properties are not well known and their behavior not well analyzed. In this paper, we propose an analysis method that allows to obtain the main characteristics of these structures.

In part II, we consider a single cylindrical shell, for which we extract the reflection and transmission coefficients when the incident wave is an outgoing cylindrical one. Using these results, we analyze in part III, the multiple layer periodic structures and give the (approximate) expression of their dispersion equation.

II -Characterization of a single cylindrical surface

Consider the structure of figure 1, where a line source is placed in the center of the cavity constituted by a cylindrical surface. The structure is characterized by the radius C, the (complex) reflection and transmission coefficients (r=|r|.exp(jϕ r ), t) of the cylindrical surface and the reflection coefficient r a of the central line. Our aim is now to obtain the characteristic parameters r and t of the cylindrical surface from the knowledge of T and R (figures 2), the "total" transmission and reflection coefficients of the cylindrical cavity for the uniform cylindrical wave. There is multiple wave reflections between the surface and the central line as in the plane structure case [START_REF] Boutayeb | Antenna inside PBG and Fabry-Perot cavities[END_REF]. The transmission coefficient T is defined as the ratio of the total waves transmitted outside the cavity, to the incident wave expressed by the first order Hankel function H 0 2 (kρ)=|H 0 2 (kρ)|.e -jη 0 (kρ) . T can be expressed in terms of (r,t) and r a (see figure 2a) :

( ) ( ) ( ) ( ) kC 2 j exp rr 1 t kC 2 jn exp r r t T 0 a 0 n 0 n a n η - - = η - = ∑ +∞ = , where ( ) ( ) ( )         = η x J x N arctan x 0 0 0 (1)
J 0 (x) and N 0 (x) are the Bessel functions of order 0, and k is the free space wave number.

Let us call R the wave at a distance D from the line source inside the cavity, normalized by the incident wave (figure 2b). It can be expressed in terms of (r,t) and r a :

( ) ( ) ( ) ( ) ( ) ( ) kC 2 j exp rr 1 kD 2 j exp kC 2 j exp . r 1 R 0 a 0 0 η - - η η - + = (2)
With expressions ( 1) and ( 2), we can calculate the reflection and transmission coefficients (r,t) if we know R and T:

1 2 a B ) B R r ( 1 R r + - = , ) B rr 1 ( T t 1 a - = (3) where ( ) ( ) kC j 2 exp B 0 1 η - = , ( ) ( ) kD j 2 exp B 0 2 η = (4)
In the example bellow the coefficients R and T are obtained by FDTD method, where the incident wave is created by an infinite line of current sources for witch r a =1 (virtual source).

Let us consider now an example of cylindrical surface composed of periodic infinitely long metallic wires of diameter a. P θ is the angular period and P t is the transversal period (P t =P θ *C). Figure 3 gives the magnitude of the reflection and transmission coefficients (r,t) of a metallic wire cylindrical surface versus normalized frequency P t /λ, where a=2mm and P t =pi/6*200mm≈105mm. The coefficients (r,t) are obtained from R and T (evaluated by FDTD method) by using formulas (3) and (4). Figure 4 shows that the resonances of |T| are well foreseen by the condition [START_REF] Boutayeb | Antenna inside PBG and Fabry-Perot cavities[END_REF]. It is also verified that the boundary condition r+1=t is satisfied. These facts validate the method of extraction of (r,t) proposed here.

( ) r 0 n 2 kC 2 ϕ = π + η (n=0,1…) in formula

III -Multiple layer periodic cylindrical structure and dispersion equation

The method of the previous part is applied to obtain the transmission coefficient t of a cylindrical surface of periodic metallic wires, for different values of the radius C (Figure 5). The transverse period P t is maintained constant by varying appropriately the angular period P θ . One can see that for C sufficiently large, surfaces with the same P t have the same coefficient |t|. In fact, the surfaces of the same Pt have the same characteristics (r,t) for sufficiently large C.

Figure 6 presents a multiple layer periodic cylindrical structure where the radial period P r is the distance between two successive surfaces and C 1 is the radius of the first layer.

In a first example, we take C 1 =P r . One the other hand, the surfaces have all the same transversal period P t (so approximately the same (r,t)). Figure 7 shows the magnitude of the transmission coefficient T i for i=1, 2 and 3 layer periodic structure of metallic wires. One can notice that the number of frequency resonance picks is equal to the number of cavities.

Figure 8 gives the transmission line model of the n th layer (n=0,1,2…) of the cylindrical periodic structure where in each side of the equivalent admittance y n of the surface, there is two non-uniform lines of length P r /2. The chain matrix of the quadripole of figure 8 can be written:

(5)

            =                         =       1 1 1 1 2 2 2 2 n 1 1 1 1 2 2 I V H G F E I V d c b a 1 y 0 1 d c b a I V
Where the coefficients a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , d 1 , d 2 are combinations of Bessel functions (see [START_REF] Marcuvitz | Waveguide Handbook[END_REF]). As an example, a 1 coefficient is given bellow:

( ) ( ) ( ) ( ) ( ) ( ) ( ( ) [ ] r r 1 r r 0 r r 0 r r 1 r r 1 P n P k N P n 2 / P k J P n 2 / P k N P n P k J 2 ) P n P ( k a + + - + + + π = ) (6)
We suppose that the outgoing cylindrical wave propagates through the infinite structure with a constant of propagation β such that V 2 =λ V V 1 and I 2 =λ I I 1 (cylindrical equivalent for Floquet theorem) :

, where The dispersion equation becomes :

            λ λ =             1 1 I V 1 1 I V 0 0 I V H G F E ( ) ( ) ( ) ( ) r r 2 0 r r 2 0 V P n 2 / P 3 H P n 2 / P H + β + β = λ , ( ) ( ) ( ) ( ) r r
EH FG E H I V I V - = λ - λ - λ λ (8)
This equation depends on the layer number n and it is not evident that there is a unique β for all n. However, as n increases, the radial line chain matrix tends to the uniform transmission line chain matrix and y n becomes constant and equal to y=2(1-t)/t. Therefore, the dispersion equation (8) tends to that of the plane structure case (9). Thus, the equation (9) becomes a first approximation for cylindrical dispersion equation : In figure 9, we plot the modulus of M=cos(kP r )+jy/2sin(kP r ) (formula (9)) and also the magnitude of the reflection coefficient r 6 of a six layer structure of metallic wires. A frequency band gap appears when |M|>1 (no real solution for β) where |r 6 | is near one. Figure 10 gives the transmission coefficient |T 6 | for the six layer structure with C 1 =P r . |T 6 | has all its frequency picks in the allowed band and it vanishes in the forbidden band. If we take, now, C 1 different from P r , say C 1 =2P r we see that a pick appears within the forbidden band due to the resonance of the first cavity (figure 11).

IV -Conclusion

Analytical expressions are proposed to study periodic cylindrical structures and to obtain there principal characteristics (reflection and transmission coefficients, dispersion equation…). The method can now be applied to analyze the properties (such as frequency or spatial filterings) of these structures. The main property of cylindrical periodic structures is there angular isotropy which allows to obtain better results in many antenna applications such as surface waves suppressing (circular high impedance surfaces), gain enhancement (circular reflectors), etc.

Figure 1 :

 1 Figure 1 : Line source in the center of a cylindrical surface

Figures 2 :

 2 Figures 2 : (a) : Partial terms of the "total" transmission coefficient T (b) : Partial terms of the "total" reflection coefficient R inside the cavity

Figure 3 :Figure 4 :

 34 Figure 3 : Cylindrical periodic surface of metallic wires (C=200mm, a=2mm, P θ =30°, P t =pi/6*200mm ≈105mm)

Figure 5 :

 5 Figure 5 : Magnitude of t for periodic surface of metallic wires for several values of C, (a=1mm, P t =pi/6*40mm).

Figure 6 :Figure 7 :Figure 8 :

 678 Figure 6 : Multiple layer periodic cylindrical structure

  to obtain the k-β diagram (or Brillouin diagram).

Figure 9 :Figure 11 :

 911 Figure 9 : Six layer cylindrical periodic structure reflection and M (9). (a=1mm, Pr=C 1 =40mm P θi =60°/i (i=1,2,…), P t constant) Figure 10 :|T 6 | for six layer structure. (a=1mm, Pr=C 1 =40mm)


Abdoulwahab M Toihir

Venkataraman Sivakumar

Hassan Bencherif

Thierry Portafaix

OZONE CLIMATOLOGY OBTAINED BY A COMBINATION OF MLS/AURA AND SHADOZ OSONESONDES PROFILES OVER THE SOUTHERN TROPIC AND SUBTROPIC

In this paper a new ozone climatology (1998-2012) is presented over 8 southern tropic and subtropic sites. The climatology has been formed by combining data from Aura -MLS (Microwave Limb Sounder) and from SHADOZ ( Southern Hemisphere ADditional OZonesondes) balloon sonde. Prior to combine the two observations, a preliminary study has been conducted to validate MLS profiles by comparison with ozonesondes. A satisfactory agreement is observed between MLS and ozonesonde measurements especially from 20 km to 30 km where the monthly relative difference between the two observations with respect to ozonesonde measurement is between 0.36% and 9.30%. The bias between the two observations decreases with the increase of altitude; from 20 km to 30 km the observed mean bias is assessed to be 3.69±1.13% with respect to SHADOZ observations. The presented climatology consists of monthly averages of ozone profiles for each site obtained from 1 km to 62 km. This climatology is established from hybrid profiles, normalized with the corresponding total column ozone (TCO) measured by Aura Ozone Monitoring Instrument, OMI satellite. Ozone below 26 km is based on ozonesonde while ozone above 29 km is based on MLS measurements. SHADOZ and MLS profiles are merged between 26 and 29 km where the best agreement (around 1%) between SHADOZ and MLS is observed. The obtained climatology captures with more accurately the principle behavior of ozone seasonal and spatial distributions; the integrated total ozone amounts from the presented climatology are consistent with the measured TCO from SAOZ (Système d'Analyse par Observation Zénitale) and Dobson at Reunion and Irene respectively.

INTRODUCTION

Climatological studies allow to analyse the ozone distribution in its annual cycle and to understand the mechanism that control ozone variability at seasonal and intra-seasonal scale. Two kinds of distributions have been used: total column ozone (TCO) and ozone profiles. The technical used to measure ozone profiles are the Lidar, instruments onboard satellite and ozonesonde. However ozone measurements by ozonesondes are mostly adopted due to its high resolution and low uncertainty. SHADOZ (Southern Hemisphere ADditional OZonesondes) is the only network providing ozonesonde profiles with precision around 5 % and vertical resolution between 50 m and 100 m (Thompson et al., 2003b). It is worth noting that balloons borne measurement rarely exceed 32 km, thus satellite observations remain the main technique to complement ozonesondes measurements on altitudes above 32 km. The MLS (Microwave Limb Sounder) is one of the powerful and recent satellite instrument dedicated to observing the Earth system. MLS is a microwave radiometer limb viewing which measures the thermal emission in infrared spectral bands where vertical profiles of ozone, temperature and pressure are retrieved. Ozone profile from SHADOZ network and MLS instrument recorded over 8 southern tropic and subtropic sites are used in this work to construct ozone climatology over the noted area. Satellite instruments offer the best way to observe the upper stratosphere, however data from satellite contain high uncertainty and low vertical resolution in comparison to in-situ measurements. Of this, it is necessary to evaluate the accuracy of satellite measurement for a given area before using them for any major scientific work. In this paper, MLS profiles recorded overpasses over the 8 stations are compared with collocated measurements from ozonesondes in order to assess the quality of MLS profiles. After the MLS profiles validation, the climatology are constructed by merging the MLS and SHAODOZ profiles. The technics used to merge profile is a coupled method based on method of McPeters and Labow (2012) and [START_REF] Sivakumar | Stratospheric ozone climatology and variability over a southern subtropical site: Reunion Island (21•S; 55•E)[END_REF]. The paper is organised as follows: after a brief description of the data source and the obtained results from their comparison in section 2, the section 3 illustrates method used for combination and data validation. Section 4 presents the main features of climatology obtained from the combined profiles. Finally, a summary and conclusion are presented in section 5.

DATA DESCRIPTION AND COMPARISON

Ozone profiles from SHADOZ and MLS are used. SHADOZ radiosondes are launched from ground over the selected 8 stations while MLS profiles are derived from overpasses over each station. A radius of 2° (longitude and latitude) are used to delimit the satellite overpass zone around each station. Figure 1 presents geo-location of the selected stations.

Ozonesonde from SHADOZ

SHADOZ is a network of stations located in the tropics and subtropics which provide high quality of ozonesondes measurement; more detail about this project is found in Thompson et al. (2003a and2003b). Ozone is measured using an electrochemical concentration cell (ECC) ozonesonde. This device is coupled with standard radiosondes for pressure, temperature and relative humidity measurements, after the balloon sonde launching, the information are transmitted to ground by radio while the vertical and spatial location of the equipment are performed through a GPS. For more details on the instrument validation and the ECC SHADOZ ozonesonde operation mode, the reader may refer to Smith et al. 2007, Thompson et al. 2003aand 2007. Ozone profile from SHADOZ are often used as reference in the southern tropic and subtropic to validate satellite measurements [START_REF] Mze | Climatology and comparison of ozone from ENVISAT/GOMOS and SHADOZ/balloon-sonde observations in the southern tropics[END_REF][START_REF] Jiang | Validation of Aura Microwave Limb Sounder Ozone by ozonesonde and lidar measurements[END_REF]). The frequency of observation varies between 2 and 6 launches per month. The s data are downloadable from http://croc.gsfc.nasa.gov/shadoz/ . Here, we have used a temporal cover of 15 years of observation. Table 1 shows the number of monthly profiles recorded during the period of observation (1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012). Considering the limited number of observation, it is necessary to achieve a pre-processing step of all the profiles aiming to identify and remove the possible anomalies. The anomaly can be observed on the complete profile or a given altitude. This step is important in order to depart the unusual events which can modify the behaviour of the climatology. An ozone concentration value is considered as anomaly in a given altitude, if it is defined beyond ± 2σ with respect to the climatological value record in the same altitude. If such case is identified, the value is removed and replaced by a NaN (Not a Number) value. 2011) the considered criterion are: the profile used are those in which the status field is an even number, the quality of field is greater than 0.6 and the convergence field is less than 1.18. MLS profiles with these criteria are available online via http://avdc.gsfc.nasa.gov/pub/data/satellite/Aura/MLS/V03/L2GPOVP/O3/. We have worked with the version 3 of Level 2 overpass product. The uncertainty of these ozone products is estimated at less than 5% in the middle and upper stratosphere [START_REF] Livesey | Earth Observing System (EOS) Aura Microwave Limb Sounder (MLS) Version 3.3 Level 2 data quality and description document[END_REF]). Prior to combine the MLS and ozonesonde profiles, a comparative study was conducted between the monthly climatological profiles of the two datasets. The main objective of this exercise is to examine and to assess the MLS observations quality with respect to SHADOZ. This exercise leads also to detect the conducive altitude level for the combination of the two datasets in order to obtain a climatology covering the troposphere and stratosphere up to 62 km. considering the large variability of MLS profile in the troposphere and the altitude level that can reach a balloon borne, the comparison exercise was performed at 15-30 km.

Profile from MLS

Comparison of ozone profiles from LMS with SHADOZ ozonesondes

The monthly profiles of MLS and SHADOZ are interpolated with a vertical resolution of 100 m and 150 altitude levels have been found between 15 km and 30 km. in each altitude level, the relative difference between MLS and ozonesondes with respect to ozonesondes was calculated as follow:

100 ) ( ) ( ) ( 3 3 3 _    Sonde O Sonde O MLS O RD Sonde MLS
Figure 2 presents the relative difference obtained between MLS and SHADOZ for individual site. The relative difference varies from station to another and from the altitude as well. High values of relative different (> 30%) are observed between 16 km and 19 km, however this high relative difference is reduced during spring and winter periods. The high relative difference observed at 16-19 km is due to the high variability of MLS measurements at the upper troposphere -lower stratosphere (result not presented here). The accuracy of MLS profiles is high in the stratosphere but decreases with the decrease of altitudes to the upper troposphere as explained by Yang et al. (2007). However the quality of MLS measurements is improved during the winter-spring period due to the high ozone mixing-ratio recorded at upper troposphere (Yang et al. 2007). That could be among other reasons of the reduction of the relative difference during the winter-spring. Good agreement between observations is recorded over all stations from 19 km to 30 km where the relative difference varies between 0.36% and 9.30%. This difference indicates that MLS measurements are 3.69±1.13% higher than SHADOZ. The best agreement is recorded at 26-29 km where the bias between the two observations is less than 1%. Thus, the 26-29km altitude band is chosen to merge the MLS and SHADOZ measurements. 3a). The obtained hybrid profile is drawn in red in figure 3b The climatology is reconstructed from a combination of MLS and SHADOZ monthly climatological profiles at 26-29 km. Figure 3a presents a comparison between the monthly climatological profiles of June obtained at Natal from MLS (black) and SHADOZ (blue). The hybrid profile is presented in red on figure 3b. This profile is constituted of 3 levels. The 0-26 km altitude level is composed by ozonesonde measurements while above 29 km the profile is formed by MLS. The 26-29 km part is formed by the median average of the two observations. Each hybrid profile was integrated to calculate the corresponding total column amount. The ratio between the TCO recorded by the Ozone Monitoring Instrument, OMI [START_REF] Bhartia | OMI Algorithm Theoretical Basis Document[END_REF] and the TCO obtained by integration of hybrid profile gives the normalization factor (NF); this factor is used to create the final normalized profile from the hybrid profile. More detail about this method is found on [START_REF] Sivakumar | Stratospheric ozone climatology and variability over a southern subtropical site: Reunion Island (21•S; 55•E)[END_REF].

COMBINATION OF MLS AND OZONESONDES PROFILES

Description of method used for the combination

Comparison of TCO obtained from the combined profile and ground based

TCO obtained from integration of the combined monthly climatological ozone profiles from 1 to 62 km are compared with those recorded from ground-based instruments as SAOZ [START_REF] Pazmiño | O3 and NO2 vertical columns using SAOZ UV-Visible spectrometer[END_REF] and Dobson (Komhyr et al, 1993) operating at Reunion and Irene respectively. The aim of this exercise is to validate the calculated profile and to assess the total column of ozone retrieved from the combined profiles. Figure 4 (top) presents the overlap of ozone annual cycle as observed by the ground-based instruments and retrieved from the combined profiles. Good agreement is found between the total column amount obtained by the integration of profiles and those measured by the ground-based instruments. The obtained correlation coefficient are higher than 0.97 for both stations. The climatology described by all instruments exhibits an annual cycle with an ascending phase in winter and descending phase in summer. The minimum annual amount is recorded in summer/autumn period and the maximum during the winter/spring period as expected at these two subtropical sites (Reunion and Irene). The relative difference (RD) between the obtained climatology and the ground-based measurement with respect to the ground-based measurement is presented in figure 4 (bottom). The obtained RD reveals that the TCO calculated from the combination of MLS + sonde is slightly low with respect to that observed by the Dobson instrument over Irene; the obtained bias is estimated at 1.63±1.088%. This bias is in part due to the systematic error according to the different processes followed to calculate the TCO in the two cases. However the expressed bias at Reunion with SAOZ is very low especially between April and December where the relative difference is within ±1%. In contrast, for the 3 first months of the year, the TCO from SAOZ exhibits a bias of 2.14% with respect to the TCO calculated from the normalized profile. [START_REF] Pastel | Construction of merged satellite total O3 and NO2 time series in the tropics for trend studies and evaluation by comparison to NDACC SAOZ measurements[END_REF] found similar results and estimated that the SAOZ measurements seem to overestimate ozone between January and March with an average relative difference fluctuating between 0.2% and 2.3%. However, the TCO calculated from the normalized profile and those observed from the Dobson and SAOZ instruments are in good agreement and the average bias between them is less than 3%. The ozone varies systematically with the latitude as expected with high concentrations at equatorial region which is located at high altitude, while the ozone maximum peak in the subtropic is located at lower altitude, with respect to those of equatorial and tropical regions. Similar results are found by McPeters and Labow (2012). The obtained climatology accurately represents the behavior of ozone in its seasonal variation on the studied region. Figure 5b presents the variation of ozone profiles in the subtropic for different months. Low ozone amount is recorded in the summer, period marked by intensive ozone formation in the tropic. Indeed, the ozone transport is limited in the ascending phase of Hadley cell circulation during the summer; thus the horizontal transport of ozone from the tropical to subtropical region is not effective. In contrast from the autumn period, a transport of ozone rich-air masses is established horizontally from tropical to subtropical following the Brewer -Dobson circulation. This could be the reason why the ozone amount recorded in March (autumn) is superior to that recorded in December (summer) and the maximum is located at approximately 26.5 km, an average altitude level where the maximum ozone concentration is observed in the tropic. The discharge and accumulation of ozone in the subtropical region is effective between July and September due to a stratospheric mechanism which takes place in this period of the year between the tropic and the subtropic [START_REF] Bencherif | Lidar observations of lower stratospheric aerosols over South Africa linked to large scale transport across the southern subtropical barrier[END_REF]and 2011[START_REF] Sivakumar | Stratospheric ozone climatology and variability over a southern subtropical site: Reunion Island (21•S; 55•E)[END_REF] and constitute the main reason why the maximum annual amount is observed in this period (June and September).

SUMMARY AND CONCLUSION

In this work, a new ozone climatology (1998-2012) of ozone is presented over the southern tropics ad subtropics. The presented climatology is obtained from a combination of MLS and SHADOZ ozone profiles recorded over 8 southern hemisphere sites. Prior to merging the two observations, a preliminary study was conducted in order to qualify the MLS profiles by comparing them with radiosonde data. Good agreement was found between the two observations between 20 km and 30 km where their relative difference varies between 0.36% and 9.30%. The best agreement is found at 26-29 km altitude range where both observations were merged. Below 26 km the ozone climatology consists of radiosonde while above 29 km the climatology is based on MLS observations. The 26-29 km part is formed by the median average of the two observations. Each hybrid profile was normalized using OMI observations basing on [START_REF] Sivakumar | Stratospheric ozone climatology and variability over a southern subtropical site: Reunion Island (21•S; 55•E)[END_REF] method. The obtained climatology accurately captures the characteristics of seasonal and spatial distribution of ozone and are consistent with McPeters and Labow (2012) results. TCO obtained by integration of the obtained climatology is in perfect agreement with that measured by SAOZ and Dobson spectrometers over Reunion Island and Irene in South Africa, respectively.

Figure 1 :

 1 Figure 1: Geolocations of the 8 stations in the southern hemisphere selected for this study.

  / Aura satellite MLS instrument was for the first time onboard the UARS satellite which was operational from September 1991 to August 2001, then for the second time onboard the EOS-Aura satellite launched on 15 July 2004 into a near polar heliosynchronous orbit at an altitude of approximately 705 km. The MLS instrument observes microwave emission from the Earth's limb in the "forward" direction of the Aura orbit[START_REF] Livesey | Interrelated variations of O3 , CO and deep convection in the tropical/subtropical upper troposphere observed by the Aura Microwave Limb Sounder (MLS) during 2004-2011[END_REF]. MLS prvide around 3500 vertical profile of ozone , temperature and other chemical traces gases per day from the troposphere to the mesosphere. The measurements are acheived every 1.5° along the sattelite orbit track which correspond to a horizontal spacing of ∼165 km. The observations are achieved from 82° S to 82°N at two fixed local solar time and cross the tropical region at around 01:30 a.m./p.m. According to the considered geographical delimitation around the station (radius of 2°), the number of profile recorded over the site varies between 0 and 6 per day. Then the daily profile is taken as the average of profiles daily retrieved. Note that the uncertainty of MLS profiles is higher than 10% in the troposphere; because of this high uncertainty and according to[START_REF] Livesey | Validation of Aura Microwave Limb Sounder O3and CO observations in the upper troposphere and lower stratosphere[END_REF] and[START_REF] Jiang | Validation of Aura Microwave Limb Sounder Ozone by ozonesonde and lidar measurements[END_REF] the profile values considered in the present work are those recorded between 215.44 hPa and 0.1hPa. Other criterions are taken in account in order to ensure the reliability of profiles. According toLivesey et al. (

Figure 2 :

 2 Figure 2: monthly distribution of the relative difference between SHADOZ and MLS with respect to SHADOZ observed on the 8 sites.

Figure 3 :

 3 Figure 3: Comparison showing the climatological profile registered in June at Natal by SHADOZ (blue) and MLS (black). The altitude rang in witch MLS and SHADOZ are blinded is given in red dot line (fig.3a). The obtained hybrid profile is drawn in red in figure3b

Figure 4 :

 4 Figure 4: Comparison between the TCO obtained from integration of the MLS + sonde climatology and the TCO measured by SAOZ and Dobson instruments over Reunion (on the right) and Irene (on the left)4. MORPHOLOGY OF THE OBTAINED CLIMATOLOGY

Figure 5 :

 5 Figure 5: (a) Climatology as annual average ozone concentration over 3 latitudinal zone and (b) some climatological monthly profiles representing the different season of the year in the subtropics.

Table 1 : monthly profile number recorded over each station during the observation period ( 1998-2012)

 1

		Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sept	Oct	Nov	Dec	sum
	Irene	17	16	22	20	21	19	17	19	24	26	23	22	246
	Reunion	44	35	43	42	43	44	33	22	41	43	40	31	461
	Fiji	25	25	34	33	19	26	33	27	24	24	29	20	319
	Samoa	40	49	51	42	42	40	48	43	39	41	44	39	518
	Ascension	51	40	52	39	49	46	44	51	47	43	47	40	549
	Java	28	26	27	23	23	26	28	33	32	25	28	24	323
	Natal	46	39	44	36	41	42	35	36	39	42	40	38	478
	Nairobi	51	47	51	44	54	58	55	56	56	57	53	45	627
	sum	302	277	324	279	292	301	293	287	302	301	304	219	3521


H Boutayeb
email: halim.boutayeb@univ-rennes1.fr

K Mahdjoubi

A C Tarot

DESIGN OF A DIRECTIVE AND MATCHED ANTENNA WITH A PLANAR EBG STRUCTURE

The purpose of this presentation is to propose a method to design a directive and matched antenna with a planar Electromagnetic Band Gap (EBG) structure. The EBG structure of our example consist on metallic wires. This study use the FDTD method and analytical formulas.

I -Introduction

Several recent works concern periodic structures associated with microwave antennas to enhance there directivity [START_REF] Thevenot | Directive Photonic Band-Gap Antennas[END_REF][2][3] [START_REF] Boutayeb | EBG Antenna analysis/Unification of frequency and angular dependencies[END_REF]. However, to our knowledge, the impedance aspect is not enough studied. The objective of this presentation is to propose a method to match an antenna put inside a planar EBG structure. Before inserting a real antenna inside the EBG structure, we present a study of the angular filtering characteristics of the EBG structure and the input impedance with an excitation by a line source.

II -Plane waves and infinite long metallic wires : angular filtering

We first interest on the angular filtering aspect due to the EBG structure composed of two layers of metallic wires on each side of the excitation (Figure 1). Let us call T 2*2 the transmission coefficient obtained with an excitation by plane waves (Figure 1). T 2*2 is function of the frequency 'f' and the angular incidence 'θ'. We obtain this coefficient (T 2*2 ) using (r, t), the complex reflection and transmission coefficients of one surface of metallic wires. To simplify the study we don't consider the angular dependency of (r, t). (r, t) are calculated by the FDTD method. The coefficient T 2*2 (kD') can be written in a simple closed form [START_REF] Boutayeb | EBG Antenna analysis/Unification of frequency and angular dependencies[END_REF][5] :

( ) ( ) ( ) ( ) ( ) ' jkD exp ' kD r 1 2 / ' jkD exp ' kD t ' kD T 2 2 2 * 2 - - - = with ( ) ( ) ( ) ( ) ( ) ' D 2 jk exp k r 1 ' jkD exp k t ' kD t 2 2 2 - - - = , ( ) ( ) ( ) ( ) ( )         - - - + = ' D 2 jk exp k r 1 ' D 2 jk exp k t 1 ) k ( r ' kD r 2 2 2
k is the free space wave number and D'=Dcos(θ). Figure 2 gives the transmission coefficient |T 2*2 | versus the frequency at the normal incidence (θ=0°). In this figure, is also represented the Gap Bands and the Pass Bands. We call f 0 the frequency corresponding to the first resonance. Figure 3 gives |T 2*2 | versus the incidence angle at the different frequencies represented in Figure 2. These curves can be assimilated as radiation patterns. Until the first resonance f 0 , the patterns have only one lobe at the normal, which is an objective for a directive antenna. At a frequency higher than f 0 the patterns have multiple lobes one each sides of the normal. Figure 3 More details on the study of angular filtering and directivity of the structure can be found in ref. [START_REF] Boutayeb | Etude des structures périodiques planaires et conformes associées aux antennes. Application aux communications mobiles[END_REF], and will be presented during the presentation.

III -Line source and infinite long metallic wires : input impedance

In this paragraph, we consider infinite long metallic wires and an infinite long line source as the excitation (Figure 5a). In Figure 5b, we give the input impedance Z I for the source with and without the EBG structure. The input impedance is calculated by the FDTD method. In the first Gap Band, the EBG structure gives a low real part which make difficult the matching of an antenna in this band. The matching is possible for frequencies higher than f 0 . In the first Pass Band, the EBG structure add an imaginary part in the input impedance : as a result, the real source must be capacitive in this zone if we want to obtain a matched antenna.

IV -Monopole and finite long metallic wires

Let us now consider finite long metallic wires and a monopole as the exciting source (Figure 6). The frequency resonance of the monopole is called f M . The matching curves for the monopole alone and for the complete structure is given for a monopole resonating before f 0 (Figure 7a) and for a monopole resonating after f 0 (Figure 7b). We know from paragraph II that if we want to obtain a directive antenna at the normal we should not go far away from f 0 . The matching of the complete structure is obtained in the second case only : the matched band (6.7% for S 11 <-10dB) (Figure 7b) is between f M and f 0 , which correspond to the intersection between the possible matching zone and the capacitive zone of the monopole (f<f M ).

V -Conclusion

We have presented a method to design a directive and matched antenna using a monopole as the exciting source. The method consist on using a monopole which resonates at a frequency slightly higher than the first resonance of the transmission coefficient of the structure (which is calculated analytically).

In a future paper we will interest on structures composed of a patch antenna as the exciting source.

Figure 1 -

 1 Figure 1 -Transmission coefficient of the structure excited by plane waves.

Figure 2 -

 2 Figure 2 -|T 2*2 | versus frequency at normal incidence (θ=0°).

  gives a cartography form of |T 2*2 | versus 'f' and 'θ'.

Figure 3 -Figure- 4 -

 34 Figure 3 -|T 2*2 (θ)| at the different frequencies represented in Figure 2.|T 2*2 |

Figure 5 -

 5 Figure 5 -(a) 2 layers of metallic wires on each side of an infinite line source (b) Input impedances Z I for the source with and without the EBG (FDTD).

Figure 6 -

 6 Figure 6 -Monopole inside the finite EBG structure

Figure 8 Figure 7 -0Figure 8 -

 878 Figure8shows the radiation pattern of the structure in the H-plane (Figure6) in the matched band (Figure7b).


Viscosity of multimodal concentrated suspensions in a

Newtonian fluid.

G. Roquier 1

1 Laboratoire Navier, 2 allée Kepler, Cité Descartes, 77420 Champs-sur-Marne (France), mail: gerard.roquier@laposte.net ABSTRACT. A new theoretical formulation about the viscosity of a multimodal concentrated suspension of non-colloidal spherical rigid particles in a Newtonian fluid is presented: the relative viscosity can be linked both to the volume fraction of the suspended particles in a total volume unity and to the solid fraction of the dry mixture. Therefore, it takes into account hydrodynamic interactions and geometrical interactions between particles. Concerning the first ones, it resorts to an iterative approach by change of scale method according to the concept of Farris. Concerning the second ones, when the volume fraction of the suspended particles reaches its critical value, the suspension is jammed and the mixture reaches the solid fraction of the solid skeleton which is predicted by the Compressible Packing Model (CPM). This one takes into account both the loosening effect on big particles by interstitial small ones and the wall effect within assemblies of small particles near a big one. Our predicted viscosities are in good agreement with experimental data of relative viscosities obtained on binary mixtures with 3 different size ratios, for glass beads in silicone oil.

RÉSUMÉ. Une nouvelle formulation théorique concernant une suspension concentrée multimodale de particules sphériques rigides non-colloïdales dans un fluide Newtonien est présentée: la viscosité relative peut être reliée à la fois à la fraction volumique des particules suspendues dans un volume total unité et à la compacité du mélange sec. Elle tient compte des interactions hydrodynamiques et des interactions géométriques entre particules. Concernant les premières, elle fait appel à une approche itérative de type changement d'échelle conformément au concept de Farris. Concernant les secondes, lorsque la fraction volumique des particules suspendues atteint sa valeur critique, la suspension devient empilement et le mélange atteint la compacité du squelette solide prédite par le Modèle d'Empilement Compressible (MEC). Celui-ci tient compte à la fois de l'effet de desserrement provoqué par les fines particules sur les grosses et de l'effet de paroi provoqué par ces dernières sur les plus petites. Nos viscosités calculées sont en bonne adéquation avec les viscosités relatives expérimentales

obtenues sur des mélanges binaires avec 3 rapports de tailles différents, pour des billes de verre dans une huile de silicone.

Introduction.

Knowledge of the viscosity of concentrated suspensions involves a significant number of sectors like biology and medicine (human red cells), manufacturing and materials science (nanopowder systems), mineral processing (drilling fluids), soil science (flow of sediments). In civil engineering, a multitude of heterogeneous materials, like concrete, bitumen binder, are easy to flow if their viscosity is low. At the same time, a dense mixture must be sought, not only to reach a good strength, but also with the aim of decreasing the permeability of the hardened material. The study of the concentrated suspensions viscosity allows to optimize the composition of the granular skeleton while assuring a good workability necessary to be placed easily. The influence of the particle size ratio and of the volume fraction of the fine class on the relative viscosity of a bimodal suspension has been recently investigated by Qi and Tanner [QI 11]. Their approach presents two inconveniences. The first one is that the curve representing the maximum volume fraction of the bidisperse system as a function of the volume fraction of the fine particles is too smooth when compared with numerical data (see for example [ROQ 16] for 0.2 and 0.4 size ratios). The second one is that their study is limited to a bimodal case. The main objective of this study is to highlight a new relation between the relative viscosity η r of a multimodal concentrated suspension and both the volume fraction of the suspended particles in a total volume unity and the solid fraction of the dry mixture. We shall focus on an ideal suspension compound of spherical, inert and rigid particles in a Newtonian fluid. The predictions of the new model are compared with available experimental data from [STO 87] for three different size ratios of glass beads in silicone oil.

The concept of Farris.

Farris (1968) [FAR 68] proposed a theoretical treatment for the prediction of the viscosity of multimodal suspensions if each class size is completely independent of the others: d 1 >> d 2 ...... >> d n . In this case, all smaller spheres added to the suspending fluid behave towards the larger ones as a viscous continuous phase. Farris extends the theory developed by [START_REF] Einstein | Eine neue Bestimmung der Moleküldimensionen[END_REF] [EIN 06] by considering that the relative viscosity depends only on the solid volume concentration in the liquid:

η r = H φ ( ) [ 1 ]
Let us carry out our reasoning (Stovall, Buil, Such [STO 87]) on a suspension constituted by two spherical classes with diameters d 1 >> d 2 occupying volumes V 1 and V 2 . V 0 represents the volume of the suspending fluid. The finest class is first introduced. Its relative viscosity η r 2 can be expressed as follows:

η r 2 = H ψ 2 , ψ 2 MAX ( ) where ψ 2 = V 2 V 0 + V 2 ( ) [ 2 ]
In this precise case, ψ 2 MAX corresponds to the maximal volume fraction of the class 2 of the packing constituted by uniform-sized spherical particles. Let us now add the component with size d 1 . As the condition of separation of scales is checked according to the Farris' assumption, the model of unimodal suspension can be used to predict the viscosity variation linked to the introduction of this new class. The final suspension has thus a relative viscosity corresponding to the product of the relative viscosities associated with each class. Assuming that the mathematical form of the law H is always valid, the viscosity of the suspension becomes:

η r1 = η r 2 H ψ 1 , ψ 1 MAX ( ) = H ψ 1 , ψ 1 MAX ( ) × H ψ 2 , ψ 2 MAX ( ) where ψ 1 = V 1 V 0 + V 1 + V 2 ( ) [ 3 ] ψ 1
MAX is the maximal volume fraction of the class 1 that can be added in the suspension before causing its jamming, taking account of the presence of the finest class 2:

ψ 1 MAX = function ψ 2 ( ) .
Let us generalize this reasoning for the n-components.

η r = H ψ i , ψ i MAX ( ) i = 1 n ∏ [ 4 ]
by pointing out that ψ i MAX depends on the presence of the finest classes:

ψ i MAX = function ψ i+1 , ....., ψ n ( ) .
As the viscosity must diverge when ψ i → ψ i MAX , it is better to write the relative viscosity as follows:

η r = H ψ i ψ i MAX # $ % % & ' ( ( i = 1 n ∏ [ 5 ]
3. Binary mixture without geometrical interactions: apollonian model.

We consider d 1 >> d 2 . Let us call β the dry solid fraction of each monosize class, supposed to be constant.

We start by identifying the "dominant coarse particles" field and the "dominant fine particles" field where each class respectively constitutes the matrix of the dry mixture.

In the first case, coarse particles fill the mold without being disturbed by the finest ones. Their volume fraction in a total volume unity is φ 1 , their critical value being φ 1 * . We have then to verify that:

φ 1 ≤ φ 1 * = β [ 6 ]
In the second case, fine particles fill the porosity of coarse particles. Concerning φ 2 and φ 2 * , we have to respect the following condition:

φ 2 ≤ φ 2 * = β 1 -φ 1 ( ) [ 7 ]
We can then write that:

ψ 1 MAX = φ 1 * = β , ψ 2 MAX = φ 2 * 1 -φ 1 ( ) = β [ 8 ]
As no geometrical interaction occurs, maximal volume fractions of classes 1 and 2 are equal to the solid fraction of each monosize class. Let us set

V j V k k=0 n=2 ∑ = φ y j
where y j≠0 is the volume fraction of the class j by reference of the total solid volume and φ the volume fraction of the suspended particles in a total volume unity. By expressing ψ 1 and ψ 2 as functions of φ , y 1 and y 2 , the relative viscosity can be calculated as follows:

η r = H ψ 1 ψ 1 MAX # $ % % & ' ( ( × H ψ 2 ψ 2 MAX # $ % % & ' ( (
with ψ 1 = φ y 1 , ψ 2 = y 2 1 φ -y 1 $ % & ' ( ) , ψ 1 MAX = β , ψ 2 MAX = β [ 9 ]
4. Binary mixture with geometrical interactions.

The objective now is to combine the concept of Farris and the Compressible Packing Model (CPM) from de Larrard [DEL 00] which takes into account the geometrical interactions. To this end, we are going to follow the scientific process used by Bournonville, Coussot, Chateau [BOU 05]. The approach adopted by Stovall, Buil, Such [STO 87] is indeed incomplete in the sense that volume fractions of each class don't change when seeking to quantify the maximal partial volume of each ones when causing the jamming. However, during this operation, a certain amount of fluid will be deleted, the solid phase remaining unchanged. The developed reasoning is thus inappropriate with the concept of Farris which proceeds by successive incorporations. The flow can then be jammed when a supplementary class is introduced if a sufficient number of particles is added. For the concept of Farris to be valid, volumes of granular classes already present in the suspension need to be maintained constant when a volume δV i of a class i , supposed to be the dominant one, is progressively added. As long as the jamming does not occur, the solid phase can not be considered as a packing. According to the CPM from de Larrard, when the class 1 (coarse particles) is dominant, we have to verify that:

φ 1 ≤ φ 1 * = β -a 12 φ 2 [ 10 ]
where a 12 is the loosening effect coefficient (Figure 1).

In this case:

φ 1 = V 1 + δV 1 ( ) V 0 + V 1 + V 2 + δV 1 ( ) and φ 2 = V 2 V 0 + V 1 + V 2 + δV 1 ( ) (dominant class 1) [ 11 ]
When the class 2 (fine particles) is dominant, the relationship can be written:

φ 2 ≤ φ 2 * = β 1 -φ 1 ( ) -1 -β ( ) b 21 φ 1 [ 12 ]
where b 21 is the wall effect coefficient (Figure 1).

In this case:

φ 1 = V 1 V 0 + V 1 + V 2 + δV 2 ( )
and

φ 2 = V 2 + δV 2 ( ) V 0 + V 1 + V 2 + δV 2 ( ) (dominant class 2) [ 13 ]
Figure 1. Illustrations of the loosening effect (left) and the wall effect (right).

From the precedent relationships, we can deduce the maximal volume ΔV i of the class i that can be added to jam the suspension:

ΔV 1 = 1 1 -β ( ) β V j -V 2 a 12 -V 1 j=0 n=2 ∑ % & ' ' ( ) * * if the class 1 is dominant [ 14 ] ΔV 2 = 1 1 -β ( ) β V j -V 1 β + 1 -β ( ) b 21 ( ) -V 2 j=0 n=2 ∑ % & ' ' ( ) * * if the class 2 is dominant [ 15 ]
The value ΔV 1 takes into account the loosening effect from fine particles of the class 2 on coarse particles of the class 1. The value ΔV 2 takes into account the wall effect from coarse particles on finer ones. When the jamming occurs, we can deduce that:

φ 1 * = V 1 + ΔV 1 ( ) V 0 + V 1 + V 2 + ΔV 1 ( ) and φ 2 = V 2 V 0 + V 1 + V 2 + ΔV 1 ( ) (dominant class 1) [ 16 ] φ 1 = V 1 V 0 + V 1 + V 2 + ΔV 2 ( ) and φ 2 * = V 2 + ΔV 2 ( ) V 0 + V 1 + V 2 + ΔV 2 ( ) (dominant class 2) [ 17 ]
It is thus possible to give a mathematical definition to the partial volumes used in the calculation of the relative viscosity:

ψ 1 = V 1 V 0 + V 1 + V 2 ( ) , ψ 2 = V 2 V 0 + V 2 ( ) , ψ 1 MAX = V 1 + ΔV 1 ( ) V 0 + V 1 + V 2 + ΔV 1 ( ) , ψ 2 MAX = V 2 + ΔV 2 ( ) V 0 + V 2 + ΔV 2 ( ) [ 18 ]
The expressions of ΔV 1 and ΔV 2 provided above lead to:

ψ 1 MAX = β V 0 + V 2 β -a 12 ( ) ( ) V 0 + V 2 1 -a 12 ( ) ( ) , ψ 2 MAX = β V 0 -V 1 b 21 1 -β ( ) ( ) V 0 -V 1 b 21 1 -β ( ) ( ) [ 19 ]
By dividing each volume by

V k k=0 n=2 ∑ , by setting V j V k k=0 n=2 ∑ = φ y j
and by taking into account that y j j=1 n=2 ∑ = 1 , we can express ψ 1 , ψ 2 , ψ 1 MAX and ψ 2 MAX as functions of φ :

ψ 1 = φ y 1 , ψ 2 = y 2 1 φ -y 1 $ % & ' ( ) , ψ 1 MAX = β 1 φ -1 % & ' ( ) * + y 2 β -a 12 ( ) % & ' ( ) * 1 φ -1+ y 2 1 -a 12 ( ) % & ' ( ) * , ψ 2 MAX = β 1 φ -1 % & ' ( ) * -y 1 b 21 1 -β ( ) % & ' ( ) * 1 φ -1 -y 1 b 21 1 -β ( ) % & ' (
) *

[ 20 ]

According to the CPM, the solid fraction γ of a dry binary mixture is calculated by using the expression of γ 1 in the "coarse dominant" field and γ 2 in the "fine dominant" field:

γ = min γ 1 , γ 2 ( ) , γ 1 = β 1 -1 -a 12 ( ) y 2 ( ) , γ 2 = β 1 -1 -β ( ) 1 -b 21 ( ) y 1 ( ) [ 21 ]
By combining precedent relationships, it is possible to obtain ψ 1 , ψ 2 , ψ 1 MAX and ψ 2 MAX as functions of β , φ y 1 , y 2 and of solid fractions γ 1 and γ 2 :

η r = H ψ 1 ψ 1 MAX # $ % % & ' ( ( × H ψ 2 ψ 2 MAX # $ % % & ' ( (
ψ 1 = φ y 1 , ψ 2 = y 2 1 φ -y 1 $ % & ' ( ) , ψ 1 MAX = β 1 φ - 1 γ 1 & ' ( ) * + + y 1 1 -β ( ) & ' ( ( ) * + + 1 φ - β γ 1 & ' ( ) * + , ψ 2 MAX = β 1 φ - 1 γ 2 & ' ( ) * + + y 2 1 -β ( ) & ' ( ( ) * + + 1 φ - β γ 2 -y 1 1 -β ( ) & ' ( ) * + [ 22 ]
The advantage of these new expressions consists in highlighting a direct link between the relative viscosity η r , even if the stiffening function is not completely defined, the volume fraction φ of the suspended particles in a total volume unity, and the solid fraction of the dry binary mixture γ 1 or γ 2 . According to the approach adopted by Bournonville, Coussot, Chateau [BOU 05], it is now possible to create a suspension whose the solid phase is composed like a dry granular material with a solid fraction predicted by the CPM. When the latter is maximal by optimization of the granular skeleton, the critical solid fraction value is reached, the flow is jammed and the suspension becomes a packing. This "solid fraction -viscosity" association could also allow to improve the formulation of a liquefied concrete, for example, by following two steps: minimizing its porosity, i.e. the water volume in a volume unity, for a set viscosity but non-infinite.

Ordinary mixture composed by n-classes.

Let us now study an ordinary mixture composed by n-classes of particles. Let us suppose the i th class be the dominant one by taking into account the wall effect from coarsest particles and the loosening effect from finest ones. Its volume fraction in a total volume unity φ i must be lower than its critical value φ i * :

φ i ≤ φ i * = β - β + b ij 1 -β ( ) ( ) j=1 i-1 ∑ φ j - a ij φ j j=i+1 n ∑ [ 23 ]
where a ij indicates the loosening effect coefficient and b ij the wall effect coefficient.

' ( ( )

* + + 1 φ - β γ i -1 -β ( ) y j j=1 i-1 ∑ & ' ( ( ) * + + [ 30 ]
The expression of the relative viscosity diverges to infinity as φ goes to γ i when the class i is dominant: the suspension becomes a packing. However, it is necessary to be conscious of the following paradox that may occur [BOU 05]: the concept of Farris assumes additions of successive classes by change of scale. The divergence can occur when the class i is introduced because this one reaches its critical volume fraction, the following classes

having not yet been incorporated. However, the volume ΔV i is calculated by taking into account geometrical interactions with all the granular classes, including those not present yet in the suspension when the divergence occurs. The expression allowing to determine the relative viscosity must then be used carefully.

Experimental verifications on binary mixtures.

Experimental data concerning binary mixtures are those presented by [STO 87]. They concern glass beads classified in rather narrow granular classes. The suspension liquid is a silicon oil with a viscosity equal to 17.9 Pa.s for a temperature of 20°C. It exhibits a Newtonian behaviour. Measurements are performed in coaxial cylinders.

The relative viscosity is calculated by a power-law relation (Krieger-Dougherty type [STO 87]):

η r = 1 - ψ i ψ i MAX $ % & & ' ( ) ) -η * + , -1 β i=1 n=2 ∏ [ 31 ]
where η " # $ % 1 = 2.5 is the first-order intrinsic viscosity for spheres [EIN 06] and β the solid fraction of each monosize class. The wall effect coefficient b 21 and the loosening effect coefficient a 12 are determined from the expressions of de Larrard [DEL 88] which are specifically adapted to spheres:

b 21 x ( ) = 1 -1 -x ( ) 1.6 , a 12 x ( ) = 1 -1 -x ( ) 3.1 -3.1 x 1 -x ( )
2.9

[ 32 ]

where x is the size ratio fine/coarse particles diameters. Predictions of the viscosity model are, for the studied suspensions, in good agreement with experimental data by taking into consideration that the solid fraction of each monosize class β is only estimated, and by considering high experimental uncertainties for the tested concentration ranges. For them, the exponent of the Krieger-Dougherty law equal to -2.5 β seems to be appropriate.

7.

Upper and lower limits for η r,th as a function of φ / φ m for a binary mixture.

If now, the theoretical relative viscosity corresponding to the three size ratios x = 0.2 , x = 0.7 , x = 0.4 is represented for y 2 = 0.30 as a function of the fraction φ / φ m , called the normalized total solid volume fraction, the results show that the curves are very close together. However, this is not a single master curve (Figure 3).

An upper limit of η r,th can be found for the mixture with the highest solid fraction: the apollonian model. In this case:

x = 0 , b 21 = 0 , a 12 = 0 , γ 1 = β 1 -y 2 ( ) , γ 2 = β 1 -1 -β ( ) y 1 ( ) [ 33 ]
The highest solid fraction is reached when γ 1 = γ 2 for which:

y 1 = 1 2 -β ( ) , y 2 = 1 -β ( ) 2 -β ( ) , φ m = β 2 -β ( ) , ψ 1 = β φ φ m , ψ 2 = 1 -β ( ) 1 β φ m φ % & ' ( ) * -1 % & ' ' ( ) * * , ψ 1 MAX = β , ψ 2 MAX = β [ 34 ]
By using a power-law relation (Krieger-Dougherty type), we can deduce that: A lower limit of η r,th can also be found for the mixture with the lowest solid fraction when small particles are as fine as possible in infinitesimal quantity:

η r,th sup = 1 -φ / φ m ( ) -η $ % & ' 1 β × 1 - 1 -β ( ) 1 φ / φ m -β * + , - . / * + , , , , , - . /
x = 0 , b 21 = 0 , a 12 = 0 , y 1 → 1 , y 2 → 0 , γ 1 → β , γ 2 → 1 , φ m → β

[ 36 ]

ψ 1 → φ , ψ 2 → 0 , ψ 1 MAX = β , ψ 2 MAX = β [ 37 ]
We can deduce that:

η r,th inf = 1 -φ / φ m ( ) -η $ % & ' 1 β [ 38 ]

Conclusion.

From a work of Bournonville, Coussot, Chateau, we have been able to develop a model of viscosity in a Newtonian fluid linking this latter to the volume fraction of the suspended particles in a total volume unity and to the solid fraction of the dry mixture. This theory takes into account both the hydrodynamic interactions, through the concept of Farris, and the geometrical interactions, through the CPM. However, to be compatible with these two approaches, the model initially proposed by Farris is modified to adopt the description of the wall effect and of the loosening effect used in the CPM. First results are positive because they give relatively good predictions about relative viscosities on suspensions of glass beads with three different size ratios. The model also allows to determine upper and lower limits for the theoretical relative viscosity as a function of φ / φ m which shows that a single master curve can not exist. Even if it requires improvements, this work could constitute an interesting comprehension tool to optimize granular skeletons for various kinds of materials where the densest mixture must be sought for a given viscosity.

  KEY WORDS: Relative viscosity, concentrated suspension, change of scale method, solid fraction, Compressible Packing Model (CPM), spheres. MOTS-CLÉS : Viscosité relative, suspension concentrée, méthode de changement d'échelle, compacité, Modèle d'Empilement Compressible (MEC), sphères.

Figure 2 .

 2 Figure 2. Bimodal suspensions: comparison between relative predicted viscosities and relative experimental values from [STO 87] with 3 size ratios (x=0.2, x=0.4, x=0.7), for glass beads in silicone oil.

Figure 3 .

 3 Figure 3. Evolution of the theoretical relative viscosity as a function of the normalized total solid volume fraction φ / φ m for a volume fraction of the fine class y 2 =0.30 (β=0.608).

φ i and φ j≠i are calculated as follows when a volume δV i of the class i is added:

and

From this system of equations, we can now deduce the maximal volume ΔV i of this class i to be added to jam the suspension.

The value ΔV i takes into account both the wall effect and the loosening effect. When the jamming occurs, we can write:

and

It is now possible to define ψ i and ψ i MAX of the class i in a mixture only constituted by finest classes, according to the concept of Farris:

We can finally express ψ i and ψ i MAX as follows:

According to the CPM, the solid fraction of a dry granular mixture composed by n-classes when the class i is dominant is:

We can then determine the relative viscosity: