François Druelle

Mélanie Berthet

Benoit Quintard

The body centre of mass in primates: is it more caudal than in other quadrupedal mammals?

Keywords: Centre of Mass, Locomotor evolution, Morphotype, Primates

Objective: Whole body centre of mass position values are lacking for a comparative sample of primates. Therefore, it still remains unknown whether the body centre of mass in primates is more posteriorly located than in other mammals. The aim of the present report is to provide data for a large sample of primate species and to compare the position of the body centre of mass in primates to nonprimate mammals.

Materials and Method:

We collected morphometrics on eight primate species belonging to various families: Hylobatidae (Nomascus grabriellae, Nomascus Siki), Cercopithecidae (Cercopithecus roloway, Cercopithecus lhoesti, Colobus guereza, Trachypithecus francoisi), Cebidae (Sapajus xanthosternos) and Atelidae (Ateles fusciceps). Using a geometric model, we assessed the position of the body centre of mass (BCoM) in a natural quadrupedal posture and in a control posture. To complete our comparative sample with a wider range of morphotypes, we added the data available in the literature for hominoids (Pan paniscus, Pan troglodytes, Gorilla gorilla, Pongo pygmaeus, Hylobates lar) and another cercopithecoid species (Papio anubis). We also evaluated the phylogenetic signal of the position of the BCoM in primates.

Results:

The variation in the position of the BCoM in primates is very large, ranging from 40% of the distance between the hip and the shoulder in Ateles fusciceps to 63% in Hylobates lar. We observed a strong phylogenetic signal for this trait: hominoid species, as well as the baboon, have a cranial BCoM relative to the midline between the hip and the shoulder, arboreal cercopithecoids and the spider monkey have a caudal BCoM, and the capuchin monkey has a BCoM positioned at mid-trunk. The variation observed in non-primate quadrupedal mammals lies inside the variation range of primates, from 51% in Felis catus to 63% in Canis familiaris.

Discussion:

The BCoM of primates is not more posteriorly located than in other quadrupedal mammals, however, there is a substantial range of variation in primates, from caudal (in arboreal quadrupeds) to cranial (in hominoids and terrestrial quadrupeds) positions. This variation is related to a phylogenetic model that suggests stabilizing selection for this trait. It seems that the BCoM position mostly depends of the size of the appendicular system (i.e. limbs) and the tail. Therefore, it may also reflect a general trend in quadrupedal mammals with arboreal species exhibiting a caudal BCoM and terrestrial species exhibiting a cranial BCoM. These results are discussed in the context of the locomotor evolution of primates including locomotor habits and gait mechanics. We also propose a new 'passive' mechanism for the explanation of the particular weight support pattern observed in primates with tails.

INTRODUCTION

It has long been assumed that the body centre of mass of primates is relatively caudal, i.e. situated close to the hip, compared to other mammals [START_REF] Rollinson | Comparative aspects of primate locomotion, with special reference to arboreal cercopithecines[END_REF]. This idea was initially based on force plate data indicating that primates commonly support more of their weight on their hind limbs [START_REF] Kimura | Kinesiological characteristics of primate walking: its significance in human walking. Environment, behavior, and morphology: Dynamic interactions in primates[END_REF]. Although the differences in the amount of weight supported by the fore-and hind limbs should indeed be related to the position of the whole body centre of mass (e.g. [START_REF] Gray | Studies in the Mechanics of the Tetrapod Skeleton[END_REF][START_REF] Manter | The dynamics of quadrupedal walking[END_REF], it is not a direct determination of its location [START_REF] Vilensky | Primate locomotion: utilization and control of symmetrical gaits[END_REF][START_REF] Reynolds | Mechanics of increased support of weight by the hindlimbs in primates[END_REF]. Furthermore, some authors have suggested that primates could actively shift their weight to their hind limbs during walking [START_REF] Reynolds | Mechanics of increased support of weight by the hindlimbs in primates[END_REF][START_REF] Schmitt | Compliant walking in primates[END_REF]. Recently, both 'passive' (the position of the limbs relative to the centre of mass) and 'active' models have found support in different primate species (Passive Model: Raichlen, Pontzer, Shapiro, & Sockol, 2009;Active Model: Larson & Stern, 2009;[START_REF] Larson | Weight support distribution during quadrupedal walking in Ateles and Cebus[END_REF]. As a result, on the one hand, a full understanding of the weight support pattern observed in primates is lacking. On the other hand, it still remains unknown whether the whole body centre of mass of primates is more posteriorly located than in other mammals.

The average location of the body centre of mass can be accurately estimated from the inertial properties (mass and centre of mass) of the body segments and from their position in space (e.g. [START_REF] Miller | Biomechanics of sport: a research approach[END_REF]. Using this methodology, [START_REF] Vilensky | Primate locomotion: utilization and control of symmetrical gaits[END_REF] tried to address the issue of whether the body centre of mass of primates is more caudal than in other quadrupedal mammals by comparing the few inertial data available at that time for 1 brown lemur [START_REF] Wells | Measurement of body segment mass, center of gravity, and determination of moments of inertia by double pendulum in Lemur fulvus[END_REF], 1 spot-nosed guenon [START_REF] Rollinson | Interspecific Comparisons of Locomotor Behavior and Prehension in Eight Species of African Forest Monkey: A Functional and Evolutionary Study[END_REF] and 15 rhesus monkeys [START_REF] Vilensky | Masses, centers-of-gravity, and moments-of-inertia of the body segments of the rhesus monkey (Macaca mulatta)[END_REF].

They compared these data to the inertial properties of 1 cat [START_REF] Manter | The dynamics of quadrupedal walking[END_REF] and concluded that these differences were certainly too minor to profess the presence of a more caudal centre of mass in primates. Nevertheless, their very small sample size did not lead to a reliable conclusion on this issue. Surprisingly, larger comparative datasets evaluating the position of the centre of mass in different primate and non-primate species have never been published. The objective of the present report is, therefore, to fill this gap of knowledge by providing an accurate estimation of the body centre of mass position in different primate species based on the inertial properties of their body segments, and to compare it to non-primate quadrupedal mammals.

Based on the aforementioned observations [START_REF] Vilensky | Primate locomotion: utilization and control of symmetrical gaits[END_REF], we hypothesize that the position of the whole body centre of mass of primates is not more posterior than in other quadrupedal mammals. To test this hypothesis, we used a sample representative of the diversity of primate body shape and size by collecting data on Hylobatidae, Atelidae, Cebidae, Cercopithecinae and Colobinae.

We also added available data from the literature on Hominidae and other Cercopithecinae (Druelle, Aerts, D'Août, Moulin, & Berillon, 2017a;[START_REF] Druelle | Segmental morphometrics of the southern yellow-cheeked crested gibbon (Nomascus gabriellae): the case study of four individuals in zoo[END_REF][START_REF] Druelle | Segmental morphometrics of bonobos (Pan paniscus): are they really different from chimpanzees (Pan troglodytes)?[END_REF][START_REF] Isler | Inertial properties of hominoid limb segments[END_REF][START_REF] Schoonaert | Morphometrics and inertial properties in the body segments of chimpanzees (Pan troglodytes)[END_REF]. First, we tested whether the phylogeny can explain the variation observed among and within clades in primates. Second, we compared these data to data available for the cat [START_REF] Manter | The dynamics of quadrupedal walking[END_REF], the dog [START_REF] Amit | Segmental inertial properties in dogs determined by magnetic resonance imaging[END_REF] and the horse [START_REF] Buchner | Inertial properties of Dutch Warmblood horses[END_REF]. Third, we compared the position of the whole body centre of mass between hominoids (described as orthograde primates) and other quadrupedal (pronograde) primate species.

MATERIALS AND METHODS

Sample

The data were collected in the Zoological Park of the Besançon Museum, France, and in the Zoological Park of Mulhouse, France. We took the opportunity of taking external measurements on eighteen primate individuals during veterinary checks while they were under anesthesia. These individuals belong to eight species: the brown-headed spider monkey, Ateles fusciceps robustus (Atelidae), the buff-headed capuchin, Sapajus xanthosternos (Cebidae), the Roloway monkey, Cercopithecus roloway (Cercopithecinae), the L'Hoest's monkey, Cercopithecus lhoesti (Cercopithecinae), the Guereza monkey, Colobus guereza (Colobinae), the Francois' leaf monkey, Trachypithecus francoisi (Colobinae), the southern yellow-cheeked crested gibbon, Nomascus gabriellae (Hylobatidae) and the southern white-cheeked crested gibbon, Nomascus siki (Hylobatidae).

The veterinary examinations did not reveal any musculo-skeletal abnormalities. All of these individuals live inside large enclosures with various enrichments. Table 1 shows the individual information of our sample and includes the additional comparative sample, i.e. the chimpanzee, Pan troglodytes [START_REF] Schoonaert | Morphometrics and inertial properties in the body segments of chimpanzees (Pan troglodytes)[END_REF], the bonobo, Pan paniscus [START_REF] Druelle | Segmental morphometrics of bonobos (Pan paniscus): are they really different from chimpanzees (Pan troglodytes)?[END_REF], the gorilla, Gorilla gorilla, the orangutan, Pongo pygmaeus, the lar gibbon, Hylobates lar [START_REF] Isler | Inertial properties of hominoid limb segments[END_REF] and the southern yellow-cheeked crested gibbon, Nomascus gabriellae [START_REF] Druelle | Segmental morphometrics of the southern yellow-cheeked crested gibbon (Nomascus gabriellae): the case study of four individuals in zoo[END_REF].

Measurement protocol

The protocol of measurements is based on the geometric model developed by [START_REF] Crompton | Segment inertial properties of primates: New techniques for laboratory and field studies of locomotion[END_REF]. It has previously been applied on hominoid [START_REF] Isler | Inertial properties of hominoid limb segments[END_REF][START_REF] Druelle | Segmental morphometrics of bonobos (Pan paniscus): are they really different from chimpanzees (Pan troglodytes)?[END_REF][START_REF] Schoonaert | Morphometrics and inertial properties in the body segments of chimpanzees (Pan troglodytes)[END_REF][START_REF] Crompton | Segment inertial properties of primates: New techniques for laboratory and field studies of locomotion[END_REF][START_REF] Druelle | Segmental morphometrics of the southern yellow-cheeked crested gibbon (Nomascus gabriellae): the case study of four individuals in zoo[END_REF] and cercopithecoid specimens (Druelle et al., 2017a;[START_REF] Raichlen | Ontogeny of limb mass distribution in infant baboons (Papio cynocephalus)[END_REF][START_REF] Raichlen | Convergence of forelimb and hindlimb Natural Pendular Period in baboons (Papio cynocephalus) and its implication for the evolution of primate quadrupedalism[END_REF] and allows for the reliable estimation of the inertial properties of the body (tested on non-human primate cadavers in [START_REF] Crompton | Segment inertial properties of primates: New techniques for laboratory and field studies of locomotion[END_REF][START_REF] Isler | Inertial properties of hominoid limb segments[END_REF]. External linear measurements were taken on eight body segments (or nine, if a tail was present): head (including neck), trunk, arm, forearm, hand, thigh, shank and foot. The landmarks used follow those of previous studies (e.g. [START_REF] Schoonaert | Morphometrics and inertial properties in the body segments of chimpanzees (Pan troglodytes)[END_REF]. We obtained individual segment dimensions (lengths and antero-posterior and medio-lateral diameters) and, according to the model, we estimated segment mass (average estimated density: 1 g.cm -3 ) and segment centre of mass (CoM; calculated from the proximal joint of each segment, from the hip for the trunk and the back for the head). Summed per individual, the mass of the different body segments yields an estimated total body mass. Figure 1 shows that the total body mass estimated by the geometric model is consistent with that directly measured with an electronic scale, confirming the reliability and accuracy of the procedure.

Based on the morphological variables, we designed an average free body diagram for each species using the mathematical and geometric software GeoGebra 5.0. Each species morphotype has been built on a minimum of two adult individuals. The morphotypes are scaled, i.e. segment masses are divided by the body mass, segment length is expressed as percentage of trunk + head (including neck) length and segment CoM is expressed as percentage of the respective segment length. We used two postures to assess the location of the whole body centre of mass (BCoM): a natural posture and a control posture. The natural posture is an average posture inspired from biomechanical studies of walking and from lateral pictures available in the literature (Table 2; [START_REF] Pontzer | Bipedal and quadrupedal locomotion in chimpanzees[END_REF][START_REF] Finestone | Great ape walking kinematics: Implications for hominoid evolution[END_REF][START_REF] Patel | Ontogenetic Scaling of Fore-and Hind Limb Posture in Wild Chacma Baboons (Papio hamadryas ursinus)[END_REF][START_REF] Helmsmüller | Ontogenetic allometry of the Beagle[END_REF][START_REF] Buchner | Inertial properties of Dutch Warmblood horses[END_REF]; figure 2 shows the average posture and the limb angles used for the chimpanzee. When there is a tail, it was positioned at a 30°angle with the vertical. The control posture is a standardized quadrupedal body posture in which the fore-and hind limbs are extended perpendicular to the trunk, the tail (if any) is placed perpendicular to the trunk and along the hind limbs, and the head is perpendicular to the forelimbs and along the trunk. For each species we calculated the position of the BCoM in these two postures. The BCoM is expressed from the hip joint as the percentage of the horizontal distance between the hip joint and the shoulder joint (i.e. the moment arm of the BCoM about the hip joint).

Phylogenetic signal Using the K of [START_REF] Blomberg | Testing for phylogenetic signal in comparative data: Behavioral traits are more labile[END_REF] and the "phylosig" function in R [START_REF] Revell | phytools: an R package for phylogenetic comparative biology (and other things)[END_REF], we estimated the phylogenetic signal for the position of the BCoM in primates. The statistical significance of K is evaluated on 1000 simulations from the randomization test. This enables to assess the amount of phylogenetic signal observed relative to the amount expected according to a Brownian motion. The phylogeny used in the present study is time-scaled and is based on a consensus chronometric tree of extant primate species. It has been downloaded from the 10kTrees WebServer (https://10ktrees.nunn-lab.org/) that provides a phylogeny sampled from a Bayesian phylogenetic analysis.

Sensitivity analysis

The effects of limbs and tail postural variations on BCoM translation have been evaluated for each primate species in our sample. In the control posture, the limbs were successively protracted and retracted by an angle of 20° while keeping other body parts static. We have also tested the effect of shifting the tail by an angle of 20° backward. We calculated the influence of each of these average postures on the position of the BCoM as follows:

𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝐵𝐶𝑜𝑀 = (𝑥 𝑖 -𝑥 𝑓 ) (𝑥 𝑖 + 𝑥 𝑓 ) × 100
where xi is the original position of the BCoM in the control posture and xf is the new position of the BCoM after shifting the forelimbs, the hind limbs or the tail. By summing the absolute values of the variation in protracted and retracted limb postures, we calculated the BCoM range of variation for the fore-and hind limbs separately.

Statistics

We tested the difference in BCoM position between hominoids and other primates using exact permutation tests for independent samples. We also tested whether there is a difference in how much fore-and hind limbs influence the BCoM translation between hominoids and other primate species. Statistical significance was set at P<0.05 and the tests were performed using StatXact 3.1 (software, Cytel, Inc., Cambridge, MA).

RESULTS

Our results indicate that the variance of the primate BCoM position is concentrated between clades rather than within clades (K=1.54; P=0.001). Because K is greater than 1, it indicates that closely related species resemble each other more than expected under Brownian motion (K≈1). Figure 3 shows the relative position of the BCoM in our comparative sample of primates and non-primate mammals in the natural and control postures. Table 3 summarizes these values. In both postures, there is an obvious overlap between the group of primates and the group of non-primate mammals. In the natural body posture, the most extreme positions of the BCoM are found in A. fusciceps which exhibits the lowest value (40%, i.e. the BCoM is closer from the hip), and the H. lar and C. familiaris which exhibit the highest value (63%, i.e. the BCoM is closer from the shoulder). In the control body posture, the spider monkey remains the species with the most caudal BCoM (43%), and the dog is the one with the most cranial BCoM (65%).

Overall, hominoid species exhibit a cranially located BCoM which is significantly different from other quadrupedal primates (natural posture: permutation test: 3.224, P=0.0006; control posture: permutation test: 3.036, P=0.0012). This is the result of huge forelimbs in terms of size and mass (one forelimb represents 7-10% of total body mass in hominoids; see supplementary material for body mass distribution data). Our sensitivity analysis also shows that the position of the BCoM is significantly more affected by the forelimbs in hominoids compared to other primates (permutation test: 3.062, P=0.0006); this is not the case for the hind limbs (permutation test: 1.647, P=0.1031). Arboreal Cercopithecinae (i.e. Cercopithecus species) and (arboreal) Colobinae exhibit a caudal BCoM. This is the result of relatively more massive hind limbs than forelimbs (mainly due to the mass of the thigh) as well as the presence of a long and relatively massive tail (3-4% of total body mass). The sensitivity analysis shows that protracting and retracting the hind limbs in the range amplitude of 40° impacts the BCoM position by 3.63% (C. roloway) to 5.27% (C. lhoesti) in these species, while moving the forelimbs in this range amplitude only affects the BCoM position between 1.5% (C. roloway) and 2.05% (T.

francoisi). Retracting the tail by an angle of 20° affects the position of the BCoM by 0.49% (C. guereza) to 0.85% (C. roloway). The (semi-)terrestrial Cercopithecinae, the olive baboon, exhibit a cranial BCoM, which is the result of more similar fore-and hind limbs, a lighter and smaller tail (2% of total body mass) and the presence of a larger head segment. The spider monkey is a Platyrrhini (i.e. New World Monkeys) and exhibits the most caudal BCoM position. Although the fore-and hind limb proportions seem to be similar to that of Cercopithecidae in terms of relative segment masses, the massive prehensile tail (7% of total body mass) of the spider monkey shifts the BCoM posteriorly. A 20°-shift of the tail translates the BCoM by 1.85% backward. The BCoM of the capuchin monkey (Platyrrhini) is located at mid-trunk (49.69% in the natural posture and 50.16% in the control posture). With regard to the two Platyrrhini species, the tail has a significant impact on the position of the BCoM as a 20°shift of the tail translates the BCoM by 1.85% in spider monkeys and by 0.98% in capuchins. Figure 4 shows the free body diagrams used for the calculation of the BCoM in the natural postures.

DISCUSSION

The present dataset offers a comparative sample of (hominoid and non-hominoid) primate and nonprimate species for the evaluation of the BCoM position. There is an important overlap in the location of the body centre of mass between hominoids and non-primate mammals. However, most nonhominoid primates have a body centre of mass closer to the hips than carnivores and ungulates. Our results thus partially support the previous hypothesis of [START_REF] Vilensky | Primate locomotion: utilization and control of symmetrical gaits[END_REF] stipulating that, overall, the BCoM of primates is not more posterior than in other quadrupedal mammals. Indeed, according to the present study, all arboreal quadrupedal primates seem to exhibit a more posterior BCoM than quadrupedal mammals. In addition, we found a phylogenetic signal that may indicate stabilizing selection for this trait (BCoM position) in primates.

BCoM and locomotor habits

Primates commonly live in arboreal environments that imply discontinuity, variability, flexibility and instability of the substrates, and although they generally rely on the quadrupedal system [START_REF] Rose | Quadrupedalism in primates[END_REF][START_REF] Schmitt | Compliant walking in primates[END_REF][START_REF] Schmitt | Adaptive value of ambling gaits in primates and other mammals[END_REF], some have evolved towards new positional (orthograde) habits involving important interlimb dissociation [START_REF] Young | Development and the evolvability of human limbs[END_REF]. For example, the group of hominoid species can be considered to be generally adapted to orthograde, suspensory locomotion (e.g. [START_REF] Hunt | Positional behavior in the Hominoidea[END_REF][START_REF] Crompton | Locomotion and posture from the common hominoid ancestor to fully modern hominins, with special reference to the last common panin/hominin ancestor[END_REF]Thorpe & Crompton, 2006;[START_REF] Fan | Locomotion behavior of cao vit gibbon (Nomascus nasutus) living in karst forest in Bangliang Nature Reserve, Guangxi, China[END_REF] and, therefore, do not belong to the typical quadrupedal primate group (e.g. [START_REF] Rose | Quadrupedalism in primates[END_REF][START_REF] Finestone | Great ape walking kinematics: Implications for hominoid evolution[END_REF]. In this context, removing the hominoid species from the present sample leaves only one pronograde (semi-)terrestrial primate (the olive baboon) exhibiting a cranial configuration of the BCoM as in non-primate quadrupedal mammals.

Therefore, it is possible that the BCoM of arboreal quadrupedal primates is more posterior than in quadrupedal mammals, although it may not be the case for arboreal non-primate mammals (e.g. [START_REF] Schmitt | Origins of primate locomotion: Gait mechanics of the woolly opossum[END_REF].

The BCoM can be considered as an average representation of the general body shape of animals.

According to our results, there is a strong phylogenetic signal for its position, which is very likely in the context of primate body shape evolution. Closely related species may resemble to each other in terms of body proportions (e.g. [START_REF] Druelle | Segmental morphometrics of bonobos (Pan paniscus): are they really different from chimpanzees (Pan troglodytes)?[END_REF]. Furthermore, the variation in the BCoM position appears to be mostly related to the size and mass of the appendicular system, i.e. fore-and hind limbs, as well as the tail. In primates, the variation in the proportions of these body segments are likely to be under strong selective pressures as limbs are strongly associated to locomotor adaptations [START_REF] Fleagle | Primate Adaptation and Evolution: 3rd Edition[END_REF][START_REF] Preuschoft | Size Influences on Primate Locomotion and Body Shape, with Special Emphasis on the Locomotion of 'Small Mammals[END_REF][START_REF] Young | Development and the evolvability of human limbs[END_REF], and the tail to balance control [START_REF] Larson | Maintenance of above-branch balance during primate arboreal quadrupedalism: coordinated use of forearm rotators and tail motion[END_REF][START_REF] Young | Tail function during arboreal quadrupedalism in squirrel monkeys (Saimiri boliviensis) and tamarins (Saguinus oedipus)[END_REF] and locomotion [START_REF] Anapol | Comparative postcranial body shape and locomotion in Chlorocebus aethiops and Cercopithecus mitis[END_REF][START_REF] Rodman | Skeletal differentiation of Macaca fascicularis and Macaca nemestrina in relation to arboreal and terrestrial quadrupedalism[END_REF][START_REF] Youlatos | Positional Behavior of Black Spider Monkeys (Ateles paniscus) in French Guiana[END_REF]. [START_REF] Raichlen | Understanding hind limb weight support in chimpanzees with implications for the evolution of primate locomotion[END_REF] suggested that the particular weight support pattern of primates could be an evolutionary by-product of the variations in limb morphology (e.g. intermembral index). The present results also corroborate that the position of the primate BCoM is a by-product of the variations in limb (and tail) size and mass. For example, due to relatively heavy hind limbs and a large and massive prehensile tail the spider monkey exhibits the most extreme caudal configuration of the BCoM. In hominoid species, the long and heavy forelimbs shift the BCoM cranially. The BCoM of the quadrupedal arboreal primates, i.e. Cercopithecus roloway, Cercopithecus Lhoesti, Colobus guereza and Trachypithecus francoisi, is slightly closer to the hip than the shoulder because of their long tail and heavy thigh; note that the Sapajus xanthosternos is also a quadrupedal arboreal primate and exhibits a BCoM at mid-trunk. This probably reflects leaping and climbing capacities (e.g. [START_REF] Workman | Positional Behavior of Delacour's Langurs (Trachypithecus delacouri) in Northern Vietnam[END_REF][START_REF] Gebo | Positional behavior in five sympatric old world monkeys[END_REF][START_REF] Mcgraw | Cercopithecid locomotion, support use, and support availability in the Tai Forest, Ivory Coast[END_REF][START_REF] Rose | Positional behavior of natural populations: some quantitative results of a field study of Colobus guereza and Cercopithecus aethiops. Environment, behavior, and morphology: Dynamic interactions in primates[END_REF], while a more caudal BCoM in these arboreal quadrupedal animals may also improve manoeuvrability for travelling in the trees [START_REF] Aerts | Bipedalism in lizards: whole-body modelling reveals a possible spandrel[END_REF]. Foreand hind limbs morphology (length and mass distribution) is more similar in baboons and the BCoM is more cranial, therefore reflecting adaptation to terrestrial quadrupedalism (e.g. [START_REF] Rose | Positional behaviour of olive baboons (Papio anubis) and its relationship to maintenance and social activities[END_REF][START_REF] Raichlen | Convergence of forelimb and hindlimb Natural Pendular Period in baboons (Papio cynocephalus) and its implication for the evolution of primate quadrupedalism[END_REF]Druelle et al., 2017a). The position of the BCoM may, therefore, be a by-product of the interlimb morphology. As a result, considering pronograde mammals in general, a more caudal BCoM should be observed in arboreal species, while a more cranial BCoM should be observed in terrestrial species.

Nevertheless, specific locomotor habits such as bounding and half-bounding locomotion in (terrestrial) lagomorphs for example (e.g. [START_REF] Young | Limb bone morphology, bone strength, and cursoriality in lagomorphs[END_REF]) may be related to a caudal position of the BCoM.

BCoM and gait mechanics

The limbs are anatomically connected to the trunk, therefore there should be a functional link between limb and trunk mechanics. In this context, it has long been hypothesized that the diagonal walking gait -the typical footfall pattern of primates -could be related to a posterior location of the BCoM [START_REF] Rollinson | Comparative aspects of primate locomotion, with special reference to arboreal cercopithecines[END_REF][START_REF] Gray | Studies in the Mechanics of the Tetrapod Skeleton[END_REF][START_REF] Tomita | A study on the movement pattern of four limbs in walking[END_REF]. [START_REF] Rollinson | Comparative aspects of primate locomotion, with special reference to arboreal cercopithecines[END_REF] observed that captive cercopithecines used a lateral sequence for descending steep inclines which is likely to bring the centre of mass forward. From this, they concluded that because monkeys maintain otherwise diagonal walking gaits, they should possess a caudal centre of mass. Although this hypothesis has been supported by ontogenetic studies showing parallel changes between the caudal migration of the BCoM and an increase use of the diagonal walking gaits (e.g. [START_REF] Young | Gait selection and the ontogeny of quadrupedal walking in squirrel monkeys (Saimiri boliviensis)[END_REF][START_REF] Nakano | Footfall patterns in the early development of the quadrupedal walking of Japanese macaques[END_REF][START_REF] Turnquist | Ontogeny of locomotion in rhesus macaques (Macaca mulatta): I. Early postnatal ontogeny of the musculoskeletal system[END_REF][START_REF] Grand | Body weight: Its relation to tissue composition, segment distribution, and motor function II. Development of Macaca mulatta[END_REF], a certain number of works were not able to find such a relationship (e.g. [START_REF] Young | Body mass distribution and gait mechanics in fattailed dwarf lemurs (Cheirogaleus medius) and patas monkeys (Erythrocebus patas)[END_REF][START_REF] Anvari | Kinematics and spatiotemporal parameters of infant-carrying in olive baboons[END_REF][START_REF] Druelle | Intrinsic limb morpho-dynamics and the early development of interlimb coordination of walking in a quadrupedal primate[END_REF]. Experimental alterations of the antero-posterior mass distribution of the trunk revealed a clear influence on the mechanics of the limbs such as contact time, positioning, angular excursion and compliance (e.g. [START_REF] Lee | Effects of mass distribution on the mechanics of level trotting in dogs[END_REF][START_REF] Young | Body mass distribution and gait mechanics in fattailed dwarf lemurs (Cheirogaleus medius) and patas monkeys (Erythrocebus patas)[END_REF][START_REF] Anvari | Kinematics and spatiotemporal parameters of infant-carrying in olive baboons[END_REF]. Nevertheless, the footfall pattern may be related to many confounding factors that require further investigation.

It is commonly assumed that primates walk with a more protracted hind limb than the forelimb is retracted [START_REF] Larson | Uniqueness of primate forelimb posture during quadrupedal locomotion[END_REF][START_REF] Larson | Limb excursion during quadrupedal walking: how do primates compare to other mammals[END_REF]. Therefore, the hind limbs are brought closer to the BCoM than the forelimbs. This is the argument used by [START_REF] Raichlen | Understanding hind limb weight support in chimpanzees with implications for the evolution of primate locomotion[END_REF], in their study on chimpanzees, to explain primates' greater hind limb weigh support (whatever the configuration of the BCoM position). Nevertheless, [START_REF] Larson | Weight support distribution during quadrupedal walking in Ateles and Cebus[END_REF], in their study on spider monkeys and capuchins, were not able to confirm the results of [START_REF] Raichlen | Understanding hind limb weight support in chimpanzees with implications for the evolution of primate locomotion[END_REF] and suggested the presence of other active mechanisms to explain the weight support asymmetry observed in primates [START_REF] Larson | Hip extensor EMG and forelimb/hind limb weight support asymmetry in primate quadrupeds[END_REF][START_REF] Larson | Weight support distribution during quadrupedal walking in Ateles and Cebus[END_REF]. According to our results, it is obvious that the results and conclusions of these two studies were driven by the very different morphotypes and BCoM positions of the species studied (cranial in the chimpanzee, caudal in the spider monkey and at mid-trunk in the capuchin). Indeed, while the chimpanzee has no tail and needs to bring the hindlimbs into a very protracted position at touch-down in order to have the foot falling under the BCoM, the spider monkeys and the capuchins may simply move the position of their heavy tail to shift the position of the BCoM posteriorly. This other "passive" mechanism would significantly increase their hind limb weight support pattern. However, no information is provided on the influence of the tail in these species in [START_REF] Larson | Weight support distribution during quadrupedal walking in Ateles and Cebus[END_REF], but according to our results, shifting the tail backward can shift the BCoM posteriorly. For example, extreme positions of the tail (i.e. a 90°-angle with the vertical) will shift the BCoM backward by 5.62% in spider monkeys and by 2.99% in capuchin monkeys. Finally, [START_REF] Larson | Weight support distribution during quadrupedal walking in Ateles and Cebus[END_REF] reported that the mean percentage of forelimb weight support is greater in capuchins compared to spider monkeys. This is in accordance with our results showing a more caudal BCoM and a relatively heavier tail in spider monkeys compared to capuchins.

The present report offers new comparative data about the BCoM position in primates. This information can be used for further investigations regarding the particular (quadrupedal) locomotor features of these animals. These data allowed us discussing a new 'passive' mechanism for the explanation of the particular weight support pattern observed in primates with tails.

Fig. 1 .

 1 Fig. 1. Relationship between the total measured body mass and the total estimated (by the geometric model) body mass for our sample. The solid line is the least-squares regression (y=1.06+0.14, R 2 =0.88). The 95% Confidence Interval [0.85-1.27] of the slope overlaps with 1 (i.e. the line of identity), therefore validating the reliability of the procedure. The green circles and triangles indicate Cercopithecus lhoesti and Cercopithecus roloway individuals, respectively, the orange circles and triangles indicate Ateles fusciceps and Sapajus xanthosternos, the pink circles and triangles indicate Trachypithecus francoisi and Colobus guereza, and the blue circles and triangles indicate Nomascus siki and Nomascus gabriellae.

Fig. 2 .

 2 Fig. 2. Average morphotype for P. troglodytes standing up in a typical quadrupedal posture. The CoM of each segment is indicated with a blue circle and the BcoM is indicated with the black and white circle. The average limb angles have been chosen following Pontzer et al. (2014) and Finestone et al. (2018a) and are presented inTable 2. Illustration: 2018, Menelia Vasilopoulou-Kampitsi.

Fig. 3 .

 3 Fig. 3. Histogram of the relative positions of the BCoM from the hip joint and expressed as a percentage of the horizontal distance between the hip and shoulder joints. The position of the BCoM in the natural average posture is represented with the grey bars, the BCoM position in the control posture is represented with the white bars. The dashed line indicates mid-trunk.

Fig. 4 .

 4 Fig. 4. Free body diagrams for the 17 (primate and non-primate) species of our comparative sample. Each diagram is positioned in the natural body posture used for the calculation of the BcoM, indicated with the black and white circle.

  Figure 1.

Figure 4 .

 4 Figure 4.

ACKNOWLEDGMENTS

We are very grateful to Céline François-Brazier and Marine Giorgiadis, veterinary interns in Mulhouse, Alexandrine Vesz, veterinarian in Besançon, Guillaume Limouzin, animal keeper in Besançon, and Bérengère Dourou, veterinary student in Besançon, for their valuable help during the protocol for measurements. We thank Gilles Berillon for allowing us to use his anthropometric material for the external measurements. We are also very grateful to Josie Meaney-Ward who revised and improved the English of the manuscript. Financial support was provided by the CNRS-INEE International Research Network n°GDRI0870 Bipedal Equilibrium.


Situation actuelle : indicateurs clés

••

  Réduction SAU (7,2 ↘ 5,8Mha (50 à 60 % STH ) -boisement • Naissance d'une valeur « socio-éco » du paysage • Population stabilisée, rurale, puis urbanisation -rurbanisation • 450 000  98 000 EA (mais 20% France, constant) • Spécialisation, agrandissement, alourdissement (Bov/Ov) • Développement d'une filière « bovins maigres » très structurée • Impact fort des politiques publiques (PAC, et évolutions) Importance ↗ des aides directes dans un revenu qui stagne Résilience des intermédiaires (1990-2010) • Industrialisation -concentration forte des outils d'abattage Développement 2 e et 3 e transformation • Achat des ménages en GMS dès 1990 • Crises de consommation aigues et répétées (ex : ESB) • Traçabilité animal  produit : qualité sanitaire produit maîtrisée • Développement récent viande hachée (++) et ses déclinaisons à l'herbe du Massif central, une référence d'excellence (en réaction à la baisse conso++) 2. Le Massif central s'adapte au sein de marchés libéralisés 3. la viande agroécologique pour tous 4. La distribution alliée aux régions : partenariat pour une viande de massif 5. La viande géopolitique Scénario « Excellence » Contexte : Exigences sociétales : place de l'animal, santé humaine, environnement. Rejet de « l'industriel intensif » Forte baisse de conso de viande en FR/UE (-60%) -autres protéines Objectif : Exporter des viandes haut de gamme, dans le monde entier grâce à une image de marque reconnue et partagée. Place de l'herbe: finition + base du marketing pour l'export Scénario « Excellence » Moyens • Organisation collective autour d'une image (herbe, montagne = naturel) • Pilotage par l'amont qui maitrise des outils d'aval et la mise en marché • Montée en gamme de produits finis garantis sur conditions d'élevage, abattage • Export vers classes riches Conséquences • Maintien des élevages, production en systèmes extensifs à cycles longs, abandon du maigre (-88%) et du lait (-58% VL) • Développement des outils de production (emplois indirects) et des volumes (+59% de ktec). • Hausse des prix Innovation centrale : Modèles de prédictions de la qualité ( morceaux) Marchés libres, forte concurrence d'autres bassins de production lait/viande Baisse tendancielle de la conso de viande (-30%) Objectifs : Rechercher des économies d'échelles dans toute la filière pour baisser les coûts dans les productions compétitives Place de l'herbe : Alimentation économique + biomasse Scénario « Libéralisation » Moyens • Recentrage sur l'élevage bovin allaitant • Foncier disponible • Valorisation d'une herbe « bon marché » pour limiter les coûts • Peu d'organisation collective des acteurs Conséquences • Extensification (600ha/EA et 0,7 UGB/ha), Automatisationprestation • Productions réduites (140 ktec) • Modification paysage : baisse SAU Rum (-31%) et Reconversion biomasse (17%) ou autres usages • Activité rurale limitée (30 000 emplois, peu d'acteurs d'aval) Innovations centrales : Gestion des grands troupeaux par l'automatisation Biomasse végétale et son cracking Scénario « Agroéco » Contexte : Attente sociétale forte pour le respect de l'environnement + réglementation incitative et collaborative. Conso FR/UE en baisse tendancielle (-30%) Objectif : Développer les synergies entre les systèmes, les animaux et les territoires pour produire selon les principes de lHausse du prix de la viande • Polyvalence des systèmes : complémentarité ovin/bovin, croisés/ mixtes -animal/ végétal (-57% de VA) • Filières locales : maillage industriel + de proximité, emplois directs et indirects ++ (120 000) • Modification du paysage : diversité, mosaïque Innovation centrale : Efficience maitrisée des systèmes via l'agroécologie (mesures) Contexte : Intervention des Régions + maintien du niveau de conso de viande en FR/UE (-5%) Objectif : Réorganisation des filières assistées par les Régions -développement et partage équitable de la valeur ajoutée. Répondre aux attentes d'un consommateur connaisseur Place de l'herbe : Usage optimal (quantité/qualité) Composante de la qualité organoleptique Garantie de la qualité organoleptique & traçabilité intégrale (champ  assiette) Instabilité dans le bassin méditerranéen (dépendance alimentaire ↗) Baisse tendancielle Conso FR/UE (-30%) Objectif: Assurer un équilibre géopolitique entre l'Europe et l'Afrique du Nord Moyen-Orient en accompagnant son développement économique et démographique. Intégration de filière entre bassins d'élevage (MC naisseur) et de finition (plaines) • Offre « maigre » diversifiée • Assurance prix reliée aux coûts de production Conséquences • Spécialisation des bassins de production en naisseurs (2,2 Mio de VA -+35% de broutards) • Territoire peu diversifié (90% d'herbe). Peu d'outils de transfo (30 000 emplois indirects) • Dépendance d'un marché garanti mais spécifique et complémentaire d'autres zones Innovations centrales : Génétique animale rapidement modulable Réactivité dans les changements des caractéristiques

Effectifs des cheptels

Impacts environnementaux

Essai de synthèse

Les points forts Les messages clés (1/2)

• Conso : modes de production et provenance importent.

Les filières viande devraient considérer les bouquets de services proposés par les territoires du massif.  Positionner, construire l'offre : quantité, qualité, image


Nan He
email: nan.he@espci.fr

Yutong Cui

David Wai

Quan Chin

Thierry Darnige

Philippe Claudin

Benoît Semin
email: benoit.semin@espci.fr

Sedimentation of a single soluble particle at low Reynolds and high Péclet numbers

We investigate experimentally the dissolution of an almost spherical butyramide particle during its sedimentation, in the low Reynolds high Péclet regime. The particle sediments in a quiescent aqueous solution, and its shape and position are measured simultaneously by a camera attached to a translation stage. The particle is tracked in real time, and the translation stage moves accordingly to keep the particle in the field of the camera. The measurements from the particle image show that the radius shrinking rate is constant with time, and independent of the initial radius of the particle. We explain this with a simple model, based on the sedimentation law in the Stokes' regime and the mass transfer rate at low Reynolds and high Péclet numbers. The theoretical and experimental results are consistent within 20%. We introduce two correction factors to take into account the non-sphericity of the particle and the inclusions of air bubbles inside the particle, and reach quantitative agreement. With these corrections, the indirect measurement of the radius shrinking rate deduced from the position measurement is also in agreement with the model. We discuss other correction factors, and explain why there are negligible in the present experiment.

We also compute the effective Sherwood number as a function of an effective Péclet number.

I. INTRODUCTION

The mass transfer from a solid soluble particle in a fluid is of major relevance in chemical engineering [START_REF] Mccabe | Unit operations of chemical engineering[END_REF], for example in food industry [START_REF] Fryer | Chemical Engineering for the Food Industry[END_REF] and in pharmaceutical industry [START_REF] Grijseels | Hydrodynamic approach to dissolution rate[END_REF][START_REF] Dokoumetzidis | A century of dissolution research: From noyes and whitney to the biopharmaceutics classification system[END_REF]. Mass transfer from particles associated to phase change also occurs in a geophysical context, for instance the melting of rocks in magma [START_REF] Mcleod | Melting of a sphere in hot fluid[END_REF], the sublimation of ice drops in the atmosphere [START_REF] Chouippe | Heat and water vapor transfer in the wake of a falling ice sphere and its implication for secondary ice formation in clouds[END_REF], or the melting of snow and hail when reaching a sea [START_REF] Vahab | Fluid dynamics of frozen precipitation at the air-water interface[END_REF].

The dynamics of dissolution is different for isolated particle and for a suspension of particles. In the latter case, the particles interact through the concentration of the solute in the fluid phase, and due to the hydrodynamic interaction between the particles [START_REF] Kriaa | Effects of particle size and background rotation on the settling of particle clouds[END_REF]. Moreover, the dissolution of the particle can affect the density of the fluid phase, inducing buoyancy effects like the formation of plumes. In the present article we focus on the case of an isolated particle.

classified following the two main dimensionless parameters which control the mass transfer from a single spherical particle. A usual choice (which will be ours, see next section) is to use the Reynolds Re and the Péclet Pe numbers. An alternative to Pe is to use the Schmidt number Sc, where these three numbers are related by Pe = Re × Sc. Both Re and Pe vanish for a motionless particle, for which the mass transfer is only due to molecular diffusion. In gases, these two numbers are usually similar, whereas in liquids the Péclet number is several orders of magnitudes higher than the Reynolds number. The dimensionless mass transfer is the Sherwood number Sh, defined here (see Eq. 5 below) as the ratio of the total mass transfer by its purely diffusive component. Many past studies have performed experiments or numerical simulations, or developed models, to relate Sh to Re and Pe. Finally, in order to better cover the subject, it is also important to consider heat transfer from a particle, which is analogous to mass transfer within some hypotheses, in particular negligible radiation. In the case of heat transfer, the analog to Pe is the thermal Péclet number defined with the thermal diffusivity, the equivalent to Sc is the Prandtl number Pr, and the equivalent to Sh is the Nusselt number Nu.

The first regime is when both Re and Pe are small. It has for example been investigated experimentally by [START_REF] So | Modeling drug dissolution in 3-dimensional space[END_REF], measuring the size of an almost spherical succinic acid particle in unstirred water as a function of time. The results are compatible with a purely diffusive mass transfer (i.e. Sh = 1 with our definition). The diffusive mass transfer has also been verified experimentally in the case of a droplet of hexadecane levitated in an eletrodynamic balance and undergoing a flow of N 2 and helium [START_REF] Zhang | Mass transfer from a single micro-droplet to a gas flowing at low Reynolds number[END_REF]. Since the fluids are at rest, the expected result is similar for a liquid or a solid particle in this regime.

The regime we are particularly interested in this work is the case of small Re and large Pe, which, as we already said, is possible in a liquid. A fundamental analytical calculation for a sphere has been performed by [START_REF] Levich | Physicochemical Hydrodynamics[END_REF], which gives Sh ∝ Pe 1/3 . This scaling law has been confirmed by numerical simulations [START_REF] Clift | Bubbles, drops and particles[END_REF][START_REF] Assunção | On the dissolution of a solid spherical particle[END_REF] as well as experiments using two rotating cylinders to impose an homogeneous flow and electrochemical measurements to obtain the mass flux [START_REF] Kutateladze | Electrochemical measurements of mass transfer between a sphere and liquid in motion at high Péclet numbers[END_REF]. A similar configuration is the sinking of small spheres in a turbulent flow, which has been investigated experimentally and numerically by [START_REF] Lawson | Mechanisms of mass transfer to small spheres sinking in turbulence[END_REF]. Finally, many experiments and simulation have been performed in the regime where both Re and Pe are large, for a fixed particle submitted to a uniform flow or a free falling particle.

In this case, one expects the scaling law Sh ∝ Re 1/2 Sc 1/3 [START_REF] Lochiel | Mass transfer in the continuous phase around axisymmetric bodies of revolution[END_REF], and most of the results are effectively compatible with a correlation Sh ∝ Re 1/2 Sc α , α is in the range 0.3-0.4, plus correction terms. Using numerous simulations, [START_REF] Melissari | Development of a heat transfer dimensionless correlation for spheres immersed in a wide range of Prandtl number fluids[END_REF] have found α = 0.36 in the case of heat transfer for 3 × 10 -3 < Pr < 10 1 and 10 2 < Re < 5 × 10 4 . The experiments include various systems: sedimentation and dissolution of urea spheres in a vertical glass column [START_REF] Petrescu | Mass transfer at solid dissolution[END_REF], dissolution of benzoic acid spheres in a flow (natural or forced convection) of water or propylene glycol [START_REF] Steinberger | Mass transfer from a solid soluble sphere to a flowing liquid stream[END_REF], dissolution of ice ball in a hydrodynamic channel with water flow [START_REF] Hao | Heat transfer characteristics of melting ice spheres under forced and mixed convection[END_REF], dissolution of hard candy submitted to a flow [START_REF] Huang | Shape dynamics and scaling laws for a body dissolving in fluid flow[END_REF]. However, some other experimental configurations with a very different flow, like large ice balls melting in a turbulent von Kármán flow [START_REF] Machicoane | Melting dynamics of large ice balls in a turbulent swirling flow[END_REF], leads to different scalings.

Spherical versus non-spherical particles have also been studied in this large Re-Pe regime.

One can in particular mention experiments on the dissolution of neutrally buoyant particles with rectangular cuboid initial shapes in isotropic turbulent environments [START_REF] Oehmke | A new particle for measuring mass transfer in turbulence[END_REF]. Numerically, simulations of heat transfer past spheres, cuboids and ellipsoids been performed by [START_REF] Richter | Drag forces and heat transfer coefficients for spherical, cuboidal and ellipsoidal particles in cross flow at sub-critical Reynolds numbers[END_REF], and empirical correlation for non-spherical ellipsoids have been obtained by [START_REF] Kishore | Momentum and heat transfer phenomena of spheroid particles at moderate Reynolds and Prandtl numbers[END_REF][START_REF] Ke | On the drag coefficient and averaged nusselt number of an ellipsoidal particle in a fluid[END_REF][START_REF] Chen | Drag and heat transfer coefficients for axisymmetric nonspherical particles: A LBM study[END_REF][START_REF] Kiwitt | Nusselt correlation for ellipsoidal particles[END_REF].

Finally, several correction factor to the mass transfer, associated with the aspect ratio of non-spherical particles, the thermal effect due to dissolution, and the finite solubility effect have been discussed in [START_REF] Assunção | On the dissolution of a solid spherical particle[END_REF][START_REF] Elperin | Effect of solute concentration level on the rate of coupled mass and heat transfer during solid sphere dissolution in a uniform fluid flow[END_REF].

In the present work, we investigate the dissolution of an almost spherical particle that sediments in an aqueous solution at rest. These experiments belong to the low Reynolds and high Péclet regime, which, in comparison to the high Reynolds regime, has been less studied.

In the next section 2, we present the theoretical framework that we need to interpret the results, showing in particular that we expect a constant reduction rate of the particle size.

We describe the experimental setup in section 3 which allows us to measure simultaneously the position and the shape of the particle during its sedimentation. We emphasise the use of particles made of butyramide, a chemical which does not change the density of water when dissolving, hence preventing any buoyancy effect in the fluid. Section 4 is devoted to the comparison of our experimental results with the model, and we show that we can make it quantitative accounting for shape and density correction factors. Finally, conclusions and perspectives are drawn in section 5. Some technical aspects are gathered in supplementary material.

II. THEORY FOR A SPHERICAL PARTICLE

We present in this section the theory of a spherical particle falling in a quiescent fluid.

We focus on the regime for which the particle Reynolds number Re = ρ f aU/η is low, where a is the particle radius, U its settling velocity, η is the dynamic viscosity of the fluid and ρ f its mass density. It means that the fluid motion around the particle can be described by the Stokes equations, where the fluid inertia can be neglected. The particle velocity then results from the balance of the three relevant forces: the downward force of gravity F g , the upward buoyant force F b , and the drag force F d , which classically express as:

F g = 4 3 πa 3 ρ p g, (1)
F b = 4 3 πa 3 ρ f g, (2)
F d = 6πηaU. ( 3
)
ρ p is density of the particle and g is gravity acceleration. The resulting settling velocity of the particle is:

U = 2 9 (ρ p -ρ f )g η a 2 . ( 4
)
This particle can dissolve in the fluid, and we assume that it does so in the regime where its Péclet number, defined as Pe = U a D , where D is the diffusion coefficient of diffusion of the dissolved matter composing the particle into the fluid, is large. Both small Re and large Pe are encountered for small soluble particles in liquids as one typically has D ≪ η/ρ f (Parameters can be found in Table I).

In this regime, following the analytical calculation of [START_REF] Levich | Physicochemical Hydrodynamics[END_REF], the mass transfer rate ṁ of the sphere, i.e. the mass the particle losses per unit time when is dissolves, can be expressed in terms of the Sherwood number as:

Sh = - ṁ 4πDac 0 ≃ 2 π Pe 1/3 , (5)
where the dot denotes time derivative, and c 0 is the concentration (in kg/m 3 ) of the dissolved matter that closes to the particle. In first approximation, c 0 is the saturated concentration of the solute. Relating the mass of the sphere to its radius m = 4 3 πρ p a 3 , the mass transfer rate can also be written as:

ṁ = 4πρ p a 2 ȧ. ( 6
)
Equating ( 5) and ( 6), we obtain ȧ = -2 π 2 9

1 3 D 2 3 c 0 ρ p (ρ p -ρ f )g η 1 3
.

All factors on the right hand side of the above expression only depend on the characteristics of the fluid and the particle. The rate at which the particle size decreases over time is thus constant, i.e. independent of the radius of the particle, resulting in a linear relationship between a and t. We will test this remarkably simple behaviour experimentally in the present article.

III. EXPERIMENTAL SET-UP

We have built an experiment to investigate the dynamics of such a particle that sediments and continuously dissolves, resulting in a reduction of its size and mass. The setup consists in an elongated tank and a particle tracking system, as illustrated in Figure 1. The tank has an inner width of 10 mm and an inner length of 150 mm. As the particles we consider are rather small (on the order of 100 µm), and thus easily disturbed by small velocity fluctuations, the tank is placed in a larger water bath (internal width of 30 mm) to avoid convective disturbances inevitably caused by small temperature differences between the two sides of the experiment. The transparency of the two tanks allows for visual observation of the particle sedimentation process. The reliability of the entire experimental device has been verified through the sedimentation of plastic beads in distilled water. The two tanks are joined at the top by a detachable connector to ensure the verticality of the inner tank and prevent temperature fluctuations in the water bath caused by evaporation. In the present study, a camera with a resolution of 1936 × 1216 pixels, manufactured by IDS industrial camera company, was used. Prior to the experiments, careful scale calibration was performed, with a typical resolution of 5 pixels/µm. The particle tracking was controlled by a self-written Labview program, inspired by [START_REF] Darnige | Lagrangian 3D tracking of fluorescent microscopic objects in motion[END_REF]. The camera was connected to a computer to measure in real-time the position of the particle in the image, enabling to move the linear stage from PI (Physical Instrument) and thus the camera to follow the particle. Thus, synchronous position information and images of the particle were obtained from the tracking system.

The density of the aqueous solution containing the dissolved matter from the solid particle is usually larger than that of the pure water -this is the case for NaCl for example. Here we use particles of butyramide, a chemical whose saturated solution has a density very close to that of pure water, which minimise the effect of density increase around the particle during its sedimentation-dissolution motion. At 21 • C, the typical temperature at which experiments were run, we indeed measured with a pycnometer and a precision scale that the density of the saturated butyramide is 0.998 ± 0.001 g cm -3 , i.e. similar to the density of water at the same temperature (see Table I). Butyramide is also very soluble in water [START_REF] Romero | Solubility of acetamide, propionamide, and butyramide in water at temperatures between (278.15 and 333.15K[END_REF],

and the crystals have a bulk density of 1.032 g cm -3 . We made the particles using a stainless steel tip with an inner diameter of around 1 mm, resulting in particles with a shape similar to a cylinder with an aspect ratio close to unity.

Before the experiment is started, two different layers are prepared in the tank. The upper layer is a saturated butyramide solution in which the particle cannot dissolve. The lower layer is water with some dissolved NaCl, in order to make it slightly heavier than the upper layer for stability. The amount of NaCl is such that the density of this lower layer is 1.014 g cm -3 . We assume that this moderate presence of NaCl does not influence the dissolution process of butyramide in water. Once prepared, the two solutions are first put in a vacuum pump to remove air bubbles before being placed in the experimental tank. At the beginning of the experiment, pure water at room temperature is injected into the water bath.

The sedimentation tank is then carefully placed and attached by the detachable connector.

After the NaCl solution is poured at the bottom of the tank, the saturated butyramide is carefully injected using a syringe with a small-sized tip, providing a more stable flow.

This process results in a narrow transition layer, and the interface between the upper and lower layers can be visually distinguished. The vertical thickness of the lower layer solution is 95 mm, and that of the upper layer solution is 45 mm. Butyramide particles that have been pre-stocked in a saturated solution are drawn into a syringe without a tip, and the syringe is then placed vertically on top of the tank. At this point, the saturated solutions in the syringe and in the tank are connected and the particles can start to sediment. Once a particle is in the field of view of the camera, the tracking system captures it immediately, and track it until its size becomes smaller than ≃3 µm.

Moreover, the different physical parameters involved in this experiment have been carefully measured or determined, the value of them shown in Table I. The dynamic viscosities of the saturated butyramide and NaCl solutions, respectively denoted η b and η n , were measured with high precision using a rheometer Anton Paar specifically for liquids with a viscosity similar to water. The saturated butyramide concentration c 0 is well calibrated as a function of temperature in [START_REF] Romero | Solubility of acetamide, propionamide, and butyramide in water at temperatures between (278.15 and 333.15K[END_REF]. For the diffusivity D, we recorded with a camera the refraction angle of the interface of a stratified solution consisting of a saturated butyramide solution and still water over time. D was deduced from the square relationship between maximum reflected angle and time. More details about these measurement can be found in the supplementary material.

Particle Saturated butyramide solution NaCl solution Water

ρ p D c 0 ρ f η ρ f η f ρ f g cm -3
m 2 s -1 g cm -3 g cm -3 mPa s g cm -3 mPa s g cm -3

1.

IV. RESULTS AND DISCUSSION

A. Simultaneous measurements of radius and position of the particle Figure 2 illustrates the dissolution process of the particle during its sedimentation. We observe that the particle shrinks over time, gradually rounding off into a shape slightly elongated in the vertical direction. We can also notice a slight rotation of the particle. More quantitatively, we estimated the volume change of the particle by image analysis: binarising the picture of the particle with a suitable gray threshold and finding boundaries after convex hull, which fills the holes inside of binarised image, we could extract a projected area of the particle. From that surface, a centroid, which we take as the effective location z of the particle, and an equivalent radius a can be defined (Figure 2, bottom line). Following these quantities picture after picture, we could this way measure z and a as functions of time, as displayed in Figure 3. After the particle is released from the syringe, it first sediments in the upper layer composed of a saturated butyramide solution. During this period (stage I), as the particle does not dissolve, its equivalent radius remains constant. Slight fluctuations can however be ob-served, caused by the rotation of the particle. By taking the average particle radius during this stage, the initial particle radius, denoted as a 0 , can be obtained. The settling velocity of the particle also remains constant, and an initial value U 0 can be similarly computed from the average slope of the particle vertical displacement z(t).

At time t = t 0 the particle enters the transition layer where the upper-layer butyramide and lower-layer NaCl solutions are mixing. This is a rather thin transition layer, but its stratification causes a significant drop of the particle settling velocity associated with an enhanced drag [START_REF] Magnaudet | Particles, drops, and bubbles moving across sharp interfaces and stratified layers[END_REF]. During this stage II, the particle begins to dissolve, and it does so with an almost constant radius shrinking rate. At time t = t 1 the particle has reached the lower layer and its velocity is back to normal settling values. As shown in Figure 3(a), throughout its sedimentation in this layer (stage III) the particle equivalent radius continues to decrease at a constant rate, which is consistent with the theoretical expectation [START_REF] Vahab | Fluid dynamics of frozen precipitation at the air-water interface[END_REF]. A linear fitting of the data a(t) gives a direct measurement of the radius shrinking rate, denoted as ȧ1 . We shall see below that this rate can be also estimated in another way.

Simultaneously, the particle velocity continuously decreases and notably reaches zero at some time t up , after which the particle motion is reversed, see Figure 3(b). This is due to some air bubbles trapped inside the particle during its preparation. As we detail in the following analysis, we will need to account for the fact that the effective density of the particle must be corrected by a factor β b , associated with the presence of these bubbles. As the density difference ρ p -ρ f is small, even a value of β b close to unity has a significant quantitative effect. Of course, such a constant correction factor cannot reproduce the particle motion reversal. Instead, close to that moment, the volume of the bubbles V b inside the particle can be assumed constant, so that, as the particle matter further dissolves, its effective density becomes less than that of the surrounding fluid solution and it eventually rises. One can one then can compute V b at that reversing time with

V b (ρ p -ρ g ) = 4 3 π (ρ p -ρ f ) a 3 (t = t up ), (8)
where ρ g = 1.2 kg/m 3 is the air density. Notice that, interestingly, the value of the rate ȧ remains unchanged during the rising stage. We define the time t 2 at which V b represents 1% of the overall particle volume. Later analysis will then be restricted to times between t 1 and t 2 , so that the effect of these bubbles in the particle sedimentation is small.

The radius shrinking rate can be alternatively obtained from the particle position. As fit of z(t) in stage III with Eq. 9, from which another estimate ȧ2 of the radius shrinking rate is obtained.

ȧ is a constant (Eq. 7), a in the expression of the settling velocity U (Eq. 4) can then be replaced by a = a 1 + ȧ(t -t 1 ), where a 1 = a(t 1 ). Integrated once, the vertical position of the particle thus writes

z = t t 1 U dt ′ = 2 9 (ρ p -ρ f )g η 1 3 ȧ2 (t -t 1 ) 3 + a 1 ȧ(t -t 1 ) 2 + a 2 1 (t -t 1 ) + z 1 , (9)
where the z 1 is particle vertical position at the beginning of stage III (time t 1 ). Fitting this expression to the data z(t) allows us to get a value of ȧ, which we denote as ȧ2 to distinguish with the more direct estimate ȧ1 . Importantly, in this fitting process, two other parameters are determined by fit: a 1 and z 1 . The precision on fitting parameters is good (the maximal error bar for ȧ2 is ±0.05 µm s -1 ), and we have for instance checked that imposing a 1 from the measurement a(t) in the fitting of z(t) leads to consistent results.

B. Radius shrinking rate ȧ

As theoretically expected and showed in Figure 4, we find both ȧ1 and ȧ2 constant, i.e.

independent of the initial size a 0 of the particle. The prediction from ( 7) is above the ȧ1 measurements by 20%. Moreover, although on the same order, ȧ2 is systematically smaller than ȧ1 by a factor of ≃ 2. These discrepancies prompt us to revisit the above theoretical expressions in order to understand where the idealised case of a homogeneous spherical particle we have considered must be corrected. We have already mentioned in the previous section that the presence of trapped air bubbles must be accounted for with an effective particle density corrected by a factor β b . Another important aspect is the geometry of the particle. Since we have only access to a projection of the particle shape, it is unlikely that the effective radius a we have introduced quantitatively works for the particle volume. This volume is key for the computation of the gravity and buoyancy forces. To account for this volume uncertainty, we introduce a correction factor β a that will multiply the radius in the expression of these forces. We will discuss later in section IV C why we do not introduce correction factors for the other variables. With these two correction factors β a and β b , the expressions for U , ȧ and z rewrite

U = 2 9 (β b ρ p -ρ f )g η β 3 a a 2 , ( 10
) ȧ = - 2 π 2 9 1 3 D 2 3 c 0 β 2 a β b ρ p (β b ρ p -ρ f )g η 1 3 , (11)
z = t t 1 U dt ′ = 2 9 (β b ρ p -ρ f )g η β 3 a 1 3 ȧ2 (t -t 1 ) 3 + a 1 ȧ(t -t 1 ) 2 + a 2 1 (t -t 1 ) + z 1 . (12)
Using the data in stage I (upper layer), where U 0 and a 0 are measured accurately and for which the density as well as the viscosity of saturated butyramide are known, [START_REF] Zhang | Mass transfer from a single micro-droplet to a gas flowing at low Reynolds number[END_REF] gives a first relationship between the correction factors β a and β b . Similarly, with the linear fit of the radius reduction in stage III (lower layer) giving the rate ȧ1 , layer in which the density, the viscosity as well as the diffusivity of NaCl solution are known, [START_REF] Levich | Physicochemical Hydrodynamics[END_REF] gives a second relationship linking β a and β b . They can be solved numerically, and, upon ensemble averaging over 13 independent experimental runs, we obtained β a = 0.921 ± 0.002 and β b = 0.988 ± 0.002.

With these values, the fit of the curve z(t) in stage III with [START_REF] Clift | Bubbles, drops and particles[END_REF] allows us to deduce a new value of ȧ2 . As shown in Figure 4, the theoretical prediction of ȧ now fits the direct measurements ȧ1 as it should, and the corrected ȧ2 are now quantitatively consistent with ȧ1 . Importantly, these corrections assume that these factors can be taken constant over the whole sedimentation process (in fact, until time t 2 ).

A value of β b so close to unity may seem surprising, but because we are dealing with a small density difference between particle and fluid, these numerical adjustments are very sensitive. In fact, trying to impose β b = 1, we were not able to reach a quantitative matching of ȧ1 , ȧ2 and theory as in Figure 4 playing with β a only. Furthermore, the value we got for β a corresponds to an actual volume of the particle about 3/4 times smaller than deduced from the surface-induced effective radius a. This is consistent with particles in the form of an ellipsoid with its major axis parallel to the vertical, as observed in the experiments (see supplementary material).

C. Other correction factors

Other correcting factors could of course be considered. First of all, if ellipsoid-shaped particles are at play, the drag force is modified by the particle aspect ratio E, defined as the ratio between the major and the minor axes lengths. Following the work of [START_REF] Loth | Drag of non-spherical solid particles of regular and irregular shape[END_REF] for

FIG.
4. Reduction rate of the particle radius ȧ from theory and data analyses for various initial particle size a 0 . Red dots: direct measurement from the linear fitting of a(t) (see Fig. 3a). These are what we denoted as ȧ1 . Green dots: indirect values obtained from the fitting of z(t) (see Fig. 3b).

These are what we denoted as ȧ2 . Light green points: fitting without any correction factors, i.e.

using Eq. 9. Dark green points: fitting accounting for correction factors, i.e. using Eq. 12. Gray dash line: uncorrected theory [START_REF] Vahab | Fluid dynamics of frozen precipitation at the air-water interface[END_REF]. Black solid line: corrected theory [START_REF] Levich | Physicochemical Hydrodynamics[END_REF]. Data dispersion shows the overall precision we can reach, but from the fitting process of a single experimental run, errors bars are smaller than the symbol size.

spheroids, the Stokes drag correction factor for a motion parallel to the major axis follows the relation

β drag = 4 5 + E 5 E -1/3 . ( 13
)
The analysis of the pictures of the sedimenting particles shows aspect ratios typically between 1 and 2, with an average around 1.3 (see supplementary material). This corresponds to a drag correction factor β drag ≃ 0.97. Accounting for it in the force balance, the above analysis of the particle's dynamics is not significantly affected: variations by less than a percent are found for β a and β b , and around 1% for ȧ. At the first order, this shape effect on the drag can then be ignored for the present problem.

The shape of the particle influences its mass and heat transfer processes as well. The heat transfer from a particle, which is analogous to mass transfer within some hypotheses, in particular negligible radiation. We use the work of [START_REF] Chen | Drag and heat transfer coefficients for axisymmetric nonspherical particles: A LBM study[END_REF] that provides Nusselt numbers for ellipsoids across a wide range of aspect ratios, E. Here with Re ≃ 0.1 and Sc ≃ 1400, which are typical values of these experiments, we obtain β m ≃ 1.008 for the Sh ratio between E = 1.3 and E = 1. Including this correction factor into the theoretical framework has a negligible impact on the results. As demonstrated in the previous paragraph, deviations of less than one percent are noted for β a , β b and ȧ.

The theory of [START_REF] Levich | Physicochemical Hydrodynamics[END_REF] which gives equation ( 5) for the value of the Sherwood number rely on the hypotheses that the concentration of the solute is infinitesimal, and that there is no thermal effects during the dissolution. These two hypotheses are not verified for butyramide [START_REF] Romero | Solubility of acetamide, propionamide, and butyramide in water at temperatures between (278.15 and 333.15K[END_REF]: the solubility of butyramide is large, and its dissolution in water is endothermic. In the following we evaluate the corresponding correction factor β sol using the results of [START_REF] Elperin | Effect of solute concentration level on the rate of coupled mass and heat transfer during solid sphere dissolution in a uniform fluid flow[END_REF].

These results are valid in the case of high Péclet and Schmidt number, which is the regime of the present experiments. The Sherwood number of equation ( 5) is modified by a factor

β sol = 1 γ -1 -J 2/3 /K . (14)
In this expression γ is a correction factor introduced by [START_REF] Elperin | Effect of solute concentration level on the rate of coupled mass and heat transfer during solid sphere dissolution in a uniform fluid flow[END_REF] which depends on the weight fraction of the solute at solid-liquid interface and in the bulk of the liquid. We estimate γ = 1.09 for butyramide using a linear fit from the data of table 2 of [START_REF] Elperin | Effect of solute concentration level on the rate of coupled mass and heat transfer during solid sphere dissolution in a uniform fluid flow[END_REF]. J is ratio of the molecular D of the solute by the thermal diffusivity of the liquid α. The value of α for water is α = 1.45 × 10 -7 m 2 s -1 [START_REF] Balderas-López | Thermal-wave resonator cavity design and measurements of the thermal diffusivity of liquids[END_REF]. K is a dimensionless number involving the specific heat of the liquid c p , the latent heat of absorption L and a coefficient d, which is the slope of the relationship of the concentration and temperature. For water, the specific heat is c p = 4.15 kJ kg -1 K -1 , d = 0.01 K -1 and L = -400 kJ kg -1 was sourced from the measurements by [START_REF] Romero | Solubility of acetamide, propionamide, and butyramide in water at temperatures between (278.15 and 333.15K[END_REF]. The positive value of d implies that heat absorbed during butyramide dissolution results in a decrease of the interfacial temperature and equilibrium concentration. The value of correction factor β sol experiences only a minor change when the two values evaluated based on water are replaced by those calculated using NaCl solution: β sol changes from 1.057 to 1.062, so that we take β sol ≃ 1.06. Incorporating this value into the theoretical analysis does not affect much the results, with, as in the above paragraph, variations by less than a percent are found for β a and β b , and around 1% for ȧ. This correction can thus be neglected at first order for the present analysis.

D. Effective Péclet and Sherwood numbers

These experimental data finally allow us to assess the scaling law relating the Sherwood to the Péclet numbers (5). Because we do not measure ṁ directly but the grain size reduction rate ȧ instead, we rather define an effective Sherwood-like number as:

Sh = β a β b ρ p a ȧ Dc 0 . (15)
For a spherical particle, for which ṁ and ȧ are simply related (Eq. 6), and setting the corrective factors β a,b to unity, both definitions of Sh and Sh coincide. Here, we not only wish to express this number with quantities we have direct access to, but also aim at accounting for the corrections we have discussed above. Similarly, the effective Péclet number writes

Pe = β a U a D (16)
It can be directly estimated along each experimental run, also accounting for the radius correction. Plotting Sh as a function of Pe for all of our data clearly provides the expected increasing trend (Fig. 5). Data scattering is important, on the order of 30%, which is similar to what is displayed in Figure 4. For comparison to theory, Sh is computed with U and ȧ from their corrected expressions [START_REF] Zhang | Mass transfer from a single micro-droplet to a gas flowing at low Reynolds number[END_REF] and [START_REF] Levich | Physicochemical Hydrodynamics[END_REF], setting the factors to the experimentallydetermined averaged values β a = 0.921 and β b = 0.988, and where a is deduced from Pe with [START_REF] Lochiel | Mass transfer in the continuous phase around axisymmetric bodies of revolution[END_REF]. The agreement is quantitative, showing self-consistency with the fit of the theory in Figure 4.

V. CONCLUSION

We have investigated the dissolution of an almost spherical particle during its sedimentation, in the low Reynolds and high Péclet regime. We use butyramide particles sedimenting in aqueous solution so that the density contrast between the particle the solution is small, and thus the sedimentation velocity. The advantage of butyramide is that the density of the saturated solution is very close to the one of water, i.e. the dissolution does not affect the density of the solution.

The particle sediments in a squared tube, where a saturated butyramide layer is placed on top of a NaCl layer. The role of the top layer is to measure the sedimentation of the particle without dissolution and to have time to focus on the particle. The shape and the We develop a simple model for a perfect sphere based on Stokes' law (hypothesis of low Reynolds number) and the mass transfer at low Reynolds and high Péclet derived in [START_REF] Levich | Physicochemical Hydrodynamics[END_REF].

We obtain a radius shrinking rate ȧ which is constant in time, and only depends on the properties of the solid and the aqueous solution. The position of the particle is a third order polynomial of the time t. In the experiment, we define an equivalent radius from the image of the particle. We find as expected by the simple model that ȧ is constant in time, and independent of the initial radius of the particle. Moreover, the theoretical and experimental results are consistent within 20% without any adjustable parameter.

In order to obtain an even more quantitative agreement, we introduce two correction factors: one to take into account the non-sphericity of the particle in the evaluation of its

FIG. 1 .

 1 FIG. 1. Schematic diagram of the experimental set-up. In the tank, the upper layer I is saturated butyramide and the lower layer III is the NaCl solution. The transition layer is numbered II.

FIG. 2 .

 2 FIG. 2. Dissolving process of the particle over time. The top line shows the original pictures, as captured by the camera during the experiment. The bottom line shows the corresponding processed images, where the white area A, obtained after convex closure of the binarised picture, representsthe projected surface of the particle. On each of these bottom images, the blue-star point indicates the centroid of the white area, and the red circle, centred on that point, has the same surface as the white area, i.e. gives the equivalent radius a = A/π of the particle.

FIG. 3 .

 3 FIG. 3. Time evolution of the equivalent radius a (a) and the vertical displacement z (b) of the particle. Blue circles: experimental data. The vertical dashed lines show the times separating the different stages: t 0 end of the saturated layer, t 1 end of the transition layer, t up motion reversal.Those times were determined using the velocity deduced from the derivative of the displacement (b), see supplementary material. Solid black line: time before which the volume of the air bubbles attached to the particle is less than 1%. Red line: linear fit in stage III of the radius decrease to deduce ȧ1 . Yellow line: linear fit in stage I of the particle position to deduce U 0 . Green curve:

FIG. 5 .

 5 FIG. 5. Effective Sherwood number vs effective Péclet number. Symbols: experimental data, corrected by their factors β a and β b computed as explained in the text after equations (10-12). The colours correspond to different runs. Gray dash line: uncorrected theory. Black solid line: theory accounting for averaged correction factors.

  volume and weight (β a ), and a correction of the density of the particle due to the inclusions of air bubbles inside the particle (β b ). The non-sphericity of the particle and the inclusion of air bubbles are visible on the images. These two correction factors are close to one (β a = 0.921±0.002 and β b = 0.988±0.002). With these corrections, both the radius shrinking rate deduced from the equivalent radius and the one deduced from the particle trajectory are in quantitative agreement with the corrected model. We discuss other correction factors, such as the correction of the drag due to the non-sphericity of the particle, the correction of the mass transfer due to the non-sphericity of the sphere and the finite solubility and nonisothermal effects in the dissolution of butyramide. We have shown that these corrections factors have a negligible effect in the present experiment, in contrast with β a and β b . Finally, we have defined an effective Sherwood number Sh and an effective Péclet number Pe, and we have displayed the corresponding curve, which show the Sh ∝ Pe 1/3 scaling.VI. ACKNOWLEDGEMENTWe gratefully acknowledge X. Benoit-Gonin, A. Fourgeaud and L. Quartier for technical support and A. Limare for technical help and scientific discussions. N.H. and Y.C. have been funded by Chinese Scholarship Council Scholarship.