Christophe Dumouchel

LIQUID ATOMIZATION AND SPRAY: A MULTI-SCALE DESCRIPTION

A multi-scale approach to investigate liquid atomization processes is introduced. It describes the liquid system by the scale distribution whose determination is inspired from the Euclidean Distance Mapping used to measure the fractal dimension of a contour. The scale distribution is introduced in 2D and in 3D and applications from previous investigations are presented. The 2D applications are performed on experimental images and the 3D applications are performed on results obtained from Direct Numerical Simulation. The multi-scale analysis allows identifying and quantifying the mechanisms responsible for the interface evolution according to the scale. Among other results, the analyses presented here demonstrate the improvement of the atomization process when an elongation mechanism contributes to the thinning of the small structures. The multi-scale tool also provides new metrics that may be used to validate simulation results. An example of this is presented and discussed. Finally, the paper evokes several approaches to implementing the scale-distribution concept to improve or build new models.

The distributions en(d) (n = 2, 3) are monotonously decreasing functions, their dimension is the inverse of a length, and, they are normalized, i.e.,

 

. Note that e3(0) is similar to the specific-surface-area introduced by Evers [1].

The scale distribution can be mathematically established for simple systems. For a sphere of diameter D, we obtain:

INTRODUCTION

In many domestic or industrial applications, flows of droplets of liquids are used. It is known that the efficiency of these applications depends on the characteristics of these drops including their size, velocity, spatial density, etc. Investigations to understand, model, simulate and control liquid spray formation are therefore very much needed.

The most encountered process to produce a spray consists of ejecting a liquid flow into a gaseous environment. As soon as the liquid flow issues from the injector nozzle, deformation appear at the liquid interface. These deformations grow in space and time and eventually result in the ejection of liquid fragments on which the whole process may reiterate. The atomization process stops when the initial flow is fully transformed into a flow of droplets small enough to be stable, i.e., a spray.

The experimental investigations of atomization processes are based on visualizations and image analysis and often report measurements of characteristic features such as a spray angle, a breakup length, a mean diameter of the drops… Although these quantities are important, they don't provide a complete description of the process. Liquid atomization is a process of energy exchange between the interface and the two fluids. The liquid-gas interface stores energy, which, per unit mass, is equal to the product of the specific-surface-area (surface area per unit mass) and the surface tension [START_REF] Evers | Analogy Between Atomization and Vaporization Based on the Conservation of Energy[END_REF]. The specific-surface-area is conveyed by the shape of the liquid system, the shape that continuously varies during the atomization process. The profusion of visualization approaches available in the literature [START_REF] Dumouchel | On the Experimental Investigation on Primary Atomization of Liquid Streams[END_REF] points out the complexity of this shape. Attempts of using the fractal concept developed to characterize the tortuosity of complex contours [START_REF] Mandelbrot | The Fractal Geometry of Nature[END_REF][START_REF] Sreenivasan | The Fractal Facets of Turbulence[END_REF][START_REF] Hunt | Kolmogorov's Contribution to the Physical and Geometrical Understanding of Small-Scale Turbulence and Recent Developments[END_REF] have been performed to characterize this complexity [START_REF] Shavit | Fractal Dimensions of Liquid Jet Interface under Break-Up[END_REF][START_REF] Dumouchel | Experimental Analysis of Liquid-Gas Interface at Low-Weber-Number: Interface Length and Fractal Dimension[END_REF][START_REF] Grout | Fractal Analysis of Atomizing Liquid Flows[END_REF]. Among other results, it was pointed out that atomizing liquid systems cannot be fully described by a single fractal dimension and that the scale range representative of the whole liquid system shape is wider than the one where selfsimilarity is observed. Thus, the traditional fractal concept must be replaced by a multi-scale description approach where the fractal dimension becomes a scale-dependent function. The notion of scale distribution has been introduced for this purpose.

The multi-scale description tool based on the measurement of the scale distribution is introduced. Coming from previous investigations, applications of this approach on experimental and simulated results are presented and discussed. They demonstrate that such a multi-scale analysis of the atomization process allows identifying the physical mechanisms responsible for the interface evolution according to the scale. Specific attention is paid on the importance of the mechanisms at small scales. This paper shows to which extend the multi-scale approach can help characterizing, understanding and modelling liquid atomization processes.

THE MULTI-SCALE DESCRIPTION

The multi-scale analysis describes the liquid system by the cumulative scale function En(d) where d is the scale of observation. This function is obtained by applying successive erosion operations to the liquid system. An erosion operation is illustrated in Fig. 1 for a 2-D system. Figure 1-left shows the initial system, which has a 2-D total surface area noted S2T. The system is eroded by circular structuring elements with a diameter d. The eroded system (dark-gray area in Fig. 1

      T S d L d d E d e 2 2 2 2 d d   ( 2
)
This derivative is equal to the ratio of the perimeter length L(d) of the eroded system at scale d on twice S2T.

The extension of this definition in 3-D is straightforward. In this case, the system is characterized by its total volume VT, the erosion operation is performed with a sphere of diameter

                     D d d e D d D d D n d e n n n for 0 for 1 1 (4)
For a cylinder of diameter D and by considering the lateral surface only, i.e., the two circular ends are not counted, we obtain:

                      D d
D(t 1 ) D(t 2 ) D(t 3 ) D(t 4 ) 1/D(t 1 ) 1/D(t 2 ) 1/D(t 3 ) 1/D(t 4 ) 0 0 Scale d (AU) e 3 (d,t) (AU -1 ) 2/D(t 1 ) 2/D(t 2 )
2/D(t 3 ) 2/D(t 4 ) It is instructive to consider the case of a thinning cylindrical ligament, i.e., a cylinder with a diameter D(t) that decreases with time. The temporal evolution of the scale distribution en(d,t) calculated from Eq. ( 5) are shown in Figs.

D(t 1 ) D(t 2 ) D(t 3 ) D(t 4 ) 0 0 Scale d (AU) e 2 (d,t) (AU -1 ) t 1 t 2 t 3 t 4 t 1 < t 2 < t 3 < t 4



and a scale independence of these two functions. e3(0,t) indicates a constant increase of the specific-surface-area.

Combining the quasione-dimensional continuity equation provided by Stelter et al. [START_REF] Stelter | Validation and Application of a Novel Elongation Device for Polymer Solutions[END_REF] together with the fact that the section of the cylinder is constant along the axial direction z, we obtain the rate of stretching   , i.e.:

    t t D t D z v d d 2        (6)
where v designates the longitudinal velocity. Considering Eqs. ( 4) and ( 5), we can express the stretching rate   of the thinning cylinder as a function of the scale distributions:

        t d e t d e t d e t d e , ' , ' , , 2 3 3 2 2       ( 7
)
where the dot indicates a temporal derivative and the prime a scale derivative. For a thinning cylindrical ligament, the stretching rate is of course independent of the scale d.

From a physical point of view, the thinning of a liquid ligament can be due to an elongation mechanism or a contraction mechanism. In the first case, the ligament is elongated by an external constraint and its volume is constant. In the second case, the contraction is driven by surface tension forces and expulses liquid out of the ligament whose volume therefore decreases. If for both mechanisms the specific-surface-area increases, the absolute amount of surface area actually increases during the elongation and decreases during the contraction. In the second case, the reduction of the ligament diameter is not compensated by an increase of its length. Both mechanisms are associated to a positive stretching rate and the only way to dissociate them from each other is to get an information on the temporal variation of the ligament volume.

APPLICATIONS ON EXPERIMENTAL ATOMIZATION PROCESS

This section presents two examples of the use of the multiscale approach to analyze experimental images of a liquid atomization process. The first example concerns the behavior of elongated liquid ligaments and the second one considers a turbulent liquid jet issuing from a gasoline direct injector (GDI).

Elongated liquid ligaments

The elongated liquid ligaments are those developing during the atomization of a turbulent liquid sheet. The experiment work (detailed in [START_REF] Dumouchel | Multi-Scale Analysis of Atomizing Liquid Ligaments[END_REF]) consisted in visualizing the behavior of the ligaments with a high-speed camera (Phantom V12, 25000 frames/s) in a shadowgraph configuration. Three liquids with a surface tension  ranging from 38 to 70 mN/m were used. For each liquid, several ligaments were detected and analyzed. Figure 3 shows an example of the temporal evolution of one ligament. The elongation of the ligament is recognized by the constant increase of its length during time. Being produced by a turbulent liquid sheet, the ligament is deformed at initial time.

Image analyzing tools are applied to measure the scale distribution e2(d,t). It first consists in determining the location of separation between the ligament and the bulk when they are still in contact. This is done by hand (see the arrow in Fig. 3). Second, the ligament contour is determined at each time and the corresponding scale distribution is measured. (Refer to [START_REF] Dumouchel | Multi-Scale Analysis of Atomizing Liquid Ligaments[END_REF] for details). Figure 4 shows an example of the temporal evolution of the scale distribution obtained for one ligament of water ( = 0.0070 N/m). As mentioned above, the distribution e2(d,t) is a continuous decreasing function in the scale space. At t = 0, e2(d,t) resembles to a smoothed step function. The similarity with a step function reveals the global ligamentary shape of the liquid system and the smoothed feature indicates that the ligament is perturbed. (Indeed, as shown in Fig. 2-left, a non-perturbed cylindrical ligament reports a stiff step function.) At the final time (280 µs in Fig. 4), the scale distribution characterizes the few drops produced after the breakup of the ligament.

Globally speaking, Fig. 4 reports an increase of e2(0,t) with time revealing the presence of a thinning mechanism as shown in Fig. 2-left. In the meantime, e2(d,t) increases with time in the large scale region (d > 130 µm) and dmax increases. This behavior demonstrates the production of bigger structures and is attributed to the presence of a thickening mechanism. Since the distribution is normalized, the combination of the thinning and thickening mechanisms imposes a decrease of e2(d,t) with time for the medium scale range. The thinning mechanism at small scales and the thickening mechanism at large scales are reported for each ligament. They are clearly dependent on the liquid surface tension. For instance, e2(0,t) increases more with time when the liquid surface tension decreases and the final scale distribution characterizing the set of droplets evolves from a linear to a convex function. Considering Eq. ( 4), this latter observation says that all drops produced have the same size when the surface tension is high.

The behavior of the ligaments actually results from the competition between two mechanisms: the thinning mechanism imposed by the initial ligament elongation and the thickening mechanism resulting from the development of a capillary instability. The multi-scale analysis allows quantifying this competition. Using Eq. ( 7), the initial elongation rate is evaluated by:

    i i i t e t e t , 0 , 0 ) , 0 ( 2 2 
  (8)
where ti is the initial time. The characteristic time te of the initial elongation mechanism is therefore te = 1/(0,ti). The capillary instability of a cylindrical ligament is a surface tension driven mechanism whose characteristic time t is given by [START_REF] Rayleigh | On the Instability of Jets[END_REF]:

     3 i L t D t  ( 9
)
where L is the liquid density and D(ti) a characteristic length of the initial ligament. This characteristic length is the inverse of the scale distribution, i.e., D(ti) = 1/e2(0,ti). The behavior of the ligaments depends on the ratio of the capillary characteristic time on the elongation characteristic time introducing a new Weber number Wei, i.e.:

        2 3 2 , 0 i i i L e i t t D t t We           (10)
It is found that when Wei < 0.03, the capillary instability dominates the ligament behavior and the surface tension is high enough to oppose any interface increase induced by the elongation mechanism. The resulting drops have similar size.

When Wei > 0.03, the initial elongation mechanism is strong and the surface tension doesn't oppose to the interface creation it induces. Breakup occurs at smaller and smaller liquid fragments and droplets with a wider range of diameter are produced. Therefore, an effective elongation mechanism damps the surface tension mechanism, delays the breakup process and increases the atomization efficiency. This agrees with Marmottant and Villermaux's findings [START_REF] Marmottant | Fragmentations of Stretched Liquid Ligaments[END_REF] on the behavior of static laminar stretched ligaments. In the same study, Marmottant and Villermaux [START_REF] Marmottant | Fragmentations of Stretched Liquid Ligaments[END_REF] also showed that the diameter of the droplets produced by the atomization of the ligaments was distributed according to the following pdf distribution: [START_REF] Marmottant | Fragmentations of Stretched Liquid Ligaments[END_REF] where D10 corresponds to the mean diameter, the parameter , which has no dimension, controls the width of the distribution (it decreases when the width of the distribution increases), and  is the Gamma function. Since one ligament produces few droplets (see Fig. 3 for instance), the use of Eq. ( 11) required a statistical experimental distribution obtained from the analysis of a high number of ligaments. This is not necessary anymore if the scale distribution is used since this function is always continuous even if the system is composed of few droplets. We can demonstrate that the scale distribution of a set of spheres whose diameters are distributed according to Eq. ( 11) is:

W -L1 -Exp WE10 -L2 -Exp WE15 -L3 -Exp W -L1 -Math WE10 -L2 -Math WE15 -L3 -Math
                10 10 1 0 exp D D D D D f      
    2 , , 1 2 10 10 10 10 2                               D d D d D d D d e ( 12
)
where (x,y) is the incomplete gamma function. The parameters D10 and  were determined for each ligament. Beside the fact that the agreements between the measurements and Eq. ( 12) were always good (see examples in Fig. 5), a strong correlation was obtained between the parameter  and the Weber number Wei; i.e.:

i We 1   [START_REF] Tanguy | Application of a Level Set Method for Simulation of Droplet Collisions[END_REF] This correlation illustrates the fact that an increasing elongation mechanism (Wei increases) promotes the production of smaller droplets ( decreases). The multi-scale description of the behavior of deformed and elongated ligaments demonstrated the importance of the initial elongation mechanism on the atomization efficiency and showed an interesting alternative to represent and model fragmented liquid system without statistical requirements.

GDI turbulent jets

The details of the experimental work described in this section are available in Dumouchel et al. [START_REF] Dumouchel | Towards an Interpretation of the Scale Diffusivity in Liquid Atomization Process: An Experimental Approach[END_REF]. The GDI device of this second experiments has three identical and cylindrical discharge orifices (diameter Dor = 250 µm). The turbulent jet issuing from one orifice is considered only. The liquid used is Shellsol D40 whose physical properties are closed to those of gasoline. In order to ensure a visible atomization process, the injection pressure is restricted to [0.1 MPa; 2.5 MPa]. The GDI device is conceived to work in transient conditions and is equipped with a needle that controls its opening and closing. The investigation concentrates on the jet behavior issuing from the injector during the fully open stage only. Contrary to the previous study, the approach is statistical, i.e., one image is taken per injection during 250 consecutive injections and the mean behavior of the jet is investigated. A shadowgraph optical arrangement with a spatial resolution equal to 5.8 µm/pixel is installed. An example of the image is shown in Fig. 6-Left. The liquid jet is not laminar because of the complex internal geometry of the injector that imposes drastic flow deflections and stresses. Some of the initial perturbations grow and structure the jet evolution. This atomization process shows the production of transverse ligaments and the transformation of the bulk as a ligament network. When the injection pressure increases, more ligaments are produced. For the highest injection pressures, the bulk flow is surrounded by droplets and the jet liquid core is more indented.

Image analyzing tools are applied to measure the scale distribution. The images are orientated so that the jet appears vertical. The temporal description of the atomization process is performed on the reoriented images by analyzing a portion of the liquid system delimited by a rectangular analyzing window, and by sliding this window from the top down to the bottom of the image. The position h of the analyzing window is converted into an equivalent time t = h/Vq where Vq is the jet mean velocity otherwise measured (Fig. 6-Right). This treatment is performed on the 250 images and a temporal evolution of the mean scale distribution is obtained for each injection pressure. Figure 7 shows the temporal evolution of the scale distribution obtained for Pi = 0.5 MPa. Similar results are found for the other injection pressures. At the smallest time, the scale distribution resembles a step function saying that the 2D perception of the jet is close to the one of a cylinder. The smooth decrease in the small scale range ([0 µm; 100 µm]) and in the large scale range ([400 µm; 600 µm]) result from the presence of textural (located on the interface) and structural (concerning the bulk flow) perturbations, respectively. As time goes, Fig. 7 shows an increase of e2(d,t) in the small scale range, a decrease of e2(d,t) in the large scale range and a continuous decrease of the scale dmax. These variations are the characteristic features of a thinning mechanism. Contrary to the case of a cylindrical ligament (see Fig. 2) this thinning mechanism is scale dependent.

Copyright    at scale d = 0 characterizes the dynamic of the interface variation imposed by the thinning mechanism. The associate characteristic time 1/(0,t) is shown as a function of time for several injection pressures in Fig. 8. This figure shows the existence of two phases in the atomization process. During the first phase, the thinning characteristic time is constant. It decreases with the injection pressure: it has been otherwise found that

  28 1 0 1 . Re t ,   
where Re is the issuing Reynolds number. This reveals the dominant influence of the liquid jet turbulence on the initial stretching mechanism. During the second stage, the thinning characteristic time linearly increases with time denoting a less and less effective mechanism, i.e., a continuously decreasing specific-interface-length creation. We see that the rate of increase in this second stage is independent of the injection pressure meaning that it is not controlled by the jet dynamic anymore. The time-dependent version T(t) of the characteristic capillary time defined by Eq. ( 9) is introduced:

       3 t D t T L  (14)
where D(t) = 1/e2(0,t). T(t) is calculated and plotted in Fig. 8. We see that the passage from phase 1 to phase 2 always occurs when the characteristic capillary time becomes smaller that the characteristic stretching time. Therefore, during the phase 2, the stretching mechanism is controlled by the surface tension forces explaining why its dynamic becomes independent of the injection pressure. In Fig. 7, the atomization process has been described as a thinning mechanism whose impact on the interface evolution follows two dynamics (Fig. 8). This thinning mechanism splits the scale space in two regions: the small scale region covering the scales for which the stretching rate is positive, and the large scale region covering the scales for which the stretching rate is negative. This space separation introduces a characteristic scale, namely, d1, which is the smallest scale for which:

  0 1 2  d d t , d e  (15)
A thinning mechanism creates small scale fragments and slims large scale structures. This continuous mechanism imposes a temporal decrease of the scale d1. This is illustrated in Fig. 9 for several pressures. The temporal decrease of d1 is stiffer when the injection pressure increases. Furthermore d1 decreases when the injection pressure increases. This illustrates the fact that a prompt thinning mechanism improves the atomization efficiency since it enhances the production of small liquid fragments before the breakup mechanism occurs. The beneficial influence of a rapid and strong initial thinning mechanism identified here is similar to the one observed on the elongated ligaments presented in the previous section.

Figure 9 also evidences a change of dynamic of the scale d1(t) (the dash lines show this change for Pi = 0.6 MPa). These dynamics correspond to the two phases of the atomization process identified in Fig. 8. Thus, the scale d1(t) is sensitive to the physical mechanism controlling the small scale thinning process.

APPLICATIONS ON SIMULATED ATOMIZATION PROCESSES

Thanks to recent developments, the direct numerical simulation (DNS) now achieves close reproduction of liquid atomization processes. Furthermore DNS allows an independent control of the initial conditions and returns a 3D information that is not reachable from experiments so far. It opens an alternative way of investigation, and, in particular, it allows performing 3D multi-scale analyses. Two examples are presented in this section, namely, the capillary instability of a ligament and the atomization of a turbulent jet.

Developed in the CORIA laboratory, the DNS code ARCHER used in this paper is mainly devoted to multi-phase flows. (Details on this code can be found in [START_REF] Tanguy | Application of a Level Set Method for Simulation of Droplet Collisions[END_REF][START_REF] Ménard | Coupling Level Set/VOF/Ghost Fluid Methods: Validation and Application to 3D Simulation of the Primary Break-Up of a Liquid Jet[END_REF].) The twophase flow is modelled by a single set of incompressible Navier-Stokes equations with variable density and viscosity, and interfacial forces. These equations are coupled with the transport equation of the phase function C:

0  U . (16)     st T F U U . g p U . U t U                       (17) 0      C . U t C (18)
In ARCHER solver, the phase function is modelled with both Volume Of Fluid and Level Set methods, in the CLSVOF framework. Jump conditions across the interface are taken into account with the Ghost Fluid approach. These methods are coupled with a projection method to solve the Navier-Stokes equations on a staggered grid. To treat the convective derivative in Eq. ( 17) we used WENO5 scheme and 2-order central difference scheme for other. The time integration is performed with RK3TVD scheme. All details of these techniques, and how Eqs. [START_REF] Bogy | Drop Formation in a Circular Liquid Jet[END_REF][START_REF] Eggers | Non-Linear Dynamics and Breakup of Free Surface Flows[END_REF][START_REF] Eggers | Physics of Liquid Jets[END_REF] are solved, were presented in previous works [START_REF] Tanguy | Application of a Level Set Method for Simulation of Droplet Collisions[END_REF][START_REF] Ménard | Coupling Level Set/VOF/Ghost Fluid Methods: Validation and Application to 3D Simulation of the Primary Break-Up of a Liquid Jet[END_REF].

The capillary instability

The multi-scale analysis of the simulated capillary instability summarized in this section is detailed in [START_REF] Dumouchel | Multi-Scale Analysis of Simulated Capillary Instability[END_REF]. The capillary instability manifests on a cylindrical ligament subjected to disturbances that induce surface displacements and generate a gradient of surface tension forces. Under certain conditions, the pressure distribution caused by these gradients generates internal flows that concentrate the liquid in certain regions to the detriment of others and rearrange the ligament as a succession of crests and necks. This process continues until the ligament diameter at the necks is so small that a breakup occurs and produces one drop for each swollen region. Several complete reviews on the investigations dedicated to this topic are available in the literature [START_REF] Bogy | Drop Formation in a Circular Liquid Jet[END_REF][START_REF] Eggers | Non-Linear Dynamics and Breakup of Free Surface Flows[END_REF][START_REF] Eggers | Physics of Liquid Jets[END_REF][START_REF] Ashgriz | Handbook of Atomization and Sprays: Chapter 1-Capillary of Free Liquid Jets[END_REF].

The capillary instability is calculated for a cylindrical ligament of water into air with a constant initial diameter Dj = 666 µm. An initial sinusoidal perturbation is applied. Its amplitude is fixed (0 = 17 µm, 0/Dj = 0.025) but eight values of the non-dimensional wavenumber k (= Dj/ where  is the perturbation wavelength) are considered, namely, 0.55, 0.60, 0.65, 0.69, 0.75, 0.80, 0.88, 0.95.

The simulation result for k = 0.55 is shown in Fig. 10. The evolution of the ligament from t = 0 to the breakup time tBU (= 8.13 ms for this case) undergoes three distinct steps. First (left column in Fig. 10), the ligament deformation increases in amplitude keeping a sinusoidal shape. Second (middle column in Fig. 10), the deformation of the ligament is not sinusoidal anymore. Two necks appear and a liquid thread develops between the main swells. Third (right column in Fig. 10), the necks do not travel anymore: they impose a local high-pressure that induces a pinch-off mechanism until breakup occurs. A similar behavior was observed for every k. These results are in agreement with previous calculations conducted in similar configurations [START_REF] Ashgriz | Temporal Analysis of Capillary Jet Breakup[END_REF]. The volume-based scale-distribution is calculated from images similar to those shown in Fig. 10 and by assuming that the ligament is axisymmetric at all times. This assumption is reasonable in the present context because axisymmetric disturbances are unstable only [START_REF] Rayleigh | On the Instability of Jets[END_REF]. The temporal evolution of the scale distribution e3(d,t) for k = 0.55 is shown in Fig. 11 where a thickening mechanism at large scales and a thinning mechanism at small scales are observed. The thickening mechanism is recognized by a continuous increase of the maximum scale dmax. appears independent of the scale, this mechanism is similar to the thinning of a cylindrical ligament. The characteristic scale d1(t) under which the stretching rate is positive is determined as a function of time. In 3D, this scale is defined as (see Eqs. ( 7) and ( 15)): Fig. 12: Temporal evolution of the characteristic scale d1(t) (Lines are the mathematical representation of the three regimes) An example of d1(t) is shown in Fig. 12 (k = 0.69). We see that d1(t) decreases according to three regimes which corresponds to the three steps identified in Fig. 10. In regime 1, d1(t) reports an exponential decrease with time similar to the one reported by the linear theory due to Rayleigh [START_REF] Rayleigh | On the Instability of Jets[END_REF]. In regime 2, d1(t) decreases as ( t 2 /L) 1/3 and in regime 3, d1(t) decreases as ( t/µL). The dynamics in regimes 2 and 3 have been reported in the literature for the pinch-off mechanism at breakup [START_REF] Lister | Capillary Breakup of a Viscous Threads Surrounded by Another Viscous Fluid[END_REF] and the liquid-bridges contraction [START_REF] Clasen | Dispensing of Rheologically Complex Fluids: The Map of Misery[END_REF]. The existence of these three regimes demonstrates an evolution of the thinning mechanism observed at small scales.

  0 1 3  d d t , d ' e  ( 19
)
The thinning mechanism at small scales can be characterized by the stretching rate   introduced by Eq. ( 7). This rate has been calculated for every wavenumber and the results are presented in Fig. 13 as a function of the ratio t/t* where t* is the time at which the regime 2 begins (see Fig. 12). Figure 13 shows a continuous increase of the stretching rate for all wavenumber. This is actually coherent with the fact that the thinning mechanism is representative of the creation of smaller and smaller liquid structures and that this phenomenon actually lasts until breakup occurs. Figure 13 also shows that the stretching rate increase when the wavenumber increases. The impact of the thinning mechanism on the specific-surface-area of the ligament can be seen in Fig. 14 that shows the temporal evolution of e3(0,t) as a function of t/t*.

The specific-surface-area e3(0,t) first increases, reaches a maximum and then decreases to a final value smaller than the initial one. This latter point is a characteristic feature of the capillary instability which is known to reduce the total amount of surface area. The increase of the specific-surface-area corresponds to regime 1 (t/t * < 1) and the maximum specificsurface-area correlates with the value of the stretching rate. The thinning mechanism during regime 1 produces interface: it is therefore an elongation mechanism. During regimes 2 and 3, the specific-surface-area decreases: the thinning mechanism becomes a contraction mechanism driven by surface tension forces. In regime 2, this contraction mechanism is controlled by inertia; in regime 3 it is controlled by viscosity. This scenario has been confirmed by a model that is not presented here. The multi-scale analysis demonstrates the existence of an elongation mechanism at small scale even for the case of a capillary instability. Once again, it is found that this elongation mechanism is beneficial for the atomization. Indeed, we see in Fig. 14 that for k = 0.88, the elongation mechanism is the more Copyright © 2017 by ASME efficient since the maximum of e3(0,t) is the highest and the corresponding loss of specific-surface-area is the smallest. It can be added here that the characteristic times t* and tBU both increase with k. The elongation mechanism therefore delays the pinch-off and breakup instant. All these conclusions are valuable for the operating conditions of this study only. They might differ for other initial perturbation amplitude or wavelength since the capillary instability is known to deeply depend on these two parameters.

Simulated turbulent jet atomization process

The atomization of a turbulent jet have been calculated by the DNS code ARCHER. The liquid properties are those of the Shellsol D40. The initial diameter of the jet is Dj = 192.5 µm. The mean velocity is equal to 25 m/s. (These conditions are close to those of the turbulent jets produced by the GDI device presented above with an injection pressure equal to Pi = 1.1 MPa.) The initial turbulence of the jet is assumed homogenous and isotropic. It is characterize by a 5% turbulent intensity and an integral length equal to 0.08Dj. The calculation domain is 128x128x1024 and the whole simulation represents an injection time of 3 ms. Figure 15 shows an image of the simulated atomization process. At the beginning we see that the jet is perturbed by large scale disturbances only. Then, smaller and smaller disturbances appear and grow when going downstream. Two multi-scale analyses are conducted. The first one is a 2D approach that intends to reproduce the analysis performed in the experimental work on the GDI jets presented above. 70 images similar to the one shown in Fig. 15 are extracted from the simulation and 2D projection images of the jet are produced. (A 2D projection image is shown in Fig. 16.) Then, the process of e2(d,t) measurement developed for the experimental images (see Fig. 6) is applied on the set of 2D images. The characteristic scale d1(t) introduced by Eq. ( 15) is determined from the temporal evolution of the surface-based scale distribution e2(d,t). The result is reproduced by the blue line in Fig. 9. We see an excellent proximity between the experimental and simulated results from the time at which the small scale disturbances have developed. This agreement is an important result. First, it somehow validates the simulation. Second, it says that even with simplified initial conditions (which is the case here for the turbulence of the jet) the simulated atomization process may resemble the actual one after the input energy has been distributed in the scale space by the simulation. Third, this result presents a different and new procedure to validate numerical simulations.

The second analysis is 3D. It is based on the same process as the 2D approach, namely, the scale distribution e3(d,t) is determined on a portion of the jet that is delimited by an analyzing window. This window is slid from the beginning to the end of the jet and this translation is associated with a temporal evolution of the scale-distribution. A module of 3D scale distribution measurement has been developed here and included in the ARCHER code. The e3(d,t) measurements have been performed on 44 positions on each image and this has been repeated on 11 images. The temporal evolution of the mean volume-based scale distribution is extracted from these measurements. This evolution is shown in Fig. At initial time, the scale distribution is linear indicating that the jet has an almost non-deformed cylindrical shape (see Fig. 2). As time goes, the simulated atomization process progresses as a thinning mechanism recognized in the small scales by a continuous increase of e3(d,t) and in the large scales by a continuous decrease of dmax. (Such a mechanism was also identified on the experimental 2D scale distributions (Fig. 7).) The stretching rate introduced by Eq. ( 7) provides a characteristic of the thinning mechanism. In the present case, the thinning mechanism in the small scales is not similar to the one of a cylindrical ligament since the volume-based scale distribution is not linear (see Fig. 2). The stretching rate becomes Copyright © 2017 by ASME a scale dependent function. It has been calculated for all scales and its temporal evolution is presented for some of them in Fig. 18. At all scales, the stretching rate shows a fast increase at earlier times followed by a sharp decrease down to a value more or less close to zero. This suggests a two phase atomization process with a decrease of the stretching during the second phase. This resembles the experimental observations except for the first phase where the stretching rate is found to increase here. This of course is due to the difference in turbulent initial conditions between the experiment and the simulation. The increase of the stretching rate in phase 1 characterizes the response of the jet to the initial homogeneous and isotropic turbulence. Figure 18 shows that the effect of the turbulence is concentrated at the very beginning of the jet history and lasts a very short time. This is of course due to the fact that this turbulence is not maintained. A complementary analysis should be conducted to evaluate the degree of elongation of the thinning mechanism as a function of the characteristics of the turbulence. Figure 18 also shows that the peak of stretching rate is higher and appears later as the scale decreases. This illustrates that the thinning mechanism operates on smaller and smaller and deformed liquid fragments only as time goes. Knowing the important influence of the thinning mechanism in small drops production, this distribution of stretching rate, which is probably connected to the decaying turbulence of the jets, might be also related to the final spray drop size distribution. This is an illustration of the new possibilities offered by the multi-scale analysis to investigate liquid atomization process and spray formation.

MODELING ATOMIZATION PROCESSES

This last short section addresses the question of the modelling of liquid atomization processes and intends to estimate the possible contribution of the multi-scale tool. Three models are rapidly introduced and discussed, namely, the model of successive fragmentations, the model of successive agglomerations and the scale-entropy diffusion model.

In the successive fragmentation model [START_REF] Gorokhovski | Analyses of Kolmogorov's Model of Breakup and its Application into Lagrangian Computation of Liquid Sprays under Air-Blast Atomization[END_REF][START_REF] Gorokhovski | Fragmentation under the Scaling Symmetry and Turbulent Cascade with Intermittency[END_REF] the atomization process is described as a succession of elementary fragmentations of spherical drops. Each mother drop breaks into daughter drops whose number-based distribution characteristics are not dependent on the diameter of the mother drop. Equating this fragmentation cascade provides the temporal evolution of the number-based diameter distribution. Using appropriate assumptions, the equation of the temporal evolution of the drop diameter distribution becomes a Fokker-Planck type equation. This model is written for high Weber number atomization process.

The model of successive fragmentations uses a diameter distribution to describe the liquid system at each time. This distribution implicitly characterizes an Equivalent System whose definition is omitted by the model. This definition must be based on a measureable characteristic of the atomization liquid system. The scale distribution could be an interesting candidate for this duty. The equivalent system could be a set of simple objects whose scale distribution is the same as the actual system. This idea is identical to the one that led to the mathematical expression for the scale distribution of the few droplets produced by the atomization of a stretched deformed ligament (Eq. ( 12)). This equation actually says that a set of spheres having the same scale distribution as the actual system exists. It is believed that we could extend such an approach to more complex situations. The model of successive agglomerations is due to Villermaux et al. [START_REF] Villermaux | Ligament-Mediated Spray Formation[END_REF] and Villermaux [START_REF] Villermaux | Fragmentation[END_REF]. It describes an atomization process as a succession of agglomerations of elementary blobs. Blobs are defined as spheres. The initial blobs diameter distribution is related to shape of the liquid system at initial time. The blobs diameter distribution evolves to the final drop diameter distribution. The temporal evolution of the blubs diameter distribution is governed by a Boltzmann-type equation. This mathematic is similar to the one encountered in the modeling of soot aggregate formation. In atomization, this model has been developed to describe the last stage of the atomization process in which liquid fragments break under the action of the surface tension forces. The drop diameter distribution reported by this model in the one given by Eq. [START_REF] Marmottant | Fragmentations of Stretched Liquid Ligaments[END_REF]. The pertinence of this distribution within the context of a multi-scale analysis has been illustrated in the investigation dedicated to the atomization of stretched deformed ligaments. It is therefore believed that the concept of elementary blob distribution used to describe the liquid system could find a concrete definition by making use of the scale distribution.

The scale entropy diffusion model has been developed by Queiros-Condé [START_REF] Queiros-Condé | A Diffusion Equation to Describe Scale-and Time-Dependent Dimensions of Turbulent Interfaces[END_REF] to address the evolution of turbulent interfaces. It has been applied to a liquid atomization process in previous works [START_REF] Dumouchel | Application of the Scale Entropy Diffusion Model to Describe a Liquid Atomization Process[END_REF][START_REF] Dumouchel | On the Scale Diffusivity of a 2-D Liquid Atomization Process Analysis[END_REF]. The atomization process is described by a diffusion equation: the scale entropy diffusion equation:

Copyright © 2017 by ASME       t t , x t , x x t , x           1 2 2 (20)
where the scale entropy (x,t) = ln(1/E2(d,t)), x = ln(d/D) (D being a characteristic scale of the system), (x,t) is the scale evolutivity function (scale-entropy flux sink function) and  is the scale diffusivity which characterizes the ability of a perturbation to diffuse in the scale space. In turbulence, Queiros-Condé [START_REF] Villermaux | Fragmentation[END_REF] reported:

      2 0 2 2 3 ln 16 9 l Re Re        ( 21
)
where l0 is the integral scale,  the kinematic viscosity and

Re is the Reynolds. In atomization, the following expression was obtained [START_REF] Dumouchel | Application of the Scale Entropy Diffusion Model to Describe a Liquid Atomization Process[END_REF]:

D * We We . L . L    2 555 0 67 ln 9 2                ( 22
)
where WeL is the liquid Weber number and µ* is an equivalent dynamic viscosity. The similarity between Eqs. ( 21) and ( 22) is notable. The Reynolds number in turbulence (Eq. ( 21)) is replaced by a liquid Weber number in atomization (Eq. 22)). Although this result may appear pertinent at first glance, it questions the capacity of the model to properly take the liquid turbulence into account. A part of the answer to this question has been given in a recent investigation [START_REF] Dumouchel | Towards an Interpretation of the Scale Diffusivity in Liquid Atomization Process: An Experimental Approach[END_REF] in which it was established that the entropy diffusion model introduced by Eq. ( 20) is valuable in the second phase of the atomization process only, i.e., where the stretching rate at small scales are controlled by surface tension forces. In other words, a different diffusion equation should be proposed for the first phase of the atomization process during which the effect of turbulence is dominant.

CONCLUSIONS

The atomization of a liquid system in a gaseous environment is a process of continuous variation of the shape of the system until a final stable state is reached. Because of the action of the surface tension forces which characterize the liquid-gas interface, this final state is a flow of small, stable and spherical drops. Having several origins, the disturbances involved in the system deformation introduce spectra of characteristic length and time scales. Any investigation of liquid atomization processes must take this into account and therefore must be multi-scale. The present approach based on the notion of scale distribution has been motivated by this necessity. The applications described in this paper demonstrate the pertinence of the approach that improves the physical understanding of the atomization process and provides alternative tools to conduct further investigations.

An atomization process is a combination of deformation and breakup mechanisms. The multi-scale analysis shows that the deformation mechanisms may create interface whereas the breakup mechanisms consume interface. To improve the efficiency of the atomization process, the creation of interfaces during the deformation mechanisms should be favored. The multi-scale analysis allows identifying the presence of a thinning mechanism at small scales during the deformation stage. It represents the formation of small structures that develop as the breakup stage is approached and is therefore visible until this breakup event occurs. Due to the multi-scale analysis, the quantification of this mechanism is possible. It is found that the thinning mechanism is due either to an elongation mechanism or a contraction mechanism. In the first case, the interface area is increased and the breakup is delayed and operates on smaller liquid fragments. The atomization is therefore improved. Thus, mechanisms creating elongation at small scales should be favored. Those mechanisms can be imposed by the dynamic of the initial liquid system. In particular, the turbulence of the liquid system can fulfil this role. However, the present analysis reveals that the effects of the turbulence on the development of the thickening mechanism at small scales are rapid and don't last long since the turbulence is not maintained. Thus, atomization processes appear as a two-phase mechanism: during phase 1, the deformation mechanism is controlled by the liquid system dynamic, during phase 2 it is controlled by the surface tension forces.

All these results demonstrate the interesting potential of the multi-scale approach that allows handling experimental and simulation results in a more pertinent and constructive way as usually done. Furthermore, the notion of scale distribution shows an interesting potential in the domain of liquid atomization modeling.

  -right) has a surface area noted S2(d). The erosion operation is applied for d varying from 0 to infinity and the cumulative surface-based scale distribution E2(d) is constructed as: d = 0, S2(d) = S2T. For a sufficiently large scale, the erosion operation removes all the system and the remaining surface area S2(d) = 0. Therefore in the range d[0; ∞[, E2(d) monotonously increases from 0 to 1. The first derivative of E2(d) with respect to the scale is the surface-based scale distribution e2(d):

  ) = (VT -V(d))/VT involves the volume V(d) of the eroded system at scale d. The first derivative of E3(d) is the volumebased scale distribution e3(d). This function is equal to the ratio of the 3-D surface area S3(d) of the eroded system divided by twice the system total volume:

  Eqs. (4) and[START_REF] Hunt | Kolmogorov's Contribution to the Physical and Geometrical Understanding of Small-Scale Turbulence and Recent Developments[END_REF], n = 2 or 3 according to the dimension of the description. The diameter D of these systems is equal to the maximum scale dmax defined as the smallest scale for which En(d) = 1 and en(d) = 0.

Fig. 1 :

 1 Fig. 1: Illustration of an erosion operation. Left: Initial 2-D system. Its total surface area is noted S2T; Right: Erosion at scale d. The light gray strip is removed. The eroded system has a surface area noted S2(d) (dark gray area)

Fig. 2 :

 2 Fig.2: Temporal evolution of the scale distribution of a thinning cylindrical ligament: Left -e2(d,t), Right -e3(d,t)

Fig. 3 :Fig. 4 :

 34 Fig. 3: Temporal evolution of an atomizing water ligament into air. Time gap between two consecutive images 80 µs. Spatial resolution 10.36 µm/pixel. The arrow indicates the point of separation used to isolate the ligament [10]

Fig. 5 :

 5 Fig. 5: Comparison between the experimental and the mathematical (Eq. (12)) scale distribution of the final droplets produced by individual ligaments (W:  = 70 mN/m; WE10:  = 44 mN/m; WE15:  = 38 mN/m)

Fig. 6 :Fig. 7 :

 67 Fig. 6: Left -Image of the turbulent jet at 0.5 MPa (6x8 mm); Right -Local measurement of the scale distribution; The rectangle represents the analyzing window. h: position of the analyzing window, Equivalent time t = h/Vq where Vq is the jet mean velocity

Fig. 8 :

 8 Fig. 8: Temporal evolution of the characteristic time of the thinning mechanism at d = 0 and comparison the characteristic capillary time (several injection pressures).

Fig. 9 :

 9 Fig. 9: Temporal evolution of the characteristic scale d1(t). (Several injection pressures. The dash lines visualize phase 1 and 2 of the atomization process for the Pi = 0.6 MPa. The blue line shows the d1(t) obtained from the DNS simulation)

Fig. 10 :Fig. 11 :

 1011 Fig. 10: DNS result: Temporal evolution of the ligament for k = 0.55

Fig. 13 :

 13 Fig.13: Stretching rate at small scales for all wavenumbers.

Fig. 14 :

 14 Fig. 14: Temporal evolution of the specific-surface-area (five wavenumbers)

Fig. 15 :Fig. 16 :

 1516 Fig. 15: Simulated atomization process of a turbulent jet. Code ARCHER

  [START_REF] Eggers | Non-Linear Dynamics and Breakup of Free Surface Flows[END_REF]

  .

Fig 17 :

 17 Fig 17: Temporal evolution of the volume-based scale distribution of the simulated turbulent jet.

Fig. 18 :

 18 Fig. 18: Stretching rate of the 3D simulated turbulent jet as a function of the scale

  The thinning mechanism in the small scales is recognized by a continuous temporal decrease of the scale

	derivate   t , d ' e 3	. Furthermore, since	' e 3	  t , d
				Copyright © 2017 by ASME

ACKNOWLEDGMENTS

The author acknowledges the financial support from the French National Research Agency (ANR) through the program Investissements d'Avenir (ANR-10 LABX-09-01), LABEX EMC3. The author would like to thank Dr. Thibaut Ménard and Dr. Wojciech Aniszewski for the ARCHER simulations.


Jean B Lasserre
email: lasserre@laas.fr

Simple formula for integration of polynomials on a simplex Simple formula for integration

Keywords: Numerical integration, simplex, Laplace transform Mathematics Subject Classification (2010) 65D30, 78M12, 44A10

We show that integrating a polynomial f of degree t on an arbitrary simplex (with respect to Lebesgue measure) reduces to evaluating t homogeneous related Bombieri polynomials of degree j = 1, 2, . . . ,t, each at a unique point ξ j of the simplex. This new and very simple formula could be exploited in finite (and extended finite) element methods, as well as in applications where such integrals must be evaluated. A similar result also holds for a certain class of positively homogeneous functions that are integrable on the canonical simplex.

Introduction

We consider the problem of integrating a polynomial f ∈ R[x] on an arbitrary simplex of R n and with respect to the Lebesgue measure. After an affine transformation this problem is completely equivalent to integrating a related polynomial of same degree on the canonical simplex ∆ = {x ∈ R n + : e T x ≤ 1} where e = (1, . . . , 1) ∈ R n . Therefore the result is first proved on ∆ and then transferred back to the original simplex. The result is also extended to positively homogeneous functions of the form ∑ α∈M f α x α 1 1 • • • x α n n where M ⊂ R n is finite and -1 < α i ∈ R, ∀i.

Background

In addition to being a mathematical problem of independent interest, integrating a polynomial on a polytope has important applications in e.g. computational geometry, in approximation theory (constructing splines), in finite element methods, to cite a few; see e.g. the discussion in Baldoni et al. [START_REF] Baldoni | How to integrate a polynomial over a simplex[END_REF]. In particular, because of applications in finite (and extended finite) element methods and also for volume computation in the Natural Element Method (NEM), there has been a recent renewal of interest in developing efficient integration numerical schemes for polynomials on convex and non-convex polytopes For instance the HNI (Homogeneous Numerical Integration) technique developed in Chin et al. [START_REF] Chin | Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra[END_REF] and based on [START_REF] Lasserre | Integration on a convex polytope[END_REF], has been proved to be particular efficient in some finite (and extended finite) element methods; see e.g. Antonietti et al. [START_REF] Antonietti | Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods[END_REF], Chin and Sukumar [START_REF] Chin | Modeling curved interfaces without element-partitioning in the extended finite element method[END_REF], Nagy and Benson [START_REF] Nagy | On the numerical integration of trimmed isogeometric elements[END_REF] for intensive experimentation, Zhang et al. [START_REF] Zhang | Natural element method for solving radiative transfer with or without conduction in three-dimensional complex geometries[END_REF] for NEM, Frenning [START_REF] Frenning | Efficient Voronoi volume estimation for DEM simulations of granular materials under confined conditions[END_REF] for DEM (Discrete Element Method), Leoni and Shokef [START_REF] Leoni | Attraction Controls the Inversion of Order by Disorder in Buckled Colloidal Monolayers[END_REF] for volume computation. For exact volume computation of polytopes in computational geometry, the interested reader is also referred to Büeler et al. [START_REF] Büeler | Exact volume computation for polytopes: A practical study[END_REF] and references therein.

For integrating a polynomial on a polytope, one possible route is to use the HNI method developed in [START_REF] Lasserre | Integration on a convex polytope[END_REF][START_REF] Chin | Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra[END_REF] and also extended in [START_REF] Antonietti | Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods[END_REF], without partitioning the polytope in simplices. Another direction is to consider efficient numerical schemes for simplices since quoting Baldoni et al. [START_REF] Baldoni | How to integrate a polynomial over a simplex[END_REF] "among all polytopes, simplices are the fundamental case to consider for integration since any convex polytope can be triangulated into finitely many simplices". In [START_REF] Baldoni | How to integrate a polynomial over a simplex[END_REF] the authors analyze the computational complexity of the latter case and describe several formulas; in particular they show that the problem is NP-hard in general. However, if the number of variables is fixed then one may integrate polynomials of linear forms efficiently (with Straight-Line program for evaluation) and if the degree is fixed one may integrate any polynomial efficiently. They also describe several formulas in closed form for integrating powers of linear forms [2, Corollary 12] and also arbitrary homogeneous polynomials [2, Proposition 18] and [START_REF] Lasserre | The multi-dimensional version of b a x p dx[END_REF], all stated in terms of a summation over vertices of the simplex.

Our main result

With f ∈ R[x], x → f (x) = ∑ α f α x α is associated the Bombieri-type polynomial f ∈ R[x]: x → f (x) = ∑ α∈N n fα x α = ∑ α∈N n α 1 ! • • • α n ! f α x α , x ∈ R n . (1.1)
We establish the following simple formula:

Theorem 1.1 Let ∆ ⊂ R n be the canonical simplex. If f ∈ R[x]
is a polynomial of total degree t ∈ N then with ξ j := e/((n + 1)

• • •(n + j)) 1/ j : ∆ f dx = 1 n! ( f0 + t ∑ j=1 f j (ξ j )), (1.2)
where f = ∑ t j=0 f j and each f j is a homogeneous form1 of degree j. Similarly, let M ⊂ R n be a finite set and let f : (0, +∞) n → R with

x → f (x) := ∑ α∈M f α x α , α ∈ R n , α i > -1, i = 1, . . . , n, (1.3)
be positively homogeneous of degree t ∈ R (i.e. ∑ i α i = t for all α ∈ M). Then

∆ f dx = 1 n! f (ξ t ), (1.4)
where ξ t := 1 θ e ∈ ∆ with θ t = Γ (1 + n + t)/Γ (1 + n), and

x → f (x) := ∑ α∈ M Γ (1 + α 1 ) • • •Γ (1 + α n ) f α x α .
Theorem 1.1 states that integrating a polynomial f of degree t on the canonical simplex ∆ can be done by evaluating each Bombieri form f j at a unique point ξ j ∈ ∆ . In addition all points (ξ j ) are on a line between the origin and the point e/n on the boundary of ∆ . To the best of our knowledge, and despite their simplicity, we have not been able to find formula (1.2) or (1.4) in the literature, even if they can be obtained in several relatively straightforward manners from previous results in the literature.

The point that we make in (1.2) is to relate ∆ f dx to point evaluation of its Bombieri-polynomial at only t very specific points of the simplex ∆ ; and similarly for positively homogeneous functions of the form (1.3).

Similarly, integrating a polynomial f of degree t on an arbitrary simplex Ω ⊂ R n can be done by evaluating related polynomials h j of degree j, each at a certain point of Ω . Indeed an arbitrary (full-dimensional) simplex can be mapped to the canonical simplex ∆ by an affine transformation y = A (xa) for some real nonsingular matrix A and vector a ∈ R n . Therefore (1.2) translates to a similar formula on Ω with ad-hoc polynomials and t aligned points x j ∈ Ω .

Hence formula (1.2) is much simpler than those in [START_REF] Baldoni | How to integrate a polynomial over a simplex[END_REF][START_REF] Lasserre | The multi-dimensional version of b a x p dx[END_REF]. In particular it is valid for an arbitrary polynomial and there are only t points involved if the degree of the polynomial is t. In contrast, for integrating a t-power of a linear form, the formula in [START_REF] Baldoni | How to integrate a polynomial over a simplex[END_REF] requires a summation at the (n + 1) vertices and in [START_REF] Lasserre | The multi-dimensional version of b a x p dx[END_REF], integrating a form of degree t requires n+t n evaluations of a multilinear form at the (n + 1) vertices. We would like to emphasize that (1.2) resembles a cubature formula but is not. A cubature formula is of the form:

∆ f dx = s ∑ j=1 w j f (x j ), ∀ f ∈ R[x] t , (1.5)
for some integer s and points (x j ) ⊂ ∆ with associated weights w j , j = 1, . . . , s. However Theorem 1.1 suggests that as long as polynomials are concerned, the simpler alternative formula (1.2) could be preferable to cubature formulas involving many points. The point of view is different. Instead of evaluating the single polynomial f of degree t at several points x j in (1.5), in (1.2) one evaluates t other polynomials f j of degree j, j = 1, . . .t (simply related to f ); each polynomial f j is evaluated at a single point only.

Our technique of proof is relatively simple. It uses (i) Laplace transform technique and homogeneity to reduce integration on ∆ with respect to (w.r.t.) Lebesgue measure to integration on the nonnegative orthant w.r.t. exponential density; this technique was already advocated in [START_REF] Lasserre | Solving a class of multivariable integration problems via Laplace Lechniques[END_REF] for computing certain multivariate integrals and in [START_REF] Lasserre | A Laplace transform algorithm for the volume of a convex polytope[END_REF] for volume computation of polytopes. Then (ii) integration of monomials w.r.t. exponential density can be done in closed-form and results in a simple formula in closed form.

Interestingly and somehow related, recently Kozhasov et al. [START_REF] Kozhasov | Positivity certificates via integral representations[END_REF] have considered integration of a "monomial" y α-e (with 0 < α ∈ R n ) with respect to exponential density on the positive orthant. In [START_REF] Kozhasov | Positivity certificates via integral representations[END_REF] the density y → ∏ i Γ (α i ) y α i -1 i is called the Riesz kernel of the monomial x -α . The Riesz kernel offers a certificate of positivity for a function to be completely monotone (a strong positivity property of functions on cones).

Main result

Notation, definitions and preliminary result

Let R[x] denote the ring of real polynomials in the variable x = (x 1 , . . . ,

x n ). With N the set of natural numbers, a polynomial f ∈ R[x] is written x → f (x) = ∑ α∈N n f α x α , (2.1)
in the canonical basis of monomials. A polynomial f is homogeneous of degree t if

f (λ x) = λ t f (x) for all λ ∈ R and all x ∈ R n . For α ∈ R n let |α| := ∑ i |α i |. Let R n + := {x ∈ R n : x ≥ 0} denotes the positive orthant of R n . A function f : (R + \ {0}) n → R is positively homogeneous of degree t ∈ R if f (λ x) = λ t f (x), ∀x ∈ (R + \ {0}) n , ∀λ > 0, and a polynomial f is homogeneous of degree t ∈ R if f (λ x) = λ t f (x), ∀λ ∈ R, ∀x ∈ R n . Denote by ∆ ⊂ R n + the canonical simplex {x ∈ R n + : e T x ≤ 1} where e = (1, . . . , 1) ∈ R n . For 0 < x ∈ R n let 1/x = ( 1 x 1 , . . . , 1 x n ) ∈ R n . With a polynomial f ∈ R[x] in (2.1) is associated the "Bombieri" polynomial: x → f (x) = ∑ α∈N n fα x α = ∑ α∈N n f α α 1 ! • • • α n ! x α , x ∈ R n . (2.2)

Main result

After an affine transformation, integrating f on an arbitrary full-dimensional simplex Ω ⊂ R n reduces to integrating a related polynomial of same degree on the canonical simplex ∆ = {x ∈ R n + : e T x ≤ 1} where e = (1, . . . , 1) ∈ R n . Therefore in this section we consider integrals of polynomials (and a certain type of positively homogeneous functions) on the canonical simplex ∆ .

Theorem 2.1 Let 0 < z ∈ R n and let ∆ z = {x ∈ R n + : z T x ≤ 1}. Let f be a positively homogeneous function of degree t ∈ R on (R + \ {0}) n such that t > -(1 + n) and ∆ z | f | dx < ∞. Then: ∆ z f (x) dx = 1 Γ (1 + n + t) R n + f (x) exp(-z T x) dx. (2.3)
If f is a homogeneous polynomial of degree t (hence t ∈ N) then

∆ z f (x) dx = 1 (n + t)! f (1/z) 1 z e , (2.4)
and in particular with z = e:

∆ f (x) dx = 1 (n + t)! f (e) = 1 n! f (ξ t ), (2.5)
where ξ t = 1 θ e ∈ ∆ and θ t = (n + 1) • • •(n + t). Proof Let 0 < z ∈ R n be fixed and let h : R → R be the function:

y → h(y) := {x∈R n + :z T x≤y} f (x) dx.
Observe that h = 0 on (-∞, 0] and in addition, h is positively homogeneous of degree n + t, so that h(y) = y n+t h(1) (well defined since h(1) is finite). As t > -(n + 1), its Laplace transform H : C → C is well defined and reads:

λ → H(λ ) = Γ (1 + n + t) λ n+t+1 h(1), λ ∈ C, ℜ(λ ) > 0. (2.6)
On the other hand, for real λ > 0:

H(λ ) = ∞ 0 h(y) exp(-λ y) dy = ∞ 0 exp(-λ y) {x∈R n + :z T x≤y} f (x) dx dy = R n + f (x) ∞ z T x exp(-λ y) dy) dx [by Fubini-Tonelli] = 1 λ R n + f (x) exp(-λ z T x) dx = 1 λ n+t+1 R n + f (x) exp(-z T x) dx .
Identifying with (2.6) yields (2.3). Next, to get (2.4) observe that for 0 < z ∈ R, ∞ 0 y k exp(-z y) dy = k!/z k+1 for all k ∈ N, and therefore:

R n + y α exp(-z T y) dy = 1 z e α 1 ! • • • α n ! (1/z) α ,
for every α ∈ N n . Summing up yields the result (2.4) and (2.5) with z = e. Finally, the last equality of (2.5) is obtained by homogeneity of f .

As the reader can see, formula (2.5) is extremely simple and only requires evaluating f at the unique point ξ t ∈ ∆ . This in contrast to the formula in [START_REF] Lasserre | The multi-dimensional version of b a x p dx[END_REF] which requires a sum of n+t n terms, each involving evaluations at the vertices of ∆ .

Remark 2.1 Notice that (2.3) can also be interpreted as follows: Let f : R n + → R + , be positively homogeneous of degree t ∈ R and as in Theorem 2.1. Define the function

h f : R n + → R + , by: z → Γ (1 + n + t) ∆ z f (x) dx.
Then h f is the multidimensional Laplace transform of f , or equivalently in the terminology of Kozhasov et al. [START_REF] Kozhasov | Positivity certificates via integral representations[END_REF], h f is completely monotone 2 (by the Bernstein-Hausdorff-Widder-Choquet theorem; see [START_REF] Kozhasov | Positivity certificates via integral representations[END_REF]Theorem 2.5]). Next, let C ⊂ R n be an open cone with dual cone C * . If p ∈ R[x], s ∈ R, and

p s (x) = C * exp(-y T x) dµ(y), ∀x ∈ C,
for some Borel measure µ on C * , then µ is called a Riesz measure. In addition if µ has a density q with respect to Lebesgue measure on C * then q is called the Riesz kernel of p s ; see [7, p. 4]. ♣

Hence from Remark 2.1, with C = (R \ {0}) n and dµ = f dx, we obtain:

Proposition 2.1 For every positively homogeneous f : (R + \ {0}) n → R + of total degree t ∈ R as in Theorem 2.1, the function h f : R n + → R + :

z → h f (z) := Γ (1 + n + t) {x∈R n + :z T x≤1} f (x) dx , (2.7
)

is completely monotone. In addition if f = x α-e with 0 < α ∈ R n then f is the Riesz kernel of the function x → x -α • ∏ i Γ (α i ) on (R + \ {0}) n .
Proposition 2.1 is in the spirit of [START_REF] Kozhasov | Positivity certificates via integral representations[END_REF][Proposition 2.7]. Next given ℓ, x ∈ R n denote by ℓ • x the scalar product ℓ T x and by ℓ ⊗ x ∈ R n the vector (ℓ 1 x 1 , . . . , ℓ n x n ). Finally let E t ∈ R[x] the homogeneous polynomial of degree t with all coefficients equal to 1, that is,

x → E t (x) := ∑ |α|=t x α .
As a consequence of Theorem 2.1 we obtain:

2 A real-valued function f : (R \ {0}) n → R is completely monotone if (-1) k ∂ k f ∂ x i 1 •••∂ x i k (x) ≥ 0 for all x ∈ (R \ {0}) n and for all index sequences 1 ≤ i 1 ≤ ••• ≤ i k ≤ n of arbitrary length k.
Corollary 2.1 (i) Let f be polynomial of degree t and write f = ∑ t j0 f j where each f j is homogeneous of degree j. Then

∆ f (y) dy = vol(∆ ) ( f0 + t ∑ j=1 f j (ξ j ) ), (2.8)
where ξ j = e/((n + 1) • • •(n + j)) 1/ j and f j is as in (2.2).

(ii) For every ℓ ∈ R n and t ∈ N:

∆ (ℓ • x) t dx = 1 (n + t)! f (e) = t! (n + t)! E t (ℓ).
(2.9)

Proof (i) As ∆ f (y) dy = ∑ t j=0 ∆ f j (y)
dy, use Theorem 2.1 for each f j and sum up. Next, (ii) is a direct consequence of Theorem 2.1 and the fact that (using

(ℓ • x) t = t! ∑ |α|=t ℓ α x α /(α 1 ! • • • α n !)), f (x) = (ℓ • x) t = t! ∑ |α|=t ℓ α x α = t! ∑ |α|=t (ℓ ⊗ x) α = t! E t (ℓ ⊗ x),
and therefore f (e) = t! E t (ℓ ⊗ e) = t! E t (ℓ).
Hence Corollary states that integrating f on ∆ reduces to evaluate each f j at the unique point ξ j ∈ ∆ , and sum up. In addition, all points ξ j , j = 1, . . . ,t, are aligned in ∆ ; they are between the origin 0 and the point e/n ∈ ∆ , on the line joining 0 to e/n. Again formula (2.8) and (2.9) are extremely simple. The former only requires evaluating f j at ξ j (t evaluations) and the latter only requires evaluating the polynomial E t at the point ℓ ∈ R n . This is contrast with [2, Corollary 12, p. 307] which is more complicated (even for a single form ℓ).

Finally, Theorem 2.1 can be extended to a class of homogeneous functions Proposition 2.2 Let M ⊂ R n be a finite set of indices and let f : (R + \ {0}) n → R be positively homogeneous of degree t ∈ R and of the form

x → ∑ α∈ M f α x α , α ∈ R n , α i > -1, ∀i = 1, . . . , n, for some real coefficients ( f α ) α∈ M . Then ∆ f (x) dx = 1 Γ (1 + n + t) f (e) = 1 n! f (ξ t ), (2.10
)

where ξ t = 1 θ e ∈ ∆ with θ t = Γ (1 + n + t)/Γ (1 + n), and x → f (x) := ∑ α∈ M Γ (1 + α 1 ) • • •Γ (1 + α n ) f α x α .
Proof Let -1 < α ∈ R n with |α| = t be fixed and let 0 < z ∈ R n . By Theorem 2.1:

∆ z x α dx = 1 Γ (1 + n + t) R n + x α exp(-z T x) dx = 1 Γ (1 + n + t) z -α-e n ∏ i=1 Γ (1 + α i ).
Summing up over all α ∈ M yields

∆ z f (x) dx = z -e Γ (1 + n + t) ∑ α∈M f α z -α n ∏ i=1 Γ (1 + α i ) = z -e Γ (1 + n + t) f (z -e ),
and with z = e one obtains (2.10). Finally, the last equality in (2.10) uses the positive homogeneity of f and Γ (1 + n) = n! for all n ∈ N.

Example 2.1 For illustration purpose consider the elementary two-dimensional example (i.e., n = 2) where ∆ = {x ≥ 0 :

x 1 + x 2 ≤ 1}. With x → f (x) := f 10 x 1 + f 11 x 1 x 2 + f 02 x 2
2 one obtains ξ 1 = e/3 and ξ 2 = e/ √ 12. The right-hand-side of (2.8) reads:

1 2 ( f 10 (ξ 1 ) 1 + f 11 (ξ 2 ) 1 (ξ 2 ) 2 + 2 f 02 (ξ 2 ) 2 2 ) = 1 2 ( f 10 3 + f 11 12 + 2 f 02 12
).

For instance with x → f (x

) := x 1 + x 1 x 2 + x 2 2 , one obtains: 1 2 ((ξ 1 ) 1 + (ξ 2 ) 1 (ξ 2 ) 2 + 2(ξ 2 ) 2 2 ) = 1 2 ( 1 3 + 1 12 + 2 1 12 ) = 7 24 ,
and indeed ∆ f dx = 7/24. In Figure 2.1 is displayed the 2D-Simplex ∆ with the points ξ 1 = e/3 and ξ 2 = e/ √ 12.

Back to an arbitrary simplex

Let Ω ⊂ R n be an arbitrary full-dimensional simplex (its n-dimensional Lebesgue volume is strictly positive). Then Ω is mapped to ∆ by some affine transformation. In a fixed basis, this is obtained by the change of variable y := A(xa) where A is one non singular n × n real matrix and where a ∈ R n is a vertex of Ω . Then:

x ∈ Ω ⇔ y := A (x -a) ∈ ∆ . Next, if f is a polynomial of degree t: Ω f (x) dx = 1 det(A) ∆ f (A -1 y + a) g(y) dy = 1 det(A) ∆ g(y) dy, ( 2.11)
where g has same degree as f . Next, writing g = ∑ t j=0 g j (y) with g j homogeneous of degree j, one has ĝ j (y) = ĝ j (A(xa)) and therefore with ξ j ∈ ∆ as in Corollary 2.1, ψ j := A -1 ξ j + a ∈ Ω and ĝ j (ξ j ) = ĝ j (A(ψ ja)). Therefore combining (2.11) and (2.8) in Corollary 2.1 yields:

Ω f (x) dx = 1 det(A) 1 n! [ ĝ0 + t ∑ j=1 ĝ j (A(ψ j -a)) ],
the analogue for Ω of (2.8) for ∆ .

In addition, let f be homogeneous of degree t with Waring-like decomposition3

x

→ f (x) = s ∑ i=1 ε i (c i • x) t ,
for finitely many c i ∈ R n and ε i ∈ {-1, 1}, i = 1, . . . , s. Then letting ℓ i = (A -1 ) T c i :

Ω f (x) dx = 1 det(A) s ∑ i=1 ε i ∆ (c i • A -1 y + c i • a) t dt = s ∑ i=1 ε i det(A) ∆ (c i • a + ℓ i • y) t dt = s ∑ i=1 ε i det(A) t ∑ k=0 t k (c i • a) t-k ∆ (ℓ i • y) k dt = s ∑ i=1 ε i det(A) t ∑ k=0 t k (c i • a) t-k k! (n + k)! E k (ℓ i ) = 1 det(A) t ∑ k=0 t k k! (n + k)! s ∑ i=1 ε i (c i • a) t-k E k (ℓ i ),
where we have used Newton binomial formula and Corollary 2.1(ii) (and recall that E k (x) = ∑ |α|=k x α ). In particular, if a = 0 and now with ℓ i := (A -1 ) T c i , i = 1, . . . , s, it simplifies to

Ω f (x) dx = 1 det(A) t! (n + t)! s ∑ i=1 ε i E t (ℓ i ).

Conclusion

We have provided a very simple closed-form expression for the integral of an arbitrary polynomial f on an arbitrary full-dimensional simplex. Remarkably if f has degree t, it consists of evaluating t polynomials (related to f ) of degree 1, 2, . . . ,t, respectively, each at a unique point of the simplex. To the best of our knowledge this simple formula is new and potentially useful in all applications where such integrals need to be computed; for instance in finite and extended finite element methods. Therefore it could provide a valuable addition to the arsenal of techniques already available for multivariate integration on polytopes.

Fig. 2 .

 2 Fig. 2.1 n = 2; Simplex ∆ and points ξ 1 = e/3 and ξ 2 = e/ √ 12

f j is the unique form of degree j which is the sum of all monomials of degree j of f (with their coefficient).

In number theory, the Waring problem consists of writing any positive integer as a sum of a fixed number g(n) of nth powers of integers, where g(n) depends only on n. It generalizes to forms as a generic form of degree d can be written as a sum of s d-powers of linear forms; s is called the Waring rank of the form.

Research funded by the European Research Council (ERC) under the European's Union Horizon 2020 research and innovation program (grant agreement 666981 TAMING)


MD, MA Romain Lutaud
email: romain.lutaud@univ-amu.fr

Pierre Verger

PhD Patrick Peretti-Watel

PhD; Carole Eldin

Diagnostic pathways of patients consulting at the infectious diseases ward for presumed Lyme disease : a qualitative descriptive study

Keywords: Lyme borreliosis, post-Lyme disease syndrome, medically unexplained symptoms, social sciences, medical uncertainty Lyme Disease COnsolidated Criteria for REporting Qualitative research

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Lyme disease (LD), which received little media coverage in France until the end of the 2000s, is now the subject of a controversy, [START_REF] Sordet | Chronic Lyme disease: fact or fiction?[END_REF] with sometimes virulent public debates. Doctors and patients represented by associations, [START_REF]Association France Lyme -Association de lutte contre les maladies vectorielles à tiques[END_REF] demand recognition of a chronic form of the disease associated with non-specific symptoms such as pain, asthenia, and concentration disorders. [START_REF] Feder | A Critical Appraisal of "Chronic Lyme Disease[END_REF] In this context, long-term antibiotic treatments are often prescribed despite the absence of proven benefits and may cause serious adverse reactions and even death in some patients. [START_REF] Marzec | Serious Bacterial Infections Acquired During Treatment of Patients Given a Diagnosis of Chronic Lyme Disease -United States[END_REF][START_REF] Marks | Antibiotic Treatment for Chronic Lyme Disease-Say No to the DRESS[END_REF][START_REF] Holzbauer | Death due to community-associated Clostridium difficile in a woman receiving prolonged antibiotic therapy for suspected lyme disease[END_REF][START_REF] Berende | Randomized Trial of Longer-Term Therapy for Symptoms Attributed to Lyme Disease[END_REF][START_REF] Sr | Misleading information about Lyme disease[END_REF] To date, there is no evidence in humans pointing towards the diagnostic criteria of a possible chronic LD. [START_REF] Melenotte | Post-bacterial infection chronic fatigue syndrome is not a latent infection[END_REF] However, media coverage of this disease has led to an increase in consultations for presumed LD in France and in Europe. [START_REF] Septfons | Epidemiology of Lyme borreliosis through two surveillance systems: the national Sentinelles GP network and the national hospital discharge database, France, 2005 to 2016[END_REF][START_REF] Wijngaard Cc Van Den | Surveillance perspective on Lyme borreliosis across the European Union and European Economic Area[END_REF] In France, annual incidence is estimated at around 33,000 cases and presents strong regional disparities, with the incidence being very low around the Mediterranean area, where the vector is rare. [START_REF] Figoni | Lyme borreliosis and other tick-borne diseases. Guidelines from the French Scientific Societies (I): prevention, epidemiology, diagnosis[END_REF] Series of patients consulting with a suspicion of LD result in a confirmed diagnosis of LD for only 10 to 20%, while significant numbers of patients (6-26%) with non-specific symptoms (arthralgia, asthenia, myalgia, headaches) remain undiagnosed at the end of the etiological investigation. [START_REF] Jacquet | Multidisciplinary management of patients presenting with Lyme disease suspicion[END_REF][START_REF] Haddad | Holistic Approach in Patients With Presumed Lyme Borreliosis Leads to Less Than 10% of Confirmation and More Than 80% of Antibiotic Failures[END_REF][START_REF] Bouiller | Consultation for Presumed Lyme Borreliosis: The Need for Multidisciplinary Management[END_REF] In a context of easier access to medical information, media coverage of many health issues, and official discourse promoting patient autonomy, [START_REF] Peretti-Watel | Lyme disease: Insight from social sciences[END_REF] this paradigm has been begin to be reversed the last decades: physicians are now confronted with patients who produce diagnoses and seek to confirm them through the use of health professionals. [START_REF] Fainzang | A deviant diagnosis? Doctors faced with a patient's diagnostic work[END_REF] Using a biographical approach, we sought to describe the diagnostic pathways of patients who initially consulted for a suspicion of LD and for whom this diagnosis had been rejected by an infectiologist at the time of their inclusion in the study. We wanted to better understand firstly the influence of the social environment on the genesis of "LD hypothesis"; secondly the role of the patient in the diagnostic investigation.

Materials and methods

Design and Study Setting

This is a qualitative single-centre study. Between 1 May and 30 June 2019, we recruited the first 30 consecutive patients from a prospective cohort consulting in infectious diseases department for presumed LD as part of regional clinical research programme dedicated to tick-borne diseases led by University Hospital Institute (IHU) Méditerranée Infections in Marseille. This project was approved by an ethical committee. The reporting of this study follows the COREQ guidelines.

Patients were eligible to participate if they were 18 years or older, French-speaking, and able to provide informed consent. They had all previously been seen in consultation with a senior infectiogist. Inclusion criteria were as follows: presenting non-specific symptoms such as fatigue, difficulty concentrating, joint, muscle or headache pain for at least six months; having a negative Lyme serology; and no evidence for an organic differential diagnosis.

Interview guide and biographical approach

The interview guide (Appendix 1) was developed and iteratively revised during pretesting with five older adults (not included in the study). The semi-structured interviews followed a biographical approach, using an interview guide covering the history of symptoms and referrals to different medical specialties, detailed diagnostic pathway, including the history of the differential diagnoses, genesis of the LD diagnosis, the patient's prioritisation of the most likely diagnostic hypotheses, and the associated diagnostic degree of certainty (low, medium, high). The interview guide also covered relationships with health professionals, in particular general practitioners (GPs), their role in conducting the diagnostic enquiry, the impact of symptoms on daily life, and finally patients' views of the disease. The biographical approach consisted of collecting patients' trajectories by combining sequences from the interviewee's life and developing themes related to the subject of study. [START_REF]Groupe de réflexion sur l'approche biographique[END_REF] It uses a life-calendar (Appendix 2), a retrospective data collection tool highlighting the chronological order and proximity of events, important transitions in health pathwat and makes it possible to jointly analyse several aspects of the patient's life.

Data collection and analysis

One male investigator (RL) trained in qualitative methods (GP with a master's degree in social sciences) conducted all interviews in person. The investigator had no direct clinical relationship with any participant. Interviews occurred in a private meeting room at the IHU after a scheduled follow-up consultation. Interviews were systematically audio recorded with the patients' agreement. We also used personal documents spontaneously provided by the patient (medical files, illness diary etc) and investigator' field notes. Interviews were fully transcribed, coded and analysed using the NVivo qualitative data software. All the collected data were systematically cross-checked. Two investigators (RL and CE, the clinician who performed the medical consultation) independently coded all transcripts. Differences were reconciled by consensus until 100% agreement was reached.

The clinical profile category was defined according to the patient's prioritisation of symptoms (see Table 2), in decreasing order of their impact on their quality of life. The category "diagnostic survey coordinator" was coded from the intersection of the following elements: the person who asked for serology test, the patient's deliberate search for a "pro-Lyme doctor" to confirm the diagnosis, spontaneous consultation of specialists (without referral by the GP), particularly infectious disease consultations, presence/absence of a referring GP (or other referring physician) and finally spontaneous statements during the qualitative interview (e.g. "I conducted the investigation"). The category "pro-Lyme caregiver" was chosen if the patient reported during their pathway at least one consultation with a "specialist in chronic LD", whether they were a doctor or other caregiver providing non-conventional medicines.

The category of "Lyme activist" was chosen if the patient was a member of an association or an active member of a forum dedicated to LD.

Results

Participants' sociodemographic and clinical characteristics:

The 30 participants were mainly women with an average age of 47.3 years, with a high education level. Interviews lasted from 44 minutes to 85 minutes. The "pain" profile was predominant. The history of the disease was long with an average symptom duration of 8.5 years (Table 1 and2).

Genesis of the Lyme Disease diagnostic hypothesis

A majority of patients [START_REF] Aronowitz | The Rise and Fall of the Lyme Disease Vaccines: A Cautionary Tale for Risk Interventions in American Medicine and Public Health[END_REF] did not report a history of tick bites. For these patients, the main types of potential exposure to the Lyme vector were the presence of ticks in their environment [START_REF] Sr | Misleading information about Lyme disease[END_REF], contact with traditionally tick-carrying animals [START_REF] Berende | Randomized Trial of Longer-Term Therapy for Symptoms Attributed to Lyme Disease[END_REF], and having spent time in a region perceived to be endemic like forests in the north-east of France [START_REF] Berende | Randomized Trial of Longer-Term Therapy for Symptoms Attributed to Lyme Disease[END_REF].

The hypothesis of "chronic LD" in the diagnostic pathway/trajectory of patients was most often triggered by their identification with other patients' clinical narratives circulating on different media and social networks [START_REF] Haddad | Holistic Approach in Patients With Presumed Lyme Borreliosis Leads to Less Than 10% of Confirmation and More Than 80% of Antibiotic Failures[END_REF]. Other circumstances triggered diagnostic investigations including presence of false positive Lyme serology during a medical check-up [START_REF] Holzbauer | Death due to community-associated Clostridium difficile in a woman receiving prolonged antibiotic therapy for suspected lyme disease[END_REF], family or close friends raising the question of LD [START_REF] Marks | Antibiotic Treatment for Chronic Lyme Disease-Say No to the DRESS[END_REF] and finally, the hypothesis evoked by their doctors [START_REF] Marks | Antibiotic Treatment for Chronic Lyme Disease-Say No to the DRESS[END_REF].

All patients had previously undergone serological testing in a laboratory. For a majority of patients [START_REF] Peretti-Watel | Lyme disease: Insight from social sciences[END_REF], the test was negative, for the others [START_REF] Haddad | Holistic Approach in Patients With Presumed Lyme Borreliosis Leads to Less Than 10% of Confirmation and More Than 80% of Antibiotic Failures[END_REF], the result was considered as a false positive by the clinician according to international and national guidelines. [START_REF] Eldin | Review of European and American guidelines for the diagnosis of Lyme borreliosis[END_REF][START_REF] Raffetin | Unconventional diagnostic tests for Lyme borreliosis: a systematic review[END_REF] Of the latter, half of them [START_REF] Berende | Randomized Trial of Longer-Term Therapy for Symptoms Attributed to Lyme Disease[END_REF] had used laboratories whose techniques were not validated by international standards: private laboratories in Germany, "alternative" private French laboratories, or via a self-test kit obtained on the internet, all recommended by the websites of various Lyme associations. In addition, nearly half of participants ( 14) had received an antibiotic therapy for "chronic Lyme disease", which was not justified on the basis of current recommendations.

Most of patients [START_REF] Jutel | Beyond the Sociology of Diagnosis[END_REF] stated that they believed the diagnosis of LD to be the main explanation for their symptoms, among which 13 considered this diagnosis to be highly certain (Table 3).

Few patients had a "Lyme activist" profile (3), and had consulted a "pro-Lyme" caregiver [START_REF] Sr | Misleading information about Lyme disease[END_REF].

Serology was mainly prescribed at the request of patients [START_REF] Fainzang | A deviant diagnosis? Doctors faced with a patient's diagnostic work[END_REF], despite the fact that their referring doctors [START_REF] Pollak | Une Identité blessée[END_REF] were sceptical about the Lyme hypothesis. (Table 3).

Diagnostic pathway

During their diagnostic trajectory, patients had consulted 3.7 different specialists on average, half of them consulted a psychiatrist. 18 used complementary therapies (naturopathy, homeopathy, kinesiology, etc.) and 14 consulted a pain relief centre.

Regarding the history of differential diagnoses, 23 patients mentioned the diagnosis of fibromyalgia, among which 21 rejecting it because they considered it to be a "psychiatrisation" of their symptoms. In the end, we could conclude majority of participants coordinated the etiological investigation [START_REF] Pollak | Une Identité blessée[END_REF]; for the others, the enquiry was conducted by their GP (4) or by their referring specialist (3) (Table 4).

Discussion

"When the patient makes the diagnosis"

A large majority of patients (with the participation of their entourage) were at the origin of the LD hypothesis. We found a significant role of narratives from other patients on social networks or in the media in the genesis of "diagnostic hypotheses". These stories are particularly valued by patients in situations marked by the absence of a satisfactory diagnostic proposal from doctors. Studies show the general public refers to this type of information source much more often, to the detriment of more "objective" and official sources. [START_REF] Peretti-Watel | Lyme disease: Insight from social sciences[END_REF] This is in line with the results of a qualitative survey performed in Connecticut (USA), which reported that patients with LD placed greater trust in the experiences of close relatives who had contracted Lyme disease than in information disseminated by health professionals and health authorities. [START_REF] Macauda | Long-Term Lyme Disease Antibiotic Therapy Beliefs Among New England Residents[END_REF] Moreover, patients could find online tools to back up their hypotheses.

A majority of patients managed to convince their GP to prescribe a Lyme serology test, illustrating that the medical decision is no longer monopolised by doctors. This reflects the contemporary role of patients claiming the legitimacy of a diagnosis based on their own experience. [START_REF] Aronowitz | The Rise and Fall of the Lyme Disease Vaccines: A Cautionary Tale for Risk Interventions in American Medicine and Public Health[END_REF][START_REF] Pollak | Une Identité blessée[END_REF] The recognition of the patients' diagnostic proactivity has recently been the subject of studies in the field of social sciences of health. [START_REF] Fainzang | A deviant diagnosis? Doctors faced with a patient's diagnostic work[END_REF][START_REF] Jutel | Beyond the Sociology of Diagnosis[END_REF][START_REF] Jutel | Putting a Name to It: Diagnosis in Contemporary Society[END_REF][START_REF] Fainzang | Self-Medication and Society: Mirages of Autonomy[END_REF] Fainzang showed that diagnostic work was more particularly exercised by patients when physicians are unable to elucidate the causes of their disorders than when they are, with patients taking charge of the entire sequence from self-diagnosis to self-prescribing. [START_REF] Fainzang | Self-Medication and Society: Mirages of Autonomy[END_REF]

"A diagnosis set in advance"

For the first time, our study explores the level of conviction associated with LD diagnosis.

A large majority of patients did not report a tick bite, but often mentioned that they may have been bitten without noticing or remembering. A negative serology was not sufficient to completely exclude the diagnosis of LD and, in the case of uncertain serologies, patients often gave more weight to the positivity of the ELISA test than to a negative Western Blot reference test. The "Lyme activist" profile, and/or an encounter with a "pro-Lyme" caregiver concerned a minority of the pathways described in this study and cannot by itself explain this high level of conviction observed among participants.

The attribution of symptoms to a well-identified external (environmental) cause is well described in the literature on LD and more generally on somatoform disorders. [START_REF] Forestier | Cheminement diagnostique et vécu des patients se pensant atteints de « maladie de Lyme chronique[END_REF][START_REF] Rebman | Living in Limbo: Contested Narratives of Patients With Chronic Symptoms Following Lyme Disease[END_REF][START_REF] Duddu | Somatization, somatosensory amplification, attribution styles and illness behaviour: A review[END_REF][START_REF] Chamoux | Modèles explicatifs des patients souffrant de Lyme chronique[END_REF] The infectious origin is often guilt-reducing for patients who often refuse, as in our study, any psychological explanation of their symptoms, what they called "a psychiatrisation of their symptoms". Moreover, the higher level of certainty about the LD hypothesis in patients leading their own diagnostic pathway than in the others suggests that patients had a preestablished etiological scenario and were seeking to put together the different elements of the medical puzzle to demonstrate this. In cognitive psychology, this phenomenon is known as "confirmation bias" and describes our tendency to seek, interpret, promote, and recall information that supports our previous personal beliefs.

Disappointment with science and scientific controversies

The current controversies over the chronic form of LD remind us of the strength of the population's contemporary disenchantment with modern science, as highlighted by Ulrich Beck. [START_REF] Peretti-Watel | Lyme disease: Insight from social sciences[END_REF][START_REF] Beck | Risk Society: Towards a New Modernity[END_REF] especially when modern science generates a multitude of highly specialised, fragmented, temporary, and often contradictory results, especially in the biomedical field. This is especially the case in the French context of the Lyme controversy. In 2018, French scientific societies and the National Academy of Medicine refused to approve the recommendations on LD published by the Haute Autorité de Santé, a French government agency. [START_REF]Borréliose de Lyme et autres maladies vectorielles à tiques : recommandations de bonnes pratiques[END_REF][START_REF] Cnge | Borréliose de Lyme et autres maladies transmises par les tiques : Pourquoi les sociétés scientifiques et professionnelles refusent de cautionner la recommandation de bonne pratique élaborée par la HAS[END_REF] Indeed, French scientific societies (including French College of General Practitioners) did not recognise the new clinical entity called "symptom/polymorphic syndrome persisting after a possible tick bite" arguing that the term was not based on scientific evidence and opened the door to over-diagnosis and inappropriate antibiotic prescriptions. [START_REF] Cnge | Borréliose de Lyme et autres maladies transmises par les tiques : Pourquoi les sociétés scientifiques et professionnelles refusent de cautionner la recommandation de bonne pratique élaborée par la HAS[END_REF][START_REF] Webber | Lyme disease overdiagnosis in a large healthcare system: a population-based, retrospective study[END_REF]

Strengths and limitations

The originality of this study lies in the population studied which consists of patients who have reached a diagnostic dead-end. In this frequent situation regarding LD, we showed that most of the patients were the driving force of the etiological investigations of their symptoms.

Despite medical evidence, almost half of them were strongly convinced that they were suffering from LD.

To our knowledge, this study was also the first to apply a biographical approach to the analysis of the diagnostic trajectories of patients consulting in infectious disease wards for a suspected LD. This approach allowed for the joint analysis of contextualised self-reported data and clinical data from medical records. The interviews, by focusing on the overlap between life, medical and clinical events, highlighted the two dimensions at work in any care pathway: biology and biography. [START_REF] Fassin | Life: A Critical User's Manual[END_REF] Our qualitative approach limits our ability to extrapolate but our sample size (n=30) was the largest to date among the international qualitative research published on the subject. [START_REF] Forestier | Cheminement diagnostique et vécu des patients se pensant atteints de « maladie de Lyme chronique[END_REF][START_REF] Rebman | Living in Limbo: Contested Narratives of Patients With Chronic Symptoms Following Lyme Disease[END_REF][START_REF] Chamoux | Modèles explicatifs des patients souffrant de Lyme chronique[END_REF][START_REF] Drew | A qualitative approach to understanding patients' diagnosis of Lyme disease[END_REF][START_REF] Ali | Experiences of patients identifying with chronic Lyme disease in the healthcare system: a qualitative study[END_REF] Comparison with existing literature A previous qualitative study involving 13 patients in the Savoy region of France reported the same results on the role of the internet and the media in the care pathways of these patients and in triggering their suspicion of chronic LD. [START_REF] Forestier | Cheminement diagnostique et vécu des patients se pensant atteints de « maladie de Lyme chronique[END_REF] However, the method of recruitment through a patient association led to an over-representation of patients who had activist attitudes, who were likely to support conspiracy theories, who were explicitly reported to be in conflict with the medical profession, and who had a relationship with "pro-Lyme caregivers", than our study population. Other international qualitative studies on the subject focused on the experience and impact of the disease in the daily life of patients. [START_REF] Rebman | Living in Limbo: Contested Narratives of Patients With Chronic Symptoms Following Lyme Disease[END_REF][START_REF] Drew | A qualitative approach to understanding patients' diagnosis of Lyme disease[END_REF][START_REF] Ali | Experiences of patients identifying with chronic Lyme disease in the healthcare system: a qualitative study[END_REF]

Implications for practice

The patients in our study were the main actors in their diagnostic enquiry. The role of physicians appears paradoxical. Although patients frequently solicited them (number of referrals), physicians were powerless, or unwilling, to offer structured care pathways. The clinician's ability to listen to the patient's disease history rarely includes consideration of the patient's diagnostic experience. [START_REF] Fainzang | A deviant diagnosis? Doctors faced with a patient's diagnostic work[END_REF] Dissatisfaction with the medical diagnosis is the classic explanation for the use of alternative medicine. [START_REF] García-Campayo | The use of alternative medicines by somatoform disorder patients in Spain[END_REF][START_REF] Lantos | Unorthodox alternative therapies marketed to treat Lyme disease[END_REF] By opposing the doctor as the sole custodian of the medical diagnosis, [START_REF] Jutel | Towards a sociology of diagnosis: Reflections and opportunities[END_REF] to patients reduced to the subjectivity of their symptoms, run the risk of seeing the development of diagnostic dead-end or parallel diagnostic pathways. Finally, patient empowerment in the diagnostic process suggest a failure of mainstream medicine to propose solutions for symptoms that it cannot explain because of lack of evidence when the limits of medical knowledge are attained. In conclusion, clinicians should systematically explore the diagnostic work and etiologic representations of their patients in a patient-centred care approach in order to create the conditions for a therapeutic alliance.

Table 1. Characteristics of the 30 patients

a The 2 other participants were also from the Mediterranean region (Occitania).

Characteristic

No a The clinical profile was defined according to patient's prioritisation of symptoms, in decreasing order of their impact on their quality of life.

b The "chronic pain" category included neuropathic, musculo-articular, diffuse, poorly characterised or headache-type pain. c The "neurological profile" category included patients whose main complaint was vertigo or sensitive motor disorders or cognitive complaints. The term "psychiatrisation" was used when the patient explicitly mentioned the refusal to be told that the cause of their pain was psychological (e.g. "it's not in my head").

Characteristic features No. (%)

Average no. of specialties used in relation to the history of symptoms (excluding infectiology and psychiatry)

d

  The "chronic fatigue syndrome" category included patients with predominant fatigue, often associated with concentration difficulties.

3. 7 [0- 9 ]

 79 Referral to > 5 medical specialists in relation to the history of symptoms (excluding infectiology and psychiatry) 10 (33) Referral to a psychiatrist in relation to the history of symptoms 15 (50) Use of alternative medicine in relation to their symptoms 18 (60) Patients treated in a pain-treatment centre 14 (47) Fibromyalgia: diagnosis evoked by a doctor 23 (77) Refusal of "psychiatrisation a of their symptoms" 21(70)Main diagnostic pathway coordinator (typological approach)

Table 2 . Clinical characteristics of the 30 patients

 2

	. (%)

Table 3 . Genesis of the Lyme Disease diagnostic hypothesis for the 30 participants

 3

	Characteristic features

Table 4 . Characteristic features of the diagnostic pathways for the 30 Participants a

 4

Acknowledgements:

We thank our colleagues, Jeremy Ward (sociologist of our research unit) and Clementine Montagnac (from the Neurology Department) for their collaboration in our study.

Funding

This work was supported by the IHU Méditerranée Infection; the National Research Agency under the programme "Investissements d'avenir" [ANR-10-AHU-03]; the Region Provence

Author Contributions:

Dr Lutaud had full access to all of the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis.

Study concept and design: RL, PV, PW, CE.

Collection of data and data analyses: RL, CE.

Drafting of the manuscript: RL, CE.

Critical revision of the manuscript for important intellectual content: RL, PV, PW, CE.

Alpes Côte d'Azur and European funding "FEDER PIRMI".

The funding sources had no role in the preparation, review, or approval of the manuscript; and decision to submit the manuscript for publication.

The authors declare that there is no conflict of interest regarding the publication of this article.