Keywords:

The goal of this paper is to derive expressions for the pressure forces and moments acting on an elongated body swimming in a quiescent fluid. The body is modelled as an inextensible and unshearable (Kirchhoff) beam, whose cross-sections are elliptic, undergoing prescribed deformations, consisting of yaw and pitch bending. The surrounding fluid is assumed to be inviscid, and irrotational everywhere, except in a thin vortical wake. The Laplace equation and the corresponding Neumann boundary conditions are first written in terms of the body coordinates of a beam treating the body as a fixed surface. They are then simplified according to the slenderness of the body and its kinematics. Because the equations are linear, the velocity potential is sought as a sum of two terms which are linked respectively to the axial movements of the beam and to its lateral movements. The lateral component of the velocity potential is decomposed further into two sub-components, in order to exhibit explicitly the role of the two-dimensional potential flow produced by the lateral motion of the cross-section, and the role played by the curvature effects of the beam on the crosssectional flow. The pressure, which is given by Bernoulli's equation, is integrated along the body surface, and the expressions for the resultant and the moment are derived analytically. Thereafter, the validity of the force and moment obtained analytically is checked by comparisons with Navier-Stokes simulations (using Reynolds-averaged Navier-Stokes equations), and relatively good agreements are observed.

Three-dimensional extension of Lighthill's large-amplitude elongated-body theory of fish locomotion

Introduction

The understanding of the mechanism of propulsion or, more precisely, the determination of the force and the moment experienced by a swimming slender fish has been a challenging topic for several decades. This problem has many fundamental aspects not only for fluid mechanicians but also for other scientific communities such as biologists or bio-inspired roboticians. As a result, the literature in this Fabien Candelier 1 , Frédéric Boyer 2 , Alban Leroyer 3 area is extensive and we shall not attempt a detailed review here. The interested reader can find numerous theoretical, experimental or numerical reviews that analyse the hydrodynamics of swimming fish such as [START_REF] Sfakiotakis | Review of fish swimming modes for aquatic locomotion[END_REF], [START_REF] Triantafyllou | Hydrodynamics of fishlike swimming[END_REF], [START_REF] Sparenberg | Survey of the mathematical theory of fish locomotion[END_REF], [START_REF] Colgate | Mechanics and control of swimming: a review[END_REF] or [START_REF] Blake | Fish functional design and swimming performance[END_REF].

At first sight, the modelling of undulatory fish motion looks rather complex, if not insoluble, due to major difficulties that must be overcome in order to develop a comprehensive theory. This probably explains why many researchers working on the topic have chosen to focus their attention on the wake generated by the swimming fish and, in particular, on the shed vorticity. This indeed is probably a promising way to provide, one day, a theory that combines both resistive (viscous) and reactive (pressure) effects. In particular, numerous studies tend to demonstrate that the high performance of swimming fish, in terms of efficiency and manoeuvrability, lies in their ability to control, through their body shape, the large-scale vortices shed in their wake [START_REF] Wolfgang | Hydrodynamics of flexible-body swimming motions[END_REF][START_REF] Triantafyllou | Hydrodynamics of fishlike swimming[END_REF][START_REF] Zhu | Three-dimensional flow structures and vorticity control in fish-like swimming[END_REF]. Owing to the continuous improvement in computational fluid dynamics methods, these phenomena have recently been extensively investigated [START_REF] Kern | Simulation of optimized anguilliform swimming[END_REF][START_REF] Eldredge | Dynamically coupled fluid-body interactions in vorticity-based numerical simulations[END_REF]) which enables more accurate and complex simulations. As a necessary complement to these numerical approaches, experimental works using the particleimage velocimetry (PIV) method [START_REF] Muller | Hydrodynamics of unsteady fish swimming and the effects of body size: comparing the flow fields of fish larvae and adults[END_REF] have emphasized the complex interactions of bound and tail vortices in the wake of swimming fish.

Nevertheless, when fast calculations are needed, such as when using the model for the on-line control of fish-like robots, full numerical approaches are still not well adapted as they are time-consuming. Therefore, finding accurate analytical expressions for the force and the moment acting on a moving and deformable body is still a challenging task for fluid mechanicians (see for instance the recent contributions of [START_REF] Kanso | Locomotion of articulated bodies in a perfect fluid[END_REF][START_REF] Kanso | Swimming due to transverse shape deformations[END_REF][START_REF] Munnier | Locomotion of deformable bodies in an ideal fluid: Newtonian versus Lagrangian formalisms[END_REF]. Historically, seminal contributions in the range of high Reynolds numbers were made by [START_REF] Wu | Swimming of the waving plate[END_REF] and [START_REF] Lighthill | Note on the swimming of slender fish[END_REF]. The first, due to [START_REF] Wu | Swimming of the waving plate[END_REF], is based on the undulating infinite height plate, while the second, due to [START_REF] Lighthill | Note on the swimming of slender fish[END_REF], is an extension of the classical slender-body theory (see for instance Batchelor 1967, § 6.9), called the elongated-body theory (EBT). In both cases, the fluid is inviscid and the modelling takes advantage of the characteristics of fish geometry in order to approximate the three-dimensional (3D) flow around the fish body by a stratification (horizontal in the first case and vertical in the second) of planar potential flows. In particular, in the EBT, it is considered that for each transverse slice of a fish, the lateral flow corresponds to the potential flow produced by the movement of an infinite cylinder of the same cross-section.

Since then, many improvements in the EBT have been proposed by several authors. For instance, Newman & Wu (1973) and Newman (1973) took into account the effects of the body fin on the pressure resultant while [START_REF] Lighthill | Aquatic animal propulsion of high hydromechanical efficiency[END_REF] proposed an extension of the EBT, called the large-amplitude elongated-body theory (hereinafter LAEBT) in order to account for large amplitude motions.

This theory remains one of the basic results of biofluiddynamics, valuable for deriving estimates of the force acting on a swimming body (Weihs 1972, 1973[START_REF] Lighthill | Mathematical Biofluiddynamics[END_REF] or fast simulation and control in robotics [START_REF] Boyer | Macro-continuous computed torque algorithm for a three-dimensional eel-like robot[END_REF]).

However, in recent years, this theory has become a topic for discussion among fluid mechanicians. In particular, numerical studies have shown that in each transverse slice of a fish, the flow differs significantly from the potential flow produced by the movement of an infinite cylinder of the same cross-section [START_REF] Wolfgang | Hydrodynamics of flexible-body swimming motions[END_REF][START_REF] Triantafyllou | Hydrodynamics of fishlike swimming[END_REF]. Therefore, the objectives of this paper are twofold: to

2 Configuration at T = 0 O Configuration at T o E 1 E 2 E 3 e 1 e 2 e 3 t 1 t 2 t 3 v 0 X ω 0 G r X = 0 (a)( b) X = 0 X = l X = l K Y K Z π
Fi gure 1. Sketch of the transformation.

provide a better understanding of the physical reality of the flow underlying the LAEBT and to generalize this theory to 3D swimming, including a possible rotation (roll) of the body around its vertebral axis.

To do this, we propose, following [START_REF] Lighthill | Note on the swimming of slender fish[END_REF], to determine analytically (at leading order) the velocity potential of the fluid surrounding the body, and then, to calculate directly the resultant (and here the moment) of the pressure forces exerted by the flow at its surface. As in the LAEBT, the elongated body considered here is made up of elliptic transverse sections whose size may vary along the body, which has no appendices. Its 3D movements are provided by prescribing an internal pitch bending in addition to the yaw motion required by planar swimming. Note that to carry out this calculation, it is essential to change the coordinates so that the body becomes a fixed surface. Usually, when we are concerned with a deformable body moving in an inviscid fluid, the classical approach is to measure the deformation fields from a mobile (floating) frame that follows the rigid motion of the body, using Kirchhoff potentials method (Galper & Miloh 1995[START_REF] Wolfgang | Hydrodynamics of flexible-body swimming motions[END_REF]. However, in the present investigation, we have chosen to make use of a different transformation, written in an inertial frame, which relies on a nonlinear beam theory introduced by [START_REF] Simo | On the dynamics in space of rods undergoing large motions: a geometrically exact approach[END_REF], and said to be geometrically exact. This transformation is presented in § 2, and after that it is applied to the equations governing the flow around the body. Section 3 is devoted to the explicit calculation of the pressure resultant and moment exerted by the fluid on the body and the results are compared with numerical simulations in § 4. Finally, in § 5, the results are discussed.

Transformation of coordinates

2.1. The body model The transformation of coordinates used in this investigation is inspired by those proposed by [START_REF] Reissner | On a one-dimensional large displacement finite-strain theory[END_REF] and [START_REF] Simo | On the dynamics in space of rods undergoing large motions: a geometrically exact approach[END_REF]. Let R 0 =(o, e 1 , e 2 , e 3 )b e the laboratory reference frame. Without losing generality, it may be assumed that, at the initial time, the body experiences no deformations, as shown in figure 1(a), the idea being to introduce a transformation that links the current position of the body to its initial position. Note that in the following, vectors which refer to the initial position will be written upper case, while lower case is used for vectors which refer to the current position. Owing to the elongated geometry of the slender body, it can be considered as a Kirchhoff (Cosserat) beam B,i . e .ao n e -d i m e n s i o n a l continuum of rigid cross-sections (here elliptic) along a curve (here the backbone). Let F b =(O, E 1 , E 2 , E 3 ) be a Cartesian (body) frame attached to the reference (initial) configuration, in which the beam is assumed to be straight and aligned along (O, E 1 ). As we can see in figure 1(a), X E 1 is the position of any cross-section's centre G(X, T = 0), so that any rigid section can be labelled only by its axial coordinate X and therefore denoted by S X .

Thereafter, when the body is moving, a continuous set of moving orthonormal frames F m (X, T )=(G, t 1 , t 2 , t 3 ) are attached to each cross-section, such that t 1 (X, T ) is the outward unit normal of the section as represented in figure 1(b). According to this parametrization, any material point, located at x, belonging to the body during an admissible motion is related to its initial position by the following one-to-one mapping:

x = r(X, T )+x ⊥ , (2.1) where x ⊥ = R(X, T ) • X ⊥ .
Here, r is the position vector of G(X, t) in the laboratory frame (located initially at

X E 1 ), R(X, t) is a rotation operator (i.e. R T = R -1 )mapping (E 1 , E 2 , E 3 )ont o(t 1 , t 2 , t 3 ), while if X = X E 1 + Y E 1 + Z E 3
is the position vector of a material point in the reference configuration, then X ⊥ = X -X E 1 and x ⊥ correspond, respectively, to the position of the same point relative to the backbone in the reference and the deformed configuration. According to (2.1), the velocity field of any material point belonging to the body is entirely specified by two fields that depend only on (X, T ),

v = ∂ r ∂T + ω × x ⊥ , (2.2)
the first field corresponding to the linear velocities of the mass centres of the rigid cross-sections (initially located at X), (2.3) and the second to their angular velocities,

∂ r ∂T = V X t 1 + V Y t 2 + V Z t 3 ,
ω = Ω X t 1 + Ω Y t 2 + Ω Z t 3 .
(2.4)

Note that the angular velocity field is linked to the tensor R by the following relation:

ω × y = ∂R ∂T • R T • y , ω • y, ∀ y, (2.5)
where the symbol , stands for 'equal by definition'.

Similarly, in this approach, the strain state of the body is also totally specified by two vector fields, also depending only on (X, T ), which correspond to the tangent field of the material axis of the body B: .6) and to the torsion and curvature vector field of the sections:

∂ r ∂X , ( 2
k = K X t 1 + K Y t 2 + K Z t 3 , (2.7)
which is also linked to R by the relation

k × y = ∂R ∂X • R T • y , k • y, ∀ y.
(2.8)

The frame field (G, t 1 , t 2 , t 3 ) defined by the transformation proposed by [START_REF] Simo | On the dynamics in space of rods undergoing large motions: a geometrically exact approach[END_REF] is materially attached to the rigid cross-sections. In particular, the frame (G, t 1 , t 2 , t 3 ) differs significantly from the more usual Frenet frame, which is defined only by the position of the centreline of the body since, in any circumstance, the local frames (G, t 1 , t 2 , t 3 ) are always principal inertial frames of the cross-sections. Furthermore, the geometrically exact approach introduces a torsion field which is not a geometrical quantity defined by the centreline only, but an actual torsion field defined as a measure of the internal torsion strain state. Consequently, this material torsion field (K X ) is never singular, even when the centreline tends to be a straight line, in contrast to the geometrical torsion field defined by the Frenet frame. Note that, physically, the other components of k,t h a ti sK Y and K Z , can be interpreted as the inverse of the radius of curvature of the body in the planes (t 1 , t 3 )a n d( t 1 , t 2 ), respectively. If we now return to the swimming of a natural fish, the variables K Y dX and K Z dX are merely the two infinitesimal angles of rotation (standing for the pitch and yaw rotations along the backbone) of an infinitesimal Cardan joint, which links any pair of contiguous cross-sections together.

In what follows, K Y and K Z are considered as time-dependent control inputs, which entirely parametrize the body shapes. Since the elongated body of a vertebrate animal admits no relative rotation of two contiguous vertebrae about the vertebral axis, it is natural to impose that the torsion strain field satisfies K X =0.

(2.9)

Strictly speaking, if we denote by K the order of approximation of K Y and K Z and by l the length of the body, the beam model presented here is valid only within the limit Kl . O(1), (2.10) which nevertheless allows large deformations to be considered, as those experienced by a swimming (manoeuvring) slender fish. In addition, it will be assumed that the body is inextensible and sufficiently slender to be considered as unshearable. Mathematically, these assumptions read (2.11) According to (2.9) and (2.11), the body can be considered now as an untwistable and inextensible Kirchhoff beam, and the corresponding gradient transformation tensor reads F(X)=R • U, (2.12) where we recognize the polar decomposition which is valuable for any non-singular transformation with U being the tensor of pure deformation, defined as .13) In this context, the metric tensor of the transformation is given by .14) and in the following, we will also be often concerned with the term |G| (where | • | stands for the matrix determinant) which corresponds physically to a measure of local volume dilatations and contractions generated by the transformation. Also, during the investigation, this analysis will be in a domain restricted to the vicinity of the body, where K Y Z ∼ K Z Y ≪ 1, so that we can write

∂ r ∂X = t 1 .
U = ⎛ ⎝ 1+K Y Z -K Z Y 00 01 0 00 1 ⎞ ⎠ . ( 2
G = F T • F = U T • U, ( 2
|G| =1+K Y Z -K Z Y, (2.15)
since there is no ambiguity about the sign of this term. So far, we have described a transformation that links any current configuration of the body to its initial position.

In what follows, it is shown that this transformation also provides relevant information about the kinematics of an inextensible and unshearable slender body.

2.2. Kinematics of an inextensible and linear beam-like body When a slender body is moving, the orders of the velocities and the angular velocities are linked together, according to the assumptions of inextensibility and unshearability. Indeed, if we derive (2.11) with respect to T and also use the fact that derivatives with respect to X and to T commute, it is simple to show that

∂ ∂X ∂ r ∂T • t i = ∂ t 1 ∂T • t i + ∂ r ∂T • ∂ t i ∂X for i = {1, 2, 3}.
(2.16)

In addition, since we also have

∂ t i ∂T = ω × t i and ∂ t i ∂X = k × t i , (2.17)
then we obtain the following three relations:

∂V X ∂X = K Z V Y -K Y V Z , (2.18
)

∂V Y ∂X = Ω Z -V X K Z , (2.19
)

∂V Z ∂X = -Ω Y + V X K Y .
(2.20)

Similar relations can also be obtained for the angular velocities. Indeed, if we use ∂(R T • R)/∂T = 0, together with the definition (2.5) of ω, then it becomes possible to show that ∂R T /∂T = -R T • ω, and subsequently, from the definition of k (see (2.8)), it follows that

∂ k ∂T = ω • k -k • ω + ∂ ω ∂X .
(2.21)

By writing (2.21) in terms of components and using (2.9), we obtain

∂Ω X ∂X = K Z Ω Y -K Y Ω Z , (2.22
) (2.23)

∂Ω Y ∂X = ∂K Y ∂T -K Z Ω X ,
∂Ω Z ∂X = ∂K Z ∂T + K Y Ω X .
(2.24)

As we can see, (2.18)-(2.20) and (2.22)-(2.24) provide us with ideas about the relative order of the derivatives with respect to X of the components of the linear and angular velocities of an inextensible and unshearable slender body. In particular, if we denote by V the order of the different components of the translational velocity

(i.e. V ∼ V X ∼ V Y ∼ V Z )a n db yΩ ⊥ the order of Ω Y and Ω Z , then, according to (2.19),
(2.20) and (2.10), we see that (2.25) and α stands for the typical length scale of the cross-sections. This result will be used in what follows.

αΩ ⊥ V . O(ε), where ε = α l ≪ 1,
2.3. Velocity potential equations in terms of the body coordinates In this problem, the surrounding fluid is assumed to be inviscid, and the flow to be irrotational everywhere except in a thin vortical wake which occurs downstream of the caudal fin and takes the form of a vortex sheet (at least just behind the trailing edge). Note that in theory, it should be expected that this vortex sheet produces an additional flow which can be determined by the Biot-Savart law. In particular, such effects are taken into account when the inviscid fluid equations are solved with the boundary element method (with the Kutta condition) (see [START_REF] Wolfgang | Hydrodynamics of flexible-body swimming motions[END_REF]). In the present investigation however, and in common with the original theory of Lighthill (EBT), the pressure perturbation due to the wake-induced flow is expected to be negligible, and as a consequence, once it is shed in the flow, the vorticity is later ignored. In this condition, it is possible to limit the analytical investigation within a domain located upstream of a flat surface π perpendicular to the caudal fin, without specifying particular boundary conditions for the potential in this plane. Thus, if φ denotes the velocity potential in this domain, its governing equations read, in a reference frame in which the fluid far from the body is at rest, as

φ( y,t)=0, (2.26) ∇φ • n = v • n, on ∂B, (2.27) ∇φ → 0, | y -x|→∞.
(2.28) Then, if the body transformation of coordinates presented in the previous section is applied to (2.26) and (2.27), we obtain

∂ ∂X 1 |G| ∂Φ ∂X + ∂ ∂Y |G| ∂Φ ∂Y + ∂ ∂Z |G| ∂Φ ∂Z = 0 (2.29) and 1 |G| ∂Φ ∂X N X + ∂Φ ∂Y N Y + ∂Φ ∂Z N Z = 1 |G| (V X + Ω Y Z -Ω Z Y )N X + (V Y -Ω X Z)N Y + (V Z + Ω X Y )N Z .
(2.30)

(Note that we have not written the transformation of (2.28) since it is not used explicitly in the following.)

In order to analyse the order of the different terms occurring in (2.29) and (2.30), we denote by N the order of N X and by N ⊥ the order of N Y or N Z . Owing to the slenderness of the body, we first have necessarily

N ∼ εN ⊥ .
(2.31) Furthermore, since the typical length scale for Y and Z is given by α (at least in the vicinity of the body) and because

Kα ∼ Klε, (2.32)
we have |G|∼1+Klε.

(2.33) In contrast with Y and Z, the typical length of the axial coordinates X is given by l. As a result, in such a problem, variations of the variables with respect to X are weak compared with those within the cross-sectional plane. (Roughly, it may be considered that ∂/∂X ∼ 1/l, whereas ∂/∂Y ∼ ∂/∂Z ∼ 1/α). If we exploit this, as well as the geometrical relations given by ( 2.31) and(2.33) andthe kinematic relation given by (2.25), then (2.29) and(2.30) can be simplified as follows: .34)

∂ 2 Φ ∂Y 2 + ∂ 2 Φ ∂Z 2 ≃ K Z ∂Φ ∂Y -K Y ∂Φ ∂Z , ( 2
∂Φ ∂Y N Y + ∂Φ ∂Z N Z ≃ V X N X + (V Y -Ω X Z)N Y + (V Z + Ω X Y )N Z , (2.35)
where the terms that scale as o(ε 2 ) compared with leading order terms have been neglected. (Note that, for simplicity, 'small' terms scale as o(ε 2 ) compared with leading order.)

Decomposition of the velocity potential

If we now use the linearity of (2.34) and (2.35), the solution of these equations may be sought in the form

Φ = Φ + Φ ⊥ , (2.36
) where Φ and Φ ⊥ are solutions respectively of

∂ 2 Φ ∂Y 2 + ∂ 2 Φ ∂Z 2 ≃ 0, (2.37)
∂Φ ∂Y N Y + ∂Φ ∂Z N Z ≃ V X N X , (2.38) and ∂ 2 Φ ⊥ ∂Y 2 + ∂ 2 Φ ⊥ ∂Z 2 ≃ K Z ∂Φ ⊥ ∂Y -K Y ∂Φ ⊥ ∂Z , (2.39
)

∂Φ ⊥ ∂Y N Y + ∂Φ ⊥ ∂Z N Z ≃ (V Y -Ω X Z)N Y +(V Z + Ω X Y )N Z .
(2.40)

Note that the right-hand-side terms of (2.37) can be neglected, since due to N ∼ εN ⊥ , Φ scales like εαV . In this context, we also observe that the potential Φ may be interpreted as a pseudo slender-body-like potential, where the velocity V X varies along X (in contrast with what happens when we deal with a classical rigid slender body), whereas Φ ⊥ accounts for the lateral motion and axial (roll) rotation of the body when it is swimming.

With now the lateral component Φ ⊥ , a serious issue occurs, since the solution of (2.39) is not easily achieved because of the non-zero right-hand side. Fortunately, this difficulty can be circumvented if we use a perturbation method to determine Φ ⊥ ,a t least at the order we are interested in. In the present investigation, according to (2.32), the solution of (2.39) and (2.40) can indeed be sought in the form

Φ ⊥ ≃ Φ 0 ⊥ + Φ 1 ⊥ , (2.41)
where .43) with the boundary conditions

Φ 1 ⊥ Φ 0 ⊥ ∼ O(αK) (2.42) and Φ 0 ⊥ is the solution of (2.39)-(2.40), when K Y = K Z =0, i. e. Φ 0 ⊥ is the solution of ∂ 2 Φ 0 ⊥ ∂Y 2 + ∂ 2 Φ 0 ⊥ ∂Z 2 =0, ( 2
∂Φ 0 ⊥ ∂Y N Y + ∂Φ 0 ⊥ ∂Z N Z =(V Y -Ω X Z)N Y +(V Z + Ω X Y )N Z .
(2.44)

Therefore, if we inject (2.41) into (2.39) and (2.40) and if the small terms are once again neglected, then the equations governing .46) Note that the explicit analytical expressions of Φ 0 ⊥ and Φ 1 ⊥ are needed in the following to determine the pressure force and the moment acting on the body when it is swimming. Their derivations are detailed in the Appendix. Instead, in what follows, we illustrate the flow produced by each part of the velocity potential in typical situations encountered in planar swimming.

Φ 1 ⊥ read ∂ 2 Φ 1 ⊥ ∂Y 2 + ∂ 2 Φ 1 ⊥ ∂Z 2 = K Z ∂Φ 0 ⊥ ∂Y -K Y ∂Φ 0 ⊥ ∂Z (2.45) and ∂Φ 1 ⊥ ∂Y N Y + ∂Φ 1 ⊥ ∂Z N Z =0. ( 2

Cross-sectional flow features

During the planar swimming of a slender fish, the propulsion is ensured by a wave of deformation which travels the body, and whose amplitude increases gradually from head to tail. Near the head, the curvature is often negligible and the lateral velocities of the cross-sections are generally small compared with the axial velocity. As a consequence, the lateral flow that is observed in such a cross-section is almost only generated by Φ . Figure 2 shows the flow produced by a pure axial motion of the slender body, in a normalized case where the axial velocity has been set equal to -1, and where the dimensions of the elliptic cross-section of the body, whose major and minor semi-axes denoted respectively α and β below, have been set equal to 1 and 0.5, and where ∂α/∂X = ∂β/∂X =0.1. When the body moves axially, it pushes the surrounding fluid laterally, as illustrated by the directions of the streamlines. Note that the pattern of the streamlines can be qualitatively compared with reasonably good agreement with the numerical results presented by [START_REF] Triantafyllou | Hydrodynamics of fishlike swimming[END_REF].

In contrast, near the tail, the local curvature of the body may become significant. If it is assumed that the cross-sectional area is almost constant with respect to X (i.e. N X = 0), then the flow that takes place in a cross-section is, at this time, only produced by Φ ⊥ . Figure 3 shows such a flow, again in a normalized case, where the lateral velocity V Z has been set equal to 1 and the normalized curvature reads K Y =0.25. As shown previously, this cross-sectional flow results from the sum of a two-dimensional (2D) potential flow (Φ 0 ⊥ ) around an infinite cylinder, which is displayed in figure 4, and a perturbation flow produced by Φ 1 ⊥ , which is due to curvature effects of the body on the flow, and is displayed in figure 5.

In order to better understand the physical origin of this perturbation flow, it is worth recalling that by definition, the flow produced by the potential Φ 0 ⊥ is in a direction perpendicular to the backbone. Let us therefore consider an infinitesimal volume of control, surrounding the body surface and lying between the abscissa X and X +dX and whose lateral sections remain perpendicular to the backbone during the deformation of the body, as illustrated in figure 6. In this example, V Z > 0 and, since K Y > 0, the surface of the volume corresponding to the entering fluid is smaller than the surface corresponding to the outgoing fluid. Hence, according to the continuity equation, the outgoing fluid velocity must be smaller than the entering fluid velocity. Because the velocity produced by Φ 0 ⊥ is symmetrical, we see therefore that an additional velocity (which is encoded in Φ 1 ⊥ ) must take place within

(V X = -1, V Z =0, K Y =0).
1 ⊥ (N X =0, V Z =1, K Y =0.25).
the cross-section, in order to satisfy the continuity equation. Also, we see that the cross-sectional flow underlying the LAEBT is actually more rich than the 2D potential flow produced by the movement of an infinite cylinder of the same cross-section involved in the EBT. Note that the 2D potential flow and the perturbation flow are both symmetrical, but the former is symmetrical in relation to the axis (G, t 3 ) whereas the latter is only symmetrical in relation to the axis (G, t 2 ), and as a result, their sum is no longer symmetrical. Again, the typical patterns of the streamlines can be qualitatively compared, with relatively good agreement, with the numerical results presented by [START_REF] Triantafyllou | Hydrodynamics of fishlike swimming[END_REF].

t 1 t 3 K Y v dX
So far, we have discussed the flow generated by the velocity potential Φ.I nw h a t follows, it is shown that this potential is also consistent with the existence of a vortical wake occurring downstream of the caudal fin of the body.

Vortex shedding

First of all, let us recall that to formulate the LAEBT, [START_REF] Lighthill | Mathematical Biofluiddynamics[END_REF] considered the rate of change of momentum within a volume of fluid enclosing the fish, whose boundary at each instant includes a flat surface π (see figure 1). In this approach, one of the main difficulties he had to overcome was to determine the resultant of the pressures generated by the fluid motion within this caudal plane. As noted by [START_REF] Lighthill | Aquatic animal propulsion of high hydromechanical efficiency[END_REF], a possible interpretation of this resultant rests on the existence of a vortical wake which occurs downstream of the caudal fin. This idea was later investigated by [START_REF] Childress | Mechanics of Swimming and Flying[END_REF], who recovered the pressure resultant within the caudal plane by considering the energy balance in a volume of control downstream of the caudal fin that involved the wake energy. Childress (1981, p. 104) also provided an analytical approach which gets around the Kutta condition by using Kelvin's circulation theorem to estimate the vorticity shed at the trailing edge. Following his statements, if we denote as δΦ(Y ), the difference of the potential on either side of the trailing edge, and at the same height of the cross-section (i.e. Y ), then when the body moves forward, the two components of the shed vorticity read .47) where V X (l) corresponds to the axial velocity of the caudal fin (see figure 7). From a practical point of view, a sharp trailing edge is nothing but an elliptic crosssection whose minor semi-axis (i.e. β) tends to zero, and therefore, in terms of elliptic coordinates (see the Appendix), the difference of the potential simply reads

γ Y = - 1 V X (l) ∂δΦ ∂T and γ X = - ∂δΦ ∂Y , ( 2
δΦ = Φ(u X =0,v ) -Φ(u X =0, 2π -v).
(2.48)

According to the analytical solutions of Φ (see Appendix), together with when u X → 0, the Cartesian coordinate Y is linked to the angular elliptic coordinate by the relation 1. 98 1 1.9 81 1.981

1 .9 8 1 1 .9 8 1 t 2 (l)

t 1 (l) V X (l ) X = l
Trailing edge (β → 0) Y = α cos(v), and the difference of the velocity potential involved in this investigation can be written in the form

δΦ = √ α 2 -Y 2 -2V Z + α 2 Ω X K Z 6 -Ω X + V Z K Z 2 Y - Ω X K Z 6 Y 2 .
(2.49)

Strictly speaking, this analytical approach provides only a model of the shed vorticity, and the actual vortical wake is probably extremely complex. Nevertheless, in order to further illustrate the flow underlying the LAEBT theory, let us assume that immediately downstream of the trailing edge, the wake is defined by a material surface of frozen vortex lines, forming a vortical sheet. Then, by extending the curvilinear body coordinates (X, Y ) into the wake, the vorticity located at a distance X from the trailing edge is linked to the time T taken by the caudal fin to travel this distance by the following relation:

X(T ) -l = T 0 -V X (l)dτ.
(2.50)

According to (2.49), if it is also assumed that the axial velocity V X (l) is constant and the lateral velocity V Z (l) of the trailing edge is known as a time-dependent function, it is possible to reconstruct the vorticity field in the wake. Figure 7 illustrates such a field in a normalized case, corresponding to planar swimming and where we have set arbitrarily V X (l)=-1a n dV Z (l)= sin(T ). Note that, in reality, once shed into the wake, such a sheet of vorticity is actually highly unstable and splits into vortex rings (see for instance the numerical simulations presented by [START_REF] Kern | Simulation of optimized anguilliform swimming[END_REF].

Force and moment acting on the body

In terms of the body coordinates X, Y and Z, it can be shown that the force and the moment acting on the beam are given by the following integrals:

f h = ∂B -P |G|F -T • N dS and m h = l 0 r × ∂ f h ∂X dX + c h , (3.1)
where c h corresponds to the integration of the density of the hydrodynamic couple applied to the beam per unit of length

c h = ∂B -P X ⊥ × |G|F -T • N dS, (3.2)
and where the pressure reads

P (X,T) -P ∞ = -ρ ∂Φ ∂T -U -1 • R T • ∂ r ∂T + ω × x ⊥ •∇ X Φ - ρ 2 (∇ X Φ) • (G -1 •∇ X Φ). (3.3)
In (3.3), if small terms are neglected, the pressure can be reorganized as follows:

P -P ∞ ≃ ρ - ∂Φ 0 ⊥ ∂T +(V Y -Ω X Z) ∂Φ 0 ⊥ ∂Y +(V Z + Ω X Y ) ∂Φ 0 ⊥ ∂Z - 1 2 ∂Φ 0 ⊥ ∂Y 2 - 1 2 ∂Φ 0 ⊥ ∂Z 2 + ρ - ∂Φ 0 ∂T + V X ∂Φ 0 ⊥ ∂X + V Y -Ω X Z - ∂Φ 0 ⊥ ∂Y ∂Φ ∂Y + V Z + Ω X Y - ∂Φ 0 ⊥ ∂Z ∂Φ ∂Z + ρ - ∂Φ 1 ⊥ ∂T + V Y -Ω X Z - ∂Φ 0 ⊥ ∂Y ∂Φ 1 ⊥ ∂Y + V Z + Ω X Y - ∂Φ 0 ⊥ ∂Z ∂Φ 1 ⊥ ∂Z , (3.4)
which we write as P -P ∞ = P 0 + P 1 + P 2 . Note that, physically, P 0 corresponds to the unsteady pressure distribution that would act on a moving infinite cylinder of the same cross-section, and clearly, this part is the leading-order part of the pressure acting on the body. Here P 1 is the pressure correction brought by the coupling between the potentials Φ (linked to the axial motion of the body) and Φ 0 ⊥ , whereas P 2 is another correction which corresponds to the coupling between the potentials Φ 1 ⊥ (encoding the curvature effects of the body on the fluid) and Φ 0 ⊥ . In practice, it can be shown that

P 1 P 0 = O(ε)a n d P 2 P 0 = O(Klε).
(3.5)

The pressure resultant

We now return to the first integral given in (3.1). By using (2.12) and (2.13) together with

t i = R • E i , it is possible to show that |G|F -T • N = N X t 1 + |G|N Y t 2 + |G|N Z t 3 (3.6)
and, since N Y dS =dZ dX and N Z dS = -dY dX, the force acting on the body is given by

f h = ∂B -P t 1 N X dS + l 0 C X -P |G|t 2 dZ - C X -P |G|t 3 dY dX, (3.7)
where C X stands for the contour of the cross-section located at X. In the present investigation, if we remember that N ∼ εN ⊥ , and according to (3.5), (3.7) can be simplified by neglecting the integrals involving the terms P 1 N X and P 2 N X , since they are not consistent with our level of approximation. Thus, if we consider only the

t 3 t 3 t 1 t 2 t 1 ds dl X X + dX Frontal surface N C X (Contour of S X ) C X+dX (Contour of S X+dX ) ǫ Figure 8. Integration surface.
terms that bring a non-negligible contribution, we obtain

f h = ∂B -P 0 t 1 N X dS + l 0 C X -P 0 t 2 dZ - C X -P 0 t 3 dY dX + l 0 C X -P 1 t 2 dZ - C X -P 1 t 3 dY dX + l 0 C X -P 2 t 2 dZ - C X -P 2 t 3 dY dX + l 0 C X -P 0 (K Y Z -K Z Y )dZ t 2 - C X -P 0 (K Y Z -K Z Y )dY t 3 dX, (3.8)
which we write as

f h = f h1 + f h2 + f h3 + f h4 + f h5 .
Actually, the first integral of (3.8) can be rewritten in such a manner that global force can also be written in the form

f h = l 0 5 i=1 ∂ f hi ∂X dX, (3.9)
where the integrand corresponds to the local cross-sectional loading acting on any body surface lying between the sections located at X and X +dX (see figure 8).

Physically, it may be interesting to emphasize the following.

(i) Here ∂ f h1 /∂X is the only axial part of the force density. This term occurs only if the section varies with respect to X. Indeed, in this case, the axial component N X is not null, and since it is coupled with P 0 (leading order), we obtain a small term of first order that must be taken into account.

(ii) Note that ∂ f h2 /∂X is the lateral force density corresponding to the unsteady motion of an infinite cylinder of the same cross-section.

(iii) Also, ∂ f h3 /∂X is the force density produced by the coupling between Φ and Φ 0 ⊥ , that is between the axial motion of a slender-like body whose cross-section varies along X and the lateral motion of the body. Note that such a coupling results from the nonlinearity of the Bernoulli equation.

(iv) In addition, ∂ f h4 /∂X is the force density produced by the coupling between the lateral 2D potential Φ 0 ⊥ and the perturbation potential Φ 1 ⊥ . This term is a consequence of the curvature effects on the surrounding flow.

(v) Note also that ∂ f h5 /∂X is the force density produced by the stretching of the body surface which is exposed to the pressure P 0 . This term is also a consequence of the curvature effects, but this time on the body.

For the sake of clarity, the detailed derivations of these five components are not given here, and the resulting expression will be written directly. For an elliptic crosssection (whose major and minor semi-axes are α and β, respectively) the added-mass coefficients read [START_REF] Milne-Thomson | Theoretical Hydrodynamics[END_REF])

M Y = ρπβ 2 ,M Z = ρπα 2 ,M X = ρπ (α 2 -β 2 ) 2 8 , (3.10)
respectively, and these coefficients are also involved in the density of linear fluid momentum expression (3.11) where P Y = M Y V Y and P Z = M Z V Z . According to these notations, the five force densities described previously read

p f = P Y t 2 + P Z t 3 ,
∂ f h1 ∂X = - 1 2 dM Y dX V 2 Y + dM Y dX V 2 Z + dM X dX Ω 2 X t 1 , (3.12) ∂ f h2 ∂X = - ∂P Y ∂T t 2 + ∂P Z ∂T t 3 -Ω X t 1 × p f , (3.13) ∂ f h3 ∂X = V X ∂P Y ∂X t 2 + ∂P Z ∂X t 3 , (3.14) ∂ f h4 ∂X ≃ + 1 2 βM X α + β ∂K Y Ω X t 2 ∂T + 1 2 αM X α + β ∂K Z Ω X t 3 ∂T + T f ∂ t 1 ∂X (3.15)
and .16)

∂ f h5 ∂X = - β 2 M X α 2 -β 2 ∂K Y Ω X t 2 ∂T + α 2 M X α 2 -β 2 ∂K Z Ω X t 3 ∂T + ∂V X ∂X p f -2T f ∂ t 1 ∂X . ( 3
We now focus on the combinations that are possible between these five equations, in order to simplify the final expression of the force. First, by summing f h2 and f h3 , we obtain

f h2 + f h3 = l 0 - ∂P Y ∂T -V X ∂P Y ∂X t 2 - ∂P Z ∂T -V X ∂P Z ∂X t 3 -Ω X t 1 × p f dX.
(3.17) Moreover, by using (2.17), ( 2 .19) and ( 2.20), it is possible to show that (3.19) and therefore, it follows that .20) Now, if we sum f h1 and f h2 + f h3 and note that .21) since the term M X Ω X ∂Ω X /∂X is negligible, as we can see by (2.22), then we obtain (3.22) This partial result actually corresponds to the force that would be obtained if we had considered only the 2D potential flow Φ 0 ⊥ (and its coupling with Φ ), which results from the movement of an infinite cylinder whose cross-section is S X , without taking into account the effect of the curvature on the cross-sectional flow (given by Φ 1 ⊥ ) nor the deformation of the body surface. In other words, (3.22) corresponds to a 3D extension of the EBT proposed by [START_REF] Lighthill | Note on the swimming of slender fish[END_REF]. This result therefore provides us with the pressure force acting on the body when the deformations of the swimming body are small, that is, when Kl ≪ O(1).

∂ t 2 ∂T -V X ∂ t 2 ∂X = - ∂V Y ∂X t 1 + Ω X t 1 × t 2 , (3.18) and ∂ t 3 ∂T -V X ∂ t 3 ∂X = - ∂V Z ∂X t 1 + Ω X t 1 × t 3 ,
f h2 + f h3 = l 0 - ∂ p f ∂T -V X ∂ p f ∂X -P Y ∂V Y ∂X + P Z ∂V Z ∂X t 1 dX. ( 3
-P Y ∂V Y ∂X + P Z ∂V Z ∂X - 1 2 dM Y dX V 2 Y + dM Z dX V 2 Z + dM X dX Ω 2 X ≃- ∂T f ∂X , ( 3
f h1 + f h2 + f h3 ≃ l 0 - ∂ p f ∂T -V X ∂ p f ∂X - ∂T f ∂X t 1 dX.
As to the part of the force coming from curvature effects, which must be taken into account when the deformations of the body are no longer negligible (i.e. Kl ∼ O(1)), it is given by

f h4 + f h5 = l 0 ∂V X ∂X p f -T f ∂ t 1 ∂X + M XY ∂K Y Ω X t 2 ∂T + M XZ ∂K Z Ω X t 3 ∂T dX, (3.23)
where we have introduced, for simplicity, the coefficients .24) Now, if we combine (3.22) and (3.23), we finally obtain the expression (3.25) where it turns out that the part of the force due to curvature effects, that is (3.23), brings exactly what is necessary to introduce V X and t 1 into the derivatives with respect to X in the different terms of (3.22). As we can see, therefore, the first integral of (3.25) rigorously corresponds to the 3D extension of the LAEBT of [START_REF] Lighthill | Aquatic animal propulsion of high hydromechanical efficiency[END_REF], whereas the second integral accounts for a possible coupling between the roll and the body curvature. Note that, for clarity, the main steps of the derivation of the final result are summarized in figure 9.

M XY = ρ πβ(α 2 -β 2 )(α -3β) 16 and M XZ = ρ πα(α 2 -β 2 )(3α -β) 16 . ( 3
f h = l 0 - ∂ p f ∂T + ∂V X p f ∂X - ∂T f t 1 ∂X dX + l 0 M XY ∂K Y Ω X t 2 ∂T + M XZ ∂K Z Ω X t 3 ∂T dX,

The moment of hydrodynamic forces acting on the body

So far, we have been interested in the pressure resultant but in order to determine totally the dynamics of the body (in particular for 3D swimming), it is also necessary to calculate the moment of forces acting on it, which is given by the second part of (3.1). As before, detailed derivations will not be given here, in order to simplify the reading. Let us simply note that if we use a very similar approach as for the resultant, Curvature effects 3D Extension of the LAEBT [START_REF] Lighthill | Aquatic animal propulsion of high hydromechanical efficiency[END_REF] 3D Extension of the EBT [START_REF] Lighthill | Note on the swimming of slender fish[END_REF]) then it can be shown that the moment acting on the body reads .26) where σ f = M X Ω X t 1 represents the density of the roll kinetic momentum per unit of body length, r is the field of position on the backbone in relation to a reference point o (see figure 1), and where we have introduced the new coefficients

P0(Φ 0 ⊥ ) P1(Φ 0 ⊥ , Φ ) P2(Φ 0 ⊥ , Φ 1 ⊥ ) KαP0(Φ 0 ⊥ ) ∂fh1 ∂X ∂fh2 ∂X ∂fh3 ∂X ∂fh4 ∂X ∂fh5 ∂X - 1 2 ∂MX ∂X Ω 2 X + ∂Mβ ∂X V 2 β t1 -Pβ ∂Vβ ∂X t1 - ∂ pf ∂T -VX ∂ pf ∂X - ∂Tf ∂X t1 - ∂ pf ∂T -VX ∂ pf ∂X ∂VX ∂X pf -Tf ∂ t1 ∂X + MXβ tβ ∂T - ∂ pf ∂T + ∂VX pf ∂X - ∂Tf t1 ∂X + MXY ∂KY ΩX t2 ∂T + MXZ ∂KZ ΩX t3 ∂T
m h = - ∂ ∂T l 0 (σ f + r × p f )dX +[V X (σ f + r × p f ) -r × T f t 1 ] l 0 + ∂ ∂T l 0 ((M XY K Y V Y + M XZ K Z V Z ) t 1 + r × (M XY K Y Ω X t 2 + M XZ K Z Ω X t 3 )) dX +[V Y M XY Ω X t 2 + V Z M XZ Ω X t 3 ] l 0 + ∂ ∂T l 0 ( MZ V Z t 2 -MY V Y t 3 )dX + l 0 (M X -M XY ) ∂V Y ∂X Ω X t 2 +(M X -M XZ ) ∂V Z ∂X Ω X t 3 dX, ( 3
MY = π 4 αβ α ∂β ∂X +3β ∂α ∂X and MZ = π 4 αβ β ∂α ∂X +3α ∂β ∂X , (3.27)
which scale like O(ε) compared with the classical added-mass coefficients. Note that concerning the moment, the physical origin of the different terms could be also identified as has been done previously for the cross-sectional forces, but it turns out that its expression does not benefit from the same fortunate arrangements, and therefore, we will not discuss this point further.

Comparisons with numerical results

4.1. Description of the numerical solver In order to check the validity of the fluid force model obtained previously, simulations using an incompressible unsteady Reynolds-averaged Navier-Stokes (RANS) solver are carried out. The in-house solver ISIS-CFD, developed at the Laboratory of Fluid Mechanics at Ecole Centrale Nantes, has been used for this task. The interested reader can find details of the numerical methodology in [START_REF] Leroyer | Numerical methods for RANSE simulations of a self-propelled fish-like body[END_REF]. In brief, this approach performs numerical simulation of the RANS equations which are integrated by means of a second-order-accurate finite volume method. The velocity field is obtained from the momentum conservation equations and the pressure field is extracted from the mass conservation constraint or continuity equation, which is transformed into a pressure equation. Also, in the case of turbulent flows, additional transport equations for modelled variables are solved in a form similar to the momentum equations. Finally, in order to deal with moving bodies, an arbitrary Lagrangian-Eulerian (ALE) approach is used. It is associated with grid deformation techniques to keep the mesh fitted to the body, as its shape varies. The positioning of the body in space is then obtained using a rigid motion of this deformed mesh.

The mesh here is composed of 120 000 cells spread in a block-structured grid (see figures 10 and 11).

General settings: shape of the body and physical parameters

Figure 10 shows the body geometry used in these simulations. It is composed of elliptical cross-sections obtained with the following process. The shape is first defined as a cylinder of diameter D =0.1 l for any X ∈ [0.05 l, 0.99 l], ended with two halfellipsoids between X =[0, 0.05 l]a n dX =[0.99 l, l]. Then, for any cross-section, the major axis (along E 2 ) is multiplied by κ 2 (X), and its minor axis (along E 3 )b yκ 3 (X), where κ 2 and κ 3 are two functions defined by κ 2 (X)=1, for 0 6 X<0.2 l, (4.1) κ 2 (X)= 3 5 , for 0.75 l 6 X<1, (4.3)

κ 2 (X)=1- sin(π X -π/2) + 1 5 , for 0.2 l 6 X<0.75 l, with X = X -0.2 l 0.75 l -0.2 l , (4.2)
κ 3 (X)=- 1 6 sin πX - π 2 +1 - X 2 8 + 1 2 , for 0 6 X<l. (4.4)
Here, the total length l of the fish is set to 1 m. As far as the fluid is concerned, the density and the dynamic viscosity are respectively set to 1000 kg m -3 and 0.001 Pa s, similar to the properties of water.

Numerical experimentations and comparisons with theoretical results

As mentioned previously, the goal of these numerical simulations is to compare the body forces and moments given by (3.25) and (3.26) with those given by the Navier-Stokes simulation (using RANS equations). These comparisons are carried out by prescribing given motions of the body and by observing the resulting forces and moments. More precisely, in the cases considered, the body is simply towed by the head, and deforms according to imposed curvature laws, the complexity of which increases in time, in order to bring into play all the degrees of freedom of the body dynamics, and consequently, all the components of the models. During these numerical investigations, two different motions are considered, which follow the same protocol.

(i) In the first step, the body is towed straight forward by the head at a constant axial velocity V 0 = -0.6ms -1 along the X -axis (leading to a Reynolds number equal to 6 × 10 5 ), and a planar deformation is imposed through the following curvature law:

K Y = A exp sX l -s s l 2 - 2π λ 2 sin 2πX λ -ω 0 T + 4πs lλ cos 2πX λ -ω 0 T , (4.5)
which is progressively applied (the establishing time is equal to one period of undulation, i.e. 2π/ω 0 ) and thereafter maintained throughout the numerical run.

Note that (4.5) was adapted from the work of [START_REF] Tytell | The hydrodynamics of eel swimming. II. Effect of swimming speed[END_REF], who proposed a law of deflection for the backbone deduced from the experimental analysis of real eels, but here it has been expressed in terms of curvature.

(ii) After 2 s, an unsteady rotation around the X-axis model. Indeed, when correcting the model using the initial value of F X given by the CFD computation, we observed a very good agreement for the EBT-like motion (motion 1), at least before the rotation was activated. This result indicates that when the deformations of the body remain small, the EBT is efficient in predicting the reactive part of the force acting on the body. However, when the unsteady rotation starts, the difference between the two responses becomes significant (as can be seen in figure 13a after T =2.5 s). As to the responses observed when the second motion is used (motion 2, which involves larger deformation), discrepancies in F X between the numerical results and the model can be observed even before the rotation has started, as can be seen in figure 13(d ). These results clearly show that the resultant of the pressure forces in the direction of the head is probably the most sensitive to the viscous-induced perturbations of flow (i.e. boundary-layer separations) which occur when the deformation of the body becomes significant.

Ω 0 = θ 0 ω 0 cos(ω 0 T ) (4.6) 024681 0 -0.15 -0.10 -0.05 0 0.05 (a)( d) (e) ( f ) (b) (c) F X (N) F Y (N) F Z (N)
For other components F Y and F Z , as well as for the three components of the moment, in contrast to the axial force, the agreement between the model and the numerical simulations is globally very good (even for motion 2 involving large deformations), as can be seen in figures 13(b, c), 13(e, f ) and 14. The dynamics are indeed well reproduced by the model, though some discrepancies are sometimes noted, as for example, when the components of the force or the moment reach their maximal values. However, in general, it seems that the added-mass coefficients are only slightly modified by the viscosity effects in such unsteady flows, induced by the two deformations proposed here. Also, note that the forces and moments induced by the viscosity in the Navier-Stokes simulations have not been added in these figures, because they are far weaker than the other ones, except for F X in which the longitudinal viscous forces play a non-negligible role.

In any case, the numerical results suggest that the extended LAEBT appears to be a convenient tool for investigating 3D transient manoeuvres.

Concluding discussion

We have provided a 3D extension of the LAEBT, obtaining the expressions of the resultant and moment of pressure exerted on a slender body moving in a perfect fluid, in response to prescribed deformations parametrized by curvature laws. In a comparison with numerical simulations, it has been shown that even though the present model fails to predict accurately the axial force when the imposed curvature becomes significant, it may be used to investigate transient manoeuvres, since all the other components of the hydrodynamic force and moment acting on the body seem to be correctly reproduced by the model.

Note that, for the sake of generality, (3.25) and (3.26) have been obtained in a general case, that is, for arbitrary motions of a slender body (going beyond the case of swimming). However, a reader acquainted with the original LAEBT could naturally wonder if these equations could have been deduced from the balance of the fluid momentum density ( p f , σ f ) applied to the domain located upstream of the plane π (see figure 1) after postulating, as did Lighthill, a stratification of the linear and angular momentum of the fluid along the backbone. This question has been addressed recently by [START_REF] Boyer | Poincaré-Cosserat equations for the Lighthill threedimensional large-amplitude elongated body theory: application to robotics[END_REF] using a geometrical approach, which gave the following results:

f h = - ∂ ∂T l 0 p f dX +[V X p f -T f t 1 ] l 0 (5.1)
and

m h = - ∂ ∂T l 0 (σ f + r × p f )dX +[V X (σ f + r × p f ) -r × T f t 1 ] l 0 .
(5.2)

As we can see, all the terms involved in these two equations are included in the more general equations (3.25) and (3.26), but these latter equations also contain additional terms. In order to analyse the contribution of these additional terms during usual swimming of a slender body, let us first specify that in this case, the lateral velocity components of the body, as well as the components of the curvature vector (i.e. K Y and K Z ), can be generally modelled with functions of the form f (X ± cT )t h a t correspond to propagating waves, and where c corresponds to the wave speed. These wave speeds scale generally like the axial velocity (for instance, it is slightly greater during planar swimming) and this enables us to estimate the order of magnitude of the different terms involving a time derivative, since

∂ ∂T ∼ c ∂ ∂X ∼ V X ∂ ∂X .
(5.3)

In addition, as regards the axial angular velocities, by integrating (2.22) with respect to X,w eo b t a i n .4) where Ω 0 (which may be a time-dependent function) stands for the axial angular velocity of the head of the body (i.e. at X = 0). Since the angular velocities Ω Y and Ω Z have been shown to be small compared with Ω X (see (2.25)), then, at leading order, the axial angular velocity of any cross-section is uniform and is given by the axial velocity of the head. During a natural swimming gait, rotation of the head is generally avoided by fish and in any case the condition αΩ 0 ≪ V is satisfied As regards the parametrization of the body deformation, since the pioneering work of [START_REF] Gray | Studies in animal locomotion. I. The movement of fish with special reference to eel[END_REF], most of the work on swimming kinematics defines the deformation through an analytical expression of the deflection: the transverse motion of the midline with respect to the axis of movement of the fish. Although this parametrization is easy to relate to the photographs of a fish swimming straight forward, when dealing with more complex gaits (e.g. turns or 3D manoeuvres), it becomes artificially complex, all the more so since the inextensibility constraint is not naturally taken into account (see for example [START_REF] Singh | The hydrodynamics of flexible body manoeuvres in swimming fish[END_REF]. However, if the shape is parametrized by strain measures (here curvatures), the inextensibility constraint is 'by construction' satisfied since then any shape is deduced from the integration of a strain field compatible with the inextensibility. In fish, this description is quite natural, since the 'curvature is directly related to muscle contractions and strength' [START_REF] Kern | Simulation of optimized anguilliform swimming[END_REF], and during classical manoeuvres it is often very simple. For instance, when a caranguiform fish is performing a turn manoeuvre, it contorts its body into an arc of circle (as a 'C-curve'), whose radius varies in time (see for instance the experimental results of [START_REF] Anderson | Vorticity control for efficient propulsion[END_REF], and, in this case, at least at leading order, the curvature, which is uniform along the body, can be written as the simple time law: K Y = K u h(T ), where h(T ) is a time-dependent function for progressively imposing and subsequently removing the desired curvature.

Ω X = Ω 0 + X 0 K Z Ω Y -K Y Ω Z dX, ( 5
Finally, it can be said that LAEBT is essentially useful for investigating the leading physical mechanisms involved during transient manoeuvres, where reactive forces and moments are expected to be dominant, and not for investigating self-propelled swimming. In the latter case, if fast simulations must be done, at present it seems that there is no other choice than to add empirical forces and moments to the models, which take into account viscous effect as is proposed for example by [START_REF] Taylor | Analysis of the swimming of long and narrow animals[END_REF] or, more recently, by Galper & Miloh (2000) or Lopes, Païdoussis & Semler (2002). Also, another improvement to the LAEBT could be to extend the results of Newman & Wu (1973), to account for the effect of the vorticity shed by the fins upstream of the caudal fin. 2.44), it appears that for each abscissa X, Φ 0

⊥ corresponds to the 2D potential in the (Y , Z) plane resulting from the movement (translation and rotation) of an infinite cylinder whose cross-section is S X . In this paper, we consider the cross-sections of the body to be elliptic, and the usual way to derive the 2D potential in such a case is to use a conformal transformation that links the elliptic section to a circular section. However, in the present investigation, and for later convenience, we prefer to use elliptic coordinates (u, v), where u > 0a n d 0 6 v 6 2π, since such coordinates are also used to obtain the expression of Φ 1 ⊥ , where conformal transformation can no longer be used (Φ 1 ⊥ is not a harmonic function). In this investigation, we are concerned with elliptic cross-sections whose major and m i n o rs e m i -a x e sa r eα and β, respectively. Furthermore, we have chosen to position both foci of the elliptic cross-section of the body on the Y -axis, the distance between the foci being 2a,w h e r ea is linked to the major and minor axes by the relation and the boundary of an elliptic cross-section simply reads u = u X , where the value of u X is linked to α and β by the following relations: α = a cosh(u X )a n dβ = a sinh(u X ) or, reciprocally u X = 1 2 ln((α + β)/(αβ)), (A 3) so that here u X depends on X since the size of the cross-section of the body may change along the body.

a = α 2 -β 2 . ( A
Elliptic coordinates are orthogonal and the tangential unit vectors, namely e u and e v , are given by e u = a J (u, v) 1/2 (sinh(u)cos(v)t 2 +cosh(u) sin(v)t 3 ), (A 4) e v = a J (u, v) 1/2 (-cosh(u) sin(v)t 2 +sinh(u)cos(v)t 3 ), (A 5) where J = a 2 sinh(u) 2 + a 2 sin(v) 2 (A 6) is the Jacobian determinant of the elliptic coordinates. We also specify that, for any function f (u, v), the 2D gradient and the Laplacian read and, necessarily, we also have

∇f = 1 J 1/
∇Φ 0 ⊥ → 0, |u|→∞. (A 10)
Since the potential Φ 0 ⊥ must be 2π-periodic with respect to v, it is convenient to seek it by using a Fourier series

Φ 0 ⊥ = ∞ -∞
F n (u) exp(i nv), (A 11) where 'i' is the imaginary number. Then, if we introduce (A 11) into (A 8) and exploit the orthogonality properties of the complex exponential, it is quite simple to show that, in terms of elliptic coordinates, the very general solution of the 2D Laplacian is

Φ 0 ⊥ = A 0 + B 0 u + ∞ n=1
(A n cos(nv)+B n sin(nv)) exp(-nu)+(C n cos(nv) + D n sin(nv)) exp(nu), (A 12) where coefficients A n , B n , C n and D n are constants that must be determined by the boundary conditions. Here, (A 10) imposes taking only decreasing harmonics into account so that C n = D n = 0, and according to (A 9), we obtain Φ 0 ⊥ = -a 2 Ω X 4 sin(2v) exp(2u X -2u) -βV Y cos(v) exp(u Xu) -αV Z sin(v) exp(u Xu), (A 13) where we have neglected a constant. Note that (A 13) is obviously not an original result and this solution can be found in many classical books on fluid mechanics (see for instance Milne-Thomson 1968, pp. 258-259). However, what is interesting here is how this solution has been derived, since the same approach will be used to determine Φ 1 ⊥ , where the expression of Φ 0 ⊥ is also involved. where J is given by (A 6). According to (A 13), it may be shown, after some algebra, that (2.45) can be written as follows: Once again, since Φ 1 ⊥ is also 2π-periodic, the solution can be sought in the form

∂ 2 Φ 1 ⊥ ∂u 2 + ∂ 2 Φ 1
C 0 = - a exp(u X ) 2 (K Z V Y β + K Y V Z α). ( A
Φ 1 ⊥ = ∞ -∞
G n (u) exp(inv). (A 22) After injecting this sum into (A 16), and after using complex exponential orthogonal properties, we obtain a set of ordinary differential equations of the form where A n and B n are constants which must be determined by the boundary conditions. According to (A 21), we obtain the solution

Φ 1 ⊥ = - a exp(-u X ) 4 (K Z βV Y + K Y αV Z ) u + 1 2 exp(2u X -2 u) + a 3 Ω X 32 exp(-u X ) (3 exp(u X -u) -exp(3u X -3u)) (K Y cos(v)+K Z sin(v))
a exp(u X ) 8 ((K Z βV Y + K Y αV Z )cos(2v)+(K Z αV Z -K Y βV Y ) sin(2v)) + a 3 Ω X 32 exp(u X ) 1 3 exp(3u X -3u)exp(u Xu) (K Y cos(3v)+K Z sin(3v)).

(A 26)

  Fi gure 2. Slender-body flow produced by Φ

Figure 3 .

 3 Figure 3. Lateral flow in a cross-section produced by Φ ⊥ (V Z =1, K Y =0.25).

  Fi gure 4. Two-dimensional potential flow (N X =0, V Z =1).

  Fi gure 5. Flow produced by Φ

Figure 6 .

 6 Figure 6. Illustration of the body deformation effects on the flow.

Figure 7 .

 7 Figure 7. Illustration of the shed vorticity during planar swimming.

Fi gure 9 .

 9 Diagram summarizing the main steps of the calculations leading to result (3.25).

Figure 10 .

 10 Figure10. View of the body and the mesh around it.

Figure 11 .

 11 Figure 11. Global view of the fluid domain.

Figure 13 .

 13 Figure 13. Body forces for (a-c)m o t i o n1a n d( d -f ) motion 2. Lines: --, pressure NS; -• -• -, model.

Figure 14 .

 14 Figure 14. Body moments for (a-c)m o t i o n1a n d( d -f ) motion 2. Lines: --, pressure NS; -• -• -, model.

  1)Elliptic coordinates(u, v), where u > 0a n d0 6 v 6 2π, are linked to the classical Cartesian coordinates by the following relations:Y = a cosh(u) cos(v)a n d Z = a sinh(u) sin(v), (A 2)

A. 2 .Φ 1 ⊥

 21 SolutionIn terms of elliptic coordinates, partial derivatives of the right-hand side of (2.45) read∂ ∂Y = a J sinh(u)cos(v) ∂ ∂u cosh(u) sin(v)

⊥

  ∂v 2 =(C -3 exp(-3iv)+C 3 exp(3iv)) exp(-u)+(C -2 exp(-2iv) + C 2 exp(2iv)) + (C -1 exp(-iv)+C 1 exp(iv)) exp(Z V Y β + K Y V Z α)+i(K Z V Z α -K Y V Y β)), C 2 =c.c.(C -2 ), (A 18) C -1 = -A -3 , C 1 =c.c.(C -1 ), (A 19) and

⊥

  20) (Here c.c. stands for complex conjugate.) Also, the boundary conditions are ∂Φ 1

d 2

 2 G n du 2n 2 G n = C n exp(-mu),n ∈ Z, (A 23)and m is equal to -1, -2o r-3. General solutions of these ordinary differential equations readG 0 (u)=A 0 + B 0 u + C 0 m 2 exp(-mu), if n =0, (A 24)orG n = A n exp(nu)+B n exp(-nu) -C n n 2m 2 exp(-mu), if n =0, (A 25)

  . If we use this condition, together with (5.3), it turns out that the additional terms existing in (3.25) and (3.26), in comparison with (5.1) and (5.2), are actually negligible in almost all the situations encountered during natural swimming by animals. And, finally, even for 3D manoeuvres, if the rotation of the head has a smooth and slow evolution leading to negligible roll dynamics, the model given by (3.25) and (3.26) has a form similar to the original LAEBT obtained for the 2D planar swimming, except that the linear fluid momentum p f contains an additional component which accounts for the pitch motion of the body.

  This work was supported by the European Commission, Information Society and Media, Future and Emerging Technologies (FET) contract 231845.

	Appendix. Analytical determination of the potentials Φ 0 ⊥ and Φ 1 ⊥ In this appendix, solutions Φ 0 ⊥ and Φ 1 ⊥ corresponding to (2.43)-(2.44) and (2.45)-(2.46), respectively, are derived. If we look at (2.43) and (

  In terms of elliptic coordinates and since in this problem N Y t 2 + N Z t 3 = e u + O(ε 2 ), (2.43) and (2.44) governing Φ 0

						A.1. Solution Φ 0 ⊥
					⊥ read
						∂ 2 Φ 0 ⊥ ∂u 2 +	∂ 2 Φ 0 ⊥ ∂v 2 =0,			(A 8)
	∂Φ 0 ⊥ ∂u u X	= βV Y cos v + αV Z sin(v)+	Ω X a 2 2	sin(2v),	(A 9)
	2	∂f ∂u	e u +	∂f ∂v	e v	and	f =	1 J	∂ 2 f ∂u 2 +	∂ 2 f ∂v 2 .	(A 7)

is progressively imposed (again during one period of undulation), and this unsteady rotating motion is maintained until the end of the run. Note that θ 0 is the angle corresponding to the rotation amplitude, and it has been arbitrarily set equal to π/6.

In the deformation law given by (4.5), four parameters are involved, namely the amplitude of tail deflection A, a shape parameter s, the wave length λ and the angular frequency ω 0 . In all runs, we have set ω 0 =4π s -1 , λ =0.75 m and s =4, and the two motions tested differ by the value of the amplitude A. For motion 1, A =0.03 m whereas for motion 2, A =0.12 m.

The typical deformations corresponding to these two prescribed motions are presented in figures 12(a) and 12(b). As we can see, deformations involved in the first motion are quite limited, so that they remain more or less inside the frame of the EBT, whereas deformations involved in the second case are more important. This second motion will enable us to check more specifically the validity of the LAEBT.

The ISIS-CFD computation starts at time T = 0 with a converged solution of the flow for the body rigidly towed at the forward speed equal to -V 0 e 1 , and the pressure forces and moments acting on the body are calculated in the head frame of the body, so that in the following, the components F X , F Y and F Z are given in the basis (t 1 (0,T), t 2 (0,T), t 3 (0,T)), and moments M X , M Y and M Z are defined in the same basis and related to the head of the body (i.e. the point X =0) .

Figures 13(a-c) and 13(d -f ) show the fluid pressure forces acting on the body obtained numerically and those obtained with the analytical model, respectively for the two motions, whereas figures 14(a-c) and 14(d -f ) compare the moments. Three dash-dotted vertical lines are also plotted in these graphs, delimiting respectively the end of the transient phase where the amplitude increases, the start of the rotation and the end of its transient phase.

We first discuss the results related to the fluid force directed along the head axis t 1 (0,T), i.e. F X obtained in response to the two motions. In figure 13(a) (it is also the case for figure 13d, but the range prevents its observation), we observe a small discrepancy at time T = 0 between the CFD simulation and the model. This is probably due to the rounded-nose geometry of the body (see figure 10), since the pressure drag acting on the head along the X-axis is not captured by the A.3. Determination of Φ In order to determine the part Φ of the velocity potential, we first need to determine analytically the expression of the component N X of the outward normal of the body surface lying between the two sections S X and S X+dX . To determine N X , we first need to determine the expression of the spatial increments dl and ds (see figure 8).

Here, we are concerned with the frontal surface (along t 1 ) lying between two elliptic cross-sections (whose major and minor axes are, respectively, α(X)a n dα(X +dX) and β(X)a n dβ(X +dX)). Then, the implicit equation of the contour of S X+dX reads

Furthermore, in terms of elliptic coordinates, we also have

where δu is the increment of the dimensionless elliptic coordinates u X that must be added to reach the contour of S X+dX from the contour of S X . If we introduce these expressions of Y and Z into (A 27), we can show, by using a classical series expansion (with respect to dX), that this increment is given by

where J (u X ,v) is the Jacobian determinant corresponding to elliptic coordinates, evaluated at the elliptic coordinates u X :

If we return to the spatial increments dl and ds, we have dl = J (u X ,v) 1/2 δu and ds = J (u X ,v) 1/2 dv. (A 31)

It is straightforward to show that

which leads to the result

In these conditions, Φ can be obtained by following the same approach (for deriving Φ 0 ⊥ )t oo b t a i n

(A 34)


Claudio Vitari

Isabelle Bourdon

Florence Rodhain

'Entreprise 1 La recherche a

Keywords: Internet, Communication, Web 2.0, Responsabilité Sociale de l Internet, Communication, Web 2.0, Corporate Social Responsibility Internet, Comunicación, Web 2.0, la responsabilidad social de la empresa

été partiellement subventionnée par la région Rhône-Alpes 2

Introduction

Les catastrophes environnementales de ces dernières décennies (accidents des centrales nucléaires de Tchernobyl en 1986, de Fukushima en 2011,…) ont déclenché un changement visible des préoccupations du public envers les questions environnementales et la performance sociale des entreprises [START_REF] Rodhain | Pour une éthique des Sciences du Management : Formation à la Connaissance de Soi[END_REF]. Les entreprises sont ainsi confrontées à des pressions fortes qui les poussent à intégrer les principes de la Responsabilité Sociale de l'Entreprise (RSE) dans leurs activités [START_REF] Ryan | Building sustainable IT[END_REF][START_REF] Melville | Information Systems Innovation For Environmental Sustainability[END_REF][START_REF] Pierrot | Des entreprises responsables : à l'égard de qui et à quel propos ? Une étude du message du Président de sociétés cotées françaises[END_REF][START_REF] Richomme-Huet | Crise et gestion de la responsabilité sociétale des entreprises du CAC 40: Analyse de leur communication sur leur politique RSE entre 2006 et 2010[END_REF]. [START_REF] Jenkin | An agenda for `Green' information technology and systems research[END_REF] montrent que ces pressions conduisent les entreprises à mettre en place des stratégies de « green IT2 » [START_REF] Siegler | A practical approach to Green IT[END_REF], Ait Daoud et al. 2010). Pourtant, bien peu de recherches s'intéressent à ces questions [START_REF] Chen | Information systems and ecological sustainability[END_REF]) et les enquêtes empiriques sur le sujet sont peu nombreuses [START_REF] Jenkin | An agenda for `Green' information technology and systems research[END_REF]. Notre travail vise à enrichir la recherche dans ce domaine à travers une étude sur une pratique de green IT spécifique : les systèmes de communication Internet sur la RSE. Notre intérêt pour l'étude des systèmes de communication Internet sur la RSE est lié à la diffusion croissante d'Internet pour diffuser l'information des entreprises [START_REF] Cormier | The use of Web sites as a disclosure platform for corporate performance[END_REF], et à l'usage privilégié d'Internet par les entreprises pour leur communication RSE [START_REF] Cho | Environmental reporting on the internet by America's Toxic 100: Legitimacy and self-presentation[END_REF]. Mais nous ne savons pas si leur communication RSE vise à instaurer un dialogue avec les différentes parties prenantes ou si elle prend plus la forme d'un discours unilatéral notamment. L'analyse des systèmes et des technologies, du Web 1.0 au Web 2.0, utilisées par les entreprises permet d'en évaluer l'orientation stratégique. C'est pourquoi, nous souhaitons répondre à la question de recherche suivante :

Quels sont les facteurs qui influencent les systèmes de communication Internet sur la RSE dans les entreprises ?

Pour répondre à cette question, nous avons mené une étude longitudinale sur les systèmes Internet de toutes les entreprises du CAC40, qui permet d'appréhender leur évolution dans le temps, à l'instar de recherches portant sur les systèmes de communication par Internet des entreprises [START_REF] Marston | Corporate reporting on the Internet by German companies[END_REF].

-Cadre théorique et modèle de recherche

Nous présentons dans une première partie le cadre théorique de notre recherche en rappelant la définition de l'objet de notre recherche, les systèmes Internet de communication sur la RSE, et en identifiant les facteurs pouvant influencer leur mise en place dans les organisations. Nous présentons également le modèle de recherche proposé.

Système Internet de communication de la RSE

Afin de répondre aux forces réglementaires, normatives et sociales, les entreprises développent des systèmes de communication de leur RSE, qui s'appuient pour part sur les Technologies Internet (TI) avec la mise en ligne de sites web institutionnels abordant les aspects sociaux et environnementaux, en plus des performances économiques [START_REF] Patten | Legitimacy and the internet: an examination of the corporate web page environmental disclosures[END_REF]. Ainsi, [START_REF] Jenkin | An agenda for `Green' information technology and systems research[END_REF] montrent que les pressions subies par les entreprises les conduisent à mettre en place des stratégies de green IT [START_REF] Siegler | A practical approach to Green IT[END_REF], et plus l'entreprise est engagée dans une démarche RSE et plus elle met en oeuvre des actions en matière de Green IT [START_REF] Bohas | Responsabilité Sociale de l'Entreprise : quels impacts sur l'adoption de pratiques de Green IT ?[END_REF]. De nombreuses études ont porté sur le contenu de la communication des entreprises en matière de RSE [START_REF] Bohas | Responsabilité Sociale de l'Entreprise : quels impacts sur l'adoption de pratiques de Green IT ?[END_REF][START_REF] Chauvey | The Normativity and Legitimacy of CSR Disclosure: Evidence from France[END_REF][START_REF] Richomme-Huet | Crise et gestion de la responsabilité sociétale des entreprises du CAC 40: Analyse de leur communication sur leur politique RSE entre 2006 et 2010[END_REF], mais peu d'attention a été portée aux technologies Internet utilisées par les entreprises pour communiquer sur leur RSE [START_REF] Esrock | Social responsibility and corporate Web pages: Self-presentation or agenda-setting[END_REF][START_REF] Maignan | Corporate Social Responsibility in Europe and the U.S.: Insights from Businesses' Self-presentations[END_REF][START_REF] Snider | Corporate Social Responsibility in the 21st Century: A View from the World's Most Successful Firms[END_REF][START_REF] Patten | Legitimacy and the internet: an examination of the corporate web page environmental disclosures[END_REF]. Pourtant, les technologies du web 2.0 (O'Reilly 2005), qui correspondent aux technologies Internet couplées à des services collaboratifs, constituent des technologies permettant une communication interactive en temps réel avec les différentes parties prenantes (Biloslavo & Trnavčevič 2009;[START_REF] Isenmann | Online reporting for sustainability issues[END_REF][START_REF] Esrock | Social responsibility and corporate Web pages: Self-presentation or agenda-setting[END_REF][START_REF] Guimarães-Costa | The atrium effect of website openness on the communication of corporate social responsibility[END_REF].

Nous considérons, dans notre cadre d'analyse, deux types de systèmes Internet de communication de la RSE : celui de type Web 1.0, quand le site internet organise des pages web statiques reliées par des liens hypertextes [START_REF] Cho | Environmental reporting on the internet by America's Toxic 100: Legitimacy and self-presentation[END_REF]

Les facteurs influençant le Système Internet de communication de la RSE

Trois principaux facteurs influencent le système de communication RSE dans la littérature : la taille, la performance sociale et environnementale et la performance économique et financière de l'entreprise 3 .

La taille de l'entreprise influence fortement le développement des politiques de communication RSE. En effet, les entreprises de grande taille communiquent plus que les plus petites entreprises sur leur politique RSE [START_REF] Branco | Factors Influencing Social Responsibility Disclosure by Portuguese Companies[END_REF][START_REF] Marston | Corporate reporting on the Internet by German companies[END_REF][START_REF] Patten | Legitimacy and the internet: an examination of the corporate web page environmental disclosures[END_REF]. Par ailleurs, les grandes entreprises sont plus portées à développer des activités de responsabilité sociale et à communiquer institutionnellement dans le but d'améliorer leur réputation [START_REF] Branco | Factors Influencing Social Responsibility Disclosure by Portuguese Companies[END_REF] [START_REF] Marston | Corporate reporting on the Internet by German companies[END_REF]. Nous proposons l'hypothèse suivante :

La performance économique et financière de l'entreprise influence positivement le système de communication Internet sur la RSE Web 1.0 (H3.1) et Web 2.0 (H3.2).

Le Système internet de communication de la RSE est enfin influencé par l'histoire passée de l'entreprise et notamment par le système Internet de communication de la RSE existant précédemment [START_REF] Marston | Corporate reporting on the Internet by German companies[END_REF] Afin de tester le modèle proposé, nous avons mené une étude longitudinale sur l'échantillon des entreprises du CAC40 entre 2007 et 2015.

Une enquête longitudinale

Afin de répondre à notre question de recherche, et tester le modèle empirique proposé, nous avons conduit une enquête longitudinale auprès de l'intégralité des entreprises du CAC 40. Ce choix est justifié par le fait que ces entreprises sont, de par leur notoriété, les plus exposées aux critiques sur leur politique sociale et environnementale [START_REF] Guthrie | Corporate Social Reporting: A Rebuttal of Legitimacy Theory[END_REF]. La caractéristique de l'échantillon nous permet également de garantir une certaine diversité sur les différents variables indépendantes retenues. Notre unité d'analyse est constituée du site web institutionnel « corporate » (www.nom_entreprise.com) en français, en tant que systèmes Internet de communication sur la RSE. Les sites des sociétés filles et les sites en autres langues n'ont pas été inclus dans notre échantillon.

Description des instruments de mesure

L'identification des instruments de mesure est issue d'une revue de la littérature complétée par une navigation préliminaire sur divers sites web analysés, afin de vérifier la complétude de nos instruments. Nous avons ainsi ajouté quelques éléments qui nous paraissaient importants mais qui étaient absents du cadre issu de la littérature. Nos variables dépendantes sont le système Web 1.0 et le système Web 2.0. Nos variables indépendantes sont la Taille, la Performance économique et financière et la Performance environnementale et sociale. Nous décrivons ci-après l'opérationnalisation de chaque variable du modèle proposé.

2.2.1. Instrument de mesure du système Web 1.0 Une large littérature existe au sujet du web 1.0 [START_REF] Chauvey | The Normativity and Legitimacy of CSR Disclosure: Evidence from France[END_REF][START_REF] Maignan | Corporate Social Responsibility in Europe and the U.S.: Insights from Businesses' Self-presentations[END_REF][START_REF] Marston | Corporate reporting on the Internet by German companies[END_REF][START_REF] Patten | Legitimacy and the internet: an examination of the corporate web page environmental disclosures[END_REF][START_REF] Cho | Environmental reporting on the internet by America's Toxic 100: Legitimacy and self-presentation[END_REF]Biloslavo & Trnavčevič 2009;[START_REF] Isenmann | Online reporting for sustainability issues[END_REF][START_REF] Esrock | Social responsibility and corporate Web pages: Self-presentation or agenda-setting[END_REF]. L'analyse des différentes études a donné forme à notre instrument de mesure du Web 1.0 qui inclut :

-le nombre de liens vers la section sur la RSE en page d'accueil du site web, adresse email de l'entreprise, formulaire en ligne, service FTP, option de marque page. Étant donné les différentes acceptations du terme RSE, nous avons également pris en compte des termes voisins 4 . En conséquence, pour les rapports, nous avons considéré la dénomination « rapport RSE » mais aussi les rapports sociétaux, les rapports sociaux, les rapports Développement Durable, etc... de même en ce qui concerne l'audit RSE. Concernant la charte, nous avons accepté les termes suivants : Code de conduite, Code éthique, Charte éthique, Feuille de route. La valeur finale de l'instrument de mesure pour chaque entreprise est donnée par l'addition des valeurs des mesures de chaque indicateur (en convertissant les « oui » en « 1 » pour les champs à réponses binaire (oui/non)).

-
2.2.2. Instrument de mesure du système Web 2.0 Concernant les technologies Web 2.0, notre instrument de mesure est fondé sur la liste des technologies Web 2.0 proposée par Andriole (2010) qui inclut : les flux RSS, l'existence d'un monde virtuel, la présence d'email à envoyer à des tiers, l'approvisionnement collectif (crowdsourcing), les forums, les blogs, l'indexation personnelle (folksonomy), les wiki, podcasts, le partage sur réseaux sociaux. La valeur finale de l'instrument de mesure pour chaque entreprise est donnée par l'addition des valeurs des mesures de chaque indicateur avec la conversion des « oui » en « 1 ».

Modalités de collecte et d'analyse des données

La saisie des données a été répartie entre les trois auteurs de cet article, assisté par un candidat à un doctorat de recherche. D'abord, pour assurer la fiabilité inter-codage entre les différents participants [START_REF] Straub | Validation Guidelines for IS Positivist Research[END_REF][START_REF] Miles | Qualitative data analysis[END_REF]

-Résultats et discussion

Nous présentons les principaux résultats obtenus concernant les systèmes internet de communication de la RSE de type Web 1.0 et 2.0 ci après.

Les Systèmes

Les Systèmes Web 2 .0

La corrélation pour échantillons appariés du score du Web 2.0 n'est pas significative entre leurs valeurs en 2007 et celles de 2009-2010 et entre les valeurs en 2009-2010 et celles de 2014-2015 ; Ainsi, aucune tendance dans le temps à l'élargissement de la couverture Internet de type 2.0 pour la communication institutionelle de la RSE n'est statistiquement significative. L'hypothèse relative (H4.2. t1 à t2, et t2 à t3) qui avançait que le système Internet Web 2.0 de la période précédente influence positivement le système Internet Web 2.0 actuel, n'est alors pas validée.

Les facteurs influençant les systèmes de communication sur la RSE

Discussion

Conclusion

Afin de répondre aux forces qui motivent la soutenabilité environnementale des organisations, les entreprises développent tout un système de communication sur leur politique RSE, le site web institutionnel en faisant partie. Dans cet article, nous nous sommes interrogés sur la question suivante : Quels sont les facteurs qui influencent les systèmes de communication Internet sur la RSE dans les entreprises du CAC40 ? Nous constatons que le développement du système Internet de communication de la RSE tend vers une standardisation. Si au départ les facteurs Taille, Performance environnementale et sociale et Performance économique et financière influencent, d'une manière ou d'une autre, les systèmes de communication Internet sur la RSE des entreprises du CAC40, leur influence s'estompe avec le temps. En 2014-15 plus aucun des trois facteurs étudiés n'influence les systèmes de communication Internet sur la RSE des entreprises du CAC40. Si les systèmes élargissent dans le temps leur offre d'information (Web 1.0), les entreprises françaises du CAC40 ne semblent pas poursuivre une stratégie de dialogue, qu'autorisent aujourd'hui les outils Internet de nouvelle génération, entre elles et leurs parties prenantes sur la question de la RSE (Web 2.0). Elles ont choisi du point de vue stratégique d'utiliser leur site web principalement comme un outil d'information à sens unique. Mais, avec de type d'information unilatéral et la normalisation des pratiques en la matière, ces mêmes entreprises prennent le risque de se voir accuser de pratiquer, en place et lieu d'une véritable communication, un discours unilatéral de propagande. Une piste de recherche qui s'ouvre est, alors, celle d'explorer si ces pratiques restent confinées aux entreprises du CAC40 ou si elles sont suivies aussi par les plus petites entreprises.

  et celui de type Web 2.0, quand le site internet utilise les technologies du Web 2.0 permettant d'instaurer une communication participative avec les parties prenantes (O'Reilly 2005).

Figure 1 :

 1 Figure 1 : Modèle de recherche proposé 2. -Une étude longitudinale des systèmes Internet de communication de la RSE des entreprises du CAC40

Tableau 1 :

 1 Les indicateurs des systèmes Web 1.0 et 2.0 et leur mesure de liens RSE directement visibles sur la page d'accueil du site nombre 1.0 Mention de l'existence du rapport RSE dans le site Oui = de lecture de la Charte RSE dans le site Oui = 1 ou Non = 0 1.0 Mention de l'existence d'indicateurs RSE dans le site Oui = total de certifications RSE dans le site nombre 1.0 nombre total de labels RSE dans le site nombre 1.0 Présence d'une adresse email de l'entreprise spécifique sur la RSE dans le site Oui un service de syndication du contenu spécifique sur la RSE par flux RSS Oui = 1 ou Non = 0 2.0 Présence d'un service d'accès à un monde virtuel spécifique sur Présence d'un service de folksonomy spécifique sur la RSE Oui = 1 ou Non = 0

  On observe une extension du système internet Web 1.0 de communication RSE durant notre étude longitudinale. Entre 2007 et 2009-2010, le score du Web 1.0 a augmenté. La corrélation (0,69) significative (au niveau de 0,0001) entre le score de 2007 et celui de 2009-2010 met en évidence une tendance dans le temps à l'élargissement de la couverture Internet. Le T-test pour échantillons appariés est égal à -9,56 et statistiquement significatif à 0,0001. La corrélation perd en significativité entre le score de 2009-10 et celui de 2014-2015. Cela signifie que le système internet Web 1.0 relatif à la RSE de la période précédente n'influence plus le système internet Web 1.0 de la période suivante. Néanmoins, le T-test pour échantillons appariés est égal à -5,32 et statistiquement significatif à 0,0001 entre 2009-2010 et 2014-2015. Cela signifie que l'effet temps produit une extension des systèmes internet Web 1.0. Notre hypothèse (H4.1) selon laquelle le système Internet Web 1.0 existant précédemment influence positivement le système Internet Web 1.0 actuel est corroborée en t2 (H4.1.t1 à 2), mais pas en t3 (H4.1.t2 à t3).

  3.3.1. Taille et communication sur la RSE Concernant la taille, en 2007, nous avons trouvé une corrélation égale à 0,41 en 2007 et significative, au niveau de 0,023, entre le web 1.0 et le poids indiciel du CAC40, mais qui ne se reproduit pas ni en 2009-2010, ni en 2014-2015 (Tableau 2). Les autres variables ne montrent aucune corrélation significative pour le système internet web 1.0 et pour le système internet web 2.0. En 2009-2010, par contre, nous avons trouvé deux corrélations significatives entre le poids indiciel du CAC40 (R 2 =0,388 et p=0,002) et le chiffre d'affaires (R 2 =0,39 et p=0,035) et le web 2.0, mais qui ne se reproduit ni en 2007 ni en 2014-2015. Le nombre de salariés n'est, à aucun moment, significatif. Nous avons également exploré la possibilité de réduire le nombre d'indicateurs pour mesurer la taille à travers une échelle sommée (summated scale). L'analyse factorielle nous a permis d'apprécier que le composant incluant les trois variables explique à lui seul 60% de la variance. Cette nouvelle variable a une fiabilité mesurée à travers l'Alpha de Cronbach égale à 0,61. La variable sommée est significative à 0,034 pour la communication Web 1.0 en 2007 (R 2 =0,388), mais elle perd en significativité en 2009-2010 et en 2014-2015 et elle n'en a pas pour le Web 2.0 à aucun moment. Nous pouvons avancer qu'il y aurait pu avoir une influence au départ (en 2007) de la taille (particulièrement dans le poids indiciel) sur le système internet Web 1.0. Ainsi, les entreprises ayant un poids plus important dans le CAC40, c'est-à-dire les entreprises ayant plus d'importance sur les marchés financiers, ont donné plus de place à la RSE sur leur site Web1.0, comme l'avaient déjà soutenu Igalens et Gond (2005). Toutefois, par mimétisme et standardisation, cette influence a disparu avec le temps. Une influence similaire aurait pu avoir lieu par contre en 2009-10 sur le développement du système internet Web 2.0 par le poids indiciel et le chiffre d'affaires. Cette influence a disparu avec le temps par mimétisme et standardisation des pratiques du web 2.0. L'hypothèse affirmant l'influence positive de la taille sur la communication Internet Web 1 .0 est corroborée pour 2007 (H1.1.t1 à t1) mais pas pour 2009-2010 (H1.1.t2 à t2), ni pour 2014-2015 (H1.1.t3 à t3). Par contre, l'hypothèse proposant l'influence positive de la taille sur la communication Internet Web 2.0 n'est vérifiée ni en 2007 (H1.2.t1 à t1), ni en 2009-2010 (H1.2.t2 à t2), ni pour 2014-2015 (H1.2.t3 à t3) (Figure 2).

  3.3.2. Performance environnementale et sociale et communication sur la RSE Concernant la performance environnementale et sociale, aucune différence de communication Internet sur la RSE, web 1.0 et web 2.0, statistiquement significative n'émerge entre la portion d'entreprises incluses dans les indices sociaux et environnementaux et la portion exclue en 2007, en 2009-2010 et en 2014-2015. Nous avons également exploré la possibilité de réduire le nombre d'indicateurs pour mesurer la performance environnementale et sociale à travers une échelle sommée. Celle-ci somme les cinq indicateurs où un point a été attribué pour chaque « oui ». L'échelle sommée a une corrélation égale à 0,305 et significative à 0,05 en 2009-2010 avec le système internet Web 1.0. Nous pouvons donc provisoirement conclure que les sociétés reconnues comme responsables socialement communiquent sur la RSE de la même manière que les entreprises non reconnues responsables. Nos hypothèses qui proposaient que la performance environnementale et sociale de l'entreprise influence négativement la communication Internet sur la RSE ne sont pas validées (H2.1.t1 à t1, H2.2.t1 à t1, H2.2.t2 à t2, H2.1.t3 à t3, H2.2.t3 à t3) à l'exception de celle sur la période de 2009-2010 pour le web 1.0 (H2.1.t2 à t2). 3.3.3. Performance économique et financière et communication sur la RSE Nous avons identifié quelques corrélations statistiquement significatives sur les indicateurs économiques et financiers individuels, comme pour le return on equity et le book to market en 2014-2015, le return on equity et le taux de marge brut en 2007, et le return on capital employed en 2009-2010. Nous avons également exploré la possibilité de réduire le nombre d'indicateurs pour mesurer la performance économique et financière à travers une échelle sommée. L'analyse factorielle nous a permis d'identifier un composant expliquant 36% de la variance ; ce dernier inclut : return on equity, taux de marge brut, taux de marge net et return on capital employed. Cette nouvelle variable a une fiabilité mesurée à travers l'Alpha de Cronbach égale à 0,61. Nous retrouvons des corrélations significatives entre l'échelle sommée sur les données de l'année civile 2006 et les systèmes internet de 2007 pour le Web 1.0 (égale à 0,25 avec p=0,038) et Web 2.0 (égale à 0,322 avec p=0,022). Par contre, nous constatons un manque de significativité statistique entre la communication Internet sur la RSE et l'échelle sommée pour les années 2009-10 et 2014-2015. D'après nos résultat, il semblerait donc que la communication Internet sur la RSE ne soit que partiellement dépendante des résultats économiques et financiers. Au final, l'hypothèse liant la performance économique et financière de l'entreprise à la communication Internet sur la RSE est corroborée, pour le Web 1.0 ainsi que pour le Web 2.0 que pour l'année 2007 (H3.1.t1 à t1, H3.2.t1 à t2), mais pas les autres hypothèses (H3.1.t2 à t2, H3.2.t2 à t2, H3.1.t3 à t3, H3.2.t3 à t3).

  Cette étude longitudinale sur les facteurs influençant les systèmes de communication Internet sur la RSE des entreprises du CAC 40 nous a conduits à explorer les sites web institutionnels de ces entreprises en s'attachant plus particulièrement à la communication en page d'accueil, sur le site dans son ensemble et aux services Internet pouvant être mobilisés pour communiquer sur la RSE. Au final, les analyses permettent de confirmer 5 hypothèses du modèle de recherche empirique. En résumé, nous avons pu prouver qu'en 2014-15 les systèmes Web 1.0 et Web 2.0 de communication RSE ne sont plus influencés par aucune des variables influençant la communication RSE. En 2007, seulement la taille de l'entreprise et la performance économique et financière influençaient le système Web 1.0 de communication RSE, comme l'avait déjà soutenu Igalens et Gond (2005). En 2009, seuls le système Web 1.0 existant précédemment (en 2007) et la performance environnementale et sociale influencent le système Web 1.0 actuel. Pour le système Web 2.0 de communication RSE, il ne semble être influencé par aucune variable indépendante, sauf par la performance économique et financière en 2007. Nous avançons alors que les facteurs motivant l'adoption de pratiques Green IT n'ont plus aucun impact sur le système Internet de communication de la RSE. En 2007, la taille et la performance économique et financière avaient un effet sur les systèmes Internet de communication ; En 2009 seuls la performance environnementale et sociale et le système Web 1.0 existant précédemment ont un effet sur les systèmes Internet de communication sur la RSE des entreprises du CAC40. En 2014-15, aucun facteur n'a d'impact sur le système Internet de communication de la RSE. Nous estimons que les stratégies RSE et les pratiques de communication des entreprises du CAC40 sont en train de converger vers un standard partagé par l'ensemble des entreprises. Cette convergence peut venir des actions de benchmarking qu'elles effectuent et qui les poussent à mettre en place ce que les autres entreprises du CAC40 mettent en oeuvre. Cette standardisation des systèmes peut venir d'un esprit de mimétisme et corporatisme entre ces entreprises qui les renforcent face aux forces externes, en faisant front commun. Chauvey et al. (2015) ont déjà montré une normativité croissante dans les discours des entreprises françaises sur la RSE durant la période 2004-2010. Seul l'historique, sous forme de système Internet de communication RSE existant précédemment, justifie partiellement les différences entre les systèmes de communication RSE actuels. Nous estimons que dans le futur, ces mêmes différences historiques s'effaceront. Nous avançons que cette tendance donnera des systèmes de communication de la RSE Web 1.0 et Web 2.0 encore plus similaires entre les différentes entreprises du CAC40. Comme Wheeler et Elkington (2001), nous constatons que les entreprises utilisent peu le potentiel des TI pour leur communication RSE, notamment le Web 2.0. La communication cybernétique (Wheeler & Elkington 2001), interactive et en temps réel (Townsend 2007). Le dialogue effectif sur la RSE dans le cadre d'un double processus de communication et de négociation (Crane & Livesey 2003; Burchell & Cook 2006) qui était nul en 2007, reste très faible en 2009-2010 et en 2014-2015. Pourtant, les organisations ne peuvent se contenter d'uniquement diffuser leurs informations sur la RSE aux parties prenantes (Web 1.0), il est également souhaitable d'instaurer un dialogue effectif sur la RSE dans le cadre d'un double processus de communication et de négociation (du type Web 2.0). Cependant, les coûts de fonctionnement des technologies Internet du web 2.0, surtout en termes de personnel nécessaire pour leur animation (Joyce 2007) et la crainte des attaques des activistes par ces moyens mis à disposition directement par l'entreprise elle-même (Crane & Livesey 2003; Burchell & Cook 2006) constituent des enjeux majeurs en terme de gestion des risques. Il en va de même concernant la relation de confiance entre les entreprises et ses parties prenantes en matière de communication RSE. En l'état de nos investigations longitudinales, la communication RSE des entreprises du CAC40 reste une vitrine peu propice aux débats et à l'interactivité. Taille [t1, t2, ou t3] Performance environnementale et sociale [t1, t2, ou t3] Performance économique et financière [t1, t2, ou t3] Comm. Internet RSE Web 1.0 [t1, t2, ou t3] Comm. Internet RSE Web 2.0 [t1, t2, ou t3] H4.1.[t1 à t2,ou t2 à t3] H4.2.[t1 à t2,ou t2 à t3] H1.1.[t1 a t1, t2 à t2 , ou t3 à t3] H1.2.[t1 a t1, t2 à t2 , ou t3 à t3] H2.1.[t1 a t1, t2 à t2 , ou t3 à t3] H2.2.[t1 a t1, t2 à t2 , ou t3 à t3] H3.1.[t1 a t1, t2 à t2 , ou t3 à t3] H3.2.[t1 a t1, t2 à t2 , ou t3 à t3]

Figure 2 :

 2 Figure 2 : Résultats avec les hypothèses confirmées en gras et soulignées

. Nous proposons cette dernière hypothèse : Le système Internet de communication de la RSE existant précédemment influence positivement le système Internet de communication Internet sur la RSE actuel Web

	Taille [t1, t2, ou t3]	H1.1.[t1 a t1, t2 à t2 , ou t3 à t3]	Comm. Internet RSE Web 1.0 [t1, t2, ou t3]
	H1.2.[t1 a t1, t2 à t2 , ou t3 à t3]
			H4.1.[t1 à t2,ou t2 à t3]
	H2.1.[t1 a t1, t2 à t2 , ou t3 à t3]
	Performance environnementale et sociale [t1, t2, ou t3]
	H2.2.[t1 a t1, t2 à t2 , ou t3 à t3]
			H4.2.[t1 à t2,ou t2 à t3]
	H3.1.[t1 a t1, t2 à t2 , ou t3 à t3]
	Performance économique et financière [t1, t2, ou t3]	H3.2.[t1 a t1, t2 à t2 , ou t3 à t3]	Comm. Internet RSE Web 2.0 [t1, t2, ou t3]
			1.0 (H4.1) et Web 2.0
	(H4.2).

  Instrument de la mesure de la taille La taille des entreprises a été mesurée à partir de trois indicateurs numériques : la valeur du poids indiciel de l'entreprise au sein du CAC 40, le chiffre d'affaires et le nombre d'employés de l'année précédant le début de la collecte de données. 2.2.4. Instrument de mesure de la performance environnementale et sociale Pour notre instrument de mesure de la performance environnementale et sociale, nous avons retenu l'ensemble des indices boursiers reconnus au niveau international dits « développement durable » ou « socialement responsables » (World Health Organization 2005) : ASPI Eurozone index de Vigeo, FTSE4GOOD Europe index de FTSE, ECPI Ethical Europe Tradable Equity index de ECPI, Dow Jones Sustainability Index (DJSI) Europe de CME Group Index Services, et Ethibel Sustainability Index (ESI) de Ethibel. Cinq valeurs dichotomiques sont ainsi obtenues pour chaque entreprise : l'entreprise est incluse ou exclue de l'indice au début de chaque collecte des données. 2.2.4. Instrument de mesure de la performance économique et financière Les huit indicateurs synthétiques économiques et financiers les plus communs ont été pris en compte : rentabilité financière, taux de marge brut, taux de marge net, levier financier, rentabilité du capital investit, variation du chiffre d'affaires, taux de rentabilité et ratio de capitalisation des fonds propres. Huit valeurs numériques résultent de cet instrument de mesure pour chaque entreprise, un pour chaque indicateur, à l'exception des 6 entreprises financières et immobilières du CAC40 pour lesquelles certains indicateurs ne sont pas comparables avec les autres entreprises du CAC40.

	LES SYSTÈMES DE COMMUNICATION INTERNET DES ENTREPRISES DU
	CAC40 : UNE ÉTUDE LONGITUDINALE (2007-2015)	11
	2.0	Présence d'un service de wiki spécifique sur la RSE	Oui = 1 ou Non = 0
	2.0	Présence d'un service de podcast spécifique sur la RSE	Oui = 1 ou Non = 0
	2.2.3.

  , les sites web institutionnels de cinq sociétés du CAC40, choisies au hasard, ont été analysés séparément par chacun des trois chercheurs. La confrontation des résultats obtenus a permis d'affiner les points problématiques du cadre d'analyse et d'homogénéiser, entre les auteurs et l'assistant, les critères d'évaluation. Ensuite, chaque site web institutionnel des restantes 35 sociétés du CAC40 ont été analysées en les répartissant entre les différents auteurs. La liste des sociétés du CAC40 est revue régulièrement. Nous avons gardé dans notre échantillon les entreprises qui en étaient exclues, nous avons ajouté les entreprises qui étaient incluses et nous avons appliqué les fusions et scissions concernant les entreprises de notre échantillon, pour un total de 47 entreprises à la dernière collecte de données. La première analyse des sites des entreprises du CAC40 s'est échelonnée entre septembre et octobre 2007, la deuxième entre octobre 2009 et janvier 2010 et la troisième entre octobre 2014 et janvier 2015. Le test des hypothèses a été effectué principalement par des mesures de corrélation entre chaque variable indépendante et chaque variable dépendante.

Le "green IT" regroupe les initiatives et programmes en matière de système d'information qui concernent directement ou indirectement la soutenabilité environnementale[START_REF] Siegler | A practical approach to Green IT[END_REF]

comme Développement Durable, Développement Soutenable, Responsabilité Sociale et Environnementale, Mécénat, Environnement, Changement Climatique, Fondation

RÉFÉRENCES BIBLIOGRAPHIQUES