Nicolas Leprovost

Eun-Jin Kim

Effect of rotation on the tachoclinic transport

Keywords: Turbulence -Sun, interior -Sun, rotation

Aims. We study the effect of rotation on sheared turbulence due to differential rotation in the solar tachocline. Methods. By solving quasi-linear equations for the fluctuating fields, we derive turbulence amplitude and turbulent transport coefficients (turbulent viscosity and diffusivity), taking into account the effects of shear and rotation on turbulence. We focus on the regions of the tachocline near the equator and the poles where rotation and shear are perpendicular and parallel, respectively. Results. For parameter values typical of the tachocline, we show that the shear reduces both turbulence amplitude and transport, more strongly in the radial direction (parallel to the shear) than in the horizontal one, resulting in anisotropic turbulence. Rotation further reduces turbulence amplitude and transport at the equator whereas it does not have much effect near the pole. The interaction between shear and rotation is shown to give rise to a novel non-diffusive flux of angular momentum (known as the Λ-effect), possibly offering a mechanism for the occurrence of a strong shear region in the solar interior. Further implications for the transport in the tachocline are discussed.

Introduction

Solar rotation, both global and differential, plays a crucial role in the dynamical processes taking place in the Sun. For example, mean-field dynamo theory [START_REF] Moffatt | Magnetic field generation in fluids (CUP) Parker, E. N[END_REF][START_REF] Krause | Mean field MHD and dynamo theory[END_REF] uses these two ingredients to explain the presence of the solar magnetic field. Firstly, the differential rotation in the tachocline is responsible for the in-situ generation of toroidal field via the Ω effect. Secondly, convection under the influence of (global) rotation leads to helical motion in the convection zone which permits the creation of poloidal fields via the α effect.

To understand the differential rotation in the convective zone, many authors investigated the structure of turbulence in rotating bodies. The main feature of this type of turbulence is the appearance of non-diffusive term in the transport of angular momentum which prevents a solid body rotation from being a solution of the Reynolds equation (Lebedinsky 1941;Kippenhahn 1963). Starting from Navier-Stokes equation, it is possible to show that these fluxes arise when there is a cause of anisotropy in the system, either due to an anisotropic background turbulence (see Rüdiger 1989, and references therein) or due to inhomogeneity such as an underlying stratification (Kichatinov 1987). To explain the observed internal differential rotation and the depletion of light elements on the surface of the Sun, it is also of prime importance to understand the influence of rotation on turbulent transport coefficients such as the turbulent heat conductivity and the turbulent diffusivity of particles.

The purpose of this Letter is to investigate the effect of (global) rotation on the tachocline transport. In our previous works, we have studied the turbulent transport in the tachocline by taking into account the crucial effect of shearing, the socalled shear stabilisation, due to a strong radial differential rotation (Kim 2005;Leprovost & Kim 2006). We have also incorporated the interaction of this sheared turbulence with different types of waves that can be excited in the Sun due to magnetic fields (Leprovost & Kim 2007) or stratification (Kim & Leprovost 2006). In this Letter, we elucidate the effect of global rotation on sheared turbulence by studying a (local) Cartesian model valid near the equator and the poles. We consider a turbulence driven by an external forcing such as plumes from the convection zone and perform a quasi-linear analysis to derive the dependence of turbulence amplitude and transport on rotation and shear. Compared to two-dimensional turbulence studied in Leprovost & Kim (2007), global rotation supports the propagation of inertial waves in three dimensions (3D), which interact with the shear flow, playing an important role in the overall turbulent transport. In particular, we report a novel result that the momentum transport can be not only due to eddy viscosity but also to non-diffusive Λ-effect in the tachocline. Non trivial Λ-effect here results from an anisotropy induced by shear flow on the turbulence even when the driving force is isotropic, in contrast to the case without shear flow where this effect exists only for anisotropic turbulence (Kichatinov 1987).

Model

To elucidate the effect of rotation on sheared turbulence, we model the tachocline by an incompressible fluid in local Cartesian coordinates (x, y, z) in a rotating frame (see Fig. 1) with average rotation rate Ω making an angle θ with a mean shear flow in the azimuthal direction: U 0 = -xA ĵ. We study the effect of this large-scale shear on the transport properties of turbulence by assuming the velocity as a sum of a radial shear (i.e. in the x-direction) and fluctuations:

u = U 0 + v = U 0 (x) ĵ + v = -xA ĵ + v.
We resort to the quasi-linear approximation [START_REF] Townsend | The structure of turbulent shear flow[END_REF][START_REF] Moffatt | Magnetic field generation in fluids (CUP) Parker, E. N[END_REF], also called first-order smoothing (FOSA) or second-order correlation approximation (SOCA), where the product of fluctuations is neglected to obtain the following equations for the evolution of the fluctuating velocity field:

∂ t v + U 0 • ∇v + v • ∇U 0 = -∇p + ν∇ 2 v + f -Ω × v , (1) ∇ • v = 0 ,
where p and f are respectively the small-scale components of the pressure and forcing, and Ω ≡ 2 Ω.

To study the influence of rotation and shear on the particle and heat transport, we have to supplement Eq. ( 1) with an advection-diffusion equation for these quantities. We here focus on the transport of particles since a similar result also holds for the heat transport. The density of particles is also written as a sum of a large-scale component N 0 and fluctuations n. Using the quasi-linear approximation, the fluctuating concentration of particles is governed by the following equation:

∂ t n + U 0 • ∇n + v • ∇N 0 = D∇ 2 n , ( 2
)
where D is the molecular diffusivity of particle. Note that, in the case of heat equation, D should be replaced by the molecular heat conductivity χ.

As the large-scale velocity is in the y (azimuthal) direction, we are mostly interested in the momentum transport in that direction. The equation for the (large-scale) azimuthal velocity U 0 is then given by Navier-Stokes equation with the contribution from fluctuations given by ∇ • R, where R = vv y is the Reynolds stress. One can formally Taylor expand R with respect to the gradient of the large-scale flow:

R i = Λ i U 0 -ν T ∂ x U 0 δ i1 + . . . = Λ i U 0 + ν T A δ i1 + . . . . (3)
where we introduced two coefficients Λ i and ν T . The effect of the turbulent viscosity ν T is simply to change the viscosity from the molecular value ν to the effective value ν + ν T . In comparison, the first term Λ i U 0 in Eq. ( 3) is proportional to the velocity rather than its gradient. This means that it does not vanish for a constant velocity field and can thus lead to the creation of gradient in the velocity field. This term is equivalent to the α-effect in dynamo theory (Parker 1955;Steenbeck & Krause 1966), and has been known as the Λ-effect [START_REF] Krause | [END_REF] or anisotropic kinetic alpha (AKA)-effect (Frisch et al. 1987).

Similarly, the transport of species in the large scales is governed by an advection diffusion equation where the molecular diffusivity is supplemented by a turbulent diffusivity, defined as

v i n = -D i j T ∂ j N 0 .
To calculate the turbulence amplitude and transport coefficients (Reynolds stress and turbulent diffusivity), we prescribe the forcing in Eq. ( 1) to be incompressible, isotropic and short correlated in time (modelled by a δ-function) with a power spectrum F. Specifically, we assume:

fi (k 1 , t 1 ) f j (k 2 , t 2 ) = τ f (2π) 3 δ(k 1 + k 2 ) δ(t 1 -t 2 ) × (4) F(k) (δ i j -k i k j /k 2 ) .
The angular brackets stand for an average over realisations of the forcing, and τ f is the (short) correlation time of the forcing. We solve Eqs. ( 1) and (2) in the case of unit Prandtl number (ν = D) for simplicity, and use the results of these and Eq. ( 4) to compute the turbulent transport. In this Letter, we focus on the case relevant to the Sun and discuss the implications of our findings for the dynamics of the tachocline. Note that in the tachocline, |A| ∼ 3 × 10 -6 s -1 by using the upper limit on the tachocline thickness (5% of the solar radius) while the average rotation rate of the interior is Ω ∼ 2.6 × 10 -6 s -1 . Therefore, the ratio of rotation to shear is very close to unity. However, if the tachocline is found to be thinner, the value of the shearing rate A becomes larger, with |Ω/A| < 1. Furthermore, with molecular viscosity of ν ∼ 10 2 cm 2 s -1 , ξ = νk 2 y /|A| is a small parameter for a broad range of reasonable length scales in the azimuthal direction, L y > 10 4 cm. Thus, the present results are valid for slow rotation (Ω ≪ |A|) and strong shear (ξ ≪ 1).

Near the equator (θ = π/2)

Near the equator, the rotation and the direction of the shear are orthogonal. Expanding the velocity in powers of Ω = Ω/A, we compute the turbulence amplitude and transport coefficients up to second order in | Ω| ≪ 1.

Turbulence intensity

In the strong shear limit (ξ ≪ 1), using similar algebra as in Kim (2005), we obtain the turbulent intensity in the radial direction as follows:

v 2 x = τ f A d 3 k (2π) 3 F(k)       L 0 (k) + Ω k 2 z k 2 y L 1 (k)       . ( 5
)
Here, k 2 H = k 2 y + k 2 z is the (squared) amplitude of the wave number in the horizontal plane, and L 0 and L 1 are two positive definite integrals depending only on the wave number k. Note that the details of L 0 and L 1 are not important for our discussions. Therefore, the turbulence intensity v 2

x in Eq. ( 5) increases for Ω > 0 whereas it decreases for Ω < 0. This is because a weak rotation destabilises sheared turbulence for Ω > 0 whereas it stabilises for Ω < 0 (Bradshaw 1969;[START_REF] Salhi | [END_REF].

For the other components of the turbulence amplitude, we obtain:

v 2 z ∼ τ f A d 3 k (2π) 3 F(k)L 2 (k) 3 2ξ 1/3 × (6)       Γ(1/3) + Ω k 2 z k 2 y Γ(4/3)(-ln ξ)       .
Here, Γ is the Gamma function and L 2 is a positive definite function of k. Here again, the turbulence amplitude can increase or decrease depending on the sign of Ω. Furthermore, the correction due to rotation has a logarithmic dependence on the shear, contrary to the case of the radial amplitude. Note that the anisotropy in the first correction is less pronounced than in the leading order term. As a result, the turbulence due to shearing becomes less anisotropic. This illustrates the tendency of rotation to lead to almost isotropic turbulence. An equation similar to ( 6) is also found for v 2 y with a slightly different function L 2 . However, performing the angular integration, we find that v 2 y is larger than v 2 z , in agreement with numerical simulations (Lee et al. 1990). This effect is present for Ω = 0 and is thus caused by the shear only.

Transport of angular momentum

In the strong shear limit (ξ ≪ 1), the transport of angular momentum can be found as a sum of two terms v x v y = ν T A+Λ x Ω, the first and second being even and odd with the rotation rate, respectively. The first term, the turbulent viscosity, takes the following form:

ν T ∼ τ f A 2 d 3 k (2π) 3 F(k) - 1 2 + L 3 (k) , (7)
where L 3 is a positive definite function of k. For Ω = 0 we recover the result of Kim (2005) showing that the turbulent viscosity is reduced proportionally to A -2 for strong shear. The turbulent viscosity can be either positive or negative depending on the relative magnitude of the two terms inside the integral.

In the 2D limit (where L 3 = 0), we can easily see that the turbulent viscosity is negative. On the contrary, in the case of an isotropic forcing in 3D, the turbulent viscosity is positive.

The correction due to the rotation is proportional to Ω and is thus odd in the rotation. This is the so-called Λ-effect, a nondiffusive contribution to Reynolds stress, which can be shown to be:

Λ x ∼ τ f A 2 d 3 k (2π) 3 F(k)L 4 (k) (-ln ξ) . ( 8
)
It is important to emphasise that this non trivial Λ-effect results from an anisotropy induced by shear flow on the turbulence even when the driving force is isotropic. This should be contrasted to the case without shear flow where non-diffusive fluxes emerge only for anisotropic forcing (Rüdiger 1980;Kichatinov 1986). In our case, the anisotropy in the velocity field is not artificially introduced in the system but is created by the shear and calculated self-consistently.

Transport of particles

In the strong shear limit (ξ ≪ 1), the computation of nv gives:

D xx T ∼ τ f A 2 d 3 k (2π) 3 F(k)L 5 (k)       1 + Ω k 2 z k 2 y -ln ξ 3       , ( 9
)
D zz T ∼ τ f A 2 d 3 k (2π) 3 F(k)L 6 (k) 3 2ξ 2 3       1 + 2 Ω k 2 z k 2 y -ln ξ 3       , ( 10
)
where L 5 and L 6 are positive definite functions. Eqs. (9-10) show that the effect of rotation on the transport of particles depends on the sign of Ω: for Ω > 0, the transport is increased whereas it is reduced when Ω < 0. This is again because a weak rotation destabilises sheared turbulence for Ω > 0 whereas it stabilises for Ω < 0. Note that a similar behaviour was also found in turbulence intensity, given in Eqs. ( 5) and ( 6). The correction term due to rotation in D xx T and D zz T in Eq. (9-10) depends weakly on the shear by a logarithmic factor | ln ξ|, which cannot be too large even for ξ ≪ 1, and is of the same order for transport in different directions. Thus, the scaling of the turbulent diffusivity is roughly the same as that in the case without rotation: the radial transport (∝ A -2 ) is more reduced than the horizontal one (∝ A -4/3 ). This result should be contrasted to the rapid rotation limit where the transport in the radial direction was larger (but only by a factor 2) than the one in the horizontal direction [START_REF] Rüdiger | Differential rotation and stellar convection (Gordon and Breach[END_REF]). These results thus highlight the crucial role of shear in transport, in particular, by introducing anisotropy.

Near the poles (θ = 0)

Near the poles, the directions of rotation and shear can be taken to be parallel. Contrary to the case at the equator, we find that the leading order correction (proportional to Ω) vanishes in the case of an isotropic forcing (due to the fact that these terms are odd in one of the components of the wave number) for all the previously calculated quantities. Consequently, the turbulent amplitude, viscosity and diffusivity are given by Eqs. (5-6), ( 7) and (9-10) with Ω = 0, respectively. Similarly, the Reynolds stress involving the radial component ( v x v y ) vanishes. However the component of the Reynolds stress involving the latitudinal velocity ( v y v z ) does not vanish and is odd in Ω. Thus, the Λ-effect appears here in v y v z (recall that at the equator, the Λ-effect was present only in v x v y ), and takes the following form:

Λ z ∼ - τ f A 2 d 3 k (2π) 3 F(k) 3 2ξ 2/3 [L 7 (k) -L 8 (k)] . (11)
Eq. ( 11) shows that Λ z is of indefinite sign, as L 7 and L 8 are two positive definite functions but appear with different signs. However, as for Eq. ( 8), in the case of an isotropic forcing, the function L 7 dominates over L 8 , leading to a negative Λ z . This Λ-effect arises even in the case of an isotropic forcing as the shear favours fluctuations in the y-direction compared to that in the z-direction (see §3.1), leading to anisotropic turbulence in the plane perpendicular to the rotation. Eq. ( 11) also shows that Λ z is larger (scaling as A -4/3 ) than Λ x in the equatorial case.

Implications for the tachocline

Table 1 summarises the scaling of our results with the shear A and rotation Ω in the limit of weak rotation (Ω ≪ |A|) and strong shear (ξ = νk 2 y /|A| ≪ 1).

Equator Pole

v 2 x A -1 1 + c Ω A -1 v 2 y ∼ v 2 z A -2/3 1 + c Ω| ln ξ| A -2/3 ν T A -2 A -2 Λ x A -2 | ln ξ| 0 Λ z 0 -A -4/3 D xx T A -2 1 + c Ω| ln ξ| A -2 D yy T ∼ D zz T A -4/3 1 + c Ω| ln ξ| A -4/3
Table 1. Scaling of turbulence amplitude, turbulent viscosity (ν T ), Λ-effect and turbulent diffusivity with the shear A and rotation Ω = Ω/A. c is a positive constant of order unity.

The first two rows show that, both near the equator and poles, the turbulence amplitude in the radial direction is more reduced by the shear than in the horizontal one, by a factor of A -1 and A -2/3 , respectively. These results thus imply an effectively stronger turbulence in the horizontal (y-z) plane than the one in the radial (x) direction. Furthermore, at the equator, Ω < 0 as the shear increases from the interior towards the convective zone. Thus, the turbulence amplitude is further reduced by the rotation in all directions, but the turbulence in the radial direction is more suppressed than the one in the horizontal direction by a factor ln ξ.

A novel result is also found for the Reynolds stress, which involves two contributions: the turbulent viscosity and the Λeffect. The latter, a source of non-diffusive flux, is present even with isotropic forcing due to the shear-induced anisotropy (Kim 2005). Note that for an isotropic forcing, the turbulent viscosity is found to be positive and therefore cannot act on its own as a source of differential rotation in the tachocline. The radial differential rotation can, however, be created by the Λ-effect: since it does not depend only on the gradient of angular velocity, it does not vanish for a uniform rotation and thus prevents the uniform rotation from being a solution of the large-scale momentum equation. To determine the sign of A as a result of the Λ-effect, we seek a solution of the large-scale turbulent equations by demanding the Reynolds stress to vanish (e.g. Kichatinov 1987). In the equatorial case, we then obtain the following equation for the shear:

ν T (A)A + Λ(A)Ω = 0 . ( 12
)
As both ν T and Λ are positive, A must be negative. Interestingly, this is exactly what the observation indicates (a rotation rate decreasing towards the interior at the equator). We cannot use such a simple equation to predict the sign of the shear at the pole as the Λ-effect now appears in the v y v z component of the Reynolds stress and is found to scale as A -4/3 , which is larger than Λ x (∝ A -2 ) near the equator. Furthermore, we find that the transport of chemical species is reduced by shear and more severely in the the radial direction than the horizontal one (by a factor A -2 and A -4/3 respectively), with a further, but weak reduction due to rotation at the equator only. As a result, the radial mixing (in the x direction) of chemicals is slightly more efficient at the pole than at the equator. These results thus suggest that turbulent mixing of light elements is weak (due to shearing) in the tachocline and also that the depletion of light elements on the solar tachocline depends on latitude. However, this latitudinal dependency is unlikely to be observed on the surface due to the rapid mixing in the convection zone (Spruit 1977). Since a similar result applies to heat transport, a faster heat transport at the pole could contribute to making hotter poles than equator at the solar surface. Finally, we note that all our results here are valid in the absence of stratification and magnetic fields (Kim & Leprovost 2006) and will extend our study to include these in future publications. Of particular interest would be the sign of ν T and Λ, and thus the resulting sign of A.

Fig. 1 .

 1 Fig. 1. The configuration in our model: A, Ω and θ are the shearing rate, the rotation rate and the co-latitude, respectively.

Acknowledgements. This work was supported by U.K. PPARC Grant No. PP/B501512/1.


Yannick Masson

Laurent Monasse

Existence of global Chebyshev nets on surfaces of absolute Gaussian curvature less than 2π

Keywords: Mathematics Subject Classification (2010), 53C21; 58G30, Keywords, Chebyshev nets, curvature bounds

We prove the existence of a global smooth Chebyshev net on complete, simply connected surfaces when the total absolute curvature is bounded by 2π. Following Samelson and Dayawansa, we look at Chebyshev nets given by a dual curve, splitting the surface into two connected half-surfaces, and a distribution of angles along it. An analogue to the Hazzidakis' formula is used to control the angles of the net on each half-surface with the integral of the Gaussian curvature of this half-surface and the Cauchy boundary conditions. We can then prove the main result using a theorem about splitting the Gaussian curvature with a geodesic, obtained by Bonk and Lang.

Introduction

In this paper we call surface a Riemannian 2-manifold, whose metric will be denoted g, and we consider complete, simply connected surfaces. A Chebyshev net Φ on a surface is a parameterization of the surface satisfying [START_REF] Ya | Chebyshev nets in manifolds of bounded curvature[END_REF] which means that length is preserved in two directions, called the primal coordinates. Let Ω(u 1 , u 2 ) be the angle between the coordinate curves at Φ(u 1 , u 2 ).

∀(u 1 , u 2 ) ∈ R 2 , ∂ 1 Φ(u 1 , u 2 ) g(Φ(u 1 ,u 2 )) = ∂ 2 Φ(u 1 , u 2 ) g(Φ(u 1 ,u 2 )) = 1,
A question of interest both for the existence of bi-Lipschitz maps [START_REF] Yu | Remarks on Chebyshev coordinates[END_REF] and for applications in textile or architectural shape design [START_REF] Ghys | Sur la coupe des vêtements: variation autour d'un thème de Tchebychev[END_REF][START_REF] Douthe | Form-finding of a grid shell in composite materials[END_REF] is the existence of a Chebyshev net on a given surface. While local existence of a Chebyshev net holds for all surfaces [START_REF] Bianchi | Lezione di geometria differenziale[END_REF], two approaches have been explored to obtain global existence. The first approach relies on a geometric proof on more general Alexandrov surfaces. The first result on global existence of a Chebyshev net on complete and simply connected Alexandrov surfaces has been obtained by Bakelman [START_REF] Ya | Chebyshev nets in manifolds of bounded curvature[END_REF]. Existence is proved on every sector Q α delimited by two geodesics crossing at an angle α such that Qα K + < α and Qα K -< π -α, where K + and K -denote, respectively, the positive and negative parts of the Gaussian curvature K. A global Chebyshev net is then obtained on the surface for α = π/2. Burago, Ivanov and Malev [START_REF] Yu | Remarks on Chebyshev coordinates[END_REF] improved this result by showing that a complete and simply connected Alexandrov surface S admits a global Chebyshev net under the relaxed constraints S K + < 2π and S K -< 2π, but this net is not necessarily smooth on a Riemannian surface as the coordinate curves might present kinks at the boundaries between adjacent sectors.

A second analytical approach was developed by Samelson and Dayawansa [START_REF] Samelson | On the existence of global Tchebychev nets[END_REF] using the dual coordinates. A geodesic cutting the surface into two connected components R + and R -is chosen as a dual curve. In each connected component, existence and uniqueness of a smooth solution to Servant's equations (see [START_REF] Douthe | Form-finding of a grid shell in composite materials[END_REF] below) in the dual coordinates is proved for any smooth distribution of angles along the dual curve. Then, choosing a constant angle π/2 along the geodesic and using an estimate on the angle Ω of the net on R ± , it is shown that the solution to Servant's equations is a diffeomorphism under the condition that R ± K + < π/2 and R ± K -< π/2. In the present paper, we add some arguments to the proof to improve this result by relaxing the constraints on the integrals of the Gaussian curvature. Our main result is the following: Theorem 1.1. Let S be a complete, simply connected, C ∞ surface. Suppose that S |K| < 2π. Then S admits a global Chebyshev net.

Let us note that the Hazzidakis' formula [START_REF] Hazzidakis | Ueber einige Eigenschaften der Flächen mit constantem Krümmungsmaass[END_REF] suggests that this result is optimal in the sense that a smooth global Chebyshev net does not exist on a complete, simply connected surface S such that | S K| > 2π.

The paper is organized as follows. In Section 2, we restate two existing results in a form that is useful for our purpose. The first one is that the estimates obtained on half the surface in [START_REF] Samelson | On the existence of global Tchebychev nets[END_REF] can be doubled using a theorem proved by Bonk and Lang [START_REF] Bonk | Bi-lipschitz parameterization of surfaces[END_REF] stating that the surface can be split by a geodesic into two connected components (called half-surfaces) such that each component contains half of the integral of K + and K -on the surface. The second result concerns the existence (and uniqueness) of a smooth solution to Servant's equations proved in [START_REF] Samelson | On the existence of global Tchebychev nets[END_REF] which we restate by specifying the boundary conditions from the choice of a dual curve and an arbitrary smooth angle distribution on that curve. Then in Section 3, we prove the existence of a Chebyshev net on each half-surface. To that end, we first observe that it is sufficient to keep the angles Ω of the net uniformly away from 0 and π for Φ to be a global diffeomorphism. This means that no nonlocal self-intersection of the parameterization can occur when Φ is locally injective everywhere. Proving an analogue to the Hazzidakis' formula in the dual coordinates, we are able to choose an adequate (constant) angle distribution on the dual geodesic curve to control the angles in each half-surface. Finally, we conclude the proof in Section 4 by assembling the intermediate results.

Preliminaries

In this section, we restate two existing results on splitting the surface into two components and on the solution to Servant's equations.

Splitting the surface into two components

We give a restatement of [3, prop. 6.1] applied to the continuous nonnegative function K -. Proposition 2.1. Let S be a smooth, complete surface homeomorphic to the plane. Suppose that S K + < 2π and S K -< ∞. Then there exists a properly embedded, complete geodesic γ, splitting S into two connected components S 1 and S 2 such that

S1 K + = S2 K + , S1 K -= S2 K -.
(2)

Solution to Servant's equations

Following Samelson and Dayawansa [START_REF] Samelson | On the existence of global Tchebychev nets[END_REF], we introduce the change of variables

D : (u 1 , u 2 ) → (x 1 , x 2 ) = (u 1 -u 2 , u 1 + u 2 ) to the dual coordinates.
We use upper case letters Φ and Ω for the primal parameterization and angle between primal coordinate curves, lower case letters ϕ = Φ • D -1 and ω = Ω • D -1 for their dual counterparts. The first fundamental form in a Chebyshev net is then

g = (du 1 ) 2 + 2 cos(Ω(u 1 , u 2 ))du 1 du 2 + (du 2 ) 2 (3) = sin 2 ω(x 1 , x 2 ) 2 (dx 1 ) 2 + cos 2 ω(x 1 , x 2 ) 2 (dx 2 ) 2 . (4)
A derivation of (1) along coordinates (u 1 , u 2 ) leads to Servant's equations, expressed here in the dual coordinates (x 1 , x 2 ),

∂ 2 22 ϕ i -∂ 2 11 ϕ i + 2 k,l=1 Γ i k,l (ϕ) ∂ 2 ϕ k ∂ 2 ϕ l -∂ 1 ϕ k ∂ 1 ϕ l = 0, for i = 1, 2, ( 5
)
where Γ i k,l denote the Christoffel symbols, together with the Cauchy boundary conditions ϕ(x 1 , 0) = γ(x 1 ),

∂ 2 ϕ(x 1 , 0) = ψ(x 1 ), ∀x 1 ∈ R, (6)
for given functions γ, ψ : R → S. The following result on smooth solutions with smooth boundary conditions is proved in [START_REF] Samelson | On the existence of global Tchebychev nets[END_REF].

Proposition 2.2. Assume that γ, ψ are C ∞ functions. Then there exists a unique C ∞ solution ϕ to (5)-( 6).

The primal parameterization Φ = ϕ • D, where ϕ is the solution to ( 5)-( 6), is then a Chebyshev net if and only if (1) is satisfied by Φ on the boundary. This is expressed, on the dual coordinates, by

   ∂ 1 ϕ(x 1 , 0), ∂ 2 ϕ(x 1 , 0) g(ϕ(x 1 ,0)) = 0, ∂ 1 ϕ(x 1 , 0) 2 g(ϕ(x 1 ,0)) + ∂ 2 ϕ(x 1 , 0) 2 g(ϕ(x 1 ,0)) = 1, ∀x 1 ∈ R. (7)
But ( 7) is equivalent to the existence of an arc length parameterized curve γ and an angle distribution ω : R → (0; π) along γ such that

     ϕ(x 1 , 0) = γ(α(x 1 )), ∂ 2 ϕ(x 1 , 0) = cos ω(x 1 ) 2 n(α(x 1 )), ∀x 1 ∈ R, (8)
where t(x 1 ) is the tangential vector to γ at x 1 , n(x 1 ) is the normal vector to this curve at x 1 such that (t(x 1 ), n(x 1 )) is positively oriented, and

α(x 1 ) = x 1 0 sin ω(s) 2 ds.
Moreover, we have ω(x 1 ) = ω(x 1 , 0) where ω is the angle between the primal coordinate curves. The Cauchy boundary conditions for Servant's equations ( 5) satisfying the Chebyshev property ( 1) are now prescribed by specifying • an arc length parametrized curve γ : R → S;

• an angle distribution ω : R → (0; π) uniformly bounded away from 0 and π.

We will say that a map ϕ : R 2 → S is a local Chebyshev net on S if it satisfies Servant's equations [START_REF] Douthe | Form-finding of a grid shell in composite materials[END_REF] with Cauchy boundary conditions [START_REF] Hazzidakis | Ueber einige Eigenschaften der Flächen mit constantem Krümmungsmaass[END_REF].

Existence of a Chebyshev net on each half-surface

In this section we prove the existence of a Chebyshev net on each halfsurface. The key ingredient is a new Hazzidakis-type formula on the dual (see Lemma 3.2) which allows us to specify a suitable angle distribution along the geodesic (see Lemma 3.4).

Global injectivity of the map ϕ

Lemma 3.1. Let S be a complete, simply connected, C ∞ surface. A mapping ϕ : R 2 → S satisfying (1) is a global Chebyshev net if it satisfies ∃ε > 0 s.t. ε < ω(x 1 , x 2 ) < π -ε, ∀(x 1 , x 2 ) ∈ R 2 (9)
where ω is the angle of the map defined in (3).

Proof. Let ds 2 be the metric (3) associated with ϕ. Since ω satisfies ( 9), (R 3.2 Hazzidakis' formula on the dual Lemma 3.1 shows that the map ϕ can be proved to be a diffeomorphism by deriving a uniform estimate on the angle ω of the net. To this purpose, we derive an equivalent to the Hazzidakis' formula in the dual coordinates. We derive the result from the classical relation (see for instance [START_REF] Stoker | Differential Geometry, volume XX of Pure and Applied Mathematics[END_REF]):

∂ 2 12 Ω(u 1 , u 2 ) = -K Φ(u 1 , u 2 ) sin(Ω(u 1 , u 2 )). ( 10
)
Lemma 3.2. Let ϕ be a local Chebyshev net of S. Define the three points A = ϕ(x 0 , y 0 ), B = ϕ(x 0 + h, y 0 ), and C = ϕ(x 0 + h 2 , y 0 + h 2 ). Consider the dual line γ = ϕ(•, y 0 ) (containing the points A and B) and let ABC be the triangle delimited by the two primal lines AC and BC and the dual line γ, depicted on Figure 1. Then the following holds:

∀(x 0 , y 0 ) ∈ R 2 , ∀h > 0, ω(x 0 + h 2 , y 0 + h 2 ) = ω(x 0 , y 0 ) + ω(x 0 + h, y 0 ) 2 + AB k γ - ABC K (11)
where ω is the angle of the net introduced in (3) and k γ is the geodesic curvature of γ defined as

k γ = - 1 |∂ 1 ϕ| g(ϕ) ∂ 1 ∂ 1 ϕ |∂ 1 ϕ| g(ϕ) , ∂ 2 ϕ |∂ 2 ϕ| g(ϕ) g(ϕ) dual curve γ primal curve primal curve π-ω A 2 π-ω B 2 ω C A B C Γ +
∂ 2 ω(•, y 0 ) = 2k γ sin ω(•, y 0 ) 2 . ( 12
)
Proof. Define ∂ x 1 := ∂ 1 ϕ and ∂ x 2 := ∂ 2 ϕ. Using the commutation of ∂ x 1 and ∂ x 2 , we obtain

∂ 1 ∂ x 1 |∂ x 1 | g(ϕ) , ∂ x 2 g(ϕ) = ∂ 1 ∂ x 1 |∂ x 1 | g(ϕ) , ∂ x 2 g(ϕ) + 1 2 |∂ x 1 | g(ϕ) ∂ 2 |∂ x 1 | 2 g(ϕ) = -|∂ x 2 | g(ϕ) |∂ x 1 | g(ϕ) k γ + 1 2 |∂ x 1 | g(ϕ) ∂ 2 sin 2 ω 2 . Moreover, ∂ x 1 , ∂ x 2 g(ϕ) = 0, |∂ x 1 | g(ϕ) = sin( ω 2 ) and |∂ x 2 | g(ϕ)
= cos( ω 2 ) due to (4), so that (12) holds.

Proof. (Hazzidakis' formula): Let (u 0 , v 0 ) = D -1 (x 0 , y 0 ). Let Γ + be the triangle delimited by the dual curve joining A to B and the two primal curves joining B to C and C to A (see Figure 1):

Γ + = ϕ (x, y) ∈ R 2 | x 0 ≤ x ≤ x 0 + h, y 0 ≤ y ≤ y 0 + h 2 -x -x 0 -h 2 = Φ (u, v) ∈ R 2 | u ≤ u 0 + h 2 , v ≤ v 0 , u + v ≥ u 0 + v 0 .
Integrating [START_REF] Stoker | Differential Geometry, volume XX of Pure and Applied Mathematics[END_REF] on Γ + leads to

- Γ + K = u0+ h 2 u0 v0 u0+v0-u ∂ 2 12 Ω(u, v)dvdu = u0+ h 2 u0 (∂ 1 Ω(u, v 0 ) -∂ 1 Ω(u, u 0 + v 0 -u)) du = Ω(u 0 + h 2 , v 0 ) -Ω(u 0 , v 0 ) - 1 2 u0+ h 2 u0 (∂ 1 -∂ 2 )Ω(u, u 0 + v 0 -u)du - 1 2 u0+ h 2 u0 (∂ 1 + ∂ 2 )Ω(u, u 0 + v 0 -u)du = Ω(u 0 + h 2 , v 0 ) -Ω(u 0 , v 0 ) - 1 2 Ω(u 0 + h 2 , v 0 -h 2 ) -Ω(u 0 , v 0 ) - 1 2 x0+y0+h x0+y0 ∂ 2 ω(-y 0 + x, y 0 )dx = Ω(u 0 + h 2 , v 0 ) - 1 2 Ω(u 0 , v 0 ) - 1 2 Ω(u 0 + h 2 , v 0 -h 2 ) - AB ∂ 2 ω 2 sin( ω 2 ) = ω(x 0 + h 2 , y 0 + h 2 ) - 1 2 ω(x 0 , y 0 ) - 1 2 ω(x 0 + h, y 0 ) - AB k γ .
This completes the proof.

K + + max i=1,2 Si K -< π, (13)
then there exists a distribution ω : R → (0; π) such that the solution of Servant's equations [START_REF] Douthe | Form-finding of a grid shell in composite materials[END_REF] given by the Cauchy boundary conditions γ and ω in (8) is a C ∞ -diffeomorphism.

Proof. Lemma 3.2 applied with x 0 = x 1 -x 2 , y 0 = 0 and h = 2x 2 leads to the following estimate:

∀(x 1 , x 2 ) ∈ R 2 , inf

Proof of the main theorem

Let γ be a geodesic splitting S into two connected components S 1 and S 2 , as resulting from Proposition 2.1. Then the hypotheses of Lemma 3.4 are satisfied upon choosing the geodesic γ and the constant distribution of angles ω = π 2 + 1 4 S K as Cauchy boundary conditions in [START_REF] Hazzidakis | Ueber einige Eigenschaften der Flächen mit constantem Krümmungsmaass[END_REF]. This proves the main theorem.

Figure 1 .

 1 Figure 1. Scheme of the Hazzidakis' formula on triangle ABC The proof of Lemma 3.2 hinges on the following lemma. Lemma 3.3. Let ϕ : R 2 → S be a local Chebyshev net of S. The following holds for all y 0 ∈ R:

3. 3

 3 Angle distribution along the dual curve Lemma 3.4. Let γ be a geodesic of S which splits this surface into two connected components S 1 and S 2 . If max i=1,2 Si

K

  + ≤ ω(x 1 , x 2 ) ≤ max i=1,2 Si K -+sup s∈R ω(s).Hence, choosing for instance the constant distribution of angles along γ such that ωChebyshev net owing to Lemma 3.1.

Aknowledgements

This work has benefited from a French government grant managed by ANR within the frame of the national program Investments for the Future ANR-11-LABX-022-01. The authors are also grateful for the support of Labex Bezout, ANR-10-LABX-58-01. The authors would like to thank Alexandre Ern (CERMICS, University Paris-Est), Laurent Hauswirth (LAMA, University Paris-Est), Olivier Baverel (NAVIER, University Paris-Est) and Arthur Lebée (NAVIER, University Paris-Est) for fruitful discussions and advice.


Angela Minzoni-Deroche

Julie Stal-Le Cardinal

OPEN -MINDEDNESS, EMPATHY AND GENDER ISSUES: GOING BEYOND NATIONAL AND CORPORATE CULTURES

Keywords: Gender, culture, empathy, human behavior, design

It has become commonplace to speak about national culture and corporate culture. In last ICED, a workshop ended up by a debate about the impact of deeply rooted national culture characteristics on designers' attitudes. We decided to focus in this paper on the evolution of the perception of the concept of culture. Does the difference made between national vs. corporate culture still make sense in today's working environments? Which are the new trends regarding the concept of culture when used by corporations in understanding knowledge sharing, diversity or design processes? We present here a gender oriented point of view of the perception of culture in design environment. This proposition is based on the analysis of cross-disciplinary research references about culture and on answers given to a survey by executive women engineers.

INTRODUCTION

In an era of intellectual and geographical globalization, binary debates about the impact of national cultures or corporate cultures on corporate vision, strategy and human relations, seem outdated. Most of the referenced sources indicate that national cultures have a stronger effect on managerial attitudes than corporate culture does or that collected evidence show that individual differences can be much more prominent than cultural differences. But is there any widely shared idea of what corporate culture means, how it emerges and is transmitted, of how long a coherent corporate culture can last? Culture and pragmatic, short lasting business practices and attitudes are they not being mixed up? Do gender approaches, linked to quantitative or qualitative thought streams, can contribute to the rise and development of professional cultures and open mindedness?

In this paper, we focus on main shifts in the definition of culture and their impact in the understanding of factors influencing -even indirectly-design, and explore the way women consider culture within this framework.

METHODOLOGY

During ICED 2007 K12 workshop debates about cognitive and cultural dimensions of collaborative design showed particular interest in questions about the impact of deeply rooted national culture characteristics on designers' attitudes and ways of learning and sharing knowledge. A difference between multi-crossed disciplinary and multi-crossed cultural approaches also emerged during the debate, through participants inputs.

In the following of this workshop's debate, led by Margareta Norell, we found useful to give a chronological perspective of the origin and evolution of the concept of culture, including a specific gender approach.

We were guided by three main research questions: First, is "corporate culture" the right concept for today's and tomorrows corporate organizations facing high design processes expectations? Our hypothesis is that "corporate culture", as it has been stated during the last thirty years, predominantly decreed by CEO's and managers, is not. New emerging trends, targeted, user centred and gender sensitive seem promising. Second, do women apprehend culture in a specific way allowing organizations to go past the binary debate on national culture and corporate culture impacts? Our hypothesis was that it was probable, given women's high empathic capabilities 1 and ease with soft, not to say purely qualitative and insightful, even if not measurable, approaches. This question seemed to us particularly important regarding design research as presented during ICED 2007 conference where almost 70% of authors were male and where some trends, discussed more over, can be identified within women and men design research orientations. Third, to what extent can the recent, less wide covering and more targeted and in depth way to look at cultural issues lead to innovation in the organization of design teams, either virtual or collocated teams? Partly we expected this orientation to be promising because professional focused culture can be learned during initial training and practiced as such , under professional's own steam more than under corporate guidelines .Do women have a particular role to play in this context; are they more prone to explicit culture and to hand it down? These questions were looked at through three kinds of data sources: Answers to a survey (27 questions) sent by email to a panel of executive women working for the most of them in companies where design is strategic. This questionnaire consisted of pre-coded questions (40%) and open questions (60%). The analysis of these data has been focused on the highest scores of similar answers as well as on discourse analysis. The participants of this survey were executive women, having responsibilities for business units with supervisory or organizational strategy responsibilities within international corporations developing high standard design practices.

As these characteristics are restrictive, the population concerned by the study was small scale : 20 women working as business manager, production engineer, consultant, project leader, marketing director, R&D director… Respondents' ages ranged from 27 to 55 years old, most of them having between 35/37 years old. They are French, working for international corporations and all of them held a high level engineering degree. The main aim of the survey was to start an ongoing process to understand women perceptions about culture in their working environment in order to assess this population awareness about research debates on the subject and find out if there are meaningful new ways in which women could put at good use their sensitivity in this area. The first part of the questionnaire concerns the way women perceive corporate culture in their working environment. The kind of questions here is: On which occasions cultural discourses on corporate culture are delivered and by whom? Does your company offer corporate culture trainings? If yes, please describe their contents. It is then suggested to respondents to position themselves on definitions of corporate culture or national culture. For that purpose, we ask them, for instance, which aspects (hierarchical distance, definition of objectives, formal communication, gender issues, attitudes towards errors…) are the most impacted by a culture rather than the other one; either, for a given definition, to say to which culture it relates the most.

The last part of the questionnaire concerns more the national culture, with questions such as: What are the elements of the national culture (where is located your company) which influence your work the most? What are the elements of the culture of your country of birth which slow down (and reinforce) the performance of the individual (and collective) work/team work? Even if a small-fire war has been going for years among researchers discussing the pros and cons of qualitative or quantitative ways of looking at corporate culture, research on this field seems to lean mostly on quantitative data, collected via questionnaires (almost entirely pre coded), issued after a very variable amount of in depth interviews. Surveys are carried out either in a same organization having subsidiaries in different countries, either in different companies of a same industrial sector and in a same country (North American, Japanese, and Danish or Dutch corporations figure among those having shown interest in this kind of comparative research). Even if sometimes observations are carried out by management or management oriented researchers, theses are not recorded in detail, contrary to observations carried out by anthropologists when studying, for example, consumers, final users, operators or directory board's attitudes (Jordan [5]). G. Hofstede et al. [START_REF] Hofstede | Measuring Organizational Cultures : A Qualitative and Quantitative Study Across Twenty Cases[END_REF] (p.289) illustrates an interesting methodological point when reporting on the large scale (116,000 survey questionnaires, 20 different language versions, 72 different national subsidiaries…) IBM study "Paradoxically, the cross-national research in IBM did not reveal anything about IBM's corporate culture, except that it engaged in a survey project of this size". Other articles by Hofstede follow similar core methods and orientation (Hofstede [START_REF] Hofstede | Culture's consequences[END_REF], Hofstede & Hofstede, [START_REF] Hofstede | Cultures and Organizations, Software of the Mind, Intercultural Cooperation and its importance for Survival[END_REF]) Another comment (Flamholtz [4] p.274) illustrates methodological pitfalls "However -replicating research-will require a research site comparable to the Banner Corporation. The research site does not require a whole industry, only the existence of a large number of relatively comparable divisions. Nevertheless, this is difficult to do in practice" Flamholtz, again, points another methodological issue: principles defining the desired culture at Banner Corporation (Flamholtz [4] Appendix A). They seem to be so universal that it becomes difficult to consider them as characteristic of Banner's corporation culture (ex: we will make business decisions with our customer's perspective in mind, open and timely communication on all issues (good and bad) is expected and encouraged, our people are Banner's most valuable assets…). A major risk hanging over understanding and reporting about corporate culture seems to be the degree to which it is really culture what is being spoken about and not only teamwork, accountability or standard business practices.

CULTURE, AN OVERLAPPED OVERLAPPING CONSTRUCT?

Culture, as an operational concept to describe and understand human learning organizations, symbols, religions, rituals, artefacts, settlements or methods of handling problems was first developed within the framework of anthropological research. Within the framework of this article, we can consider three main stages in the evolution of the concept's definition: the first one, starting around 19302 , was quite diversified, including descriptive definitions (Benedict [6], Boas [START_REF] Boas | [END_REF]), historical (Mead, Malinowski [8]) or normative (Kluckhohn & Kelly [9])3 one's.

"that complex whole which includes all the habits acquired by man as a member of a society" (Benedict [START_REF] Benedict | The Science of Custom[END_REF] )

"culture embraces all the manifestations of social habits of a community, the reactions of the individual as affected by the habits of the group in which he lives, and the products of human activities as determined by these habits (Boas [START_REF] Boas | [END_REF]) "culture comprises inherited artefacts, goods, technical processes, ideas, habits and values... culture is a reality sui generis and must be studied as such". (Malinowski [8])

"by culture we mean all those historically created designs for living explicit and implicit, rational, irrational and non rational, which exist at any given time as potential guides for behaviour of men" (Kluckhohn & Kelly [START_REF] Kluckhohn | The Concept of Culture[END_REF])

The second significant stage, starting at the beginning of the 70', attempted to apply the concept of culture to corporate organizations and management, developed a variety of typologies, questionnaires, indicators and measures, embedding the concept of culture in management vocabulary and introducing the debate, on a binary basis, about national and corporate cultures: which of the one or the other would influence the most people at work, their motivation and their willingness and capacity to change. Authors like Pettigrew [START_REF] Pettigrew | On studying organizational cultures[END_REF], Hofstede [START_REF] Hofstede | Measuring Organizational Cultures : A Qualitative and Quantitative Study Across Twenty Cases[END_REF][START_REF] Hofstede | Culture's consequences[END_REF][START_REF] Hofstede | Cultures and Organizations, Software of the Mind, Intercultural Cooperation and its importance for Survival[END_REF], Schein [START_REF] Schein | Organizational culture and leadership[END_REF], Pizam & Reichel [START_REF] Pizam | Culture determinants of managerial behaviour[END_REF], Terprstra & David [START_REF] Terpstra | The Cultural Environment of International Business[END_REF], Deal & Kennedy [START_REF] Deal | Corporate Cultures: The Rites and Rituals of Corporate Life[END_REF] belong to this thought stream.

"culture is the collective programming of the human mind that distinguishes the members of one human group from those of another » (Hofstede [START_REF] Hofstede | Culture's consequences[END_REF]) "a pattern of basic assumptions -invented, discovered, or developed by a given group as it learns to cope with its problems of external adaptation and internal integration -that has worked well enough to be considered valid and, therefore, to be taught to new members as the correct way to perceive, think, and feel in relation to those problems" (Schein [START_REF] Schein | Organizational culture and leadership[END_REF]) Besides personal definitions, authors of this second stage have created multiple typologies, contributing to enlarge vision but also to make culture commonplace. For example, can be cited Hofstede's [START_REF] Hofstede | Culture's consequences[END_REF] initially four and then five dimensions (power distance, collectivism versus individualism, femininity versus masculinity, uncertainty avoidance, long-term versus short-term orientation), Deal & Kennedy's [START_REF] Deal | Corporate Cultures: The Rites and Rituals of Corporate Life[END_REF] four cultural types according to the degree of risk and the feedback speed :work hard play hard culture, tough-guy macho culture, process culture, bet-yourcompany culture, Rousseau's [START_REF] Rousseau | Assessing organizational culture: The case for multiple methods[END_REF] five rings : artefacts, patterns of behaviour, behavioural norms, values, fundamental assumptions, Harrison's (see Hampden-Turner, [START_REF] Hampden-Turner | Charting the Corporate Mind: From Dilemma to Strategy[END_REF]) quadrant model based on the twin axis of Formulation (high-low) and Centralisation (high-low) to give the four cultures of Role, Task, Atomistic (Person) and Power, Kono's [START_REF] Kono | Corporate culture and long-range planning[END_REF] 5 culture types: Vitalised, Follow the leader and vitalised, Bureaucratic, Stagnant, Follow the leader and stagnant, O'Reilly et al.'s [START_REF] O'reilly | People and organizational culture : a profile comparison approach to assessing person-organization fit[END_REF] seven dimensions: innovation and risk taking, attention to detail, outcome orientation, people orientation, individual vs team orientation, aggressiveness, stability… According to Wilkins & Ouchi [START_REF] Wilkins | Efficient cultures : exploring the relationship between culture and organizational performance[END_REF], using the term "culture" for both national and organizational cultures is misleading, both being different phenomena. Despite this insightful statement, both are still called cultures and be compared as if their nature was the same.

It shall be noticed that research about factors influencing virtual teams concentrates on four factors: social, technology, business and government 4 , focalising more on behavior than on culture but, when culture is concerned, despite its youthfulness, research seems to follow the second stage trend, with binary understanding of corporate and national cultures 5 . Main subjects of interest being how to built trust and how to manage temporal virtual coordination and conflicts. In addition, there seems to be an implicit hope that technology (software constraints) will help virtual teams to "erase" what is perceived to be the negative impacts of intercultural communications 6 , not to say physical differences or pronunciation singularities, while keeping "identity" differences supposed to be a source of creativity. However, this field is very recent one and more research initiatives will be needed before cultural virtual patterns can be designed.

The third stage, starting in the mid 90's, focuses on more targeted and "situated" aspects like project management, financial performance, design processes, consumer attitudes, gender…. Authors like Kotter & Haskett [START_REF] Kotter | Corporate Culture and Performance[END_REF], Anastasopoulos [START_REF] Anastasopoulos | Women on boards : not just the right thinking but the bright thinking[END_REF], Bierema [START_REF] Bierema | How Executive Women Learn Corporate Culture[END_REF], Petermann [START_REF] Petermann | Cross Cultural Collaboration in Design Engineering. Influence Factors and their Impact on Work Performance[END_REF], Gautam [START_REF] Gautam | Cultural Influences On the Design Process[END_REF] or Strickfaden & Heylighen [START_REF] Strickfaden | Exploring the Cultural Capital of Design Educators[END_REF], belong to this trend 7 .

The term "design anthropology" can be included into this stage. "Design anthropology seeks to answer the question how do the processes and artefacts of design help come define what it means to human. It explores a wide range of interests related to design practice: how interfaces can be developed based on values of shared learning versus individual study; how the adoption of technologies can lead to greater social equality and inequalities over time; and how not just the words but the meanings behind words change as you design for one culture versus another. These are all issues of the human context that has grown more complex. Design anthropology is the field to help you feel confident in your design decisions by showing you the global ramifications of past, current, and potential communications, artefacts, and experiences as they affect the human context". (Quotation from: www.adobe.com/designcenter/thinktank/tt_tunstall.html). Design anthropology fights preconceptions, using anthropologist's ability to step back and look at the issues being addressed in a larger and deeper cultural context 8 .

Authors like Leonard & Rayport [START_REF] Leonard | Spark Innovation Through Empathic Design[END_REF] (p.113), base their research on observation, in depth practices and view empathic design as a culture shift: "empathic design pushes innovation beyond producing the same thing only better. So for example, computer company managers who have been exposed to a deep cultural understanding of mobility no longer think only of making lighter, faster and more durable laptops. Instead, they are challenged to consider other communication needs as a portable computer might meet." On their side, Strickfaden & Heylighen [START_REF] Leonard | Spark Innovation Through Empathic Design[END_REF] (p.265), also call for orienting further work about cultural capital to targeted approaches : "Subsequently, we will zoom in on specialized areas of design, such as design for sustainability or design for all, to investigate how the cultural capital of designers and/or design educators relates to their design processes and artefacts". Bierema [START_REF] Petermann | Cross Cultural Collaboration in Design Engineering. Influence Factors and their Impact on Work Performance[END_REF], too, opens a new field of research investigating how the learning processes of corporate cultures occur as well as male's and female's roles in this learning process.

The most striking differences between the three stages mentioned above can be resumed as follows:

First researches about culture were more oriented towards a holistic understanding of small, quite homogeneous, slow changing human groups in which all features were interrelated, without the ambition of making people change and without any value judgment of rightness or wrongness or usefulness. On the business side, this period was characterized by business policies including government regulation, patriotism, World War II.

During the second stage, culture went besides the field of anthropology, gave up holism and became an operational corporate tool among others to lead to change, expressing value judgments and attempting to measure, even if artificially, the non measurable. Quantitative, more superficial and wide covering approaches, took the upper hand over in depth qualitative studies, questionnaires replaced much of the observation practices, subjectivity changed hands: from the observer's subjectivity to the subjectivity of the people answering questionnaires as well as that of people analyzing them. This period coincided with the deregulation of industries, a growing focus on customers, statistical process control for quality, managerial grids and human potential development. In this business environment, corporate culture took place as a mean to strengthen leadership and increase employee commitment.

The third and actual stage coincides with a quite different international policy and business environment where internet, virtual organization, international mergers, flexible organizations, sustainable development, supply chain management and knowledge capital measures are key features making the debate about national or corporate culture less meaningful. On the other hand, there is an increasing awareness that firms' changing pace has become too quick for cultural traits to emerge and be transmitted. This stage opens the way to the study and understanding of professional cultures, less wide covering than attempts to define core universal values and practices within an organization are but more in depth and long lasting. As an hypothesis, we can state that more the corporate life rhythm is high and without boundaries, the less there is appropriate time and structure to share thoughts, ideals or attitudes in order to create and preserve corporate "culture". On their side, anthropologists have also changed their thinking about culture to accommodate to new economically and culturally linked areas that cross cut old cultural boundaries and contribute to understand how cultural groupings interface.

Literature about culture gives the impression to make no link with open mindedness. However, this topic is sometimes presented in training oriented literature as the key to advancement in new ideas and critical to teach future generations (Blount [28]). Hare [29], defines open-mindedness as follows: "an intellectual virtue that involves a willingness to take relevant evidence and argument into account in forming or revising our beliefs and values, especially when there is some reason why we might resist such evidence and argument, with a view to arriving at true and defensible conclusions". Even if this description incorporates cultural issues, no mention is explicitly made to culture. We put forward the hypothesis that open-mindedness is a fundamental cultural point to take into account in nowadays business, political and intellectual context. 13 panel participants on 20 put forward open mindedness as a key factor influenced by corporate culture (vs. national culture).

WOMEN AT THE CUTTING EDGE

Survey's panel women, not particularly involved in research results concerning culture, intuitively consider national culture as an influent life lasting basis where management and working human relations and behaviours are embedded, whatever these are called "corporate culture" or not.

According to them 9 (see figure 1), gender issues (Q8a), success perceptions (Q8f), attitudes towards errors (Q8j) figure among the key, in depth, structural issues stemming from national culture and having an important impact on professional issues.

Figure 1. Men and women main features

Hierarchical distance (Q8b), formal communication (Q8r), and organisational features (Q8u) are perceived as more contextual, circumstantial elements upon which corporations can have an influence -as is the case, according to verbal qualitative responses-with open mindedness, objectives definition and decision making processes. Preliminary insights coming from a men population (9 people with similar profile to the women panel) answering to the questions reported in figure 1, show that men and women do both consider gender issues to be deeply embedded in national culture but men consider success representations (Q8f) to be less influenced by national inherited culture than women do. Men also seem to differ from women in regard to formal communication (Q8r) and organizational features (Q8u), thinking that these are culturally embedded vs. contextual, circumstantial, corporate driven characteristics.

Q1 and Q2 show similarities between men and women answers: both think that it is mainly chairmen and CEO's (vs, for example, stakeholders or project managers) who carry responsibility about corporate culture. Chairmen and CEO's being men in the majority of cases, we can induce that corporate culture's official arguments and statements have been until now mainly the fact of men.

As for Q2 (occasions on which cultural discourses on corporate culture are delivered), again, men and women think that it is mainly when corporate objectives are announced, the most often by Chairmen or CEO's -mostly men-that cultural issues are explicitly put forward (vs, for example, reorganisation processes, evaluation or project objectives, where culture is no or little put forward).

On the purely discursive, qualitative side, the survey results showed women give particular attention to participating in official and non official situations related with culture re-definition and transmission, they report being often involved in training, hiring policies and practices as well as new collaborators integration monitoring, knowledge management, cross-sector teams management and client relations. Cultural transmission and learning is also related by the women panel to every day attitude, more than to official statements. They report feeling quite free to speak about cultural issues formally as well as informally. Striking sensitivity towards discrepancies is mentioned by the women panel while referring to gaps between the publicly stated "corporate culture" and the day to day, task oriented attitudes and arguments. This fact nurtures the initial hypothesis concerning the concept of culture for corporate practices and values: an inconsistent, short lasting, opportunistic and problem driven system does not appear to fit in with a more in depth, holistic and structuring perception of culture.

Cultural environments where panel's women appreciate to work in are characterized by collaborative practices, confidence and independence, where the chance to keep attentive towards common good and diversity do exist. Women also report perceiving themselves as having a particular capacity in terms of cultural transmission, through their commitment in training as well as the importance they declare giving to functional aspects, to final users, to empathy and emotion, to iterative processes. On their side, men do not seem to perceive this capacity.

Interesting differences between men and women perceptions of culture impacts are mainly shown through verbal answers to the survey's open questions. For example, in women words we can find expressions like "indépendance", "respect", "confiance dans les jeunes", exigence", "éthique", "qualité", "innovation", "collaboration avec les clients", diversité des profiles", "partage des savoirs" as cultural elements favouring individual and collective work where, in men words we find: "bilan performances annuelles", "primes individuelles", "délégation de responsabilités", "outils technologiques de travail collaboratif", "respect", "contrôle"… Data presented in this paper constitute a non representative -from a quantitative point of viewpreliminary exploration and intend to encourage further research on this new topic at an international level. Nevertheless, in regard to the initial hypothesis stated in this paper, we think women's perceptions about culture are not binary, going beyond traditional debate about national or corporate culture, looking at culture as a mix of national cultures, individual perceptions and corporate practices, in line with nowadays intellectual and geographical globalization. It also seems that women will be particularly inclined to choose research topics considering, for example, ethics, learning of culture or empathy's role in design and innovation. Inclined as well, in their professional practice to be acute observers of subjectivity and consider explicitly differences as well as similarities among team members trying to create value out of it in design processes and being aware that culture goes beyond the official top down chairman's yearly strategic speech to nest in every day attitudes and ways to foster cultural momentum.

CONCLUSIONS

We have presented until here a mainly feminine point of view that should be challenged and enriched by men's perceptions in regard of culture. These perceptions will be explored through focus groups and surveys in a near future. If the concept of "corporate culture" originally comes out from anthropology, our point of view is that little by little, key aspects of culture like heritage (and the necessary span of time to give sense to it) and apprenticeship, have been left apart to the profit of a "hard", rapid changing normative approach focusing on values, rules and whatever measurable best practices having been successful -even if just for a few weeks-in an organization. New trends looking at culture more as the soft, interwoven whole set of practices and beliefs in targeted professional situations represent, on our point of view, a promising way to take advantage of the concept of culture. We feel that binary debates on national culture or "corporate culture", the first supposed to stem from values and the second stemming mostly from practices will not yield reliable results in expanded, open and global business contexts. In women's view, culture is more a matter of understanding diversity, contexts, situations and introducing new ideas than a matter of indexes, scores, winners and losers.

It is generally admitted that corporate ways of looking at finance and profit, of organizing work or deciding about planning or evaluation grids are mostly the fact of upper management, functions occupied in a large part by men. As far as design activities are concerned, they also seem to be male driven. A look at 2007 ICED articles, allows setting out that some subjects like training, ethics, networking, cultural capital and cultural influences, intuitive use, participative innovation, collaboration, are mainly developed by women whereas men would rather focus on aspects like concept mapping, quantitative representation, automatization, modelling, competition, optimization, decision support… It can also be noticed that the 14 authors signing more than 5 articles (up to14 for some) are men, within only one woman exception. These considerations coincide with referenced research and survey's results on women's role in culture where they have shown to be empathic, astute observers, active learners through other people and networking, valuing experiential and informal learning, the latter being essential to transmit culture through time.

Paris, mainly involved in cultural impacts on training.

Dr. Julie Stal-Le Cardinal is a researcher and assistant professor at Ecole Centrale Paris. Her field concerns mechanical engineering. She works on team diversity and project management. She coaches project managers in companies concerning the management of their project and the choice of actors in teams.

Empathy involves both affective arousal and cognitive structuring; flexibility of ego boundaries is shaped by prevailing cultural mores. Having the same gender nurturing figure significantly influences the quality of empathy that develops in females essential to the process of empathy(Jordan [5] p.2). Generally females are more empathic than males because of socialization experiences, early childhood identification, and sex-role identification-all shaped by prevailing cultural mores.

Some early descriptions of culture can be found at the end of the XIX century (Taylor, 1871 "The Science of Culture". Primitive Culture, pp. 1-9. Reprinted in Herbert Applebaum (ed.), Perspectives in Cultural Anthropology, pp. 47-59 (SUNY Press, 1987). They can be classified as belonging to the « descriptive » type.

More than 200 definitions of culture are described and analyzed by A.L. Kroeber and C. Kluckhohn[START_REF] Kroeber | Culture, a critical review of concepts and definitions[END_REF]

http://www.seanet.com/~daveg/articles.htm

http://www.leadingvirtually.com

More radically, the 1995 OECD report of the study group on connections among scientists, agencies and governments in decision making for regional and/or global mega projects, suggests that national cultures are a problem in international collaborative projects.

Alongside with this new, more focused trend, the pre existing binary orientation is still developed.

The field of design anthropology is extensively explored by women (Marietta Baba, Rita Denny, Elizabeth Tunstall, Christina Wasson, Lucy Suchman, Susan Squires…)

Only answers scoring between 13 and 20 out of 20 are taken into account here.

We suggest that women's gender sensibility towards culture be explicitly taken into account in design researches and practices given the important changes in demographics, the flexible and more and more diverse but nevertheless integrated world we live in and the stressing need for design ethics. This is directly illustrated in this international congress by the fact that for the first time in ICED history a topic called Culture & Empathy is proposed, where our paper subscribes, with a focus on Gender. As for the theme, we feel that Human Behaviour in Design is the most appropriate one.


Vincent Strubel

Nicolas Fillot

Fabrice Ville

Philippe Vergne

Alexandre Mondelin

Yves Maheo

Etude du piégeage de contaminants solides dans des contacts EHD

Keywords: piégeage de contaminants, roulements hybrides, contacts EHD, modélisation numérique, machine bi-disques

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

Résumé

Cette étude porte sur l'analyse des phénomènes de piégeage de particules survenant dans des roulements à billes soumis à une lubrification polluée. Des outils à la fois numériques et expérimentaux ont permis de mieux appréhender un problème récurrent aux conséquences désastreuses. Des simulations numériques aussi bien que des tests sur machine bi-disques ont permis de mettre en évidence des paramètres clefs influant directement sur le taux de piégeage de contaminants. Des tests avec différents matériaux ainsi que différentes géométries ont permis de quantifier l'impact des paramètres d'un contact élastohydrodynamique sur le piégeage de particules.

Introduction

Lubrifier un système mécanique, c'est assurer son bon fonctionnement et garantir une durée de vie du mécanisme acceptable. Tout contact lubrifié s'établit alors au travers d'une interface solide ou liquide qui préserve les surfaces du mécanisme. Un lubrifiant exempt de tout contaminant est cependant impossible à garantir et tout approvisionnement en lubrifiant équivaut nécessairement à l'introduction de contaminants solides dans les zones de contacts. Comme souligné par des études précédentes [START_REF] Wan | The behavior of suspended solid particles in rolling and sliding elastohydrodynamic contacts[END_REF], ces contaminants bien que micrométriques pénètrent alors dans les zones sub-micrométriques des contacts élastohydrodynamiques (EHD) avec pour conséquence des endommagements conséquents des surfaces. De nombreuses études ont été réalisées ces dernières décennies [START_REF] Dwyer-Joyce | The entrainment of solid particles into rolling elastohydrodynamic contacts[END_REF][START_REF] Ville | An Experimental Study on the Concentration and Shape of Dents Caused by Spherical Metallic Particles in EHL Contacts[END_REF][START_REF] Nikas | Mathematical analysis of the entrapment of solid spherical particles in non-conformal contacts[END_REF] pour analyser et quantifier l'impact des différents paramètres des conditions opératoires (charge, glissement, taille …) sur le piégeage de particules solides au sein d'un contact ponctuel. Le but de cette étude est de se focaliser sur l'influence des paramètres propres au contact sur le piégeage de particules, telles que la nature des matériaux en contact (contacts hybrides) ou la géométrie des contacts générés (contacts circulaires ou elliptiques). Des simulations numériques ainsi que des essais sur machine bi-disques ont été réalisés, pour d'une part simuler les trajectoires de contaminants en amont d'un contact lubrifié et d'autre part étudier le piégeage effectif de contaminants solides dans un modèle de contact en lubrification polluée.

Simulations numériques

L'étude numérique a porté sur la modélisation de l'écoulement bi-phasique du lubrifiant contaminé. Ainsi, le comportement du lubrifiant confiné a d'abord été considéré avant de prendre en compte la dynamique propre aux contaminants. Les particules sont ainsi suivies depuis leur entrée dans le domaine fluide en amont de l'aire de contact EHD jusqu'au contournement ou leur piégeage au sein de la zone du contact, comme représenté dans la Figure 1 : Une fois les essais réalisés, l'observation des surfaces des disques en contact (Figure 2) permet de quantifier et qualifier le piégeage des particules. La quantification de ces phénomènes passe par un comptage du nombre total des marques d'indentation laissées sur les disques en contact. Pour qualifier le type d'indentation, des relevés sont réalisés avec un rugosimètre optique afin de déterminer la nature et les dimensions caractéristiques de ces marques d'indentation.

Figure 1 :Figure 2 :

 12 Figure 1 : Simulations numériques en amont du contact

Fig. 2

 2 Fig.2

Etude du piégeage de contaminants solides dans des contacts EHD

	V. STRUBEL a , N. FILLOT a , F. VILLE a , P. VERGNE a ,
	A. MONDELIN b , Y. MAHEO b
	a. Université de Lyon, INSA-Lyon, CNRS, LaMCoS, UMR5259,
	F-69621, Villeurbanne, France
	b. SKF Aerospace,
	F-26300, Châteauneuf-sur-Isère, France