Jérôme Bikard
email: jerome.bikard@ensmp.fr

Pierre Menard

Edith Peuvrel-Disdier

Tatiana Budtova

3D numerical simulation of the behaviour of a spherical particle suspended in a Newtonian fluid and submitted to a simple shear

Keywords: hydrodynamic stresses, suspended sphere, size effect, 3D flow, finite element method, multidomain approach

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3D numerical simulation of the behaviour of a spherical particle suspended in a Newtonian fluid and submitted to a simple shear Jérôme Bikard, Pierre Menard, Edith Peuvrel-Disdier, Tatiana Budtova

INTRODUCTION

The comprehension and the prediction of the behaviour of objects (rigid particle, agglomerated fillers, liquid drop, gel particles or capsules, etc) suspended in a matrix and submitted to a flow require a good knowledge of the mechanical fields (velocity, hydrodynamic stresses) around this object. For example, the dispersion of agglomerated fillers (such as carbon black, silica, calcium carbonate) in a polymer matrix, which is performed in an internal mixer or an extruder, depends on the local hydrodynamic stresses applied on the pellet. For example, dispersion occurs as soon as the hydrodynamic stress overcomes the agglomerates' cohesion [START_REF] Bohlen | Intensive mixing[END_REF][START_REF] Mackelvey | Mixing chapt. 12, Polymer Processing[END_REF]. During mixing, the size of the pellet and the flow cell can be of similar dimension. This may affect the local hydrodynamic stresses around the pellet as compared with their distribution in an infinite medium. As a result, the behaviour of the particle will be modified. It is thus important to know the local stresses applied on the pellet and to consider the potential wall effect due to the finite cell dimensions.

The behaviour of objects of basic shapes (spheres) in an infinite Newtonian matrix is well described by analytical solutions in the case of pure shear [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF][START_REF] Bagster | The stresses within a sphere in simple flow fields[END_REF]. The particle rotates with a constant period which is inversely proportional to the shear rate and the theoretical tensile stress around the particle is maximal at 45° and 225° [START_REF] Brenner | The stockes resistance of an arbitrary particle III. Shear fields[END_REF].

However this analysis does not take into account the finite dimensions of the cell around the object nor their consequences on the sphere behaviour. More complex flows (for example, with several spheres) were considered analytically first by [START_REF] Bartok | Particle motions in sheared suspensions. VII Internal circulation in fluid droplets (theorical)[END_REF] and then completed by [START_REF] Kao | Streamlines around single spheres and trajectories of pairs of spheres in two dimensional creeping flows[END_REF]. But the effect of the restricted flow field dimension was not explicitly taken into account in their analyses. Recently, [START_REF] Pomchaitaward | Investigation of the dispersion of carbon-black agglomerates of various sizes in simple-shear flows[END_REF] used a 3D numerical simulation in order to calculate the effect of the finite dimension of the shear cell on the rotation period of a sphere. The sphere rotation is imposed by considering a condition of zero torque on the sphere surface. The rotation rate of the particle is determined by assuming that the global torque applied on the sphere is also equal to zero. In both cases it was shown that streamlines are influenced by the sphere size and the rotation period differs from the one obtained by Jeffery. No results on the stress or strain fields were reported.

The objective of this work is to use a finite element method to solve the motion equation for a rigid sphere suspended in a Newtonian matrix and submitted to simple shear and to study the effect of the proximity of the cell walls on the behaviour of the sphere. The Rem3D ® software is used to calculate the flow and the hydrodynamic stresses around the sphere. A multidomain approach and a sticking condition at the sphere/matrix interface are considered.

The equations of the problem and the numerical method to solve them are described in section 2. The direct results of the three-dimensional simulation of the flow around the sphere in a finite matrix (impact of the size effect on the streamlines, hydrodynamic stresses and rotation period) are presented in section 3 and compared with the ones known from literature.

THEORETICAL BACKGROUND

In this study, a numerical simulation is used to calculate the behaviour of a rigid sphere suspended in a Newtonian fluid and submitted to a constant shear rate in a restricted geometry. The Rem3D ® software initially dedicated to the injection of the polymers [START_REF] Pichelin | Finite element solution of the 3D mold filling problem for viscous incompressible fluid[END_REF][START_REF] Pichelin | A taylor discontinuous Galerkin method for the thermal solution in 3D mold filling[END_REF][START_REF] Coupez | 3D anisotropic mesh generation and adaptation with applications[END_REF] is used to calculate the flow and the hydrodynamic stresses around the rigid sphere. The shear flow is imposed by two planes rotating in the opposite direction. The infinite medium is restricted by the finite gap between the translating planes (see Fig. 1). The shear cell is considered as a multiphase domain consisting of the sphere and the Newtonian fluid.

Figure 1

Mechanical equations and boundary conditions

The rigid sphere is approximated by a "droplet" of a Newtonian fluid with the viscosity part η much higher and density ρ part being the same than the ones of matrix η f and ρ f , respectively (e.g. 1). The viscosity difference ensures rigid body behaviour of the sphere. Equal densities allow neglecting gravity effect. The behaviour of the fluid is assumed to follow a Newtonian law, though viscoelastic cases could be considered (forthcoming study). The contact between the fluid and the sphere (described by the surface Σ ) is a perfect sticking. The system can be described by the following set of equations:

       Σ = ⋅ Σ = ⋅ - η = - η = on 0 on 0 sphere in the p ) ( 2 fluid in the p ) ( 2 part f t [v] n [v] I I I I v v v v ε ε ε ε σ σ σ σ I I I I v v v v ε ε ε ε σ σ σ σ (1)
where v, ε(v) and p are the velocity, the deformation rate and the local pressure, respectively, [v] represents the jump of the velocity field on the particle surface Σ and I is the identity tensor.

Table 1

The schematic presentation of the shear cell with a control volume Ω = L*h*e, where L is its length, e is the width and h is the distance between the upper and lower plates (the gap size) is shown in Fig. 1. e and L dimensions are assumed to be equal and h is much smaller than L which reflects experimental conditions [START_REF] Seyvet | Collision-Induced Dispersion of Agglomerate Suspensions in a shear Flow[END_REF]). For the sake of symmetry, the sphere is located in the middle of the gap. The boundary conditions on the border Ω ∂ are given by the following relations:

φ = Ω ∂ ∩ Ω ∂ Ω ∂ = Ω ∂ ∪ Ω ∂    Ω ∂ Ω ∂ V p V p V 2 1 p 2 1 ; where on v - = v on p = p (2)
The cell dimensions relatively to the sphere size were chosen in such a way that the "non disturbed" kinematics of the fluid far from the sphere corresponds to pure shear. Inertial effects are neglected, thus the non disturbed flow is quasi twodimensional (in the (E,G) plane). A zero constant pressure is imposed on the vertical walls (i.e. p 1 =p 2 =0), and thus the calculated pressure will directly correspond to the perturbation caused by the presence of a rotating sphere.

Numerical considerations

In order to solve equations (1) and the evolution of the surface Σ as a function of time, a characteristic function C f is defined as a new field in the control volume Ω :

   = particle in the is if 0 fluid in is if 1 ) , ( x x t x f C (3)
where x represents the Cartesian coordinates of a material point in the control volume Ω .

The local balance equations in the sphere and in the fluid can be written in the volume Ω with the virtual work principle as follows:

[ ]          =         ⋅ + ∂ ∂ = ⋅ - = - - + ∫ ∫ ∫ ∫ ∫ Ω Ω Ω ∂ Ω Ω 0 d 0 d ) ( ds d ) ( d ) ( : ) ( ) 1 ( 2 * * 0 蔔 蔕 d i v d i v f f inc f f f t p p p C grad C n C C v v v v v v v v v v v v v v v v v v v v ε ε ε ε v v v v ε ε ε ε * * * * * * * * * * * * ϕ η η (4) where ( * * , , ϕ p * * * * v v v v
) are functions chosen in adapted Sobolev [START_REF] Pichelin | Finite element solution of the 3D mold filling problem for viscous incompressible fluid[END_REF][START_REF] Pichelin | A taylor discontinuous Galerkin method for the thermal solution in 3D mold filling[END_REF]. The first equation in the set (4) expresses the mechanical quasi-static balance, the second -the incompressibility of fluids and the third -the convection of the characteristic function.

This last equation means that the only one allowed degree of freedom for the sphere is a rotation around the axis V (in agreement with the boundary conditions). The surface conditions are trivially satisfied in these equations and thus the formulation of the problem is completed.

The interface Σ between the sphere and the fluid mathematically corresponds to the surface of discontinuity of the characteristic function C f through the elements of mesh [START_REF] Béraudo | A finite element method for computing the flow of multi-mode viscoelastic fluids : comparison with experiments[END_REF]. In terms of distributions, the interface can be defined by:

{ } n C grad x f x x δ = Ω ∈ = ∑ )) ( ( / (5)
where δ x is the Dirac distribution on x.

The equations ( 4) are discretised by the discontinuous Galerkin method with mixed pressure/velocity finite elements (see e.g. [START_REF] Pichelin | Finite element solution of the 3D mold filling problem for viscous incompressible fluid[END_REF][START_REF] Pichelin | A taylor discontinuous Galerkin method for the thermal solution in 3D mold filling[END_REF][START_REF] Béraudo | A finite element method for computing the flow of multi-mode viscoelastic fluids : comparison with experiments[END_REF][START_REF] Lesaint | Mathematical aspects of finite elements in partial differential equations[END_REF][START_REF] Hulbert | Space-time finite element formulation for incompressible viscous flows[END_REF]). The choice of a non structured mesh with tetrahedral elements with a bubble stabilisation ensures the numerical stability of the method (with respect to the Ladyszhenskaya, Babuska and

Brezzi criterion [START_REF] Babuska | Adaptive approaches and reliability estimations in finite element analysis[END_REF][START_REF] Hughes | A new finite element formulation for computational fluid dynamics: VIII the Galerkin/least-square method for advective-diffusive equations[END_REF]. The mesh of the control volume Ω is based on adaptive techniques of anisotropic meshes and detection of interfaces [START_REF] Coupez | 3D anisotropic mesh generation and adaptation with applications[END_REF] and uses the GCM ® software associated to Rem3D ® .

As shown in Fig. 2, the use of adaptive mesh and algorithms is essential for a correct location of the interface fluid/sphere. Fig. 2a shows the initial interface between the sphere and the fluid and Fig. 2b -a large numerical diffused interface due to a too coarse mesh. Fig. 2c shows an adapted mesh around the interface and Fig. 2d -the excellent calculation of the characteristic function. The mesh adaptation is correlated to the detection of the interface. This technique enables to minimize the numerical error on the location of the interface. Figure 3 shows a cross section of the meshed volume with the anisotropic adapted mesh.

RESULTS AND DISCUSSION

The following dimensions of the control volume are considered: e = L = 20 mm and h = 0.1 mm. The velocity of the walls is equal to 0.25 mm.s -1 leading to a macroscopic shear rate of 5 s -1 . The applied shear rate is fixed constant in the numerical simulation. The other parameters are given in Table 1. The Reynolds number for the flow is equal to R e = 1.2 10 -7 which means that the flow is purely laminar. The dimension of the sphere (diameter = d) as compared with the distance between the translating walls (gap = h) is described by the ratio:

h d r = (6)
All simulations are performed in 3D. A sensibility analysis towards the variation of particle viscosity was performed. It showed that the results do not depend on the viscosity of the particle if the ratio (particle viscosity)/(fluid viscosity) is greater than 10 3 . The calculated streamlines, stress fields and resulting rotation periods are described in the following sections.

3.1. Velocity fields and streamlines around the sphere Figure 4 shows the calculated velocity fields around the sphere in the (E,G) median plane which contains the equator of the sphere for several (sphere diameter)/(gap size) ratios. The results exhibit a recirculation zone with strong shearing.

The streamlines that are in this recirculation zone make a half-turn before reaching the sphere and the local flow direction (in the vicinity of the sphere) becomes opposite to the direction of the main flow. This zone of high shear rate corresponds to a zone of high shear stress and thus of viscous dissipation. The presence of a singularity point in such a flow was also predicted by [START_REF] Kao | Streamlines around single spheres and trajectories of pairs of spheres in two dimensional creeping flows[END_REF].

Figure 4

The effect of the r ratio on the dimensions of the recirculation zone is depicted in Fig. 4a-d for r = 0.2, 0.4, 0.6 and 0.8, respectively. The greater is the r ratio (the larger is the sphere relatively to the gap size and thus the closer to the sphere are the walls), the larger is the recirculation zone. Moreover, the recirculation zone moves closer to the sphere with r increase. As it will be shown in part 3.2, this can be correlated to the normal stress increase and the tangential stress decrease with the growth of r value (sphere diameter becoming comparable to the gap size).

In order to better characterise the importance of this recirculation, the main recirculation zone is described by its thickness l(x) (see Fig. 4a), x being the distance from the sphere centre. The variation of this thickness reduced to the sphere diameter l(x)/d versus x/d reduced distance is plotted on Fig. 5 for different r ratios. This shows that the maximal size of the main recirculation zone is of the order of the sphere dimension and increases with the ratio r increase. This can also qualitatively be seen from Fig. 4.

Figure 5

The position of l(x)/d local minimum indicates the location of the zone of the highest shear rate due to the opposite flows and thus of high shear stress. For example, for r = 0.2, l(x) = 0 when x = d (see Fig. 5, curve 1); this means that the zone of high shear stress is located at the distance from sphere centre equal to its diameter. The corresponding area is shown by a circle in Fig. 4a. For r = 0.4, l(x) = 0 when x < d (see Fig. 5, curve 2): the point of singularity is moving closer to the sphere surface(x/d < 1).

While the value of the minimum of l(x) is equal to zero for r = 0.2 and r = 0.4, this minimum value increases for r > 0.4. For r = 0.8, the zone of high stress due to the opposite flows touches the sphere surface (l(x) is minimal for x < d/2). This means that for r > 0.4 the zone of high stress is not reduced to a singularity point but becomes larger and larger and encircles part of the sphere (see Fig 4d). A similar behaviour was also predicted by [START_REF] Kao | Streamlines around single spheres and trajectories of pairs of spheres in two dimensional creeping flows[END_REF] in their analytical solution. As it will be shown in the part 3.2, the presence of high shear rate zone in the vicinity of the sphere can be correlated with the evolution of the angle at which the stress is maximal as the dimension of the sphere becomes comparable to the gap size.

The effect of the third dimension (along the V axis) on the velocity fields is shown in Figure 6. An example of six cross sections in the (E,G) plane cutting the sphere at different distances from its centre at V = 0 (equatorial plane), d/4, d/2, d (tangential to the sphere surface), 2d and 4d is shown for the case r = 0.6. Fig. 6a is equivalent to Fig. 4c but represented in 3D. Figure 6b demonstrates that the recirculation area is not confined in the equatorial plane but also has a thickness in the third dimension V. This is confirmed in Figure 6c where a deflection of velocity lines appears around the sphere (with swirls in V direction). The streamlines do not remain in the Figures 6a-c shows that there is a second recirculation area around the sphere (closed streamlines). The closed loops around the sphere correspond to a layer in which an object orbits around the sphere. These closed streamlines exist only in the vicinity of the sphere (see Figure 6d and6e). At V = 2d only the main recirculation area exists (no closed streamlines around the sphere), and the warping effect is less pronounced. At V > 3-4d, streamlines are not disturbed by the presence of the sphere anymore and are parallel (corresponding to a shear field) (Fig. 6f).

Figure 6

Fig. 6 clearly shows the importance of a 3D modelling. The appearance of a second recirculation zone could not be noticed in 2D calculations, because the flow field is too constrained. This difference between normal and tangential stress values obtained in 2D and 3D modelling will be discussed in part 3.2.3.

Stress and velocity fields around the sphere

This part demonstrates the effect of the sphere diameter relatively to the gap size on the normal and tangential stresses and velocity profiles around the sphere. The results concerning the velocity and stress fields around the sphere are presented in Figs.

7 and 8, respectively.

Consequence of the proximity of the walls on the velocity field

Fig. 7 shows the velocity field inside and outside the sphere for r = 0.2 and r = 0.8, in an equatorial cross-section in (E,G) plane. The results are as follows:

• the velocity field inside the sphere (which is approximated by a very viscous fluid) corresponds to the rotation field of a solid; • in the case of quasi-infinite medium (r = 0.2, Fig 7a) the velocity field outside the sphere corresponds to a pure shear (the velocity is proportional to the distance from the sphere). This result correspond to the classical theory; • in the case of finite medium (r = 0.8, Fig 7b), the velocity field corresponds to a Poiseuille shear (with parabolic velocity profiles). This result can be explained by the fact that the proximity of the plates generates a flow around the sphere with a strong pressure gradient (Poiseuille flow) as shown in Fig. 4.

Figure 7

Consequence of the proximity of the plates on the stress field

For normal and tangential stresses a known analytical solution was developed for a sphere in an infinite medium. In this case, normal and tangential stresses are given by Brenner equations [START_REF] Brenner | The stockes resistance of an arbitrary particle III. Shear fields[END_REF]:

       = ⋅ ⋅ = = ⋅ ⋅ = • • ) 2 cos( 2 5 ) 2 sin( 2 5 θ γ η θ γ η f f n σ t F n σ n F t n (7)
The theoretical maximal normal stress F n is obtained at a θ angle equal to 45° (curve 1 in Fig. 8a). For a 200 Pa.s viscosity and a 5 s -1 shear rate, the value of the maximal theoretical normal stress is equal to 2.5 10 3 Pa.

Figure 8

Fig. 8 shows the calculated dependence of normal and tangential stresses on angle θ for different r ratios. Two cases can be clearly distinguished: r ≤ 0.2 (quasiinfinite medium) and r > 0.2. a) r ≤ 0.2

• The maximal normal stress obtained for r = 0.2 is at θ = 45.03° (see Fig. 8a), which corresponds to Brenner theory prediction. The difference between the calculated and theoretical values of θ is less than 0.1%. The value of the calculated maximal normal stress is 2513 Pa (less than 1% difference with the value obtained via Brenner equations, which can be explained by the fact that the assumption of quasi-infinite medium is not fully satisfied with r = 0.2). This explains why the curves of the normal stress versus orientation angle for the analytical solution and for the numerical simulation at r = 0.2 are practically coinciding and the case of r ≤ 0.2 can be considered as a quasi-infinite medium. • The zero tangential stress obtained for r = 0.2 is at θ = 45.02° (see Fig. 8b). The difference between the calculated and the theoretical values is also lower than 0.1%. The variation of the orientation angle leads to the change in the sign of the tangential stress: it is positive in the flow direction (θ → 0°) and negative being perpendicular to the flow

(θ → 90°) b) r > 0.2
The analytical prevision in an infinite medium [START_REF] Bagster | The stresses within a sphere in simple flow fields[END_REF] is not valid in this case. The 3D simulation shows that: • the angle θ at which the normal stress is maximal is displaced from 45° to higher values with the r ratio increase (curves 3-5 in Fig. 8a) and the curves become asymmetrical (the angle is moving towards the closest plate). The maximal value of the normal stress is also increased with the r ratio. For example, with r variation from 0.4 to 0.9 the maximal normal stress evolves from about 3000 Pa to about 30000 Pa, which corresponds to a high super-pressure in the fluid near the sphere;

• the value of θ at which the tangential stress becomes equal to zero is also shifted from 45° to 65° (curves 3-5 in Fig. 8b). In other words, for r between 0.4 and 0.9, the angle at which the tangential stress is zero moves towards the direction of the wall and the maximal tangential stress increases from 2500 Pa to about 13000 Pa. The curves of the tangential stresses become asymmetrical as well as the ones for the normal stresses.

The global net torque applied on the sphere was also calculated. In the case of an infinite medium (r ≤ 0.2), the torque is equal to zero, as predicted by Brenner (see Eq. ( 7)). For the case of a finite medium, the calculated torque remains equal to zero, as assumed by [START_REF] Pomchaitaward | Investigation of the dispersion of carbon-black agglomerates of various sizes in simple-shear flows[END_REF] in their simulation.

Discussion: comparison between the stress field in 2D and 3D

The angle values at which the normal stresses are maximal and the tangential stresses are equal to zero were calculated in 2D and 3D approximation. The comparison is depicted in Fig. 9a,b where the difference between numerical and analytical (quasiinfinite medium) stresses calculated 2D and 3D modelling is shown. The main result is that 2D calculation is overestimating these angles. The difference between 2D and 3D simulations is increasing with the r ratio increase. As shown in Part 3.1, 2D simulation cannot correctly predict all the aspects of the flow around the sphere. The constrained flow in 2D is at the origin of the overestimated stress values in this geometry. 2D simulations should not be used to interpret flow fields around spheres in a finite medium.

Figure 9

The results obtained with Rem3D ® modelling are shown by curve 2 in Fig. 10. In this case, the motion of the sphere occurs due to the sticking condition between the sphere and the suspending fluid.

Figure 10

The results of both numerical simulations coincide with Jeffery period for r < 0.2, which is given by the following relation [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF]:

• = γ π 4 Jeffery T ( 8
)
The sticking condition predicts a deceleration of sphere rotation as the sphere diameter relatively to the gap size ratio increases. This is in contradiction with the simulation assuming a condition of zero torque (Pomchaitaward et al. model). The recirculation area and the point of singularity are not reported in this work [START_REF] Pomchaitaward | Investigation of the dispersion of carbon-black agglomerates of various sizes in simple-shear flows[END_REF]. In the present modelling, the deceleration is explained by the presence of zones of high shear rates (opposite flows in a narrow dimension) corresponding to a zone of strong mechanical energy dissipation as identified on Fig 4a-d. This dissipation, increasing with r, is correlated to a reduction of the mechanical energy in the flow and thus of kinetic energy resulting in a decrease of velocity. This decrease of velocity in the flow then results in an increase of the rotation period of the sphere. Above r > 0.9 calculations cannot be performed because of convergence problems when r approaches 1, even by refining the meshes between the sphere and the plates. However, when the sphere diameter is equal to the gap size (r = 1), the period of rotation should be T = • γ π / 2 according to the sticking contact between the sphere and the walls, which gives half of Jeffery period.

Using a method of least square approximation, the dependence of the rotation period T on r for ratios r in the [0; 0.9] interval was determined as follows:

T(r) = T Jeffery (1 -2.212r + 0.642r 2 ) (9)
The choice of a parabolic approximation has been made for simplicity reasons, and the coefficients in equation ( 9) were estimated for a fixed shear rate of 5 s -1 . This formula was checked to be valid for a Newtonian liquid for shear rates between 0.5 s -1 and 10s -1 .

The overall results obtained show the importance of considering the proximity of the particle and the wall in the shear cell. The ratio (particle diameter/gap size) has to be taken into account during experiments, especially if willing to have a simple shear and to avoid swirls and fluid recirculation. This problem concerns different types of systems and phenomena, for example, shear-induced rupture of agglomerates [START_REF] Collin | Etude rhéo-optique des mécanismes de dispersion du noir de carbone dans des élastomères[END_REF] or release of solvent from a swollen micro-gel or micro-capsule [START_REF] Zanina | Hydrogel under shear: a rheo-optical study of the particle deformation and solvent release[END_REF][START_REF] Vervoort | Evidence of shear-induced polymer release from a swollen gel[END_REF][START_REF] Zanina | Shear-induced solvent release from gel particles: application to drug-delivery systems[END_REF].

CONCLUSIONS

The behaviour of a sphere suspended in a Newtonian fluid and submitted to a simple shear was modelled by 3D numerical simulation using the Rem3D ® software based on finite elements approximation. The classical Kao, Cox and Mason approach with sticking condition between the sphere and the fluid was used as a basis of the modelling. The effect of the sphere diameter relatively to the gap size on the velocity pattern, stress distribution around the sphere, streamlines and period of rotation was considered and compared with results obtained in the framework of other approaches.

For a sphere diameter/gap size ratio lower than 0.2 the numerical results correspond to the analytical solutions of [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF] and [START_REF] Brenner | The stockes resistance of an arbitrary particle III. Shear fields[END_REF]. For diameter/gap ratios greater than 0.2, the effects of the proximity of the walls towards the sphere are significant:

• the period of rotation increases as the sphere and the gap become of a comparable size, • a recirculation zone appears leading to an increase of the maximal normal stresses around the sphere and a shift of the angle where this stress is maximum towards higher values, • the appearance of swirls of streamlines in the V axis which shows the importance of the modelling in 3D.

The calculated flow with Rem3D ® showed a good agreement with [START_REF] Kao | Streamlines around single spheres and trajectories of pairs of spheres in two dimensional creeping flows[END_REF] analytical results (case of a complex shear generated by the proximity of the plates). However, the results obtained are in disagreement with the rotation period of the sphere predicted by [START_REF] Pomchaitaward | Investigation of the dispersion of carbon-black agglomerates of various sizes in simple-shear flows[END_REF]. The two simulations differ in their hypothesis to impose the motion of the sphere. In the present case, the evolution of the calculated period has been correlated with the velocity and stress fields and the mechanical dissipation in the flow.

FIGURE CAPTIONS

Schematic presentation of the flow cell of volume Ω (a) and of the particle with the surface Σ (b). The trihedral (E, G, V) corresponds to the global referential, whereas (n, t, V) corresponds to the local one.

Figure 2

Comparison between interface calculations without and with mesh adaptation (a and c), and corresponding numerical diffusion of the characteristic function as a result of the simulation (b and d).

Figure 3

Cross-section of the volume Ω in (E,G) plane: anisotropic mesh in a non disturbed area, adapted around the particle.

Figure 8

Normal (a) and tangential (b) stresses as a function of angle θ for the infinite medium [START_REF] Bohlen | Intensive mixing[END_REF] and several values of ratio r = 0.2 (2), 0.6 (3), 0.8 (4) and 0.9 [START_REF] Brenner | The stockes resistance of an arbitrary particle III. Shear fields[END_REF].

Figure 9

Difference between numerical and theoretical (in a quasi-infinite medium) maximum normal (a) and tangential forces (b) in 2D and 3D.

Figure 10

Rotation periods as a function of (particle diameter)/(gap size) ratio: Jeffery reference (1), present modelling in Rem3D ® (2) and zero torque approach [START_REF] Pomchaitaward | Investigation of the dispersion of carbon-black agglomerates of various sizes in simple-shear flows[END_REF] (3).

Table 1. Geometrical and material characteristics used in calculations

Figure 10

Period calculated in quasi-infinite medium

Figure 4

 4 Figure 4

Figure 5

 5 Figure 5

Figure 6 3D

 6 Figure 6

Figure 7 A

 7 Figure 7

  Figure 1

ACKNOWLEDGEMENTS

Authors are grateful to Veronique Collin (present address: Lafarge, France) and Sylvie Vervoort, former PhD students of CEMEF-Ecole des Mines de Paris, for the stimulating discussions.

Figure 2

Figure 3


Khaled Khanbari
email: kkhanbari@hotmail.com

Philippe Huchon

Paleostress analysis of the volcanic margins of Yemen

The western part of Yemen is largely covered by Tertiary volcanics and is bounded by volcanic margins to the west (Red Sea) and the south (Gulf of Aden). The oligo-miocene evolution of Yemen results from the interaction between the emplacement of the Afar plume, the opening of the Red Sea and the westward propagation of the Gulf of Aden. Structural and microtectonic analyses of fault slip data collected in the field reveal that the volcanic margins of Yemen are affected by three main extensional tectonic events. The chronological order of these events is as follows: first E-W extension was associated with the emplacement of volcanic traps of Yemen, then NE-SW extension was related to the Red Sea rifting and finally the volcanic margin was submitted to N160°E extension, perpendicular to the overall trend of the Gulf of Aden, that we interpret as induced by the westward propagation of the oceanic ridge of the Gulf of Aden.

ﺍﻟﺒﺮﻛﺎﻧﻴﺔ

Introduction

The movement of the Arabian plate away from Africa in a north-east direction since 30 Ma [START_REF] Beydoun | Southern Arabia and northern Somalia: comparative geology[END_REF][START_REF] Pichon | A plate tectonic analysis of the Red Sea-Gulf of Aden area[END_REF] led to opening of two young oceanic basins: the Red Sea, between Africa (Nubia) and Arabia, and the Gulf of Aden between Somalia and Arabia (see a review in [START_REF] Bosworth | The Red Sea and Gulf of Aden basins[END_REF]) (Fig. 1). These two oceanic basins are connected in a complicated pattern within the area of the Afar triple junction. The Sheba ridge of the Gulf of Aden initiated from the Carlsberg ridge in the Indian Ocean and progressed westwards into Afar region [START_REF] Laughton | The evolution of the gulf of Aden, Phil[END_REF][START_REF] Manighetti | Propagation of rifting along the Arabia-Somalia plate boundary: the gulfs of Aden and Tadjoura[END_REF][START_REF] Fournier | Arabia-Somalia plate kinematics, evolution of the Aden-Owen-Carlsberg triple junction, and opening of the Gulf of Aden[END_REF]. Its general trend is WSW-ENE (N70°E).

The Gulf of Aden is thus oblique to the motion of Arabia toward the north-east [START_REF] Dauteuil | Propagation of an oblique spreading center: the western Gulf of Aden[END_REF]. The oldest magnetic lineations are identified in the eastern part of the Gulf of Aden and correspond to chron 5D (17.8 Ma) [START_REF] Sahota | Geophysical investigations of the Gulf of Aden Continental Margins : Geodynamic implications for the Development of the Afro-Arabian Rift System[END_REF]; [START_REF] Leroy | From rifting to spreading in the eastern Gulf of Aden: a geophysical survey of a young oceanic basin from margin to margin[END_REF][START_REF] Fournier | Arabia-Somalia plate kinematics, evolution of the Aden-Owen-Carlsberg triple junction, and opening of the Gulf of Aden[END_REF]) and chron 6 (20 Ma) [START_REF] Fournier | Arabia-Somalia plate kinematics, evolution of the Aden-Owen-Carlsberg triple junction, and opening of the Gulf of Aden[END_REF]. The oceanic floor becomes younger in the westernmost part of the Gulf of Aden, where the oldest magnetic anomaly corresponds to chron 2 or possibly 2A (3 Ma) [START_REF] Cochran | The Gulf of Aden: Structure and Evolution of a Young Ocean Basin and Continental Margin[END_REF][START_REF] Audin | Pénétration de la dorsale d'Aden dans la dépression Afar entre 20 et 4 Ma[END_REF]). The oceanization would then have started around 20 Ma, and then the ridge propagated toward the west into Afar [START_REF] Courtillot | Propagation of an accreting plate boundary: a discussion of new aeromagnetic data in the Gulf of Tadjurah and Southern Afar[END_REF][START_REF] Manighetti | Propagation of rifting along the Arabia-Somalia plate boundary: the gulfs of Aden and Tadjoura[END_REF].

The Red Sea is a narrow oceanic rift oriented NW-SE which extends from the Sinai Peninsula in the north to the strait of Bab-El Mandeb in the south. Only its central part is characterized by organized magnetic lineations [START_REF] Cochran | A model for development of the Red Sea[END_REF]. The southern and northern parts show no sign of seafloor spreading [START_REF] Ehrhardt | Conrad Deep, Northern Red Sea: Development of an early stage ocean deep within the axial depression[END_REF]. However, Le [START_REF] Pichon | The rotation of Arabia and the Levant fault system[END_REF] proposed that the Red Sea spreading started at about 13 Ma although magnetic anomalies indicate a younger age at 5 Ma [START_REF] Izzeldin | Seismic, gravity and magnetic surveys in the central part of the Red Sea: their interpretation and implications for the structure and evolution of the Red Sea[END_REF]. The age of the onset of rifting is still disputed [START_REF] Hughes | Evidence for middle Oligocene rifting of the Gulf of Aden and for late Oligocene rifting of the southern Red Sea[END_REF] but is generally considered to be late Oligocene [START_REF] Bosworth | The Red Sea and Gulf of Aden basins[END_REF].

Yemen is located on the southwestern corner of the Arabian plate on the eastern margin of the Red Sea and on the northern margin of the Gulf of Aden, opposite to Afar depression (Fig. 1). The western part of Yemen represents a typical volcanic margin resulting from the magmatic activity of the Afar plume since 30 Ma [START_REF] Baker | A brief Oligocene period of flood volcanism in Yemen: implications for duration and rate of continental flood volcanism at the Afro-Arabian triple junction[END_REF][START_REF] Hoffmann | Timing of the Ethiopian flood basalt event and implications for plume birth and global change[END_REF]. The Tertiary basin in the western part of Yemen (Tihama coastal plain) is also related to rifting of the Red Sea [START_REF] Beydoun | Sedimentary basins of the Republic of Yemen: their structural evolution and geological characteristics[END_REF].

The aim of this paper is to study the extensional deformations of the volcanic margins of Yemen and to discuss the relationship between extension, magmatism and kinematics associated to the rifting and subsequent spreading of the Red Sea and the Gulf of Aden. We first briefly describe the geology of Yemen, then summarize previous work on the link between extension and magmatism in Yemen before presenting new structural data in three selected areas of western Yemen.

Geological setting of Yemen

Geology of Yemen extends from the Archean to the Cenozoic [START_REF] Geukens | Contribution à la géologie du Yémen[END_REF][START_REF] Beydoun | The stratigraphy and structure of the eastern Aden protectorate[END_REF]Greenwood and Blackley 1967) (Fig. 2). The Proterozoic basement, largely metamorphic [START_REF] Whitehouse | Precambrian basement character of Yemen and correlations with Saudi Arabia and Somalia[END_REF]) is unconformably overlain by Ordovician ferrigeneous sandstones, Permian black shales and early Jurassic marls and sandstones (Kohlan Formation) [START_REF] Ziegler | Late Permian to Holocene paleofacies evolution of the Arabian plate and its hydrocarbon occurrences[END_REF]. The late Jurassic is characterized by well-developed transgressive series composed of marls and limestones (Amran Group). These platform series are overlain by Cretaceous continental sandstones (Tawilah Group). Then, during the Cenozoic, the geological evolution of Yemen is dominated by (1) the impingement of the Afar plume beneath the Afro-Arabian lithosphere and (2) the development of the Red Sea and Gulf of Aden rifts and their subsequent oceanization.

The western part of Yemen is characterized by the Tertiary volcanics (Traps of Yemen) related to the Afar plume [START_REF] Civetta | K-Ar ages of the Yemen plateau[END_REF][START_REF] Capaldi | Tertiary anorogenic granites of the western border of the Yemen Plateau[END_REF][START_REF] Chiesa | Geological and structural outline of Yemen Plateau, Yemen Arab Republic[END_REF][START_REF] Chiesa | The Yemen trap series: genesis and evolution of a continental flood basalt province[END_REF], whereas the eastern part of Yemen is characterized by Paleocene-Eocene sedimentation of limestone, marl and gypsum (Hadramout Group) and Oligocene-Miocene (syn-rift) sedimentation of conglomerates, sandstones, marls, gypsum, limestones and shales (Shihr Group) [START_REF] Beydoun | The stratigraphy and structure of the eastern Aden protectorate[END_REF][START_REF] Haitham | The Gulf of Aden Rift : Hydrocarbon Potential of the Arabian Sector[END_REF][START_REF] Bott | The tectonic framework and regional hydrocarbon prospectivity of the Gulf of Aden[END_REF]. Therefore the western part of Yemen represents a volcanic margin along the Red Sea coast and the western part of the coast of the Gulf of Aden, while the eastern part is a non-volcanic margin. [START_REF] Tard | The volcanic passive margin in the vicinity of Aden, Yemen[END_REF] shows that the southern part of the volcanic margin (Aden margin) presents offshore the characteristics features of a volcanic passive margin, in particular seaward dipping reflectors.

The Yemen Volcanics cover an area of approximately 45000 km2. Their thickness is more than 2500 meters in the western part but they thin eastward and are only a few tens of meters thick near the contact with the pre-Mesozoic basement (Fig. 2). These Tertiary volcanics consist of both lava flows and intrusions. The lava flows are typical bimodal series, which include alkaline and transitional rocks [START_REF] Civetta | K-Ar ages of the Yemen plateau[END_REF][START_REF] Capaldi | Tertiary anorogenic granites of the western border of the Yemen Plateau[END_REF][START_REF] Chiesa | The Yemen trap series: genesis and evolution of a continental flood basalt province[END_REF][START_REF] Huchon | Extensional deformations in Yemen since Oligocene and the Afar triple junction[END_REF] while the intrusions consist of granites that occur mainly along the western border of the plateau [START_REF] Capaldi | Tertiary anorogenic granites of the western border of the Yemen Plateau[END_REF][START_REF] Geoffroy | Did Yemeni Tertiary granites intrude neck zones of a stretched continental upper crust?[END_REF]. The granitic bodies are generally alkaline or peralkaline granites and are produced by fractional crystallization from basic magmas [START_REF] Capaldi | Tertiary anorogenic granites of the western border of the Yemen Plateau[END_REF][START_REF] Chazot | Genesis of silicic magmas during Tertiary continental rifting in Yemen[END_REF]. All lava series were fed by dykes emplaced throughout the magmatic activity.

The Tertiary volcanics and intrusives of Yemen are related to the Afar mantle plume that impacted the Arabia-Africa area during the Oligocene. They may be also partly related to the opening of the Red Sea and the Gulf of Aden. The lava flows were erupted into two stages [START_REF] Chiesa | Geological and structural outline of Yemen Plateau, Yemen Arab Republic[END_REF]: early basaltic traps (>2 km thick) were produced between 31-29 Ma and 26 Ma [START_REF] Civetta | K-Ar ages of the Yemen plateau[END_REF][START_REF] Chiesa | The Yemen trap series: genesis and evolution of a continental flood basalt province[END_REF][START_REF] Baker | A brief Oligocene period of flood volcanism in Yemen: implications for duration and rate of continental flood volcanism at the Afro-Arabian triple junction[END_REF] while the second stage is comprised of acidic rocks dated between 23-18 Ma [START_REF] Civetta | K-Ar ages of the Yemen plateau[END_REF][START_REF] Mohr | Structure of Yemen Miocene dike swarms, and emplacement of coeval granite plutons[END_REF][START_REF] Zumbo | 40 Ar-39 Ar geochronology of the Tertiary magmatic activity in south-western Yemen during the early Red Sea-Gulf of Aden rifting[END_REF]. [START_REF] Riisager | Paleomagnetism and 40Ar / 39Ar Geochronology of Yemeni Oligocene volcanics: Implications for timing and duration of Afro-Arabian traps and geometry of the Oligocene paleomagnetic field[END_REF] has further refined these ages on the basis of Ar-Ar geochronology and magnetostratigraphy, evidencing a first and short phase of basaltic volcanism between 30.3 Ma and 26.6 Ma followed by acidic and bimodal volcanics until 27.7 Ma. Late Miocene to Quaternary volcanics also occurs in different parts of Yemen [START_REF] Huchon | Extensional deformations in Yemen since Oligocene and the Afar triple junction[END_REF]. The Quaternary volcanics are characterized by a basaltic activity with formation of volcanic cones [START_REF] Chazot | Pre-, syn-and post-rift volcanism on the southwestern margin of the Arabian plate[END_REF].

The geological map of Yemen (Fig. 2) shows striking structural directions in the basement, especially NE-SW directions. During the Mesozoic, several rift basins formed (Fig. 2): the NW-trending Marib-Shabwa basins (Brannan et al., 1997) and the WNWtrending Sayun-Al Masila and Jiza-Qamar basins further east [START_REF] Redfern | The interior rifts of the Yemen. Analysis of basin structure and stratigraphy in a regional plate tectonic context[END_REF][START_REF] Beydoun | Sedimentary basins of the Republic of Yemen: their structural evolution and geological characteristics[END_REF][START_REF] Bosence | Mesozoic rift basins of Yemen[END_REF]. These basins have been affected by three successive phases of rifting: during the Late Jurassic, associated with the fragmentation of Gondwana, then during Early to mid-Cretaceous, corresponding to the separation of India from Madagascar and finally during Cenozoic [START_REF] Ellis | A tectono-stratigraphic framework for Yemen and its implications for hydrocarbon potential[END_REF][START_REF] Brannan | Geological Evolution of the Central Marib-Shabwa Basin, Yemen[END_REF][START_REF] Schüppel | The development of Tertiary in the Habban-Al Mukalla area, PDR Yemen[END_REF][START_REF] Watchorn | Rift-related sedimentation and stratigraphy, southern Yemen (Gulf of Aden). In "Sedimentation and tectonics in the Rift Basins Red Sea-Gulf of Aden[END_REF][START_REF] Ziegler | Late Permian to Holocene paleofacies evolution of the Arabian plate and its hydrocarbon occurrences[END_REF]. Cenozoic NW-trending normal faulting affects both the sedimentary cover and the Tertiary volcanics. This trend is parallel to the Red Sea rift. The eastern part of Yemen is affected by ENE-WSW to WNW-ESE trending faults which do not, in general, parallel the trend of the Gulf of Aden but rather display a pattern typical of oblique rifting [START_REF] Fantozzi | Transition from continental to oceanic rifting in the Gulf of Aden: structural evidence from field mapping in Somalia and Yemen[END_REF][START_REF] Fournier | Oblique rifting and segmentation of the NE Gulf of Aden passive margin[END_REF]Bellhasen et al. 2006).

Extension and magmatism: former studies

The existence of Afar plume under the southern Red Sea at 30 Ma is responsible for the initiation of a period of a major flood volcanism. Therefore, the Yemen volcanic margin is a key area to understand the geodynamics of the triple junction, the emplacement of the Afar plume and the relationships between extension and magmatism. The Yemen margin has clearly undergone extension since the beginning of rifting of the Red Sea and the Gulf of Aden, but the relationship between volcanism and extension is not simple.

Two types of extensional margins are classically distinguished: (1) active rifted margins where uplift and magmatism precedes rifting, and (2) passive rifted margins which start by rifting then uplift and limited magmatism until the oceanic spreading starts. The passive model thus requires that extension predates any uplift and magmatism whereas in the active model uplift predates magmatism and extension.

In terms of relations between plume magmatism and extensional tectonics, most of research work focused on the active rift system of the Afar region (Gaulier andHuchon 1991, Manighetti et al. 1998). In the southern margin of Yemen, [START_REF] Menzies | The timing of magmatism, uplift and crustal extension : preliminary obervations from Yemen. In : "Magmatism and the causes of continental Beak-Up[END_REF] first suggested that magmatism was followed by synchronous crustal extension and uplift.

On the southern Red Sea margin in Yemen, [START_REF] Davison | Geological evolution of the southeastern Red Sea Rift margin, Republic of Yemen[END_REF] recognized a major episode of flood volcanism between 30 and 20 Ma, with extensional faulting beginning after the eruption of volcanic rocks. They also recognized that the regional direction of extension was mainly NE-SW and they suggest that there is no direct spatial link between crustal extension and volcanism in that part of the Red Sea margin. As most important part of the lava pile is affected by the NE-SW (N40°E) extension, [START_REF] Davison | Geological evolution of the southeastern Red Sea Rift margin, Republic of Yemen[END_REF] argued that the rifting episode responsible for the opening of the Red Sea postdated the mantle plume activity in Yemen.

From the analysis of faulting-slip data sets and dykes in western Yemen, [START_REF] Huchon | Extensional deformations in Yemen since Oligocene and the Afar triple junction[END_REF] recognized an early phase of E-W extension (30-26 Ma) which was associated with the main period of emplacement of the volcanic traps. They also have shown that the upper most part of Yemen volcanics was emplaced under a N-S extension until 18 Ma. Finally, Yemen was affected by NE-SW extension between 18 and 10 Ma, the former age being the age of the latest tilted volcanic flows and the later corresponding to Late Miocene tabular basaltic flows overlying tilted blocks. This NE-SW extension is responsible for the major structures in the Yemen volcanics. Therefore, both Huchon et al. (1991) and [START_REF] Davison | Geological evolution of the southeastern Red Sea Rift margin, Republic of Yemen[END_REF] recognized that the N40°E extension during the opening of the Red Sea was the major structuring event affecting the Yemeni crust. This extension is marked in the upper crust by the development of tilted blocks limited by normal faults and was probably contemporaneous with a rapid uplift of Yemen area [START_REF] Menzies | The timing of magmatism, uplift and crustal extension : preliminary obervations from Yemen. In : "Magmatism and the causes of continental Beak-Up[END_REF]. [START_REF] Menzies | Evolution of the Red sea volcanic margin, western Yemen. In Large igneous provinces: continental, oceanic and planetary flood volcanism[END_REF] suggested that the extension was largely post-volcanism because the volcanic stratigraphy is devoid of major faults. They propose that the extension occurred in the Late Oligocene-Early Miocene (after 26 Ma) with contemporaneous plutonic hypabyssal rocks while the initiation of magmatism occurred at 31 Ma. [START_REF] Geoffroy | Did Yemeni Tertiary granites intrude neck zones of a stretched continental upper crust?[END_REF] further demonstrated that the Miocene alkali granites of Yemen, dated at 21 Ma [START_REF] Capaldi | Tertiary anorogenic granites of the western border of the Yemen Plateau[END_REF][START_REF] Huchon | Extensional deformations in Yemen since Oligocene and the Afar triple junction[END_REF] are contemporaneous with the major Miocene extension post-dating the basaltic lava pile. Therefore, most of the previous work show that the main phase of extension related to rifting postdated the main mantle plume activity in Yemen.

Analysis of paleostresses

Paleostress fields in Yemen were first analyzed by [START_REF] Huchon | Extensional deformations in Yemen since Oligocene and the Afar triple junction[END_REF], who described successive phases of extension. Their work was limited to former North Yemen and was extended to former South Yemen by [START_REF] Huchon | Rotation of the syn-rift stress field of the northern Gulf of Aden margin, Yemen[END_REF]. Paleostress analysis has also been performed further east in Dhofar (Sultanate of Oman) [START_REF] Lepvrier | Cenozoic extension in coastal Dhofar (southern Oman): implications on the oblique rifting of the Gulf of Aden[END_REF][START_REF] Fournier | Oblique rifting and segmentation of the NE Gulf of Aden passive margin[END_REF]Bellhasen et al. 2006) and in Socotra island [START_REF] Fournier | Segmentation and along-strike asymmetry of the passive margin in Socotra, eastern Gulf of Aden: are they controlled by detachment faults?[END_REF]).

Here we present the results of field studies in the volcanic part of Yemen, in three areas:

the Dhala graben and the Taizz area in the south (Aden margin) and the Al-Mahwit area in the north (Red Sea margin). Our field data consist in measurements of faults, including slickenside lineations, and dikes. Most of these measurements were performed within the Tertiary volcanics. However, some outcrops in Jurassic limestones and Cretaceous sediments were also studied. Data analysis has been conducted using various methods [START_REF] Angelier | Tectonic analysis of fault slip data sets[END_REF], including stress tensor computation by direct inversion of data [START_REF] Angelier | Inversion of field data in fault tectonics to obtain the regional stress-III. A new rapid direct inversion method by analytical means[END_REF] for each sufficiently documented site.

Dhala area (Fig. 3)

The Dhala area is located in the southern part of Yemen, between longitudes 44°30'E and 45°30'E, and latitudes 13°N and 14°N (Fig. 3). In the east of the Dhala area, the dominant trend of basement structures is NE-SW. The western part of the basement is characterized by NW-trending faults that bound the Dhala graben [START_REF] Beydoun | Sedimentary basins of the Republic of Yemen: their structural evolution and geological characteristics[END_REF], Thoué 1993). These faults separate Jurassic and Cretaceous blocks tilted towards the southwest.

The analysis of these NW-trending faults (Fig. 3 Taiz area (Fig. 4) TAIZ area is located in the southern part of Yemen (Fig. 4), between longitudes 43°30'E and 44°30'E, and latitudes 13°N and 14°N. This area is characterized mainly by Tertiary volcanics, granitic intrusions and some outcrops of Cretaceous sandstones and Jurassic limestones overlying basement, which occurs mainly in the south-east part of the area. The analysis of faults shows that the direction of extension is chiefly N-S (sites TAIZ-1a, TAIZ-3, TAIZ-6a, TAIZ-7a). This extension corresponds to the faults trending E-W, ENE and WNW. Some faults oriented ENE originated from an extension oriented NNW (sites TAIZ-1b, TAIZ-2). At sites TAIZ-6b and TAIZ-5, the dip-slip slickensides of the faults shows NE extension, while some faults trending NW in the site TAIZ-6a have an oblique slip corresponding to a direction of extension N-S. The N-S phase of extension is thus younger than the NE one. In the northern part of this area, the dip-slip slickensides on conjugate faults trending N-S shows that the direction of extension is E-W (site TAIZ-4, TAIZ-7b). Most of the phases of extension are recorded in the Tertiary volcanics except the site TAIZ-7a where the extension affected the cretaceous sandstones. Taiz area was thus affected by three Tertiary extensions: E-W extension, NE extension and N-S to NNW extension. [START_REF] Khanbari | Study of Structures and Tectonic Evolution of Yemen Tertiary Granites, by Using Remote Sensing Technique[END_REF]. Dykes intrude theses blocks and the density of the tilted dykes increases towards the granites [START_REF] Davison | Geological evolution of the southeastern Red Sea Rift margin, Republic of Yemen[END_REF][START_REF] Geoffroy | Did Yemeni Tertiary granites intrude neck zones of a stretched continental upper crust?[END_REF]. [START_REF] Geoffroy | Did Yemeni Tertiary granites intrude neck zones of a stretched continental upper crust?[END_REF] proposed that the granites strongly control the geometry of tilted blocks. were tilted with the crustal blocks. The dominantly SW and SE grouping of poles of dykes reveals NE and NW extension (Fig. 6). Some dykes have N-S trending and the grouping of their poles maybe related to the E-W extension that was recognized in Yemen by [START_REF] Huchon | Extensional deformations in Yemen since Oligocene and the Afar triple junction[END_REF] and [START_REF] Mohr | Structure of Yemen Miocene dike swarms, and emplacement of coeval granite plutons[END_REF]. The NW-SE trending dykes are mechanically compatible with the extension responsible for the tilting, while the NE-SW dykes are perpendicular and may represent a later phase of magma injection. In summary, the Al-Mahwit area is affected by three extensional stress regimes: NE extension, NW extension and WNW to E-W extension.

Relative chronology of faulting events

Our field survey and the analysis of the data indicate that the volcanic margin of Yemen is affected by three main Oligo-Miocene extensions: E-W, NE and NNW. Most of the normal faults have been observed in Tertiary volcanics, which confirms the Oligomiocene age of faulting. The relative chronology of these phases is based on our faults analysis and the previous works in the area. [START_REF] Huchon | Extensional deformations in Yemen since Oligocene and the Afar triple junction[END_REF] recognized in the west of Yemen an early (30-26? Ma) E-W extension, which was associated with the main period of emplacement of the volcanics traps. The absence of this extension in the eastern part of Yemen [START_REF] Huchon | Rotation of the syn-rift stress field of the northern Gulf of Aden margin, Yemen[END_REF]) confirms its relation with the emplacement of the Tertiary volcanics as the result of the increasing influence of Afar plume towards the west. The tilting of N-S trending dykes in Al-Mahwit area is another argument in favor of the early character of the E-W extension.

The NE extension had been identified by [START_REF] Huchon | Extensional deformations in Yemen since Oligocene and the Afar triple junction[END_REF] and [START_REF] Davison | Geological evolution of the southeastern Red Sea Rift margin, Republic of Yemen[END_REF] as the major structuring event in western part of Yemen. This extension affected the lava pile, thus it is post dated the main mantle plume activity in Yemen. NNW to N-S extension was also observed in the western part of Yemen by [START_REF] Huchon | Extensional deformations in Yemen since Oligocene and the Afar triple junction[END_REF].

The NE and NNW extensions are also recorded in the eastern part of Yemen, along the Gulf of Aden, as Oligo-Miocene syn-rift extensions [START_REF] Thoué | Quantification par imagerie 3D d'un champ de déplacements sur la marge sud du Yémen : implications sur l'ouverture du golfe d'Aden[END_REF][START_REF] Huchon | Rotation of the syn-rift stress field of the northern Gulf of Aden margin, Yemen[END_REF]. [START_REF] Thoué | Quantification par imagerie 3D d'un champ de déplacements sur la marge sud du Yémen : implications sur l'ouverture du golfe d'Aden[END_REF] proposed that the NNW extension predates the NE extension, while [START_REF] Huchon | Extensional deformations in Yemen since Oligocene and the Afar triple junction[END_REF] proposed that NE extension predates the NNW extension. However, Thoué's observations were made in Jurassic limestones, and there is a possibility that the early NNW extension they observed is actually a Late Jurassic to Cretaceous one.

In our structural analysis, some oblique slip slickensides of NW trending faults shows that the NE extension is older than NNW one (site TAIZ-7c in Taiz area and site DH-2c in Dhala area), where the oblique slickensides on NW-SE trending faults corresponds to NNW extension. This indicates that the faults have been reactivated under a younger NNW extension. This observation is consistent with the chronology established further east by [START_REF] Huchon | Rotation of the syn-rift stress field of the northern Gulf of Aden margin, Yemen[END_REF].

Discussion and conclusion

Our analysis suggests that there is a strong link between magmatism and extension and that the evolution of the volcanic margin is closely related to Afar plume, which is responsible for the generation of basic and acidic traps. The presence of syn-volcanic structures indicates that some extension did occur during the emplacement of the early volcanics around 30 Ma. However, most crustal extension postdated this early magmatic phase.

We propose two main stages of evolution of the Yemen volcanic margin in relation with the hotspot and plate movement (Table 1). The first stage is an E-W syn-magmatic extension, coeval with the emplacement of most volcanic pile. The presence of this extension in the western part of Yemen and its absence in the southern-eastern Yemen [START_REF] Huchon | Rotation of the syn-rift stress field of the northern Gulf of Aden margin, Yemen[END_REF] resulted from its position near the influence of the Afar plume. The second stage of evolution of the volcanic margin is a post-magmatic extension, which associated with NE-SW and NNW-SSE extensions. Our structural analysis indicates that the NE extension predates the NNW extension. We propose that the major NE extension is related to the rifting of the Red Sea and the Gulf of Aden (starting also at about 30 Ma) and to the movement to the Arabian plate. The NE extension is perpendicular to the Red Sea rifting and was marked by NW-SE trending faults; it is also parallel to the Arabian plate movement towards the NE. On the other hand, the NNW extension which is perpendicular to the general trend of the Gulf of Aden (N70°E), is related to the opening and the propagation of the Sheba ridge toward the west.

The volcanic margin of Yemen is therefore complex and presents a combination of both active and passive processes. At 30 Ma, the volcanic margin was part of an active rift where the E-W extension was driven by the Afar plume. The post-volcanic NE and NNW extensions are consistent with a passive model of rifting, where the extension results from far-field forces related to the plate motion as well as the local effect of the westward propagation of the Sheba ridge. and Huchon (1991) and [START_REF] Huchon | Rotation of the syn-rift stress field of the northern Gulf of Aden margin, Yemen[END_REF], respectively. The dashed line shows the propagation of the oceanic ridge in the Gulf of Aden. (1) (2a) (2b)

YEMEN

2 B Q v v Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q J J J J J k Q k k B B
Q Q Q v G G Q G Q Q Q Q B B B B G G G k k k k k k k k k k k k k k k G G G G G G G G G G G G G G G G G G G G G G k k G G k k k k k k k k k B k B B B B k k k k k B B B B k k k k k Q Q Q B B k k k k k k k k k k k k k k k k k v G G Q Q k k Q Q Q Q Q Q Q Q Q Q k G G k k k k k v v k k k k k k k k k Q Q k Q Q 43°30'E 44°30'E 13°30'N 13°N R e d S

  ) indicates ENE-WSW (~N70°E) extension (sites DH-1b, DH-2a, DH-5), NNE-SSW (N20°E) extension (site DH-2b) and NE-SW (N40°E) extension (site DH-6). Some NW faults have an oblique slip corresponding to a N-S direction of extension (site DH-1a and DH-2c). In the site DH-3, the faults that have NW-SE trend, correspond to NNE-SSW (~N35°E) extension. The faults analysis in site DH-4 shows that the NW-SE and NE-SW faults indicate a direction of extension NNW-SSE. Three phases of extension have thus been recognized in this area: ENE (N70°E), NE (N40°E) and the last one varies from NNE to NNW. The faults of sites DH-4, DH-3, DH-2 and DH-6 affect the Tertiary volcanics, while the faults of sites DH-1 and DH-5 affect the Cretaceous sandstones.

  Al-Mahwit area (Fig. 5) Al-Mahwit area is located in the northwestern part of Yemen. The simplified geological map of Al-Mahwit area (Fig. 5) shows that tertiary volcanics and granitic intrusions outcrop mainly in the area. The Jurassic limestones and the Cretaceous sandstones overlie the basement, which occurs in the north. The Quaternary sediments cover the coastal area (west of the study area) and some of outcrops of Jurassic limestone and Cretaceous sandstone occur in the coastal area (Tihama plain). Quaternary volcanics outcrop in the eastern part of the area. Most of normal faults are dominantly oriented NW-SE. The granites which intruded flood basalts, are bounded by long, NW-trending normal faults at the NE and SW borders of the granites and by shorter, NE to ENE-trending faults on their northern and southern edges. Tilted blocks of Cretaceous sandstones and Tertiary volcanics dip towards the granites

  The results of stress inversion from faults indicate that NW-SE conjugated normal faults are associated with the NE extension (sites MAHW-11, MAHW-10, MAHW-9, MAHW-16), while the NE-SW conjugated faults at site MAHW-25 is related to NWextension. At sites MAHW-19 and MAHW-17, the NE extension results almost from faults trending NW-SE. The oblique slip of some N-S to NNE-SSW trending faults at site MAHW-9 is associated with NW extension. At site MAHW-14, the NE-SW to N-S oriented faults are corresponding a WNW extension, while at site MAHW-22, the dip slip of NW faults and oblique slip of NNE to N-S and NNW faults indicate to NE extension. Most of the observed faults affect the Tertiary volcanics (sites MAHW-17, MAHW-19, MAHW-20, MAHW-16 and MAHW-22), but some of them are also observed in Cretaceous sandstones (MAHW-11) and in Jurassic limestones (site MAHW-10). Most of the dykes in the study area are oriented NE-SE and NW-SE. Many of them

Fig. 1 :

 1 Figure Captions:

Fig. 2 :Fig. 3 :Fig. 4 :

 234 Fig.2: Simplified geological and structural map of Yemen and location of the three study areas.

Fig. 5 :Fig. 6 :

 56 Fig.5: Geological and structural map of Al-Mahwit area (location on fig.2) and stereographic plots of faults. Black arrows: direction of extensional stress. 3-, 4-and 5branches stars are the minimum, intermediate and maximum principal stress axes, respectively.

  Figure 1

  Fig.4Fig.3

Figure

  Figure 2

  Figure3

  Figure4

Table 1 .

 1 Summary of reconstructed paleostresses with their inferred age in relation to the main geodynamics events. Directions for Djibouti and Eastern Yemen are from Gaulier

			Volcanic margin		Non-volc.
	AGE	EVENTS	Red Sea		Gulf of Aden
		Djibouti	Al-Mahwit	Taiz	Dhala	East Yemen
	3 Ma	Tadjoura
		spreading
	5 Ma	Red Sea spr.
	9 Ma	Western Gulf
		of Aden spr.
	20 Ma	Eastern Gulf
		of Aden spr.
	21 Ma	Red Sea uplift
	30 Ma	Afar trapps			?

Acknowledgements

The French Foreign Affairs Ministry and CNRS have funded the fieldwork in Yemen, with the logistical support of the Sana'a University. We thank Jean-Michel Gaulier and Laurent Geoffroy for their contribution to the fieldwork. We also thank F. Roure for giving us the opportunity to publish these data and an anonymous reviewer for his constructive remarks on a previous version of this paper.

ﺃﺣﺪﺍﺙ ﺭﺋﻴﺴﻴﺔ.


Metabolic Paediatrician/ Marije Hogeveen
email: hogeveen@cukz.umcn.nl

MD, PhD Martin Den Heijer

Yvonne Schonbeck

MD Marloes Ijland

Diny Van Oppenraaij

Jacqueline Kleingunnewiek

PhD Henk J Blom

PhD Jacqueline Klein Gunnewiek

The effect of folinic acid supplementation on homocysteine concentrations in newborns

Keywords: Cerebrovascular accident, stroke, newborns, homocysteine, folinic acid, folic acid

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The overall incidence of cerebrovascular accident (CVA) in childhood is very low (approximately 3/100.000 per year) [START_REF] Raju | Ischemic perinatal stroke: summary of a workshop sponsored by the National Institute of Child Health and Human Development and the National Institute of Neurological Disorders and Stroke[END_REF]. In strong contrast, CVA's occurring in the perinatal period have a much higher incidence [START_REF] Raju | Ischemic perinatal stroke: summary of a workshop sponsored by the National Institute of Child Health and Human Development and the National Institute of Neurological Disorders and Stroke[END_REF]. It is estimated that the incidence of ischemic perinatal stroke is 1 in 2300 to 5000 births and accounts for 30% of children affected with cerebral palsy [START_REF] Raju | Ischemic perinatal stroke: summary of a workshop sponsored by the National Institute of Child Health and Human Development and the National Institute of Neurological Disorders and Stroke[END_REF]. Of major importance is the fact that about 10% of newborns admitted to a neonatal intensive care unit suffer from a CVA. Known potential risk factors of the latter are prematurity, asphyxia, hypoglycemia, polycythemia, catheter-related complications, infectious factors, persistent fetal circulation, intra-uterine growth restriction and maternal factors such as diabetes or antiphospholipid syndrome [START_REF] Raju | Ischemic perinatal stroke: summary of a workshop sponsored by the National Institute of Child Health and Human Development and the National Institute of Neurological Disorders and Stroke[END_REF][START_REF] Grabowski | Prothrombotic risk factors in the evaluation and management of perinatal stroke[END_REF]. Independent of the possible cause CVA can result in significant morbidity, varying from mild hemiplegia to severe cerebral palsy, sensory impairments, mental retardation and even death. Yet aspects of the multifactorial pathophysiology leading to CVA in the newborn remain unclear and in about one third of cases no obvious cause or risk factors can be diagnosed [START_REF] Akar | Common mutations at the homocysteine metabolism pathway and pediatric stroke[END_REF]. Elevated total homocysteine (tHcy) concentrations have been found to be a risk factor in adults for development of cardiovascular disease and venous and arterial thrombosis including cerebrovascular accident. Elevated tHcy concentrations can, amongst others, be caused by genetic polymorphisms of the methylenetetrahydrofolate reductase gene and/or cystathionine B synthase gene (van der Linden et al., 2006). Non-genetic factors such as folate, vitamin B12 and/or B6 intake, age, sex, estrogen, alcohol consumption and smoking can also influence tHcy [START_REF] Gellekink | Genetic determinants of plasma total homocysteine[END_REF]Verhoef et al., 2005;[START_REF] Ueland | Plasma homocysteine, a risk factor for vascular disease: plasma levels in health, disease, and drug therapy[END_REF].

In adults, tHcy can be easily, safely and quickly lowered by 0.2 to 5 mg folic acid each day, even in the absence of elevated tHcy concentrations [START_REF] Den Heijer | Hyperhomocysteinemia and venous thrombosis: a meta-analysis[END_REF]van Oort et al., 2003;[START_REF] Brattstrom | Folic acid--an innocuous means to reduce plasma homocysteine[END_REF]. Current intervention trials show negative results from secondary prevention regarding the cardiovascular risk reduction. Meta-analyses on CVA and elevated tHcy concentrations however suggest that B-vitamin supplementation may lower the risk of CVA by approximately 20-25% [START_REF] Herrmann | Hyperhomocysteinaemia: a critical review of old and new aspects[END_REF]Homocysteine Lowering Trialists' Collaboration, 2000). Wang et al performed a meta-analysis on folic acid supplementation in relation to stroke and observed an overall decrease of 18%. Furthermore, their results suggested that folic acid supplementation can significantly reduce the risk of stroke in primary prevention (RR 0.75[0.62;0.90] (Wang et al., 2007).

Van Beynum et al demonstrated that elevated tHcy concentrations increased the risk of ischemic stroke in Dutch children aged 0-18 years (van Beynum et al., 1999). Sirachainan et al found an odds ratio of 8.2 [95% CI 1.4-47.2] for developing ischemic stroke in patients aged ≤ 18 years with elevated tHcy concentrations [START_REF] Sirachainan | Homocysteine, MTHFR C677 T, vitamin B12, and folate levels in Thai children with ischemic stroke: a case-control study[END_REF]. In a previous study, we described significantly higher mean tHcy concentrations in 24 newborn infants with stroke compared to 94 healthy newborn infants ] [START_REF] Hogeveen | Hyperhomocysteinemia as risk factor for ischemic and hemorrhagic stroke in newborn infants[END_REF]. These studies suggest that elevated tHcy concentrations are also a risk factor for CVA in children and newborns. Since the incidence of CVA in patients admitted to a neonatal intensive care unit is relatively high, we conducted the first prospective randomized folinic acid intervention study in newborns to study whether this risk factor can be modified in this age category.

Subjects and Methods

Subjects

We conducted a prospective randomised intervention study. All newborns admitted to the neonatal intensive care unit in the University Medical Centre Nijmegen, the Netherlands in 2003 were eligible to the study. Those with midline-defects, ECMO-treatment, overt renal failure or blood transfusion were excluded. After written informed consent was obtained, the subjects were randomised for receiving daily folinic acid supplement (intervention group) or no supplement (control group). No supplement was given because a placebo-effect is unlikely in this age category. Parents, nurses and physicians were not blinded to group assignment. Patients received normal medical care during this research project. Evaluations of cerebral ultrasounds and/or scans were performed by physicians not involved in this project and unaware of supplementation. Randomization was centralized but not computerized. This means that for each participant an envelope was drawn by the research assistant at the time of inclusion. These envelopes, filled with "intervention" or "control" notes were mixed thoroughly before they were stored in a box. This study was approved by the local ethics committee (Committee on Ethics in Research with Human Subjects Nijmegen).

Data collection

Clinical data from the newborn were obtained daily from the medical record. We obtained data on gestational age, birth weight, newborn sex, days on ventilator, laboratory results, sepsis/meningitis, intraventricular bleeding or stroke. When newborns were transferred to another hospital during the study period daily folinic acid supplementation (intervention group) was continued and blood samples were drawn as planned.

Blood sampling

Blood was drawn either capillary or arterially during routine blood sampling. Samples were taken within 48 hours after birth (baseline), at day 8 ±1 day (one week) and at day 15 ±1 day (two weeks). EDTA blood (0.5-1.0 ml) was placed on ice immediately and centrifuged within 2 hours. Plasma and cell pellets were separated and stored at -30 0 C until analysis.

Folinic Acid supplementation

Folinic acid was given for two weeks as 5-formyltetrahydrofolate (10 mg/ml) (Pharmachemie bv, Haarlem, the Netherlands). This solution was administered either intravenously (first week) or orally. To lower tHcy in adults 5 mg/day folic acid is frequently used. Using an average bodyweight of 70 kg for adults we calculated a daily dose of 70 µgram/kg/day for our newborns.

Biochemical analyses

Plasma tHcy was determined in EDTA plasma by an automated high-performance liquid chromatography method with reverse-phase separation and fluorescent detection (Gilson 232-401 sample processor, Spectra Physics 8800 solvent delivery system and Spectra Physics LC 304 fluorometer) as described by Poele-Pothof (te Poele-Pothoff et al., 1995).

Plasma folate was determined in EDTA plasma in a competitive immunoassay (Immulite 2000, Siemens Medical Solutions Diagnostics).

Statistical analysis

Results were presented as median values with their 25 th and 75 th percentiles. Differences between groups were determined by Mann-Whitney test or independent samples T-test where appropriate. Statistical analyses were all performed using SPSS 14.0 for Windows.

Results

We have included 37 newborns in our study (19 intervention group, 18 control group).

Characteristics of both groups are presented in Table I. Newborns were admitted because of prematurity, respiratory, infectious or circulatory problems. They were between 26 and 40 weeks of gestational age and between 600 and 3500 grams. Parameters were not significantly different at baseline between the intervention and control group.

Figure 1 shows individual plasma tHcy and folate concentrations at baseline and after one and two weeks from both groups. Due to transfer to another hospital, death, loss of samples and/or too little amount of blood sampled data were not complete for all newborns.

In table II median tHcy and folate concentrations at three time points in both groups and mean change between baseline and 2 weeks are shown. Neither plasma tHcy nor folate concentrations were correlated to gestational age, birth weight or creatinine at the three time points. Plasma tHcy concentrations did not differ significantly at the three time points in the intervention group nor the control group. Plasma tHcy concentrations in the intervention and control group were not different. Plasma folate concentrations at two weeks in the intervention group were significantly higher compared to concentrations at two weeks in the control group.

Discussion

In contrast to our expectation we could not demonstrate a homocysteine lowering effect of folinic acid administration in newborns in this prospective randomized intervention study. This unexpected finding might be influenced by folinic acid dose, time of blood sampling, way of administration or other factors. However, it could also indicate that homocysteine metabolism in the neonate may differ from metabolism later in life.

We are fully aware that this is a small study with only 37 patients included. Still, since data on folate and homocysteine in newborns are scarce, we believe that our results add information on one carbon metabolism in this age category. We have performed an intervention study without placebo group because a placebo-effect is unlikely in this age category. We strongly do believe that this did not influence our results since every newborn received routine neonatal intensive care. Furthermore, cerebral ultrasounds and/or scans were judged by physicians who were not aware of this study.

Fuller et al routinely supplemented preterm infants with 1 mg folate orally (pteroylglutamic acid) to encourage growth rates in erythroblastotic newborns without gross side effects. This is, on a body-weight basis, around 250 times the recommended amount for adults and around 5-20 times the amount recommended at that time for (premature) infants.

Based on results from other studies, they extrapolated that 0.05-0.2 mg/day folate appear safe for preterm infants which is comparable to our dose [START_REF] Fuller | Plasma folate levels in preterm infants, with and without a 1 mg daily folate supplement[END_REF]. In adults, folic acid supplementation (0.4-5mg per day) reduces tHcy between 13 and 25% even in the absence of folate deficiency or elevated tHcy concentrations (Homocysteine Lowering Trialists' Collaboration, 2005). In a meta-analysis by the Homocysteine Lowering Trialists' Collaboration the effect of folic acid supplementation was more profound with incremental doses of folic acid, higher pretreatment tHcy concentrations or lower pretreatment folate concentrations (Homocysteine Lowering Trialists' Collaboration, 2005). In the present study, we supplemented the newborns with 0.07mg/kg/day folinic acid which is comparable to 5 mg folic acid a day in adults. We considered this dose to be high enough to expect an effect in newborns since in adults 0.2 mg folic acid or higher a day decreases tHcy concentrations.

Blood samples were obtained at baseline and one and two weeks after birth. Since several authors have described a significant tHcy concentration reduction with folic acid within 14 days in adults we considered this a sufficient period of time to study if folinic acid decreases tHcy concentrations [START_REF] Brattstrom | Folic acid--an innocuous means to reduce plasma homocysteine[END_REF]Wilcken et al., 1988).

Folic acid supplementation was given as 5-formyltetrahydrofolate (folinic acid), an active form of folate containing a one carbon group for metabolism. Several studies show that folinic acid given either intravenously or orally was as effective or even better than folic acid in lowering plasma tHcy concentrations [START_REF] Hauser | Efficacy of folinic versus folic acid for the correction of hyperhomocysteinemia in hemodialysis patients[END_REF][START_REF] Buccianti | Reduction of the homocysteine plasma concentration by intravenously administered folinic acid and vitamin B(12) in uraemic patients on maintenance haemodialysis[END_REF]Yango et al., 2001;[START_REF] Touam | Effective correction of hyperhomocysteinemia in hemodialysis patients by intravenous folinic acid and pyridoxine therapy[END_REF]. Ducloux et al showed a similar tHcy lowering effect of folinic acid administered intravenously compared to orally supplemented folinic acid. They also observed a similar tHcy lowering effect of intravenously administered folinic acid compared to folic acid in adults [START_REF] Ducloux | Hyperhomocysteinaemia therapy in haemodialysis patients: folinic versus folic acid in combination with vitamin B6 and B12[END_REF]. Based on these results and the possibility of intravenous administration we considered folinic acid as the best way to study its lowering effect on tHcy in newborns.

Data on the pattern of neonatal tHcy and plasma folate concentrations are scarce.

Median tHcy between 6.3-6.8 µmol/L and mean tHcy between 6.1-7.8 µmol/L have been described in umbilical cord blood or the first days after birth [START_REF] Minet | Assessment of vitamin B-12, folate, and vitamin B-6 status and relation to sulfur amino acid metabolism in neonates[END_REF][START_REF] Karademir | Vitamin B12, folate, homocysteine and urinary methylmalonic acid levels in infants[END_REF][START_REF] Couto | Folate, vitamin B12 and total homocysteine levels in neonates from Brazil[END_REF][START_REF] Guerra-Shinohara | Relationship between total homocysteine and folate levels in pregnant women and their newborn babies according to maternal serum levels of vitamin B12[END_REF][START_REF] Monsen | Homocysteine and methylmalonic acid in diagnosis and risk assessment from infancy to adolescence[END_REF][START_REF] Refsum | Screening for serum total homocysteine in newborn children[END_REF][START_REF] Hongsprabhas | Plasma homocysteine concentrations of preterm infants[END_REF]. At day 7 Jyothi et al describe median tHcy concentrations of 6.3 µmol/L [p25-p75:2.8-8.4 µmol/L] whereas Fokkema et al observed mean tHcy concentrations of 6.4±2.6 µmol/L at day 10 and 6.7±2.4 µmol/L at day 20 [START_REF] Jyothi | Red cell folate and plasma homocysteine in preterm infants[END_REF][START_REF] Fokkema | Plasma total homocysteine increases from day 20 to 40 in breastfed but not formula-fed low-birthweight infants[END_REF]. Our data are comparable with these results.

Hongsprabhas et al describe an increase in mean tHcy concentrations in the first two weeks of life [START_REF] Hongsprabhas | Plasma homocysteine concentrations of preterm infants[END_REF]. In our study, tHcy concentrations do not differ between baseline and one and two weeks after birth neither in the control group nor in the intervention group.

With respect to folate, mean neonatal plasma concentrations of about 25 nmol/L have been described [START_REF] Minet | Assessment of vitamin B-12, folate, and vitamin B-6 status and relation to sulfur amino acid metabolism in neonates[END_REF][START_REF] Couto | Folate, vitamin B12 and total homocysteine levels in neonates from Brazil[END_REF][START_REF] Guerra-Shinohara | Relationship between total homocysteine and folate levels in pregnant women and their newborn babies according to maternal serum levels of vitamin B12[END_REF][START_REF] Schulpis | Maternalneonatal folate and vitamin B12 serum concentrations in Greeks and in Albanian immigrants[END_REF][START_REF] Refsum | Screening for serum total homocysteine in newborn children[END_REF] which is comparable to our results. These relatively high plasma folate concentrations in newborns compared to adults might explain the loss of correlation with tHcy concentrations in this age category.

The past few years, several authors describe a stronger relationship in infants between cobalamin and tHcy than between plasma folate and tHcy [START_REF] Minet | Assessment of vitamin B-12, folate, and vitamin B-6 status and relation to sulfur amino acid metabolism in neonates[END_REF][START_REF] Bjorke Monsen | Determinants of cobalamin status in newborns[END_REF][START_REF] Guerra-Shinohara | Relationship between total homocysteine and folate levels in pregnant women and their newborn babies according to maternal serum levels of vitamin B12[END_REF][START_REF] Molloy | Maternal and fetal plasma homocysteine concentrations at birth: the influence of folate, vitamin B12, and the 5,10-methylenetetrahydrofolate reductase 677C-->T variant[END_REF][START_REF] Fokkema | Plasma total homocysteine increases from day 20 to 40 in breastfed but not formula-fed low-birthweight infants[END_REF]. Bjorke-Monsen et al describe even an absence of correlation between plasma folate and tHcy (r= -0.06) in 173 apparently healthy newborns at day 4 while correlation between cobalamin and tHcy was high (r=-0.52)(Bjorke [START_REF] Bjorke Monsen | Determinants of cobalamin status in newborns[END_REF]. Minet et al also describe a better correlation between cobalamin and tHcy in comparison to plasma folate and tHcy (r=-0.64 and r= 0.30 respectively) [START_REF] Minet | Assessment of vitamin B-12, folate, and vitamin B-6 status and relation to sulfur amino acid metabolism in neonates[END_REF]. Van Beynum et al suggest that homozygosity for MTHFR 677C->T polymorphism only influences homocysteine concentrations in children (0-19 years) with low folate status [START_REF] Van Beynum | Total homocysteine and its predictors in Dutch children[END_REF].

Taken together, cobalamin might be a more important determinant of tHcy than plasma folate in infants and thus cobalamin treatment may be a better choice to decrease tHcy concentrations in infants and newborns (Bjorke- [START_REF] Bjorke-Monsen | Common metabolic profile in infants indicating impaired cobalamin status responds to cobalamin supplementation[END_REF].

During our study period, four children suffered from a CVA. Although this study was not designed to explore the effect of folinic acid supplementation on the incidence of CVA, only one patient from the intervention group had a CVA compared to three newborns in the control group.

In conclusion: we could not observe a homocysteine lowering effect of folinic acid supplementation in a two weeks intervention study in newborns. This observation was unexpected because similar conditions in adults substantially lower tHcy concentrations. Our findings indicate that one carbon metabolism in newborns differs from adults. Recent data suggest that cobalamin is a stronger determinant of tHcy concentrations in newborns. Thus, There are no conflicts of interest. We do not have to declare any competing interests. There is no funding to report.

Figure 1 .

 1 Figure 1. Two weeks follow-up of homocysteine and folate concentrations in newborns in the control and intervention group.Individual data are shown. Concentrations at baseline, after 1 and 2 weeks are depicted.

Figure 1a .

 1a Figure 1a. Plasma homocysteine concentrations in the control group.

Figure 1b .

 1b Figure 1b. Plasma homocysteine concentrations in the intervention group.

Figure 1c .

 1c Figure 1c. Plasma folate concentrations in the control group.

Figure 1d .

 1d Figure 1d. Plasma folate concentrations in the intervention group.

Folic

  acid and reduction of plasma homocysteine concentrations in older adults: a dose-response study. Am J Clin Nutr, 77, 1318-1323. Verhoef P & de Groot LC (2005). Dietary determinants of plasma homocysteine concentrations. Semin Vasc Med, 5, 110-123. Wang X, Qin X, Demirtas H, Li J, Mao G, Huo Y et al. (2007). Efficacy of folic acid supplementation in stroke prevention: a meta-analysis. Lancet, 369, 1876-1882. Wilcken DE, Dudman NP, Tyrrell PA, and Robertson MR (1988). Folic acid lowers elevated plasma homocysteine in chronic renal insufficiency: possible implications for prevention of vascular disease. Metabolism, 37, 697-701. Yango A, Shemin D, Hsu N, Jacques PF, Dworkin L, Selhub J et al. (2001). Rapid communication: L-folinic acid versus folic acid for the treatment of hyperhomocysteinemia in hemodialysis patients. Kidney Int, 59, 324-327.

Table I :

 I Population characteristics.

	Data are expressed as median [25th-75 th percentile] or number (percentage). CVA =
	cerebrovascular accident. tHcy baseline=total homocysteine concentration at baseline.
	Folate concentration=folate concentration at baseline. Independent samples t-test was used
	to compare groups. P values are calculated using Mann-Whitney U test or Fisher exact test
	where appropriate.

Table I . Population characteristics of 37 newborns participating in this folinic acid intervention study Intervention group Control group pvalue

 I

	Number of children	19	18
	Gestational age (weeks)	29.7 [28.4;31.3]	29.7 [29.6;34.9]	0.55
	Birth weight (grams)	1349 [1108;1640]	1256 [956;1712]	0.63
	Male sex, n(%)	9 (47%)	11 (61%)	0.51
	Days on ventilator, n [range]	1 [0;2.9]	1.5 [0.5;3.0]	0.90
	Sepsis, n(%)	7 (37%)	11 (61%)	0.19
	CVA, n(%)	1 (5%)	3 (17%)	0.34
	tHcy baseline (µmol/L)	5.6 [3.9;15.1]	7.0 [5.8;13.9]	0.43
	Folate baseline (nmol/L)	23 [16;44]	45 [35;53]	0.09

Table II : Changes in homocysteine and folate concentrations in folinic acid supplemented and control newborns during the first two weeks of life.

 II

	tHcy=total homocysteine concentration (µmol/L). Fol=folate (nmol/L). Values are expressed
	as median and 25 th and 75 th percentile. Intervention consisted of 70 µgram/kg/dag folinic acid
	intravenously or orally. Differences between intervention and control group are tested using
	Mann-Whitney test. P value < 0.05 was considered significant.

TABLE II : Changes in homocysteine and folate concentrations in folinic acid supplemented and control newborns during the first two weeks of life

 II Smeitink JA, den Heijer M, te Poele Pothoff MT, and Blom HJ (1999).Afman LA, Heil SG, and Blom HJ (2006). Genetic variation in genes of folate metabolism and neural-tube defect risk. Proc NutrSoc, 65,[204][205][206][207][208][209][210][211][212][213][214][215] van Oort FV, Melse-Boonstra A, Brouwer IA, Clarke R, West CE,Katan MB et al. (2003).

	van Beynum I, Hyperhomocysteinemia: a risk factor for ischemic stroke in children. Circulation, 99,
	2070-2072.
	van der Linden I,
		Median tHcy	Median tHcy	Median tHcy	Mean % change
		[p25-p75]	[p25-p75]	[p25-p75]	(95%CI)
		baseline	1 week	2 weeks
	placebo	7.0 [5.8;14.1]	5.3 [1.4;20.4]	6.1 [2.2;17.1]	2.6% (-82;87)
	intervention	5.6 [4.0;16.2]	7.3 [1.9;26.8]	6.3 [2.7,11.9]	111% (-373;597)
	p value	0.43	0.33	0.79	0.62
		Median fol	Median fol	Median fol	Mean % change
		[p25-p75]	[p25-p75]	[p25-p75]	(95%CI)
		baseline	1 week	2 weeks
	placebo	46 [36;53]	31 [15;61]	40 [12;134]	-12% (-132;108)
	intervention	23 [16;44]	35 [13;97]	78 [33;184]	167% (-291;625)
	p value	0.09	0.44	0.01	0.03