Astrid Decoene
email: astrid.decoene@math.u-psud.fr

Sébastien Martin
email: sebastien.martin@math.u-psud.fr

Bertrand Maury
email: bertrand.maury@math.u-psud.fr

Microscopic modelling of active bacterial suspensions 1

Keywords: Stokes flow, fluid-particle flows, self-propulsion, finite element method, penalty method, chemotaxis. AMS subject classification: 12A34, 56B78

We present two-dimensional simulations of chemotactic self-propelled bacteria swimming in a viscous fluid. Self-propulsion is modelled by a couple of forces of same intensity and opposite direction applied on the rigid bacterial body and on an associated region in the fluid representing the flagellar bundle. The method for solving the fluid flow and the motion of the bacteria is based on a variational formulation written on the whole domain, strongly coupling the fluid and the rigid particle problems: rigid motion is enforced by penalizing the strain rate tensor on the rigid domain, while incompressibility is treated by duality. This model allows to achieve an accurate description of fluid motion and hydrodynamic interactions in moderate to concentrated active suspensions. A mesoscopic model is also used, in which the size of the bacteria is supposed to be much smaller than the elements of fluid: the perturbation of the fluid due to propulsion and motion of the swimmers is neglected, and the fluid is only subjected to the buoyant forcing induced by the presence of the bacteria, which are denser than the fluid. Although this model does not accurately take into account hydrodynamic interactions, it is able to reproduce complex collective dynamics observed in concentrated bacterial suspensions, such as bioconvection. From a mathematical point of view, both models lead to a minimization problem which is solved with a standard Finite Element Method. In order to ensure robustness, a projection algorithm is used to deal with contacts between particles. We also reproduce chemotactic behaviour driven by oxygen: an advection-diffusion equation on the oxygen concentration is solved in the fluid domain, with a source term accounting for oxygen consumption by the bacteria. The orientations of the individual bacteria are subjected to random changes, with a frequency that depends on the surrounding oxygen concentration, in order to favor the direction of the concentration gradient.

1. Individual and collective dynamics of self-propelled chemotactic bacteria

Individual swimming and chemotactic behaviour

We focus on flagellated bacteria that are able to swim in a viscous fluid by means of self-propulsion, as Escherichia coli (E. coli) and Bacillus subtilis (B. subtilis). These are very common bacteria that can be found in digestive tract of animals, or in soil and water. They are so small that, on their scale, the fluid is essentially viscous and inertial effects can be neglected when describing their locomotion. In fact, these microorganisms live in the low Reynolds regime (Re ∼ 10 -2 ) (see [START_REF] Berg | Coli in Motion[END_REF], [START_REF] Lauga | The hydrodynamics of swimming microorganisms[END_REF] and references therein). E. coli and B. subtilis are among the most studied and well-understood bacteria, and they represent important model organisms in many fields of research, particularly molecular biology, genetics, biochemistry. Individual cells of these bacteria are rod-shaped (length about 2 µm for E. coli, about 4 µm for B. subtilis, diameter of the order of 1 µm for both), and their surface is peritrichously flagellated. The helical flagella, responsible for propulsion, are significantly longer than the cell's body: approximately 20 nm in diameter, and 10 -15 µm long. For E. coli, the flagellar filaments are four or five. They are distributed randomly over the cell's body, attached to motors that are fixed within the cell membrane [START_REF] Lauga | The hydrodynamics of swimming microorganisms[END_REF][START_REF] Turner | Real-time imaging of fluorescent flagellar filaments[END_REF]. When the motor turns counter-clockwise, the filaments wrap into a bundle that pushes the cell along at speeds of 25 to 35 µm s -1 . When one or more of the motors reverse, the swimming direction of the cell can change.

As described by Berg [START_REF] Berg | Random walks in biology[END_REF][START_REF] Berg | Coli in Motion[END_REF], these bacteria alternate two basic behavioral modes: a more or less linear motion called a run, and a highly erratic motion called tumble, the purpose of which is to reorient the cell. During a run the bacterium moves in the most recently chosen direction. Then it tumbles: it randomly chooses a new direction and starts moving straightforward again. That is how these bacteria perform chemotaxis, which means swimming up a chemoattractant gradient [START_REF] Berg | Random walks in biology[END_REF][START_REF] Kessler | Hydrodynamic phenomena in suspensions of swimming microorganisms[END_REF]. Run times are typically much longer than the time spent tumbling, and they are closely related to the surrounding environment. In fact, when bacteria move in a favorable direction (i.e. in the direction of a nutrient concentration gradient) the run times are increased; on the contrary, they are decreased when the direction is not favorable. These bacteria are too small to detect instantaneously spatial differences in the concentration of an attractant at their own length scale. Therefore they measure if they are going in the right direction by doing several successive evaluations of the concentration in time. The effect of alternating these two modes of behavior, and in particular, of increasing the run duration when moving in a favorable direction, is that a bacterium executes a three-dimensional random walk with drift in a favorable direction when observed on a sufficiently long time scale.

Note that many flagellated bacteria can also move on the surface of certain solid media. Their rapid and organized propagation on these surfaces is known as swarming, and strongly depends on cell-to-cell contact and reversal of direction (see for instance [START_REF] Darnton | Dynamics of bacterial swarming[END_REF] and [START_REF] Kaiser | Bacterial swarming, a re-examination of cell movement patterns[END_REF]). We will not consider this behaviour in this work since we focus on hydrodynamic interactions, but there are interesting common aspects in both swimming and swarming motility.

Collective dynamics in bacterial suspensions

The individual swimming mechanism and the chemotactic behaviour of self-propelled bacteria give rise to interesting and complex collective dynamics and large-scale motions in bacterial suspensions. This has been highlighted through many experiments, as for example those in [START_REF] Cisneros | Fluid dynamics of self-propelled microorganisms. from individual to concentrated populations[END_REF][START_REF] Dombrowski | Self-concentration and large-scale coherence in bacteria dynamics[END_REF][START_REF] Lauga | The hydrodynamics of swimming microorganisms[END_REF][START_REF] Sokolov | Enhanced mixing and spatial instability in concentrated bacterial suspensions[END_REF][START_REF] Wu | Particle diffusion in a quasi-two-dimensional bacterial bath[END_REF], and has created important concern during the last decade. In particular, there has been a great interest in understanding if and how small scale swimming could generate large scale correlation. It is believed that the essential ingredients in the formation of large-scale patterns are hydrodynamic interactions between individual swimmers and the environment, including boundary effects and cell-to-cell interaction, as well as gravity and tumbling.

The easiest observable patterns in a bacterial suspension, especially in simulations (see Section 4), are recirculations and whirls at a scale which is much larger than the individual bacterium. This can already be observed at moderate concentration. But when the bacterial cells are very concentrated, or nearly close-packed, they form codirectionally swimming structures that move chaotically, giving the appearance of turbulence [START_REF] Cisneros | Fluid dynamics of self-propelled microorganisms. from individual to concentrated populations[END_REF][START_REF] Wu | Particle diffusion in a quasi-two-dimensional bacterial bath[END_REF]. The velocity of the flow can also be significantly enhanced by collective dynamics. In [START_REF] Sokolov | Enhanced mixing and spatial instability in concentrated bacterial suspensions[END_REF], experimental studies have revealed collective flow speeds that exceed the speed of the individual bacterium by an order of magnitude. This phenomenon can be essential to the survival of the organisms, since they allow an enhanced molecular transport into the bacterial suspension. In the case of oxygen, for example, which is consumed by the bacteria, diffusion may not be sufficient in absence of transport to supply the suspension.

Besides, the presence of microscopic swimmers can strongly modify the rheological properties of a fluid. This has been well established in the case of passive inclusions. But experimental [START_REF] Rafai | Effective viscosity of microswimmer suspensions[END_REF][START_REF] Sokolov | Reduction of viscosity in suspension of swimming bacteria[END_REF] and numerical studies [START_REF] Lefebvre | Apparent viscosity of a mixture of a Newtonian fluid and interacting particles. Fluid-solid interactions: modeling, simulation, bio-mechanical applications[END_REF][START_REF] Kessler | Hydrodynamic phenomena in suspensions of swimming microorganisms[END_REF] have shown that suspensions of active inclusions, i.e. inclusions which are submitted to some internal force, may have completely different properties. In [START_REF] Lefebvre | Apparent viscosity of a mixture of a Newtonian fluid and interacting particles. Fluid-solid interactions: modeling, simulation, bio-mechanical applications[END_REF], the apparent viscosity of a mixture of a Newtonian fluid and interacting particles has been evaluated through numerical simulations. This has shown that the viscosity of the fluid can be significantly enhanced or decreased, depending on the interaction force between the particles. In [START_REF] Sokolov | Reduction of viscosity in suspension of swimming bacteria[END_REF], experiments show that self-propelled bacteria can reduce the viscosity of the suspension up to five times compared to passive bacteria. The rheology properties of suspensions of self-propelled bacteria have recently been extensively studied by Berlyand and coauthors in [START_REF] Haines | Effective viscosity of dilute bacterial suspensions: a two-dimensional model[END_REF][START_REF] Gyrya | Effective shear viscosity and dynamics of suspensions of micro-swimmers from small to moderate suspensions[END_REF]: they have considered different models for the individual bacteria, and analysed the resulting suspensions analytically in the dilute regime and numerically in the moderate concentration regime. These studies show that the decrease in the effective viscosity of the suspension is due not only to self-propulsion, but also to hydrodynamical interactions, which can lead to an alignment of bacteria with the flow. In fact, alignment is essential to the decrease of the effective viscosity (see for instance [START_REF] Kessler | Hydrodynamic phenomena in suspensions of swimming microorganisms[END_REF]).

Bioconvection is another important issue of collective dynamics in suspensions of swimming microorganisms. In a shallow suspension, due to chemotaxis, the cells swim upward toward the oxygen-rich air/fluid interface. Since bacteria are approximately 10 percent denser than water, accumulation at the interface results in an unstable gradient of mean fluid density, creating convective dynamics. In a horizontal meniscus, these unstable stratifications evolve through a Rayleigh Taylor instability in falling plumes, dragging the swimmers from the surface into the bulk fluid [START_REF] Kessler | Hydrodynamic phenomena in suspensions of swimming microorganisms[END_REF]. We have focused on an experiment by Dombrowski and coauthors [START_REF] Dombrowski | Self-concentration and large-scale coherence in bacteria dynamics[END_REF], in which bioconvec-tive dynamics in a suspension of B. subtilis in a sessile drop are displayed. The experiment shows in particular the influence of the geometry of the drop on these dynamics. Bacteria swim up to the free surface and, under the influence of gravity, slide down the slanted meniscus. This generates a persistent hydrodynamic vortex, which confines the bacteria near the contact line and promotes the advection of oxygen into the suspension. Besides, instabilities in the accumulation layer appear in the flatter part of the drop, consistent with the instability of bioconvection in fluids with a horizontal meniscus [START_REF] Kessler | Hydrodynamic phenomena in suspensions of swimming microorganisms[END_REF].

In [START_REF] Tuval | Bacterial swimming and oxygen transport near contact lines[END_REF], a macroscopic model for chemotactic self-propelled bacteria was introduced, that reproduces numerically the bioconvective dynamics observed in the experiment described previously. The model consists of three coupled equations describing the dynamics of the oxygen concentration, cell concentration and fluid velocity. The first one is an advection-diffusion equation with a sink term accounting for consumption of oxygen by the bacteria, and the second one consists of the Navier-Stokes equations with buoyant forcing, accounting for the density variations caused by bacteria. The equation describing the dynamics of the cell concentration is as follows :

∂ρ ∂t + u • ∇ρ = D n ∆ρ -∇ • (f (c)ρ∇c) , (1.1)
where ρ is the cell concentration, u is the fluid velocity, D n is the bacterial diffusion constant and f (c) is the chemosensitivity function that modulates motility, so that f (c)∇c represents the chemotactic velocity of the cells. The present approach is based on a microscopic description of those phenomena. By microscopic we mean that bacteria are considered individually, and not represented as a global density. A wide hierarchy of such microscopic models can actually be considered (see e.g. [START_REF] Hernandez-Ortiz | Transport and col lective dynamics in suspensions of confined swimming particles[END_REF][START_REF] Peruani | Self-propelled particles with fluctuating speed and direction of motion in two dimensions[END_REF][START_REF] Saintillan | Orientational order and instabilities in suspensions of selflocomoting rods[END_REF]), depending on the way the hydrodynamic interactions are handled. We also refer to [START_REF] Wu | Periodic reversal of direction allows Myxobacteria to swarm[END_REF] for a microscopic model describing swarming dynamics. To allow for suitable balancing between accuracy in the description of hydrodynamic effects and computational cost constraints, we propose here two models, at both ends of this hierarchy. The first one (see Section 2.1.) is fully microscopic: the bacteria are embodied in the computational domain, and direct simulation of Stokes flow is performed at their very scale. In order to capture alignment phenomena, and to give a realistic description of the shape of the bacteria, we have modelled bacteria as elongated elliptic rigid particles. As we shall see this fully microscopic model, which is computationally expensive, is necessary to reproduce low turbulence phenomena which are observed in dense suspensions. The second approach (see Section 2.2.), which we shall refer to as mesoscopic, lies inbetween the microscopic and the macroscopic models. The swimmers are supposed to be so small that the perturbation of the fluid due to their propulsion and motion is neglected, and the fluid is only subjected to buoyant forcing due to the presence of bacteria. This model, based on a cruder description of entities and therefore less expensive, allows to qualitatively reproduce bioconvective dynamics, as those displayed in the experiment by Dombrovski and coauthors (see Section 4). This model is mesoscopic in the sense that fluid computations shall be performed at a scale which is considered much larger than the size of the perturbation induced by the motion of a single particle. Yet it keeps some microscopic features: in particular, each bacterium is described individually, as the model integrates the run-and-tumble cycles for individual particles. Besides, it could be enriched to account more accurately for hydrodynamic interactions (e.g. by integrating Stokesian dynamics tools more sophisticated than the simple Faxen law).

Outline of the paper

In the next section, we describe the different models used for our simulations: first, the fully microscopic model (Section 2.1), then the mesoscopic model (Section 2.2), the stochastic model used to represent the run and tumble cycles (Section 2.3), and finally the model for oxygen dynamics (Section 2.4). In Section 3 we present the numerical procedure used to solve the fluid-particle interaction problem, including the way we handle contacts beween particles, and the discrete version of the stochastic model for tumbling. Finally, Section 4 is devoted to the presentation of the simulation results, obtained using both the fully microscopic and the mesoscopic models, coupled (or not) to the oxygen dynamics in the fluid and the run and tumble cycles.

Mathematical models 2.1. Fully microscopic model

We consider a connected bounded and regular domain Ω ⊂ R 2 and we denote by (B i ) i=1,...,N the rigid particles, strongly included in Ω (see right in Fig. 1). B denotes the whole rigid domain: and p is the pressure. For the sake of simplicity we will consider homogeneous Dirichlet conditions on ∂Ω. On the other hand, viscosity imposes a no-slip condition on the boundary ∂B of the rigid domain.

B = ∪ i B i . The domain Ω \ B is
In [START_REF] Decoene | Simulation of self-propelled chemotactic bacteria in a Stokes flow[END_REF], following [START_REF] Haines | Effective viscosity of dilute bacterial suspensions: a two-dimensional model[END_REF], we modelled bacteria as spheres, with an associated point force representing the action of flagella. Here, in order to better take into account the elongated shape of the swimmers, we model the bacterial bodies as ellipses. Self-propulsion is modelled by a couple of forces of same intensity and opposite direction applied on the rigid bacterial body and on a region in the fluid next to the body representing the flagellar bundle. In fact, self-propulsion is characterized by a balance of forces, unlike the case of propulsion due to external forces, which are unbalanced. It is important to note that, while the bacterial bodies are handled as rigid elongated particles moving in the fluid, the flagella are not materialized in our model. We only take into account the force they exert on the particles and on the fluid. In that framework, we have chosen to model the force exerted by the flagellar bundle on the fluid as a volume force density f p supported in a very elongated ellipse P , placed at a constant distance from the bacterial body. Spreading out the Dirac force in that way makes sense from a modelling point of view: the actual force is exerted by a bunch of flagella which move in a complex way, and that results in an effective force acting on the fluid in a zone downstream the bacterium with non-zero volume. The propulsion force is directed outward from the center of B, parallel to the major semiaxes of B and P , and has some orientation angle θ, measured from the x-axis, as shown in Fig. 1. The total force exerted on the fluid by the flagella associated to the ith particle has a constant magnitude f p and will be denoted by F i p :

F i p = -f p τ i = P i f i P dx, with f i p = - f p meas(P i ) τ i ,
where τ i is given by the orientation angle θ i and meas(P i ) denotes the measure of the ellipse P i . The total force exerted by the flagella on their associated particle B i will be denoted by F i b and is equal to -F i p , so that we can write:

F i b = f p τ i = B i f i b dx, with f i b = f p meas(B i ) τ i .
At the initial time the particles are distributed randomly over the fluid (without overlapping). The position of the center of the ith particle is denoted by x i , and by v i and ω i its translational and angular velocities.

We have to find the velocity u = (u 1 , u 2 ) and the pressure field p defined in Ω \ B, as well as the velocities of the particles

V := (v i ) i=1,...,N ∈ R 2N and ω := (ω i ) i=1,...,N ∈ R N such that:        -µ∆u + ∇p = f f in Ω \ B, ∇ • u = 0 in Ω \ B, u = 0 on ∂Ω, u = v i + ω i × (x -x i ) on ∂B i , ∀i ∈ {1, ..., N }. (2.1)
The forces considered on the fluid are the forces exerted by the flagella, so that

f f = i=1,...,N f i p χ i p ,
where χ i p is the characteristic function associated to the flagellar bundle P i . Finally, Newton's second law of motion, written here in the non-inertial regime, couples the equations:

       B i ρ g + B i f i b - ∂B i σ • n = 0 ∀i, ∂B i (x -x i ) × σ • n = 0 ∀i , (2.2)
where ρ denotes the buoyant density of bacteria, which is positive as bacteria are denser than the fluid, and g = (0, -g) is gravity.

The motion of each bacterium B i is then set by its instantaneous velocity u(x, t) = v i (t) + ω i (t) × (xx i (t)) defined on ∂B i . On one hand, the center x i of the particle B i follows the differential equation

ẋi (t) = v i (t). (2.3)
On the other hand the relative position of its corresponding flagellar bundel P i rotates with angular

velocity ω i , i.e. θi (t) = ω i (t). (2.4)
This model describes the dynamics of self-propelled particles in a Stokes flow, and thus it describes the hydrodynamic interactions between particles (i.e. their interaction through the fluid). One can wonder what happens when they get very close to each other, especially in the case of dense suspensions. Assuming, as we have in this model, that particle surfaces are smooth and that Stokes model is valid at any scale, it is known that contacts are not supposed to happen (see [START_REF] Gérard-Varet | Regularity Issues in the Problem of Fluid Structure Interaction[END_REF][START_REF] Hillairet | Lack of collision between solid bodies in a 2D constant-density incompressible flow[END_REF]). In fact, two passive or active particles can get infinitely close to each other, but they will never collide. Therefore, solving the equations of this model exactly would lead to interactions between particles without contact. However, this is no longer true when we solve the time and space approximation of these equations, and therefore we have to deal with possible collisions in the numerical simulations. We send the reader to section 3.4 for a description of the method used to avoid contact between particles numerically.

A mesoscopic model

We consider here that the swimmers are much smaller than the scale at which fluid flow is described. The solid phase is represented as:

ρ = i=1,...,N m δ x i ,
where m is the buoyancy mass of one bacterium and δ x i is the delta function related to the position of i-th cell. The equilibrium of forces applied on the body of each bacterium i writes:

F i b + β(u(x i ) -v i ) + mg = 0 , (2.5)
where β denotes the coefficient relating the hydrodynamic force exerted on the body to its relative velocity, u(x i ) is the velocity of the fluid at the position of the bacterium i, and v i is the velocity of the swimmer. We recall that following Faxen's law (see for instance [START_REF] Happel | Low Reynolds Number Hydrodynamics[END_REF][START_REF] Kim | Microhydrodynamics: Principles and Selected Applications[END_REF]), for a sphere in an unbounded Stokes flow, β = 6πµR b , where R b is the radius of the sphere. Since the size of bacteria is very small compared to the fluid elements, we neglect the effect of the propulsion force and the swimmer's motion on the fluid: the latter is only subjected to buoyant forcing induced by the presence of the bacteria. Thus, the model consists in solving the following problem in the fluid domain:

       -µ∆u + ∇p = i=1,...,N mgδ x i in Ω, ∇ • u = 0 in Ω, u = 0 on ∂Ω .
(2.6) Following (2.5), the velocity of each particle i is then given by :

v i = u(x i ) + 1 β (F i b + mg) . (2.7)
The position of each particle B i will follow the differential equation (2.3). Let us make it clear that this model is somewhat ambiguous, for its level of accuracy strongly depends on the way discretization parameters are chosen (i.e. typical mesh step size), compared to the typical interparticle distance. We shall consider here coarse discretizations: the mesh diameter shall not be small compared to this distance, so that the fluid actually sees the solid phase as a macroscopic density. The same model could be used in a different manner (closer to the full microscopic model) by using a fine discretization : in this case, local perturbations induced by single bacteria can be reproduced by the numerical model, only perturbations due to the swimming process (which are an order of magnitude smaller) are neglected. To make a step further toward full microscopic description, yet keeping a punctual description of entities, one could account for these perturbations by adding a doublet force term F i b (δ x -δ y ) for each bacterium, y denoting some average position of the flagella.

Yet, as we shall see, despite the crude assumptions upon which it is based, the model just described allows to reproduce the basic features of the collective behaviour observed in bioconvection experiments.

Modelling tumbling and chemotaxis

The bacteria we consider perform chemotaxis by alternating running and tumbling (see Section 1.1), and biasing the duration of the runs. The run durations are exponentially distributed with a rate that depends on the environment. In the case of E. coli, for example, the tumbling rate is of average λ ∼ 1 s -1 in isotropic medium (see [START_REF] Locsei | Run and Tumble in Chemotaxis in a Shear Flow; The Effect of Temporal Comparisons, Persistence, Rotational Diffusion, and Cell Shape[END_REF]). Therefore, we model tumbling through a Poisson process with variable intensity. The time spent tumbling is neglected, and the run time T run of each bacterium, i.e. the time elapsed between two tumbles, follows an exponential law of parameter λ, which represents the tumbling frequency. This frequency varies depending on the surrounding environment of the bacterium, that is depending on if it is going in a favorable direction or not.

Let c be the concentration of the chemical attractant considered. We use a simple model in which the tumbling frequency is determined by the sign of the Lagrangian derivative Dc/Dt of the attractant concentration along the trajectory of the bacterium :

λ(t) = λ 0 (1 -ε φ Dc Dt ) , (2.8)
were λ 0 is the average tumbling rate and φ is the sign function. Note that this model doesn't take into account the memory of bacteria relatively to the concentration of attractant along their trajectory. Finer models have been used to deal with memory (see for instance [START_REF] Locsei | Run and Tumble in Chemotaxis in a Shear Flow; The Effect of Temporal Comparisons, Persistence, Rotational Diffusion, and Cell Shape[END_REF] and [START_REF] Segall | Temporal comparisons in bacterial chemotaxis[END_REF]).

Although the swimming directions before and after a tumble are correlated, with a mean angle between both directions of ∼ 60 • [1, 24], we neglect this fact and, in our model, bacteria choose a new orientation angle θ which is uniformly distributed over (0, 2π) after each run. However, persistence in the re-orientation seems to have non-neglectable effects, as analysed by Locsei and Pedley in [START_REF] Locsei | Run and Tumble in Chemotaxis in a Shear Flow; The Effect of Temporal Comparisons, Persistence, Rotational Diffusion, and Cell Shape[END_REF].

Oxygen dynamics

In this work we consider oxygen as the only chemoattractant in the bacteria's environment. The dynamics of the oxygen concentration c are described by the following equation:

∂ t c + u • ∇c -D c ∆c = -g(c, B), (2.9)
where D c is the oxygen diffusion coefficient, u the fluid velocity and -g(c, B) is the sink term accounting for the oxygen consumption by the bacteria. Following [START_REF] Tuval | Bacterial swimming and oxygen transport near contact lines[END_REF], this term is defined through

g(c, B) = κ f (c) χ B ,
where κ is the volumic consumption rate (in oxygen molecules per second per volume of bacteria), χ B is the characteristic function for the rigid ("bacterial") domain, and c → f (c) modulates this rate, equal to 1 when c is large, but vanishing as c is smaller than a critical value. This modulation allows to verify the minimum principle on the oxygen concentration. The motion of the bacteria is also related to the oxygen concentration, since bacteria which do not consume enough oxygen loose their capacity of self-propulsing. Therefore, the intensity of the self-propulsion force f p is modulated by a function called the motility function r(c) (see again [START_REF] Tuval | Bacterial swimming and oxygen transport near contact lines[END_REF]), which is equal to one at large c and vanishes rapidly for small c. In our numerical simulations we have taken r(c) = arctan (kc/c * ) arctan k , where c * is the maximal value of the concentration in the domain and k = 10.

Coupling the hydrodynamic model with tumbling and oxygen dynamics

The hydrodynamic model is coupled to chemotaxis through the individual reorientation (tumbling) process just described. Concretely, each bacterium has a translation and rotation movement determined by the hydrodynamic model (see equations (2.3)-(2.4)). But it additionally tumbles at a frequency λ(t) which is given by equation (2.8) and depends on the sign of the oxygen concentration gradient on the bacterium trajectory. When it tumbles, a bacterium changes its position following equation (2.3) and randomly chooses a new orientation instead of rotating following equation (2.4).

Numerical procedure

Variational formulation for the fully microscopic model

We want to treat the fluid-particle interaction problem (2.1)-(2.2) in a strongly coupled way. In addition, we want to avoid working on the time dependent fluid domain, what would necessitate remeshing. Therefore, following [START_REF] Janela | A penalty method for the simulation of fluid-rigid body interaction[END_REF][START_REF] Lefebvre | Fluid-particle simulations with Freefem++[END_REF], we write a variational formulation of the continuous problem involving functions which are defined on the whole domain Ω.

Let us first introduce the following constrained functional spaces:

K ∇ = {u ∈ H 1 0 (Ω), ∇ • u = 0} , K B = {u ∈ H 1 0 (Ω), ∀i ∃(v i , ω i ) ∈ R 2 × R; u(x) = v i + ω i × (x -x i ) a.e. in B i } .
K ∇ is the space of divergence free functions defined on Ω, and K B is the space of functions which do not deform B. We will rather use the equivalent characterisation D(u) = 0 for rigid motion (see for instance [START_REF] Temam | Mathematical modeling in continuum mechanics[END_REF]), so that:

K B = u ∈ H 1 0 (Ω), D(u) = 0 a.e. in B .
The solution to the initial problem, defined on Ω \ B, can be extended on the whole domain Ω by a function in

K B : u(x, •) = v i + ω i × (x -x i ) in B i ∀i,
and we still denote this extension by u. Let (u, p) be the extended solution of the problem. On one hand, we multiply the momentum equation (2.1) by a test function u in K B and integrate it by parts over Ω \ B. Using the fact that ∇ • u = 0 and D(u) = 0 in B, the integrals can be extended over Ω. Finally, using the boundary conditions of the problem and writing the incompressibility equation in its usual variational form, we obtain the following variational formulation:

         2µ Ω D(u) : D( u) - Ω p ∇ • u = Ω f • u, ∀ u ∈ K B , Ω q ∇ • u = 0, ∀q ∈ L 2 0 (Ω), (3.1)
where

f = N i=1 f i b χ i b + f i p χ i p ,
and L 2 0 (Ω) stands for the set of L 2 functions over Ω with zero mean value. Functions χ i p and χ i b are, respectively, the characteristic function associated to the flagellar bundle P i and to the particle B i .

Penalty method to enforce the rigid motion

As mentioned before, we do not want to mesh the fluid domain: the complexity of the mesh due to the presence of inclusions would no allow to use standard fast solvers, and the computational cost of remeshing could become prohibitive. Therefore, we choose the fictitious domain approach, which has been used for fluid-particle interactions by several authors. One possibility is to use an iterative algorithm on an auxiliary field, composed by Lagrange multipliers, which warrants the rigid motion constraint of the particles (see for instance [START_REF] Glowinski | A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow[END_REF][START_REF] Glowinski | Finite element methods for incompressible viscous flow[END_REF]). An alternative to this are penalty methods: the rigid motion constraint is obtained by relaxing a term in the variational formulation, what amounts to replacing rigid zones by highly viscous ones (see [START_REF] Vincent | An adaptative augmented Lagrangian method for three-dimensional multimaterial flows[END_REF][START_REF] Janela | A penalty method for the simulation of fluid-rigid body interaction[END_REF][START_REF] Lefebvre | Fluid-particle simulations with Freefem++[END_REF]). In this work we choose a penalty method already used in [START_REF] Lefebvre | Fluid-particle simulations with Freefem++[END_REF] for the simulation of passive particles in a fluid. Formulation (3.1) involves a solution and test functions defined in the constrained space of rigid motions on B. We consider the minimization problem over a constrained domain associated to (3.1) and relax the constraint by introducing a penalty term in the minimized functional (see [START_REF] Lefebvre | Fluid-particle simulations with Freefem++[END_REF]). The added term is the following:

1 ε D(u) : D(u)
so that D(u) goes to zero when ε goes to zero and u tends to a rigid motion in B.

The variational formulation obtained is:

         2µ Ω D(u ǫ ) : D( u) + 2 ε B D(u ǫ ) : D( u) - Ω p ǫ ∇ • u = Ω f • u, ∀ u ∈ H 1 0 (Ω), Ω q ∇ • u ǫ = 0, ∀q ∈ L 2 0 (Ω), (3.
2) It has been proven in [START_REF] Janela | A penalty method for the simulation of fluid-rigid body interaction[END_REF] that, in this framework, the penalty method converges linearly in ε: the solution of problem (3.2) u ǫ converges to the solution u of problem (3.1) as ε vanishes, and the convergence is of order 1 in ε. We refer to [START_REF] Maury | Numerical Analysis of a Finite Element / Volume Penalty Method[END_REF] for a detailed analysis of a scalar version of this problem, which provides an error estimate for the space-discretized problem at the order ε + h 1/2 , where h is the typical size of the space discretization.

Variational formulation for the mesoscopic model

The intermediate model described in Section 2.2 consists of the Stokes equations with a particular source term, and we use the following classical variational formulation:

         2µ Ω D(u) • D( u) - Ω p ∇ • u = Ω f • u, ∀ u ∈ H 1 0 (Ω), Ω q ∇ • u = 0, ∀q ∈ L 2 0 (Ω), (3.3)
where f = i=1,...,N mgχ i b .
Note that, for regularity reasons, the delta function in (2.6) has been replaced by the characteristic function associated to a non-zero volume representing the swimmer.

Contact handling

As mentioned previously, if we consider particles with smooth surfaces in a Stokes flow, there should be no contact between them. Yet in actual simulations, collisions between particles are likely to occur because of the discretization in time. From a numerical point of view, it means that the particles may overlap when their positions are updated after the velocity field computation. The treatment of possible overlaps is even more important in the case of dense suspensions, if we want to avoid small time steps. For that purpose we use a numerical method proposed in [START_REF] Maury | A time-stepping scheme for inelastic collisions[END_REF], where inelastic collisions between rigid particles are computed. It consists of projecting the velocity field onto some convex admissible set depending on the current configuration, so that particles do not overlap.

Let us detail the method in the case of spherical particles: we denote by X n := (x n i ) i=1,...,N the position of N particles (more precisely, the position of their gravity centre) at time t n , and by Vn their a priori velocity. As stated before, the a priori updated position of the particles, defined as

X n+1 = X n + ∆t • Vn ,
may lead to non-admissible configuration, in the sense that the particles overlap. To avoid this, we project the velocities onto the following set

K(X n ) = V ∈ R 2N , D ij (X n ) + ∆t G ij (X n ) • V ≥ 0, ∀i < j , (3.4)
where

D ij (X n ) = x n i -x n j -2R b
denotes the signed distance between two spheres B i and B j and

G ij (X n ) = ∇D ij = (..., 0, -e ij , 0, ..., 0, e ij , 0, ...), e ij = x j -x i |x j -x i |
is the gradient of the distance.

Remark 1. Denote E(X n ) = V ∈ R 2N , D ij (X n + ∆t V)
≥ 0, ∀i < j the set of velocity fields V such that particles, at position X n at time t n and with velocity V, do not overlap at the next time step. The constraint

D ij (X n ) + ∆t G ij (X n ) • V ≥ 0 is the linearized form of the constraint D ij (X n + ∆t V) ≥ 0 and, furthermore, it can be shown that K(X n ) ⊂ E(X n ).
It means in particular that particles with admissible velocities at time t n do not overlap at time t n+1 .

In order to avoid overlapping, the following splitting procedure is proposed: in a first step, we solve the variational problem without taking into account the possible overlapping of the particles (thus defining the a priori velocity of the spheres), then compute the projection of this a priori velocity onto the set of admissible velocities defined by (3.4). The constrained problem is formulated as a saddle-point problem, by using the introduction of Lagrange multipliers:

Find (V n , Λ n ) ∈ R 2N × R N (N -1)/2 + such that J (V n , λ) ≤ J (V n , Λ n ) ≤ J (V, Λ n ), ∀(V, λ) ∈ R 2N × R N (N -1)/2 +
, with the following functional:

J (V, λ) = 1 2 V -Vn 2 - 1≤i<j≤N λ ij (D ij (X n ) + ∆t G ij (X n ) • V) .
Notice that the number of Lagrange multipliers corresponds to the number of possible contacts. In particular, if there is no contact between particles i and j, then Λ ij = 0 and the Lagrange multiplier is not activated; conversely, if there is a contact between the two spheres, then Λ ij may be positive and the corresponding auxiliary field allows the velocity field to satisfy the nooverlapping constraint. The approximate reaction fields Λ n = (Λ n ij ) is the dual component of a solution to the associated saddle-point problem. This problem is solved by an Uzawa algorithm (see, e.g., [START_REF] Ciarlet | Introduction à l'analyse numérique matricielle et à l'optimisation[END_REF]).

The interest in the procedure relies on the possibility to use any suitable solver for the computation of the dynamics. Contacts are handled at a second stage, without any consideration of the proper dynamics. At some point, it allows the use of any solver for the resolution of the dynamics problem: then the so-called predicted velocity field is projected onto the set of admissible velocity fields.

Remark 2. In the mesoscopic model, the bacterial bodies are modelled by spheres, and we apply the contact algorithm just described in order to prevent their overlapping. In the fully microscopic model, the bacterial bodies are modelled as elongated ellipses. In order to handle contacts, one should adapt the algorithm to the case of ellipses; that involves computing the distance between two elliptic bodies, which is not straightforward. In order to simplify computations, we rather handle contacts between the smallest spheres containing the ellipses.

Discrete tumbling

As described in 2.3, the chemotactic behaviour of the bacteria is modelled as a Poisson process with variable intensity (the frequency of tumbling λ). We time-discretise this process as follows: let ∆t tumb > 0 be a fixed small time-step, P tumb is the probability of a swimmer to tumble at least one time during ∆t tumb . This probability can be computed and depends on the values of ∆t tumb and λ:

P tumb (λ, ∆t tumb ) = 1 -exp(-λ ∆t tumb ) ∼ λ ∆t tumb . (3.5)
Now, for each bacterium, the frequency of tumbling λ varies in time following (2.8). We discretize the lagrangian derivative of the oxygen concentration Dc/Dt evaluated by the i-th bacterium along its trajectory as follows:

Dc Dt (t) i ≈ c(t) i -c(t -∆t tumb ) i ∆t tumb , (3.6)
where c(t) i denotes the oxygen concentration at time t, at the position occupied by the i-th bacterium at time t.

Thus the value of P tumb also varies in time, depending on the sign of the discrete lagrangian derivative. At each time t m = m ∆t tumb , bacterium i performs a random reorientation (a tumble) with probability:

(P tumb ) m i ∼ λ 0 (1 -ε φ (c(t m ) i -c(t m -∆t tumb ) i )) ∆t tumb ,
where we recall that φ is the sign function. Note that with this procedure we neglect the fact that there can be more than one reorientation during ∆t tumb . But this has no influence on the result as long as we neglect correlation between the directions before and after tumbling.

In the simulations presented in the next section, we have used the following values for the parameters of the tumbling model : ǫ = 1 and λ 0 ∆t tumb = 0.5. Note that this amounts to use a tumbling probability of P tumb = 0 if the direction is favorable, and P tumb = 1 if it is not. More realistic simulations can be performed by taking a smaller value of ǫ and of λ 0 ∆t tumb , which can be interpreted as the ratio between the time-step ∆t tumb used to discretize the Poisson process and the typical tumbling period 1/λ 0 .

Numerical results

We have simulated individual and collective dynamics in moderate to dense bacterial suspensions, using both models described in Section 2. The results in Sections 4.1 and 4.2 have been obtained with the fully microscopic model, whereas for the results in Section 4.3 we have used the mesoscopic model. In Section 4.1 we analyse the instantaneous velocity field obtained around one bacterium in the absence of interactions, and after that we focus on the simulation of hydrodynamic interactions: between a bacterium and a wall, and between two and several bacteria. Then, we show the velocity field obtained in moderate to dense suspensions. In Section 4.2, we concentrate on the effect of chemotaxis in a moderately concentrated suspension using the microscopic model coupled with the model for chemotaxis. Finally, Section 4.3 is devoted to the simulation of bioconvective dynamics in dense suspensions using the mesoscopic model described in 2.2.

Numerical results have been obtained with data provided in Tables 1 and2.

Fluid boundary conditions

Upper / lower walls u = 0 id. id. id. id. Lateral walls periodicity id. id. id. id.

Table 1: Parameters for the numerical simulations (I): hydrodynamic interactions.

Simulation of individual and collective motion with the complete model Velocity field induced by a single bacterium

As pointed out previously, the key feature in the dynamics of a single bacterium is that the net propulsive force of the flagella must equal the opposing drag force of the body connected to the flagella. Thus, Fig. 2 We address the swimming of individual bacteria near surfaces. Although the specific interactions between helical flagella and a boundary are crucial, our model does not focus on such a complex description of the helical flagella, but aims at describing some phenomenological features. Fig. 3 describes the velocity fluid flow generated by a bacterium placed close to a boundary, and the contour lines around it. Fig. 4 reproduces the trajectory of a bacterium which is initially located with an angle π/4 with respect to the boundary ; the motion of the self-propelled bacterium illustrates the reflection to the wall. Note that, from a numerical point of view, the flagella may be located outside the domain: the propulsive force is simply spread over the part of the flagellum which is located inside the domain, thus preserving global force balance (of course, the flagellum should not be entirely located outside the domain).

Pairwise interactions

Consider two bacteria next to each other and prescribed to move in a parallel direction. The flow patterns shown on Fig. 5 suggest that the flow tends to push the bacteria toward each other, as there is a component of the fluid flow pointing into the direction of the neighbouring bacterium, thus revealing the attraction of the fluid flow. We emphasize that Fig. 6 evidences a behaviour which has been described in [START_REF] Gyrya | Effective shear viscosity and dynamics of suspensions of micro-swimmers from small to moderate suspensions[END_REF] and in [START_REF] Lauga | The hydrodynamics of swimming microorganisms[END_REF]: trajectories of initially parallel bacteria in the mirror image configuration attract and rotate outward. Then when rotated sufficiently outward, the bacteria swim off. Note that the flagella may overlap in our case, as they only model the phenomenological response of the flagella over the fluid.

Interactions between five ellipsoidal bacteria Fig. 7 shows the velocity fluid flow and streamlines around five bacteria. The velocity distribution shows the attraction of most of the cells to each other. Transverse flows between the body of a follower and the tail of a preceder can be observed. As pointed out in [START_REF] Cisneros | Fluid dynamics of self-propelled microorganisms. from individual to concentrated populations[END_REF], the numerical results show that there is little front-to-back penetration of fluid. The exchange is almost lateral: the leading heads push water forward whereas the tail-end cells push water backward, generating much of the collective forward propulsion. Thus the associated interior flows are weak, but vortical regions can significantly enhance transport of suspended particles.

Motion in moderate to dense suspensions Fig. 8 shows a moderate suspension including 100 bacteria, what represents a volumic concentration of 4.25%. In this configuration, the velocity of a single bacterium (in absence of other swimmers) is of magnitude 2.5 times the length of the bacterium per second. In the suspension, the maximal fluid velocity is about 15 times the length of the bacteria per second: the velocity of the flow is thus signicantly enhanced by collective dynamics, as observed in experimental studies (see [START_REF] Sokolov | Enhanced mixing and spatial instability in concentrated bacterial suspensions[END_REF] and [START_REF] Cisneros | Fluid dynamics of self-propelled microorganisms. from individual to concentrated populations[END_REF]). This feature is even stronger in simulations of dense suspensions: Fig. 9 shows a dense suspension including 1000 bacteria, what represents a volumic concentration of 28%. In this case, the maximal fluid velocity reaches 25 times the length of the bacterium per second, consistent with the experimental study in [START_REF] Sokolov | Enhanced mixing and spatial instability in concentrated bacterial suspensions[END_REF], which has revealed collective flow speeds that exceed the speed of the individual bacterium by an order of magnitude. We also emphasize that the microscopic model is able to reproduce the low turbulence dynamics observed in dense suspensions (see [START_REF] Cisneros | Fluid dynamics of self-propelled microorganisms. from individual to concentrated populations[END_REF][START_REF] Wu | Particle diffusion in a quasi-two-dimensional bacterial bath[END_REF]).

Chemotactic behaviour

Bacteria may require oxygen for swimming. In this case, chemotactic behaviour based on run and tumble cycles allows them to swim up the oxygen gradient on the long time scale. Meanwhile, oxygen transport into the bacterial suspension from the adjacent air involves molecular diffusion and also advection by collectively generated streaming. Fig. 10 qualitatively reproduces this main feature of chemotactic response linked to hydrodynamic interactions. We may observe screenshots of the oxygen distribution and position of the microstructure (100 bacteria) at different time steps in a square domain, for which the upper wall represents the air-fluid interface. At initial time, the oxygen distribution is homogeneous in space and the bacteria have been placed randomly in the domain. After one time step, we observe a local consumption of oxygen, which is larger where the density of bacteria is high. After 20-50 time steps, an oxygen gradient has appeared through the conjunction of i) consumption of oxygen by the bacteria, ii) oxygen supply from the fluid-air interface and iii) diffusion of oxygen into the whole domain. After 1000 time steps, most of the bacteria have migrated towards the supply boundary, following the oxygen gradient. Notice also that, after 3400 time steps, the bacteria are still mainly located at the fluid-air interface and, besides, the oxygen concentration far from the supply boundary has increased because of the low density of bacteria. From a numerical point of view, numerical data, such as diffusion parameter of oxygen in water and rate of oxygen consumption by the bacteria, have been fitted in order to reproduce "just in time" oxygen delivery (as mentioned in [START_REF] Cisneros | Fluid dynamics of self-propelled microorganisms. from individual to concentrated populations[END_REF]). Note that gravity effects have been neglected in this experiment: the bacteria motion is driven by the run and tumble dynamics governed by the fluid flow and oxygen distribution, which also depends on the bacteria consumption.

Simulation of bioconvective dynamics with the mesoscopic model coupled to chemotaxis

The response to oxygen gradients can initiate a collective behavior that creates striking hydrodynamic flows. In particular, we aim at reproducing bioconvection patterns in bacterial suspensions as those described in [START_REF] Dombrowski | Self-concentration and large-scale coherence in bacteria dynamics[END_REF] and [START_REF] Tuval | Bacterial swimming and oxygen transport near contact lines[END_REF] (see Section 1.2). Arising from upward oxygentaxis and downward gravitational forcing, these dynamics are explained through the analysis of oxygen diffusion and consumption, chemotaxis, buoyant forcing and viscous fluid dynamics.

We have first simulated bioconvective dynamics arising in a domain with a horizontal upperboundary representing the fluid-air interface (see Table 2, rectangular domain). Gravity is taken into account in the computation through a buoyant forcing term in the Stokes equation (see the description of the mesoscopic model in Section 2.2). Fig. 11 reproduces the main observable features of the simulation. It can be observed that, starting from a homogeneous configuration (see Fig. 11 (t)), respiration of the bacteria depletes dissolved oxygen in the fluid medium in the following way (see Fig. 11 (m)): first, bacteria swim up the resultant gradient of oxygen concentration toward the air and, second, bacteria which are located at the oxygen low-concentration part (i.e. at the bottom) are trapped, because the propulsion force is directly related to the oxygen consumption which has become very poor. Then, since bacteria are denser than water, accumulation of bacteria at the interface results in an unstable gradient of mean fluid density, leading to the emergence of falling plumes of bacteria (see Fig. 11 (b)). Note that these results are consistent with the instability of bioconvection in fluids with a horizontal meniscus [START_REF] Kessler | Hydrodynamic phenomena in suspensions of swimming microorganisms[END_REF]. The instabilities lead to a highly concentrated population at the bottom of the domain. Besides, they play a major role in the renewal of oxygen uptake for bacteria which were trapped in depleted regions, as convection not only concerns the fluid flow which has strong influence on the bacteria but also concerns the oxygen transport towards lower concentration regions.

We have also simulated bioconvective dynamics in a drop-like geometry: the results are shown in Figs. 12 and 13. This simulation qualitatively reproduces the experiment presented in [START_REF] Dombrowski | Self-concentration and large-scale coherence in bacteria dynamics[END_REF], in which the evolution of a B. subtilis suspension in a sessile drop has been observed. The domain that has been considered for the numerical simulation resembles the sessile drop in the following sense: the domain is composed of a lower flat boundary (width 1.10), an upper flat boundary (width 1.10) and two lateral curved boundaries connecting the flat walls. In this setting, the lateral walls and the upper wall represent the air-fluid interface. As in Fig. 11, numerical results presented on Figs. 12 and 13 show similar issues such as the formation of a depletion zone due to oxygen consumption and the emergence of falling plumes due to bacterial accumulation at the fluid-air interface. It has been observed numerically that these instabilities do not occur in the absence of a flat fluid-air interface. On the contrary, increasing the width of the flat interface leads to the formation of an increasing number of instability patterns. Besides, we observe the generation of a persistent hydrodynamic vortex near the three-phase contact line, as observed in the experiment in [START_REF] Dombrowski | Self-concentration and large-scale coherence in bacteria dynamics[END_REF]. This vortex traps cells near the contact line (see in particular Fig. 13 (m)), but it also enhances the uptake of oxygen into the suspension through advection.

Conclusion

The fully microscopic model presented in Section 2.1 has proved to achieve a qualitatively accurate description of the fluid motion and hydrodynamic interactions. In order to reproduce the low-turbulence motion field in a suspension, or to compute the effective viscosity, one needs to solve accurately hydrodynamic interactions, and therefore the fully microscopic model should be used. But when focusing on bioconvection phenomena, the mesoscopic model (Section 2.2), combined with the simple stochastic model for chemotaxis (Section 2.3), is sufficient to reproduce the dynamics described in Section 1.2.

  filled with Newtonian fluid governed by the Stokes equations. Here, µ is the viscosity of the fluid and f f the external forces exerted on it. Since we consider a Newtonian fluid, the stress tensor σ writes σ = 2µD(u) -p I, where D(u) = ∇u + (∇u) T 2

Figure 1 :

 1 Figure 1: Left: single bacterium, right: whole domain with inclusions

Figures 2

 2 Figures 2, 3, 5, 7 4 6 8 9 Domain unit square id. id. id. id. Mesh Vertices 26765 47646 47646 26765 26765 Triangles 52928 94490 94490 52928 52928 Bacteria Number of individuals 1 ∼ 5 1 2 100 1000 Head semi-major axis * 10 3 54.0 27.0 20.0 20.0 14.0 Head semi-minor axis * 10 3 20.0 10.0 7.0 7.0 7.0 Flagellum semi-major axis * 10 3 80.0 54.0 40.0 40.0 27.0 Flagellum semi-minor axis * 10 3 7.0 3.5 3.5 3.5 3.5 Distance head-flagellum * 10 3 150.0 95.0 75.0 75.0 55.0 Propulsion intensity 13.5 6.5 1.0 1.0 1.0

  (t) describes the velocity fluid flow generated by a single bacterium and the contour lines of the velocity. The contours reveal the extent of the fluid disturbance created by the bacterium. Besides, Fig. 2 (b) provides the streamlines of the instantaneous velocity field, revealing recirculations patterns.

Figures

  Figures 10 11 12, 13

Figure 2 :

 2 Figure 2: Velocity profiles ( (t) velocity field (b) streamlines ) in the fluid flow in the neighbourhood of an ellipsoidal bacterium

Figure 3 :

 3 Figure 3: Velocity profiles ((t) velocity field (b) streamlines) in the fluid flow in the neighbourhood of an ellipsoidal bacterium located near a solid wall

Figure 4 :

 4 Figure 4: Trajectory of an ellipsoidal bacterium moving towards a rigid wall

Figure 5 :

 5 Figure 5: Velocity profiles ((t) velocity field (b) streamlines) in the fluid flow in the neighbourhood of two ellipsoidal bacteria interacting through the fluid flow

Figure 6 :Figure 7 :

 67 Figure 6: Hydrodynamic interactions between two close bacteria

Figure 8 :Figure 9 :Figure 10 :

 8910 Figure 8: Moderate suspension of 100 ellipsoidal bacteria in a flow at rest, at a given time step

Figure 11 :

 11 Figure 11: Dense suspension and oxygen distribution in a rectangular domain (l) and modulus of the velocity fluid (r) at different time steps t n = n∆t. From top to bottom n = 3, 99, 1231.

Figure 12 :

 12 Figure 12: Dense suspension and oxygen distribution in a drop domain at different time steps t n = n∆t. From top to bottom : n = 3, 200, 350.

Figure 13 :

 13 Figure 13: Modulus of the fluid velocity for a dense suspension in a drop domain at different time steps t n = n • ∆t, for n = 3, 200, 350

Table 2 :

 2 Parameters for the numerical simulations (II): chemotactic results.

 [START_REF] Berg | Random walks in biology[END_REF]This work was supported by the ANR project MOSICOB. 1


To be published in J. De Munck, C. Didry, J. I. Ferreras, A. Jobert, Civil Dialogue, Social Dialogue : A New connection to Change our Model of Development, Bruxelles, Peter Lang, 2011 Social Dialogue and Deliberation: a New Dimension in European Industrial Relations

Claude Didry and Annette Jobert 1

In comparing social and civil dialogue, social dialogue is often presented as a form of negotiation, that is to say, as the confrontation between actors with diverging interests and preferences, each of which aims at reaching agreements maximising its own interests at the expense of others. Taking the analysis of Dunlop, this corresponds to the operation of a "system of industrial relations" articulating three social forces taken as given: the workers' representatives, the employers and the state. The system of market-based competitive confrontation between supply and demand of individuals has been replaced by a kind of duopoly arbitrated by the state in which the debates are primarily on wages and working time and are accompanied by conflicts or potential conflicts (in this case, by strikes of wage earners). In contrast, some of the manifestations of civil dialogue are presented as the defence of "causes" that represent the general interest above those of the particular parties involved. In this perspective, civil dialogue is seen as more innovative and open in its ability to pose new problems and to reach consensual solutions through exchanges of arguments accepted as valid by the participants in public debate. In this sense, we often tend to associate civil dialogue with deliberation.

From the perspective of Habermas, this contrast, between negotiation on pre-identified issues and open discussion to identify common problems, is insurmountable. Indeed, social dialogue is fundamentally characterised by negotiations which correspond to a balance of power relations defined by an economic system based on monetary transaction. In contrast to this focus on the monetary medium, civil dialogue owes its creativity to the fact that it sees problems "relating to real life" as going beyond economic determinism and reflecting a genuine communicative action. In this contraposition, civil dialogue encourages social dialogue to leave the beaten path of negotiations and to consider other issues related to the general problems of society.

Our hypothesis is that social dialogue is not confined to negotiation in a context which, according to [START_REF] Dunlop | Industrial Relations Systems[END_REF]Dunlop ( [1958]]), is considered by its actors as given. For us, social dialogue includes a new orientation in which negotiation involves not only a clash of interests but also the defining of its corresponding context and general objectives. Moreover, it is initially based on configuring unitary demands through the assimilation of a multitude of individual viewpoints. As underlined by Lallement (2007, p. 450) 2 , "Trade unions must manage heterogeneous demands ranging from raising wages to improving the quality of life. In addition, the priorities that shape the agenda of demands develop on the basis of more or less democratic deliberative procedures." Faced with this fundamental heterogeneity of employees' demands, employers find themselves in a position of favouring economic calculation aimed at increased profitability. However, because of the specificity of their production, employers do not form a homogenous block.

In this sense, social dialogue is linked to a form of deliberation referring to the diverse fields which work involves. As human activity, work is part of a broad questioning of the "quality of life." As an activity aimed at the production of products, it leads to a broad questioning of the economic processes that in particular constitute globalisation of trade or of innovation. If the immobilisation of "industrial relations systems" led for a time to a well-established practice of negotiations of wages and working time, particularly in the context of what has been called "neo-corporatism," current developments tend to open wider space for deliberation implying sharing information and critical discussion on their content, that is to say, what Sen calls "informational bases." The extent of deliberations is related to an increasing ability to understand the interconnections between the specific challenges of work and of other social spheres. This opening of deliberations joined the perspective, advanced by [START_REF] Sen | Development as Freedom[END_REF] of a "development of capabilities" involving an analysis of development beyond the growth of GNP and integrating the contributions to a broader freedom for individuals to choose their own life.

European social dialogue is the first typical case of this growing openness to a deliberative dimension. Indeed, it is from the identification of a common perspective-"the achievement of the Single Market"-that an "institutional" deliberation, under the auspices of the Commission presided by Jacques Delors, was undertaken. This deliberation was initiated by the public actor through a "reflexive incentive" engaging the social partners to diagnose problems and to consider their solutions in proposals and common opinions. In other forms, such processes of deliberation are found in both the dynamics of professional sectors at company level as well as that of the territories. Collective actors are more or less permanently crystallised in the very course of the deliberative process and the first cognitive productions to which it leads. A review of this new process highlights the different forms of coordination and cooperation between the actors involved as well as opening up to themes related to the balance between work and the personal lives of employees (work-life balance). Moreover, deliberation does not lead exclusively to the production of new rules (e.g., through collective bargaining agreements in the strict sense of the word), but to a broader set of exchanges and various devices.

After considering the paradigmatic figure represented by the emergence of European social dialogue, we will report on the establishment and operation of these new spaces of deliberation both in the workplace as well as in the territories.

European social dialogue as a model of a "reflexive incentive"

European construction is not limited to the creation of a continental market based on the removal of customs barriers. From the inception of the European Coal and Steel Community (ECSC) in the 1950s, it has involved the integration of trade unions in discussions on the organisation of the future European market and its implications for business and employment. This orientation is reflected in Article 2 of the Treaty of Rome, linking the establishment of a "Common Market" to the intention "to promote throughout the Community a harmonious development of economic activities, a continuous and balanced expansion, an increased stability, an accelerated raising of the standard of living and closer relations between its Member States." After the obstacles to European integration of the 1960s, the revival of the process in the 1970s was reopened with a summit meeting on employment bringing together the main unions of the member states and focusing on the issue of restructuring. It continued with the presentation of a "Social Action Program" by the Commission in 1974, including considering the principle of "workers participation in management" leading in 1975 to a directive requiring the consultation of workers representatives in cases of redundancy, and then to a directive requiring the maintenance of employment contracts and the information and consultation of employee representatives in the event of the transfer of undertakings In the 1970s this dynamic confronted the increasing tension between opposing positions due to rising unemployment. The European Trade Union Confederation (ETUC), founded in 1974, asserted itself through an economic recovery program based on the reduction of working time linking the demand for the 35-hour week to that of five weeks of paid leave in a work-sharing perspective. Conversely, the employers' association UNICE (now known as BusinessEurope) adhered to the "supply-side policies" initiated soon after Margaret Thatcher came to power in Britain. The scope of deliberation was initially linked to the concept of community regulations but gave way to a clash of competing interests. Thus, deliberation was gradually replaced by a more conventional pattern of negotiations between different positions in which the public body-the Commission-acted as an arbitrator. In this perspective, negotiation concerned general economic mechanisms, those of employment policies to be implemented at the level of national economies, and was caught up in the tension between stimulus and recovery policies and those of supply-side economics.

European social dialogue launched at the Val Duchesse talks under the leadership of the Commission headed by Jacques Delors in 1985, introduced a break with this logic of negotiation. In fact, it was based on the concerns of involving the social partners in a broad goal, that of the "achievement of the Single Market" which Jacques Delors put at the centre of his program for his presidency of the Commission [START_REF] Didry | Le moment Delors, les syndicats au coeur de l'Europe sociale[END_REF]. The result was a process that gradually evolved away from the initial plan of what [START_REF] Walton | A Behavioral Theory of Labor Negotiations. An Analysis of a Social Interaction System[END_REF] presented as a "distributive negotiation". In this scheme, the actors are endowed with strictly conflicting preferences: in the case of the social partners, stimulus versus supply-side policies. Finding agreement can only be accomplished through the acceptance of mutual concessions leading players to a strategy minimising losses. The program of "achievement of the Single Market" transformed the exchanges between social partners, bringing them closer to a form of "integrative negotiation" (Walton and MacKersie, op. cit.) in which actors follow a problem solving approach, thus moving towards a form of deliberation. Indeed, this program starts with the identification of a new economic reality, that of the Internal Market that is shaping the development of intra-European trade and its consequent reconstructions. Its "achievement" corresponds to the identification of a common problem, that of the regulations ensuring its functioning and development. In this view, before reflecting on the content of the rules to be adopted3 and their modality (legislation or collective agreement), the social partners' contributions are first solicited in order to identify the field covered by this regulatory activity. 4 The establishment of "working groups" by the summit meetings is a defining moment in the process. Indeed, the "common opinions" elaborated by these groups correspond to the setting up of a shared diagnosis before launching a process, sometimes long, of elaborating solutions.

The concept of "integrative negotiation" provides a first insight into the process that starts with the first meetings between the social partners. However, it suggests that the "integrative" dimension tends to erase any confrontation, in favour of support for the goals set by the Commission. It reinforces an analysis in which the social partners abandon their specificity to rally, by consensus, to the Commission's policies. Such an analysis obscures the tensions that arise, no longer in the form of confrontation but rather as a refusal to take part in the deliberations and of confrontations within the organisations involved. During this process, major internal conflicts have developed between the members of UNICE which on many occasions has found it difficult to participate. The ETUC is also subject to such pressures being impatient concerning the outcome of meetings whose results in terms of regulations seem to be far in the future. These tensions have also led to a transformation of the actors themselves, with a major reorganisation of the social partners taking place in parallel with the meetings of the 1980s. Thus, the ETUC has become structured as a true confederation at the European level, integrating the sectoral trade union committees into federations and clarifying the decision-making methods of its various organs during the Congress of Luxembourg in 1991.

The Social Protocol annexed to the Maastricht Treaty reflects the institutional recognition of this deliberative activity, involving the European Social Partners in legislative activity. European social dialogue (at the intersectoral level) is often understood through the agreements to which it has given rise: parental leave (1995), part-time work (1997), fixed-term contracts (1999), telework( 2002), work-related stress (2004), harassment and violence at work (2007), and inclusive labour markets (2010). It is not, however, limited only to negotiated agreements. But whether there is an agreement of the European social partners or a directive adopted in the absence of agreement, the latter participate in a norms-producing process which was generally initiated by the Commission. This initiative takes the form of a "reflexive incentive" addressed to the social partners. The resulting social dialogue helps to clarify the areas of future contractual and legislative interventions. It has led to the writing of numerous common documents, opinions, recommendations and diagnoses affecting social issues and, more broadly, the dimensions of economic policy. Such activity is particularly intense in the framework of sectoral dialogue with a reflection on the future at the European level of the Didry_et_Jobert_Social_dialogue_and_deliberation-final-1.doc page 5/1 sectors concerned, taking the form, for example, of "common positions" for the Commission (Dufresne, Degryse and Pochet, 2006).

Three moments can be identified in this complex process of deliberation. The first concerns the common views and positions which establish a common diagnosis of an issue. The second is to identify areas of possible legislative action. The third concerns the final trade-offs to be made in conventional and legislative activity. In this process the public actor, the Commission, is called on to play an important role, particularly in initiating the studies, and in providing the "informational bases" that feed into the social partners' discussions. Indeed, the latter have proven to be decisive, including in the work programs that the social partners today adopt independently. One cause of the current weakening of European social dialogue is the blurring of the Commission's role, whose initiatives now tend to place themselves in the field of liberalisation starting from a conception of the market as an economic mechanism that allows national economies to achieve a Pareto optimum.

Within the existing EU rules, we must emphasise the importance of those concerning the right of workers to information and consultation which try to involve workers' representatives in company decisions. Since the first directives on collective redundancies (1975) and the transfer of undertakings (1977), this right has been considerably expanded with the European Works Councils Directive (1994), the directive on the involvement of workers in European companies ( 2001) and the directive establishing a general framework for information and consultation for companies with more than 50 employees (2002). These rules help to open a particularly important European space for deliberation for companies concerning issues of employment and restructuring. They are, if not an injunction, at least a strong incentive to open a thorough debate on corporate activity and strategy.

Social dialogue in the workplace

The institutions of "industrial democracy" which have become part of the "industrial relations systems" centred on negotiations have found new meaning in the debates implied by the deliberations on restructuring, discrimination and strategic workforce planning. This evolution leads to breaking away from the focus on unions as the central actors of "industrial democracy", identified by Beatrice and Sidney Webb beginning with the case of the British trade unions. It also involves considering the functioning of the fields of information and consultation encouraged by European legislation starting from the experience of countries such as Germany and France, where union activity is linked to that of employee representative bodies. As suggested by Negrelli in his contribution to this volume, the institutions of industrial democracy thus appear, in adopting the perspective of Sen, as "conversion factors" ensuring the desire for social representation and recognition that work implies. Thus, in France, the traditional demarcation between negotiations, as the responsibility of the unions, and of information and consultation, as being in the realm of activities of the elected bodies, tends to be challenged in these debates which involve the management of the company and its future, and which are based on expectations which are difficult to establish. This evolution is reflected in a transformation in the relations between these two forms of representation and Didry_et_Jobert_Social_dialogue_and_deliberation-final-1.doc page 6/1 between the collective representation of employees and the employees themselves. It corresponds to the emergence of a "new field of social dialogue" (Didry and Jobert, 2010).

This orientation is primarily linked to the expansion of the themes and issues of institutionalised collective bargaining. The classic themes of wages and working time have themselves become the subject of a renewed approach arising from the deliberative activity that raises the questions of employment, discrimination, etc. The negotiation on the reduction of working time in France in the late 1990's is a good illustration. In addition to the requirements for business operations, it took into account elements relating to particular situations of employees, including their life outside work, their modes of transport, etc. The multidimensional nature of enterprise bargaining has led to the greater complexity of negotiations and a more intense phase of deliberation. This complexity also justifies a greater use of experts.

In the perspectives opened by the directives providing for a dimension of information and consultation, the orientation of negotiations towards deliberation only becomes effective as new issues of social dialogue become involved. Article 5 of the 2002 Information and Consultation Directive opens a space for negotiations on the deliberation process itself, both in content as well as in form. 5 Thus, it opens consideration of negotiations on matters subject to information and consultation of employee elected bodies, where they exist. It also leads to negotiations on the implementation of the rights to information and consultation in countries such as the United Kingdom, where, outside the unions, there is no permanent structure of information and consultation similar to the German or French works councils. Workers become involved in these new structures, often established at the initiative of employers trying to evade the more stringent provisions of the implementing law of 2006, from the moment when employment is threatened. 6 In France, legislation on collective redundancies has focused on information and consultation of employee representative bodies. It is oriented towards the search for a dialogue between representatives of employers and of employees in order to examine the economic reasons justifying the proposed job cuts and related matters provided by the employer in the framework of a "social plan."

In recent years, the legislature has given greater weight to negotiations in collective redundancy situations with the laws of 2003 and 2005 providing for the possibility of entering into "agreements of method." These include negotiating methods for information and consultation of the works council, alternative proposals to management projects and the 5 Article 5 Information and consultation deriving from an agreement Member States may entrust management and labour at the appropriate level, including at undertaking or establishment level, with defining freely and at any time through negotiated agreement the practical arrangements for informing and consulting employees. These agreements, and agreements existing on the date laid down in Article 11, as well as any subsequent renewals of such agreements, may establish, while respecting the principles set out in Article 1 and subject to conditions and limitations laid down by the Member States, provisions which are different from those referred to in content of the employment plan (Didry and Jobert, op. cit.). The negotiation and implementation of agreements of method in practice bring union representatives and elected representatives to works councils closer together. This new link between collective bargaining and the information and consultation of works councils is part of a redefinition of systems of representation and social dialogue. This redefinition is based on a blurring of boundaries between union responsibilities and those of elected employee representatives. This is done simultaneously around the definition and implementation of a project which has been crystallised through the agreement. It particularly relies on the proximity between the union actors and elected officials as well as on the construction of new communities involving employees, stabilised in specific instances (working groups and steering committees as well as those of consultation, negotiation and monitoring). This activity, combining information and consultation with negotiations in the context of restructuring or re-organising working time, highlights a logic of commitment that develops between the employee representatives, company management and the employees themselves. This logic of commitment represents a new level of exchange between them, registering continuity in the discussions. The latter initially includes new interest in the economic situation of the company, a new look by all the actors at the company itself as an entity which is part of a group with shifting boundaries. Subsequently, a new look at employment beyond the quantitative aspects has been added to this cognitive dimension including career trajectories of individual employees and their conditions of life in and outside the company. This development of social dialogue in the company echoes the current debate on career security and strategic workforce planning. It also provides material for thinking about "a policy of capabilities" carried out through methods and collective bodies that make possible an evaluation and increase in the real freedom of individuals in their work [START_REF] Salais | Europe and the Politics of Capabilities[END_REF].

The new rules adopted in France in 2008 concerning union representation and the validity of agreements reflect the decision to more firmly anchor union activity in the workplace and reach out to employees. Before this reform, five unions enjoyed a virtual monopoly of representation that had been assigned to them by the state after the war. Their representation was therefore independent of their membership and the number of votes they received in elections. It established a number of rights, including the right to conclude agreements to be applied erga omnes. According to the 2008 Act (which incorporates the terms of an agreement between the two social partners and was signed on the union side by the CGT and the CFDT, the two main French organisations), the representation of trade unions is based on seven criteria of which the most important is the "support confirmed by the results of workplace elections." It is therefore a major change that has disrupted the social rules of the game. It introduced greater union democracy which may reinforce the legitimacy of unions and the agreements they negotiate.

The evolution of workplace dialogue has revealed a new dynamic resulting just as much in the new rules of union representation as in the emerging sense of commitment in which the interests of employees as people are more closely related to these collective bodies and to the future of the company as a whole. But the logic of electoral participation and commitment leads to considering work in a broad way, as a social activity, a "functioning" in the vocabulary of Sen, interrelated to all social activities through which individual "capabilities" are expressed. It involves going out beyond the company and, in particular, the company of more than fifty employees, to consider the field of "capabilities". This opening is also based on calling the company itself into question as a player in economic life. The development of flexibilitysometimes leading to situations of exclusion, that of outsourcing,7 of SMEs, of the reorganisation of large companies into "business units" leading to a distance between the decision-making centres and the "company," generates a return to the workplace. The emergence of a "territorial social dialogue" appears as a complex dynamic, involving both the identification of multiple challenges, the search for relevant actors8 and the development of solutions that address identified problems. In this process, unions have only gradually been finding their place through contact with other participants.

Territorial Social Dialogue

Territorial social dialogue corresponds to a number of experiences observed in various European countries consisting of a dialogue on economic and social dynamics on the level of a territory. In this category we find both "territorial pacts" [START_REF] Heidenreich | Regional capabilities and the European Employment Strategy[END_REF][START_REF] Serafino | The territorial pacts in Italy: the competitive corporatism assumption in question[END_REF], as well as sectoral or inter-sectoral committees set up by the social partners [START_REF] Frédéric | Les dynamiques locales de la négociation collective[END_REF], and initiatives which are broadly related to the economic and social future of a territory (Jobert, 2008;[START_REF] Jobert | Le dialogue social territorial en Europe : perspectives comparatives[END_REF]. The observed cases show a specificity of this diffuse dialogue in relation to social dialogue at the company or the European level. Certainly, certain forms of this dialogue constitute a continuation of social dialogue and institutionalised public action. But others reflect the existence of more autonomous processes, echoing what some refer to as a "territorial governance" (Le [START_REF] Galès | Régulation, gouvernance et territoire[END_REF][START_REF] Pasquier | La gouvernance territoriale. Pratiques, discours et théories[END_REF]. This form of social dialogue leads to questioning its features and how it might influence social dialogue in general.

From deliberated diagnosis to collective project

Territorial social dialogue covers very diverse fields, regional or local (provincial, county, municipality, local labour market), only some of which represent administrative and political units. Territory is not defined a priori. It is determined by the nature and objectives of the project, by the actors who support it and by the collective action they develop to bring it to fruition. Each of them is in a relation of mutual dependence on the other: it is the project and collective action that bind the actor and, somehow, gives him life.

It is also characterised by the variety of topics it addresses: local economic development, competitive clusters, technological innovation, restructuring, relations between businesses and sub-contractors, housing for seasonal workers, regulation of labour markets, integration of vulnerable groups, strategic workforce planning, vocational training or career security. Government regulations and professional traditions [START_REF] Méhaut | Recomposition de l'action publique et émergence de nouvelles scènes de négociation sociale ? L'exemple de la formation professionnelle[END_REF] do not take these themes into account-or only do so in an erroneous way. Corporate restructuring, which has accelerated over the last twenty years, has shown the need to involve the "stakeholders" of a territory in a very broad way in assessing the impact of decisions to restructure the company, in finding new job solutions for redundant employees, in creating new activities which could "revitalise" the territories affected by the decisions of the company [START_REF] Freyssinet | Quels acteurs et quels niveaux pertinents de représentation dans un système productif en restructuration ?[END_REF] and, in general, in anticipating economic and social evolution (Didry and Jobert, 2010). The situation of seasonal workers in tourist areas provides another example of the value of a "territorial" approach involving government, employers' organisations and trade unions, and managers of transportation and social housing, because of the difficulties that these workers face not only in respect to work related conditions but also in relation to housing and transportation. Thus, a number of "homes for seasonal workers" were created. Strategic workforce planning as well as career security in cases of employee mobility, which have become major themes of company policies, of collective bargaining and of public debate in France, provide two other examples of approaches that address the community of employees including those of small businesses and subcontractors who often escape the mesh of occupational regulations. Actors' consciousness is demonstrated by the growing commitment of European trade union organisations in favour of local dialogue. In France, the CGT and CFDT take part in many territorial initiatives. As one member of the CGT's Economic, Social and Environmental Committee (2009) pointed out during the 7 July 2009 plenary discussion of the draft opinion on the territorial social dialogue, "the time has come to take seriously all dimensions of social dialogue."

Much of this experience is part of a dominant logic of action, of projects and programming which aims at solving problems before dealing with the extension of rights. The public-private partnership is another characteristic of these experiences involving multiple actors: workplaces, labour and business organisations, economic groups (such as chambers of commerce), associations, training institutions, professional agencies, cooperatives, banks, universities, local authorities and decentralised state services. The collaborative conception of a local "integrated" strategy of employment and development focuses on the exploitation of assets and resources of the territory. Knowledge of its strengths and weaknesses is, therefore, essential. Creating shared economic and social-territorial diagnoses is thus a very cognitive question. This identification work is an integral part of the process and defines in common both priorities and references for action. Here we find, in the view of Sen, the constitution of "informational bases" in the very course of action for the common good. Deliberated diagnoses are effective if they break down the barriers which separate each partner's practices to achieve a common definition of a strategy which is no longer sectoral (based only on data on employment, for example) but territorial. By integrating both the economic and the social development of a territory, a collective project can be established. It is on such cooperation that renewed local governance can be defined. This dialogue is also based on organisational devices such as steering committees or working groups, and various forms of contracts and agreements that bind local authorities and public services to businesses, associations, trade unions or other public and private entities. "Development agencies" play an essential role among the support mechanisms as intermediaries between the different categories of actors and as an interface between these devices and institutions. They are the place of exploitation par excellence of territories' "strategic skills." The economic development discussed here does not correspond to the implementation of a previously defined economic model. As Sen suggests, we are in a situation where the identification of a "development model" accompanies the social dialogue or the development itself.

Territorial social dialogue sets a new path for social regulation, more open than that of conventional social dialogue but also more uncertain, unstable and no doubt more fragile. In this sense, employee and employer representatives can play a very important role, even being at the initiative of exchanges and projects. But this is not always the case and, most importantly, actors share with others the responsibility for conducting and monitoring the actions undertaken. This kind of dialogue evokes a form of "civil dialogue" within the meaning of public action, involving citizens in a broad sense and going beyond what is called "organised civil society" (European Economic and Social Committee, 2003).

Territorial social dialogue: between social and civil dialogue

The development of territorial social dialogue and especially the growing involvement of trade unionists in this type of dialogue have aroused misgivings and criticisms. In particular, there is concern that the promotion of territorial social dialogue will be at the expense of collective bargaining and the negotiation of work rules. The practice of social dialogue, whatever its diversity, is based on a conception of partnership that values participation and consultation much more than negotiation between participants. These negotiations, backed by recognized and established institutional arrangements, offer more guarantees for the parties concerned. Territorial social dialogue, which is often quite informal, frequently leads to arrangements rather than to perennial rules. The aspect of constraint contained in the rules is also largely absent from the process of territorial social dialogue despite widespread use of alternative forms of contracting and agreements. The concept of "soft law" which refers to non-binding processes, often cognitive, resulting from the dissemination of guidelines, of content and of tools for public policies (Hassenteufel 2005, p. 125), could be applied to these local rules. The extension of territorial social dialogue would be part of a general movement tending to weakening labour law with its source in legislation and collective agreements. Support for such dialogue, especially by European authorities, accompanies that for flexible methods of social regulation as an open method of coordination-often considered to be an alternative to European directives which are deemed comparable to laws. With the decentralisation of collective bargaining in local workplaces, it would contribute to a fragmentation of negotiations, a source of increased inequality.

But the clear-cut opposition between territorial social dialogue on one side and collective bargaining and agreements on the other, does not stand up to the examination of the two forms of dialogue. We have seen that the European social dialogue as well as that which develops in the workplace do not necessarily lead to the adoption of binding rules, but also have led to common positions, joint opinions etc. The field of deliberation as well as the importance of cognitive issues that characterise territorial social dialogue are also asserted in all areas where social dialogue takes place. Territorial social dialogue can be likened to an "integrative negotiation" oriented towards common goals in which the parties' interests are convergent or complementary. As noted, collective bargaining is expanding to new topics and also involves actors in addition to the trade unions and employers such as elected employee representatives, experts, persons who have been appointed and sometimes direct representatives of work collectives which are made up of non-union members. This evolution of classical dialogue establishes a form of convergence with the territorial social dialogue even if the latter, as yet, has seldom resulted in setting new standards.

Territorial social dialogue today is no longer a marginal phenomenon filling the gaps and shortcomings of traditional regulation. It has asserted itself as a mode of regulation in itself, likely to meet new economic challenges and requirements of efficiency and justice in the workplace and in society. In its approach from the field focusing on deliberation from shared diagnoses; the intervention of a plurality of actors, public and private; the use of innovative devices to fit the action; and finding consensus solutions, this dialogue questions the established forms of public action and occupational regulations It simultaneously contributes to the evolution of public policy and social dialogue by giving substance to the concepts of interested parties and deliberation, and by encouraging social dialogue to broaden its themes. As such, it appears as a link between civil and social dialogue.

Conclusion

Social dialogue includes the intertwined dimensions of negotiation and deliberation. Without excluding attention to content and the execution of employees' work contracts, it includes a reflection on the development and future of economic activities, of workers and territories. It is thus linked to a search for models of development based on a form of democratization of the economy, echoing the thoughts of Sen on relations between democracy and development.

The analysis of the exercise of dialogue both at the European level as well as that of local businesses and territories reveals developments in the relationship between the two constitutive dimensions of social dialogue, that is, between negotiation and deliberation. The scope of deliberation tends to increase in all forms of social dialogue. Moreover, the classical distinction between information and consultation on one side and negotiation of the other has been fading. By opening up a dimension of negotiation, the information and consultation procedures tend to be transformed: the pure confrontation gives way to a collective work directed towards identifying and solving problems of common interest. Conversely, deliberation tends to enrich the negotiation by identifying new questions, new investigations and new themes. This development of social dialogue highlights the importance of the commitment capacity of the actors who take part in the various areas in which it is utilised. Such a trend reflects the existence of an overlapping relationship between social and civil dialogue. With greater attention to issues of deliberation, to a cognitive approach, to the specific trajectories of individuals while taking into account both their professional position and their living conditions, social dialogue takes forms and aims at objectives which are frequently those of a "civil dialogue." Indeed, what is presented as "civil dialogue" is the preoccupation of clarifying conflicts of interest in specific fields of economic and social life Didry_et_Jobert_Social_dialogue_and_deliberation-final-1.doc page 12/ which do not directly affect the framework of business and labour. In this dynamic, social dialogue is able to appropriate in a specific way the subjects opened by civil dialogue, such as those related to the issues of sustainable development and environment, access to public services or urban development. This specificity is a consequence of the ability to rethink these issues from a strong grounding in the work and the future of economic activity. It also takes into account the existence of representative actors whose legitimacy rests on professional elections and union membership, as well as institutional procedures that support the permanence of the deliberative activity and scope of its results.

  Article 4. . (Directive 2002/14/EC of the European Parliament and of the Council of 11 March 2002 establishing a general framework for informing and consulting employees in the European Community). 6 Cf. the contribution by Simon Deakin and Aristea Koukiadaki in this collection.

Referring to the thesis of a "dual logic of collective action" developed by[START_REF] Offe | Two Logics of Collective Action[END_REF].

That is, finding solutions to identified problems.

Didry_et_Jobert_Social_dialogue_and_deliberation-final-1.doc page 8/1

As in the case, for example, of the Saint-Nazaire shipyards or that of Airbus in Toulouse.

Which, at least initially, are not necessarily the unions but could also be local elected officials, associations or more transient groups.Didry_et_Jobert_Social_dialogue_and_deliberation-final-1.doc page

9/1