Mustapha Hellou

Dong Trong

Pascal Nguyen

Dupont

DNAPL Infiltration in a Two-Dimensional Porous Medium-Influence of the Shape of the Solid Particles

come

The multiphase flows in the porous media are numerous in nature and various industrial processes. While being limited to the immiscible transport of pollutant with water known under the name of DNAPL (Dense Not Aqueous Liquid Phase), these flows meet in the geomaterials ones, the processes of remediation of the grounds, the biofilters employed in waste water treatment, etc… The properties of these flows are complex and often unknown because of a large number of physical parameters. At the present time, the influence of the density, the viscosity or the interfacial tension of DNAPL on its infiltration in the porous medium is studied much by several authors like [START_REF] Pennell | Influence of Viscous and Buoyancy Forces on the Mobilization of Residual Tetrachloroethylene during Surfactant Flushing[END_REF][START_REF] Dawson | Influence of Viscous, Gravitational, and Capillary Forces on DNAPL Saturation[END_REF][START_REF] Hofstee | Removal of DNAPL contamination from the saturated zone by the combined effect of vertical upward flushing and density reduction[END_REF][START_REF] Jeong | Force analysis and visualization of NAPL removal during surfactant-related floods in a porous medium[END_REF][START_REF] Tartakovsky | Heterogeneity Effects on Capillary Pressure-Saturation Relations Inferred from Pore-Scale Modeling[END_REF][START_REF] Meakin | Modeling and simulation of pore scale multiphase fluid flow and reactive transport in fractured and porous media[END_REF] Most of these studies are realized on a macroscopic scale which does not make it possible to understand the retention of small drops of DNAPL in the pores of the medium. However, few studies carried out at the level of the interstices showed the fractionation of the DNAPL as well in experiments as by numerical modeling [START_REF] Krishna | Rise characteristics of gaz bubbles in a 2D rectangular column : VOF simulations vs experiments[END_REF][START_REF] Storr | Comparisons between experiment and numerical simulation using a free surface technique of free-falling liquid jets[END_REF][START_REF] Harvie | A parametric study of droplet deformation through a microfluidic contraction : Low viscosity Newtonian droplets[END_REF].

In this work, we highlight the influence of a parameter which is currently still less known -geometry of the solid particles of the porous medium -on the infiltration and the fractionation of a drop of DNAPL. Numerical modeling is carried out in two-dimensional configuration in order to decrease the computational time.

Problem Position

The fluids used in this study are water (carrier fluid) and the DNAPL (the drop). The density and the viscosity of water are of 998 kg.m -3 and 1.10 -3 Pa.s. Those of the DNAPL are respectively equal to 1623 kg.m -3 and 0,89 Pa.s. As an example, Perchlorethylen (PCE) has these properties (INERIS, 2002). We imposed a contact angle of 65° in order to consider a wetting DNAPL because most of them being in nature are wetting.

Figure 1 presents one of the three structures of the porous medium used. This porous medium is represented by a regular assembly of similar and non deformable solid particles. The circle apart from the medium represents a drop of DNAPL at the initial time. It is injected without initial velocity within the carrying fluid which is at rest. Under the effect of the difference of the density between the carrying fluid and the drop, the latter infiltrates through the porous medium. The domain of calculation is rectangular. Its dimensions relative to the radius R of a solid particle are 12,5x18,1. It is composed of 3 zones. From top to bottom, zone1 contains the drop at t = 0, zone 2 contains the porous medium composed by 64 particles, zone 3 contains the wall of the bottom which is used to recover the infiltrated DNAPL.

The shape of a solid particle is described by the following equation which enables us to pass from the circular form to the almost square form by increasing the value of the parameter m :

2 2 1 m m x y R R     + =        
In this paper, we present three models of the porous medium which differ by the form of the solid particles by varying m: m = 1 (circle), m = 2 and m → ∞ (square). In view to conserve the same porosity (0.36), we vary the side a of the square mesh (see figure 2) while preserving R constant. In this study, we use a numerical simulation based on algorithm VOF (Volume of Fluid). This algorithm is approached for the first time by [START_REF] Hirt | Volume of Fluid (VOF) method for the dynamics of free boundaries[END_REF]. Besides solving the Navier-Stokes equation, these two authors introduced the definition of the volume fraction χ of two immiscible fluids (1 and 2) into the mesh. The volume fraction is equal to 0 in fluid 1, equal to 1 in fluid 2 or between 0 and 1 within the interface of the fluids. Here fluid 1 is the main fluid. The volume fraction with the interface is controlled by the equation:

χ . χ 0 R a R R 2R = 2,5 mm m = 1 m = 2 m → ∞ Figure 1 Description of the computational domain, case m → ∞
The surface tension model, known as continuum surface force (CSF) model, proposed by [START_REF] Brackbill | A continuum method for modeling surface tension[END_REF] is used to track the interface.

Results and discussion

Figure 3 shows the infiltration of a drop in each of the models presented in figure 2. The results are presented in function of time. For generalization, time can ce normalized by the time of settling of the drop in an infinite static continuous flow field. The settling velocity found by [START_REF] Hadamard | Mouvement permanent lent d'une sphère liquide et visqueuse dans un liquide visqueux[END_REF][START_REF] Rybczynski | On the translatory motion of a fluid sphere in a viscous medium[END_REF] for small Reynolds number is

) 9 6 ( ) 1 ( 2 gd U c 2 d λ λ µ ρ ∆ + + = ∞
Where ∆ρ is the difference of densities, λ is the ratio of the drop viscosity and the continuous fluid viscosity, µ c is the viscosity of the continuous fluid. For water as continuous fluid and PCE as the drop, the value of the settling velocity is 31.7 mm.s -1 and the reference is thus t 0 =d/U d∞ =0.24s.

We observe that for the three models of the porous medium studied, the drop is spread out initially horizontally and then infiltrates vertically. The drop tries to maintain its volume constant but the more deeply it infiltrates, the more it is divided into several droplets. By comparing the infiltration in these three models, we note that when m increases, separation of the principal drop onto droplets is important. The droplets resulting from the fractionation of the principal drop are retained in these porous medium either between two solid particles rows vertically, or between two solid particles rows horizontally, or between four solid particles. We note that surface of the liquid/solid contact increases with increasing the parameter m. This surface is minimum for the case where the solid particles are circles. The more surface of the liquid/solid contact increases, the more adherence on the area of this contact increases. That explains why for the cases m > 1, the drops are retained between the solid particles with increasingly large volume as m increases.

We compare then the velocity of the infiltration of the DNAPL in the different models of the porous medium. For each model, calculation is carried out until the moment when the immobility of the drop becomes permanent. We present in figure 5a the depth of the infiltration (vertical

σ σ σ σ σ σ σ σ F F α = γ β = 90 -γ α β 90 -γ γ (a) (b)
position of the lower face of the drop relatively to the surface of the porous medium) according to time for the three models. We find that the velocity of the infiltration of DNAPL decreases when m increases. Figure 5b shows the volume of retention of DNAPL in the different models of the porous medium. The parameter V r /V t defines the ratio of the volume of DNAPL retained in the medium on the total volume of DNAPL injected. In the case of the particles having the circular form, most of the drop passes through the porous medium. There is only 30% of the volume of the drop which is retained in the pores. When the shape of the particles tends towards the square form, the volume of DNAPL retained in the pores increases. It reaches 64% and 72% respectively for the case m = 2 and the case of the square. .

Conclusion

The analysis of the influence of the shape of the solid particles on the infiltration of the DNAPL is carried out for several forms: circular, square, intermediate form enter the circle and the square. This study highlights that as the shape of the particles passes from the circle shape to the square shape, the speed of infiltration decreases while the retention of the DNAPL increases. This retention is the result of the fractionation of the drop.

DNAPL

  Infiltration in a porous medium -Influence of the shape of the solid particles Mustapha HELLOU, Trong Dong NGUYEN, Pascal DUPONT Université Européenne de Bretagne Laboratoire de Génie Civil et de Génie Mécanique (LGCGM)-EA 3913 INSA de Rennes, 20 Av. des Buttes de Coësmes, 35043 Rennes Cedex ABSTRACT The infiltration with atmospheric pressure of Dense Non Aqueous Phase Liquid (DNAPL) in a model of porous media saturated by another liquid is studied when the DNAPL liquid has a contact angle characterizing wetting liquid. The model of porous media consists of an assembly of solid particles for various forms. The influence of the shape of the particles is studied. KEYWORDS: DNAPL; Multiphase flow; Wetting liquid; Solid particle; VOF corresponding Author : Mustapha.Hellou@insa-rennes.fr

Figure 2

 2 Figure 2 Shape of the solid particles

Figure 3

 3 Figure 3 Behavior of a DNAPL drop during infiltration in a model of porous medium

Figure 4

 4 Figure4explains the mechanism of retention of the droplets in contact with two solid particles of an horizontal row.

Figure 4

 4 Figure 4 Illustration of the mechanism of drop retention represents the contact force ; represents the apparent weight of the drop

  Figure 5 (a) DNAPL infiltration velocity ; (b) DNAPL retention volume


Mustapha Hellou
email: mustapha.hellou@insa-rennes.fr

Geometric Inversion of two-dimensional Stokes flows -Application to the flow between parallel planes

Keywords: inversion transformation, geometric inversion, Stokes flow, viscous eddies, flow around a cylinder, microflow

HAL is a

Introduction

Geometric inversion is a type of transformation of the Euclidean plane. This transformation preserve angles and map generalized circles into generalized circles, where a generalized circle means either a circle or a line (a circle with infinite radius). One of the main properties of this method is the transformation of a straight line to a circle. Many difficult problems in geometry become much more tractable when an inversion is applied.

In engineering, the geometric inversion could be very useful to solve complex problems. For example, in fluid mechanics, the equation of two-dimensional Stokes flow remains valid in the new coordinates system obtained by inversion. Thus two-dimensional Stokes flow around certain bodies presenting circular shape appears less difficult to calculate by inversion of flow in channels of parallel walls than by direct calculation using polar coordinates. Although this method is rather general, we will apply it to the case of cellular flows (recirculation flow) presenting viscous eddies. In fact these flows are characterized by the presence of dividing streamlines (separating streamlines) which also give by inversion in the new geometry dividing streamlines.

Much attention has been paid to the steady viscous flow between parallel plates at low Reynolds number (Stokes flow) because of its theoretical importance and also its engineering applications. The particular case of the flow with vanishing velocity to zero when ∞ → x r (i.e. flow with mean rate equal zero) has been widely studied by authors motivated among others by separation phenomena. Thus, after the pioneering work of [START_REF] Moffatt | Viscous and resistive eddies near a sharp corner[END_REF], which presented predictions of cellular motion in Stokes regime between parallel walls, theoretical works like those of [START_REF] O'neill | On angles of separation in Stokes Flow[END_REF] and [START_REF] Bourot | Sur la structure cellulaire des écoulements plans de Stokes, à debit moyen nul, en canal indéfini à parois parallèles[END_REF] demonstrate the existence of such flow. They showed that, independently of the motion source, any two-dimensional flow with mean rate null in a channel presented necessarily cellular motion composed by successive counter rotating eddies bounded by separating streamlines reattaching the walls. In order to examine the influence of the motion source, accurate computations for various motion sources have been performed [START_REF] Shen | Low Reynolds number flow over cavities[END_REF][START_REF] Bourot | Sur l'utilisation de la série cellulaire pour le calcul d'écoulements plans de Stokes en canal indéfini : Application au cas d'un cylindre circulaire en translation[END_REF][START_REF] Carbonaro | Transient Stokes flow in a channel driven by moving sleeves[END_REF][START_REF] Hellou | Cellular Stokes flow induced by rotation of a cylinder in a closed channel[END_REF][START_REF] Moreau | Ecoulements cellulaires de Stokes produits en canal plan illimité par la rotation de deux cylindres[END_REF][START_REF] Shankar | Fluid mechanics in the driven cavity[END_REF][START_REF] Jeong | Slow viscous flow in a partitioned channel[END_REF][START_REF] Hellou | Structures d'écoulements de Stokes dans une jonction bidimensionnelle de canaux[END_REF][START_REF] Wang | Slow viscous flow between hexagonal cylinders[END_REF][START_REF] Wang | The recirculating flow due to a moving lid on a cavity containing a Darcy -Brinkman medium[END_REF][START_REF] El Naby | The flow separation through peristaltic motion of power-law fluid in uniform tube[END_REF][START_REF] Van Der Woude | Stokes flow in a rectangular cavity by rotlet forcing[END_REF][START_REF] Zabarankin | Asymetric three-diemensional Stokes flows about two fused equal spheres[END_REF]. Stokes flows and particularly cellular flows could be encountered in numerous applications in physics, biophysics, chemistry and MEMS (Micro and ElectroMechanical Systems) where microflows appear. The particularity of these applications is that they use microchannels [START_REF] Brody | Biotechnology at low Reynolds numbers[END_REF][START_REF] Yeom | Low Reynolds number flow across an array of cylindrical microposts in a microchannel and figure-of-merit analysis of micropost-filled microreactors[END_REF][START_REF] Wang | Flow through a finned channel filled with a porous medium[END_REF]. Thus, several theoretical and numerical results are available. They could be useful to obtain by inversion transformation the structure and the features of Stokes flows around bodies of circular shape. This transformation is also useful to obtain flow around bodies with complex shape for which the direct calculation could be tiresome.

Geometric inversion-Definitions and properties

Inversion is the process of transforming points M to a corresponding set of points N known as their inverse points. Two points M and N are said to be inverses with respect to an inversion circle having inversion centre O and inversion radius R 0 if N is the perpendicular foot of the altitude of the triangle OQM, where Q is a point on the circle such that OQ ┴QM.

Figure 1 Geometric inversion -definition

If M and N are inverse points, then the line L through M and perpendicular to OM is called a "polar" with respect to point N, known as the "inversion pole". In addition, the curve to which a given curve is transformed under inversion is called its inverse curve (or more simply, its "inverse").

From similar triangles, it immediately follows that the inverse points M and N obey to:

ON R R OM 0 0 = or ON OM R 2 0 × = (1)
where the quantity 2 0 R is known as the circle power or inversion power [20].

Q M N Inversion centre Pole R 0 L Polar 3 2 0 2 0 0 2 0 0 ) y y ( ) x x ( ) x x ( R x ' x - + - - + = , 2 0 2 0 0 2 0 0 ) y y ( ) x x ( ) y y ( R y ' y - + - - + = (2)
Note that a point on the circumference of the inversion circle is its own inverse point. In addition, any angle inverts to an opposite angle.

Treating lines as circles of infinite radius, all circles invert to circles. Furthermore, any two nonintersecting circles can be inverted into concentric circles by taking the inversion centre at one of the two so-called limiting points of the two circles [START_REF] Coxeter | Introduction to Geometry[END_REF], and any two circles can be inverted into themselves or into two equal circles. Orthogonal circles invert to orthogonal circles. The inversion circle itself, circles orthogonal to it, and lines through the inversion centre are invariant under inversion. Furthermore, inversion is a conformal map, so angles are preserved. Note that a point on the circumference of the inversion circle is its own inverse point. In addition, any angle inverts to an opposite angle.

The property that inversion transforms circles and lines to circles or lines (and that inversion is conformal) makes it an extremely important tool of plane analytic geometry. Figure 2 shows a simple example of application of geometric inversion. In figure 3 we present the classical example of inversion of a square relatively to a circle inscribed in this square. This inversion image becomes more complex if the number of squares is increased (see Mathematica Notebook presenting inversion of a grid).

Inversion transformation of Stokes flow

Let

) z ( ψ be the stream function of a plane Stokes flow denoted F 0 where

) i exp( r z θ =
. This flow is governed by the biharmonic equation :

0 = ψ ΔΔ ( 3 ) The function ) z ( ψ is necessarily expressed by: ) z ( X ) z ( X ) z ( z ) z ( z ) z ( + + + = ϕ ϕ ψ (4) R 0
(see [START_REF] Laurentiev | Les méthodes de la théorie des fonctions de la variable complexe, traduit du russe par[END_REF]). Introducing the complex variable z a Z 2

=

, it follows that the function

) M ( a ) N ( 2 2 ψ ρ φ = (6)
The velocity components in polar coordinates framework of centre O on the homologous points are readily found to be:

) M ( V ) N ( V r = ρ ; r ) M ( 2 ) M ( V ) N ( V ψ θ θ - - = ( 7 )
Hereafter, flow on which we apply the inversion method (Flow F 0 ) is called reference flow. Now let F 0 be a reference flow around a fixed body C 0 and F 1 is the equivalent flow in the proximity of the inverse fixed body C 1 . Assuming that the body C 1 is at rest, by virtue of Eq. ( 7), the non slip conditions on C 1 are satisfied only for 0 = ψ on C 0 . However, we can add a constant which yields 0 = ψ on this body without modifying F 0 . In this condition, an additional velocity is therefore added to F 1 in order to ensure the non slip conditions on C 1 . Hence, the knowledge of the stream function field of F 0 permits to determine the features of F 1 . Nevertheless, the inversion of a streamline of F 0 does not give a streamline of F 1 unless this streamline is a circle or of value 0 = ψ . Thus in order to draw the streamlines, it would be necessary to calculate the stream function field of F 1 by using Eq. ( 6) and one can deduce straightforward the desired streamlines of F 1 . It is worth noting that the relations of equivalence between F 0 and F 1 for the other physical quantities (velocity, pressure, vorticity,…) can be determined without any particular problem.

Characteristics of the cellular flow between parallel plane walls

The stream function of two dimensional Stokes flow with zero mean rate between parallel walls of a long rectangular channel has been found by [START_REF] Bourot | Sur la structure cellulaire des écoulements plans de Stokes, à debit moyen nul, en canal indéfini à parois parallèles[END_REF], [START_REF] Bourot | Sur l'utilisation de la série cellulaire pour le calcul d'écoulements plans de Stokes en canal indéfini : Application au cas d'un cylindre circulaire en translation[END_REF], [START_REF] Hellou | Cellular Stokes flow induced by rotation of a cylinder in a closed channel[END_REF] and [START_REF] Moreau | Ecoulements cellulaires de Stokes produits en canal plan illimité par la rotation de deux cylindres[END_REF] to be :

∑ - = n n x n ) y ( G e C ) y , x ( n α ψ (8)
where the functions G n (y) have the following expressions according to the flow is antisymmetric Each term of the stream function of Eq. ( 8) denoted n ψ is null infinitely many times as

) y , x ( ) y , x ( - =ψ ψ (Eq. ( 9
∞ → x
whatever the motion source may be. There is an infinity of dividing streamlines of value ψ n = 0 which attach the parallel walls where the stream function is likewise zero and divide thus the flow on successive eddies. The equation of these dividing streamlines is given by:

) y ( Q ) y ( P x tan n n n = μ ( 13
)
where P n (y) and Q n (y) represent the real and imaginary parts of G n (y). The equation [START_REF] Wang | The recirculating flow due to a moving lid on a cavity containing a Darcy -Brinkman medium[END_REF] means that the location of the dividing streamlines is periodic in the longitudinal direction. Hence, the axial length of each class n of eddies is constant and readily given by:

n n L μ π = ( 1 4 )
Furthermore, angle at which the separating streamlines detach from the walls, i.e. the separation angle (angle T indicated in figure 5), satisfies the equation:

n n 2 tan 3 tan λ α - = (15)
For n=1, the separation angle is 58°61 for an antisymmetric structure and 46°25 for a symmetric one.

The ratio of the velocity on homologous points, i.e. points as S(x, y) and S'(x+L n , y) (for example points S and S'indicated in figure 5), is given by:

n n n exp K μ λ π = (16)
For the same value of n, the velocity of the symmetric flow decays more than the velocity of the antisymmetric one. Thus, practically an arbitrary Stokes flow of zero mean rate between parallel plates becomes antisymmetric far from the motion source except when this source is strictly symmetric. Of course, numerical calculation of this arbitrary flow in the proximity of the motion source requires to conserve a sufficient number of terms n ψ of the stream function, n=20 or more ([5], [START_REF] Hellou | Cellular Stokes flow induced by rotation of a cylinder in a closed channel[END_REF] and [START_REF] Moreau | Ecoulements cellulaires de Stokes produits en canal plan illimité par la rotation de deux cylindres[END_REF]).

Figure 5 shows the streamlines of the theoretical flow corresponding to n=1 for antisymmetric and symmetric conditions between parallel plates. The sequences of the dividing streamlines drawn in figure 5 Consequently, the infinite cellular flow between parallel plates leads by inversion transformation to a finite cellular flow within the corners between the transformed bodies which the shape depends closely on the position of the inversion centre.

Applications

Antisymmetric and symmetric Stokes flows between parallel walls can be concretely realized by the uniform rotation respectively the uniform longitudinal translation of a cylinder midway between the parallel walls of a long rectangular channel filled with a viscous oil. The boundary conditions are the non slip conditions on the cylinder boundary and the matching conditions at x=0 between the two semi-infinite domains. Precisely, these conditions are written as following:

Antisymmetric flow induced by a rotating cylinder

θ θ cos V ) y , x ( v , sin V ) y , x ( u 0 C C 0 C C 0 0 0 0 = - = with 2 0 2 C 2 C R y x 0 0 = + , 2 0 π θ ≤ ≤ , 0 0 y y R , 0 ) y , 0 ( v < < = , 0 0 y y R , 0 ) y , 0 ( p ≤ ≤ = ( 17
)
where R 0 represents the cylinder radius, V 0 the cylinder velocity, u v the Cartesian components of the velocity and p the pressure (see figure 6 for the notations).

Symmetric flow induced by a translating cylinder along the channel axis

0 ) y , x ( v , V ) y , x ( u 0 0 0 0 0 C C C C = = , 2 0 π θ ≤ ≤ 0 0 y y R , 0 ) y , 0 ( v < < = , 0 0 y y R , 0 y ) y , 0 ( p ≤ ≤ = ∂ ∂ (18)
These flows, produced by rotating or translating cylinder, have been previously computed and examined by [START_REF] Bouard | Etude théorique et expérimentale de l'écoulement engendré par un cylindre en translation uniforme dans un fluide visqueux en régime de Stokes[END_REF] and [START_REF] Moreau | Ecoulements cellulaires de Stokes produits en canal plan illimité par la rotation de deux cylindres[END_REF].

Figure 6 Delineation of the domain of reference composed by two parallel walls and a rotating circle of radius R 0 (dashed lines ---) and the inverse domain composed by two circles in contact set in a rotating circle of radius R 1 (solid lines -). In this sketch, the power of the transformation is

0 y a = , thus 0 2 0 1 R y R =
Now let apply the positive inversion to the domain bounded by parallel infinite walls. The centre of the cylinder is chosen as the inversion centre. The choice of the power a is not important. This parameter modify only the scale of the inverse field; let take a=y 0 here. The transformation of the lines b 0 and d 0 representing the parallel walls leads to the tangent circles b 1 and d 1 (Figure 6).

The rotating circle C 0 of radius R 0 gives a circle C 1 of radius

0 2 0 1 R y R =
which rotates with the velocity

0 0 0 1 R 2 V V ψ - - =
where 0 ψ is the stream function on C 0 . In the case of translating cylinder on the axis or transversally to this axis with velocity 0 V the stream function on C 0 is

θ ψ sin R V 0 0 0 = respectively θ ψ cos R V 0 0 0 =
. By virtue of Eq. ( 7), we see that the translation motion of a cylinder is invariant when applying the inversion transformation relatively to the centre. Consequently, we obtain by inversion a circle C 1 moving with the same velocity. Note that inversion of a circle animated by both rotation and translation motions relatively to its centre leads likewise to a circle with the same motion.

Using results of the reference flows F 0 , published in [START_REF] Hellou | Cellular Stokes flow induced by rotation of a cylinder in a closed channel[END_REF] and [START_REF] Moreau | Ecoulements cellulaires de Stokes produits en canal plan illimité par la rotation de deux cylindres[END_REF], the purpose is now to obtain by the inversion method, the flow around two cylinders in contact set in a rotating or translating one. In the interest of brevity, we focused our attention to only the streamlines patterns.

The flow depicted in Figure 7b induced by a rotating cylinder around two fixed cylinders in contact is obtained by inversion of the flow presented in Figure 7a. It exhibits an important cellular motion which is theoretically composed by an infinite sequence of closed viscous eddies, as demonstrated by [START_REF] Moffatt | Viscous and resistive eddies near a sharp corner[END_REF] for a sharp corner. Their size diminishes as we approach the cusp whereas the size of eddies between parallel walls remains constant. The velocity decay is not constant but is important comparatively to the reference flow, about 4400 between the first and the second eddy and decreases to the limit of 358 in the neighbourhood of the contact line. Furthermore, the angles are invariant in the inversion transformation hence the angles subtended by the dividing streamlines are equivalent to those of the antisymmetric flow between parallel plates and found to be equal to 46° for the first line and 58°61for the next lines [START_REF] O'neill | On angles of separation in Stokes Flow[END_REF], [START_REF] Bourot | Sur la structure cellulaire des écoulements plans de Stokes, à debit moyen nul, en canal indéfini à parois parallèles[END_REF]. The first value of the contact angle is less than 58°61 because it is influenced by the motion source.

In the case of the translating cylinder, two configurations seem to be possible depending on whether the framework is absolute or relative. The inversion of the stream visualized by an observer attached to the absolute framework gives an instantaneous image of the flow around two cylinders in contact placed in the centre of the fluid domain bounded by a cylinder translating in the direction parallel to the contact plane, Figure 8a. We see a symmetric cellular flow with an extent relatively small compared to the antisymmetric one. Note that adding the constant x V 0 r -, leads to the more practical problem of two cylinders in contact translating in a fixed cylindrical enclosure whose the streamlines are presented in Figure 8b. The relative reference flow regarded by an observer attached to the translating cylinder centre is equivalent to the flow around a fixed cylinder induced by translation of the walls. The inversion of this flow leads to the flow produced by two counter rotating cylinders in a fixed cylindrical enclosure, Figure 8c. There is no separating streamlines because the corresponding reference stream is not of zero mean rate. ). The point of contact is located at the enclosure centre.

Conclusion

For a two dimensional Stokes field corresponds theoretically by inversion an infinity of Stokes flows because the inversion centre can be any point of the ) y , x , O ( r r plane. For example, in the case of the flow between parallel plates, choosing this centre position on the y r axis normal to the parallel plates, provide solutions to flows around two unequal cylinders in contact internally or externally or around a cylinder in contact with a plane. Nevertheless, this centre would be suitably chosen otherwise the motion source obtained by inversion can be out of physical sense. For example, the inversion of the flow relatively to a point which is not the centre of a rotating cylinder leads to a flow across the inverse cylinder. Thus, useful configurations seem to be obtained essentially when the inversion centre coincides with the cylinder centre. From a general point of view, the inversion transformation is an interesting method which can be wide with various problems even apart from the fluid mechanics (electricity, chemistry, engineering processes, …) but a precaution in the choice of the position of the centre of inversion is necessary.

Figure 2 2 0R

 22 Figure 2 Inversion of two parallel straight lines

Figure 3

 3 Figure 3 Inversion of a square. The inversion centre coincides with the square centre.

Figure 4

 4 Figure 4 illustrates the inversion of parabola. This transformation leads to a cardiod defined in figure 4c. It's worth to note that the change of the inversion centre leads to other figures of inversion.

Figure 4

 4 Figure 4 Inversion of a parabola (1996 © 2010 by Xah Lee).

  be written in a form similar to Eq. (3). Consequently ) Z ( φ is likewise biharmonic and represents therefore the stream function of an other Stokes flow denoted thereafter F 1 . Let ) , r ( M θ be a point of F 0 and ) , ( N θ ρ its homologous in the flow field F 1 obtained by the positive inversion transformation of power a and centre O defined by Eq. (1). It follows: And it's easy to show that :

  of these basic flows can compose the stream function of a more general flow. The coefficientsC n are arbitrary complex coefficients to be determined by the boundary conditions and 2y 0 represents the width of the channel. The parameters the following equations coming from the non slip conditions on the channel walls. n λ and n μ have been accurately computed by Bourot & Moreau[START_REF] Bourot | Sur l'utilisation de la série cellulaire pour le calcul d'écoulements plans de Stokes en canal indéfini : Application au cas d'un cylindre circulaire en translation[END_REF] ; their sign is chosen here such as 0 → ψ for x ∞ → .

Figure 5 θ

 5 Figure5shows the streamlines of the theoretical flow corresponding to n=1 for antisymmetric and symmetric conditions between parallel plates. The sequences of the dividing streamlines drawn in figure5are located arbitrary since the motion source is not yet considered.

Figure 7 aFigure 8

 78 Figure 7 a Streamline patterns of the flow between parallel plates induced by a rotating cylinder of radius 0 0 y 9 . 0 R = b Streamline patterns of the flow around two cylinders in contact of radius

Figure 9a present anFigure 9

 9 Figure 9a present an example of a flow obtained by a rotating cylinder decentred from the axis of a channel of parallel plates with the distance δ. If the inversion centre coincides with the cylinder centre, the inversion of the parallel walls leads to two cylinders in contact of different radius. The inversion of the flow leads to the flow drawn in figure 9b.


Mustapha Hellou

Thi Diep

Phuong Bach

Stokes Flow in a Junction of two-dimensional Orthogonal Channels

Keywords: Channels, Junction, Flow branches, Stokes eddies, Recirculation flow, Low Reynolds number, Microflows

The purpose of this study is to calculate Stokes flow structures in relation to flow rate distribution in a junction of four orthogonal channels. Particular attention was paid to the existence of recirculation flow inside the channels for low Reynolds numbers. The presence of this recirculation flow may be observed, but only when the flow rate is relatively low in at least one of the branches.

Introduction

The problem of viscous flow in a junction of flow branches is encountered in many fundamental and applied situations. Therefore, an understanding of the multiscale transport mechanisms at work at different stages in channel networks is of fundamental interest. Dispersion phenomena in a fracture network is a large-scale example for which the impact of the junction is considered to be a heterogeneity. [START_REF] Adler | Transport processes in spatially periodic capillary networks-Geometrical description and linear flow hydrodynamics[END_REF]; Adler and Thovert (1999); [START_REF] Berkowitz | Mass transfer at fracture intersections:An evaluation of mixing models[END_REF]; Bruderer and Barnabé (2001); [START_REF] Bryden | Multiple-timescale analysis of Taylor dispersion in converging and diverging flows[END_REF]; [START_REF] Dentz | Transport behavior of a passive solute in continuous time random walks and multirate mass transfer[END_REF]; [START_REF] Mourzenko | Solute transport at fracture intersections[END_REF]; [START_REF] Park | Analytical Solutions for Solute Transfer Characteristics at Continuous Fracture Junctions[END_REF][START_REF] Park | Effects of junction transfer characteristics on transport in fracture networks[END_REF][START_REF] Park | Transport behavior in three-dimensional fracture intersections[END_REF]]. On a microscopic scale, the junction is continuous but microhydrodynamic effects can cause recirculation, and the presence of these local hydrodynamic eddies causes nonuniform convective mass transport, as shown in a number of studies of wet chemical etching, and in microelectronics [START_REF] Occhialini | Convective mass transport from rectangular cavities in viscous flow[END_REF]; Kondo and Fukui (1998); [START_REF] Driesen | Mass transport in a partially covered fluid-filled cavity[END_REF]; Georgiado et al. (2000)]. Flow in channel networks: has numerous applications; larger-scale examples include pollutant transport, predictive tools for modelling water supply resources, and surface waters, whereas small-scale examples include flow in capillary networks (human body, microfluidics), the cooling of electronic compounds, and chemical etching for the process of fabrication of electronic circuits. This paper focuses on the problem of flow in channel networks on a microscopic scale, with particular attention accorded to the conditions under which eddies arise in Stokes regime.

Recirculation motion in viscous flow is a problem of long-standing interest [START_REF] Moffatt | Viscous and resistive eddies near a sharp corner[END_REF], [START_REF] Taneda | Visualizationof separating Stokes flow[END_REF], [START_REF] O'neill | On angles of separation in Stokes flow[END_REF], [START_REF] Bourot | Sur la structure cellulaire des écoulements plans de Stokes, à débit moyen nul, en canal indéfini à parois parallèles[END_REF], [START_REF] Hellou | Cellular Stokes flow induced by rotation of a cylinder in a closed channel[END_REF], [START_REF] Pozrikidis | Boundary integral and singularity methods for linearized viscous flow[END_REF], [START_REF] Hajjam | Contribution à l'étude de différents types d'écoulements en régime de Stokes[END_REF], [START_REF] Wang | Stokes flow through an array of rectangular fibers[END_REF][START_REF] Wang | Slow viscous flow between hexagonal cylinders[END_REF], [START_REF] Shankar | Oscillatory eddy structure in a container[END_REF]]. Most of these works concern individual channels or cavities, whereas the recirculation flow in a network of channels has received much less attention. [START_REF] Jeong | Slow viscous flow in a partitioned channel[END_REF] analyzed the flow in a compartmented channel; comprising an infinite channel in which a semi-infinite wall is placed on its axis generating three semi-infinite channels. He studied the structure of the flow over a wide range of flow rates, showing the formation of recirculation motion in the channel of low flow rate, and illustrated how the fluid flows out by surrounding this motion.

In this study, we investigate the possible Stokes flow structures in a fluid domain formed by four orthogonal channels or flow branches. We present a complete study covering a variety of flow rate distributions, with particular analysis of cases which exhibited eddies.

Fluid flow formulation

The fluid flow domain indicated in figure 1a is formed by orthogonal flow channels of the same width 2y 0 , connected by walls whose equation is expressed in a local frame by the following equation:

1 b Y a X m 2 L m 2 L = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ (1)
where X L and Y L are the coordinates in the local frame indicated in figure 1. The parameters a and b represent the dimensions shown in figure 1 which permit to calculate the shape of the curve of connection given by equation ( 1). The values of a and b used are a=b=4y 0 . To avoid errors in calculation, the shape described by this equation contains no sharp corners. In our case, we chose m=4, which leads to the shape described in figure 1.

A viscous fluid of flow rates q 1, α |q 1 |, β|q 1 | and γ|q 1 | respectively, flows through the four branches; where α, β and γ are dimensionless flow rates in branches 2, 3 and 4 with respect to the real flow rate |q 1 | in branch 1.

The sign of α, β and γ must be consistent with the sign of

i i n U r r
. , where i U r represents the mean normal velocity vector at the inlet of every branch, and the external normal is i n r

. Furthermore, the values α , β and γ must respect the law of conservation as defined by the following equation:

0 1 = + + + - γ β α (2)
The fluid is assumed to be Newtonian and isothermal, with viscosity μ and density ρ constant. Let U i be a characteristic value of the mean velocity in each branch. The local Reynolds number is defined as:

μ ρ 0 y U 2 Re i i = (3)
It is assumed to be less than unity in order to meet the requirements for Stokes regime. More detailed information about Stokes flow can be found in the following references: [START_REF] Hasimoto | Stokeslests and eddies in creeping flow[END_REF]; [START_REF] Sangani | Slow flow past periodic arrays of cylinders with application to heat transfer[END_REF]; [START_REF] Happel | Low Reynolds number hydrodynamics. Mechanics of fluids and transport processes[END_REF]; [START_REF] Pozrikidis | Boundary integral and singularity methods for linearized viscous flow[END_REF]; [START_REF] Hellou | Cellular Stokes flow induced by rotation of a cylinder in a closed channel[END_REF][START_REF] Hajjam | Contribution à l'étude de différents types d'écoulements en régime de Stokes[END_REF]; [START_REF] Higdon | Permeability of the three-dimensional models of fibrous porous media[END_REF]; [START_REF] Wang | Stokes flow through an array of rectangular fibers[END_REF][START_REF] Wang | Slow viscous flow between hexagonal cylinders[END_REF][START_REF] Wang | The recirculating flow due to a moving lid on a cavity containing a Darcy -Brinkman medium[END_REF][START_REF] Wang | Flow through a finned channel filled with a porous medium[END_REF]].

Since it is assumed that the flow is both slow and two dimensional, we chose to solve the problem using the stream function. When inertia forces are negligible compared to viscous forces, the equation of two dimensional viscous flow known as Stokes equation is:

0 = ΔΔΨ (4)
where Ψ is the stream function of the flow. In the case of a parallel-walled channel, the solutions to equation ( 4

i P Ψ , i A Ψ , i S Ψ .
Furthermore, a solution 0 Ψ is found for the flow in the junction of the flow branches; it is also given in appendix A; equation A9.

To determine the unknown coefficients of the stream function solutions ( 0

Ψ , i A Ψ , i S
Ψ ) corresponding to each branch, and the stream function 0 Ψ in the junction, the following boundary conditions must be satisfied: for branches 1 and 3 for branches 2 and 4

) x , 0 ( ) 1 , x ( ) x , 0 ( p ) 1 , x ( p ) x , 0 ( u ) 1 , x ( v ) x , 0 ( v ) 1 , x ( u 1 x 1 - 3 i , 1 i i 0 i 0 i 0 i 0 ξ ξ = ± = ± = ± = ± ≤ ≤ = = ) y , 0 ( ) y , 1 ( ) y , 0 ( p ) y , 1 ( p ) y , 0 ( v ) y , 1 ( v ) y , 0 ( u ) y , 1 ( u 1 y 1 - 4 i , 2 i i 0 i 0 i 0 i 0 ξ ξ = ± = ± = ± = ± ≤ ≤ = = (5)
where the velocity components u and v, the pressure p and the vorticity ξ are calculated using the following relationhips:

y u ∂ ∂ = Ψ , x v ∂ ∂ - = Ψ , ΔΨ ξ - = , y x p ∂ ∂ = ∂ ∂ ξ (6)
It is worth noting that the stream function i P Ψ is entirely known when the flow rate in the corresponding channel is done. These boundary conditions result, on the one hand, from the non-slip conditions on the walls of the flow branches, and on the other hand, from matching the flow branches to the flow junction. The matching conditions correspond to the continuity of the velocity vectors and the stress components, and it is easy to show that the continuity of the stress components leads to equality of pressure and vorticity. In addition to the conditions represented by equations ( 6), we must write the non-slip conditions on the curved parts of the boundaries (corners on figure 1) denoted by c wi .

These conditions are:

0 ) y , x ( v , 0 ) y , x ( u i i i i cw cw 0 cw cw 0 = = (7)
The quadratic minimisation method described in appendix B is applied to satisfy all the boundary conditions.

The resulting equations are also developed in appendix B. The number of coefficients to be calculated is where N 0 , and are the number of terms retained for the functions

) N N ( N N i i S 4 1 i A 0 ∑ = + + = i A N i S N 0 Ψ , i A Ψ and i S Ψ respectively.
The width of the branches being equal, it was simple to take the same number of points on their interfaces with the central domain. Furthermore, the same number of terms in the series (A7) is considered for each branch. The system of equations resulting from quadratic minimization is solved using gauss elimination. The choice of , , and

40 m i = 40 m 0 = 20 N i A = 20 N i S =
is considered to be satisfactory to calculate the motion with accuracy.

Flow structures

The conservation of mass and the hypothesis of two-dimensional Stokes flow lead to three independent configurations for the flow directions shown in figure 2. A parametric study is conducted by varying the flow rates in the channels. However, in order to limit the cases studied, particular flow rate distributions were analysed : α =γ and β variable, or α = 1 and β = ± γ variable. Moreover, other original structures were selected.

Configuration 1 (α=γ >0, β ≥ 0 )

For β = 0, a flow of symmetrical cellular structure is established in branch 3, as illustrated in figure 3.1. The fluid flows with open streamlines through branches 1, 2 and 4, however, the structure of the flow in branch 3 is radically different because of the occurrence of a recirculation flow as successive cells, symmetric to the axis, are bounded by separating streamlines, which are attached to the walls of this branch. By increasing β slightly, the separating streamlines detach from the walls and the symmetrical cellular flow is consequently made up of only two recirculation cells, which are now located near the axis rather than being attached to the walls. The fluid flows out of branch 3 by circumventing these cells. The recirculation flow disappears when β is greater than approximately 0.001. In branch 3, a structure was obtained in which all streamlines were open. Hence, the flow in the overall domain is composed of three flows, distinguished by two separation lines in the vicinity of the vertices of the junction, attached to the walls of branch 3; the separation lines are attached to these vertices for β=1/3. When β approaches 0.98 (in this case α and γ approach 0.01), two cells appear, each of which is attached to the lower wall of both branches 2 and 4. This cell grows as β increases. Finally for β = 1 (α =γ = 0) we obtain purely recirculation flow in branches 2 and 4. This flow is composed of a succession of antisymmetric eddies, whose properties were reported by [START_REF] Hellou | Cellular Stokes flow induced by rotation of a cylinder in a closed channel[END_REF].

In order to show other possible structures for configuration 1, we have selected two examples given in figure 3.2. Respectively, these examples show a case in which recirculation occurs in adjacent channels, and a case of dissymmetric structure (α ≠β ≠γ ).

Flow reversal in branches 2 or 4 (inversion of the sign of α or γ) leads to configuration 2, as seen in figure 2, while inversion of flow β leads to configuration 3. However, simultaneous reversing of flows α and β or β and γ or α and γ results in structures conforming to configuration 1. Figure 3.3 shows examples of structures when α and γ are reversed. It is worth noting that in this situation β is necessarily greater than 1. Thus, the structures obtained in figure 3.1 for β between 0 and 1, are now obtained for β∈ [1, ∞[. For β=1, the structure in figure 3.1 remains unchanged when the flows α and γ are reversed. Nevertheless, the structures obtained when β approach zero exhibiting symmetric cells in branch number 3 (figure 3.1), can be obtained for flow reversal only for large values of |α| and |γ | as shown in figure 3.3 where flow rates are 1000 times greater than in branch 1. This example shows a structure closely similar to the structure shown in figure 3.1 corresponding to β=0.0005 but this situation would not be realistic under Stokes regime conditions. Finally, simultaneous inversion of the sign of α, β and γ leads to an impossible situation under 2D conditions. Consequently, the communication between these branches is suppressed. For |β | in the order of 0.02, a cell attached to the other wall of each of these branches (walls connected respectively with branches 1 and 2) appears, this cell will grow as | β |decreases. At the limit (β=0), an antisymmetric sequence of eddies is formed.

Configuration 2 (α

As was the case for the previous configurations, we complete the illustrations with the cases presented in figure 5.2 corresponding to β = -1 and a ratio γ /α variable. Increasing this ratio leads to displacement of the saddle point along the axis of branches 2 and 4 towards the branch of small flow rate. In addition, we observe an original structure in the branch of small flow rate exhibiting two cells which each of them is attached separately to a wall of this branch. It is obvious that these cells will grow when γ /α more and more . A coalescing process would happen resulting in one cell of symmetric structure.

Conclusion

We have conducted a study of the Stokes flow structures in flow branches linked by a junction, taking into consideration various flow rate distributions. The structures presented correspond to three possible configurations under 2D conditions, and particular streamline patterns exhibiting eddies were found. Such eddies develop within branches characterized by low or zero flow. For zero flow rate, these structures exhibit antisymmetric or symmetric closed streamlines, occupying the overall width of the channels in which they appear. For small flow rates, these eddies form original structures, occupying a part of the channel branches in which they develop, thus forcing the streamlines to surround them, creating positive pressure gradients.

Although these eddy structures appear only in a limited range of flow rate, such structures may have consequences on transport processes. = ΔΔΨ

A1. General solution

The separation of variables method is used to solve the following equations 0 1 = ΔΨ and . Thus,

writing 1 Ψ ΔΨ = ) y ( ) x ( 1 Θ Φ Ψ =
the following equations are obtained:

( A 1 ) 0 ) y ( ) y ( ' ' 0 ) x ( ) x ( ' ' 2 2 = + = - Θ σ Θ Φ σ Φ
where the constant σ may be either real or complex. The solution of this system leads to the following

+ + + = Ψ (A2) ) e D e C )( e B e A ( ) y i y i x x 1 σ σ σ σ σ σ σ σ σ σ Ψ - - + + = ∑ (A3)
where A, B, C, D are real coefficients and are either real or complex coefficients. The function

σ σ σ σ D , C , B , A σ Ψ Ψ Ψ 1 1 1 0 + =
is harmonic, and consequently also biharmonic. We can easily show that the functions are biharmonic. The sought stream function

1 2 2 1 1 ) y (x , y , x Ψ Ψ Ψ + Ψ which satisfies is thus: 0 = ΔΔΨ 1 2 2 1 1 1 ) y (x y x Ψ Ψ Ψ Ψ Ψ + + + + = (A4)

A2. Solution in the case of a parallel-walled channel

For a parallel-walled channel i, the stream function (A4) is written as:

i i i S A P i Ψ Ψ Ψ Ψ + + = ( A 5 )
which reveals, on the one hand, a flow with non-zero mean rate, represented by

) y , x ( ) y , x ( i i A i i i A i - =Ψ Ψ ) while the stream function i S Ψ is symmetric ( ) y , x ( ) y , x ( i i i S i i S i - - = Ψ Ψ
). The co-ordinates (x i, y i ) are dimensionless local co-ordinates (x i along the axis of the channel i).

To ensure non-slip wall conditions, we must suppress the terms factor of x in equations (A2) and (A4). When the flow rate is non-zero, the combination of the equations A2 and A4 gives the following solution:

) y 3 y ( q 4 3 A i 3 i * i i P i - - = Ψ ( A 6 )
which represents the dimensionless stream function of a flow of parabolic velocity profile (plane Poiseuille flow) corresponding to the dimensionless flow rate ; A * i q i is an arbitrary constant.

When the flow rate is zero, the expressions of

σ σ Ψ Ψ 1 1 y + . Writing μ λ σ i + =
and after setting in real form, one finds:

{ } i n i n i i n i n i n i n i n i i n i n i n i i i i i n n i n n n i n n i n n n n i n i S i A S sinh S cos D D sinh D cos S Q 2 , S cosh S sin D D cosh D sin S P 2 y 1 S , y - 1 D ) x sin P x cos Q ( C ) x sin Q x cos P ( B ) x exp( ) (or μ λ μ λ μ λ μ λ μ μ μ μ λ Ψ Ψ ± = ± = + = = + - - = ∑ (A7)
The upper and lower signs of (A7) relate to the functions We consider the pair

) 0 , 0 ( n n > > μ λ
for the negative semi-infinite field were calculated up to n=40 with a high degree of accuracy [START_REF] Bourot | Sur l'utilisation de la série cellulaire pour le calcul d'écoulements plans de Stokes en canal indéfini : Application au cas d'un cylindre circulaire en translation[END_REF].

0 x i ≤ < ∞ - and 1 y 1 i ≤ ≤ -
Because equations (A8) are periodic, the stream functions i A Ψ and i S Ψ of each channel are zero on certain lines joining the walls of this branch. These lines are located periodically along this branch for each value of n. Thus, a flow with zero mean rate in a branch presents an infinite succession of lines known as separation lines delimiting recirculation cells [START_REF] O'neill | On angles of separation in Stokes flow[END_REF][START_REF] Bourot | Sur la structure cellulaire des écoulements plans de Stokes, à débit moyen nul, en canal indéfini à parois parallèles[END_REF][START_REF] Hellou | Cellular Stokes flow induced by rotation of a cylinder in a closed channel[END_REF][START_REF] Hellou | Sensitivity of cellular Stokes flow to geometry[END_REF].

Previously, [START_REF] Moffatt | Viscous and resistive eddies near a sharp corner[END_REF] showed that slow flow in a sharp corner presents this phenomenon when the angle is smaller than 146°3.

A3. Solution in the junction of the channels

In the junction, called domain 0, the flow rate is non-zero. In addition, it is assumed that no separation solutions can exist. Therefore, according to (A2) and (A4), the stream function representing the flow in this field is given by:

p m p , m p , m 0 y x A ∑ = Ψ (A9)
where A m,p are real coefficients, m and p are integers ranging from 0 to 3. The pairs (2, 2), (2, 3), (3, 2) and (3, 3) are not permitted because they do not satisfy equation (A9).

Appendix B: Quadratic minimisation method

Let g i and g 0 represent the expressions of velocity components, pressure or vorticity in channels 1 to 4 and in the junction domain respectively. The expressions of cumulative quadratic errors committed on the matching conditions (equation ( 5)), and on the boundary conditions in the junction domain, equation ( 7), are respectively:

( B 1 ) ∑∑ = - = 4 1 i 2 i m 0 1 ) g g ( S i ( B 2 ) ∑ = 0 0 m 2 2 g S
where m i and m 0 respectively are the number of points on each interface and on the walls of the junction domain.

The minimization of these errors is expressed by:

(B4) i i i 0 i i S A 0 i N 1 N k i i k k i i * i i N N N N 4 ..., , 1 i ) y , x ( f A ) y ( h q g + + = = - = ∑ + = (B5) ∑ = = 0 0 N 1 k k 0 k 0 ) y , x ( f A g
Of course in this setting, conveniently used to condense the expressions of velocity components, pressure and vorticity, the functions , and are different according to which of the aforementioned expressions they represent. They are obtained without any particular difficulty using equations (A6), (A7) and the relationships (7). Applying minimization leads to writing the system of equations in the following condensed form:

0 k f i h i k f (B6) N ..., 1, j , f h q A f A 4 1 i j i * i m N 1 k j k k = = ∑ ∑∑ = =
where m is the total number of minimization points. In this system, indices 0 and i, corresponding to the five domains, do not appear because of the classification of all the coefficients from 1 to N. In the calculation, this classification requires the cancellation on each interface of the functions concerning the other interfaces.

Notation

i i * i y U 2 q q = non dimensioned flow rate α, β, γ
values of the flow rates in the channels relatively to the flow rate of reference ( e.g. q 1 )

1 3 1 3 1 2 q q , q q , q q = = = γ β α Δ Laplacian operator λ n , μ n

  >0, γ = -β, β ≥ 0 ) parts either side of the straight line joining the opposite vertices common to branches 2-3 and branches 1-4 (figure 4.1). When β decreases, this line maintains a curved shape and moves towards the branches with the lowest flow rate. Recirculation cells appear in the channels 2 and 3 for β= 0.02 (or γ=-0.02). These cells grow in size as β decreases, producing tortuous streamlines in the outflow. If β is greater than unity, the structures obtained for β∈ [0, 1] are now found for β∈ [1,∞].

Figure 4 .

 4 Figure 4.1 is completed by the structures given in figure 4.2 obtained when α = -γ and β = 1. In this case, the recirculation flow is found to be developed in the opposite channels 2 and 4 for flow rate α  less than 0.02 and the corresponding cells are attached to opposite walls.

Ψ

  are determined by considering the combination

  parameters λ n and μ n are the eigenvalues of λ and μ which satisfy the following equations: It is understood that equations (A7) are made non dimensional using y 0 and the absolute value of the flow rate

  positive semi-infinite field. As coefficients B B B n and C n are arbitrary, the pairs )

  where A j are the unknown coefficients. Let us write g i and g o in the following forms:

Figure 1 .Figure 2 .

 12 Figure 1. Delineation of the junction of four flow branches. The parabolic dashed lines

Figure 3 . 1

 31 Figure 3.1 Streamline patterns for flow rates corresponding to configuration 1 under the following conditions : -1, α > 0, β > 0,γ=α.

Figure 3 . 2 Figure 3 . 3 Figure 4 . 1 .

 323341 Figure3

Figure 4 .

 4 Figure41/2

Figure 4 . 2 Figure 5 . 1 .Figure 5 . 2 .

 425152 Figure 4.2 Streamline patterns for flow rates corresponding to configuration 2 under the following conditions : -1, α > 0, β = 1, α = -γ