Numerical simulations of three dimensional micro flows Charles-Henri Bruneau, Thierry Colin and Sandra Tancogne University of Bordeaux, 351 Cours de la Liberation, 33405 Talence (France). Inria Bordeaux Sud Ouest team MC2
[email protected]
1 Introduction Microfluidics deal with the manipulation and the control of liquids in channels about a hundred of microns. The consideration of various experimental configurations leads to several regimes of flows: jets, droplets or plugs [3]. Indeed, the use of coflows or drippings find his interest in various applications ([6], [4]): ink jet printing or spray atomization for example. So, it is necessary to control the evolution of a diphasic jet in a view to produce droplets of different shapes and volumes. The created microdroplets are often employed for their internal dynamic to mix products that are generally toxic and expensive. In this work, numerical results of diphasic flows in square micro channels are presented. At the scale, the flow are generally laminar and the movement of the interface between the two fluids is controlled by the effect of the surface tension. As the breaking jet, due to the Rayleigh-Plateau instability, is only observable thanks to a tridimensional modeling. All numerical simulations are done in tridimensional cartesian meshes. So, the aim is first to study the breaking jet phenomenon, when confinement and effects due to the surface tension are predominant. Then, the second point is to analyze the internal dynamic of the created droplets. Finally, a numerical result corresponding to the coalescence of microdroplets is shown. The interface liquid-liquid is followed thanks to the Level Set method coupled to the one-fluid formulation of Stokes equation for diphasic flows.
2 Modeling 2.1 The Stokes equations for diphasic flows in microfluidic We consider the Stokes equations for two fluids in a bounded domain Ω ∈ R3 . The two fluids, respectively called internal (i) and external (e), occupy at each
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Charles-Henri Bruneau, Thierry Colin and Sandra Tancogne
time t the domains Ωi (t) and Ωe (t) such that Ω = Ωi (t) ∪ Ωe (t). The interface ¯i (t) ∩ Ω ¯e (t). Γ (t) between the two fluids is defined like Γ (t) = Ω So, the hydrodynamic model is the following div(2ηD(U )) = ∇P + γκδΓ nΓ in Ω (1) ∇.U = 0 in Ω where D(U ) is the deformation rate tensor given by D(U ) = the dynamic viscosity such that ηi in the internal flow η= ηe in the external flow
∇U +(∇U )T 2
, η is
(2)
and γκδΓ nΓ denotes the surface tension contribution at the interface with γ the surface tension coefficient between the two fluids, κ the mean curvature of the interface Γ , δΓ is the Dirac mass on Γ 1 at the interface, δΓ = (3) 0 elsewhere, and nΓ is the unit vector normal to the interface Γ . 2.2 The Level Set method: parametrization of the interface Our objective is to follow the evolution in the time interval (0, T ) of the interface between the two fluids. Several methods could have been chosen (VOF, Lagrangian, Level Set ..). In our work, the interface is modeled by the level function φ(t, x, y, z) [7]. At the initial time, φ is zero at the interface, negative in one phase and positive in the other: < 0 in flow i, (4) φ(0, x, y, z) = > 0 in flow e, 0 at the interface Γ. Its motion is governed by an advection equation ∂t φ + (U.∇) φ = 0 in Ω × (0, T ) φ(t = 0) = φ(0, x, y, z) in Ω
(5)
Such a modeling implies the properties of the Level Set function to be respected at each time step. In particular, the fact that the interface is represented by the zero value of the function φ: ∀t ≥ 0,
Γ (t) = {(x, y, z), φ(t, x, y, z) = 0} .
(6)
When φ is known, the unit normal nΓ at the interface and the curvature κ are computed as follow, ∇φ ∇φ nΓ = and κ = ∇. . (7) |∇φ| φ=0 |∇φ| φ=0
Numerical simulations of three dimensional micro flows
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3 The numerical method Now, we proceed to the discretization of the equations introduced in the previous section. For the presentation of the whole algorithm, we refer the reader to [8]. 3.1 The advection equation The time discretization of the advection equation (5) is explicit and a classical Euler scheme is used: φn+1 = φn − ∆t(U n .∇)φn
(8)
with ∆t the time step, n+1 the new iteration at tn+1 = (n+1)∆t. This choice is associated to a restriction on the time step (the classical CFL condition) in order to ensure the stability and so the convergence. The space discretization is made with a five order WENO scheme [5]. 3.2 The hydrodynamic part The discretization of the incompressible Stokes equations is classical. The finite volume method on structured staggered grids is considered (Patankar, 1980). Although, the Stokes equation is a stationary equation, an explicit time discretization of the capillary unknowns is proposed with the following scheme: ∇.(2η n D(U n+1 )) = ∇P n+1 + γκn δ(φn )∇φn , (9) ∇.U n+1 = 0. In the Stokes equation, the explicit treatment of the term associated to the surface tension required a stability criterion to maintain the convergence of the method. Commonly, the criterion proposed by Brackbill, Kote and Zemach is used [1]. Recently, a less restrictive stability condition was proposed by Galusinski and Vigneaux [2].
4 The Rayleigh-Plateau instability 4.1 Experimental considerations The numerical simulations proposed are based on the following experimental configuration [3], the jet is generated with a cylindrical capillary centred in a square capillary . In this configuration, several kinds of microdroplets can be observed varying the internal or the external flow rate of the fluids.
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Charles-Henri Bruneau, Thierry Colin and Sandra Tancogne
4.2 Jets, droplets and plugs Numerically, the same configuration is adopted. The external microchannel has a square section Sc = 550 µm and the radius of the cylindrical tube is Ri = 105 µm. The inner fluid has a viscosity ηi = 55mP a.s and the outer one ηe = 235mP a.s. The surface tension between the two fluids is γ = 24mN/m. In Fig. 1, the flow rates are the following: (Qi = 7500µL/h, Qe = 6000µL/h), (Qi = 2500µL/h, Qe = 3000µL/h) and (Qi = 2500µL/h, Qe = 5500µL/h). According to these flow rates, differents regimes are respectively observed as in the experiments: an oscillating jet, droplets and a plug.
Fig. 1. From left to right: an oscillating jet, a succession of droplets and a plug.
In microfluidic experiments, the droplets are confined and the surface tension drives their shapes. These droplets are used as microreactors or micromixers. To understand their internal dynamic, it is interesting to know the droplets velocity field in their own referential. The shape of the previous plug is plotted on (Fig.2). It shows the effects due to the square section of the external capillary. In the plane (x,y), the shape of the plug is not anymore circular (Fig. 2 on the right) and the external flow circulates only by the corners of the microchannel.
Fig. 2. Example of the use of droplets as micromixers (shape of the droplets in different slices, velocity field in the droplet frame of reference and few streamlines). Left: shape and velocity field in the plane(x,z); right: view of the plug in the plane(x,y).
Numerical simulations of three dimensional micro flows
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4.3 Discussions In [8], it is proposed an approach based on the linear theory of stability to introduce a stability criterion. Thanks to this one, the flow regime (jet or droplet) can be predicted knowing the dimension of the external capillary and the properties of the two fluids. In addition, the stability length corresponding to the length of the jet just before the creation of the microdroplet can be computed. However, the study is based on the knowledge of the steady state and the perturbation of this one. In order to be as realistic as possible, it could be necessary to take into account the radius of the internal injector. In Fig. 3, three representations of the flow are plotted. The properties of the two fluids are defined in subsection 4.2. The external capillary is about 650µm of section and the diameter of the injector is respectively 60µm, 120µm and 273µm. These numerical results show that there exists a stability area since the stability length is approximately the same in each case. Generally, for this kind of configuration, the jet tends towards the steady state solution before the break-up. In addition, the smallest injector creates the smallest droplet that it is explained by the higher velocity and the fact that the stability length is a little bit higher.
Fig. 3. Relation between the section of the injector and the volume of the created droplet: representation of three different configurations.
5 The particular case of a T-junction The analysis of the break-up of a diphasic jet shows that the Level Set method manages well the topological changes. At the stage, the study of the microdroplets is expanded to an other configuration frequently employed for bench by scientists: the coalescence of the droplets. The technical points (meshing generation and implementation) are presented in [8]. In a T-junction (Fig.4), the continuous phase is injected by two branches of the T and is ejected by the third one.
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Charles-Henri Bruneau, Thierry Colin and Sandra Tancogne
Fig. 4. Coalescence of two microdroplets in a T-junction
6 Conclusion The Level Set method proposed by Osher and Sethian is used in order to follow the interface between two flows such that their movement is mostly governed by the pressure gradient and the surface tension. This method gives results in good agreement with the experiments because the main quantities like the curvature and the unit normal are well computed. This study allows to analyze the breaking jet phenomenon and gives access to quantities like the pressure and the velocity of the droplet when it is created. We take a special care on the representation of the internal capillary employed as an injector and we can in this way compared the volumes of the microdroplets for several radius of injectors. First results concerning the coalescence of two microdroplets in a T-junction are proposed.
References 1. J.U. Brackbill and D.B. Kothe and C. Zemach: A Continuum Method for Modeling Surface Tension, Journal of Computational Physics, Volume 100 , Issue 2, 1992. 2. Galusinski, C.and Vigneaux, P.: Level-Set method and stability condition for curvature-driven flows. C. R. Acad. Sci. Paris, Ser. I 344(11), 703–708, 2007. 3. P. Guillot and A. Colin: Stability of jet in confined pressure-driven biphasic flows at low Reynolds number, Phys. Rev. Lett. 99, 104502 ,2007. 4. P. Guillot, P. Panizza, J.-B. Salmon, M. Joanicot, A. Colin, C.-H. Bruneau and T. Colin: Viscosimeter on a Microfluidic Chip, Langmuir 22 , 2006. 5. G.S. Jiang and C.W Shu: Efficient Implementation of Weighted ENO Schemes. Journal of Computational Physics 126,1996. 6. F. Sarrazin, L. Prat, G. Casamatta, M. Joanicot, C. Gourdon et G. Cristobal: Micro-drops approach in micro-reactors: mixing characterisation, La Houille Blanche, 2006. 7. M. Sussman, P. Smereka and S. Osher: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys textbf114, p. 146– 159, 1994. 8. S. Tancogne: Calcul num´erique et Stabilit´e d’´ecoulements diphasiques tridimensionnels en Microfluidique. Th`ese de Doctorat - Univ. Bordeaux 1 (2007)
