Microfluics
By a complex fluid, we mean a fluid
containing some mesoscopic objects, that is to say structures whose size is intermediate between the
microscopic size and the macroscopic size of the experiment. This study began with a collaboration
between researchers from the MAB
and the Centre de Recherches Paul
Pascal which is a
laboratory for physical chemistry. Our contacts there were A. Colin, D.
Monin, D. Roux, A.-S.
Wunnenburger.
The aim was to study complex fluids containing surfactants in large quantities. It modifies the viscosity properties of the fluids and surface-tension phenomena can become
predominant. We have worked on foams drainage and on instability of lamellar phases.
A new lab (the LOF) has been built recently in Bordeaux. It is a common lab between the CNRS and Rhodia. One of its goals is to develop
experimental tools in order to use microfluidics in Chemistry. microfluidics is the study of fluids in very small quantities, in micro-channels (a micro-channel is typically 1 cm long with a section of
50 X50 micrometers). They are many advantages of using such channels. First, one needs only a small quantity of liquid to analyze. Furthermore, one can observe very stable flows and quite unusual regimes that allow to make more precise measurements. The idea is to couple numerical simulations with experiments to understand the phenomena, to predict the flows and compute some quantities like viscosity coefficients for example. Flows in micro-channels are often at low Reynolds numbers. The hydrodynamical parts is therefore stable. However, the main problem is to produce real 3 D simulations covering a large range of situations. For example one wants to describe diphasic flows with surface tension and sometimes surface viscosity. Surface tension enforces the stability of the flow. The size of the channel implies that one can observe some very stable phenomena. For example, using a "T" junction, a very stable interface between two fluids can be observed. In a cross junction, one can also have formation of droplets that travel along the channel. Some numerical difficulties arise from the surface tension term. With an explicit discretization of this term, a restrictive stability condition appears for very slow flows.
One of the main point is the wetting phenomena at the boundary. Note that the boundary conditions are fundamental for the description of the flow since the channels are very shallow. The wetting properties can not be neglected at all. Indeed, for the case of a two non-miscible fluids system, if one considers no-slip boundary conditions, then since the interface is driven by the velocity of the fluids, it shall not move on the boundary. The experiments are showing that this is not the case: the interface is moving and in fact all the dynamics starts from the boundary and then propagates in the whole volume of fluids. Even with low Reynolds numbers, the wetting effects can induce instabilities and are responsible of hardly predictable flows. Moreover, the fluids that are used are often visco-elastic and exhibit "unusual" slip length. Therefore, we can not use standard numerical codes and we have to adapt the usual numerical methods to our case to take into account the specificities of our situations. Moreover, we want to obtain reliable models and simulations that can be as simple as possible and that can be used by our collaborators. As a summary, the main specific points of the physics are: the multi-fluid simulations at low Reynolds number, the wetting problems and the surface tension that are crucial, the 3D characteristic of the flows, the boundary conditions that are fundamental due to the size of the channels. We need to handle complex fluids.
are
As an example, a very simple project done with LOF is the micro-rheometer. It deals with a very simple numerical situation. In a "T"-junction, one uses two different fluids. One measures the position of the interface. The fluids are supposed to be Newtonian. Knowing the viscosity of one of the fluids and the flow rates, one would like to deduce the viscosity of the other fluid. To do so, we use an iterative procedure: we start by giving both viscosities, then we compute the velocity fields and we compare the flow rates with the experimental ones. From the mathematical point of view the situation is quite simple: we consider a rectangular channel. The velocity field is supposed to be longitudinal, depending only on the transverse variables. The bifluid problem therefore reduces to a simple transmission problem for an elliptic equation with non continuous diffusion coefficients.
The situation discussed above is obvious from the numerical and mathematical point of view. However, the challenge is to be able to predict the range of parameters in which the coflow will be stable, that is the range of validity of the rheometer. This implies to perform 3 D, time depend simulations involving visco-elastic fluids in "T" junctions, in cross junctions and in "Y" junctions. Once the coflow becomes unstable, droplets are created and they can be used in order to measure some reaction rates or to measure some mixing properties. Microchanels can also be used to simulate experimentally some porous media. The evolution of non-newtonian flows in webs of micro-channels are therefore useful to understand the mixing of oil, water and polymer for enhanced oil recovery for example. Complex fluids arising in cosmetics are also of interest. We also need to handle mixing processes.
My collaborators are G. Cristobal, J.-B. Salmon, M. Joanicot, A. Colin (Rhodia-LoF), C.-H. Bruneau, M. Colin, C. Galusinski, O. Saut.
Recirculation inside a dropplet in a microchannel: A typical profil of a coflow in a microchannel: Simulation of a plug:
Section of a plug in the transverse direction: and in the logitudinal direction:
Tumor growth
The growth of a tumor is also a low Reynolds number flow. Several kind of interfaces are present (membranes, several populations of cells,...) The biological nature of the tissues impose to use different models in order to describe the evolution of tumor growth. The complexity of the geometry, of the rheological properties and the coupling with multi-scale phenomena is high but not far away from those encountered in microfluidics and the models and methods are close.
What are the challenges in this direction? The first one is probably to understand the complexity of the coupling effects between the different levels (cellular, genetic, organs, membranes, molecular). Trying to be exhaustive is of course hopeless, however it is possible numerically to isolate some parts of the evolution in order to better understand the interactions. Another strategy is to test in silico some therapeutic innovations. We have tested the efficacy of radiotherapy of anti-invasive agents is investigated. It is therefore useful to model a tumor growth at several stage of evolution. The macroscopic continuous model is based on Darcy's law which seems to be a good approximation to describe the flow of the tumor cells in the extra-cellular matrix. It is therefore possible to develop a two-dimensional model for the evolution of the cell densities. We formulate mathematically the evolution of the cell densities in the tissue as advection equations for a set of unknowns representing the density of cells with position (x,y) at time t in a given cycle phase. Assuming that all cells move with the same velocity given by Darcy's law and applying the principle of mass balance, one obtains the advection equations with a source term given by a cellular automaton. We assume diffusion for the oxygen and the diffusion constant depends on the density of cells. The source of oxygen corresponds to the spatial location of blood vessels. The available quantities of oxygen interact with the proliferation rate given by the cellular automaton.
One of the main issue is then to couple the system with an angiogenesis process. Of course realistic simulations will be 3D. The 3D model consists of a Stokes system coupled with some transport equations describing the populations of cells. We consider several populations of cells evolving in a cell-cycle model describing mitosis. The evolution inside the cell-cycle gives rise to a non divergence-free velocity field. Again, the system has to be coupled with diffusion of oxygen, but also with membranes that can be degraded biologically. These elastic membranes are handled by a level set version of the immersed boundary method of C. Peskin given by Cottet-Maître. The perspective of development in this direction are of course to increase the biological complexity but also to use more realistic models to describe the mechanics of living tissues and to make comparison with real medical cases. One can think to elasto-visco-plastic models for example.
My collaborators are E. Grenier, D. Bresch, B. Ribba, O. Saut.
Effects of an ellastic membrane on a tumor: Effect of hypoxia:
T. Colin et P. Fabrie, A free boundary problem modeling a foam drainage, Mathematical Models and Methods in Applied Sciences, Vol. 10, No 6 (2000) 945-961, (preprint)
The aim was to study complex fluids containing surfactants in large quantities. It modifies the viscosity properties of the fluids and surface-tension phenomena can become
predominant. We have worked on foams drainage and on instability of lamellar phases.
A new lab (the LOF) has been built recently in Bordeaux. It is a common lab between the CNRS and Rhodia. One of its goals is to develop
experimental tools in order to use microfluidics in Chemistry. microfluidics is the study of fluids in very small quantities, in micro-channels (a micro-channel is typically 1 cm long with a section of
50 X50 micrometers). They are many advantages of using such channels. First, one needs only a small quantity of liquid to analyze. Furthermore, one can observe very stable flows and quite unusual regimes that allow to make more precise measurements. The idea is to couple numerical simulations with experiments to understand the phenomena, to predict the flows and compute some quantities like viscosity coefficients for example. Flows in micro-channels are often at low Reynolds numbers. The hydrodynamical parts is therefore stable. However, the main problem is to produce real 3 D simulations covering a large range of situations. For example one wants to describe diphasic flows with surface tension and sometimes surface viscosity. Surface tension enforces the stability of the flow. The size of the channel implies that one can observe some very stable phenomena. For example, using a "T" junction, a very stable interface between two fluids can be observed. In a cross junction, one can also have formation of droplets that travel along the channel. Some numerical difficulties arise from the surface tension term. With an explicit discretization of this term, a restrictive stability condition appears for very slow flows.
One of the main point is the wetting phenomena at the boundary. Note that the boundary conditions are fundamental for the description of the flow since the channels are very shallow. The wetting properties can not be neglected at all. Indeed, for the case of a two non-miscible fluids system, if one considers no-slip boundary conditions, then since the interface is driven by the velocity of the fluids, it shall not move on the boundary. The experiments are showing that this is not the case: the interface is moving and in fact all the dynamics starts from the boundary and then propagates in the whole volume of fluids. Even with low Reynolds numbers, the wetting effects can induce instabilities and are responsible of hardly predictable flows. Moreover, the fluids that are used are often visco-elastic and exhibit "unusual" slip length. Therefore, we can not use standard numerical codes and we have to adapt the usual numerical methods to our case to take into account the specificities of our situations. Moreover, we want to obtain reliable models and simulations that can be as simple as possible and that can be used by our collaborators. As a summary, the main specific points of the physics are: the multi-fluid simulations at low Reynolds number, the wetting problems and the surface tension that are crucial, the 3D characteristic of the flows, the boundary conditions that are fundamental due to the size of the channels. We need to handle complex fluids.
are
As an example, a very simple project done with LOF is the micro-rheometer. It deals with a very simple numerical situation. In a "T"-junction, one uses two different fluids. One measures the position of the interface. The fluids are supposed to be Newtonian. Knowing the viscosity of one of the fluids and the flow rates, one would like to deduce the viscosity of the other fluid. To do so, we use an iterative procedure: we start by giving both viscosities, then we compute the velocity fields and we compare the flow rates with the experimental ones. From the mathematical point of view the situation is quite simple: we consider a rectangular channel. The velocity field is supposed to be longitudinal, depending only on the transverse variables. The bifluid problem therefore reduces to a simple transmission problem for an elliptic equation with non continuous diffusion coefficients.
The situation discussed above is obvious from the numerical and mathematical point of view. However, the challenge is to be able to predict the range of parameters in which the coflow will be stable, that is the range of validity of the rheometer. This implies to perform 3 D, time depend simulations involving visco-elastic fluids in "T" junctions, in cross junctions and in "Y" junctions. Once the coflow becomes unstable, droplets are created and they can be used in order to measure some reaction rates or to measure some mixing properties. Microchanels can also be used to simulate experimentally some porous media. The evolution of non-newtonian flows in webs of micro-channels are therefore useful to understand the mixing of oil, water and polymer for enhanced oil recovery for example. Complex fluids arising in cosmetics are also of interest. We also need to handle mixing processes.
My collaborators are G. Cristobal, J.-B. Salmon, M. Joanicot, A. Colin (Rhodia-LoF), C.-H. Bruneau, M. Colin, C. Galusinski, O. Saut.
Recirculation inside a dropplet in a microchannel: A typical profil of a coflow in a microchannel: Simulation of a plug:
Section of a plug in the transverse direction: and in the logitudinal direction:
Tumor growth
The growth of a tumor is also a low Reynolds number flow. Several kind of interfaces are present (membranes, several populations of cells,...) The biological nature of the tissues impose to use different models in order to describe the evolution of tumor growth. The complexity of the geometry, of the rheological properties and the coupling with multi-scale phenomena is high but not far away from those encountered in microfluidics and the models and methods are close.
What are the challenges in this direction? The first one is probably to understand the complexity of the coupling effects between the different levels (cellular, genetic, organs, membranes, molecular). Trying to be exhaustive is of course hopeless, however it is possible numerically to isolate some parts of the evolution in order to better understand the interactions. Another strategy is to test in silico some therapeutic innovations. We have tested the efficacy of radiotherapy of anti-invasive agents is investigated. It is therefore useful to model a tumor growth at several stage of evolution. The macroscopic continuous model is based on Darcy's law which seems to be a good approximation to describe the flow of the tumor cells in the extra-cellular matrix. It is therefore possible to develop a two-dimensional model for the evolution of the cell densities. We formulate mathematically the evolution of the cell densities in the tissue as advection equations for a set of unknowns representing the density of cells with position (x,y) at time t in a given cycle phase. Assuming that all cells move with the same velocity given by Darcy's law and applying the principle of mass balance, one obtains the advection equations with a source term given by a cellular automaton. We assume diffusion for the oxygen and the diffusion constant depends on the density of cells. The source of oxygen corresponds to the spatial location of blood vessels. The available quantities of oxygen interact with the proliferation rate given by the cellular automaton.
One of the main issue is then to couple the system with an angiogenesis process. Of course realistic simulations will be 3D. The 3D model consists of a Stokes system coupled with some transport equations describing the populations of cells. We consider several populations of cells evolving in a cell-cycle model describing mitosis. The evolution inside the cell-cycle gives rise to a non divergence-free velocity field. Again, the system has to be coupled with diffusion of oxygen, but also with membranes that can be degraded biologically. These elastic membranes are handled by a level set version of the immersed boundary method of C. Peskin given by Cottet-Maître. The perspective of development in this direction are of course to increase the biological complexity but also to use more realistic models to describe the mechanics of living tissues and to make comparison with real medical cases. One can think to elasto-visco-plastic models for example.
My collaborators are E. Grenier, D. Bresch, B. Ribba, O. Saut.
Effects of an ellastic membrane on a tumor: Effect of hypoxia:
T. Colin et P. Fabrie, A free boundary problem modeling a foam drainage, Mathematical Models and Methods in Applied Sciences, Vol. 10, No 6 (2000) 945-961, (preprint)
A.-S. Wunnenburger, A. Colin, T. Colin, D. Roux, Undulation instability under shear: a model to explain the different orientations of a lamellar phase under shear? Eur. Phys. J. E 2 (2000) 3, 277-283.
P. Guillot, A. Colin, S. Quiniou, G. Cristobal, M. Joanicot, C.-H. Bruneau et T. Colin, Un rhéomètre sur puce microfluidique, La Houille Blanche, No3 2006, (preprint).
B. Ribba, Th. Colin, S. Schnell, A multiscale mathematical model of cancer growth and radiotherapy efficacy: The role of cell cycle regulation in response to irradiation, Theoretical Biology and Medical Modelling 2006, 3:7 (10 Feb 2006) (preprint).
P. Guillot; A. Colin; P. Panizza ; S. Rousseau; Ch. Masselon; M. Joanicot; Ch.-H. Bruneau; Th. Colin, A rheometer on a chip, Spectra-analyse. 2005; 34 (247) : 50-53.
B. Ribba, O. Saut, T. Colin, D. Bresch, E. Grenier, J.P. Boissel, A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents, Journal of Theoretical Biology 243 (2006) 532–541(preprint).
P. Guillot, P. Panizza, J.-B. Salmon, M. Joanicot, A. Colin, C.-H. Bruneau and T. Colin, Viscosimeter on a Microfluidic Chip, Langmuir 2006, 22, 6438-6445.
D. Bresch, T. Colin, E. Grenier, B. Ribba, O. Saut, Modèles avec paramètre d'ordre, level set et interface diffuse : application à la cancérologie, à paraître dans ESAIM:proc, (preprint).
D. Bresch, T. Colin, E. Grenier, B. Ribba, O. Saut, Computation modeling of solid tumor growth: the avascular stage, (preprint).
A. Colin, P. Guillot, C.-H. Bruneau, Th. Colin, S. Tancogne, Stabilité d’écoulements bifluides dans un microcanal, proceeding du congrès français de mécanique 2007 (preprint).
S. Tancogne, C.-H. Bruneau, Th. Colin, A. Colin, P. Guillot, M. Joanicot, Some computational problems in microfluidics, proceeding of the NIMS international workshop on fluid dynamics, june 2007 (preprint).
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