The problem of the splitting of a suspension in bifurcating channels dividing into two branches o... more The problem of the splitting of a suspension in bifurcating channels dividing into two branches of non equal flow rates is addressed. As observed for long, in particular in blood flow studies, the volume fraction of particles generally increases in the high flow rate branch and decreases in the other one. In the literature, this phenomenon is sometimes interpreted as the result of some attraction of the particles towards this high flow rate branch. In this paper, we focus on the existence of such an attraction through microfluidic experiments and two-dimensional simulations and show clearly that such an attraction does not occur but is, on the contrary, directed towards the low flow rate branch. Arguments for this attraction are given and a discussion on the sometimes misleading arguments found in the literature is proposed. Finally, the enrichment in particles in the high flow rate branch is shown to be mainly a consequence of the initial distribution in the inlet branch, which sho...
1 - Introduction Red Blood Cells (RBCs) are one of the important components of blood and whose ma... more 1 - Introduction Red Blood Cells (RBCs) are one of the important components of blood and whose main role is to ensure the delivery of oxygen throughout the organism. Their study leads to better understand some cardiovas- cular diseases such as atherosclerosis and drepanocytosis. Thus, it is important to model blood at the cellular level which is, in this particular case, more relevant than to consider blood as a homogeneous fluid (Newtonian or non-Newtonian). In this context, recall that the blood can be seen as a mixture of a Newtonian fluid – the plasma – and several kinds of cells (also called formed elements) : Red Blood Cells, White Blood Cells and Platelets. The global behavior of this complex fluid is obviously non-Newtonian. Among the various blood cells, RBCs are the most numerous as they represent 45% of the total blood volume while white cells and platelets represent less than 1%. Thus, it is reasonable to model blood as dense suspension of RBCs immersed in the plasma. Th...
We present a numerical framework for the simulation of three-dimensional multi-fluid flows based ... more We present a numerical framework for the simulation of three-dimensional multi-fluid flows based on a finite element/level-set approach. The method allows a full Eulerian " tracking " of the interfaces between the fluids, and the properties of the interfaces can be directly taken into account as surface forces. The resolution of the fluid equations and the advection of the interface can be easily decoupled, which enables the use of efficient solving strategies. We also present a 3D benchmark of the rise of a drop in a viscous fluid. We use two different setups and compare our results to previous results obtained with other approaches to validate our method.
In this work, we develop a framework dedicated to the simulation of multi-fluid systems. We prese... more In this work, we develop a framework dedicated to the simulation of multi-fluid systems. We present validations and verifications of these methods on interface capture problems and viscous bubbles simulations.We then show that this framework is well fitted for the simulation of the rigid bodies flow.Next, we extend these methods to the simulation of deformable objects reproducing the behavior of red blood cells: the vesicles. We also validate these simulations.Finally, we apply the previous models to open micro-fluidic problems such as the splitting of a suspension at a bifurcation and the rheology in a confined environment.
Brain perfusion imaging techniques rely on the measurement of spatio-temporal concentration field... more Brain perfusion imaging techniques rely on the measurement of spatio-temporal concentration fields of various endogenous or exogenous tracers in the brain tissue. Their resolution is typically between 1 mm3 Magnetic Resonance Imaging) and (10 mm)3 Positron Emission Tomography). This is much coarser than the diameters of most arterioles and venules, which are typically below 100 μm, and, of course, of capillaries, whose diameters are tenfold smaller. This implies that methods to deduce the regional blood flow rate out of these large-scale concentration fields should rely on upscaled models, i.e. models describing the macro-scale behavior of the vascular system with effective properties taking into account its microstructure. To derive such models, the Volume Averaging Technique, which has been previously developed for upscaling mass transfer in heterogeneous porous media, can be applied to the advection-diffusion equations. Capillary networks indeed exhibit a space filling mesh-like ...
Although direct simulations of the whole cerebral microcirculation are computationally cumbersome... more Although direct simulations of the whole cerebral microcirculation are computationally cumbersome, we will show that they can be carried out on a smaller scale (about 50 to 10000 vessels). For that purpose, we introduce a numerical framework able to solve the fully coupled problem: tissue-diffusion and vessels- advection/diffusion.
We present a numerical method to simulate vesicles in fluid flows. This method consists of writin... more We present a numerical method to simulate vesicles in fluid flows. This method consists of writing all the properties of the membrane as interfacial forces between two fluids. The main advantage of this approach is that the vesicle and the fluid models may be decoupled easily. A level set method has been implemented to track the interface. Finite element discretization has been used with arbitrarily high order polynomial approximation. Several polynomial orders have been tested in order to get a better accuracy. A validation on equilibrium shapes and "tank treading" motion of vesicle have been presented.
1 - Introduction Red Blood Cells (RBCs) are one of the important components of blood and whose ma... more 1 - Introduction Red Blood Cells (RBCs) are one of the important components of blood and whose main role is to ensure the delivery of oxygen throughout the organism. Their study leads to better understand some cardiovas- cular diseases such as atherosclerosis and drepanocytosis. Thus, it is important to model blood at the cellular level which is, in this particular case, more relevant than to consider blood as a homogeneous fluid (Newtonian or non-Newtonian). In this context, recall that the blood can be seen as a mixture of a Newtonian fluid – the plasma – and several kinds of cells (also called formed elements) : Red Blood Cells, White Blood Cells and Platelets. The global behavior of this complex fluid is obviously non-Newtonian. Among the various blood cells, RBCs are the most numerous as they represent 45% of the total blood volume while white cells and platelets represent less than 1%. Thus, it is reasonable to model blood as dense suspension of RBCs immersed in the plasma. This simplification is justified by the predominance of RBCs from the rheological point of view of the blood and its mechanical behavior. As RBCs are very complex entities, it is difficult to model them by taking into account all their details. One approach is to use simpler models such as vesicles which are systems of two fluids separated by a bi-layer membrane of phospholipids. It has been shown shown that vesicles have a mechanical behavior close to the one of RBCs that’s why they have been accepted for many years as a good model for RBCs and they have been widely studied whatsoever experimentally, theoretically or numerically. We present in this work our framework to simulate vesicles by a level set method using finite element approximations and its implementation using FEEL++ library. 2 - Level set description of a two fluids system We have recently developed a new framework to simulate different two fluids systems which can be applied to bubbles, drops or vesicles. Let's define a bounded domain Ω in two or three dimension decomposed into two subdomains Ω1 and Ω2. We denote Γ the interface between the two partitions (This interface will designate later the vesicle membrane). The goal of the level set method is to track implicitly the interface Γ(t) moving at a velocity u. The level set function φ is chosen to be positive in Ω1, negative in Ω2 and zero on Γ. The motion of the interface is then described by the advection of the level set function with a divergence free velocity field u : ∂φ+u·∇φ=0, ∇·u=0. 3 - Vesicles modeling The model usually admitted for vesicle membrane assumes three properties : (i) the membrane has a bending energy, called the Canham, Helfrich energy (ii) the inner fluid is incompressible, so the total surface of the vesicle is conserved and finally (iii) the membrane is quasi inextensible, so the local perimeter is conserved over the time. • The bending energy : it has been shown that the bending energy is proportional to the square of the curvature of the membrane. The bending force is then derived from the bending energy and used as a right hand side for the fluid equations (Stokes or Navier-Stokes). • The membrane inextensibility is equivalent numerically to impose that the surface divergence of the velocity vanishes on the membrane. One can impose this constraint by using a Lagrange multiplier. To do so, we add a Lagrange multiplier to the fluid equations to impose this constraint. 4 - How to properly calculate the curvature The Helfrich energy describing the cost to bend a vesicle is proportional to the square of the curvature of the membrane. The force associated to this energy depends on the 2nd order derivative of the curvature. Recalling that the curvature in level set methods is calculated as the 2nd derivative of the level set field, it appears that this field has to be at least four times differentiable. Thus, a good control on the error associated to this quantity is crucial for this application. Two strategies are presented. The first one consists in doing a smooth projection. All derivatives are computing using L2 projections on which we add a smoothing term. This artificial diffusion makes the field obtained differentiable. We will show that, choosing properly the diffusive parameter coefficient, the L2 error associated to this method is order one regardless of the initial polynomial approximation of the level set field. Thus it is only useful for low polynomial order approximation of φ. The second method consists in expressing φ initially in a high order polynomial basis which makes the derivative possible. We will show that high order derivative is then achieved. However, for a 4th order derivative, φ has to be taken as P5 which can lead to high computational cost. 5 - Numerical simulations and validations In a first part of this work, we present the validation of our model in the case of a single vesicle in a shear flow by exhibiting some known dynamics such as Tank-Treading and…
The problem of the splitting of a suspension in bifurcating channels divided into two branches of... more The problem of the splitting of a suspension in bifurcating channels divided into two branches of non-equal flow rates is addressed. As has long been observed, in particular in blood flow studies, the volume fraction of particles generally increases in the high-flow-rate branch and decreases in the low-flow-rate branch. In the literature, this phenomenon is sometimes interpreted as the result of some attraction of the particles towards this high-flow-rate branch. In this paper, we focus on the existence of such an attraction through microfluidic experiments and two-dimensional simulations and show clearly that such an attraction does not occur but is, on the contrary, directed towards the low-flow-rate branch. Arguments for this attraction are given and a discussion on the sometimes misleading arguments found in the literature is given. Finally, the enrichment of particles in the high-flow-rate branch is shown to be mainly a consequence of the initial distribution in the inlet branc...
... Session GL: Biofluids: Physiological Circulatory I. 8:00 AM–10:10 AM, Monday, November 22, 20... more ... Session GL: Biofluids: Physiological Circulatory I. 8:00 AM–10:10 AM, Monday, November 22, 2010. Long Beach Convention Center Room: 202A Chair: Richard Braun, University of Delaware Abstract ID: BAPS.2010.DFD.GL.2. ... Mourad Ismail (UJF Grenoble). ...
The problem of the splitting of a suspension in bifurcating channels dividing into two branches o... more The problem of the splitting of a suspension in bifurcating channels dividing into two branches of non equal flow rates is addressed. As observed for long, in particular in blood flow studies, the volume fraction of particles generally increases in the high flow rate branch and decreases in the other one. We focus here on the possibility of an attraction of the particles towards one branch through microfluidic experiments and two-dimensional simulations and show that it is directed towards the low flow rate branch. The enrichment in particles in the high flow rate branch is shown to be mainly a consequence of the initial distribution in the inlet branch, which shows necessarily some depletion near the walls.
Level set method seems to be one of the most suitable solution for simulation of two phase flow s... more Level set method seems to be one of the most suitable solution for simulation of two phase flow systems. It can be implemented with different numerical methods (the commonly used one is Finite Differences Method FDM). In this work we chose to use a Finite Elements Method (FEM) to have more flexibility since it opens the way to high order discretization, irregular meshes and even high order geometry to handle large deformations. A model of vesicle using a level set method has already been developed in [1] where the membrane of the vesicle is described only by adding a forcing term to the fluid equations. This model has the advantage to be coupled easily with any fluid solver accepting a forcing term as input. In this paper, we present a new implementation of this model using a finite element approach. Firstly we show a validation of our two phase flow model on a well-known two phase flow system : the rising of a bubble. Then we show that we are able to recover the known equilibrium shapes of vesicles in a fluid at rest.
The problem of the splitting of a suspension in bifurcating channels dividing into two branches o... more The problem of the splitting of a suspension in bifurcating channels dividing into two branches of non equal flow rates is addressed. As observed for long, in particular in blood flow studies, the volume fraction of particles generally increases in the high flow rate branch and decreases in the other one. In the literature, this phenomenon is sometimes interpreted as the result of some attraction of the particles towards this high flow rate branch. In this paper, we focus on the existence of such an attraction through microfluidic experiments and two-dimensional simulations and show clearly that such an attraction does not occur but is, on the contrary, directed towards the low flow rate branch. Arguments for this attraction are given and a discussion on the sometimes misleading arguments found in the literature is proposed. Finally, the enrichment in particles in the high flow rate branch is shown to be mainly a consequence of the initial distribution in the inlet branch, which sho...
1 - Introduction Red Blood Cells (RBCs) are one of the important components of blood and whose ma... more 1 - Introduction Red Blood Cells (RBCs) are one of the important components of blood and whose main role is to ensure the delivery of oxygen throughout the organism. Their study leads to better understand some cardiovas- cular diseases such as atherosclerosis and drepanocytosis. Thus, it is important to model blood at the cellular level which is, in this particular case, more relevant than to consider blood as a homogeneous fluid (Newtonian or non-Newtonian). In this context, recall that the blood can be seen as a mixture of a Newtonian fluid – the plasma – and several kinds of cells (also called formed elements) : Red Blood Cells, White Blood Cells and Platelets. The global behavior of this complex fluid is obviously non-Newtonian. Among the various blood cells, RBCs are the most numerous as they represent 45% of the total blood volume while white cells and platelets represent less than 1%. Thus, it is reasonable to model blood as dense suspension of RBCs immersed in the plasma. Th...
We present a numerical framework for the simulation of three-dimensional multi-fluid flows based ... more We present a numerical framework for the simulation of three-dimensional multi-fluid flows based on a finite element/level-set approach. The method allows a full Eulerian " tracking " of the interfaces between the fluids, and the properties of the interfaces can be directly taken into account as surface forces. The resolution of the fluid equations and the advection of the interface can be easily decoupled, which enables the use of efficient solving strategies. We also present a 3D benchmark of the rise of a drop in a viscous fluid. We use two different setups and compare our results to previous results obtained with other approaches to validate our method.
In this work, we develop a framework dedicated to the simulation of multi-fluid systems. We prese... more In this work, we develop a framework dedicated to the simulation of multi-fluid systems. We present validations and verifications of these methods on interface capture problems and viscous bubbles simulations.We then show that this framework is well fitted for the simulation of the rigid bodies flow.Next, we extend these methods to the simulation of deformable objects reproducing the behavior of red blood cells: the vesicles. We also validate these simulations.Finally, we apply the previous models to open micro-fluidic problems such as the splitting of a suspension at a bifurcation and the rheology in a confined environment.
Brain perfusion imaging techniques rely on the measurement of spatio-temporal concentration field... more Brain perfusion imaging techniques rely on the measurement of spatio-temporal concentration fields of various endogenous or exogenous tracers in the brain tissue. Their resolution is typically between 1 mm3 Magnetic Resonance Imaging) and (10 mm)3 Positron Emission Tomography). This is much coarser than the diameters of most arterioles and venules, which are typically below 100 μm, and, of course, of capillaries, whose diameters are tenfold smaller. This implies that methods to deduce the regional blood flow rate out of these large-scale concentration fields should rely on upscaled models, i.e. models describing the macro-scale behavior of the vascular system with effective properties taking into account its microstructure. To derive such models, the Volume Averaging Technique, which has been previously developed for upscaling mass transfer in heterogeneous porous media, can be applied to the advection-diffusion equations. Capillary networks indeed exhibit a space filling mesh-like ...
Although direct simulations of the whole cerebral microcirculation are computationally cumbersome... more Although direct simulations of the whole cerebral microcirculation are computationally cumbersome, we will show that they can be carried out on a smaller scale (about 50 to 10000 vessels). For that purpose, we introduce a numerical framework able to solve the fully coupled problem: tissue-diffusion and vessels- advection/diffusion.
We present a numerical method to simulate vesicles in fluid flows. This method consists of writin... more We present a numerical method to simulate vesicles in fluid flows. This method consists of writing all the properties of the membrane as interfacial forces between two fluids. The main advantage of this approach is that the vesicle and the fluid models may be decoupled easily. A level set method has been implemented to track the interface. Finite element discretization has been used with arbitrarily high order polynomial approximation. Several polynomial orders have been tested in order to get a better accuracy. A validation on equilibrium shapes and "tank treading" motion of vesicle have been presented.
1 - Introduction Red Blood Cells (RBCs) are one of the important components of blood and whose ma... more 1 - Introduction Red Blood Cells (RBCs) are one of the important components of blood and whose main role is to ensure the delivery of oxygen throughout the organism. Their study leads to better understand some cardiovas- cular diseases such as atherosclerosis and drepanocytosis. Thus, it is important to model blood at the cellular level which is, in this particular case, more relevant than to consider blood as a homogeneous fluid (Newtonian or non-Newtonian). In this context, recall that the blood can be seen as a mixture of a Newtonian fluid – the plasma – and several kinds of cells (also called formed elements) : Red Blood Cells, White Blood Cells and Platelets. The global behavior of this complex fluid is obviously non-Newtonian. Among the various blood cells, RBCs are the most numerous as they represent 45% of the total blood volume while white cells and platelets represent less than 1%. Thus, it is reasonable to model blood as dense suspension of RBCs immersed in the plasma. This simplification is justified by the predominance of RBCs from the rheological point of view of the blood and its mechanical behavior. As RBCs are very complex entities, it is difficult to model them by taking into account all their details. One approach is to use simpler models such as vesicles which are systems of two fluids separated by a bi-layer membrane of phospholipids. It has been shown shown that vesicles have a mechanical behavior close to the one of RBCs that’s why they have been accepted for many years as a good model for RBCs and they have been widely studied whatsoever experimentally, theoretically or numerically. We present in this work our framework to simulate vesicles by a level set method using finite element approximations and its implementation using FEEL++ library. 2 - Level set description of a two fluids system We have recently developed a new framework to simulate different two fluids systems which can be applied to bubbles, drops or vesicles. Let's define a bounded domain Ω in two or three dimension decomposed into two subdomains Ω1 and Ω2. We denote Γ the interface between the two partitions (This interface will designate later the vesicle membrane). The goal of the level set method is to track implicitly the interface Γ(t) moving at a velocity u. The level set function φ is chosen to be positive in Ω1, negative in Ω2 and zero on Γ. The motion of the interface is then described by the advection of the level set function with a divergence free velocity field u : ∂φ+u·∇φ=0, ∇·u=0. 3 - Vesicles modeling The model usually admitted for vesicle membrane assumes three properties : (i) the membrane has a bending energy, called the Canham, Helfrich energy (ii) the inner fluid is incompressible, so the total surface of the vesicle is conserved and finally (iii) the membrane is quasi inextensible, so the local perimeter is conserved over the time. • The bending energy : it has been shown that the bending energy is proportional to the square of the curvature of the membrane. The bending force is then derived from the bending energy and used as a right hand side for the fluid equations (Stokes or Navier-Stokes). • The membrane inextensibility is equivalent numerically to impose that the surface divergence of the velocity vanishes on the membrane. One can impose this constraint by using a Lagrange multiplier. To do so, we add a Lagrange multiplier to the fluid equations to impose this constraint. 4 - How to properly calculate the curvature The Helfrich energy describing the cost to bend a vesicle is proportional to the square of the curvature of the membrane. The force associated to this energy depends on the 2nd order derivative of the curvature. Recalling that the curvature in level set methods is calculated as the 2nd derivative of the level set field, it appears that this field has to be at least four times differentiable. Thus, a good control on the error associated to this quantity is crucial for this application. Two strategies are presented. The first one consists in doing a smooth projection. All derivatives are computing using L2 projections on which we add a smoothing term. This artificial diffusion makes the field obtained differentiable. We will show that, choosing properly the diffusive parameter coefficient, the L2 error associated to this method is order one regardless of the initial polynomial approximation of the level set field. Thus it is only useful for low polynomial order approximation of φ. The second method consists in expressing φ initially in a high order polynomial basis which makes the derivative possible. We will show that high order derivative is then achieved. However, for a 4th order derivative, φ has to be taken as P5 which can lead to high computational cost. 5 - Numerical simulations and validations In a first part of this work, we present the validation of our model in the case of a single vesicle in a shear flow by exhibiting some known dynamics such as Tank-Treading and…
The problem of the splitting of a suspension in bifurcating channels divided into two branches of... more The problem of the splitting of a suspension in bifurcating channels divided into two branches of non-equal flow rates is addressed. As has long been observed, in particular in blood flow studies, the volume fraction of particles generally increases in the high-flow-rate branch and decreases in the low-flow-rate branch. In the literature, this phenomenon is sometimes interpreted as the result of some attraction of the particles towards this high-flow-rate branch. In this paper, we focus on the existence of such an attraction through microfluidic experiments and two-dimensional simulations and show clearly that such an attraction does not occur but is, on the contrary, directed towards the low-flow-rate branch. Arguments for this attraction are given and a discussion on the sometimes misleading arguments found in the literature is given. Finally, the enrichment of particles in the high-flow-rate branch is shown to be mainly a consequence of the initial distribution in the inlet branc...
... Session GL: Biofluids: Physiological Circulatory I. 8:00 AM–10:10 AM, Monday, November 22, 20... more ... Session GL: Biofluids: Physiological Circulatory I. 8:00 AM–10:10 AM, Monday, November 22, 2010. Long Beach Convention Center Room: 202A Chair: Richard Braun, University of Delaware Abstract ID: BAPS.2010.DFD.GL.2. ... Mourad Ismail (UJF Grenoble). ...
The problem of the splitting of a suspension in bifurcating channels dividing into two branches o... more The problem of the splitting of a suspension in bifurcating channels dividing into two branches of non equal flow rates is addressed. As observed for long, in particular in blood flow studies, the volume fraction of particles generally increases in the high flow rate branch and decreases in the other one. We focus here on the possibility of an attraction of the particles towards one branch through microfluidic experiments and two-dimensional simulations and show that it is directed towards the low flow rate branch. The enrichment in particles in the high flow rate branch is shown to be mainly a consequence of the initial distribution in the inlet branch, which shows necessarily some depletion near the walls.
Level set method seems to be one of the most suitable solution for simulation of two phase flow s... more Level set method seems to be one of the most suitable solution for simulation of two phase flow systems. It can be implemented with different numerical methods (the commonly used one is Finite Differences Method FDM). In this work we chose to use a Finite Elements Method (FEM) to have more flexibility since it opens the way to high order discretization, irregular meshes and even high order geometry to handle large deformations. A model of vesicle using a level set method has already been developed in [1] where the membrane of the vesicle is described only by adding a forcing term to the fluid equations. This model has the advantage to be coupled easily with any fluid solver accepting a forcing term as input. In this paper, we present a new implementation of this model using a finite element approach. Firstly we show a validation of our two phase flow model on a well-known two phase flow system : the rising of a bubble. Then we show that we are able to recover the known equilibrium shapes of vesicles in a fluid at rest.
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Papers by Vincent Doyeux