Mathematical and numerical study of nonlinear problems in fluid mechanics
Mathematical modeling and simulation of flow in domains separated by leaky semipermeable membrane including osmotic effect
In this paper, we propose a mathematical model for flow and transport processes of diluted solutions in domains separated by a leaky semipermeable membrane. We formulate transmission conditions for the flow and the solute concentration across the membrane which take into account the property of the membrane to partly reject the solute, the accumulation of rejected solute at the membrane, and the influence of the solute concentration on the volume flow, known as osmotic effect. The model is solved...
Mathematical properties of flows of incompressible power-law-like fluids that are described by implicit constitutive relations.
Mathematical theory for the Ginzburg-Landau approximation in semilinear pattern forming systems with time-periodic forcing applied to electro-convection in nematic liquid crystals
Maximal regularity and viscous incompressible flows with free interface
We consider a free interface problem for the Navier-Stokes equations. We obtain local in time unique existence of solutions to this problem for any initial data and external forces, and global in time unique existence of solutions for sufficiently small initial data. Thanks to global in time maximal regularity of the linearized problem, we can prove a global in time existence and uniqueness theorem by the contraction mapping principle.
Mean field limit for the one dimensional Vlasov-Poisson equation
We consider systems of particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in [Tro86]. Here we shall give a simpler proof of this result, and explain why it implies the so-called “Propagation of molecular chaos”. More precisely, both results will...
Measure attractors for stochastic Navier-Stokes equations.
Medical image – based computational model of pulsatile flow in saccular aneurisms
Saccular aneurisms, swelling of a blood vessel, are investigated in order (i) to estimate the development risk of the wall lesion, before and after intravascular treatment, assuming that the pressure is the major factor, and (ii) to better plan medical interventions. Numerical simulations, using the finite element method, are performed in three-dimensional aneurisms. Computational meshes are derived from medical imaging data to take into account both between-subject and within-subject anatomical...
Medical image – based computational model of pulsatile flow in saccular aneurisms
Saccular aneurisms, swelling of a blood vessel, are investigated in order (i) to estimate the development risk of the wall lesion, before and after intravascular treatment, assuming that the pressure is the major factor, and (ii) to better plan medical interventions. Numerical simulations, using the finite element method, are performed in three-dimensional aneurisms. Computational meshes are derived from medical imaging data to take into account both between-subject and within-subject anatomical...
Method of straight lines for a Bingham problem as a model for the flow of waxy crude oils.
Méthodes multipas pour des équations paraboliques non linéaires.
Méthodes numériques pour les équations de Navier-Stokes instationnaires des fluides visqueux incompressibles
Mixed Finite Element Approximation of the Vector Potential.
Modèles stratifiés en mécanique des fluides géophysiques
Modelling of Cancer Growth, Evolution and Invasion: Bridging Scales and Models
Since cancer is a complex phenomenon that incorporates events occurring on different length and time scales, therefore multiscale models are needed if we hope to adequately address cancer specific questions. In this paper we present three different multiscale individual-cell-based models, each motivated by cancer-related problems emerging from each of the spatial scales: extracellular, cellular or subcellular, but also incorporating relevant information from other levels. We apply these hybrid...
Modelling of convective phenomena in forest fire.
We present a model coupling the fire propagation equations in a bidimensional domain representing the surface, and the air movement equations in a three dimensional domain representing an air layer. As the air layer thickness is small compared with its length, an asymptotic analysis gives a three dimensional convective model governed by a bidimensional equation verified by a stream function. We also present the numerical simulations of these equations.
Modelling of miscible liquids with the Korteweg stress
When two miscible fluids, such as glycerol (glycerin) and water, are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients exist during some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical model consisting of the diffusion equation with convective terms and of the Navier-Stokes equations with the Korteweg stress. We prove the global existence...
Modelling of Miscible Liquids with the Korteweg Stress
When two miscible fluids, such as glycerol (glycerin) and water, are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients exist during some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical model consisting of the diffusion equation with convective terms and of the Navier-Stokes equations with the Korteweg stress. We prove the global...
Mulit-Grid Methods for Stokes and Navier-Stokes Equations. Transforming Smoothers: Algorithms and Numerical Results.
Multimodels for incompressible flows : iterative solutions for the Navier-Stokes / Oseen coupling
In a recent paper [4] we have proposed and analysed a suitable mathematical model which describes the coupling of the Navier-Stokes with the Oseen equations. In this paper we propose a numerical solution of the coupled problem by subdomain splitting. After a preliminary analysis, we prove a convergence result for an iterative algorithm that alternates the solution of the Navier-Stokes problem to the one of the Oseen problem.