Méthodes numériques pour l'écoulement de Stokes 3D: fluides à viscosité variable en géométrie complexe mobile ; application aux fluides biologiques

This work intends to provide numerical methods to solve the Stokes problem in a complex geometry for nonhomogeneous fluids. This model describes highly viscous flows, incompressible, whose viscosity is not uniform, depending on a certain agent concentration. From a mathematical point of vue, it is an elliptic problem coupled to a convection-diffusion equation, generating a non linear dynamics. The computational algorithm is based on an hybrid grid-particles discretization. A fractional-steps algorithm enable to separate the computation of the different phenomena involved, to take advantage of both these discretizations: Lagrangian methods are adapted to solve convection operators and Eulerian methods are used for diffusion. A penalization method enables an efficient treatment of the interaction between the fluid and the domain's complex moving geometry. An iterative projection method is developed for this quasi-static problem, it permits the use of fast solvers compatible with large dimension problems. Several test are presented to validate the convergence, the conservation and the performances of the algorithm in 3D. The context of this work is the mucus flow in the lung, around the epithelium ciliated cells covering bronchus. The study focuses on the efficiency of mucus transport, which captures and expectorate pathogen agents, with respect to biological parameters. Other simulations of a micro-swimmer and a porous media flow supplement this study.Ce travail propose des méthodes numériques pour la résolution du problème de Stokes en géométrie complexe pour des fluides non homogènes. Ce modèle décrit l'écoulement d'un fluide très visqueux, incompressible, dont la viscosité n'est pas uniforme mais dépend de la concentration d'un certain agent. D'un point de vue mathématique, il s'agit de résoudre un problème elliptique couplé à une équation de convection-diffusion, qui génèrent une dynamique non linéaire. L'algorithme de résolution est basé sur une discrétisation hybride utilisant une grille et des particules. Des algorithmes à pas fractionnaires permettent de séparer la résolution des différents phénomènes pour profiter des avantages spécifiques à ces discrétisations: méthodes lagrangiennes adaptées à la convection et méthodes eulériennes pour la diffusion. Une méthode de pénalisation permet de gérer efficacement l'interaction entre le fluide et la géométrie mobile du domaine. Une méthode de projection itérative est développée pour ce problème quasi-statique, cela permet d'utiliser des solveurs rapides propices aux calculs en grande dimension. Plusieurs cas tests viennent valider la convergence, la conservation et les performances de l'algorithme en 3D. Ce travail s'inscrit dans le contexte de l'étude de l'écoulement du mucus pulmonaire autour des cellules épithéliales ciliées tapissant les bronches. L'efficacité du transport du mucus, assurant la capture et l'expectoration des agents pathogènes, est étudiée en fonction des paramètres biologiques. D'autres simulations d'un micro-nageur et d'écoulements en milieux poreux complètent cette étude

Mucus and ciliated cells of human lung : splitting strategies for particle methods and 3D stokes flows

Lung walls are covered by a film of mucus, whose motility is fundamental for a healthy behavior. Indeed, mucus traps inhaled aerosols (bacteria, dust, ...), and moves from smallest to largest airways, until it reaches esophagus where is it swallowed or expectorated. A lot of biological parameters are responsible for mucus motion [6], such as the vibrations of ciliated cells covering lung walls (cilia height, frequency, ...), mucus/air interaction, water saturation in mucin network, mucus thickness

Effective viscosity of a random mixture of fluids

International audienceWe propose an estimation of the effective viscosity of a random mixture of Newtonian fluids that ignores capillary effects. The local viscosity of the mixture is assumed to be a random function of the position. Using perturbation expansions up to the second order, the resulting formula can be recast under the form of a simple power averaging mixing low. Numerical tests are used to assess the validity of the formula and the range of its applicability

Numerical and experimental investigation of mucociliary clearance breakdown in cystic fibrosis

The human tracheobronchial tree surface is covered with mucus. A healthy mucus is a heterogeneous material flowing toward the esophagus and a major defense actor against local pathogen proliferation and pollutant deposition. An alteration of mucus or its environment such as in cystic fibrosis dramatically impacts the mucociliary clearance. In the present study, we investigate the mechanical organization and the physics of such mucus in human lungs by means of a joint experimental and numerical work. In particular, we focus on the influence of the shear-thinning mucus mobilized by a ciliated epithelium for mucociliary clearance. The proposed robust numerical method is able to manage variations of more than 5 orders of magnitude in the shear rate and viscosity. It leads to a cartography that allows to discuss major issues on defective mucociliary clearance in cystic fibrosis. Furthermore, the computational rheological analysis based on measurements shows that cystic fibrosis shear-thinning mucus tends to aggregate in regions of lower clearance. Yet, a rarefaction of periciliary fluid has a greater impact than the mucus shear-thinning effects

Une stabilisation efficace de l'élément fini P1/P1 en grandes transformations

L'objectif de ce travail est de proposer un stabilisation robuste de l'élément fini P1/P1 en grandes transformations. La première partie est consacrée à l'étude des propriétés de l'élément fini P1/P1 connu pour ne pas être LBB-stable. Ensuite, une nouvelle formulation stabilisée simple et robuste est proposée afin d'éviter les modes de pression parasites pour l'analyse de structures métalliques élasto-plastiques ou élasto-viscoplastiques. Enfin, des exemples sont présentés pour illustrer l'efficacité de l'approche développée

Hybrid grid–particle methods and Penalization: A Sherman–Morrison–Woodbury approach to compute 3D viscous flows using FFT

International audienceParticle methods are very convenient to compute transport equations in fluid mechanics as their computational cost is linear and they are not limited by convection stability conditions. To achieve large 3D computations the method must be coupled to efficient algorithms for velocity computations, including a good treatment of non-homogeneities and complex moving geometries. The Penalization method enables to consider moving bodies interaction by adding a term in the conservation of momentum equation. This work introduces a new computational algorithm to solve implicitly in the same step the Penalization term and the Laplace operators, since explicit computations are limited by stability issues, especially at low Reynolds number. This computational algorithm is based on the Sherman-Morrison-Woodbury formula coupled to a GMRES iterative method to reduce the computations to a sequence of Poisson problems: this allows to formulate a penalized Poisson equation as a large perturbation of a standard Poisson, by means of algebraic relations. A direct consequence is the possibility to use fast solvers based on Fast Fourier Transforms for this problem with good efficiency from both the computational and the memory consumption point of views, since these solvers are recursive and they do not perform any matrix assembling. The resulting fluid mechanics computations are very fast and they consume a small amount of memory, compared to a reference solver or a linear system resolution. The present applications focus mainly on a coupling between transport equation and 3D Stokes equations, for studying biological organisms motion in a highly viscous flows with variable viscosity

A parametric study of mucociliary transport by numerical simulations of 3D non-homogeneous mucus

International audienceMucus 3D computations of non-homogeneous flows Fluid-structure interaction in complex geometry a b s t r a c t Mucociliary clearance is the natural flow of the mucus which covers and protects the lung from the outer world. Pathologies, like cystic fibrosis, highly change the biological parameters of the mucus flow leading to stagnation situations and pathogens proliferation. As the lung exhibits a complex dyadic structure, in-vivo experimental study of mucociliary clearance is almost impossible and numerical simulations can bring important knowledge about this biological flow. This paper brings a detailed study of the biological parameters influence on the mucociliary clearance, in particular for pathological situations such as cystic fibrosis. Using recent suitable numerical methods, a non-homogeneous mucus flow (including non-lin-earities) can be simulated efficiently in 3D, allowing the identification of the meaningful parameters involved in this biological flow. Among these parameters, it is shown that the mucus viscosity, the stiffness transition between pericilliary fluid and mucus, the pericilliary fluid height as well as both cilia length and beating frequency have a great influence on the mucociliary transport

Hybrid grid–particle methods and Penalization: A Sherman–Morrison–Woodbury approach to compute 3D viscous flows using FFT

International audienceParticle methods are very convenient to compute transport equations in fluid mechanics as their computational cost is linear and they are not limited by convection stability conditions. To achieve large 3D computations the method must be coupled to efficient algorithms for velocity computations, including a good treatment of non-homogeneities and complex moving geometries. The Penalization method enables to consider moving bodies interaction by adding a term in the conservation of momentum equation. This work introduces a new computational algorithm to solve implicitly in the same step the Penalization term and the Laplace operators, since explicit computations are limited by stability issues, especially at low Reynolds number. This computational algorithm is based on the Sherman-Morrison-Woodbury formula coupled to a GMRES iterative method to reduce the computations to a sequence of Poisson problems: this allows to formulate a penalized Poisson equation as a large perturbation of a standard Poisson, by means of algebraic relations. A direct consequence is the possibility to use fast solvers based on Fast Fourier Transforms for this problem with good efficiency from both the computational and the memory consumption point of views, since these solvers are recursive and they do not perform any matrix assembling. The resulting fluid mechanics computations are very fast and they consume a small amount of memory, compared to a reference solver or a linear system resolution. The present applications focus mainly on a coupling between transport equation and 3D Stokes equations, for studying biological organisms motion in a highly viscous flows with variable viscosity

A Hybrid Grid-Particle Method for Moving Bodies in 3D Stokes Flow with Variable Viscosity

International audienceThis article presents a new approach for the resolution of large three-dimensional Stokes equations with variable viscosity fluids, coupled with transport equations. After building the model, we will write these equations in the context of highly viscous flows and penalization, in order to consider complex geometries moving in a fluid. From a mathematical point of view, the solutions show nonlinear dynamics. Beside the use of standard tools such as finite differences and staggered grids, we have built a new methodology based on large three-dimensional simulations, including operators splitting for an efficient use of fast solvers, multi-index fixed point methods, Lagrangian methods with fast and accurate grid-particle transfers, and the multiresolution description of variables. Among the main original aspects of this method, both accurate incompressibility and variable viscosity are treated in the same fixed point. Hence the computation costs for variable and constant viscosity flows are similar. Several examples are given to validate the order of convergence and conservation rates. Such models are used, among other examples, in biological computing at the cellular scale. The present article eventually describes the ciliated epithelium cells covering a mammal's lungs, beating in a mucus film. This study of human lung diseases explores the efficiency of the mucociliary clearance, a challenging problem in health sciences, especially for the investigation of cystic fibrosis and various chronic obstructive pulmonary diseases

Hybrid grid–particle methods and Penalization: A Sherman–Morrison–Woodbury approach to compute 3D viscous flows using FFT

International audienceParticle methods are very convenient to compute transport equations in fluid mechanics as their computational cost is linear and they are not limited by convection stability conditions. To achieve large 3D computations the method must be coupled to efficient algorithms for velocity computations, including a good treatment of non-homogeneities and complex moving geometries. The Penalization method enables to consider moving bodies interaction by adding a term in the conservation of momentum equation. This work introduces a new computational algorithm to solve implicitly in the same step the Penalization term and the Laplace operators, since explicit computations are limited by stability issues, especially at low Reynolds number. This computational algorithm is based on the Sherman-Morrison-Woodbury formula coupled to a GMRES iterative method to reduce the computations to a sequence of Poisson problems: this allows to formulate a penalized Poisson equation as a large perturbation of a standard Poisson, by means of algebraic relations. A direct consequence is the possibility to use fast solvers based on Fast Fourier Transforms for this problem with good efficiency from both the computational and the memory consumption point of views, since these solvers are recursive and they do not perform any matrix assembling. The resulting fluid mechanics computations are very fast and they consume a small amount of memory, compared to a reference solver or a linear system resolution. The present applications focus mainly on a coupling between transport equation and 3D Stokes equations, for studying biological organisms motion in a highly viscous flows with variable viscosity