Irreversibility and Chaos in Active Particle Suspensions

Active matter has been the object of huge amount of research in recent years for its important fundamental and applicative properties. In this paper we investigate active suspensions of micro-swimmers through direct numerical simulation, so that no approximation is made at the continuous level other than the numerical one. We consider both pusher and puller organisms, with a spherical or ellipsoidal shape. We analyse the velocity and the characteristic scales for an homogeneous two-dimensional suspension and the effective viscosity under shear. We bring evidences that the complex features displayed are related to a spontaneous breaking of the time-reversal symmetry. We show that chaos is not a key ingredient, whereas a large enough number of interacting particles and a non-spherical shape are needed to break the symmetry and are therefore at the basis of the phenomenology. Our numerical study also shows that pullers display some collective motion, though with different characteristics from pushers

A smooth extension method for transmission problems

In this work, we present a numerical method for the resolution of transmission problems with non-conformal meshes which preserves the optimal rates of convergence in space. The smooth extension method is a fictitious domain approach based on a control formulation stated as a minimization problem, that we prove to be equivalent to the initial transmission problem. Formulated as a minimization problem, the transmission problem can be solved with standard finite element function spaces and usual optimization algorithms. The method is applied to different transmission problems (Laplace, Stokes and a fluid-structure interaction problem) and compared to standard finite element methods

Existence and uniqueness for a quasi-static interaction problem between a viscous fluid and an active structure

We consider a quasi-static fluid-structure interaction problem where the fluid is modeled by theStokes equations and the structure is an active and elastic medium. More precisely, the displacementof the structure verifies the equations of elasticity with an active stress, which models the presenceof internal biological motors in the structure. Under smallness assumptions on the data, we provethe existence of a unique solution for this strongly coupled system

Active structures in a viscous fluid : model, mathematical analysis and numerical simulations

Le transport de micro-organismes et de fluides biologiques au moyen de cils et flagelles est un phénomène universel que l’on retrouve chez presque tous les êtres vivants. Le but de cette thèse est la modélisation, l’analyse mathématique et la simulation numérique de problèmes d’interaction fluide-structure qui font intervenir des structures actives, capables de se déformer d’elles-mêmes grâce à des contraintes internes, et un fluide à faible nombre de Reynolds, modélisé par les équations de Stokes. Le Chapitre 2 traite de la modélisation de ces structures actives en considérant la loi de Saint Venant-Kirchhoff dans les équations de l’élasticité et en ajoutant un terme d’activité au second tenseur de contraintes de Piola-Kirchhoff. Les équations fluide et structures sont couplées à l’interface fluide-structure et l’étude mathématique d’un problème linéarisé et discrétisé en temps est ensuite réalisée. Une reformulation sous forme d’un problème point-selle est proposée et utilisée pour la simulation numérique du problème. Le Chapitre 3 s’intéresse à l’analyse du problème d’interaction fluide-structure quasi-statique avec une structure active, pour lequel nous montrons l’existence et l’unicité, pour des données petites, d’une solution forte localement en temps. Le Chapitre 4 présente une nouvelle méthode de type domaine fictif (la méthode de prolongement régulier ) pour la résolution numérique de problèmes de transmission. La méthode est d’abord développée pour un problème de transmission de Laplace, puis étendue aux problèmes de transmission de Stokes et d’interaction fluide-structure.The transport of microorganisms and biological fluids by means of cilia and flagella is an universal phenomenon found in almost all living beings. The aim of this thesis is to model, analyze and simulate mathematical fluid-structure interaction problems involving active structures, capable of deforming themselves through internal stresses, and a low Reynolds number fluid, modeled by Stokes equations. In Chapter 2, these active structures are modeled as elastic materials satisfying Saint Venant-Kirchhoff law for elasticity whose activity comes from the addition of an activity term to the second Piola-Kirchhoff stress tensor. Elasticity and Stokes equations are coupled on the fluid-structure interface and the mathematical study of the linearized problem discretized in time is realized. Then, the problem is formulated as a saddle-point problem which isused for numerical simulations. Chapter 3 focuses on the analysis of a quasi-static fluid-structure with an active structure, for which we show existence and uniqueness, for small data, of a strong solution locally in time. Chapter 4 presents a new fictitious domain method (the smooth extension method) for the numerical resolution of transmission problems. The method is first developed for a Laplace transmission problem and further extended to Stokes transmission and fluid-structure interaction problems

Structures actives dans un fluide visqueux : modélisation, analyse mathématique et simulations numériques

The transport of microorganisms and biological fluids by means of cilia and flagella is an universal phenomenon found in almost all living beings. The aim of this thesis is to model, analyze and simulate mathematical fluid-structure interaction problems involving active structures, capable of deforming themselves through internal stresses, and a low Reynolds number fluid, modeled by Stokes equations. In Chapter 2, these active structures are modeled as elastic materials satisfying Saint Venant-Kirchhoff law for elasticity whose activity comes from the addition of an activity term to the second Piola-Kirchhoff stress tensor. Elasticity and Stokes equations are coupled on the fluid-structure interface and the mathematical study of the linearized problem discretized in time is realized. Then, the problem is formulated as a saddle-point problem which isused for numerical simulations. Chapter 3 focuses on the analysis of a quasi-static fluid-structure with an active structure, for which we show existence and uniqueness, for small data, of a strong solution locally in time. Chapter 4 presents a new fictitious domain method (the smooth extension method) for the numerical resolution of transmission problems. The method is first developed for a Laplace transmission problem and further extended to Stokes transmission and fluid-structure interaction problems.Le transport de micro-organismes et de fluides biologiques au moyen de cils et flagelles est un phénomène universel que l’on retrouve chez presque tous les êtres vivants. Le but de cette thèse est la modélisation, l’analyse mathématique et la simulation numérique de problèmes d’interaction fluide-structure qui font intervenir des structures actives, capables de se déformer d’elles-mêmes grâce à des contraintes internes, et un fluide à faible nombre de Reynolds, modélisé par les équations de Stokes. Le Chapitre 2 traite de la modélisation de ces structures actives en considérant la loi de Saint Venant-Kirchhoff dans les équations de l’élasticité et en ajoutant un terme d’activité au second tenseur de contraintes de Piola-Kirchhoff. Les équations fluide et structures sont couplées à l’interface fluide-structure et l’étude mathématique d’un problème linéarisé et discrétisé en temps est ensuite réalisée. Une reformulation sous forme d’un problème point-selle est proposée et utilisée pour la simulation numérique du problème. Le Chapitre 3 s’intéresse à l’analyse du problème d’interaction fluide-structure quasi-statique avec une structure active, pour lequel nous montrons l’existence et l’unicité, pour des données petites, d’une solution forte localement en temps. Le Chapitre 4 présente une nouvelle méthode de type domaine fictif (la méthode de prolongement régulier ) pour la résolution numérique de problèmes de transmission. La méthode est d’abord développée pour un problème de transmission de Laplace, puis étendue aux problèmes de transmission de Stokes et d’interaction fluide-structure

A continuum active structure model for the interaction of cilia with a viscous fluid

International audienceCilia and flagella are motile elongated structures, involved in swimming and/or transport mechanisms that arise in many living organisms. Flagella are used by micro-swimmers such as sperm-cells or bacteria for motility purpose at low Reynolds number, while cilia are involved in the transport of proteins, nutrients or dust inside bigger organisms. At the origin of all these mechanisms are two essential ingredients: the capacity for cilia and flagella to modify their shapes by generating internal stresses and the strong reciprocal interaction with the surrounding fluid. Both aspects have been studied in several works, with very different strategies. Cilia can either be modeled as 1D elastic structures with self-oscillatory [1] and sliding regulation mechanisms [2] or as 3D structures with a discrete representation of their internal biological components [3]. In the first case, the coupling with the surrounding 3D fluid is often taken into account (numerically) with the slender body theory [4]. In the second case, the fluid-structure interaction is well resolved but the (discrete) model for cilia is not suitable for the mathematical analysis and introduces many parameters that may not be accessible in experiments. Unlike all previous works on cilia and flagella, we propose a model that fits in the framework of continuum mechanics. In the context of 2D or 3D elasticity, the model is based upon the definition of a suitable Piola-Kirchoff tensor mimicking the action of the internal components that induce the motility of the structure. Moreover, the framework of continuum mechanics enables to fully consider the strong interaction with the surrounding fluid. During this presentation, we will show that the present model is suitable for both the mathematical study and the numerical simulation of fluid-structure interaction problems involving active structures and low Reynold number flows. We shall also discuss the question of the identification of the internal activity

A continuum active structure model for the interaction of cilia with a viscous fluid

International audienceThis paper presents a modeling, analysis and simulation of a fluid-structure interaction model with an active thin structure, reproducing the behaviour of cilia or flagella immersed in a viscous flow. In the context of linear or nonlinear elasticity, the model is based upon the definition of a suitable internal Piola-Kirchoff tensor mimicking the action of the internal dyneins that induce the motility of the structure. In the subsequent fluid-structure interaction problem, two difficulties arise: on the one hand the internal activity of the structure which leads to more restricted well-posedness conditions and, on the other hand, the coupling conditions between the fluid and the structure that require a specific numerical treatment. In the context of numerical simulation, a weak formulation of the time-discretized problem is derived in functional frameworks that include the coupling conditions but, for numerical purposes, an equivalent formulation using Lagrange multipliers is introduced in order to get rid of the constraints in the functional spaces: this new formulation allows for the use of standard (fluid and structure) solvers, up to an iterative procedure. Numerical simulations are presented, including the beating of one or two cilia in 2d, discussing the competition between the magnitude of the internal activity and the viscosity of the surrounding fluid