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

Abstract : 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.
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Fabien Vergnet. Structures actives dans un fluide visqueux : modélisation, analyse mathématique et simulations numériques. Equations aux dérivées partielles [math.AP]. Université Paris-Saclay, 2019. Français. ⟨NNT : 2019SACLS169⟩. ⟨tel-02194265⟩

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