Asymptotic study for Stokes-Brinkman model with jump embedded transmission conditions

International audienceIn this paper, one considers the coupling of a Brinkman model and Stokes equations with jump embedded transmission conditions. In this model, one assumes that the viscosity in the porous region is very small. Then we derive a Wentzel–Kramers–Brillouin (WKB) expansion in power series of the square root of this small parameter for the velocity and the pressure which are solution of the transmission problem. This WKB expansion is justified rigorously by proving uniform errors estimates

Integral potential method for a transmission problem with Lipschitz interface in R^3 for the Stokes and Darcy–Forchheimer–Brinkman PDE systems

The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the non-linear Darcy-Forchheimer-Brinkman system and the linear Stokes system in two complementary Lipschitz domains in R3, one of them is a bounded Lipschitz domain with connected boundary, and the other one is the exterior Lipschitz domain R3 n. We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces

Modélisation mathématique de l'allongement de l'axe antéro-postérieur de l'embryon vertébré

Au cours du développement de l'embryon de vertébré, des milliers de cellules s'auto-organisent suivant une chorégraphie très précise et complexe pour former les tissus et les organes du futur adulte. L'objectif de cette thèse est de comprendre les mécanismes régissant la morphogenèse des tissus qui constituent la partie caudale de l'embryon, en combinant des approches mathématiques, numériques et expérimentales. Dans la première partie, nous développons des modèles mathématiques hydrodynamiques multi-espèces basés sur des équations aux dérivées partielles, pour comprendre comment les tissus neuraux et mésodermiques se forment pendant l'allongement de l'embryon. Ces modèles prennent en compte les propriétés biologiques observées dans l'embryon telles que la croissance et la viscosité des tissus, ainsi que l'interaction inter-tissulaire. En dérivant des problèmes à frontières libres et des problèmes de transmission, nous analysons la dynamique des tissus et obtenons des propriétés qualitatives des modèles mathématiques : saut de pression, effet fantôme. Il émerge de notre analyse théorique des modèles de nouvelles hypothèses biologiques sur les mécanismes cellulaires et tissulaires gouvernant la dynamique tissulaire, que nous confirmons ensuite expérimentalement en intégrant les données biologiques aux modèles mathématiques. Dans la deuxième partie, nous examinons de plus près le comportement des cellules pendant la croissance des embryons d'oiseau. Nous réalisons des expériences biologiques sur des cellules souches pour étudier l'effet de deux protéines dans la spécification et la migration cellulaire. Nous découvrons une zone où les cellules sont spatialement hétérogènes dans leur expression en protéines. Nous développons ensuite un modèle agent-centré qui prend en compte la spécification et la migration des cellules pour comprendre le rôle de cette hétérogénéité. Le modèle met en évidence un paradoxe surprenant : le chaos (l'hétérogénéité) peut soutenir un processus apparemment très ordonné (la morphogenèse). Grâce à une approche interdisciplinaire, notre analyse multi-échelle de la croissance des tissus révèle les mécanismes cellulaires sous-jacents qui régissent la formation des tissus. Les résultats de cette thèse reposent sur le développement d'outils mathématiques adéquats (limite incompressible, problèmes à frontières libres) qui dépassent le cadre de la biologie du développement (mécanique des fluides visqueux, croissance des tissus et oncologie).During vertebrate embryo development, thousands of cells self-organize in a very precise and complex choreography to form the tissues and organs of the future adult. The objective of this thesis is to understand the mechanisms orchestrating the morphogenesis of the tissues which form the caudal part of the embryo, by combining mathematical, numerical, and experimental approaches. In the first part, we develop hydrodynamic multi-species mathematical models based on partial differential equations, to understand how the neural and mesodermal tissues form during embryo elongation. These models consider biological properties observed in the embryo such as tissue growth and viscosity, and inter-tissue interaction. By deriving free boundary problems and transmission problems, we analyse tissue dynamics and obtain qualitative properties of the mathematical models: pressure jump, ghost effect. Biological hypotheses on the cellular and tissular mechanisms governing tissue dynamics emerge from our theoretical analysis of the models, which we then confirm experimentally by integrating the biological data to the mathematical models. In the second part, we take a closer look at the cell behaviour during the growth of bird embryos. We conduct biological experiments on stem-like cells to investigate the effect of two proteins in cell specification and migration. We discover a zone where cells are spatially heterogeneous in their protein expression. We then develop an agent-based model that considers cell specification and migration to understand the role of this heterogeneity. The model highlights a surprising paradox: chaos (heterogeneity) can sustain an apparently very ordered process (morphogenesis). Through an interdisciplinary approach, our multi-scale analysis of tissue growth reveals the underlying cellular mechanisms governing tissue formation. The results in this thesis rely on the development of adequate mathematical tools (incompressible limit, free boundary problems) which go beyond the scope of developmental biology (viscous fluid mechanics, tissue growth and oncology)

Fictitious domain methods to solve convection-diffusion problems with general boundary conditions

International audienceSince a few years, fictitious domain methods have been arising for Computational Fluid Dynamics. The main idea of these methods consists in immersing the original physical domain in a geometrically bigger and simply-shaped other one called fictitious domain. As the spatial discretization is then performed in the fictitious domain, simple structured meshes can be used. The aim of this paper is to solve convection-diffusion problems with fictitious domain methods which can easily simulate free-boundary with possibly deformations of the boundary without increasing the computational cost. Two fictitious domain approaches performing either a spread interface or a thin interface are introduced. These two approaches require neither the modification of the numerical scheme near the immersed interface nor the use of Lagrange multipliers. Several ways to impose general embedded boundary conditions (Dirichlet, Robin or Neumann) are presented. The spread interface approach is computed using a finite element method as a finite volume method is used for the thin interface approach. The numerical schemes conserve the first- order accuracy with respect to the discretization step as observed in the numerical results reported here. The spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vicinity of the immersed boundary. The results obtained are full of promise, more especially as convection-diffusion equations are the core of the resolution of Navier-Stokes equations

Buoyancy-induced heat and mass transport in a porous medium near a buried pipe.

The heat and mass transport in a porous medium induced by buoyancy from a buried heated pipe has been examined in this study. Due to the complexity and irregularity of geometry involved, body-fitted coordinate systems along with finite difference scheme were employed. First, the solutions for conduction and natural convection in a homogeneous porous medium were obtained and compared with the results available in the literature.Another area of interest is to predict how heat and mass transport when there is a breakage in the pipe. Numerical solutions are thus obtained for combined heat and mass transfer by mixed convection induced from a buried pipe with leakage. Two locations of leakage are considered in this study: one is on top of the pipe and the other is at the bottom of the pipe. The effects of Rayleigh number, Peclet number, Lewis number, and buoyancy ratio on the heat and mass transfer results have been examined. The results suggest that both the Nusselt number and Sherwood number increase for the aiding flows and decrease for the opposing flows. For aiding flows, Sherwood number increases with an increase in the Lewis number but Nusselt number behaves otherwise.Realizing that the properties of porous medium immediate around the pipe are usually different from those of the surrounding medium, the objective of this particular study is to investigate how a step change in the permeability of the backfill would affect the flow patterns and heat transfer results. Numerical solutions have been obtained for natural convection in a heterogeneous porous medium induced by a buried heated pipe. The concept of imaginary nodal points has been used to derive the interface conditions. A wide range of governing parameters (e.g., base Rayleigh number and permeability ratio) for various backfill thicknesses have been covered in the computations. It is found that a more permeable backfill can minimize the heat loss and confine the flow to a region near the pipe.Flow visualization experiments were conducted using two Hele-Shaw cells, which simulated a porous medium with distinct permeabilities, subjected to different pipe temperature for both permeable and impermeable top boundaries. Using time-elapsed photographs, it revealed that the flow fields for permeable and impermeable top boundaries displayed distinct characteristics. The flow fields predicted by numerical work for the impermeable top boundary were in good agreement with those observed in experiment

Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare?

A wide variety of techniques have been developed to homogenize transport equations in multiscale and multiphase systems. This has yielded a rich and diverse field, but has also resulted in the emergence of isolated scientific communities and disconnected bodies of literature. Here, our goal is to bridge the gap between formal multiscale asymptotics and the volume averaging theory. We illustrate the methodologies via a simple example application describing a parabolic transport problem and, in so doing, compare their respective advantages/disadvantages from a practical point of view. This paper is also intended as a pedagogical guide and may be viewed as a tutorial for graduate students as we provide historical context, detail subtle points with great care, and reference many fundamental works

Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare?

A wide variety of techniques have been developed to homogenize transport equations in multiscale and multiphase systems. This has yielded a rich and diverse field, but has also resulted in the emergence of isolated scientific communities and disconnected bodies of literature. Here, our goal is to bridge the gap between formal multiscale asymptotics and the volume averaging theory. We illustrate the methodologies via a simple example application describing a parabolic transport problem and, in so doing, compare their respective advantages/disadvantages from a practical point of view. This paper is also intended as a pedagogical guide and may be viewed as a tutorial for graduate students as we provide historical context, detail subtle points with great care, and reference many fundamental works