The Generalized Graetz Problem in Finite Domains

We consider the generalized Graetz problem associated with stationary convection-diffusion inside a domain having any regular three-dimensional translationally invariant section and finite or semi-infinite extent. Our framework encompasses any previous “extended” and “conjugated” Graetz generalizations and provides theoretical bases for computing the orthogonal set of generalized two-dimensional Graetz modes. The theoretical framework includes both heterogeneous and possibly anisotropic diffusion tensors. In the case of semi-infinite domains, the existence of a bounded solution is shown from the analysis of two-dimensional operator eigenvectors which form a basis of L2 . In the case of finite domains a similar basis can be exhibited, and the mode’s amplitudes can be obtained from the inversion of newly defined finite domain operator. Our analysis includes both the theoretical and practical issues associated with this finite domain operator inversion as well as its interpretation as a multireflection image method. Error estimates are provided when numerically truncating the spectrum to a finite number of modes. Numerical examples are validated for reference configurations and provided in nontrivial cases. Our methodology shows how to map the solution of stationary convection-diffusion problems in finite three-dimensional domains into a two-dimensional operator spectrum, which leads to a drastic reduction in computational cost

Numerical computation of 3D heat transfer in complex parallel heat exchangers using generalized Graetz modes

We propose and develop a variational formulation dedicated to the simulation of parallel convective heat exchanger that handles possibly complex input/output conditions as well as connection between pipes. It is based on a spectral method that allows to re-cast three-dimensional heat exchangers into a two-dimensional eigenvalue problem, named the generalized Graetz problem. Our formulation handles either convective, adiabatic, or prescribed temperature at the entrance or at the exit of the exchanger. This formulation is robust to mode truncation, offering a huge reduction in computational cost, and providing insights into the most contributing structure to exchanges and transfer. Several examples of heat exchangers are analyzed, their numerical convergence is tested and the numerical efficiency of the approach is illustrated in the case of Poiseuille flow in tubes

Imaging by modification: numerical reconstruction of local conductivities from corresponding power density measurements

International audienceWe discuss the reconstruction of the impedance from the local power density. This study is motivated by a new imaging principle which allows to recover interior measurements of the energy density by a non invasive method. We discuss the theoretical feasibility in two dimensions, and propose numerical algorithms to recover the conductivity in two and three dimension. The efficiency of this approach is documented by several numerical simulation

Analyse de la convection-diffusion entre deux tubes parallèles plongés dans un domaine cylindrique

Nous étudions la convection-diffusion tri-dimensionelle entre tubes parallèles par une formulation théorique bi-dimensionnelle précédement proposée. L’implémentation de cette formulation bi-dimensionnelle par éléments finis permet de calculer une vaste classe de configurations physique, hydrodynamiques et géométriques. Nous nous attachons à l’étude du champ de température et de l’évolution des flux en fonction du nombre de Péclet Pe, l’écart entre les deux tubes d, le rayon des tubes r et les vitesses des écoulements au sein des tubes

Shape optimization for the generalized Graetz problem

We apply shape optimization tools to the generalized Graetz problem which is a convection-diffusion equation. The problem boils down to the optimization of generalized eigen values on a two phases domain. Shape sensitivity analysis is performed with respect to the evolution of the interface between the fluid and solid phase. In particular physical settings, counterexamples where there is no optimal domains are exhibited. Numerical examples of optimal domains with different physical parameters and constraints are presented. Two different numerical methods (level-set and mesh-morphing) are show-cased and compared

Chapitre 8. Nobles et ignobles

A partir du XIe siècle, les textes deviennent plus explicites sur les différentes classes sociales. L’une des deux notices de Clairvaux passe même en revue l’ensemble de la société réunie en 1060 autour du pèlerin anglais Ælfwin : seniores... principes amborum castrorum sive submilitones, pauperes, divites, nobiles, ignobiles sed et mulieres... nobiles feminae. Nobles ou ignobles : désormais les choses sont claires, la société laïque est divisée en deux grandes classes qui ne correspondent pl..

Conclusion

Sos bas paratges sobrissitz Sai que fenira coma lais E tornara lai don se trais : A Meillau et en Carlades. Ces vers d’un sirventes de Bertrand de Bom visaient Alfonse II d’Aragon, descendant de Gerbert de Millau, mais ils auraient fort bien pu prendre pour cible la maison rivale, celle de Toulouse-Rouergue. En contribuant largement à la grande guerre méridionale, l’héritage millavois de Douce devait affaiblir les deux ..

Introduction

“Au nom de notre Seigneur et Dieu. Au lieu sacré qui est situé dans le pays rutène (...) dont le vocable est Conques (…) moi, Leutad (…), je fais écrire et confirmer cette petite charte de donation, à savoir : à l’intérieur du pagus de la cité de Rodez, dans la vallée du Tarn, aux lieux ou villae où s’étend le vocable de Pris, une maison de maître avec ses parties supérieures et inférieures ou ses dépendances, avec les cours, les jardins et les autres manses (…) ce que je suis vu posséder par..