Analyse vectorielle

Résumé

L'analyse vectorielle est une branche des mathématiques qui étudie les champs de scalaires et de vecteurs suffisamment réguliers des espaces euclidiens, c'est-à-dire les applications différentiables d'un ouvert d'un espace euclidien à valeurs respectivement dans et dans . Du point de vue du mathématicien, l'analyse vectorielle est donc une branche de la géométrie différentielle. Cette dernière inclut l'analyse tensorielle qui apporte des outils plus puissants et une analyse plus concise entre autres des champs de vecteurs. Mais l'importance de l'analyse vectorielle provient de son utilisation intensive en physique et dans les sciences de l'ingénieur. C'est de ce point de vue qu'elle sera présentée dans l'article, et c'est pourquoi elle sera le plus souvent limitée au cas où est l'espace usuel à trois dimensions. Dans ce cadre, un champ de vecteurs associe à chaque point de l'espace un vecteur (à trois composantes réelles), tandis qu'un champ de scalaires y associe un réel. Par exemple, dans le cas de l'eau d'un lac, la donnée de sa température en chaque point forme un champ de scalaires, et celle de sa vitesse en chaque point, un champ de vecteurs (pour une approche plus théorique, voir Géométrie différentielle). Le calcul vectoriel et l'analyse vectorielle furent développés à la fin du par J. Willard Gibbs et Oliver Heaviside à partir de la théorie des quaternions (due à Hamilton) ; la plupart des notations et de la terminologie furent établies par Gibbs et Edwin Bidwell Wilson dans leur livre de 1901, Vector Analysis (Analyse vectorielle). Le gradient, la divergence et le rotationnel sont les trois principaux opérateurs différentiels linéaires du premier ordre. Cela signifie qu'ils ne font intervenir que des dérivées partielles (ou différentielles) premières des champs, à la différence, par exemple, du laplacien qui fait intervenir des dérivées partielles du second ordre. On les rencontre en particulier : en mécanique des fluides (équations de Navier-Stokes) ; en électromagnétisme, où ils permettent d'exprimer les propriétés du champ électromagnétique.

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Source officielle

https://fr.wikipedia.org/wiki/Analyse_vectorielle

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Catégories associées (65)

Concepts associés (40)

Cours associés (23)

Séances de cours associées (213)

Publications associées (3)

Blow-up, partial regularity and turbulence in incompressible fluid dynamics

Silja Noëmi Aline Haffter

Weak solutions arise naturally in the study of the Navier-Stokes and Euler equations both from an abstract regularity/blow-up perspective and from physical theories of turbulence. This thesis studies the structure and size of singular set of such weak solutions to equations of incompressible fluid dynamics from two opposite directions. First, it aims to single-out new mechanisms which allow to break the typically supercritical scaling of the equations and, in this way, prevent the formation of singularities either globally or locally in spacetime. Second, in the absence of such mechanisms, we seek to quantify how singular (in terms of dimension of the singular set, for instance) the solutions that we are actually able to construct are. This thesis collects four results pointing in the two directions outlined above which have been obtained in several collaborations during the Ph.D. studies:- a global regularity result for the fractional Navier-Stokes equation slightly blow the critical fractional order,- a global well-posedness result for the defocusing wave equation with slightly supercritical power nonlinearity,- an a.e. smoothness / partial regularity result for the supercritical surface quasigeostrophic (SQG) equation,- an estimate (and a discussion of its sharpness) on the dimension of the singular set of wild Hölder continuous solutions of the incompressible Euler equations.All results presented in the thesis have either been published or are submitted for publication.

EPFL2022

Extension of miniature neutron scintillators features in CROCUS: novel acquisition scheme and extended flux range

Colin Rochat

The goal of this work was the improvement of the capabilities of a miniature neutron detector, for easing its use, and extending its range of applicability. Firstly, the Red Pitaya STEMlab 125-14 was the object of an exploratory study in order to propose an alternative method to the two current acquisition electronics based on photon counting, using analog and digital means. The study allowed to establish a new methodology based on the direct use of the SiPM pre-amplifier output, requiring the hypothesis that neutron events can be discriminated directly by pulse amplitude. It was first tested in the low neutron flux of the CARROUSEL facility, and the results were compared to the analog electronics. The agreement in count rate, spectrum, and linearity demonstrates the validity of this new methodology. Moreover, its application in CROCUS allowed demonstrating that it is working in reactor condition up to a flux of 1.13∗108 s−1cm−2. Secondly, an array of six miniature neutron detectors with different sensitivities was designed and tested. The qualification study in CARROUSEL allowed to show that all detectors behave normally, and that the sensitivity of the detectors increases as a function of the scintillator’s volume at low neutron flux, as expected. The conclusion is limited by the knowledge in effective dimensions of each scintillator. In a first experiment in CROCUS using the same setup, including digital threshold (50), a loss of linearity was observed simultaneously for all detectors at a power ofabout 4W(i.e. 7∗107s−1cm−2). As the cause was expected to be a higher gamma background, the experiment was repeated with a higher threshold (120): consistently with the CARROUSEL test, count rate and loss of linearity depends on the scintillation’s volume. Further studies are needed to conclude on the phenomenon at hand, but since the main parameter for gamma discrimination proved efficient, the gamma contribution increased at higher powers seems a good candidate. In conclusion, the results are globally satisfying for the new methodology with Red Pitaya STEMlab 125-14. However, the experimental campaign of this work was only a feasibility study. The results have to be confirmed with further studies. Moreover, this system could be improved. In particular, it can be adapted firstly for both RF inputs of Red Pitaya and secondly for dynamical measurements. The results of the multi-sized array of miniature neutron detector were also encouraging. However, the behaviour of certain detectors prevent to deduce with certainty some expected trends. Further studies have to be carried out by multiplying the number of detectors of each dimension.

2021

Controlling the fate of stem cells in vitro is a key challenge towards using these cells in clinical applications. Adult stem cells (ASC) are known to reside in complex microenvironments called niches in vivo. These niches regulate stem cell fate providing a variety of signals that control in a balanced way self-renewal and differentiation. In vitro ASC culture is challenging using standard culture methods because these are not able to recreate an adequate environment mimicking niche signals. The use of bioengineered polymer surfaces may thus be a solution for controlling stem cell fate in a predictable and scalable manner. Surface initiated polymerisation (SIP) is a method allowing controlled free radical polymerisation. This technique can be used to achieve highly controlled polymer architectures. Combining this versatile technology with a microfluidic gradient maker allowed the patterning of surface gradient arrays of desired tethered biomolecules. We have used monomers carrying functional groups that can be used in highly specific and high-yielding reactions such as biotin methacrylate and alkyne methacrylate to achieve surface gradients of Streptavidin as well as azide-functionalized RGD peptides. Surface characterisation was carried out by X-ray photoelectron spectroscopy (XPS) as well as fluorescence microscopy to verify surface modification steps and gradient patterns. The quantification of immobilized biomolecules was carried out using Europium labelling assays. As a proof of concept gradients of RGD peptides were patterned on polymer brushes expecting dose responding attachment of adherent cells in the presence of these strong attachment motifs. The results showed a dose dependent attachment of fibroblast on gradients of azide-RGD. However this outcome could not be achieved with biotinylated RGD as a result of a failure in binding between the biotinylated peptide and streptavidin. Nevertherless, these experiments demonstrate the suitability of the overall strategy as a mechanism for surface patterning of specific molecules. Improvements of the method established here have the potential to result in a sophisticated platform for the screening of cell-material interactions.

2011