Trigonométrie | EPFL Graph Search

vignette|droite|Un triangle rectangle sur lequel est indiqué un angle Â, le côté adjacent à cet angle, le côté opposé à celui-ci, l'hypoténuse du triangle, et son angle droit. vignette|Cercle trigonométrique et angles remarquables vignette|droite|Planche sur la Trigonométrie, 1728 Cyclopaedia. La trigonométrie (du grec τρίγωνος / trígonos, « triangulaire », et μέτρον / métron, « mesure ») est une branche des mathématiques qui traite des relations entre distances et angles dans les triangles et des fonctions trigonométriques telles que sinus, cosinus, tangente. Présentation Histoire de la trigonométrie Premières techniques de mesure du triangle Les origines de la trigonométrie remontent aux civilisations d’Égypte antique, de Mésopotamie et de la vallée de l’Indus, il y a plus de . Il semblerait que les Babyloniens aient basé la trigonométrie sur un système numérique à base 60. La tablette paléo-babylonienne Plimpton 322 (ca -1800) présenterait des ru

The tip region of a hydraulic fracture driven by a power law fluid

Brice Tanguy Alphonse Lecampion, Fatima-Ezzahra Moukhtari

We investigate the tip region of a power law fluid driven fracture propagating at constant velocity V in an impermeable linear elastic medium with the presence of fluid lag of a priori unknown length. The fracture is loaded by the internal fluid pressure pf and by the far field confining stress. The power law fluid is characterized by a fluid index n and a consistency index M. Interest in the tip region is the key to know the correct structure of the solution and to recognize the strong coupling fluid-solid which is mainly shown in a small region near the tip. The scheme is organized as follows. First, we formulate the governing equations and derive a dimensionless form of these equations which only depends on the dimensionless toughness. The nonlinear system of equations is discretized using the Gauss-Chebyshev polynomials [2,3]. This technique uses trigonometric values for the abscissas and the colocation points as made when using the Gauss-Chebyshev for solving singular integral equations corresponding to finite fractures. We transform the coordinate in order to evaluate integrals in infinite integration interval. The numerical results will include the fracture opening w, the fluid pressure pf and the relationship between the fluid lag and the toughness. We show that the solution is not only consistent with the square root singularity of linear elastic fracture mechanics, but that its asymptotic behaviour in the far field is given by the solution of a semi-infinite hydraulic fracture constructed on the assumption of zero toughness solution [1]. The two asymptotes define the two limiting regimes : in the viscosity-dominated regime, the toughness of the solid is small enough that the solution can be approximated by the zero toughness solution; while in the toughness-dominated regime the fluid can be assumed to be inviscid. The intermediate part of the solution between the two asymptotes is obtained numerically.

2016

3D Turbulence Measurements Using Three Synchronous Wind Lidars: Validation against Sonic Anemometry

Fernando Carbajo Fuertes, Giacomo Valerio Iungo, Fernando Porté Agel

This paper presents a technique to measure the time series of the three components of the wind vector at a point in space from synchronous measurements of three scanning Doppler wind lidars. Knowing the position of each lidar on the ground and the orientation of each laser beam allows for reconstructing the three components of the wind velocity vector. The laser beams must intersect at the desired point in space and their directions must be noncoplanar, so that trigonometric relationships allow the reconstruction of the velocity vector in any coordinate system. This technique has been tested during a measurement campaign carried out at Cabauw's Experimental Site for Atmospheric Research (CESAR) in the Netherlands and compared against measurements from sonic anemometers installed in a meteorological mast. The spatial resolutions of both measurement techniques differ by one order of magnitude. Therefore, in order to properly compare the results, a pseudospatial filter that mimics the attenuation induced by the lidar technology at small scales of turbulence has been applied to the velocity time series provided by the sonic anemometer. Good agreement between both measurement systems is found in terms of the measured instantaneous velocity vector, turbulence statistics, Reynolds stresses, and the spectra of the three components of the velocity and the turbulent kinetic energy. These results provide a successful validation of the proposed technique.

Amer Meteorological Soc2014

Reagent-modulated optional site selectivities: the metalation of o-, m- and p-halobenzotrifluorides

Chloro(trifluoromethyl)benzenes and bromo(trifluoromethyl)benzenes underwent deprotonation at a postion adjacent to the single halogen substituent when treated with alkyllithiums (at -75 DegC) and, resp., lithium 2,2,6,6-tetramethylpiperidide (at -100 DegC) in THF. Positional ambiguities, if existing, were be exploited to establish optional site selectivities. Thus, the reaction of butyllithium with 1-chloro-3-(trifluoromethyl)benzene led to hydrogen/metal interconversion at the 2-position whereas the reaction of sec-butyllithium led exclusively to a reaction at the 6-position. The latter mode of regioselectivity was also exhibited by 1-bromo-3-(trifluoromethyl)benzene in the presence of lithium 2,2,6,6-tetrmethylpiperidide, and only 2-bromo-4-(trifluoromethyl)phenyllithium was produced. 2-Bromo-6-(trifluoromethyl)phenyllithium was directly inaccessible, but was formed when 2-bromo-3-(trifluoromethyl)phenyllithium, (generated at -100 DegC) was allowed to isomerize at -75 DegC. [on SciFinder (R)]

1996

The tip region of a hydraulic fracture driven by a power law fluid

Brice Tanguy Alphonse Lecampion, Fatima-Ezzahra Moukhtari

2016