Equidistribution of Heegner Points and Ternary Quadratic Forms

We prove new equidistribution results for Galois orbits of Heegner points with respect to reduction maps at inert primes. The arguments are based on two different techniques: primitive representations of integers by quadratic forms and distribution relations for Heegner points. Our results generalize one of the equidistribution theorems established by Cornut and Vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow. Moreover, for fixed fundamental discriminant and variable conductor, we deduce an effective surjectivity theorem for the reduction map from Heegner points to supersingular points at a fixed inert prime. Our results are applicable to the setting considered by Kolyvagin in the construction of the Heegner points Euler system

Experiments performed with bubbly flow in vertical pipes at different flow conditions covering the transition region: Simulation by coupling Eulerian, Lagrangian and 3D random walks models

[EN] Two phase flow experiments with different superficial velocities of gas and water were performed in a vertical upward isothermal cocurrent air-water flow column with conditions ranging from bubbly flow, with very low void fraction, to transition flow with some cap and slug bubbles and void fractions around 25%. The superficial velocities of the liquid and the gas phases were varied from 0.5 to 3 m/s and from 0 to 0.6 m/s, respectively. Also to check the effect of changing the surface tension on the previous experiments small amounts of 1-butanol were added to the water. These amounts range from 9 to 75 ppm and change the surface tension. This study is interesting because in real cases the surface tension of the water diminishes with temperature, and with this kind of experiments we can study indirectly the effect of changing the temperature on the void fraction distribution. The following axial and radial distributions were measured in all these experiments: void fraction, interfacial area concentration, interfacial velocity, Sauter mean diameter and turbulence intensity. The range of values of the gas superficial velocities in these experiments covered the range from bubbly flow to the transition to cap/slug flow. Also with transition flow conditions we distinguish two groups of bubbles in the experiments, the small spherical bubbles and the cap/slug bubbles. Special interest was devoted to the transition region from bubbly to cap/slug flow; the goal was to understand the physical phenomena that take place during this transition A set of numerical simulations of some of these experiments for bubbly flow conditions has been performed by coupling a Lagrangian code, that tracks the three dimensional motion of the individual bubbles in cylindrical coordinates inside the field of the carrier liquid, to an Eulerian model that computes the magnitudes of continuous phase and to a 3D random walk model that takes on account the fluctuation in the velocity field of the carrier fluid that are seen by the bubbles due to turbulence fluctuations. Also we have included in the model the deformation that suffers the bubble when it touches the wall and it is compressed by the forces that pushes it toward the wall, provoking that the bubble rebound like a ball.The authors of this paper are indebted to the National Plan of I+D by the support of the coordinated projects REMOD-ERN ENE2010-21368-C02-01/CON and ENE2010-21368-C02-02/CON to perform the experiments.Muñoz-Cobo, JL.; Chiva, S.; Ali Abdelaziz Essa, M.; Mendez, S. (2012). Experiments performed with bubbly flow in vertical pipes at different flow conditions covering the transition region: Simulation by coupling Eulerian, Lagrangian and 3D random walks models. Archives of Thermodynamics. 33(1):3-39. https://doi.org/10.2478/v10173-012-0001-4S33933

Modelling of Boiling Heat Transfer in a Turbulent Channel Flow

International audienceA numerical study of boiling heat transfer in a turbulent liquid-vapor flow inside a heated channel is carried out for various flow conditions. The model used for the flow simulation is the Volume Of Fluid model (VOF) for liquid-vapor interface tracking coupled with a k−ε low Reynolds model to predict the effect of turbulence. Boiling and condensation phenomena are included in the model based on the general laws of phase change. The obtained results are compared with available experimental measurements in the literature, where the effect of subcooling, channel inclination and flow velocity on the vapor distribution were investigated. The computational predictions of the vapor bubble development along the heated wall are in satisfactory agreement with experimental results

Representations of integers by ternary quadratic forms

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