Exploration de l'effet Klinkenberg à différentes échelles

L'introduction du phénomène de glissement pour des écoulements fortement raréfiés conduit, pour les milieux poreux, à une modification du débit en fonction de la perte de charge. Cette modification, introduite par Klinkenberg, aboutit, pour un milieu saturé, à une loi de Darcy dont le coefficient de conductivité dépend du nombre de Knudsen Kn. Cette dépendance avec le nombre de Knudsen, pour un écoulement de Poiseuille, peut être déduite de la vitesse de glissement proposée par Cercignani. Il n'en est pas nécessairement de même pour un réseau de canaux. Pour mieux cerner le domaine de validité de la loi de Klinkenberg, nous analysons l'écoulement faiblement raréfié (de continu à transitionnel) dans un réseau élémentaire de micro-canaux pour lequel le nombre de Knudsen est multiplié par 2 ou 4 par division de la hauteur H d'un canal générateur. L'écoulement est simulé avec le modèle BGK-Hermite présenté, en particulier lors des journées CFM de 2009 et 2011 [1]. Ce modèle a été développé pour les écoulements à grand spectre de Knudsen. Le fluide considéré est un gaz parfait et les conditions aux limites sont de type diffuses sur les parois et périodiques entre l'entrée et la sortie. Les simulations sont réalisées à débit constant. Les résultats sont comparés à ceux obtenus à l'aide des équations de Navier-Stokes périodiques avec vitesse de glissement du second ordre en Kn aux parois. La transition entre zones à différents Knudsen a été modélisée pour imposer une vitesse continue. L'écoulement est généré par une force volumique constante. La communication proposée discutera la dépendance de la conductivité hydraulique en fonction du nombre de Knudsen. [1] L. de Izarra, J.L. Rouet, B. Izrar,2011, High orders lattice Boltzmann models for gas flows on a wide range of Knudsen number, Phys. Rev. E 84, 06670

Exploring the Klinkenberg effect at different scales

International audienceSimulations of microflows usually require sophisticated numerical tools. Nevertheless in the slip regime, the hydrodynamic equation with slip boundary condition may be sufficient to account for the so-called Klinkenberg effect. We propose to visit this effect using a basic network of microchannels in which the Knudsen number is multiplied by two or four by introducing successive derivations to the channel. We derived an equivalent hydraulic conductivity up to second order. Theoretical results are compared both with the results of the Navier-Stokes equations with slip condition and with those obtained using a Bhatnagar-Gross-Krook–Hermite model developed especially for flows with a large spectrum of Knudsen numbers (typically 10−4<Kn<10). A criterion is provided in order to distinguish the slip regime from the transitional one in this multiscale network

On the Voting Power of an Alliance and the Subsequent Power of its Members

Even, and in fact chiefly, if two or more players in a voting gamehave on a binary issue independent opinions, they may haveinterest to form a single voting alliance giving an average gainof influence for all of them. Here, assuming the usualindependence of votes, we first study the alliance voting powerand obtain new results in the so-called asymptotic limit for whichthe number of players is large enough and the alliance weightremains a small fraction of the total of the weights. Then, wepropose to replace the voting game inside the alliance by a randomgame which allows new possibilities. The validity of theasymptotic limit and the possibility of new alliances are examinedby considering the decision process in the Council of Ministers ofthe European Union.Voting Power; Alliance

Shadowing effects for continuum and discrete deposition models

We study the dynamical evolution of the deposition interface using both discrete and continuous models for which shadowing effects are important. We explain why continuous and discrete models implying both only shadowing deposition do not give the same result and propose a continuous model which allow to recover the result of the discrete one exhibiting a strong columnar morphology

Modification and modeling of water ingress in limestone after application of a biocalcification treatment

International audienceWater transfers have been recognized as the main vectors of alteration and are responsible for pore network modifications in building stone. Among the techniques used to limit or stop the penetration of water into the stone, the calcification properties of bacteria have been investigated and used to treat buildings. In this article we study the effect of such a treatment following a protocol used in situ. The effects of this biotreatment on limestone (here tuffeau) were measured over a large number of drying-imbibition cycles. As the imbibition curves did not follow the usual Washburn law, a model based on a space-dependent permeability coefficient is proposed. It leads to a non-linear diffusion model which accounts for the deviation from the standard Washburn model

High-order lattice Boltzmann models for gas flow for a wide range of Knudsen numbers

The authors benefited from the use of the cluster at the Centre de Calcul Scientifique en R'egion Centre.International audienceThe lattice Boltzmann methods (LBMs) have successfully been applied to microscale flows in the hydrodynamic regime, such as flows of liquids in porous media. However, the LBM in its standard formulation does not produce correct results beyond the hydrodynamic regime, i.e., for slip and transitional ones. Following the work of Shan and He [ Phys. Rev. Lett. 80 65 (1998)], we propose to extend the LBM to these regimes in which nonequilibrium effects are obvious and require us to include a larger number of distribution-function moments

Ewald Sums for One Dimension

We derive analytic solutions for the potential and field in a one-dimensional system of masses or charges with periodic boundary conditions, in other words Ewald sums for one dimension. We also provide a set of tools for exploring the system evolution and show that it's possible to construct an efficient algorithm for carrying out simulations. In the cosmological setting we show that two approaches for satisfying periodic boundary conditions, one overly specified and the other completely general, provide a nearly identical clustering evolution until the number of clusters becomes small, at which time the influence of any size-dependent boundary cannot be ignored. Finally we compare the results with other recent work with the hope of providing clarification over differences these issues have induced. We explain that modern formulations of physics require a well defined potential which is not available if the forces are screened directly.Comment: 2 figures added references expanded discussion of algorithm corrected figures added discussion of screened forc

Textural characterization using P. Gy heterogeneity functions.

International audienceA solid can be regarded as a set of contiguous elementary units. The distribution within the solid of any properties, measurable within each elementary unit, can be characterized using two parameters. These parameters are built using the constitution and distribution heterogeneities of P. Gy (1982, 1988). The former account for the granularity of the elementary units, whereas the latter assess the spatial distribution of the property. A texture which definition involves several properties can be described using a diagram where both parameters work as variables. Potential applications encompass: (i) the textural classification of soils, ore, breccia and concrete and (ii) the monitoring of textural transformation during process like dolomitization, metamorphism, weathering, deformation or annealing

Fractal geometry in an expanding, one-dimensional, Newtonian universe

International audienceObservations of galaxies over large distances reveal the possibility of a fractal distribution of their positions. The source of fractal behavior is the lack of a length scale in the two body gravitational interaction. However, even with new, larger, sample sizes from recent surveys, it is difficult to extract information concerning fractal properties with confidence. Similarly, three-dimensional N-body simulations with a billion particles only provide a thousand particles per dimension, far too small for accurate conclusions. With one-dimensional models these limitations can be overcome by carrying out simulations with on the order of a quarter of a million particles without compromising the computation of the gravitational force. Here the multifractal properties of two of these models that incorporate different features of the dynamical equations governing the evolution of a matter dominated universe are compared. For each model at least two scaling regions are identified. By employing criteria from dynamical systems theory it is shown that only one of them can be geometrically significant. The results share important similarities with galaxy observations, such as hierarchical clustering and apparent bifractal geometry. They also provide insights concerning possible constraints on length and time scales for fractal structure. They clearly demonstrate that fractal geometry evolves in the µ (position, velocity) space. The observed patterns are simply a shadow (projection) of higher-dimensional structure