Volcan islandais Eyjafjöll : mais où vont les particules ?

International audienceNous entendions ce matin à la radio qu'il devait y avoir une erreur dans les modèles mathématiques, puisque les vols d'essais menés par plusieurs compagnies aériennes semblent avoir laissé les moteurs dans un état intact. Nous allons essayer de décrypter cette affirmation

Viscoplastic Free-Surface Flows: The Herschel-Bulkley Case

International audienceIn this paper, we will describe consistent numerical methods for power-law viscoplastic free-surface flows. From the rheological viewpoint, associated models are of Herschel-Bulkley type, which is a generalization of the Bingham model. On the one hand, Bingham model is the simplest model when it comes to describe viscoplasticity. On the other hand, power-law model is a natural extension of a rate-of-shear dependant viscosity, as opposed to the canical case of the (often) constant viscosity used in the Navier-Stokes equations. After describing a shallow-water asymptotics of a 3D Navier-Stokes-Herschel-Bulkley model with free surface, we will end up with a model which has various mathematical difficulties. We will show how to handle optimization problems arising from the variational inequalities associated to the model, as well as their coupling with finite-volume discretization. Several numerical tests will be shown, including a comparison with an analytic solution, to confirm the well balanced property and the ability to cope with the various rheological regimes associated with the Herschel-Bulkley constitutive law. See : http://www.iccfd.org/iccfd7/assets/pdf/papers/ICCFD7-3302_paper.pd

On stability condition for bifluid flows with surface tension : application to microfluidics

Accepted 22 February 2008 ; Available online 5 March 2008. Accepté pour publication le 22/08/2007 ; disponible en ligne le 05/04/2008 ; 41 pages.International audienceModels for incompressible immiscible bifluid flows with surface tension are here considered. Since Brackbill, Kothe and Zemach (J. Comput. Phys. 100, pp 335-354, 1992) introduced the Continuum Surface Force (CSF) method, many methods involved in interface tracking or capturing are based on this reference work. Particularly, the surface tension term is discretized explicitly and therefore, a stability condition is induced on the computational time step. This constraint on the time step allows the containment of the amplification of capillary waves along the interface and puts more emphasis on the terms linked with the density in the Navier-Stokes equation (i. e. unsteady and inertia terms) rather than on the viscous terms. Indeed, the viscosity does not appear, as a parameter, in this stability condition. We propose a new stability condition which takes into account all fluid characteristics (density and viscosity) and for which we present a theoretical estimation. We detail the analysis which is based on a perturbation study - with capillary wave - for which we use energy estimate on the induced perturbed velocity. We validate our analysis and algorithms with numerical simulations of microfluidic flows using a Level Set method, namely the exploration of different mixing dynamics inside microdroplets

Efficient numerical schemes for viscoplastic avalanches. Part 2: the 2D case

This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non-trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated with the yield threshold: finite volume schemes are used together with duality methods (namely Augmented Lagrangian and Bermúdez–Moreno) to discretize the problem. To be able to accurately simulate the stopping behavior of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. Here we derive such schemes in 2D as the follow up of the companion paper treating the 1D case. Numerical tests include in particular a generalized 2D benchmark for Bingham codes (the Bingham–Couette flow with two non-zero boundary conditions on the velocity) and a simulation of the avalanche path of Taconnaz in Chamonix—Mont-Blanc to show the usability of these schemes on real topographies from digital elevation models (DEM)

L'Institut Courant de Sciences Mathématiques : 75 ans d'Excellence

International audienceCet article est le fruit d'une série d'entretiens avec douze membres du Courant Institute, qui fête son 75e anniversaire au cours de l'année universitaire 2010-2011. Nous en donnons ici une traduction en français qui permet de découvrir comment Richard Courant a créé un institut dont les mathématiques appliquées devinrent, et soixante-quinze ans plus tard, continuent à être considérées comme les meilleures au monde. http://images.math.cnrs.fr/L-Institut-Courant-de-Sciences.htm

A metamodel for confined yield stress flows and parameters' estimation

With the growing demand of mineral consumption, the management of the mining waste is crucial. Cemented paste backfill (CPB) is one of the techniques developed by the mining industry to fill the voids generated by the excavation of underground spaces. The CPB process is the subject of various studies aimed at optimizing its implementation in the field. In this article, we focus on the modelling of the backfill phase where it has been shown in [Vigneaux et al., Cem. Concr. Res. 164 (2023) 107038] that a viscoplastic lubrication model can be used to describe CPB experiments. The aim here is to propose an accelerated method for performing the parameters' estimation of the properties of the paste (typically its rheological properties), with an inverse problem procedure based on observed height profiles of the paste. The inversion procedure is based on a metamodel built from an initial partial differential equation model, thanks to a Polynomial Chaos Expansion coupled with a Principal Component Analysis.Comment: Rheologica Acta, 202

Level Set method for Curvature-driven Flows in Microfluidics

http://www.springerlink.com/content/w0730416876v7655/International audienceA Level Set method for bifluid flows in Microfluidics is presented. After a stability analysis which conducts to a restrictive stability condition on the computational time step due to surface tension, we introduce a splitting to shorten simulation times. Numerical results of these curvature-driven flows are in good agreement with physical experiments and are used to explore mixing processes inside micro-droplets

Méthodes level set pour des problèmes d'interface en microfluidique

Ce travail est consacré à la modélisation d’écoulements de deux fluides immiscibles et son application en microfluidique. Pour cela, nous mettons en œuvre des méthodes Level Set actuelles permettant un suivi précis de l’interface, dont le mouvement est induit par des champs de vitesse vérifiant les équations de Stokes ou de Navier-Stokes munies d’un terme de tension de surface. Dans une première partie, nous abordons la problématique du suivi d’interface et présentons en détail les composantes de la méthode Level Set. En particulier, nous détaillons les approches ENO et WENO pour discrétiser les équations de Hamilton-Jacobi ainsi que les diverses méthodes existantes de redistanciation. Dans la deuxième partie, nous traitons de l’analyse et de la résolution numérique des écoulements bifluides incompressibles pilotés par la tension de surface. Après avoir décrit les modèles mathématiques ainsi que leurs discrétisations et solveurs, nous apportons une contribution nouvelle en dérivant théoriquement une condition de stabilité valable pour les nombres de Reynolds faibles à modérés, caractéristiques des configurations microfluidiques. De plus, on introduit une méthode de décomposition de l’écoulement qui permet de diminuer les temps de simulation. Enfin, la troisième partie est consacrée à l’application des outils évoqués précédemment pour simuler la dynamique de gouttes dans des microcanaux. Nous présentons les résultats numériques obtenus avec d’une part, un code bidimensionnel cartésien et d’autre part, avec un code tridimensionnel axisymétrique que nous avons entièrement développés. Une bonne adéquation est obtenue relativement aux expériences microfluidiques du laboratoire LOF (Rhodia – CNRS). En particulier, avec nos simulations, nous mettons à jour différentes dynamiques de mélange au sein des gouttes

Multilayer models for hydrostatic Herschel-Bulkley viscoplastic flows

This is an open access article under the CC BY-NC-ND licenseStarting from Navier-Stokes’ equation we derive two shallow water multilayer models for yield stress fluids, depending on the asymptotic analysis. One of them takes into account the normal stress contributions, making possible to recover a pseudoplug layer instead of a purely plug zone. A specific numerical scheme is designed to solve this model thanks to a finite volume discretization. It involves well-balancing techniques to be able to compute accurately the transitions between yielded and unyielded (or pseudoplug) zones, an important feature of the original partial differential equations’ model. We perform numerical simulations on various test cases relevant to these physics: analytical solution of a uniform flow, steady solutions for arrested state, and a viscoplastic dam break. Simulations agree well when we perform comparisons with physical experiments of the group of Christophe Ancey (EPFL) and we make a comparative study including shallow water models and lubrication models that they present in Ancey et al. (2012) [3]. Thanks to the multilayer structure of our model, we can go further on the description of the vertical structure associated to the (bottom) sheared layer and the top (pseudo-)plug layer

Level-Set method and stability condition for curvature-driven flows

International audienceWe consider models for the simulation of curvature-driven incompressible bifluid flows, where the surface tension term is discretized explicitly. From this formulation a numerical stability condition arises for which we present a new theoretical estimation for low and medium Reynolds numbers. We illustrate our analysis with numerical simulations of microfluidic flows using Level Set method. Finally, we propose a method to reduce computational cost induced by this stability condition for low flow velocities