Le but de cette journée est de proposer quelques exposés introductifs à la modélisation des fluides complexes et des écoulements granulaires en particulier. Les exposés sont destinés à un public large et présenteront divers aspects de leur étude, que ce soit d'un point de vue physique et expérimental, de modélisation ou d'analyse mathématique et numérique.

• Sébastien Boyaval (LSV, École des Ponts ParisTech)
• Yoel Forterre (IUSTI, CNRS - Aix-Marseille Université)
• Bertrand Maury (LM, Université Paris-Sud)
• Ewelina Zatorska (Department of Mathematics, University College of London)

L'inscription à la journée est gratuite mais obligatoire, par envoi de mail à jthemcf@sciencesconf.org

La journée se déroulera à la Frumam sur le campus Saint-Charles de l'université : http://frumam.math.cnrs.fr/spip.php?article7

9h30 : Accueil autour d'un café

10h-11h : Yoël Forterre

11h15-12h15 : Sébastien Boyaval

Buffet

14h-15h : Bertrand Maury

15h-15h30 : Café

15h30-16h30 : Ewelina Zatorska

Sébastien Boyaval : Maxwell rencontre Saint-Venant à propos de fluides viscoélastiques

La modélisation des écoulements fluides non-Newtoniens reste un sujet de recherche actif depuis les travaux pionniers de Maxwell en viscoélasticité. Dans l'exposé, on discutera de modèles récents, dans le cadre des écoulements hydrostatiques à surface libre introduit historiquement par Saint-Venant. Puis on compare numériquement deux modèles qui diffèrent uniquement dans l'équation d'évolution du tenseur des efforts (par des termes de la dérivée d'un tenseur utilisant le gradient des vitesses). Ce sont deux systèmes hyperboliques nonlinéaires qu'on simule avec des approximations volumes finis grâce à une nouvelle méthode de relaxation. Un solveur de Riemann entropique, de type Suliciu, a été construit dans ce but.

Yoël Forterre : Physical modeling of dense granular flows and suspensions

Over the past two decades, important progresses have been made in our understanding of the rheology of granular and suspension flows, especially in the concentrated regime that is the most relevant for geophysical applications. In this talk, I will review some of these advances, focusing first on dry granular flows. The success and limits of a simple constitutive law describing the medium as a visco-plastic frictional liquid will be discussed, with examples ranging from avalanche flows to silo discharged.  In the second part of the talk, the extension of this approach to granular material immersed in a liquid (dense suspensions) will be discussed. We will see that the interstitial fluid affects the steady state rheology but can also strongly modify the transient dynamics of the flow, due to the coupling between packing fraction changes (Reynolds dilatancy) and pore pressure (Darcy flow). Finally, we will see how the concepts introduced so far can be used to model more complex suspensions, such as debris flows or shear thickening fluids.

Bertrand Maury : Micro-macro issues in the modeling of granular suspension

We are interested in identifying the similarities and discrepancies between micro and macro descriptions of granular flows, in the nonelastic and frictionless setting. At the microscopic level, the model takes the form of a differential inclusion together with a (nonelastic) collision law, which expresses the after-collisional velocity in terms of the pre-collisional one. The so-called pressureless Euler equations with maximal density constraint are a natural macroscopic counterpart of this microscopic model, as suggested by a quasi-perfect equivalence between both settings in the one-dimensional case. Yet, in dimension higher than 1, both descriptions greatly differ. We shall describe this discrepancy by focusing on an underlying Laplace operator which appears at both levels. While its macroscopic instance is the standard Laplace operator, the microscopic version reflects the local arrangements of hard spheres, which confers to it some pathological properties, in particular the loss of maximum principle, which can explain some phenomena which are typical of the microscopic setting, like the clogging phenomenon.

Can the fluid equation be used to model pedestrian motion or traffic? In this talk, I will present the compressible-incompressible two phase system describing the flow in the free an in the congested regimes. I will show how to approximate such system  by the compressible Navier-Stokes equations with singular pressure for the fixed barrier densities, together with some recent developments for the barrier densities varying in the space and time. At the end, I will present a couple of numerical results showing that our macroscopic system captures some features characteristic for microscopic models of collective behaviour.