The goal of this workshop is to give a few introductory talks around the modelling of complex fluids and granular flows in particular. The talks are aimed at a large audience and will present various aspect of their study, be it from a physics and experimental viewpoint, from a mathematical modelling viewpoint or from the mathematical and numerical analysis of some models.

• 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)

Registration is free but mandatory. Send a mail to jthemcf@sciencesconf.org

The workshop will take place at the Frumam on the Saint-Charles Campus of the university :  http://frumam.math.cnrs.fr/spip.php?article7

9h30 : Coffee greeting

10h-11h : Yoël Forterre

11h15-12h15 : Sébastien Boyaval

Buffet

14h-15h : Bertrand Maury

15h-15h30 : Coffee break

15h30-16h30 : Ewelina Zatorska

Sébastien Boyaval : Maxwell meets Saint-Venant to talk about viscoelastic fluids

Modelling the flow of a non-Newtonian fluid remains an active research field since the pionneering work of Maxwell on viscoelasticity. In this talk, we will discuss recent model in the framework of hydrostatic free-surface flows historically introduced by Saint-Venant. Then we compare numerically two models whose only difference is in the evolution equation of the stress tensor (by terms using the derivatives of a tensor which used the velocity gradient). Those are two nonlinear hyperbolic systems which are simulated with finite volumes approximations taking advantage of a new relaxation method. An entropic Riemann solver of Suliciu-type has been built for this purpose.

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.