Section: New Results

Complex fluids

Participants : David Benoit, Sébastien Boyaval, Claude Le Bris, Tony Lelièvre.

In his PhD under the supervision of Claude Le Bris and Tony Lelièvre, David Benoit studies models of aging fluids developed at the ESPCI (Ecole supérieure de physique et de chimie industrielles) and designed to take into account phenomena such as shear thinning, aging and shear banding in falling sphere experiments. The work consists in studying on the one hand the mathematical well-posedness of some macroscopic models, see [10] , and, on the other hand, in trying to understand the link between such macroscopic models and microscopic models which have been proposed to describe such fluids, see [34] .

Let us also mention that the paper [28] on a parareal algorithm to efficiently simulate micro-macro models which has been published this year.

Related to the mathematical modelling of free-surface complex flows under gravity, a new reduced model for thin layers of a viscoelastic upper-convected Maxwell fluid was derived by S. Boyaval in collaboration with François Bouchut. Possibly discontinuous solutions were numerically simulated with a new finite-volume scheme of relaxation type that satisfies a discrete counterpart of the natural dissipation [13] .

This work has been pursued for other fluid models and other flow regimes, with a view to better understanding the reduction mechanism leading from a physically detailed model to a useful one for numerical simulations at large (geophysical) scales [35] .

On the other hand, note that it is often possible to consider only models for incompressible fluids (at low Mach numbers). Now, it is both important and delicate to understand how to numerically discretize the incompressibility constraint, a long-standing issue in numerical fluid mechanics. In collaboration with M. Picasso (EPFL), S. Boyaval has thus investigated the possibility to numerically quantify a posteriori the quality of a well-known, "simple" numerical method discretizing the incompressibility constraint, in a simple case [36] . This is part of another effort toward useful numerical simulations of complex flows, inline with current questions focused on discretization methods..