Section: New Results

Complex fluids

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

In [48] , Claude Le Bris and Tony Lelièvre have reviewed the state-of-the-art of numerical and mathematical results on micro-macro models for viscoelastic fluids.

Following previous works, Claude Le Bris and Tony Lelièvre together with Lingbing He have analyzed in [43] the longtime behaviour of nematic polymeric fluids (liquid crystals). The longtime asymptotics for such models is much richer than for flexible polymers, which were considered in previous analysis. Indeed, for these models, periodic in time behaviours are expected.

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 on the one hand in studying the mathematical well-posedness of some macroscopic models 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.

In the line of his former work [15] , Sébastien Boyaval has pursued his work about the mathematical modelling of viscoelastic fluid flows. A new reduced model for thin layers of elastic gravity flows with a free surface was derived in collaboration with François Bouchut, following similar hydrostatic assumptions to those which lead to the Saint-Venant equations as a usual reduced model for shallow water flows. The model is naturally endowed with an energy but the conservative part is non-standard: the energy is not convex with respect to the conservative variables. It is convex with respect to other (more physical) variables. For the numerical simulation of possibly discontinuous solutions, a relaxation scheme is proposed in order to ensure that the numerical approximation mimicks the natural energy dissipation.

In [58] , Tony Lelièvre together with Giovanni Samaey and Vincent Legat explored some numerical techniques to get closed macroscopic equations from microscopic models. The proposed method can be seen as a way to justify and extend techniques based on the so-called quasi-equilibirium approximation.

In [40] , the effect of a topography on a free surface flow is studied using the free-surface Navier Stokes equations and ALE method for discretization.