Numerical simulation of viscous flows

We are concerned with systems of PDEs describing the evolution of mixture flows. The fluid is described by the density, the velocity and the pressure. These quantities obey mass and momentum conservation. On the one hand, when we deal with the 2D variable density incompressible Navier-Stokes equations, we aim to study some instabilities phenomena such as the Raileigh-Taylor instability. On the other hand, diffuse interface models have gained renewed interest for the last few years in fluid mechanics applications. From a physical viewpoint, they allow to describe some phase transition phenomena. If the Fick's law relates the divergence of the velocity field to derivatives of the density, one obtain the so called Kazhikhov-Smagulov model. Here, the density of the mixture is naturally highly non homogeneous, and the constitutive law accounts for diffusion effects between the constituents of the mixture. Furthermore, a surface tension force can be added to the momentum equation introducing a specific stress tensor, proposed for the first time by Korteweg. The first phenomena that we try to reproduce are the powder-snow avalanches, but we can also model flows where species (like salt or pollutant) are dissolved in a compressible or incompressible fluid. Other similar hydrodynamic models arise in combustion theory.

Flow control strategies using passive or active devices are crucial tools in order to save energy in transports (especially for cars, trucks or planes), or to avoid the fatigue of some materials arising in a vast amount of applications. Nowadays, shape optimization needs to be completed by other original means, such as porous media located on the profiles, as well as vortex generator jets in order to drive active control.

From polymer physics to rubber elasticity

Our aim is to rigorously derive nonlinear elasticity theory from polymer physics. The starting point is the statistical physics description of polymer-chains. Under some proper rescaling, this discrete model converges to continuum nonlinear elasticity models in the sense of Gamma-convergence. The long-term goal of our approach is to derive practical constitutive laws for rubbers (to be used in nonlinear elasticity softwares) from the discrete model.