## Section:
Application Domains2>
### Fluid mechanics3>
#### Numerical methods for viscous flows4>

#### Numerical methods for viscous flows4>

We are concerned with systems of PDEs describing the evolution of mixture flows. The fluid is described by its density, its velocity and its 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 at studying some instabilities phenomena such as the Raileigh-Taylor instability.

Furthermore, diffuse interface models have gained renewed interest in the last few years in fluid mechanics applications. From a physical viewpoint, they allow to describe phase transition phenomena. In this case, a specific stress tensor, introduced originally by Korteweg, must be added to the momentum equation.

If we use in addition the Fick law to relate the divergence of the velocity to derivatives of the density, one obtains the so-called KS model. In this case, the density of the mixture is naturally highly heterogeneous, and may model powder-snow avalanches or specific pollutants. Similar models also appear in combustion theory.

#### Control4>

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.