Section: New Results

Modeling and numerical simulation of complex fluids

Recently, C. Calgaro et al. compared some very recent numerical schemes for the resolution of incompressible variable density flows; namely an Hybrid Finite Volume/Finite Element scheme, and a Discrete Duality Finite Volume one [34] . This work was performed in collaboration with the Inria team COFFEE (Inria Nice Sophia-Antipolis). In addition to this original and attentive comparison, our main contribution has been to improve the way to implement the Neumann boundary condition on the density, when a second-order accurate scheme is considered in space. Indeed, for some critical situations such as the simulation of Rayleigh-Taylor instabilities using unstructured meshes, this point is crucial to avoid unphysical numerical instabilities in the vicinity of the boundaries corresponding to symmetric axis. The obtained results are very promising, and constitute an important step towards the simulation of more complex models on which we are working at the moment.

In [25] , M. Gisclon and I. Lacroix-Violet consider the barotropic compressible quantum Navier-Stokes equations with a linear density dependent viscosity and its limit when the scaled Planck constant vanish. Following recent works on degenerate compressible Navier-Stokes equations, we prove the global existence of weak solutions by the use of a singular pressure close to vacuum. With such singular pressure, we can use the standard definition of global weak solutions which also allows to justify the limit when the scaled Planck constant denoted by $\epsilon$ tends to 0.

The H-theorem, originally derived at the level of the Boltzmann nonlinear kinetic equation for a dilute gas undergoing elastic collisions, strongly constrains the velocity distribution of the gas to evolve irreversibly towards equilibrium. As such, the theorem could not be generalized to account for dissipative systems: the conservative nature of collisions is an essential ingredient in the standard derivation. The work [24] gives the first strong numerical evidences, along with a proof for a simplified model, of dissipation of the Boltzmann entropy (the so-called H-theorem) for solutions to the granular gases equation. This dissipative kinetic equation describes the non-equilibrium behavior of a gas composed of macroscopic particles, namely complex fluids such as avalanches, pollens flows or planetary rings.