Fluid-particle interaction systems

Fluid-particle interaction systems have been proposed to describe the behavior of sprays,aerosols or more generically two phase flows where one phase (disperse) can be consideredas a suspension of particles onto the other one (dense) thought as a fluid. In many of theseapplications, the assumption that particles are solid non-deformable spheres suspended onthe fluid leads to simplified but meaningful models [2]. This kind of systems have been used in sedimentation of solid grain by external forces [1], for fuel-droplets in combustiontheory [28] and biosprays in medicine [3, 22] for instance.The particles behavior is obtained via the evolution of the statistical distribution ofparticles in phase space, where the only forces taken into account on particles are thefriction forces due to the fluid and an stochastic term of fluctuations around the fluidvelocity. This kinetic modelling of the particle phase, as in [6, 4, 7], leads to the Vlasov-Fokker-Planck equation coupled with some fluid equation with a friction term due tothe action/reaction principle. Here, we will model the fluid by the incompressible Eulersystem. The resulting system reads as