Transport of solid particles in fluids

A theoretical framework is presented for the description of active transport of solid particles in turbulent fluids. Active transport exists when large enough particle concentration is present in the fluid and when the mass of particles is large enough for inertia effects to be non negligible. In such situations, particles trajectories dif-fer from fluid flow lines, the reaction forces of particles on the fluid are important and modify the Navier-Stokes equation, collisions between solid particles are non negligible and are affected by the turbulent fluctuations of the fluid velocity field. As a consequence the Navier-Stokes turbulent hierarchy is coupled to an inelastic Boltzmann-like kinetic equation for the phase-space one particle distribution of the solid particles. Effects of fluid velocity field fluctuations on binary collisions of par-ticles are discussed and shown to modify the collision term of the kinetic equation. Moreover, we show that the Lagrangian equations of motion for the particles in the fluid are fractional differential equations. This is due to the non-Markovian character of the hydrodynamical forces acting on particles dispersed in unsteady flows