  We consider systems of particles coupled with fluids. The particles are
described by the evolution of their density, and the fluid is described by the
Navier-Stokes equations. The particles add stress to the fluid and the fluid
carries and deforms the particles. Because the particles perform rapid random
motion, we assume that the density of particles is carried by a time average of
the fluid velocity. The resulting coupled system is shown to have smooth
solutions at all values of parameters, in two spatial dimensions.
