  Hydrodynamic interactions are transmitted by viscous diffusion and sound
propagation: the temporal evolution of hydrodynamic interactions by both
mechanisms is studied by direct numerical simulation in this paper. The
hydrodynamic interactions for a system of two particles in a fluid are
estimated by the velocity correlation of the particles. In an incompressible
fluid, hydrodynamic interactions propagate instantaneously at the infinite
speed of sound, followed by the temporal evolution of viscous diffusion. On the
other hand, in a compressible fluid, sound propagates at a finite speed, which
affects the temporal evolution of the hydrodynamic interactions by the order of
magnitude relation between the time scales of viscous diffusion and sound
propagation. The hydrodynamic interactions are characterized by introducing the
ratio of these time scales as an interactive compressibility factor.
