How sedimenting droplets grow and stir the fluid

We study sedimenting droplets in binary fluid mixture in the droplet-spinodal regime using direct numerical simulations of the Cahn-Hilliard-Navier-Stokes equation (model H without noise). In zero-gravity, the characteristic length scale follows t^1/3 law where t is time. But, we are neither in the regime of Lifshitz-Sloyzov- Wagner scaling nor in the Binder-Stauffer-Siggia scaling. In non-zero gravity, the scaling crosses over to a t^2/3 scaling at late times. During the same late times, the sedimenting droplets stir the fluid consequently the kinetic energy of the fluid grows as t^5/3 . Both in absence and presence of gravity the energy spectra of the fluid shows a k^-4 scaling at intermediate range of scales although in the former case the energy decays in time and in the latter case it grows with time. We construct a scaling theory that help us understand these numerical results.