  We present dynamic density functional theory (DDFT) incorporating general
inhomogeneous, incompressible, time dependent background flows and inertia,
describing externally driven passive colloidal systems out of equilibrium. We
start by considering the underlying nonequilibrium Langevin dynamics, including
the effect of the local velocity of the surrounding liquid bath, to obtain the
nonlinear, nonlocal partial differential equations governing the evolution of
the (coarse--grained) density and velocity fields describing the dynamics of
colloids. Additionally, we show both with heuristic arguments, and by numerical
solution, that our equations and solutions agree with existing DDFTs in the
overdamped (high friction) limit. We provide numerical solutions that model the
flow of hard spheres, in both unbounded and confined domains, and compare to
previously--derived DDFTs with and without the background flow.
