  The coupling between the transport equation for the density and the Stokes
equation is considered in a periodic channel. More precisely, the density is
advected by pure transport by a velocity field given by the Stokes equation
with source force coming from the gravity due to differences in the density.
Dirichlet boundary conditions are taken for the velocity field on the bottom
and top of the channel, and periodic conditions in the horizontal variable. We
prove that the affine stratified density profile is stable under small
perturbations in Sobolev spaces and prove convergence of the density to another
limiting stratified density profile for large time with an explicit algebraic
decay rate. Moreover, we are able to precisely identify the limiting profile as
the decreasing vertical rearrangement of the initial density. Finally, we study
boundary layers formation to precisely characterize the long-time behavior
beyond the constant limiting profile and enlighten the optimal decay rate.
