  We study a hydrodynamic limit of the Vlasov-Navier-Stokes system with
external gravity force, following the framework introduced by Han-Kwan and
Michel in [Han-Kwan - Michel, arXiv:2103.06668]. We answer a question raised in
the latter concerning the derivation of a Boussinesq-Navier-Stokes type system
for arbitrarily large times, starting from the previous fluid-kinetic system.
To do so, we consider a particular spatial geometric setting corresponding to
the half-space case. Our proof is based on an absorption effect at the boundary
which leads to crucial decay in time estimates.
