  We consider a chemotaxis-Navier-Stokes system modelling cellular swimming in
fluid drops where an exchange of oxygen between the drop and its environment is
taken into account. This phenomenon results in an inhomogeneous Robin-type
boundary condition. Moreover, the system is studied without the logistic growth
of the bacteria population. We prove that in two dimensions, the system has a
unique global classical solution, while the existence of a global weak solution
is shown in three dimensions. In the latter case, we show that the energy is
bounded uniformly in time. A key idea is to utilise a boundary energy to derive
suitable {\it a priori} estimates. Moreover, we are able to remove the
convexity assumption on the domain.
