  We consider two dimensional Keller-Segel equations coupled with the
Navier-Stokes equations modelled by Tuval et al.[32]. Assuming that the
chemotactic sensitivity and oxygen consumption rate are nondecreasing and
differentiable, we prove that there is no blow-up in a finite time for
solutions with large initial data to chemotaxis-Navier-Stokes equations in two
dimensions. In addition, temporal decays of solutions are shown, as time tends
to infinity.
