  We consider a chemotaxis-fluid system involving nonlinear cell diffusion of
porous medium type, signal consumption by cells, and rather general, possibly
matrix-valued, chemotactic sensitivities. It is shown that if the corresponding
diffusion exponent $m$ satisfies $m>7/6$, then for all reasonably regaular
initial data an associated initial-boundary value problem in smoothly bounded
three-dimensional domains possesses a globally defined weak solution which is
bounded. Under a mild additional assumption on the signal consumption rate, it
is moreover shown that any nontrivial of these solutions stabilizes toward a
spatially homogeneous equilibrium in the large time limit.
