<p>The present work deals with a Keller-Segel-Navier-Stokes system in two-dimensional domains, which involves a cell density, an attractive chemical signal consumed by the cells, and a repulsive one produced by the cells. Potential consumption and production rates jointly with a generalized logistic law for the cells are considered, under non-flux boundary conditions for cell and chemical variables and a Dirichlet boundary condition for the velocity field. We establish the existence of global classical solutions for the system under some constraints related to the rates of attraction, consumption, and logistic competition with chemotactic sensitivities.</p>