We present a partitioned algorithm aimed at extending the capabilities of existing solvers for the simulation of coupled advection-di↵usion-reaction systems and incompressible, viscous flow. The space discretization of the governing equations is based on mixed finite element methods defined on unstructured meshes, whereas the time integration hinges on an operator splitting strategy that exploits the di↵er-ences in scales between the reaction, advection, and di↵usion processes, considering the global system as a number of sequentially linked sets of partial di↵erential, and algebraic equations. The flow solver presents the advantage that all unknowns in the system (here vorticity, velocity, and pressure) can be fully decoupled and thus turn the overall scheme very attractive from the computational perspective. The robustness of the proposed method is illustrated with a series of numerical tests in 2D and 3D, relevant in the modelling of bacterial bioconvection and Boussinesq systems.