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In this paper, we propose a new nonlinear model describing the dynamical interaction of two species within a viscous flow. The proposed model is a cross‐diffusion system coupled with the Brinkman problem written in terms of velocity fluid, vorticity, and pressure and describing the flow patterns driven by an external source depending on the distribution of species. In the first part, we derive macroscopic models from the kinetic‐fluid equations by using the micro‐macro decomposition method. On the basis of the Schauder fixed‐point theory, we prove the existence of weak solutions for the derived model in the second part. The last part is devoted to developing a one‐dimensional finite volume approximation for the kinetic‐fluid model, which is uniformly stable along the transition from kinetic to macroscopic regimes. Our computation method is validated with various numerical tests. [ABSTRACT FROM AUTHOR]