LAUSR

In this paper, we propose a new nonlinear model describing dynamical interaction of two species within viscous flow. The proposed model is a cross-diffusion system coupled with 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 a macroscopic models from the kinetic-fluid equations by using the micro-macro decomposition method. Basing on 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 develop a one dimensional finite volume approximation for the kinetic-fluid model, which are uniformly stable along the transition from kinetic to macroscopic regimes. Our computation method is validated with various numerical tests.