This work intends to provide numerical methods to solve the Stokes problem in a complex geometry for nonhomogeneous fluids. This model describes highly viscous flows, incompressible, whose viscosity is not uniform, depending on a certain agent concentration. From a mathematical point of vue, it is an elliptic problem coupled to a convection-diffusion equation, generating a non linear dynamics. The computational algorithm is based on an hybrid grid-particles discretization. A fractional-steps algorithm enable to separate the computation of the different phenomena involved, to take advantage of both these discretizations: Lagrangian methods are adapted to solve convection operators and Eulerian methods are used for diffusion. A penalization method enables an efficient treatment of the interaction between the fluid and the domain's complex moving geometry. An iterative projection method is developed for this quasi-static problem, it permits the use of fast solvers compatible with large dimension problems. Several test are presented to validate the convergence, the conservation and the performances of the algorithm in 3D. The context of this work is the mucus flow in the lung, around the epithelium ciliated cells covering bronchus. The study focuses on the efficiency of mucus transport, which captures and expectorate pathogen agents, with respect to biological parameters. Other simulations of a micro-swimmer and a porous media flow supplement this study.
