Modelling of flow and transport of non-Newtonian fluids interacting with poroelastic media

We propose and analyze a model for solving the coupled problem arising in the interaction of a free fluid with a poroelastic structure. The flow in the fluid region is described by Stokes equations and in the poroelastic medium by the quasi-static Biot model. The focus of the model is on the quasi- Newtonian fluids that exhibit a shear-thinning property. We establish existence and uniqueness of the solution for two alternative formulations of the proposed model. Then we establish and show the existence and uniqueness of the solution of semidiscrete continuous-in-time formulation. We present complete stability and error analysis, as well as results of numerical simulations showing optimal rates of convergence for all variables. After that, the modeling of a transport equation in a non-linear Biot-Stokes flow will be analyzed. We use discontinuous Galerkin method to solve the transport equation. Several numerical tests are presented illustrating theoretical results and the capabilities of the method