Stokes–Brinkman–Darcy models for fluid–porous systems: derivation, analysis and validation

Flow interaction between a plain-fluid region in contact with a porous layer attracted significant attention from modelling and analysis sides due to numerous applications in biology, environment and industry. In the most widely used coupled model, fluid flow is described by the Stokes equations in the free-flow domain and Darcy’s law in the porous medium, and complemented by the appropriate interface conditions. However, traditional coupling concepts are restricted, with a few exceptions, to one-dimensional flows parallel to the fluid–porous interface.

In this work, we use an alternative approach to model interaction between the plain-fluid domain and porous medium by considering a transition zone, and propose the full- and hybrid-dimensional Stokes–Brinkman–Darcy models. In the first case, the equi-dimensional Brinkman equations are considered in the transition region, and the appropriate interface conditions are set on the top and bottom of the transition zone. In the latter case, we perform a dimensional model reduction by averaging the Brinkman equations in the normal direction and using the proposed transmission conditions. The well-posedness of both coupled problems is proved, and some numerical simulations are carried out in order to validate the concepts.