On the lid-driven problem in a porous cavity. A theoretical and numerical approach

The purpose of this paper is to analyze the motion of an incompressible viscous fluid through a porous medium located in a two-dimensional square cavity, i.e., the lid-driven flow problem described by a generalized Darcy–Brinkman model. First, we study the Dirichlet boundary value problem for a generalized Darcy–Brinkman system in a n−dimensional bounded Lipschitz domain (n ∈ {2, 3}), when the given data belongs to some L2−based Sobolev spaces. We obtain an existence and uniqueness result for this problem, under the assumption of sufficiently small given data. An existence and uniqueness result for the Robin boundary value problem associated to the same nonlinear system is also formulated. Further, we consider a special Dirichlet boundary value problem for the generalized Darcy–Brinkman system, i.e., the lid-driven problem associated with such a nonlinear system and we obtain the streamlines of the fluid flow for different Reynolds and Darcy numbers. Moreover, we consider an additional sliding parameter that corresponds to a mixed Dirichlet–Robin boundary value condition imposed on the upper moving wall and analyze the behavior of the flow with respect to such a parameter.