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The purpose of this paper is the mathematical analysis of the weak solutionof the mixed Dirichlet-Robin boundary value problem for the nonlinearDarcy-Forchheimer- Brinkman system in a bounded, two-dimensional Lipschitzdomain, and the application of the corresponding results to the study of thelid-driven flow problem of an incompressible viscous fluid located in a squarecavity filled with a porous medium. First we obtain a well-posedness result forthe linear Brinkman system with Dirichlet-Neumann boundary conditions,employing a variational approach for the corresponding boundary integralequations. The result is extended afterwards to the Poisson problem for theBrinkman system and to Dirichlet- Robin boundary conditions. Further, we studythe nonlinear Darcy-Forchheimer- Brinkman boundary value problem of Dirichletand Robin type. Using the Dual Reciprocity Boundary Element Method (DRBEM), wenumerically investigate a special Dirichlet-Robin boundary value problemassociated to the nonlinear Darcy-Forchheimer-Brinkman system, i.e., thelid-driven cavity flow problem. The physical properties of such a flow arediscussed by the geometry of the streamlines of the fluid flow for differentReynolds and Darcy numbers. Moreover, an additional sliding parameter isimposed on the moving wall and we show its importance by segmenting the upperlid into two opposite moving segments.

BIE and BEM approach for the mixed Dirichlet-Robin boundary value problem for the nonlinear Darcy-Forchheimer-Brinkman system