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

Abstract: The purpose of this paper is the mathematical analysis of the weak solution of the mixed Dirichlet-Robin boundary value problem for the nonlinear Darcy-Forchheimer- Brinkman system in a bounded, two-dimensional Lipschitz domain, and the application of the corresponding results to the study of the lid-driven flow problem of an incompressible viscous fluid located in a square cavity filled with a porous medium. First we obtain a well-posedness result for the linear Brinkman system with Dirichlet-Neumann boundary conditions, employing a variational approach for the corresponding boundary integral equations. The result is extended afterwards to the Poisson problem for the Brinkman system and to Dirichlet- Robin boundary conditions. Further, we study the nonlinear Darcy-Forchheimer- Brinkman boundary value problem of Dirichlet and Robin type. Using the Dual Reciprocity Boundary Element Method (DRBEM), we numerically investigate a special Dirichlet-Robin boundary value problem associated to the nonlinear Darcy-Forchheimer-Brinkman system, i.e., the lid-driven cavity flow problem. The physical properties of such a flow are discussed by the geometry of the streamlines of the fluid flow for different Reynolds and Darcy numbers. Moreover, an additional sliding parameter is imposed on the moving wall and we show its importance by segmenting the upper lid into two opposite moving segments.