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

TL;DR

This paper analyzes the mathematical well-posedness of a mixed boundary value problem for the nonlinear Darcy-Forchheimer-Brinkman system, applying boundary element methods to study lid-driven cavity flow in porous media.

Contribution

It extends the analysis of the Brinkman system to nonlinear Darcy-Forchheimer flows with mixed boundary conditions and employs DRBEM for numerical simulation of cavity flow.

Findings

Well-posedness established for linear and nonlinear systems.

Numerical results illustrate flow behavior with different parameters.

Sliding boundary conditions significantly affect flow patterns.

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.