Mixed Dirichlet–Robin problem for coupled anisotropic Navier–Stokes-type systems

Abstract This paper is concerned with the analysis of mixed Dirichlet–Robin boundary value problems for the anisotropic Brinkman and Darcy–Forchheimer–Brinkman systems, as well as a system of $$d \ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> coupled anisotropic Darcy–Forchheimer–Brinkman equations in bounded Lipschitz domains in $$\mathbb {R}^n, n = 2,3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . These systems provide general models for flows of anisotropic viscous incompressible fluids in heterogeneous and multidisperse porous media. By combining a variational approach and fixed-point techniques, we establish the existence and uniqueness of weak solutions in $$\textrm{L}^2-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mtext>L</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>-</mml:mo> </mml:mrow> </mml:math> based Sobolev spaces under suitable smallness conditions on the given data. To demonstrate the applicability of the theoretical results, we carry out a numerical investigation of the lid-driven flow problem in a square monodisperse or bidisperse porous cavity containing a solid circular obstacle, employing a Robin boundary condition on the moving lid and examining the effect of key physical parameters on the flow behavior.