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The aim of this paper is twofold. The first goal is to give well‐posedness results for a Robin–transmission problem and a Robin–Dirichlet problem for the Brinkman system in bounded Lipschitz domains in Rn,n≥2$$ {\mathbb{R}}^n,n\ge 2 $$. The Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by employing the layer potential theoretical methods. The second goal is to provide a well‐posedness result for a Robin–transmission problem and an existence result for the Robin–Dirichlet problem for a Navier–Stokes type system in bounded Lipschitz domains in Rn,n=2,3$$ {\mathbb{R}}^n,n=2,3 $$. Again, the Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by using the well‐posedness results in the linear case combined with a fixed point argument. In order to illustrate our theoretical results, we also consider a special case of boundary value problems of Robin–Dirichlet type, which models a fluid flow in a porous square cavity. We solve numerically this problem. [ABSTRACT FROM AUTHOR]