From the steady Stokes and Navier-Stokes models, a penalization method has been considered by several authors for approximating those fluid equations around obstacles. In this work, we present a justification for using fictitious domains to study obstacles immersed in incompressible viscous fluids through a simplified version of Brinkman's law for porous media. If the scalar function $\psi$ is considered as the inverse of permeability, it is possible to study the singularities of $\psi$ as approximations of obstacles (when $\psi$ tends to $\infty$) or of the domain corresponding to the fluid (when $\psi = 0$ or is very close to $0$). The strong convergence of the solution of the perturbed problem to the solution of the strong problem is studied, also considering error estimates that depend on the penalty parameter, both for fluids modeled with the Stokes and Navier-Stokes equations with inhomogeneous boundary conditions. A numerical experiment is presented that validates this result and allows to study the application of this perturbed problem simulation of flows and the identification of obstacles.

翻译：从稳定的 Stokes 和 Navier- Stokes 模型中, 数位作者审议了一种惩罚方法, 以近似于障碍周围的流体方程式。 在这项工作中, 我们提出使用虚构域的理由, 通过简化版的布林克曼法则, 研究渗漏媒体中浸泡在不可压缩的粘结液中的障碍。 如果将卡路里函数 $\ psi 视为反渗透性参数, 可以研究 $\ psi 的奇数, 即障碍近似值( 当$=psi 以 $= infty$为单位时) 或与液体对应的域( 当$\psi= $= 0或非常接近 $ ) 。 正在研究如何将周遭问题的解决办法与强烈问题的解决方案紧密一致起来。 同时考虑取决于罚款参数的误差估计值, 两者都是用斯托克斯 和 纳维- Stokes 等同无色边界条件的液体的近似值。 一项数字实验展示了这个模拟结果, 并允许对每张床障碍进行研究。