  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 Brinkham'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 non-homogeneous 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.
