Analysis of Obstacles Immersed in Viscous Fluids Using Brinkman's Law for Steady Stokes and Navier--Stokes Equations

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, for fluids modeled with both the Stokes and Navier-Stokes equations with inhomogeneous boundary conditions. A numerical experiment is presented that validates this result and allows us to study the application of this perturbed problem simulation of flows and the identification of obstacles.