Analysis of obstacles immersed in viscous fluids using Brinkman's law for steady Stokes and Navier-Stokes equations

TL;DR

This paper justifies using Brinkman's law with fictitious domains to model obstacles in viscous fluids, providing convergence analysis, error estimates, and numerical validation for Stokes and Navier-Stokes equations.

Contribution

It introduces a simplified Brinkman's law approach with a singular permeability function to approximate obstacles, including convergence proofs and error bounds.

Findings

Strong convergence of solutions with obstacle approximation

Error estimates depending on penalty parameter

Numerical validation of obstacle simulation method

Abstract

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.