Approximation of rigid obstacle by highly viscous fluid

TL;DR

This paper investigates approximating a rigid obstacle in 2D Navier-Stokes flows by using a highly viscous fluid, establishing regularity results and numerical evidence for the approximation's effectiveness.

Contribution

It introduces a novel approach to approximate rigid obstacles with highly viscous fluids and proves regularity and gradient estimates for the solutions.

Findings

Pointwise gradient estimates for velocity are established.

Numerical results show effective approximation with small penalty parameters.

The method provides a feasible way to simulate obstacles in fluid flows.

Abstract

In this paper, we study the problem concerning the approximation of a rigid obstacle for flows governed by the stationary Navier-Stokes equations in the two-dimensional case. The idea is to consider a highly viscous fluid in the place of the obstacle. Formally, as the fluid viscosity goes to infinity inside the region occupied by the obstacle, we obtain the original problem in the limit. The main goal is to establish a better regularity of approximate solutions. In particular, the pointwise estimate for the gradient of the velocity is proved. We give numerical evidence that the penalized solution can reasonably approximate the problem, even for relatively small values of the penalty parameter.