Asymptotic behaviour of solutions to the stationary Navier-Stokes equations in two dimensional exterior domains with zero velocity at infinity

TL;DR

This paper analyzes the asymptotic behavior of stationary solutions to the 2D Navier-Stokes equations in exterior domains, revealing a non-scale-invariant wake structure and providing numerical validation and improved simulation techniques.

Contribution

It characterizes the asymptotic decay and wake structure of solutions with non-zero net force, extending understanding beyond the zero-force case in two dimensions.

Findings

Velocity decays like |x|^{-1/3} within the wake

Outside the wake, velocity decays like |x|^{-2/3}

Numerical results confirm the asymptotic expansion accuracy

Abstract

We investigate analytically and numerically the existence of stationary solutions converging to zero at infinity for the incompressible Navier-Stokes equations in a two-dimensional exterior domain. More precisely, we find the asymptotic behaviour for such solutions in the case where the net force on the boundary of the domain is non-zero. In contrast to the three dimensional case, where the asymptotic behaviour is given by a scale invariant solution, the asymptote in the two-dimensional case is not scale invariant and has a wake. We provide an asymptotic expansion for the velocity field at infinity, which shows that, within a wake of width $|\boldsymbol{x}|^{2/3}$, the velocity decays like $|\boldsymbol{x}|^{-1/3}$, whereas outside the wake, it decays like $|\boldsymbol{x}|^{-2/3}$. We check numerically that this behaviour is accurate at least up to second order and demonstrate how to use this information to significantly improve the numerical simulations. Finally, in order to check the compatibility of the present results with rigorous results for the case of zero net force, we consider a family of boundary conditions on the body which interpolate between the non-zero and the zero net force case.