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

摘要

We investigate analytically and numerically the existence of stationarysolutions converging to zero at infinity for the incompressible Navier-Stokesequations in a two-dimensional exterior domain. More precisely, we find theasymptotic behaviour for such solutions in the case where the net force on theboundary of the domain is non-zero. In contrast to the three dimensional case,where the asymptotic behaviour is given by a scale invariant solution, theasymptote in the two-dimensional case is not scale invariant and has a wake. Weprovide an asymptotic expansion for the velocity field at infinity, which showsthat, 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 accurateat least up to second order and demonstrate how to use this information tosignificantly improve the numerical simulations. Finally, in order to check thecompatibility of the present results with rigorous results for the case of zeronet force, we consider a family of boundary conditions on the body whichinterpolate between the non-zero and the zero net force case.