  Given an obstacle in $\mathbb{R}^3$ and a non-zero velocity with small
amplitude at the infinity, we construct the unique steady Boltzmann solution
flowing around such an obstacle with the prescribed velocity as $|x|\to
\infty$, which approaches the corresponding Navier-Stokes steady flow, as the
mean-free path goes to zero. Furthermore, we establish the error estimate
between the Boltzmann solution and its Navier-Stokes approximation. Our method
consists of new $L^6$ and $L^3$ estimates in the unbounded exterior domain, as
well as an iterative scheme preserving the positivity of the distribution
function.
