Smooth solutions of the stationary Navier–Stokes equations in an infinitely long pipe, equipped with the Navier-slip or Navier–Hodge–Lions boundary condition, are considered in this paper. Three main results are presented. First, when equipped with the Navier-slip boundary condition, it is shown that, \(W^{1,\infty }\) axially symmetric solutions with zero flux at one cross section, must be swirling solutions: \(u=(- C x_2, C x_1,0)\) , and \(x_3\) -periodic solutions must be helical solutions: \(u=(-C_1x_2,C_1x_1,C_2)\) . Second, also equipped with the Navier-slip boundary condition, if the swirl or vertical component of the axially symmetric solution is independent of the vertical variable \(x_3\) , solutions are also proven to be helical solutions. In the case of the vertical component being independent of \(x_3\) , the \(W^{1,\infty }\) assumption is not needed. In the case of the swirl component being independent of \(x_3\) , the \(W^{1,\infty }\) assumption can be relaxed extensively such that the horizontal radial component of the velocity, \(u_r\) , can grow exponentially with respect to the distance to the origin. Also, by constructing a counterexample, we show that the growing assumption on \(u_r\) is optimal. Third, when equipped with the Navier–Hodge–Lions boundary condition, we can show that if the gradient of the velocity grows sublinearly, then the solution, enjoying the Liouville-type theorem, is a trivial shear flow: (0, 0, C).
