  In this paper, we investigate the Liouville-type theorems for axisymmetric
solutions to steady Navier-Stokes system in a layer domain. The both cases for
the flows supplemented with no-slip boundary and Navier boundary conditions are
studied. If the width of the outlet grows at a rate less than
$R^{\frac{1}{2}}$, any bounded solution is proved to be trivial. Meanwhile, if
the width of the outlet grows at a rate less than $R^{\frac{4}{5}}$, every
D-solution is proved to be trivial. The key idea of the proof is to establish a
Saint-Venant type estimate that characterizes the growth of Dirichlet integral
of nontrivial solutions.
