  In the recent paper \cite{LPY2024SCM}, authors proved the existence,
uniqueness, regularity and exponential decay property of the solution to the
generalized Leray's problem in a distorted infinite-long pipe with the
Navier-slip boundary condition, where the friction ratio $\alpha>0$ and the
flux $\Phi$ is no bigger than a critical flux
$\Phi_0=\frac{C\alpha}{1+\alpha}$.
  In this paper, we consider the generalized Leray's problem with the
Navier-slip boundary condition in a straight pipe
$\mathcal{D}=\Sigma\times\mathbb{R}$. We show that if the flux of the solution
is no larger than a critical value that is independent with $\alpha$, the
solution to the problem must be the Poiseuille flow with the given flux. This
smallness condition of the flux is weaker compared with the previous result,
particularly when the friction ratio is small (close to the total slip Navier
boundary condition).
  Our proof relies primarily on a refined gradient estimate of the Poiseuille
flow with the Navier-slip boundary condition. Additionally, we provide an exact
lower bound of the critical flux $\Phi_0$ for flow in an infinite unit
cylindrical pipe. Some discussions on the essential differences between 2D and
3D problems are also given.
