Characterization of smooth solutions to the Navier-Stokes equations in a pipe with two types of slip boundary conditions

TL;DR

This paper characterizes smooth stationary solutions of the Navier-Stokes equations in an infinite pipe with slip boundary conditions, identifying conditions under which solutions are swirling, helical, or trivial shear flows, and establishing optimal growth bounds.

Contribution

It provides a detailed classification of solutions under slip boundary conditions, including conditions for swirling, helical, and shear flows, and introduces optimal growth bounds for velocity components.

Findings

Axially symmetric solutions with zero flux are swirling solutions.

Solutions with vertical or swirl components independent of the vertical variable are helical.

Sublinear growth of velocity gradient implies the solution is a trivial shear flow.

Abstract

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: $(0,0,C)$.