On the Steady Navier-Stokes system with Navier slip boundary conditions in two-dimensional channels

TL;DR

This paper studies the steady Navier-Stokes equations with Navier slip boundary conditions in 2D channels, establishing existence, uniqueness, and asymptotic behavior of solutions under certain geometric and flux conditions.

Contribution

It provides new existence and uniqueness results for steady flows with Navier slip conditions in channels with slowly growing cross-sections, and analyzes boundary behavior.

Findings

Existence of solutions for arbitrary flux in channels with slow cross-section growth.

Uniqueness of solutions for small flux in unbounded channels.

Solutions tend to shear flows at far fields.

Abstract

In this paper, we investigate the incompressible steady Navier-Stokes system with Navier slip boundary condition in a two-dimensional channel. As long as the width of cross-section of the channel grows more slowly than the linear growth, the existence of solutions with arbitrary flux is established. Furthermore, if the flux is suitably small, the solution is unique even when the width of the channel is unbounded, and approaches to the shear flows at far field where the channels tend to be straight at far fields. One of the major difficulties for the analysis on flows with Navier boundary conditions is that the tangential velocity may not be zero on the boundary so that we have to study the behavior of solutions near the boundary carefully. The crucial ingredients of analysis include the construction of an appropriate flux carrier, and the detailed analysis for the flow behavior near boundary via combining a Hardy type inequality for normal component of velocity and the divergence free property of the velocity.