Prandtl Boundary Layers in An Infinitely Long Convergent Channel

TL;DR

This paper analyzes the behavior of solutions to the 2D steady Navier-Stokes equations in an infinitely long convergent channel at high Reynolds numbers, confirming the validity of Prandtl's boundary layer theory and exploring singular asymptotics.

Contribution

It establishes the validity of Prandtl boundary layer theory for general convergent nozzles with negative mass flux and analyzes singular behaviors near the nozzle vertex and at infinity.

Findings

Prandtl boundary layer theory holds in the convergent nozzle setting.

Singular asymptotics depend on the mass flux and boundary curvature.

The curvature-decreasing condition ensures pressure-favorable flow and stability.

Abstract

This paper concerns the large Reynold number limits and asymptotic behaviors of solutions to the 2D steady Navier-Stokes equations in an infinitely long convergent channel. It is shown that for a general convergent infinitely long nozzle whose boundary curves satisfy curvature-decreasing and any given finite negative mass flux, the Prandtl's viscous boundary layer theory holds in the sense that there exists a Navier-Stokes flow with no-slip boundary condition for small viscosity, which is approximated uniformly by the superposition of an Euler flow and a Prandtl flow. Moreover, the singular asymptotic behaviors of the solution to the Navier-Stokes equations near the vertex of the nozzle and at infinity are determined by the given mass flux, which is also important for the constructions of the Prandtl approximation solution due to the possible singularities at the vertex and non-compactness of the nozzle. One of the key ingredients in our analysis is that the curvature-decreasing condition on boundary curves of the convergent nozzle ensures that the limiting inviscid flow is pressure favourable and plays crucial roles in both the Prandtl expansion and the stability analysis.