Small scales in inviscid limits of steady fluids

TL;DR

This paper investigates the inviscid limit of steady 2D incompressible Navier-Stokes flows in a channel, revealing a new small-scale structure in the vertical velocity component near boundaries.

Contribution

The authors construct steady Navier-Stokes solutions with a novel small-scale profile in the vertical velocity, extending classical boundary layer theory in the inviscid limit.

Findings

Constructed solutions with a small scale of order  in the vertical velocity.

Revealed an  small scale in the vertical velocity component near boundaries.

Extended classical Prandtl boundary layer analysis to include new small-scale features.

Abstract

In this article, we study the 2D incompressible steady Navier-Stokes equation in a channel $(-L,0)\times(-1,1)$ with the no-slip boundary condition on $\{Y = \pm 1\}$, and consider the inviscid limit $\varepsilon \to 0$. In the special case of Euler shear flow $(u_e(Y),0)$, we construct a steady Navier-Stokes solution for $\varepsilon \ll1$, $$\left\{ \begin{aligned} &u^\varepsilon \sim u_e + u_p + O(\sqrt{\varepsilon}),\\ &v^\varepsilon \sim h(Y) \exp\{Xu_e(Y)/\varepsilon\} + O(\sqrt{\varepsilon}), \end{aligned}\right. $$ where $u_p$ represents the classical Prandtl layer profile, and $h(Y)$ is an arbitrary smooth, compactly-supported function with small magnitude. While the classical Prandtl boundary layer $u_p$ exhibits a small scale of order $\sqrt{\varepsilon}$ in $Y$ near $Y = \pm 1$, the profile we construct reveals an $\varepsilon$ small scale of $Xu_e(Y)$ in the vertical velocity component.