A mathematical clue to the separation phenomena on the two-dimensional Navier-Stokes equation

TL;DR

This paper provides a mathematical insight into reverse flow phenomena near boundaries in 2D Navier-Stokes flows, showing how boundary curvature influences flow separation and the impossibility of stationary parallel laminar flow near curved boundaries.

Contribution

It introduces a mathematical perspective on reverse flow phenomena, linking boundary curvature to flow separation and demonstrating the non-existence of stationary parallel laminar flow near curved boundaries.

Findings

Reverse flow direction opposes initial flow near boundaries.

Effect of boundary curvature increases reverse flow.

Parallel laminar flow cannot be stationary near curved boundaries.

Abstract

In general, before separating from a boundary, the flow moves toward reverse direction near the boundary against the laminar flow direction. Here in this paper, a clue to such reverse flow phenomena (in the mathematical sense) is observed. More precisely, the non-stationary two-dimensional Navier-Stokes equation with an initial datum having a parallel laminar flow (we define it rigorously in the paper) is considered. We show that the direction of the material differentiation is opposite to the initial flow direction and effect of the material differentiation (inducing the reverse flow) becomes bigger when the curvature of the boundary becomes bigger. We also show that the parallel laminar flow cannot be a stationary Navier-Stokes flow near a portion of the boundary with nonzero curvature.