On the Existence of Boundary Layer Separation for Incompressible Fluid Flow in the Half-Space

TL;DR

This paper analyzes boundary layer separation in incompressible fluid flow within a half-space, establishing conditions for its occurrence and exploring the dynamics of the separation point.

Contribution

It provides a criterion based on a singular integral for boundary layer separation and extends the analysis to Navier--Stokes solutions with similar behavior.

Findings

Boundary layer separation occurs if a certain integral is negative.

Separation does not occur if the integral is positive.

Constructed Navier--Stokes solutions exhibit similar separation behavior.

Abstract

We consider the Stokes system in the half-space with localized boundary data. We prove that a boundary layer separation point exists provided that a certain singular integral determined by the boundary data is negative. On the other hand, if this integral is strictly positive, then boundary layer separation does not occur. When boundary layer separation occurs, we also investigate the dynamics of the separation point and the sign of the pressure gradient. Furthermore, by a perturbation argument, we construct solutions to the Navier--Stokes equations in the half-space that exhibit the same qualitative behavior as in the Stokes case.