The Feynman-Lagerstrom criterion for boundary layers

TL;DR

This paper investigates the Feynman-Lagerstrom criterion for boundary layers in viscous flows, proving its sufficiency for the existence of periodic boundary layer solutions and developing a novel iterative scheme to handle the implicit vorticity selection.

Contribution

It establishes the sufficiency of the Feynman-Lagerstrom criterion for boundary layer existence and introduces a new iterative method to manage implicit vorticity selection in complex domains.

Findings

The Feynman-Lagerstrom criterion is sufficient for periodic boundary layer existence.

A new iterative scheme effectively handles implicit vorticity selection.

The method applies to domains with non-constant curvature.

Abstract

We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (1904) and Batchelor (1956), any Euler solution arising in this limit and consisting of a single ``eddy" must have constant vorticity. Feynman and Lagerstrom (1956) gave a procedure to select the value of this vorticity by demanding a \textit{necessary} condition for the existence of a periodic Prandtl boundary layer description. In the case of the disc, the choice -- known to Batchelor (1956) and Wood (1957) -- is explicit in terms of the slip forcing. For domains with non-constant curvature, Feynman and Lagerstrom give an approximate formula for the choice which is in fact only implicitly defined and must be determined together with the boundary layer profile. We show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. Due to the quasilinear coupling between the solution and the selected vorticity, we devise a delicate iteration scheme coupled with a high-order energy method that captures and controls the implicit selection mechanism.