The Feynman–Lagerstrom Criterion for Boundary Layers

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 (in: International mathematical congress, Heidelberg, 1904; see Gesammelte Abhandlungen II, 1961) and Batchelor (J Fluid Mech 1:177–190, 1956), any Euler solution arising in this limit and consisting of a single “eddy” must have constant vorticity. Feynman and Lagerstrom (in: Proceedings of IX international congress on applied mechanics, 1956) gave a procedure to select the value of this vorticity by demanding a 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 (J Fluid Mech 2:77–87, 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.