 Three-layer long-wave shear-flow matching

A high-Reynolds number long-wave mode near a wall has a viscous layer of thickness $\delta =(\alpha R{)}^{-1/3}$, an order-one region following the base profile, and a pressure-decay region of thickness $1/\alpha$. The lower Airy function solution fixes velocity displacement and fluid pressure. The middle Rayleigh equation for inviscid shear flow transports these to the outer constant-velocity region. Matching its decaying pressure produces an Airy wall-layer matching determinant. For order-one amplitudes, constant leading middle pressure requires ${\alpha }^2\ll \delta$; the distinguished balance $\alpha \sim \delta$ gives $\alpha \sim R^{-1/4}$.