= Three-layer long-wave shear-flow matching
{title2=$\delta=(\alpha R)^{-1/3},\ \Delta=\alpha^{-1}$}
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 $\alpha\sim R^{-1/4}$.
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