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

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 $\alpha \sim R^{-1/4}$.

With the Airy displacement integrals, the lower pressure-to-velocity ratio is $-iq\mathrm{Ai}\prime (-qC)/I_0(C)$. The decaying upper mode gives that ratio as $\alpha /\delta$. Hence for $\alpha =k{R}^{-1/4}$ the dispersion relation is $k^{4/3}{I}_0(C)+iq\mathrm{Ai}\prime (-qC)=0$. Writing it without division remains meaningful if an individual moment vanishes. The wavespeed is $c=\delta C$ and a positive wavenumber grows temporally when $\mathrm{Im}C>0$.