= Uniform high-viscosity limit of viscous-layer relaxation
{title2=$s\sim\frac{\Delta\rho gh_0}{2\mu}\frac{(2/3)\kappa^2}{1+(2/3)\lambda\kappa^3}$}
For $\kappa\ll1$ and $\lambda\kappa\gg1$, the displayed formula retains both inner transverse <shear stress> and outer normal-velocity resistance. When $\kappa^{-1}\ll\lambda\ll\kappa^{-3}$, almost immobile tangential interfaces create a pressure-driven parabolic flow and $s\sim\Delta\rho gh_0\kappa^2/(3\mu)$. When $\lambda\kappa^3\gg1$, the exterior's normal <viscous dissipation> dominates and $s\sim\Delta\rho g/(2\lambda\mu|k|)$. The exact symmetric three-layer <Stokes flow> dispersion relation is
$$
s=\frac{\Delta\rho gh_0}{2\mu\kappa}\frac{\sinh^2\kappa+\lambda(\sinh\kappa\cosh\kappa-\kappa)}{\lambda\cosh(2\kappa)+(\sinh\kappa\cosh\kappa+\kappa)+\lambda^2(\sinh\kappa\cosh\kappa-\kappa)}.
$$
The term proportional to $\lambda^2\kappa^3$ in its denominator prevents extrapolating a fixed-$\lambda$ long-wave expansion uniformly to every large-$\lambda$ scaling.
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