Uniform high-viscosity limit of viscous-layer relaxation ($s\sim \frac{\Delta \rho g{h}_0}{2\mu }\frac{(2/3){\kappa }^2}{1+(2/3)\lambda {\kappa }^3}$)

For $\kappa \ll 1$ and $\lambda \kappa \gg 1$, 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 $s\sim \Delta \rho g{h}_0{\kappa }^2/(3\mu )$. When $\lambda {\kappa }^3\gg 1$, 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 isThe term proportional to ${\lambda }^2{\kappa }^3$ in its denominator prevents extrapolating a fixed-$\lambda$ long-wave expansion uniformly to every large-$\lambda$ scaling.