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Applying the stated long-wave expression formally in the intended high-viscosity regime, the numerator is dominated by $2\lambda {\kappa }^2/3$ and the displayed denominator by $\lambda$. It givesThe mechanism is inner transverse shear: the outer fluids almost immobilize the interfaces tangentially, so the middle layer has a pressure-driven parabolic velocity instead of the plug-like extensional velocity of part (b). Indeed, with nearly stationary faces at $z=\pm h$, its integrated flux is $-2h^3{p}_x/(3\mu )$. With $p_x=\Delta \rho g{h}_x$, conservation of mass gives precisely the displayed decay rate. The leading viscous dissipation is then in the middle layer, even though its neighbours have much greater dynamic viscosity.

There is a nonuniformity in taking this conclusion literally for every $\lambda \gg {\kappa }^{-1}$. A long-wave remainder written for fixed $\lambda$ cannot be assumed uniform as $\lambda$ diverges. Here this matters physically because sufficiently viscous outer fluids also resist normal interface motion. The distinction can be resolved directly, without an unspecified remainder, from the Stokes equation. For a symmetric thickness mode, the middle vertical velocity has the formIt is odd in $z$, and the middle horizontal velocity is even. At $z=h_0$, the exact outer traction map is ${\sigma }_{xz}^{+}=-2\lambda \mu ku$ and ${\sigma }_{zz}^{+}=-2\lambda \mu kw$. Tangential continuity and the normal hydrostatic jump therefore giveThe latter uses $p=\mu (w_{zzz}-k^2{w}_z)/k^2$, obtained from horizontal force balance. The kinematic condition is $w(h_0)=-s\eta$. Writing $S=\sinh \kappa$ and $C=\cosh \kappa$, tangential continuity fixesEliminating $A,B$ in the remaining two equations gives the exact symmetric-layer rateFor $\kappa \ll 1$ and $\lambda \kappa \gg 1$, this proves the uniform high-viscosity limit of viscous-layer relaxationThus the intended inner-shear answer is valid in the distinguished range ${\kappa }^{-1}\ll \lambda \ll {\kappa }^{-3}$. If $\lambda {\kappa }^3\gg 1$, the main resistance is instead outer normal motion andBoth ranges satisfy the printed $\lambda \gg {\kappa }^{-1}\gg 1$. For example $\lambda ={\kappa }^{-4}$ lies in the latter range and has $s\sim [\Delta \rho g{h}_0/(2\mu )]{\kappa }^3$, rather than a rate proportional to ${\kappa }^2$. The exact denominator contains a ${\lambda }^2{\kappa }^3$ contribution hidden by a fixed-parameter remainder. The inner-shear limit is the intended result, but an additional upper restriction on $\lambda$ is needed to make it a uniform conclusion.