Relaxation of a symmetric viscous layer ($s=\Delta \rho g{h}_0\kappa /[2\mu (\lambda +2\kappa )]$)

A layer of half-thickness $h_0$, dynamic viscosity $\mu$ and density $\rho$, between semi-infinite fluids of viscosity $\lambda \mu$ and densities $\rho \mp \Delta \rho$, relaxes a symmetric thickness Fourier mode. In the plug-like extensional viscosity regime, mass conservation and $4\mu (h{u}_x{)}_x=h\Delta \rho g{h}_x-{\sigma }_{xz}^{+}$ together with the Fourier traction map for a viscous half-space give the displayed rate, where $\kappa =|k|h_0$. Inner longitudinal extension dominates for $\lambda \ll \kappa$; outer tangential resistance dominates for $\kappa \ll \lambda$ while the plug approximation remains valid.

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.