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Take pressure positive in compression, outward normal on the upper interface, and curvature $\kappa \simeq -h_{xx}$. The external-minus-internal traction convention gives the base pressure $p_0=V/h_0^3$. For the symmetric Fourier perturbation, the linearized upper-surface conditions areIsotropic base stress cancels all terms due merely to rotating the normal. Velocity $u$ is even and $v$ odd in $y$.

Use the same Papkovich–Neuber representation as in Q1, with $\mathbf{\Phi }=(0,\varphi )$ and symmetric harmonic potentialsSuppress the common exponential. They giveThe zero tangential stress is $A{h}_0\cosh K+B\sinh K=0$, where $K=k{h}_0>0$. Hence $B=-A{h}_0\coth K$, $v(h_0)=A\sinh K/2$ and the normal stress becomes $\mu kA(\cosh K+K/\sinh K)$. Eliminating $A$ using the kinematic condition gives the Van der Waals rupture instability of a viscous sheet rateAttraction destabilizes thickness variations; curvature pressure opposes them.

For the clean-sheet long-wave rupture plateau, writing $s_0=3V/(\mu h_0^3)$, the rate is $s/s_0\to 1/4$. Without capillarity, the curve decreases toward $1/(2K)$ at large $K$; with $\Gamma =1$ it changes sign at $K=1$ and behaves as $-K/2$. Thus short waves are weakly unstable without surface tension, but stabilized by it. The largest clean-sheet rate is approached at long wavelength. The strictly uniform $k=0$ thickness change is not an admissible fixed-volume Fourier mode; this is the limit of nonzero waves. Stress-free surfaces permit extensional, nearly plug-like motion whose mobility compensates the weak long-wave pressure gradient. Unsteady inertia eventually matters in that limit; finite sheet extent, drainage, gravity, surface elasticity/viscosity, surfactant transport and molecular-scale corrections can also modify wavelength selection.

For the comparison with surfactant, keep the same equilibrium tension ${\gamma }_0$. Flow away from a thinning neck stretches the surface, diluting its surfactant and raising tension there. Convergence into thicker regions concentrates surfactant and lowers tension. The resulting Marangoni stress points back toward the neck, opposing the draining surface motion. It therefore reduces mobility and the rupture rate relative to the clean constant-${\gamma }_0$ sheet. This mechanism needs surface transport to sustain concentration gradients; instantaneous concentration equilibration would remove it.

For the supplied strong-surfactant rate, unstable waves satisfy $K<{\Gamma }^{-1/2}$. When $\Gamma \gg 1$, all of them have $K\ll 1$. Expanding its mobility factor givesMaximizing in $K^2$ yields the strong-surfactant long-wave rupture maximumHere $\gamma$ is the equilibrium tension ${\gamma }_0$ in the soap-film comparison. Surface immobilization removes the clean long-wave plateau and selects a finite long wavelength. The growth timescale is lengthened by a factor about $3\Gamma$ compared with the clean maximum. A linear rupture-time estimate is $s_{\max }^{-1}\log (h_0/|{\eta }_{\mathrm{initial}}|)$, not a universal bubble lifetime: drainage and changing thickness, initial defects and nonlinear rupture also matter.