= Solution
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 are
$$
\boxed{v(h_0)=s\eta,\qquad
\mu(u_y+v_x)(h_0)=0,\qquad
-p'+2\mu v_y=\left(\frac{3V}{h_0^4}-\gamma k^2\right)\eta.}
$$
Isotropic 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 $\boldsymbol\Phi=(0,\phi)$ and symmetric harmonic potentials
$$
\phi=A\sinh(ky)e^{ikx},\qquad \chi=B\cosh(ky)e^{ikx}.
$$
Suppress the common exponential. They give
$$
u=-\frac{ik}{2}(Ay\sinh ky+B\cosh ky),\qquad
v=\frac A2(\sinh ky-ky\cosh ky)-\frac{kB}{2}\sinh ky,\qquad
p'=-\mu kA\cosh ky.
$$
The zero tangential stress is $Ah_0\cosh K+B\sinh K=0$, where $K=kh_0>0$. Hence $B=-Ah_0\coth K$, $v(h_0)=A\sinh K/2$ and the normal stress becomes $\mu kA(\cosh K+K/\sinh K)$. Eliminating $A$ using the kinematic condition gives \b[the <Van der Waals rupture instability of a viscous sheet> rate]
$$
\boxed{s=\frac{3V}{\mu h_0^3}
\frac{(1-\Gamma K^2)\sinh^2K}{K(2K+\sinh2K)},\qquad
\Gamma=\frac{\gamma h_0^2}{3V}.}
$$
Attraction destabilizes thickness variations; curvature pressure opposes them.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-75-rupture-rates.png]
{title=Clean-sheet rupture growth rates without capillarity and with Gamma equal to one}
For the <clean-sheet long-wave rupture plateau>, writing $s_0=3V/(\mu h_0^3)$, the rate is $s/s_0\to1/4$. 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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-75-surfactant-neck.png]
{title=Surfactant dilution at a thinning neck and opposing Marangoni surface stresses}
For the supplied strong-surfactant rate, unstable waves satisfy $K<\Gamma^{-1/2}$. When $\Gamma\gg1$, all of them have $K\ll1$. Expanding its mobility factor gives
$$
\frac{\sinh2K-2K}{4K\cosh^2K}=\frac{K^2}{3}+O(K^4),\qquad
s=\frac{s_0}{3}K^2(1-\Gamma K^2)[1+O(K^2)].
$$
Maximizing in $K^2$ yields \b[the <strong-surfactant long-wave rupture maximum>]
$$
\boxed{K_m^2=\frac1{2\Gamma}[1+O(\Gamma^{-1})],\qquad
s_{\max}=\frac{s_0}{12\Gamma}[1+O(\Gamma^{-1})]
=\frac{3V^2}{4\mu\gamma h_0^5}[1+O(\Gamma^{-1})].}
$$
Here $\gamma$ 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 $s_{\max}^{-1}\log(h_0/|\eta_{\rm initial}|)$, not a universal bubble lifetime: drainage and changing thickness, initial defects and nonlinear rupture also matter.
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