= Strong-surfactant long-wave rupture maximum
{title2=$K_m^2\sim1/(2\Gamma),\quad s_{\max}\sim3V^2/(4\mu\gamma h_0^5)$}
Consider a long-wave film-rupture model whose strong-<Marangoni stress> asymptotic growth law is
$$
s(K)\sim\frac{V}{\mu h_0^3}K^2(1-\Gamma K^2),\qquad
\Gamma=\frac{\gamma h_0^2}{3V}\gg1.
$$
Here $h_0$ is the base film thickness, $\mu$ is <dynamic viscosity>, $\gamma>0$ is the base <surface tension>, $V>0$ is the model's destabilizing interaction coefficient with dimensions of <energy>, and $K=kh_0$ is dimensionless <wavenumber>. The precise relation of $V$ to a microscopic interaction depends on the chosen film model.
Set $y=K^2$. Differentiating $y(1-\Gamma y)$ gives $1-2\Gamma y$, so the unique unstable-band maximum is
$$
K_m^2\sim\frac1{2\Gamma},\qquad
s_{\max}\sim\frac{V}{4\mu h_0^3\Gamma}=\frac{3V^2}{4\mu\gamma h_0^5}.
$$
The second <derivative> is negative. Since $\Gamma\gg1$, the selected $K_m\ll1$ is consistent with the long-wave assumption. The growth timescale is not itself a rupture time or bubble lifetime: a perturbation starting at <amplitude> $\delta h_0$ must grow to a specified nonlinear scale, giving an additional logarithmic dependence on the initial disturbance, and the base thickness may evolve before that happens.
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