Strong-surfactant long-wave rupture maximum ($K_m^2\sim 1/(2\Gamma ),s_{\max }\sim 3V^2/(4\mu \gamma h_0^5)$)

Consider a long-wave film-rupture model whose strong-Marangoni stress asymptotic growth law isHere $h_0$ 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 $K=k{h}_0$ 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 isThe second derivative is negative. Since $\Gamma \gg 1$, the selected $K_m\ll 1$ 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.