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

For strong Marangoni stress and large $\Gamma =\gamma h_0^2/(3V)$, the unstable band has $K\ll 1$. Its rate is asymptotically $[3V/(\mu h_0^3)]K^2(1-\Gamma K^2)/3$. Maximizing this quadratic in $K^2$ gives the displayed finite-wavelength maximum. The growth timescale is not alone a bubble lifetime: initial disturbance size and thickness evolution also enter.