= Strong-surfactant long-wave rupture maximum
{title2=$K_m^2\sim1/(2\Gamma),\quad s_{\max}\sim3V^2/(4\mu\gamma h_0^5)$}
For strong <Marangoni stress> and large $\Gamma=\gamma h_0^2/(3V)$, the unstable band has $K\ll1$. 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.
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