= Surfactant stabilization of an annular liquid lining
{title2=$\beta^2+q[\alpha+\lambda(q-1)]\beta+\alpha\lambda q^2(q-1)/4=0$}
For <insoluble surfactant> with $\sigma=\sigma_0-A\Gamma$ and positive base tension, let $q=k^2a^2$, $\lambda=h_0^3(\sigma_0-A\Gamma_0)/(3\mu a^4)$ and $\alpha=h_0A\Gamma_0/(\mu a^2)$. The coupled film-height and advected-<surfactant> amplitudes give the displayed growth equation. The unstable band remains $0<q<1$. Weak <surfactant> elasticity gives $\beta_{\max}=\lambda/4-3\alpha/8+O(\alpha^2/\lambda)$; strong elasticity gives $\beta_{\max}\sim\lambda/16$. The <Marangoni stress> reduces tangential surface motion; strong elasticity reduces lubrication mobility to the value for an effectively immobile interface.
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