Capillary instability of an annular liquid lining ($\beta =\frac{\sigma h_0^3}{3\mu }{k}^2(a^{-2}-k^2)$)

A thin liquid lining inside a rigid cylindrical tube has pressure perturbation $p_1=-\sigma (a^{-2}{h}_1+h_{1,xx})$ at leading order in $h_0/a$. Lubrication theory gives flux $q=-h_0^3{p}_x/(3\mu )$, so mass conservation gives the displayed growth rate. Long waves with $0<k^2{a}^2<1$ grow, while short waves are stabilized by curvature. The maximum occurs at $k^2{a}^2=1/2$.

For insoluble surfactant with $\sigma ={\sigma }_0-A\Gamma$ and positive base tension, let $q=k^2{a}^2$, $\lambda =h_0^3({\sigma }_0-A{\Gamma }_0)/(3\mu a^4)$ and $\alpha =h_{0}A{\Gamma }_0/(\mu a^2)$. 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.