 Capillary instability of an annular liquid lining

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 $k^2{a}^2=1/2$.