 Thin liquid film on a vertical cylinder

For an exterior thin film, circumferential and axial curvature together give $h_t+(h^3[h_{zzz}+h_z+G]{)}_z/3=0$ in appropriate dimensionless variables. The $h_z$ term destabilizes long waves through the Rayleigh–Plateau instability, whereas $h_{zzz}$ damps short waves. The linear growth rate is $(k^2-k^4)/3-iGk$, with fastest growth at $|k|=1/\sqrt{2}$. The gravity parameter is $G=\rho g{a}^2/(\gamma ϵ)$ when film thickness is scaled as $ϵah$ and axial distance by cylinder radius $a$.