= 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 $|k|=1/\sqrt2$. The gravity parameter is $G=\rho ga^2/(\gamma\epsilon)$ when film thickness is scaled as $\epsilon a h$ and axial distance by cylinder radius $a$.
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