Volume-constrained capillary energy of a cylindrical coating ($\mathcal{E}=\frac{1}{2}\int (h_z^2-h^2)dz$)

Expand the area of an axisymmetric cylindrical free surface to second order in film thickness and eliminate the first-order area term using conservation of the exact liquid volume. Up to positive factors and constants this gives the displayed surface energy. Its variation is $\delta \mathcal{E}/\delta h=-h-h_{zz}$. The thin-film equation is the conserved gradient flow $h_t={\partial }_z[h^3{\partial }_z(\delta \mathcal{E}/\delta h)]$. For periodic boundaries, or boundaries with vanishing energy flux,The decrease in capillary surface energy supplies viscous dissipation. On an infinite coating with unequal far films, use a local energy balance including boundary fluxes rather than an unsubtracted infinite energy integral.