= Volume-constrained capillary energy of a cylindrical coating
{title2=$\mathcal E=\tfrac12\int(h_z^2-h^2)\,dz$}
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,
$$
\frac{d\mathcal E}{dt}=-\int h^3(h_z+h_{zzz})^2\,dz\le0.
$$
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.
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