Solution of Driven Thin Film Equations on Curved Substrates by the Helmholtz Minimum Dissipation Principle

The dynamical behavior of thin viscous films on curved substrates is critically important to a range of processes fundamental to the coating industry, micro-lithography and biological flows. Substrate curvature can strongly affect film shape and stability, especially when the local film thickness couples to an external field. For thin viscous films on planar domains, accurate solutions can be obtained by exploiting the gradient flow structure of the governing equation and appealing to the Helmholtz minimum dissipation principle. Here we show how this minimization principle can be extended to include thin films on curved substrates in which the local film thickness is actively coupled to an external electric field and mitigated only by capillary forces. Accurate approximate solutions are obtained by invoking a variational principle and restricting trial solutions to polynomial functions in the direction normal to the substrate. We demonstrate this approach for a thin dielectric film coating a cylindrical conductor using a boundary/finite element method. We find that this solution method offers keen physical insight into allowable film configurations not accessible to planar geometries.