  The dynamics of evolving fluid films in the viscous Stokes limit is relevant
to various applications, such as the modeling of lipid bilayers in cells. While
the governing equations were formulated by Scriven in 1960, solving for the
flow of a deformable viscous surface with arbitrary shape and topology has
remained a challenge. In this study, we present a straightforward discrete
model based on variational principles to address this long-standing problem. We
replace the classical equations, which are expressed with tensor calculus in
local coordinate, with a simple coordinate-free, differential-geometric
formulation. The formulation provides a fundamental understanding of the
underlying mechanics and directly translates to discretization. We construct a
discrete analogue of the system using the Onsager variational principle, which,
in a smooth context, governs the flow of a viscous medium. In the discrete
setting, instead of term-wise discretizing the coordinate-based Stokes
equations, we construct a discrete Rayleighian for the system and derive the
discrete Stokes equations via the variational principle. This approach results
in a stable, structure-preserving variational integrator that solves the system
on general manifolds.
