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Zusammenfassung:
Thin viscous liquid films driven by capillarity arewell described in the lubrication theory
through the thin film equation. In this article, we present an analytical solution of
this equation for a particular initial profile: a stepped perturbation. This initial condition
allows a linearization of the problem making it amenable to Fourier analysis. The
solution is obtained and characterized. As for a temperature step in the heat equation,
self-similarity of the first kind of the full evolution is demonstrated and a long-term
expression for the excess free energy is derived. In addition, hydrodynamical fields
are described. The solution is then compared to experimental profiles from a model
system: a polystyrene nanostep above the glass transition temperature which flows
due to capillarity. The excellent agreement enables a precise measurement of the capillary
velocity for this polymeric liquid, without involving any numerical simulation.
More generally, as these results hold for any viscous system driven by capillarity, the
present solution may provide a useful tool in hydrodynamics of thin viscous films.
