The surface-tension-driven retraction of a viscida

We consider the surface-tension-driven evolution of a thin two-dimensional sheet of viscous fluid whose ends are held a fixed distance apart. We find that the evolution is governed by a nonlocal nonlinear partial differential equation, which may be transformed, via a suitable change of time variable, to a simple linear equation. This possesses an interesting dispersion relation which indicates that it is well posed whether solved forwards or backwards in time, enabling us to determine which initial shapes will evolve to a given shape at a later time. We demonstrate that our model may be used to describe the global evolution of a viscida containing small regions of high curvature, and proceed to investigate the evolution of a profile which contains a corner. We show that the corner is not smoothed out but persists for forward and inverse time. The introduction of a pressure differential across the free surfaces is shown to provide a method of controlling the shape evolution. © by SIAM.