The Viscous Catenary

A filament of an incompressible highly viscous fluid that is supported at its ends sags under the influence of gravity. Its instantaneous shape resembles that of a catenary, but evolves with time. At short times, the shape is dominated by bending deformations. At intermediate times, the effects of stretching become dominant everywhere except near the clamping boundaries where bending boundary layers persist. Finally, the filament breaks off in finite time via strain localization and pinch-off. 1. Introduction In 1690, Jacob Bernoulli issued the following challenge: determine the shape of a heavy filament or chain hanging between two points on the same horizontal line. Leibniz, Huygens and Johann Bernoulli independently discovered the equation for the shape of the catenary (derived from the Latin for chain), providing one of the first exact solutions of a nonlinear problem in continuum mechanics. Here we consider the fluid analogue of a catenary, shown in figure 1(a–f). When a thin filament of an incompressible highly viscous fluid forms a bridge across the gap between two solid vertical walls, it will sag under the influence of gravity. After a very short initial transient when it falls quickly, it slows down in response to the resistance to extension. During this process, it starts to thin, and thus speeds up, albeit inhomogeneously, eventually leading to necking and pinch-off, even in the absence of interfacial tension. The richness of this problem is paradigmatic of the free-boundary problems that are common in materials processing in such instances as filament spinning, glass manufacture etc. In addition the problem illustrates an application, to geometrically nonlinear problems, of the classical analogy between the equations of equilibrium for a linear elastic (Hookean) solid and the equations of motion for the creeping flow of a (Newtonian) fluid, due to Stokes (1845) and Rayleigh (1922).