The Shape and Stability of a Viscous Thread

When a viscous fluid, like oil or syrup, streams from a small orifice and falls freely under gravity, it forms a long slender thread, which can be maintained in a stable, stationary state with lengths up to several meters. We shall discuss the shape of such liquid threads and their surprising stability. It turns out that the strong advection of the falling fluid can almost outrun the Rayleigh-Plateau instability. Even for a very viscous fluid like sirup or silicone oil, the asymptotic shape and stability is independent of viscosity and small perturbations grow with time as $\exp({{\rm C} t^{{1/4}}})$, where the constant is independent of viscosity. The corresponding spatial growth has the form $\exp({(z/L)^{{1/8}}})$, where $z$ is the down stream distance and $L \sim Q^2 \sigma^{-2} g$ and where $\sigma$ is the surface tension, $g$ is the gravity and $Q$ is the flux. However, the value of viscosity determines the break-up length of a thread $L_{\nu} \sim \nu^{1/4}$ and thus the possibility of observing the $\exp({{\rm C} t^{{1/4}}})$ type asymptotics.