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 discuss the shape of such liquid threads and their surprising stability. The stationary shapes are discussed within the long-wavelength approximation and compared to experiments. It turns out that the strong advection of the falling fluid can almost outrun the Rayleigh-Plateau instability. The asymptotic shape and stability are independent of viscosity and small perturbations grow with time as exp (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 approximately Q(2) sigma(-2) g and where sigma is the surface tension divided by density, g is the gravity, and Q is the flux. We also show that a slow spatial increase of the gravitational field can make the thread stable.