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 $\exp \left(\mathrm{C}{t}^{1∕4}\right)$, where the constant is independent of viscosity. The corresponding spatial growth has the form $\exp \left[{(z∕L)}^{1∕8}\right]$, where $z$ is the down stream distance and $L\sim Q^2{\sigma }^{-2}g$ 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.

Figure 1

The phase plane for Eq. (11) with $\Gamma =40$ and $q=1$. It is seen that, upon backward integration, trajectories quickly converge to a well-defined “unstable manifold” (solid line) for the fixed point $(v,v_z)=(0,0)$. The asymptotic solution for large $z$, $v\sim \sqrt{z}$, is shown as the dot-dashed line and is governed by inertia and gravity. The asymptotic solution for small $z$, $v\sim z^2$, is shown as the dashed line and is obtained by neglecting inertia.

Figure 2

Plot of the numerical solution of Eq. (11) for different values of $\Gamma$ and $q=1$. The viscous (18) and inertial (14) asymptotic are shown, respectively, by the straight lines.

Figure 3

General view of a viscous thread, $8\mathrm{cm}$ down from the outlet.

Figure 4

Typical segment of a thread, photographed by charge-coupled-device (CCD) camera. The thread is seen as the dark black area. The vertical bands of different shadings are caused by the illumination. The frame shown is $4\mathrm{mm}$ high and $5\mathrm{mm}$ wide. The edge of the thread is estimated as the contour level with 95% of the brightness of the background.

Figure 5

Experimental data points (with error bars) vs numerical solution for a stationary thread, $Q=0.46{\mathrm{cm}}^3∕\mathrm{s}$, $q=0.65$.

Figure 6

Experimental data points (with error bars) vs numerical solution for a stationary thread, $Q=0.32{\mathrm{cm}}^3∕\mathrm{s}$, $q=0.45$.