Fluid ‘rope trick’ investigated

Abstract

Buckling instabilities can arise from competition between axial compression and bending in slender objects. These are not restricted to solids, but also occur with fluids with free surfaces¹,⁴, in geophysics⁵ and in materials processing⁶. Here we consider a classic demonstration of fluid buckling⁷.

Main

When honey is poured from a sufficient height, it approaches one's toast as a thin filament which whirls steadily around the vertical forming a regular helical coil (illustrated with silicone oil in Fig. 1), a behaviour reminiscent of the coiling of a falling flexible rope⁸. We derive a scaling law that predicts the coiling frequency in terms of the filament radius and the flow rate.

Figure 1

FELICE FRANKEL

Photograph of a coiling filament of silicone oil (PDMS, μ=1.019×10³ g cm⁻¹ s, ρ=1 g cm⁻³; Gelest), highlighting the constant-diameter coils of diameter of approximately 0.5 cm near the surface.

The physical parameters governing the phenomenon include the fluid density ρ, viscosity μ (kinematic viscosity ν =μ/ρ), the flow rate Q, gravity g, a characteristic filament radius r, and the height, h, over which the filament falls. As h is gradually increased, the axial stagnation flow becomes unstable to bending disturbances and the filament is steadily laid out in a circular coil of radius R at a frequency Ω.

The onset of the instability⁹,¹¹ is determined by the relative magnitude of the gravitational timescale (h/g)^1/2 and the viscous timescale r²ρ/μ ,characterized by a slenderness ratio ε = r/h, or a Reynolds number Re = gr³/ν². We note that a similar parameter ρgh³/B, where B is the bending stiffness, occurs in the elastic analogue⁸. Only below a critical value of ε or Re is the axisymmetric stagnation flow unstable. Far from onset, when ε becomes sufficiently small, the flow is still mainly an axial stretching flow. However, in a small neighbourhood of the flat surface there is a highly nonlinear coiling region, which persists over a range of falling heights.

In this region, the filament radius is constant (Fig. 1), and the rotatory inertial forces due to the whirling dominate gravity and are balanced by the viscous forces due to the differential velocities between the inner and outer segments of a curved filament. The characteristic radius of curvature of the filament scales with the coil radius R, as for a coiling rope⁸. Then the differential velocity scales as Ur/R, where U is the axial velocity. The Newtonian viscous force per unit volume scales as μ Ur/R³. The force per unit volume due to centripetal and Coriolis accelerations scales as ρΩ² R. Balancing the two forces yields

A filament of almost constant diameter is laid out in a steady circular coil near the flat surface U ∼ ΩR, whereas continuity of the axial stretching flow yields Ur² ∼ Q. Substituting these relations into equation (1) yields a scaling law for the coiling frequency

To test these predictions, we studied the coiling of silicone oil, flowing at a constant rate out of circular holes in a steel plate at the base of a large reservoir of fluid. We established the dependence between the coiling frequency and filament radius by varying the height of the fall. The coiling frequency was determined by measuring the intensity fluctuations of a refracted laser aimed at the top of the coiling region. A horizontal microscope with a charge-coupled device camera was used to measure the filament radius with an accuracy of 1%. All measurements were made in a parameter regime far from the onset of coiling and also far from the regime when the coiled column itself is unstable and collapses periodically under its own weight. The data are best fitted by a power law Ω/Q^1.5 ∼ r^{− 3.58±1.6} for each flow rate (Fig. 2), and agree well with the theoretically predicted scaling law².

Figure 2: Log-log plot of the normalized coiling frequency (Ω/Q^1.5) against filament radius (r).

Triangle: hole radius, 3.2 mm, Q=0.56 g s⁻¹; diamond: hole radius, 4.0 mm, Q =0.96 g s⁻¹; circle: hole radius, 4.8 mm, Q=1.28 g s⁻¹. The data are fitted by the relation Ω/Q^1.5 ∼ r^−3.58±1.6 for each flow rate, and confirm equation (2).

Several effects such as surface tension, the relaxation of Poiseuille flow to plug flow in the neighbourhood of the orifice, air drag, non-Newtonian effects and the effect of gravity in the coiling region, have been neglected here, as they are relatively unimportant. They can be accounted for in a quantitative long-wavelength theory similar to that used to describe coiling of falling ropes⁸.