Bending-filament model for the buckling and coiling instability of a viscous fluid rope

A simple model is proposed for the buckling and coiling instability of a viscous “fluid rope” falling on a plane. By regarding a fluid rope as a one-dimensional flow, this model accounts for only the axial and shared viscous forces. Our model successfully reproduces several experiments with no adjustable parameters, such as the existence of three distinct coiling regimes reported in the paper by Maleki et al. [Phys. Rev. Lett. 93, 214502 (2004)]. Our model allows for the discussion of unsteady motion. An expression for the critical fall height at which the coiling frequency changes from a decrease to increase was phenomenologically derived. It was found that the coil-uncoil transition shows remarkable hysteresis only for weak gravity condition.

Figure 1

(Color online) A schematic view of a fluid rope coiling.

Figure 2

Trajectories of the bottom of the model rope start at $t=0$: (a) Trajectory in the case that axial flow is stable at the steady state ($\mathrm{Re}=10.0$, $\mathrm{Fr}=100.0$, $H∕d=10$). (b) Trajectory of circular coiling ($\mathrm{Re}=1.0$, $\mathrm{Fr}=1.0$, $H∕d=10$).

Figure 3

(Color online) Transition from viscous to gravitational coiling for $\mathrm{Fr}=100$, 200, and 400. (a) Dimensionless frequency $\Omega d∕w_{\mathrm{in}}$ versus height in the condition of $\mathrm{Re}=3$. (b) Frequency-height curve rescaled using the ${\Omega }_V$ and ${\Omega }_G$. The inset shows the experimental result by Maleki et al. [4].

Figure 4

(Color online) Dimensionless frequency $\Omega \sqrt{d∕g}$ versus height obtained from the present model. Parameters are $\nu Q∕g{d}^4=0.2$ and $g{d}^3∕{\nu }^2=0.2$ (open circle), $\nu Q∕g{d}^4=0.3$ and $g{d}^3∕{\nu }^2=0.3$ (triangle), and $\nu Q∕g{d}^4=0.2$ and $g{d}^3∕{\nu }^2=0.5$ (rectangle). The inset shows the experimental result by Maleki et al. [4].

Figure 5

(a) Numerically calculated value of $z_c$ for $\mathrm{Re}=3$ and $\mathrm{Fr}=100$. The solid line is the plot of $\zeta =H$. (b) Same data with Fig. 3 for comparison.

Figure 6

(Color online) (a) Numerically calculated coiling radius for $\mathrm{Fr}={10}^4$ and $H=15$ with increasing and decreasing Re. (b) The coiling regime in the Re-Fr plane for decreasing (△) and increasing (▽) Reynolds number.