Multiple coexisting states of liquid rope coiling

A thin ‘rope’ of viscous fluid falling from a sufficient height onto a surface forms a series of regular coils. Here we investigate theoretically and experimentally a curious feature of this instability: the existence of multiple states with different frequencies at a fixed value of the fall height. Using a numerical model based on asymptotic ‘thin rope’ theory, we determine curves of coiling frequency $\Omega$ vs. fall height $H$ as functions of the fluid viscosity $\nu$, the diameter $d$ of the injection hole, the volumetric injection rate $Q$, and the acceleration due to gravity $g$. In addition to the three coiling modes previously identified (viscous, gravitational and inertial), we find a new multivalued ‘inertio-gravitational’ mode that occurs at heights intermediate between gravitational and inertial coiling. In the limit when the rope is strongly stretched by gravity and $\Pi_1\,{\equiv}\, (\nu^5/g Q^3)^{1/5}\,{\gg}\, 1$, inertio-gravititational coiling occurs in the height range $O(\Pi_1^{-1/6})\,{\leq}\, H(g/\nu^2)^{1/3}\,{\leq}\, O(\Pi_1^{-5/48})$. The frequencies of the individual branches are proportional to $(g/H)^{1/2}$, and agree closely with the eigenfrequencies of a whirling liquid string with negligible resistance to bending and twisting. The number of coexisting branches scales as $\Pi_1^{5/32}$. The predictions of the numerical model are in excellent agreement with laboratory experiments performed by two independent groups using different apparatus and working fluids. The experiments further show that interbranch transitions in the inertio-gravitational regime occur via an intermediate state with a ‘figure of eight’ geometry that changes the sense of rotation of the coiling.