Dynamics of liquid rope coiling.

Institute for Advanced Studies in Basic Sciences, Zanjan 45195-1159, Iran.
Physical Review E (Impact Factor: 2.33). 01/2007; 74(6 Pt 2):066306. DOI: 10.1103/PhysRevE.74.066306
Source: PubMed

ABSTRACT We present a combined experimental and numerical investigation of the coiling of a liquid "rope" falling on a solid surface, focusing on three little-explored aspects of the phenomenon: The time dependence of "inertio-gravitational" coiling, the systematic dependence of the radii of the coil and the rope on the experimental parameters, and the "secondary buckling" of the columnar structure generated by high-frequency coiling. Inertio-gravitational coiling is characterized by oscillations between states with different frequencies, and we present experimental observations of four distinct branches of such states in the frequency-fall height space. The transitions between coexisting states have no characteristic period, may take place with or without a change in the sense of rotation, and usually (but not always) occur via an intermediate "figure of eight" state. We present extensive laboratory measurements of the radii of the coil and of the rope within it, and show that they agree well with the predictions of a "slender-rope" numerical model. Finally, we use dimensional analysis to reveal a systematic variation of the critical column height for secondary buckling as a function of (dimensionless) flow rate and surface tension parameters.

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    ABSTRACT: Supercoiling is the large-scale secondary coiling or buckling of a structure that is already coiled at a smaller scale. Here, we show experimentally that a fluid-mechanical analog of supercoiling can occur when a thin “rope” of viscous fluid falls vertically from a great height onto a surface. For appropriate values of the viscosity ν, the flow rate Q, and the fall height H, a primary coiling instability of the rope forms a hollow coiled cylinder that then experiences a secondary buckling instability in the form of periodic folding accompanied by slow rotation of the folding plane. To delineate the conditions under which this supercoiling state appears, we carry out systematic laboratory experiments over wide ranges of Q and H using several fluids with different viscosities. We find that five different states of the rope are possible: supercoiling (SC), periodic collapse of the fluid cylinder formed by a primary coiling instability (PC), periodic folding (F), and steady coiling (C) of the rope itself, and axisymmetric stagnation flow (S). Up to three of these states can be realized for a given set of experimental conditions, and we determine detailed state diagrams showing which combinations are observed as a function of ν, Q, and H. The selection of the states is controlled by the dimensionless parameter gHQ 2/ν4 (g is the gravitational acceleration), which is directly related to the ratio of the rope radius a to the coil radius R in steady primary coiling with the parameters ν, Q, and H.
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    ABSTRACT: The behaviour of a viscous thread as it falls onto a moving belt is analysed in the asymptotic limit of a slender thread. While the bending resistance of a slender thread is small, its effects are dynamically important near the contact point with the belt, where it changes the curvature and orientation of the thread. Steady flows are shown to fall into one of three distinct regimes, depending on whether the belt is moving faster than, slower than or close to the same speed as the free-fall velocity of the thread. The key dynamical balances in each regime are explained and the role of bending stresses is found to be qualitatively different. The asymptotic solutions exhibit the ‘backward-facing heel’ observed experimentally for low belt speeds, and provide the leading-order corrections to the stretching catenary in theory previously developed for high belt speeds. The asymptotic stability of the thread to the onset of meandering is also analysed. It is shown that the entire thread, rather than the bending boundary layer alone, governs the stability. A balance between the destabilising reaction forces near the belt and the restoring force of gravity on the remainder of the thread determines the onset of meandering, and an analytic estimate for the meandering frequency is thereby obtained. At leading order, neutral stability occurs with the belt moving a little more slowly than the free-fall velocity of the thread, not when the lower part of the thread begins to be under compression, but when the horizontal reaction force at the belt begins to be slightly against the direction of belt motion. The onset of meandering is the heel ‘losing its balance’.
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