Article
Dynamics of liquid rope coiling
University of Amsterdam, Amsterdamo, North Holland, Netherlands
Physical Review E (Impact Factor: 2.29). 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 littleexplored aspects of the phenomenon: The time dependence of "inertiogravitational" 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 highfrequency coiling. Inertiogravitational coiling is characterized by oscillations between states with different frequencies, and we present experimental observations of four distinct branches of such states in the frequencyfall 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 "slenderrope" 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.

 "Hence the 9 boundary conditions (2.9) fully determine the solution. We note that these boundary conditions are directly analogous to those used ← in the analysis of coiling and they give solutions that are in excellent agreement with experimental observations both of coiling (Habibi et al. 2006) and of a dragged thread (Ribe et al. 2006b). "
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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 freefall 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 ‘backwardfacing heel’ observed experimentally for low belt speeds, and provide the leadingorder 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 freefall 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’. 
Article: Coiling of Elastic Ropes
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ABSTRACT: A rope falling onto a solid surface typically forms a series of regular coils. Here, we study this phenomenon using laboratory experiments (with cotton threads and softened spaghetti) and an asymptotic "slenderrope" numerical model. The excellent agreement between the two with no adjustable parameters allows us to determine a complete phase diagram for elastic coiling comprising three basic regimes involving different force balances (elastic, gravitational, and inertial) together with resonant "whirling string" and "whirling shaft" eigenmodes in the inertial regime.  [Show abstract] [Hide abstract]
ABSTRACT: The coiling of a thin filament of viscous fluid falling onto a surface is a common and easily reproducible hydrodynamic instability. Here we report for the first time that this instability can generate regular spiral patterns, in which air bubbles are trapped in the coil and then advected horizontally by the fluid spreading on the surface. We present a simple model that explains how these beautiful patterns are formed, and how the number of spiral branches and their curvature depends on the coiling frequency, the frequency of rotation of the coiling center, the total flow rate, and the thickness of the spreading fluid film.
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