Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-06-12T06:19:26.913Z Has data issue: false hasContentIssue false

The charmed string: self-supporting loops through air drag

Published online by Cambridge University Press:  20 August 2019

Adrian Daerr*
Matière et Systèmes Complexes, UMR 7057 Université de Paris – CNRS, 10 rue Alice Domon et Léonie Duquet, 75013 Paris, France
Juliette Courson
Matière et Systèmes Complexes, UMR 7057 Université de Paris – CNRS, 10 rue Alice Domon et Léonie Duquet, 75013 Paris, France
Margaux Abello
Matière et Systèmes Complexes, UMR 7057 Université de Paris – CNRS, 10 rue Alice Domon et Léonie Duquet, 75013 Paris, France
Wladimir Toutain
Matière et Systèmes Complexes, UMR 7057 Université de Paris – CNRS, 10 rue Alice Domon et Léonie Duquet, 75013 Paris, France
Bruno Andreotti
Laboratoire de Physique de l’ENS, UMR 8550 Ecole Normale Supérieure – CNRS – Université de Paris – Sorbonne Université, 24 rue Lhomond, 75005 Paris, France
Email address for correspondence:


The string shooter experiment uses counter-rotating pulleys to propel a closed string forward. Its steady state exhibits a transition from a gravity-dominated regime at low velocity towards a high-velocity regime where the string takes the form of a self-supporting loop. Here we show that this loop of light string is not suspended in the air due to inertia, but through the hydrodynamic drag exerted by the surrounding fluid, namely air. We investigate this drag experimentally and theoretically for a smooth long cylinder moving along its axis. We then derive the equations describing the shape of the string loop in the limit of vanishing string radius. The solutions present a critical point, analogous to a hydraulic jump, separating a supercritical zone where the wave velocity is smaller than the rope velocity, from a subcritical zone where waves propagate faster than the rope velocity. This property could be leveraged to create a white hole analogue similar to what has been demonstrated using surface waves on a flowing fluid. Loop solutions that are regular at the critical point are derived, discussed and compared to the experiment. In the general case, however, the critical point turns out to be the locus of a sharp turn of the string, which is modelled theoretically as a discontinuity. The hydrodynamic regularisation of this geometrical singularity, which involves non-local and added mass effects, is discussed on the basis of dimensional analysis.

JFM Rapids
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Biggins, J. S. 2014 Growth and shape of a chain fountain. Eur. Phys. Lett. 106, 44001.Google Scholar
Brun, P.-T., Audoly, B., Goriely, A. & Vella, D. 2016 The surprising dynamics of a chain on a pulley: lift off and snapping. Proc. R. Soc. Lond. A 472 (2190), 20160187.10.1098/rspa.2016.0187Google Scholar
Calzavarini, E., Volk, R., Lévêque, E., Pinton, J.-F. & Toschi, F. 2012 Impact of trailing wake drag on the statistical properties and dynamics of finite-sized particle in turbulence. Physica D 241 (3), 237244.Google Scholar
Collomb, D., Vanovskiy, V., Lysenko, N., Fava, S., Goux, P., Pessoa, M., Avdeev, N., Harazi, M., Magnusson, J. & Rolandi, A.2019 International Physicists’ Tournament Scholar
De Langre, E., Païdoussis, M. P., Doaré, O. & Modarres-Sadeghi, Y. 2007 Flutter of long flexible cylinders in axial flow. J. Fluid Mech. 571, 371389.10.1017/S002211200600317XGoogle Scholar
Doaré, O. & De Langre, E. 2002 The flow-induced instability of long hanging pipes. Eur. J. Mech. (A/Solids) 21 (5), 857867.10.1016/S0997-7538(02)01221-4Google Scholar
van Driest, E. R. 1956 On turbulent flow near a wall. J. Aero. Sci. 23 (11), 10071011.Google Scholar
Flack, K. A. & Schultz, M. P. 2010 Review of hydraulic roughness scales in the fully rough regime. Trans. ASME J. Fluids Engng 132 (4), 041203.Google Scholar
Gazzola, M., Argentina, M. & Mahadevan, L. 2015 Gait and speed selection in slender inertial swimmers. Proc. Natl Acad. Sci. USA 112 (13), 38743879.10.1073/pnas.1419335112Google Scholar
Gould, J. & Smith, F. S. 1980 Air drag on synthetic-fibre textile monofilaments and yarns in axial flow at speeds of up to 100 metres per second. J. Textile Inst. 71 (1), 3849.10.1080/00405008008631631Google Scholar
McMillen, T. & Goriely, A. 2003 Whip waves. Physica D 184 (1–4), 192225.Google Scholar
Mordant, N. & Pinton, J.-F. 2000 Velocity measurement of a settling sphere. Eur. Phys. J. B 18 (2), 343352.10.1007/PL00011074Google Scholar
Païdoussis, M. P. 2016 Fluid–Structure Interactions: Volume 2: Slender Structures and Axial Flow. Academic Press.Google Scholar
Rousseaux, G., Maïssa, P., Mathis, C., Coullet, P., Philbin, T. G. & Leonhardt, U. 2010 Horizon effects with surface waves on moving water. New J. Phys. 12 (9), 095018.Google Scholar
Schlichting, H. & Gersten, K. 2006 Grenzschicht-Theorie. Springer.Google Scholar
Schützhold, R. & Unruh, W. G. 2002 Gravity wave analogues of black holes. Phys. Rev. D 66 (4), 044019.Google Scholar
Selwood, A. 1962 The axial air-drag of monofilaments. J. Textile Inst. Trans. 53 (12), T576T576.10.1080/19447026208659907Google Scholar
Uno, M. 1972 A study on an air-jet loom with substreams added. J. Textile Machinery Soc. Japan 18 (2), 3744.10.4188/jte1955.18.37Google Scholar
Walton, W. & Mackenzie, C. F. 1854 Solutions of the Problems and Riders Proposed in the Senate-house Examination. Macmillan and Co.Google Scholar