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A discrete geometric approach for simulating the dynamics of thin viscous threads. (English) Zbl 1349.76024
Summary: We present a numerical model for the dynamics of thin viscous threads based on a discrete, Lagrangian formulation of the smooth equations. The model makes use of a condensed set of coordinates, called the centerline/spin representation: the kinematic constraints linking the centerline’s tangent to the orientation of the material frame is used to eliminate two out of three degrees of freedom associated with rotations. Based on a description of twist inspired from discrete differential geometry and from variational principles, we build a full-fledged discrete viscous thread model, which includes in particular a discrete representation of the internal viscous stress. Consistency of the discrete model with the classical, smooth equations for thin threads is established formally. Our numerical method is validated against reference solutions for steady coiling. The method makes it possible to simulate the unsteady behavior of thin viscous threads in a robust and efficient way, including the combined effects of inertia, stretching, bending, twisting, large rotations and surface tension.

76A99 Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena
74K05 Strings
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Full Text: DOI arXiv
[1] Forest, M. G.; Zhou, H., Unsteady analyses of thermal Glass fibre drawing processes, European Journal of Applied Mathematics, 12, 479-496, (2001) · Zbl 1055.76020
[2] Pearson, J. R.A., Mechanics of polymer processing, (1985), Applied Science Publishers New York, NY, USA
[3] Andreassen, E.; Gunderson, E.; Hinrichsen, E. L.; Langtangen, H. P., A mathematical model for the melt spinning of polymer fibers, (Daehlem, M.; Tveito, A., Numerical Methods and Software Tools in Industrial Mathematics, (1997), Birkhäuser Boston), 195-212 · Zbl 0887.76078
[4] Shimozuru, D., Physical parameters governing the formation of pele hair and tears, Bulletin of Volcanology, 56, 3, 217-219, (1994)
[5] Herczynski, A.; Cernuschi, C.; Mahadevan, L., Painting with drops, jets, and sheets, Physics Today, 31, 31, 32-36, (2011)
[6] Barnes, G.; Woodcock, R., Liquid rope-coil effect, American Journal of Physics, 26, 205-209, (1958)
[7] Taylor, G. I., Instability of jets, threads, and sheets of viscous fluid, (Proceedings of the 12th International Congress on Applied Mechanics, Stanford, (1968), Springer), 382
[8] Mahadevan, L.; Ryu, W. S.; Samuel, A. D.T., Fluid ‘rope trick’ investigated, Nature, 392, 140, 502, (1998)
[9] Ribe, N. M., Coiling of viscous jets, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 460, 2051, 3223-3239, (2004) · Zbl 1070.76019
[10] Ribe, N. M.; Huppert, H. E.; Hallworth, M. A.; Habibi, M.; Bonn, D., Multiple coexisting states of liquid rope coiling, Journal of Fluid Mechanics, 555, 1, 275-297, (2006) · Zbl 1092.76029
[11] Ribe, N. M.; Habibi, M.; Bonn, D., Stability of liquid rope coiling, Physics of Fluids, 18, 8, 084102, (2006) · Zbl 1185.76642
[12] Chiu-Webster, S.; Lister, J. R., The fall of a viscous thread onto a moving surface: a ‘fluid-mechanical sewing machine’, Journal of Fluid Mechanics, 569, 89-111, (2006) · Zbl 1104.76007
[13] Morris, S. W.; Dawes, J. H.P.; Ribe, N. M.; Lister, J. R., Meandering instability of a viscous thread, Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 77, 6, 066218, (2008)
[14] Trouton, F. R.S., On the coefficient of viscous traction and its relation to that of viscosity, Proceedings of the Royal Society of London, A, 77, 426-440, (1906)
[15] Buckmaster, J. D.; Nachman, A.; Ting, L., The buckling and stretching of a viscida, Journal of Fluid Mechanics, 69, 01, 1-20, (1975) · Zbl 0321.76014
[16] Hirt, C. W.; Shannon, J. P., Free-surface stress conditions for incompressible-flow calculations, Journal of Computational Physics, 2, 403-441, (1968) · Zbl 0197.25901
[17] Nichols, B. D.; Hirt, C. W., Improved free surface boundary conditions for numerical incompressible-flow calculations, Journal of Computational Physics, 8, 434-448, (1971) · Zbl 0227.76048
[18] Tome, M. F.; McKee, S., GENSMAC: A computational marker and cell method for free surface flows in general domains, Journal of Computational Physics, 110, 1, 171-186, (1994) · Zbl 0790.76058
[19] Tome, M. F.; McKee, S., Numerical simulation of viscous flow: buckling of planar jets, International Journal for Numerical Methods in Fluids, 29, 705-718, (1999) · Zbl 0940.76072
[20] Oishi, C. M.; Tomé, M. F.; Cuminato, J. A.; McKee, S., An implicit technique for solving 3D low Reynolds number moving free surface flows, Journal of Computational Physics, 227, 16, 7446-7468, (2008) · Zbl 1141.76045
[21] Arne, W.; Marheineke, N.; Meister, A.; Wegener, R., Numerical analysis of Cosserat rod and string models for viscous jets in rotational spinning processes, Mathematical Models and Methods in Applied Sciences, 11, 20, (2013)
[22] Marheineke, N.; Wegener, R., Asymptotic model for the dynamics of curved viscous fibres with surface tension, Journal of Fluid Mechanics, 622, 345-369, (2009) · Zbl 1165.76328
[23] Skorobogatiy, M.; Mahadevan, L., Folding of viscous sheets and filaments, Europhysics Letters, 52, 5, 532-538, (2000)
[24] Ribe, N. M., Periodic folding of viscous sheets, Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 68, 3, 036305, (2003)
[25] Blount, M. J.; Lister, J. R., The asymptotic structure of a slender dragged viscous thread, Journal of Fluid Mechanics, 674, 489-521, (2011) · Zbl 1241.76169
[26] Habibi, M.; Rahmani, Y.; Bonn, D.; Ribe, N. M., Buckling of liquid columns, Physical Review Letters, 104, 074301, (2010)
[27] Habibi, M.; Møller, P. C.F.; Ribe, N. M.; Bonn, D., Spontaneous generation of spiral waves by a hydrodynamic instability, Europhysics Letters, 81, 38004, (2008)
[28] Maddocks, J. H.; Dichmann, D. J., Conservation laws in the dynamics of rods, Journal of elasticity, 34, 83-96, (1994) · Zbl 0808.73042
[29] Goriely, A.; Tabor, M., New amplitude equations for thin elastic rods, Physical Review Letters, 77, 17, 3537-3540, (1996)
[30] Goriely, A.; Tabor, M., Nonlinear dynamics of filaments II. nonlinear analysis, Physica D: Nonlinear Phenomena, 105, 1-3, 45-61, (1997) · Zbl 0962.74514
[31] Clauvelin, N.; Audoly, B.; Neukirch, S., Matched asymptotic expansions for twisted elastic knots: a self-contact problem with non-trivial contact topology, Journal of the Mechanics and Physics of Solids, 57, 1623-1656, (2009) · Zbl 1371.74165
[32] Hou, T. Y.; Klapper, I.; Si, H., Removing the stiffness of curvature in computing 3-D filaments, Journal of Computational Physics, 143, 2, 628-664, (1998) · Zbl 0917.76063
[33] Weiss, H., Dynamics of geometrically nonlinear rods: II - numerical methods and computational examples, Nonlinear Dynamics, 30, 383-415, (2002) · Zbl 1102.74030
[34] Goyal, S.; Perkins, N. C.; Lee, C. L., Nonlinear dynamics and loop formation in Kirchhoff rods with implications to the mechanics of DNA and cables, Journal of Computational Physics, 209, 1, 371-389, (2005) · Zbl 1329.74154
[35] Bergou, M.; Wardetzky, M.; Robinson, S.; Audoly, B.; Grinspun, E., Discrete elastic rods, ACM Transactions on Graphics, 27, 3, 63, (2008)
[36] Lim, S.; Ferent, A.; Wang, X. S.; Peskin, C. S., Dynamics of a closed rod with twist and bend in fluid, SIAM Journal on Scientific Computing, 31, 1, 273-302, (2008) · Zbl 1404.92138
[37] Jung, P.; Leyendecker, S.; Linn, J.; Ortiz, M., A discrete mechanics approach to Cosserat rod theory. part 1: static equilibria, International Journal for Numerical Methods in Engineering, (2013)
[38] Benjamin, T. B.; Mullin, T., Buckling instabilities in layers of viscous liquid subjected to shearing, Journal of Fluid Mechanics, 195, 523-540, (1988) · Zbl 0653.76072
[39] da Silveira, R.; Chaïeb, S.; Mahadevan, L., Rippling instability of a collapsing bubble, Science, 287, (2013)
[40] Radovitzky, R.; Ortiz, M., Error estimation and adaptive meshing in strongly nonlinear dynamic problems, Computer Methods in Applied Mechanics and Engineering, 172, 203-240, (1999) · Zbl 0957.74058
[41] Bergou, M.; Audoly, B.; Vouga, E.; Wardetzky, M.; Grinspun, E., Discrete viscous threads, Transactions on Graphics, 29, 4, 116, (2010)
[42] Brun, P.-T.; Ribe, N. M.; Audoly, B., A numerical investigation of the fluid mechanical sewing machine, Physics of Fluids, 24, 4, 043102, (2012)
[43] Entov, V. M.; Yarin, A. L., The dynamics of thin liquid jets in air, Journal of Fluid Mechanics, 140, 91-111, (1984) · Zbl 0551.76039
[44] Buckmaster, J., The buckling of thin viscous jets, Journal of Fluid Mechanics, 61, 3, 449-463, (1973) · Zbl 0298.76028
[45] Buckmaster, J. D.; Nachman, A., The buckling and stretching of a viscida II. effects of surface tension, The Quarterly Journal of Mechanics and Applied Mathematics, 31, 2, 157-168, (1978) · Zbl 0382.76018
[46] Panda, S.; Marheineke, N.; Wegener, R., Systematic derivation of an asymptotic model for the dynamics of curved viscous fibers, Mathematical Methods in the Applied Sciences, 31, 10, 1153-1173, (2008) · Zbl 1149.41013
[47] Bechtel, S. E.; Forest, M. G.; Bogy, D. B., A one-dimensional theory for viscoelastic fluid jets, with application to extrudate swell and draw-down under gravity, Journal of Non-Newtonian Fluid Mechanics, 21, 3, 273-308, (1986) · Zbl 0617.76009
[48] Bechtel, S. E.; Cao, J. Z.; Forest, M. G., Practical application of a higher order perturbation theory for slender viscoelastic jets and fibers, Journal of Non-Newtonian Fluid Mechanics, 41, 3, 201-273, (1992) · Zbl 0747.76016
[49] Nguyen, Q.-S., Stability and nonlinear solid mechanics, (2000), John Wiley & Sons Ltd.
[50] Audoly, B.; Pomeau, Y., Elasticity and geometry: from hair curls to the nonlinear response of shells, (2010), Oxford University Press · Zbl 1223.74001
[51] Călugăreanu, G., Lʼintégrale de Gauss et lʼanlyse des nœuds tridimensionnels, Revue de Mathématiques Pures et Appliquées, 4, 5, 5-20, (1959) · Zbl 0134.43005
[52] Pohl, W., The self-linking number of a closed space curve, Indiana University Mathematics Journal, 17, 975-985, (1968) · Zbl 0164.54005
[53] White, J. H., Self-linking and the Gauss integral in higher dimensions, American Journal of Mathematics, 91, 3, 693-728, (1969) · Zbl 0193.50903
[54] Fuller, F. B., The writhing number of a space curve, PNAS, 68, 4, 815-819, (1971) · Zbl 0212.26301
[55] Fuller, F. B., Decomposition of the linking number of a closed ribbon: a problem from molecular biology, PNAS, 75, 8, 3557-3561, (1978) · Zbl 0395.92010
[56] Aldinger, J.; Klapper, I.; Tabor, M., Formulae for the calculation and estimation of writhe, Journal of Knot Theory and Its Ramification, 4, 3, 343-372, (1995) · Zbl 0840.57003
[57] Langer, J.; Singer, D. A., Lagrangian aspects of the Kirchhoff elastic rod, SIAM Review, 38, 4, 605-618, (1996) · Zbl 0859.73040
[58] Tanaka, F.; Takahashi, H., Elastic theory of supercoiled DNA, The Journal of Chemical Physics, 83, 11, 6017, (1985)
[59] White, J. H.; Bauer, W. R., Calculation of the twist and the writhe for representative models of DNA, Journal of Molecular Biology, 189, 2, 329-341, (1986)
[60] Tobias, I.; Coleman, B. D.; Olson, W. K., The dependence of DNA tertiary structure on end conditions: theory and implications for topological transitions, The Journal of Chemical Physics, 101, 12, 10990-10996, (1994)
[61] Klenin, K.; Langowski, J., Computation of writhe in modeling of supercoiled DNA, Biopolymers, 54, 5, 307-317, (2000)
[62] Maggs, A. C., Writhing geometry at finite temperature: random walks and geometric phases for stiff polymers, The Journal of Chemical Physics, 114, 13, 5888-5896, (2001)
[63] Goldstein, R. E.; Powers, T. R.; Wiggins, C. H., Viscous nonlinear dynamics of twist and writhe, Physical Review Letters, 80, 23, 5232-5235, (1998)
[64] Wolgemuth, C. W.; Powers, T. R.; Goldstein, R. E., Twirling and whirling: viscous dynamics of rotating elastic filaments, Physical Review Letters, 84, 7, 1623-1626, (2000)
[65] Levitt, M., Protein folding by restrained energy minimization and molecular dynamics, Journal of Molecular Biology, 170, 3, 723-764, (1983)
[66] Torby, B., Advanced dynamics for engineers, HRW Series in Mechanical Engineering, (1984), Holt Rinehart and Winston
[67] Batty, C.; Bridson, R., Accurate viscous free surfaces for buckling, coiling, and rotating liquids, (Proceedings of the 2008 ACM/Eurographics Symposium on Computer Animation, (2008)), 219-228
[68] Yarin, A. L., Free liquid jets and films: hydrodynamics and rheology, (1993), Longman New York, NY · Zbl 0872.76002
[69] Otaduy, M. A.; Tamstorf, R.; Steinemann, D.; Gross, M., Implicit contact handling for deformable objects, Computer Graphics Forum, 28, 2, 559-568, (2009)
[70] Doedel, E.; Champneys, A. R.; Fairgrieve, T. F.; Kuznetsov, Y. A.; Sandstede, B.; Wang, X. J., AUTO97: continuation and bifurcation software for ordinary differential equations, (2002), available at
[71] Ribe, N. M.; Habibi, M.; Bonn, D., Liquid rope coiling, Annual Review of Fluid Mechanics, 44, 249-266, (2012) · Zbl 1353.76006
[72] Tabuteau, H.; Mora, S.; Porte, G.; Abkarian, M.; Ligoure, C., Microscopic mechanisms of the brittleness of viscoelastic fluids, Physical Review Letters, 102, 15, 155501, (2009)
[73] Majmudar, T. S.; Varagnat, M.; Hartt, W.; McKinley, G. H., Nonlinear dynamics of coiling in viscoelastic jets, (2010), preprint
[74] Bonito, A.; Picasso, M.; Laso, M., Numerical simulation of 3D viscoelastic flows with free surfaces, Journal of Computational Physics, 215, 2, 691-716, (2006) · Zbl 1173.76303
[75] Rafiee, A.; Manzari, M. T.; Hosseini, M., An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows, International Journal of Non-Linear Mechanics, 42, 10, 1210-1223, (2007) · Zbl 1200.76148
[76] Bishop, R., There is more than one way to frame a curve, The American Mathematical Monthly, 83, 246-251, (1975) · Zbl 0298.53001
[77] Wald, R. M., General relativity, (1984), University of Chicago Press · Zbl 0549.53001
[78] de Vries, R., Evaluating changes of writhe in computer simulations of supercoiled DNA, The Journal of Chemical Physics, 122, 064905, (2005)
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