Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-21T17:53:37.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic model for the dynamics of curved viscous fibres with surface tension

Published online by Cambridge University Press:  10 March 2009

NICOLE MARHEINEKE  and
RAIMUND WEGENER
NICOLE MARHEINEKE*
Affiliation:
Technische Universität Kaiserslautern, Fachbereich Mathematik, Germany
RAIMUND WEGENER
Affiliation:
Fraunhofer-Institut für Techno- und Wirtschaftsmathematik, Kaiserslautern, Germany
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we derive and investigate an asymptotic model for the dynamics of curved viscous inertial Newtonian fibres subjected to surface tension, as they occur in rotational spinning processes. Accordingly, we extend the slender body theory of Panda, Marheineke & Wegener (Math. Meth. Appl. Sci., vol. 31, 2008, p. 1153) by including surface tension and deducing boundary conditions for the free end of the fibre. The asymptotic model accounts for the inner viscous transport and places no restrictions on either the motion or the shape of the fibre centreline. Depending on the capillary number, the boundary conditions yield an explicit description for the temporal evolution of the fibre end. We study numerically the behaviour of the fibre as a function of the effects of viscosity, gravity, rotation and surface tension.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 622 , 10 March 2009 , pp. 345 - 369
Copyright
Copyright © Cambridge University Press 2009

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.)

References

REFERENCES

Antman, S. S. 2006 Nonlinear Problems of Elasticity. Springer.Google Scholar
Bechtel, S. E., Forest, M. G., Holm, D. D. & Lin, K. J. 1988 One-dimensional closure models for three-dimensional incompressible viscoelastic free jets: von Karman flow geometry and elliptical cross-section. J. Fluid Mech. 196, 241–262.CrossRefGoogle Scholar
Bishop, R. L. 1975 There is more than one way to frame a curve. Am. Math. Monthly 82 (3), 246–251.CrossRefGoogle Scholar
Buckmaster, J. D., Nachman, A. & Ting, L. 1975 The buckling and stretching of a viscida. J. Fluid Mech. 69 (1), 1–20.Google Scholar
Caroselli, R. F. 1999 Glass textile fibers. In Man-Made Fibers, Science and Technology (ed. Mark, H. F., Atlas, S. M. & Cernia, E.), vol. 3, pp. 361–389. Interscience.Google Scholar
Chiu-Webster, S. & Lister, J. R. 2006 The fall of a viscous thread onto a moving surface: a ‘fluid-mechanical sewing machine’. J. Fluid Mech. 569, 89–111.CrossRefGoogle Scholar
Cummings, L. J. & Howell, P. D. 1999 On the evolution of non-axisymmetric viscous fibres with surface tension inertia and gravity. J. Fluid Mech. 389, 361–389.Google Scholar
Decent, S. P., King, A. C. & Wallwork, I. M. 2002 Free jets spun from a prilling tower. J. Engng Math. 42, 265–282.CrossRefGoogle Scholar
Decent, S. P., Simmons, M., Parau, E., Wong, D., King, A. & Partridge, L. 2004 Liquid jets from a rotating orifice. In Proceedings of the 5th International Conference on Multiphase Flow, Yokohama, Japan.Google Scholar
Dewynne, J. N., Howell, P. D. & Wilmott, P. 1994 Slender viscous fibers with inertia and gravity. Q. J. Mech. Appl. Math. 47, 541–555.CrossRefGoogle Scholar
Dewynne, J. N., Ockendon, J. R. & Wilmot, P. 1992 A systematic derivation of the leading-order equations for extensional flows in slender geometries. J. Fluid Mech. 244, 323–338.CrossRefGoogle Scholar
Dewynne, J. N. & Wilmott, P. 1993 Slender axisymmetric fluid jets. Math. Comput. Model. 18 (10), 69–82.CrossRefGoogle Scholar
Do Carmo, M. P. 1998 Differentialgeometrie von Kurven und Flächen. Vieweg.Google Scholar
Eggers, J. 1997 Nonlinear dynamics and breakup of free-surface flow. Rev. Mod. Phys. 69, 865–929.CrossRefGoogle Scholar
Eggers, J. & Dupont, T. 2001 Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J. Fluid Mech. 262, 205–221.Google Scholar
Entov, V. M. & Yarin, A. L. 1984 The dynamics of thin liquid jets in air. J. Fluid Mech. 140, 91–111.CrossRefGoogle Scholar
Finnicum, D. S., Weinstein, S. J. & Ruschak, K. J. 1993 The effect of applied pressure on the shape of a two-dimensional liquid curtain falling under the influence of gravity. J. Fluid Mech. 255, 647–665.Google Scholar
Forest, M. G. & Wang, Q. 1994 Dynamics of slender viscoelastic free jets. SIAM J. Appl. Math. 54 (4), 996–1032.CrossRefGoogle Scholar
Forest, M. G., Wang, Q. & Bechtel, S. E. 2000 1-D models for thin filaments of liquid crystalline polymers: coupling of orientation and flow in the stability of simple solutions. Physics D 99 (4), 527–554.Google Scholar
Forest, M. G. & Zhou, H. 2001 Unsteady analyses of thermal glass fibre drawing process. Eur. J. Appl. Math. 12, 497–496.Google Scholar
Geyling, F. T. & Homsey, G. M. 1980 Extensional instabilities of the glass fiber drawing process. Glass Technol. 21, 95–102.Google Scholar
Gospodinov, P. & Roussinov, V. 1993 Nonlinear instability during the isothermal drawing of optical fibers. Intl J. Multiph. Flow 19, 1153–1158.CrossRefGoogle Scholar
Götz, T., Klar, A., Unterreiter, A. & Wegener, R. 2008 Numerical evidence for the non-existencing of solutions of the equations describing rational spinning. Math. Models Meth. Appl. Sci. 18 (10), 1–16.Google Scholar
Howell, P. D. 1994 Extensional thin layer flows. PhD thesis, St. Catherine's College, Oxford.Google Scholar
Howell, P. D. & Siegel, M. 2004 The evolution of a slender non-axisymmetric drop in an extensional flow. J. Fluid Mech. 521, 155–180.CrossRefGoogle Scholar
Marheineke, N. & Wegener, R. 2007 Dynamics of curved viscous fibers with surface tension. Berichte des Fraunhofer ITWM 115.Google Scholar
Matovich, M. A. & Pearson, J. R. A. 1969 Spinning a molten threadline. Steady-state isothermal viscous flows. Ind. Engng Chem. Fundam. 8 (3), 512–520.Google Scholar
Panda, S. 2006 The dynamics of viscous fibers. PhD thesis, Technische Universität Kaiserslautern.Google Scholar
Panda, S., Marheineke, N. & Wegener, R. 2008 Systematic derivation of an asymptotic model for the dynamics of curved viscous fibers. Math. Meth. Appl. Sci. 31, 1153–1173.Google Scholar
Partridge, L., Wong, D. C. Y., Simmons, M. J. H., Parau, E. I. & Decent, S. P. 2005 Experimental and theoretical description of the break up of curved liquid jets in the prilling process. Chem. Engng Res. Des. 83 (11), 1267–1275.Google Scholar
Pearson, J. R. A. 1985 Mechanics of Polymer Processing. Elsevier.Google Scholar
Pearson, J. R. A. & Matovich, M. A. 1969 Spinning a molten threadline. Stability. Ind. Engng Chem. Fundam. 8 (3), 605–609.CrossRefGoogle Scholar
Ribe, N. M. 2004 Coiling of viscous jets. Proc. R. Soc. Lond. A 2051, 3223–3239.Google Scholar
Ribe, N. M., Habibi, M. & Bonn, D. 2006 a Stability of liquid rope coiling. Phys. Fluids 18, 084102.Google Scholar
Ribe, N. M., Lister, J. R. & Chiu-Webster, S. 2006 b Stability of a dragged viscous thread: onset of ‘stitching’ in a fluid-mechanical ‘sewing machine’. Phys. Fluids 18, 124105.Google Scholar
Schultz, W. W. & Davis, S. H. 1982 One-dimensional liquid fibres. J. Rheol. 26, 331–345.Google Scholar
Shah, F. T. & Pearson, J. R. A. 1972 On the stability of non-isothermal fibre spinning. Ind. Engng Chem. Fundam. 11, 145–149.CrossRefGoogle Scholar
Sierou, A. & Lister, J. R. 2003 Self-similar solutions for viscous capillary pinch-off. J. Fluid Mech. 497, 381–403.Google Scholar
Stokes, Y. M. & Tuck, E. O. 2004 The role of inertia in extensional fall of viscous drop. J. Fluid Mech. 498, 205–225.Google Scholar
Stokes, Y. M., Tuck, E. O. & Schwartz, L. W. 2000 Extensional fall of a very viscous fluid drop. Q. J. Mech. Appl. Math. 53 (4), 565–582.CrossRefGoogle Scholar
Trouton, F. R. S. 1906 On the coefficient of viscous traction and its relation to that of viscosity. Proc. R. Soc. Lond. A 77, 426–440.Google Scholar
Wallwork, I. M., Decent, S. P., King, A. C. & Schulkes, R. M. S. M. 2002 The trajectory and stability of a spiralling liquid jet. Part 1. Inviscid theory. J. Fluid Mech. 459, 43–65.CrossRefGoogle Scholar
Wong, D. C. Y., Simmons, M. J. H., Decent, S. P., Parau, E. I. & King, A. C. 2004 Break up dynamics and drop size distributions created from curved liquid jets. Intl J. Multiph. Flow 30, 499–520.Google Scholar
Yarin, A. L. 1993 Free Liquid Jets and Films: Hydrodynamics and Rheology. Longman.Google Scholar
Yarin, A. L., Gospodinov, P., Gottlieb, O. & Graham, M. D. 1999 Newtonian glass fiber drawing: chaotic variation of the cross-sectional radius. Phys. Fluids 11 (11), 3201–3208.Google Scholar