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3 - General quasi-one-dimensional equations of dynamics of free liquid jets, capillary and bending instability

Published online by Cambridge University Press:  05 June 2014

Alexander L. Yarin ,
Behnam Pourdeyhimi  and
Seeram Ramakrishna
Alexander L. Yarin
Affiliation:
University of Illinois, Chicago
Behnam Pourdeyhimi
Affiliation:
North Carolina State University
Seeram Ramakrishna
Affiliation:
National University of Singapore
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Summary

This chapter introduces the fundamental general equations of the dynamics of free liquid jets in Sections 3.1 and 3.2. The applications of these equations encompass practically all fiber-forming processes from melt spinning, as was sketched out in Section 1.2 in Chapter 2 to melt- and solution blowing, and electrospinning of polymer micro- and nanofibers discussed in detail in Chapters 4 and 5, or drawing of optical fibers outlined in Section 6.5 in Chapter 6. They also form the framework for description of several types of instabilities characteristic of the hydrodynamics of free liquid jets in general and of fiber-forming processes in particular. These include capillary instability (Section 3.3), aerodynamically driven bending instability (Section 3.4) and buckling of liquid impinging onto a wall (Section 3.5).

Mass, momentum and moment-of-momentum balance equations

The dynamics of free liquid jets moving in air, which are characteristic of fiber-forming processes, involve growth of various perturbations. The most notable are driven by surface tension and the dynamic interaction with the surrounding air, as well as electrically driven effects. Theoretical/numerical description of the jet evolution in general, and of perturbed jets in particular, is hindered by the fact that such problems typically involve a three-dimensional, time-dependent evolution of flows with free surfaces, the locations in time of which should also be established. Solving such problems in the framework of the rigorous equations of fluid mechanics, say the Navier–Stokes equations, in most cases would be prohibitively time-consuming, even using super-computers. Additional complicating factors arise due to the rheological complexity of polymer solutions used in fiber forming, as well as the temperature-dependent variation of material properties in nonisothermal situations. Accounting for all these factors together in the framework of the rigorous equations of non-Newtonian fluid mechanics would be tremendously difficult. However, these difficulties can be relatively easily overcome in the framework of a quasi-one-dimensional description of liquid motion in the bending jets. In the works of Entov and Yarin (1980, 1984a) and Yarin (1983, 1993) the general quasi-one-dimensional equations of the straight and bending jets were derived from the integral balances of mass, momentum and moment of momentum, as well as by averaging the three-dimensional equations of hydrodynamics over the jet cross-section.

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Fundamentals and Applications of Micro- and Nanofibers , pp. 63 - 88
Publisher: Cambridge University Press
Print publication year: 2014

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References

Ambravaneswaran, B., Phillips, S. D., Basaran, O. A., 2000. Theoretical analysis of dripping faucet. Phys. Rev. Lett. 85, 5332–5335.CrossRefGoogle ScholarPubMed
Anna, S. L., McKinley, G. H., 2001. Elasto-capillary thinning and breakup of model elastic liquids. J. Rheol. 45, 115–138.CrossRefGoogle Scholar
Ashgriz, N., Mashayek, F., 1995. Temporal analysis of capillary jet breakup. J. Fluid Mech. 291, 163–190.CrossRefGoogle Scholar
Ashgriz, N., Yarin, A. L., 2011. Chapter 1. Capillary instability of free liquid jets. Springer Handbook of Atomization and Sprays, pp. 3–53, Springer, Heidelberg.CrossRefGoogle Scholar
Basaran, O. A., Suryo, R., 2007. The invisible jet. Nature Physics 3, 679–680.CrossRefGoogle Scholar
Batchelor, G. K., 2002. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge.Google Scholar
Bazilevsky, A. V., Entov, V. M., Rozhkov, A. N., 2001. Breakup of an Oldroyd liquid bridge as a method for testing the rheological properties of polymer solutions. Polym. Sci. Ser. A 43, 716–726.Google Scholar
Bazilevsky, A. V., Voronkov, S. I., Entov, V. M., Rozhkov, A. N., 1981. Orientational effects in capillary breakup of jets and threads of dilute polymer solutions. Sov. Phys. Doklady 257, 336–339 (the English version in Vol. 26).Google Scholar
Bechtel, S. E., Cao, J. Z., Forest, M. G., 1992. Practical application of a higher-order perturbation-theory for slender viscoelastic jets and fibers. J. Non-Newton. Fluid Mech. 41, 201–273.CrossRefGoogle Scholar
Bogy, D. B., Shine, S. J., Talke, F. E., 1980. Finite difference solution of the Cosserat fluid jet equations. J. Comput. Phys. 38, 294–326.CrossRefGoogle Scholar
Bousfield, D. W., Keunings, R., Marrucci, G.Denn, M. M., 1986. Nonlinear analysis of the surface tension driven breakup of viscoelastic filaments. J. Non-Newtonian Fluid Mech. 21, 79–97.CrossRefGoogle Scholar
Brenner, M. P., Shi, X. D., Nagel, S. R., 1994. Iterated instabilities during droplet fission. Phys. Rev. Lett. 73, 3391–3394.CrossRefGoogle ScholarPubMed
Brenner, M. P., Lister, J., Stone, H. A., 1996. Pinching threads, singularities and the number 0.0304. Phys. Fluids 8, 2827–2836.CrossRefGoogle Scholar
Brenner, M. P., Eggers, J., Joseph, K., Nagel, S. R., Shi, X. D., 1997. Breakdown of scaling in droplet fission at high Reynolds number. Phys. Fluids 9, 1573–1590.CrossRefGoogle Scholar
Chang, H. C., Demekhin, E. A., Kalaidin, E., 1999. Iterated stretching of viscoelastic jets. Phys. Fluids 11, 1717–1737.CrossRefGoogle 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
Clasen, C., Eggers, J., Fontelos, M. A., Li, J., McKinley, G. H., 2006. The beads-on-string structure of viscoelastic threads. J. Fluid Mech. 556, 283–308.CrossRefGoogle Scholar
Cline, H. E., Anthony, T. R., 1978. The effect of harmonics on the capillary instability of liquid jets. J. Appl. Phys. 49, 3203–3208.CrossRefGoogle Scholar
Cruickshank, J. O, Munson, B. R., 1981. Viscous fluid buckling of plane and axisymmetric jets. J. Fluid Mech. 113, 221–239.CrossRefGoogle Scholar
Debye, P., Daen, J., 1959. Stability considerations of nonviscous jets exhibiting surface or body tension. Phys. Fluids 2, 416–421.CrossRefGoogle Scholar
Donnelly, R. J., Glaberson, W., 1966. Experiments on the capillary instability of a liquid jet. Proc. Roy. Soc. London A 290, 547–556.CrossRefGoogle Scholar
Eggers, J., 1993. Universal pinching of three-dimensional axisymmetric free-surface flow. Phys. Rev. Lett. 72, 3458–3460.CrossRefGoogle Scholar
Eggers, J., 1995. Theory of drop formation. Phys. Fluids 7, 941–953.CrossRefGoogle Scholar
Eggers, J., 1997. Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865–929.CrossRefGoogle Scholar
Eggers, J., Villermaux, E., 2008. Physics of liquid jets. Reports on Progress in Physics 71, 036601.CrossRefGoogle Scholar
Entov, V. M., Kestenboim, Kh. S., 1987. Mechanics of fiber formation. Fluid Dynamics 22, 677–686.CrossRefGoogle Scholar
Entov, V. M., Yarin, A. L., 1980. Dynamical equations for a liquid jet. Fluid Dynamics 15, 644–649.CrossRefGoogle Scholar
Entov, V. M., Yarin, A. L., 1984a. The dynamics of thin liquid jets in air. J. Fluid Mech. 140, 91–111.CrossRefGoogle Scholar
Entov, V. M., Yarin, A. L., 1984b. Influence of elastic stresses on the capillary breakup of jets of dilute polymer solutions. Fluid Dynamics 19, 21–29.CrossRefGoogle Scholar
Goedde, E. F., Yuen, M. C., 1970. Experiments on liquid jet instability. J. Fluid Mech. 40, 495–511.CrossRefGoogle Scholar
Goldin, M., Yerushalmi, J., Pfeffer, R., Shinnar, R., 1969. Breakup of a laminar capillary jet of viscoelastic fluid. J. Fluid Mech. 38, 689–711.CrossRefGoogle Scholar
Gordon, M., Yerushalmi, J., Shinnar, R., 1973. Instability of jets of non-Newtonian fluids. Trans. Soc. Rheol. 17, 303–324.CrossRefGoogle Scholar
Goren, S., Gavis, J., 1961. Transverse wave motion on a thin capillary jet of a viscoelastic liquid. Phys. Fluids 4, 575–579.CrossRefGoogle Scholar
Goren, S. L., Gottlieb, M., 1982. Surface-tension-driven breakup of viscoelastic liquid threads. J. Fluid Mech. 120, 245–266.CrossRefGoogle Scholar
Grant, R. P., Middleman, S., 1966. Newtonian jet stability. AIChE Journal 12, 669–678.CrossRefGoogle Scholar
Han, T., Reneker, D. H., Yarin, A. L., 2007. Buckling of jets in electrospinning. Polymer 48, 6064–6076.CrossRefGoogle Scholar
Han, T., Yarin, A. L., Reneker, D. H., 2008. Viscoelastic electrospun jets: initial stresses and elongational rheometry. Polymer 49, 1651–1658.CrossRefGoogle Scholar
Hinch, E. J., 1977. Mechanical models of dilute polymer solutions in strong flows. Phys. Fluids 20, 22–30.CrossRefGoogle Scholar
Joseph, D. D., 1990. Fluid Dynamics of Viscoelastic Liquids. Springer, New York.CrossRefGoogle Scholar
Kase, S., Matsuo, T., 1965. Studies on melt spinning. I. Fundamental equations on the dynamics of melt spinning. J. Polym. Sci. A, 3, 2541–2554.Google Scholar
Kroesser, F. W., Middleman, S., 1969. Viscoelastic jet stability. AIChE Journal 15, 383–386.CrossRefGoogle Scholar
Lafrance, P., Ritter, R. C., 1977. Capillary breakup of a liquid jet with a random initial perturbation. Trans ASME: J. Appl. Mech. 44, 385–388.CrossRefGoogle Scholar
Lamb, H., 1959. Hydrodynamics. Cambridge University Press, Cambridge.Google Scholar
Landau, L. D., Lifshitz, E. M., 1970. Theory of Elasticity. Pergamon Press, Oxford.Google Scholar
Landau, L. D., Lifshitz, E. M., 1987. Fluid Mechanics. Pergamon Press, New York.Google Scholar
Lee, H. C., 1974. Drop formation in a liquid jet. IBM J. Res. Dev. 18, 364–369.CrossRefGoogle Scholar
Lee, W. K., Yu, K. L., Flumerfelt, R. W., 1981. Instability of stationary and uniformly moving cylindrical fluid bodies. Int. J. Multiphase Flow 7, 385–400.CrossRefGoogle Scholar
Li, J., Fontelos, M. A., 2003. Drop dynamics on the beads-on-string structure for viscoelastic jets: A numerical study. Phys. Fluids 15, 922–937.CrossRefGoogle Scholar
Loitsyanskii, L. G., 1966. Mechanics of Liquids and Gases. Pergamon Press, Oxford (the English translation of the 2nd Russian edition), and the 3rd Russian edition published by Nauka, Moscow, 1970.Google Scholar
Rayleigh, Lord, 1878. On the instability of jets. Proc. Lond. Math. Soc. 10, 4–13.CrossRefGoogle Scholar
Mansour, N. N., Lundgren, T. S., 1990. Satellite formation in capillary jet breakup. Phys. Fluids A 2, 1141–1144.CrossRefGoogle Scholar
Marheineke, N., Wegener, R., 2009. Asymptotic model for the dynamics of curved viscous fibers with surface tension. J. Fluid Mech. 622, 345–369.CrossRefGoogle Scholar
Matovich, M. A., Pearson, J. R. A., 1969. Spinning a molten threadline. Steady-state viscous flows. Ind. and Eng. Chem. Fundam. 8, 512–520.CrossRefGoogle Scholar
Munson, B. R., 1982. Viscous buckling of slender horizontal jets. Phys. Fluids 24, 1780–1783.CrossRefGoogle Scholar
Oliveira, M. S. N., McKinley, G. H., 2005. Iterated stretching and multiple beads-on-a-string phenomena in dilute solutions of highly extensible flexible polymers. Phys. Fluids 17, 071704.CrossRefGoogle Scholar
Papageorgiou, D. T., 1995. On the breakup of viscous liquid threads. Phys. Fluids 7, 1529–1544.CrossRefGoogle Scholar
Pimbley, W. T., Lee, H. C., 1977. Satellite droplet formation in a liquid jet. IBM J. Res. Dev. 21, 21–30.CrossRefGoogle Scholar
Plateau, J.Statique Experimentale et Theorique des Liquides Soumis aux Seules Forces Moleculaires. Gauthier Villars, Paris, 1873.Google Scholar
Renardy, M., 1994. Some comments on the surface-tension driven break-up (or lack of it) of viscoelastic jets. J. Non-Newton. Fluid Mech. 51, 97–107.CrossRefGoogle Scholar
Renardy, M., 1995. A numerical study of the asymptotic evolution and breakup of Newtonian and viscoelastic jets. J. Non-Newton. Fluid Mech. 59, 267–282.CrossRefGoogle Scholar
Reneker, D. H., Yarin, A. L., Fong, H., Koombhongse, S., 2000. Bending instability of electrically charged liquid jets of polymer solutions in electospinning. J. Appl. Phys. 87, 4531–4547.CrossRefGoogle Scholar
Reneker, D. H., Yarin, A. L., Zussman, E., Xu, H., 2007. Electrospinning of nanofibers from polymer solutions and melts. Adv. Appl. Mech. 41, 43–195.CrossRefGoogle Scholar
Ribe, N. M., 2002. A general theory for the dynamics of thin viscous sheets. J. Fluid Mech. 457, 255–283.CrossRefGoogle Scholar
Ribe, N. M., Lister, J. R., Chiu-Webster, S., 2006. Stability of a dragged viscous thread: Onset of ‘stitching’ in a fluid-mechanical ‘sewing machine’. Phys. Fluids 18, 124105.CrossRefGoogle Scholar
Rubin, H., 1971. Breakup of viscoelastic liquid jet. Isr. J. Technol. 9, 579–581.Google Scholar
Rubin, H., Wharshavsky, M., 1972. A note on the breakup of viscoelastic liquid jets. Isr. J. Technol. 8, 285–288.Google Scholar
Sagiv, A., Rubin, H., Takserman-Krozer, R., 1973. On the breakup of cylindrical liquid jets. Isr. J. Technol. 11, 349–354.Google Scholar
Sagiv, A., Takserman-Krozer, R., 1975. Capillary breakup of viscoelastic liquid jet of variable cross-section. Rheol. Acta 14, 420–426.CrossRefGoogle Scholar
Scarlett, B., Parkin, C. S., 1977. Droplet production on controlled jet breakup. Chem. Eng. J. 13, 127–141.CrossRefGoogle Scholar
Schümmer, P., Tebel, K. H., 1983. A new elongational rheometer for polymer solutions. J. Non-Newton. Fluid Mech. 12, 331–347.CrossRefGoogle Scholar
Skorobogatiy, M., Mahadevan, L., 2000. Folding of viscous sheets and filaments. Europhys. Lett. 52, 532–538.CrossRefGoogle Scholar
Stelter, M., Brenn, G., Yarin, A. L., Singh, R. P., Durst, F., 2000. Validation and application of a novel elongational device for polymer solutions. J. Rheol. 44, 595–616.CrossRefGoogle Scholar
Taylor, G. I., 1969a. Instability of jets, threads and sheets of viscous fluid. In: Proceedings of the 12th International Congress on Applied Mechanics, Stanford, 1968. Stanford, pp. 382–388.Google Scholar
Taylor, G. I., 1969b. Electrically driven jets. Proc. Roy. Soc. London A 313, 453–475.CrossRefGoogle Scholar
Tchavdarov, B., Yarin, A. L., Radev, S., 1993. Buckling of thin liquid jets. J. Fluid Mech. 253, 593–615.CrossRefGoogle Scholar
Weber, C., 1931. Zum Zerfall eines Flussigkeitsstrahles. Z. Angew. Math. und Mech. 11, 136–154.CrossRefGoogle Scholar
Yarin, A. L., 1982a. Dynamics of bending disturbances of nonlinear viscous liquid jets in air. J. Applied Mechanics and Technical Physics 23, No. 1, 39–43.CrossRefGoogle Scholar
Yarin, A. L., 1982b. A numerical investigation of the bending instability of thin jets of liquid. J. Applied Mechanics and Technical Physics 23, No. 4, 498–502.CrossRefGoogle Scholar
Yarin, A. L., 1983. On the dynamical equations for liquid jets. Fluid Dynamics 18, 134–136.CrossRefGoogle Scholar
Yarin, A. L., 1986. Flexural perturbations of free jets of Maxwell and Doi-Edwards liquids. J. Applied Mechanics and Technical Physics 27, 828–836.CrossRefGoogle Scholar
Yarin, A. L., 1991. Strong flows of polymeric liquids: 2. Mechanical degradation of macromolecules. J. Non-Newton. Fluid Mech. 38, 127–136.CrossRefGoogle Scholar
Yarin, A. L., 1993. Free Liquid Jets and Films: Hydrodynamics and Rheology. Longman Scientific & Technical and John Wiley & Sons, Harlow, New York.Google Scholar
Yarin, A. L., 2011. Chapter 2. Bending and buckling instabilities of free liquid jets: experiments and general quasi-one-dimensional model. Springer Handbook of Atomization and Sprays, pp. 55–73, Springer, Heidelberg.CrossRefGoogle Scholar
Yarin, A. L., Koombhongse, S., Reneker, D. H., 2001. Bending instability in electrospinning of nanofibers. J. Appl. Phys. 89, 3018–3026.CrossRefGoogle Scholar
Yarin, A. L., Tchavdarov, B., 1996. Onset of folding in plane liquid films. J. Fluid Mech. 307, 85–99.CrossRefGoogle Scholar
Ziabicki, A. 1976. Fundamentals of Fibre Formation. John Wiley & Sons, London.Google Scholar

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