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Extension of a viscous thread with temperature-dependent viscosity and surface tension. (English) Zbl 1445.76040

Summary: We consider the evolution of a long and thin vertically aligned axisymmetric viscous thread that is composed of an incompressible fluid. The thread is attached to a solid wall at its upper end, experiences gravity and is pulled at its lower end by a fixed force. As the thread evolves, it experiences either heating or cooling by its environment. The heating affects the evolution of the thread because both the viscosity and surface tension of the thread are assumed to be functions of the temperature. We develop a framework that can deal with threads that have arbitrary initial shape, are non-uniformly preheated and experience spatially non-uniform heating or cooling from the environment during the pulling process. When inertia is completely neglected and the temperature of the environment is spatially uniform, we obtain analytic solutions for an arbitrary initial shape and temperature profile. In addition, we determine the criteria for whether the cross-section of a given fluid element will ever become zero and hence determine the minimum stretching force that is required for pinching. We further show that the dynamics can be quite subtle and leads to surprising behaviour, such as non-monotonic behaviour in time and space. We also consider the effects of non-zero Reynolds number. If the temperature of the environment is spatially uniform, we show that the dynamics is subtly influenced by inertia and that the location at which the thread will pinch is selected by a competition between three distinct mechanisms. In particular, for a thread with initially uniform radius and a spatially uniform environment but with a non-uniform initial temperature profile, pinching can occur either at the hottest point, at the points near large thermal gradients or at the pulled end, depending on the Reynolds number. Finally, we show that similar results can be obtained for a thread with initially uniform radius and uniform temperature profile but exposed to a spatially non-uniform environment.

MSC:

76D45 Capillarity (surface tension) for incompressible viscous fluids
Full Text: DOI

References:

[1] Argyros, A., Microstructures in polymer fibres for optical fibres, THz waveguides, and fibre-based metamaterials, ISRN Optics, (2013)
[2] Ashby, M. & Jones, R. H. D.2013Engineering Materials 2: An Introduction to Microstructures and Processing, 4th edn, pp. 393-473. Butterworth-Heinemann.
[3] Bansal, N. P.; Doremus, R. H., Handbook of Glass Properties. Materials Engineering Department Rensselaer Polytechnic Institute Troy, (1986), Academic
[4] Bingham, P. A.2010Design of new energy-friendly compositions. In Fiberglass and Glass Technology: Energy-Friendly Compositions and Applications (ed. Wallenberger, F. T. & Bingham, P. A.), pp. 267-351. Springer. doi:10.1007/978-1-4419-0736-3_7
[5] Blyth, M. G.; Bassom, A. P., Flow of a liquid layer over heated topography, Proc. R. Soc. Lond. A, 468, 4067-4087, (2012) · Zbl 1371.76010 · doi:10.1098/rspa.2012.0409
[6] Bradshaw-Hajek, B. H.; Stokes, Y. M.; Tuck, E. O., Computation of extensional fall of slender viscous drops by a one-dimensional Eulerian method, SIAM J. Appl. Maths, 67, 1166-1182, (2007) · Zbl 1134.35085 · doi:10.1137/050646743
[7] Chen, J. C.; Sheu, J. C.; Lee, Y. T., Maximum stable length of nonisothermal liquid bridges, Phys. Fluids, 2, 1118-1123, (1990) · doi:10.1063/1.857611
[8] Chen, Y. J.; Abbaschian, R.; Steen, P. H., Thermocapillary suppression of the Plateau-Rayleigh instability: a model for long encapsulated liquid zones, J. Fluid Mech., 485, 97-113, (2003) · Zbl 1070.76028 · doi:10.1017/S0022112003004373
[9] Chinnov, E. A.; Shatskiy, E. N., Thermocapillary instabilities in a falling liquid film at small Reynolds numbers, Tech. Phys. Lett., 40, 7-9, (2014) · doi:10.1134/S1063785014010039
[10] Craster, R. V.; Matar, O. K., Dynamics and stability of thin liquid films, Rev. Mod. Phys., 81, 1131-1198, (2009) · doi:10.1103/RevModPhys.81.1131
[11] Cummings, L. J.; Howell, P. D., On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity, J. Fluid Mech., 389, 361-389, (1999) · Zbl 0953.76022 · doi:10.1017/S0022112099005030
[12] D’Alessio, S. J. D.; Pascal, J. P.; Jasmine, H. A., Film flow over heated wavy inclined surfaces, J. Fluid Mech., 665, 418-456, (2010) · Zbl 1225.76039 · doi:10.1017/S0022112010004003
[13] Denn, M. M., Continuous drawing of liquids to form fibers, Annu. Rev. Fluid Mech., 12, 365-387, (1980) · Zbl 0466.76010 · doi:10.1146/annurev.fl.12.010180.002053
[14] Denn, M. M., Polymer Melt Processing: Foundations in Fluid Mechanics and Heat Transfer. (Part of Cambridge Series in Chemical Engineering), (2014), Cambridge University Press
[15] Dewynne, J. N.; Howell, P. D.; Wilmott, P., Slender viscous fibers with inertia and gravity, Q. J. Mech. Appl. Maths, 47, 541-555, (1994) · Zbl 0820.76035 · doi:10.1093/qjmam/47.4.541
[16] Dewynne, J. N.; Ockendon, J. R.; Wilmott, P., A systematic derivation of the leading-order equations for extensional flows in slender geometries, J. Fluid Mech., 244, 323-338, (1992) · Zbl 0760.76020 · doi:10.1017/S0022112092003094
[17] Dijkstra, H. A.; Steen, P. H., Thermocapillary stabilization of the capillary break up of an annular film of fluid, J. Fluid Mech., 229, 205-228, (1991) · Zbl 0850.76214 · doi:10.1017/S0022112091003002
[18] Eggers, J., Universal pinching of 3D axisymmetric free-surface flow, Phys. Rev. Lett., 71, 3458-3460, (1993) · doi:10.1103/PhysRevLett.71.3458
[19] Eggers, J.; Villermaux, E., Physics of liquid jets, Rep. Prog. Phys., 71, (2008) · doi:10.1088/0034-4885/71/3/036601
[20] Fitt, A. D.; Furusawa, K.; Monro, T. M.; Please, C. P., Modeling the fabrication of hollow fibers: capillary drawing, J. Lightwave Technol., 19, 1924-1931, (2001) · doi:10.1109/50.971686
[21] Fluegel, A., Glass viscosity calculation based on a global statistical modeling approach, Glass Technol.: Eur. J. Glass Sci. Technol., 48, 13-30, (2007)
[22] Forest, M. G.; Zhou, H., Unsteady analysis of thermal glass fiber drawing processes, Eur. J. Appl. Maths, 12, 479-496, (2001) · Zbl 1055.76020
[23] Gallacchi, R.; Kölsch, S.; Kneppe, H.; Meixner, A. J., Well-shaped fibre tips by pulling with a foil heater, J. Microsc., 202, 182-187, (2001) · doi:10.1046/j.1365-2818.2001.00869.x
[24] Gospodinov, P.; Yarin, A. L., Draw resonance of optical microcapillaries in non-isothermal drawing, Intl J. Multiphase Flow, 23, 967-976, (1997) · Zbl 1135.76426 · doi:10.1016/S0301-9322(97)00016-5
[25] Goussis, D. A.; Kelly, R. E., Surface wave and thermocapillary instability in a liquid film flow, J. Fluid Mech., 223, 25-45, (1991) · Zbl 0717.76038 · doi:10.1017/S0022112091001313
[26] Griffiths, I. M.; Howell, P. D., Mathematical modelling of non-axisymmetric capillary tube drawing, J. Fluid Mech., 605, 181-206, (2008) · Zbl 1145.76018
[27] Gupta, G.; Schultz, W. W., Non-isothermal flows of Newtonian slender glass fibers, Intl J. Non-Linear Mech., 33, 151-163, (1998) · Zbl 0885.76004 · doi:10.1016/S0020-7462(96)00143-6
[28] Hu, J.; Ben Hadid, H.; Daniel, H., Linear temporal and spatio-temporal stability analysis of a binary liquid film flowing down an inclined uniformly heated plate, J. Fluid Mech., 599, 269-298, (2008) · Zbl 1151.76479 · doi:10.1017/S0022112007000110
[29] Huang, H.; Miura, R. M.; Ireland, W.; Puil, E., Heat-induced stretching of a glass tube under tension: application to glass microelectrodes, SIAM J. Appl. Maths, 63, 1499-1519, (2003) · Zbl 1049.76022 · doi:10.1137/S0036139901393469
[30] Huang, H.; Miura, R. M.; Wylie, J. J., Optical fiber drawing and dopant transport, SIAM J. Appl. Maths, 69, 330-347, (2008) · Zbl 1366.76031 · doi:10.1137/070700176
[31] Huang, H.; Wylie, J. J.; Miura, R. M.; Howell, P. D., On the formation of glass microelectrodes, SIAM J. Appl. Maths, 67, 630-666, (2007) · Zbl 1116.76027 · doi:10.1137/050640722
[32] Kabova, Y. O.; Kuznetsov, V. V.; Kabov, O. A., Temperature dependent viscosity and surface tension effects on deformations of non-isothermal falling liquid film, Intl J. Heat Mass Transfer, 55, 1271-1278, (2012) · Zbl 1262.80025 · doi:10.1016/j.ijheatmasstransfer.2011.09.020
[33] Kaye, A., Convected coordinates and elongational flow, J. Non-Newtonian Fluid Mech., 40, 55-77, (1991) · Zbl 0726.76004 · doi:10.1016/0377-0257(91)87026-T
[34] Kalpakjian, S.; Schmid, S. R., Manufacturing Processes for Engineering Materials, (2007), Prentice Hall
[35] Kalliadasis, S.; Kiyashko, A.; Demekhin, E. A., Marangoni instability of a thin liquid film heated from below by a local heat source, J. Fluid Mech., 475, 377-408, (2003) · Zbl 1081.76028 · doi:10.1017/S0022112002003014
[36] Kalliadasis, S.; Demekhin, E. A.; Ruyer-Quil, C., Thermocapillary instability and wave formation on a film falling down a uniformly heated plane, J. Fluid Mech., 492, 303-338, (2003) · Zbl 1063.76030 · doi:10.1017/S0022112003005809
[37] Kostecki, R.; Ebendorff-Heidepriem, H.; Warren-Smith, S. C.; Monro, T. M., Predicting the drawing conditions for microstructured optical fiber fabrication, Opt. Mater. Express, 4, 29-40, (2014) · doi:10.1364/OME.4.000029
[38] Kuhlmann, H. C.; Rath, H. J., Hydrodynamic instability in cylindrical thermocapillary liquid bridges, J. Fluid Mech., 247, 247-274, (1993) · Zbl 0767.76020 · doi:10.1017/S0022112093000461
[39] Mashayek, F.; Ashgriz, N., Nonlinear instability of liquid jets with thermocapillarity, J. Fluid Mech., 283, 97-123, (1995) · Zbl 0833.76026 · doi:10.1017/S0022112095002242
[40] Matovich, M. A.; Pearson, J. R. A., Spinning a molten threadline steady-state isothermal viscous flows, Ind. Engng Chem. Fundam., 8, 512-520, (1969) · doi:10.1021/i160031a023
[41] Miladinova, S.; Slavtchev, S.; Lebon, G., Long-wave instabilities of non-uniformly heated falling films, J. Fluid Mech., 453, 153-175, (2002) · Zbl 1053.76024 · doi:10.1017/S0022112001006814
[42] Pearson, J. R. A.; Shah, Y. T., Stability analysis of the fibre spinning process, Trans. Soc. Rheol., 16, 519-533, (1973) · doi:10.1122/1.549263
[43] Roe, R. J., Surface tension of polymer liquids, J. Phys. Chem., 72, 2013-2017, (1968) · doi:10.1021/j100852a025
[44] Samanta, A., Stability of liquid film falling down a vertical non-uniformly heated wall, Physica D, 237, 2587-2598, (2008) · Zbl 1149.76024
[45] Scheid, B.; Ruyer-Quil, C.; Thiele, U.; Kabov, O. A.; Legros, J. C.; Colinet, P., Validity domain of the Benney equation including the Marangoni effect for closed and open flows, J. Fluid Mech., 527, 303-335, (2005) · Zbl 1142.76380 · doi:10.1017/S0022112004003179
[46] Seward, T. P. III & Vascott, T. (Eds) 2005High Temperture Glass Melt Property Database for Process Modeling. The American Ceramic Society, Wiley.
[47] Shah, Y. T.; Pearson, J. R. A., On the stability of nonisothermal fibre spinning, Ind. Engng Chem. Fundam., 11, 145-149, (1972) · doi:10.1021/i160042a001
[48] Shah, Y. T.; Pearson, J. R. A., On the stability of nonisothermal fibre spinning-general case, Ind. Engng Chem. Fundam., 11, 150-153, (1972) · doi:10.1021/i160042a002
[49] Stokes, Y. M.; Tuck, E. O.; Schwartz, L. W., Extensional fall of a very viscous fluid drop, Q. J. Mech. Appl. Maths, 53, 565-582, (2000) · Zbl 0969.76019 · doi:10.1093/qjmam/53.4.565
[50] Stokes, Y. M.; Tuck, E. O., The role of inertia in extensional fall of a viscous drop, J. Fluid Mech., 498, 205-225, (2004) · Zbl 1065.76044 · doi:10.1017/S0022112003006682
[51] Stokes, Y. M.; Bradshaw-Hajek, B. H.; Tuck, E. O., Extensional flow at low Reynolds number with surface tension, J. Engng Maths, 70, 321-331, (2011) · Zbl 1254.76039 · doi:10.1007/s10665-010-9443-3
[52] Stokes, Y. M.; Buchak, P.; Crowdy, D. G.; Ebendorff-Heidepriem, H., Drawing of micro-structured fibres: circular and non-circular tubes, J. Fluid Mech., 755, 176-203, (2014) · Zbl 1330.76043 · doi:10.1017/jfm.2014.408
[53] Suman, B.; Kumar, S., Draw ratio enhancement in nonisothermal melt spinning, AIChE J., 55, 581-593, (2009) · doi:10.1002/aic.11707
[54] Tilley, B. S.; Bowen, M., Thermocapillary control of rupture in thin viscous fluid sheets, J. Fluid Mech., 541, 399-408, (2005) · Zbl 1082.76035 · doi:10.1017/S0022112005006269
[55] Taroni, M.; Breward, C. J. W.; Cummings, L. J.; Griffiths, I. M., Asymptotic solutions of glass temperature profiles during steady optical fibre drawing, J. Engng Maths, 80, 1-20, (2013) · Zbl 1367.76019 · doi:10.1007/s10665-013-9623-z
[56] Vasilyev, O. V.; Ten, A. A.; Yuen, D. A., Temperature-dependent viscous gravity currents with shear heating, Phys. Fluids, 13, 3664-3674, (2001) · Zbl 1184.76570 · doi:10.1063/1.1416501
[57] Vlachopoulos, J., The Role of Rheology in Polymer Extrusion, New Technologies for Extrusion Conference, (2003), SPIE Press
[58] Vlachopoulos, J. & Polychronopoulos, N.2012Basic concepts in polymer melt rheology and their importance in processing. In Applied Polymer Rheology: Polymeric Fluids with Industrial Applications (ed. Kontopoulou, M.), pp. 1-27. Wiley.
[59] Wang, J. S.; Porter, R. S., On the viscosity temperature behavior of polymer melts, Rheol. Acta, 34, 496-503, (1995) · doi:10.1007/BF00396562
[60] Wei, H. H., Thermocapillary instability of core-annular flows, Phys. Fluids, 17, (2005) · Zbl 1188.76174
[61] Wu, S. H., Surface and interfacial tensions of polymer melts: I. Polyethylene, polyisobutylene, and polyvinyl acetate, J. Colloid Interface Sci., 31, 153-161, (1969) · doi:10.1016/0021-9797(69)90321-X
[62] Wu, S. H., Surface and interfacial tensions of polymer melts. II. Poly(methyl methacrylate), poly(n-butyl methacrylate), and polystyrene, J. Phys. Chem., 74, 632-638, (1970) · doi:10.1021/j100698a026
[63] Wilson, S. D. R., The slow dripping of a viscous fluid, J. Fluid Mech., 190, 561-570, (1988) · Zbl 0642.76116 · doi:10.1017/S0022112088001454
[64] Wylie, J. J.; Huang, H., Extensional flows with viscous heating, J. Fluid Mech., 571, 359-370, (2007) · Zbl 1105.76023 · doi:10.1017/S0022112006003338
[65] Wylie, J. J.; Huang, H.; Miura, R. M., Thermal instability in drawing viscous threads, J. Fluid Mech., 570, 1-16, (2007) · Zbl 1105.76029 · doi:10.1017/S0022112006002709
[66] Wylie, J. J.; Huang, H.; Miura, R. M., Stretching of viscous threads at low Reynolds numbers, J. Fluid Mech., 683, 212-234, (2011) · Zbl 1241.76151 · doi:10.1017/jfm.2011.259
[67] Wylie, J. J.; Huang, H.; Miura, R. M., Asymptotic analysis of a viscous thread extending under gravity, Physica D, 313, 51-60, (2015) · Zbl 1364.76049
[68] Yarin, A. L., Effect of heat removal on nonsteady regimes of fiber formation, J. Engng Phys., 50, 569-575, (1986) · doi:10.1007/BF00870716
[69] Yarin, A. L.; Gospodinov, P.; Roussinov, V. I., Stability loss and sensitivity in hollow fiber drawing, Phys. Fluids, 6, 1454-1463, (1994) · Zbl 0829.76039 · doi:10.1063/1.868260
[70] Yin, Z. L.; Jaluria, Y., Thermal transport and flow in high-speed optical fiber drawing, Trans. ASME J. Heat Transfer, 120, 916-930, (1998) · doi:10.1115/1.2825911
[71] Yin, Z. L.; Jaluria, Y., Neck down and thermally induced defects in high-speed optical fiber drawing, Trans. ASME J. Heat Transfer, 122, 351-362, (2000) · doi:10.1115/1.521488
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