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Self-similar breakup of polymeric threads as described by the Oldroyd-B model

Published online by Cambridge University Press:  28 January 2020

J. Eggers Open the ORCID record for J. Eggers [Opens in a new window] ,
M. A. Herrada  and
J. H. Snoeijer
J. Eggers*
Affiliation:
School of Mathematics, University of Bristol, Fry Building, Bristol BS8 1UG, UK
M. A. Herrada
Affiliation:
E.S.I., Universidad de Sevilla, Camino de los Descubrimientos s/n 41092, Spain
J. H. Snoeijer
Affiliation:
Physics of Fluids Group, Faculty of Science and Technology, Mesa+ Institute, University of Twente, 7500 AE Enschede, The Netherlands
*
†Email address for correspondence: Jens.Eggers@bristol.ac.uk
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Abstract

When a drop of fluid containing long, flexible polymers breaks up, it forms threads of almost constant thickness, whose size decreases exponentially in time. Using an Oldroyd-B fluid as a model, we show that the thread profile, rescaled by the thread thickness, converges to a similarity solution. Using the correspondence between viscoelastic fluids and nonlinear elasticity, we derive similarity equations for the full three-dimensional axisymmetric flow field in the limit that the viscosity of the solvent fluid can be neglected. Deriving a conservation law along the thread, we can calculate the stress inside the thread from a measurement of the thread thickness. The explicit form of the velocity and stress fields can be deduced from a solution of the similarity equations. Results are validated by detailed comparison with numerical simulations.

JFM classification

Drops and Bubbles: Drops Interfacial Flows (free surface): Capillary flows Non-Newtonian Flows: Polymers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 887 , 25 March 2020 , A19
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Amarouchene, Y., Bonn, D., Meunier, J. & Kellay, H. 2001 Inhibition of the finite-time singularity during droplet fission of a polymeric fluid. Phys. Rev. Lett. 86, 3558–3561.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
Basaran, O. A. 2002 Small-scale free surface flows with breakup: drop formation and emerging applications. AIChE J. 48, 1842–1848.CrossRefGoogle Scholar
Bazilevskii, A. V., Entov, V. M. & Rozhkov, A. N. 1990 Liquid filament microrheometer and some of its applications. In Proceedings of the Third European Rheology Conference (ed. Oliver, D. R.), p. 41. Elsevier Applied Science.Google Scholar
Bazilevskii, A. V., Voronkov, S. I., Entov, V. M. & Rozhkov, A. N. 1981 Orientational effects in the decomposition of streams and strands of diluted polymer solutions. Sov. Phys. Dokl. 26, 333–335.Google Scholar
Bhat, P. P., Appathurai, S., Harris, M. T., Pasquali, M., McKinley, G. H. & Basaran, O. A. 2010 Formation of beads-on-a-string structures during break-up of viscoelastic laments. Nat. Phys. 6, 625–631.CrossRefGoogle Scholar
Bhat, P. P., Basaran, O. A. & Pasquali, M. 2008 Dynamics of viscoelastic liquid filaments: low capillary number flows. J. Non-Newtonian Fluid Mech. 150, 211–225.CrossRefGoogle Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids Volume I: Fluid Mechanics; Volume II: Kinetic Theory. Wiley.Google Scholar
Chang, H.-C., Demekhin, E. A. & Kalaidin, E. 1999 Iterated stretching of viscoelastic jets. Phys. Fluids 11, 1717–1737.CrossRefGoogle Scholar
Chen, Y.-J. & Steen, P. H. 1997 Dynamics of inviscid capillary breakup: collapse and pinchoff of a film bridge. J. Fluid Mech. 341, 245–267.CrossRefGoogle Scholar
Clasen, C., Eggers, J., Fontelos, M. A., Li, J. & McKinley, G. H. 2006 The beads-on-string structure of viscoelastic jets. J. Fluid Mech. 556, 283–308.CrossRefGoogle Scholar
Cohen, I., Brenner, M. P., Eggers, J. & Nagel, S. R. 1999 Two fluid drop snap-off problem: experiment and theory. Phys. Rev. Lett. 83, 1147–1150.CrossRefGoogle Scholar
Coronado, O. M., Arora, D., Behr, M. & Pasquali, M. 2007 A simple method for simulating general viscoelastic fluid flows with an alternate log-conformation formulation. J. Non-Newtonian Fluid Mech. 147, 189–199.CrossRefGoogle Scholar
Day, R. F., Hinch, E. J. & Lister, J. R. 1998 Self-similar capillary pinchoff of an inviscid fluid. Phys. Rev. Lett. 80, 704–707.CrossRefGoogle Scholar
Deblais, A., Herrada, M. A., Hauner, I., Velikov, K. P., van Roon, T., Kellay, H., Eggers, J. & Bonn, D. 2018a Viscous effects on inertial drop formation. Phys. Rev. Lett. 121, 254501.CrossRefGoogle Scholar
Deblais, A., Velikov, K. P. & Bonn, D. 2018b Pearling instabilities of a viscoelastic thread. Phys. Rev. Lett. 120, 194501.CrossRefGoogle Scholar
Eggers, J. 1993 Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 71, 3458–3460.CrossRefGoogle ScholarPubMed
Eggers, J. 2014 Instability of a polymeric thread. Phys. Fluids 26, 033106.CrossRefGoogle Scholar
Eggers, J. 2018 The role of singularities in hydrodynamics. Phys. Rev. Fluids 3, 110503.CrossRefGoogle Scholar
Eggers, J. & Courrech du Pont, S. 2009 Numerical analysis of tip singularities in viscous flow. Phys. Rev. E 79, 066311.Google Scholar
Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J. Fluid Mech. 262, 205–221.CrossRefGoogle Scholar
Eggers, J. & Fontelos, M. A. 2005 Isolated inertialess drops cannot break up. J. Fluid Mech. 530, 177–180.CrossRefGoogle Scholar
Eggers, J. & Fontelos, M. A. 2015 Singularities: Formation, Structure, and Propagation. Cambridge University Press.CrossRefGoogle Scholar
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.CrossRefGoogle Scholar
Entov, V. M. 1988 Elastic effects in flows of dilute polymer solutions. In Progress and Trends in Rheology II (ed. Giesekus, H. & Hibberd, M. F.), pp. 260–261. Steinkopf.CrossRefGoogle Scholar
Entov, V. M. & Hinch, E. J. 1997 Effect of a spectrum of relaxation times on the capillary thinning of a filament of elastic liquids. J. Non-Newtonian Fluid Mech. 72, 31–53.CrossRefGoogle Scholar
Entov, V. M. & Yarin, A. L. 1984 Influence of elastic stresses on the capillary breakup of dilute polymer solutions. Fluid Dyn. 19, 21–29.CrossRefGoogle Scholar
Eshelby, J. D. 1975 The elastic energy-momentum tensor. J. Elast. 5, 321–335.CrossRefGoogle Scholar
Etienne, J., Hinch, E. J. & Li, J. 2006 A Lagrangian–Eulerian approach for the numerical simulation of free-surface flow of a viscoelastic material. J. Non-Newtonian Fluid Mech. 136, 157–166.CrossRefGoogle Scholar
Fattal, R. & Kupferman, R. 2004 Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech. 123, 281–285.CrossRefGoogle Scholar
Fontelos, M. A. 2003 Break-up and no break-up in a family of models for the evolution of viscoelastic jets. Z. Angew. Math. Phys. 54, 84–111.CrossRefGoogle Scholar
Fontelos, M. A. & Li, J. 2004 On the evolution and rupture of filaments in Giesekus and FENE models. J. Non-Newtonian Fluid Mech. 118, 1–16.CrossRefGoogle Scholar
Herrada, M. A. & Montanero, J. M. 2016 A numerical method to study the dynamics of capillary fluid systems. J. Comput. Phys. 306, 137–147.CrossRefGoogle Scholar
Ingremeaux, F. & Kellay, H. 2013 Stretching polymers in droplet-pinch-off experiments. Phys. Rev. X 3, 041002.Google Scholar
Kistler, S. F. & Scriven, L. E. 1984 Coating flow theory by finite element and asymptotic analysis of the Navier–Stokes system. Intl J. Numer. Meth. Fluids 4, 207–229.CrossRefGoogle Scholar
Larson, R. G. 1988 Constitutive Equations for Polymer Melts and Solutions. Butterworth Publishers.Google Scholar
Larson, R. G. 1999 The Structure and Rheology of Complex Fluids. Oxford University Press.Google Scholar
Markovich, P. & Renardy, M. 1985 A finite difference study of the stretching and break-up of filaments of polymer solutions. J. Non-Newtonian Fluid Mech. 17, 13–22.CrossRefGoogle Scholar
McKinley, G. H. 2005 Visco-elasto-capillary thinning and break-up of complex fluids. In Rheology Reviews, pp. 1–49. British Society of Rheology.Google Scholar
McKinley, G. H. & Tripathi, A. 2000 How to extract the Newtonian viscosity from capillary breakup measurements in a filament rheometer. J. Rheol. 44, 653–670.CrossRefGoogle Scholar
Mitsoulis, E. & Tsamopoulos, J. 2017 Numerical simulations of complex yield-stress fluid flows. Rheol. Acta 56, 231–258.CrossRefGoogle Scholar
Morozov, A. & Spagnolie, S. E. 2015 Introduction to complex fluids. In Complex Fluids in Biological Systems (ed. Spagnolie, S. E.), pp. 3–52. Springer.CrossRefGoogle Scholar
Morrison, N. F. & Harlen, O. G. 2010 Viscoelasticity in inkjet printing. Rheol. Acta 49, 619–632.CrossRefGoogle Scholar
Negahban, M. 2012 The Mechanical and Thermodynamical Theory of Plasticity. CRC Press.Google Scholar
Oliveira, M. S. N. & McKinley, G. H. 2005 Iterated stretching and multiple beads-on-a-string phenomena in dilute solutions of high extensible flexible polymers. Phys. Fluids 17, 071704.CrossRefGoogle Scholar
Pasquali, M. & Scriven, L. E. 2002 Free surface flows of polymer solutions with models based on the conformation tensor. J. Non-Newtonian Fluid Mech. 108, 363–409.CrossRefGoogle Scholar
Ponce-Torres, A., Montanero, J. M., Herrada, M. A., Vega, E. J. & Vega, J. M. 2017 Influence of the surface viscosity on the breakup of a surfactant-laden drop. Phys. Rev. Lett. 118, 24501.CrossRefGoogle ScholarPubMed
Renardy, M. 1995 A numerical study of the asymptotic evolution and breakup of Newtonian and viscoelastic jets. J. Non-Newtonian Fluid Mech. 59, 267–282.CrossRefGoogle Scholar
Renardy, M. 2001 Self-similar breakup of a Giesekus jet. J. Non-Newtonian Fluid Mech. 97, 283–293.CrossRefGoogle Scholar
Renardy, M. 2002a Self-similar jet breakup for a generalized PTT model. J. Non-Newtonian Fluid Mech. 103, 161–169.CrossRefGoogle Scholar
Renardy, M. 2002b Similarity solutions for jet break-up for various models of viscoelastic fluids. J. Non-Newtonian Fluid Mech. 104, 65–74.CrossRefGoogle Scholar
Renardy, M. 2004 Self-similar breakup of non-Newtonian fluid jets. Rheology Rev. 2, 171–196.Google Scholar
Renardy, M. & Losh, D. 2002 Similarity solutions for jet breakup in a Giesekus fluid with inertia. J. Non-Newtonian Fluid Mech. 106, 17–27.CrossRefGoogle Scholar
Sattler, R., Gier, S., Eggers, J. & Wagner, C. 2012 The final stages of capillary break-up of polymer solutions. Phys. Fluids 24, 023101.CrossRefGoogle Scholar
Sattler, R., Wagner, C. & Eggers, J. 2008 Blistering pattern and formation of nanofibers in capillary thinning of polymer solutions. Phys. Rev. Lett. 100, 164502.CrossRefGoogle ScholarPubMed
Snoeijer, J. H., Pandey, A., Herrada, M. A. & Eggers, J.2019 The relationship between viscoelasticity and elasticity. arXiv:1905.12339.Google Scholar
Suo, Z.2013a Elasticity of rubber-like materials. http://imechanica.org/node/14146.Google Scholar
Suo, Z.2013b Finite deformation: general theory. http://imechanica.org/node/538.Google Scholar
Szady, M. J., Salamon, T. R., Liu, A. W., Bornside, D. E., Armstrong, R. C. & Brown, R. A. 1995 A new mixed finite element method for viscoelastic flows governed by differential constitutive equations. J. Non-Newtonian Fluid Mech. 59, 215–243.CrossRefGoogle Scholar
Torres, M. D., Hallmark, B., Wilson, D. I. & Hilliou, L. 2014 Natural Giesekus fluids: shear and extensional behavior of food gum solutions in the semidilute regime. AIChE J. 60, 3902–3915.CrossRefGoogle Scholar
Turkoz, E., Lopez-Herrera, J. M., Eggers, J., Arnold, C. B. & Deike, L. 2018 Axisymmetric simulation of viscoelastic filament thinning with the Oldroyd-B model. J. Fluid Mech. 851, R2.CrossRefGoogle Scholar
Wagner, C., Amarouchene, Y., Bonn, D. & Eggers, J. 2005 Droplet detachment and satellite bead formation in visco-elastic fluids. Phys. Rev. Lett. 95, 164504.CrossRefGoogle Scholar
Wagner, C., Bourouiba, L. & McKinley, G. H. 2015 An analytic solution for capillary thinning and breakup of FENE-P fluids. J. Non-Newtonian Fluid Mech. 218, 53–61.CrossRefGoogle Scholar
Xuan, C. & Biggins, J. 2017 Plateau–Rayleigh instability in solids is a simple phase separation. Phys. Rev. E 95, 053106.Google ScholarPubMed
Yao, M. & McKinley, G. H. 1998 Numerical simulation of extensional deformations of viscoelastic liquid bridges in filament stretching devices. J. Non-Newtonian Fluid Mech. 74, 47–88.CrossRefGoogle Scholar
Yao, M., McKinley, G. H. & Debbaut, B. 1998 Extensional deformation, stress relaxation and necking failure of viscoelastic filaments. J. Non-Newtonian Fluid Mech. 79, 469–501.CrossRefGoogle Scholar
Yao, M., Spiegelberg, S. H. & McKinley, G. H. 2000 Dynamics of weakly strain-hardening fluids in filament stretching devices. J. Non-Newtonian Fluid Mech. 89, 1–43.CrossRefGoogle Scholar
Zhang, W. W. & Lister, J. R. 1999 Similarity solutions for capillary pinch-off in fluids of differing viscosity. Phys. Rev. Lett. 83, 1151–1154.CrossRefGoogle Scholar
Zhou, J. & Doi, M. 2018 Dynamics of viscoelastic filaments based on Onsager principle. Phys. Rev. Fluids 3, 084004.CrossRefGoogle Scholar