Skip to main content Accessibility help
×
Home

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

  • J. Eggers (a1), M. A. Herrada (a2) and J. H. Snoeijer (a3)

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.

Copyright

Corresponding author

Email address for correspondence: Jens.Eggers@bristol.ac.uk

References

Hide All
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, 35583561.
Anna, S. L. & McKinley, G. H. 2001 Elasto-capillary thinning and breakup of model elastic liquids. J. Rheol. 45, 115138.
Basaran, O. A. 2002 Small-scale free surface flows with breakup: drop formation and emerging applications. AIChE J. 48, 18421848.
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.
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, 333335.
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, 625631.
Bhat, P. P., Basaran, O. A. & Pasquali, M. 2008 Dynamics of viscoelastic liquid filaments: low capillary number flows. J. Non-Newtonian Fluid Mech. 150, 211225.
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids Volume I: Fluid Mechanics; Volume II: Kinetic Theory. Wiley.
Chang, H.-C., Demekhin, E. A. & Kalaidin, E. 1999 Iterated stretching of viscoelastic jets. Phys. Fluids 11, 17171737.
Chen, Y.-J. & Steen, P. H. 1997 Dynamics of inviscid capillary breakup: collapse and pinchoff of a film bridge. J. Fluid Mech. 341, 245267.
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, 283308.
Cohen, I., Brenner, M. P., Eggers, J. & Nagel, S. R. 1999 Two fluid drop snap-off problem: experiment and theory. Phys. Rev. Lett. 83, 11471150.
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, 189199.
Day, R. F., Hinch, E. J. & Lister, J. R. 1998 Self-similar capillary pinchoff of an inviscid fluid. Phys. Rev. Lett. 80, 704707.
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.
Deblais, A., Velikov, K. P. & Bonn, D. 2018b Pearling instabilities of a viscoelastic thread. Phys. Rev. Lett. 120, 194501.
Eggers, J. 1993 Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 71, 34583460.
Eggers, J. 2014 Instability of a polymeric thread. Phys. Fluids 26, 033106.
Eggers, J. 2018 The role of singularities in hydrodynamics. Phys. Rev. Fluids 3, 110503.
Eggers, J. & Courrech du Pont, S. 2009 Numerical analysis of tip singularities in viscous flow. Phys. Rev. E 79, 066311.
Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J. Fluid Mech. 262, 205221.
Eggers, J. & Fontelos, M. A. 2005 Isolated inertialess drops cannot break up. J. Fluid Mech. 530, 177180.
Eggers, J. & Fontelos, M. A. 2015 Singularities: Formation, Structure, and Propagation. Cambridge University Press.
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.
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. 260261. Steinkopf.
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, 3153.
Entov, V. M. & Yarin, A. L. 1984 Influence of elastic stresses on the capillary breakup of dilute polymer solutions. Fluid Dyn. 19, 2129.
Eshelby, J. D. 1975 The elastic energy-momentum tensor. J. Elast. 5, 321335.
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, 157166.
Fattal, R. & Kupferman, R. 2004 Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech. 123, 281285.
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, 84111.
Fontelos, M. A. & Li, J. 2004 On the evolution and rupture of filaments in Giesekus and FENE models. J. Non-Newtonian Fluid Mech. 118, 116.
Herrada, M. A. & Montanero, J. M. 2016 A numerical method to study the dynamics of capillary fluid systems. J. Comput. Phys. 306, 137147.
Ingremeaux, F. & Kellay, H. 2013 Stretching polymers in droplet-pinch-off experiments. Phys. Rev. X 3, 041002.
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, 207229.
Larson, R. G. 1988 Constitutive Equations for Polymer Melts and Solutions. Butterworth Publishers.
Larson, R. G. 1999 The Structure and Rheology of Complex Fluids. Oxford University Press.
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, 1322.
McKinley, G. H. 2005 Visco-elasto-capillary thinning and break-up of complex fluids. In Rheology Reviews, pp. 149. British Society of Rheology.
McKinley, G. H. & Tripathi, A. 2000 How to extract the Newtonian viscosity from capillary breakup measurements in a filament rheometer. J. Rheol. 44, 653670.
Mitsoulis, E. & Tsamopoulos, J. 2017 Numerical simulations of complex yield-stress fluid flows. Rheol. Acta 56, 231258.
Morozov, A. & Spagnolie, S. E. 2015 Introduction to complex fluids. In Complex Fluids in Biological Systems (ed. Spagnolie, S. E.), pp. 352. Springer.
Morrison, N. F. & Harlen, O. G. 2010 Viscoelasticity in inkjet printing. Rheol. Acta 49, 619632.
Negahban, M. 2012 The Mechanical and Thermodynamical Theory of Plasticity. CRC Press.
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.
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, 363409.
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.
Renardy, M. 1995 A numerical study of the asymptotic evolution and breakup of Newtonian and viscoelastic jets. J. Non-Newtonian Fluid Mech. 59, 267282.
Renardy, M. 2001 Self-similar breakup of a Giesekus jet. J. Non-Newtonian Fluid Mech. 97, 283293.
Renardy, M. 2002a Self-similar jet breakup for a generalized PTT model. J. Non-Newtonian Fluid Mech. 103, 161169.
Renardy, M. 2002b Similarity solutions for jet break-up for various models of viscoelastic fluids. J. Non-Newtonian Fluid Mech. 104, 6574.
Renardy, M. 2004 Self-similar breakup of non-Newtonian fluid jets. Rheology Rev. 2, 171196.
Renardy, M. & Losh, D. 2002 Similarity solutions for jet breakup in a Giesekus fluid with inertia. J. Non-Newtonian Fluid Mech. 106, 1727.
Sattler, R., Gier, S., Eggers, J. & Wagner, C. 2012 The final stages of capillary break-up of polymer solutions. Phys. Fluids 24, 023101.
Sattler, R., Wagner, C. & Eggers, J. 2008 Blistering pattern and formation of nanofibers in capillary thinning of polymer solutions. Phys. Rev. Lett. 100, 164502.
Snoeijer, J. H., Pandey, A., Herrada, M. A. & Eggers, J.2019 The relationship between viscoelasticity and elasticity. arXiv:1905.12339.
Suo, Z.2013a Elasticity of rubber-like materials. http://imechanica.org/node/14146.
Suo, Z.2013b Finite deformation: general theory. http://imechanica.org/node/538.
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, 215243.
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, 39023915.
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.
Wagner, C., Amarouchene, Y., Bonn, D. & Eggers, J. 2005 Droplet detachment and satellite bead formation in visco-elastic fluids. Phys. Rev. Lett. 95, 164504.
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, 5361.
Xuan, C. & Biggins, J. 2017 Plateau–Rayleigh instability in solids is a simple phase separation. Phys. Rev. E 95, 053106.
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, 4788.
Yao, M., McKinley, G. H. & Debbaut, B. 1998 Extensional deformation, stress relaxation and necking failure of viscoelastic filaments. J. Non-Newtonian Fluid Mech. 79, 469501.
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, 143.
Zhang, W. W. & Lister, J. R. 1999 Similarity solutions for capillary pinch-off in fluids of differing viscosity. Phys. Rev. Lett. 83, 11511154.
Zhou, J. & Doi, M. 2018 Dynamics of viscoelastic filaments based on Onsager principle. Phys. Rev. Fluids 3, 084004.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

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

  • J. Eggers (a1), M. A. Herrada (a2) and J. H. Snoeijer (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed