Skip to main content Accessibility help
×
Home

Viscoelastic liquid curtains: experimental results on the flow of a falling sheet of polymer solution

  • A. Gaillard (a1), M. Roché (a1), S. Lerouge (a1), C. Gay (a1), L. Lebon (a1) and L. Limat (a1)...

Abstract

We experimentally investigate the extensional flow of a sheet – or curtain – of viscoelastic liquid falling freely from a slot at constant flow rate under gravity. Extruded liquids are aqueous solutions of flexible polyethylene oxide (PEO) and of semi-rigid partially hydrolysed polyacrylamide (HPAM) with low shear viscosities. Velocimetry measurements reveal that the mean velocity field $U(z)$ (where $z$ is the distance from the slot exit) does not reduce to a free fall. More precisely, we show that the liquid falls initially with sub-gravitational accelerations up to a distance from the slot which scales as $g\unicode[STIX]{x1D70F}_{fil}^{2}$ (where $g$ is gravity and $\unicode[STIX]{x1D70F}_{fil}$ is the extensional relaxation time of the liquid) due to the stretching of polymer molecules. Beyond this elastic length, inertia dominates and the local acceleration reaches the asymptotic free-fall value $g$ . The length of the sub-gravitational part of the curtain is shown to be much larger than the equivalent viscous length $((4\unicode[STIX]{x1D702}/\unicode[STIX]{x1D70C})^{2}/g)^{1/3}$ for Newtonian liquids of density $\unicode[STIX]{x1D70C}$ and dynamic viscosity $\unicode[STIX]{x1D702}$ which is usually small compared to the curtain length. By analogy with Newtonian curtains, we show that the velocity field $U(z)$ rescales on a master curve. Besides, the flow is shown to be only weakly affected by the history of polymer deformations in the die upstream of the curtain. Furthermore, investigations on the curtain stability reveal that polymer addition reduces the minimum flow rate required to maintain a continuous sheet of liquid.

Copyright

Corresponding author

Email address for correspondence: antoine0gaillard@gmail.com

References

Hide All
Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55. Courier Corporation.
Aidun, C. K. 1987 Mechanics of a free-surface liquid film flow. J. Appl. Mech. 54 (4), 951954.10.1115/1.3173144
Alaie, S. M. & Papanastasiou, T. C. 1991 Film casting of viscoelastic liquid. Polym. Engng Sci. 31 (2), 6775.10.1002/pen.760310203
Allain, C., Cloitre, M. & Perrot, P. 1997 Experimental investigation and scaling law analysis of die swell in semi-dilute polymer solutions. J. Non-Newtonian Fluid Mech. 73 (1-2), 5166.10.1016/S0377-0257(97)00051-7
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 (16), 35583561.10.1103/PhysRevLett.86.3558
Anna, S. L. & McKinley, G. H. 2001 Elasto-capillary thinning and breakup of model elastic liquids. J. Rheol. 45 (1), 115138.10.1122/1.1332389
Becerra, M. & Carvalho, M. S. 2011 Stability of viscoelastic liquid curtain. Chem. Engng Process. 50 (5), 445449.10.1016/j.cep.2010.11.011
Bird, R. B., Armstrong, R. C., Hassager, O. & Curtiss, C. F. 1987 Dynamics of Polymeric Liquids: Kinetic Theory, vol. 2. Wiley.
Boger, D. V. & Walters, K. 2012 Rheological Phenomena in Focus, vol. 4. Elsevier.
Brandrup, J., Immergut, E. H., Abe, E. A. G. A. & Bloch, D. R. 1989 Polymer Handbook, vol. 7. Wiley.
Brown, D. R. 1961 A study of the behaviour of a thin sheet of moving liquid. J. Fluid Mech. 10 (2), 297305.10.1017/S002211206100024X
Brunet, P., Flesselles, J.-M. & Limat, L. 2007 Dynamics of a circular array of liquid columns. Eur. Phys. J. B 55 (3), 297322.10.1140/epjb/e2007-00057-y
Campo-Deano, L. & Clasen, C. 2010 The slow retraction method (SRM) for the determination of ultra-short relaxation times in capillary breakup extensional rheometry experiments. J. Non-Newtonian Fluid Mech. 165 (23‐24), 16881699.10.1016/j.jnnfm.2010.09.007
Cartalos, U. & Piau, J. M. 1992 Creeping flow regimes of low concentration polymer solutions in thick solvents through an orifice die. J. Non-Newtonian Fluid Mech. 45 (2), 231285.10.1016/0377-0257(92)85005-H
Casanellas, L., Alves, M. A., Poole, R. J., Lerouge, S. & Lindner, A. 2016 The stabilizing effect of shear thinning on the onset of purely elastic instabilities in serpentine microflows. Soft Matt. 12 (29), 61676175.10.1039/C6SM00326E
Chen, E. B., Morales, A. J., Chen, C. C., Donatelli, A. A., Bannister, W. W. & Cummings, B. T. 1998 Fluorescein and poly(ethylene oxide) hose stream additives for improved firefighting effectiveness. Fire Technol. 34 (4), 291306.10.1023/A:1015362426844
Chen, P., Yao, L., Liu, Y., Luo, J., Zhou, G. & Jiang, B. 2012 Experimental and theoretical study of dilute polyacrylamide solutions: effect of salt concentration. J. Mol. Model. 18 (7), 31533160.10.1007/s00894-011-1332-9
Chiba, K., Sakatani, T. & Nakamura, K. 1990 Anomalous flow patterns in viscoelastic entry flow through a planar contraction. J. Non-Newtonian Fluid Mech. 36, 193203.10.1016/0377-0257(90)85009-N
Chiba, K., Tanaka, S. & Nakamura, K. 1992 The structure of anomalous entry flow patterns through a planar contraction. J. Non-Newtonian Fluid Mech. 42 (3), 315322.10.1016/0377-0257(92)87016-5
Clanet, C. & Lasheras, J. C. 1999 Transition from dripping to jetting. J. Fluid Mech. 383, 307326.10.1017/S0022112098004066
Clarke, N. S. 1966 A differential equation in fluid mechanics. Mathematika 13 (1), 5153.10.1112/S0025579300004198
Clarke, N. S. 1968 Two-dimensional flow under gravity in a jet of viscous liquid. J. Fluid Mech. 31 (3), 481500.10.1017/S0022112068000297
Clasen, C., Bico, J., Entov, V. M. & McKinley, G. H. 2009 Gobbling drops: the jetting–dripping transition in flows of polymer solutions. J. Fluid Mech. 636, 540.10.1017/S0022112009008143
Clasen, C., Plog, J. P., Kulicke, W.-M., Owens, M., Macosko, C., Scriven, L. E., Verani, M. & McKinley, G. H. 2006 How dilute are dilute solutions in extensional flows? J. Rheol. 50 (6), 849881.10.1122/1.2357595
Crooks, R. & Boger, D. V. 2000 Influence of fluid elasticity on drops impacting on dry surfaces. J. Rheol. 44 (4), 973996.10.1122/1.551123
Culick, F. E. C. 1960 Comments on a ruptured soap film. J. Appl. Phys. 31 (6), 11281129.10.1063/1.1735765
Daerr, A. & Mogne, A. 2016 Pendent_drop: an ImageJ plugin to measure the surface tension from an image of a pendent drop. J. Open Res. Softw. 4 (1), e3.
De Gennes, P.-G. 1974 Coil-stretch transition of dilute flexible polymers under ultrahigh velocity gradients. J. Chem. Phys. 60 (12), 50305042.10.1063/1.1681018
Delvaux, V. & Crochet, M. J. 1990 Numerical simulation of delayed die swell. Rheol. Acta 29 (1), 110.10.1007/BF01331795
Dombrowski, N. & Johns, W. R. 1963 The aerodynamic instability and disintegration of viscous liquid sheets. Chem. Engng Sci. 18 (3), 203214.10.1016/0009-2509(63)85005-8
Dontula, P., Macosko, C. W. & Scriven, L. E. 1998 Model elastic liquids with water-soluble polymers. AIChE J. 44 (6), 12471255.10.1002/aic.690440603
Eggers, J. 2014 Instability of a polymeric thread. Phys. Fluids 26 (3), 033106.10.1063/1.4869721
Entov, V. M. & Hinch, E. J. 1997 Effect of a spectrum of relaxation times on the capillary thinning of a filament of elastic liquid. J. Non-Newtonian Fluid Mech. 72 (1), 3153.10.1016/S0377-0257(97)00022-0
Ewoldt, R. H., Johnston, M. T. & Caretta, L. M. 2015 Experimental challenges of shear rheology: how to avoid bad data. In Complex Fluids in Biological Systems, pp. 207241. Springer.10.1007/978-1-4939-2065-5_6
Fermigier, M., Limat, L., Wesfreid, J. E., Boudinet, P. & Quilliet, C. 1992 Two-dimensional patterns in Rayleigh–Taylor instability of a thin layer. J. Fluid Mech. 236, 349383.10.1017/S0022112092001447
Gaillard, A.2018 Flow and stability of a viscoelastic liquid curtain. PhD thesis, Université Sorbonne Paris Cité.
Graessley, W. W. 1980 Polymer chain dimensions and the dependence of viscoelastic properties on concentration, molecular weight and solvent power. Polymer 21 (3), 258262.10.1016/0032-3861(80)90266-9
Graham, M. D. 2003 Interfacial hoop stress and instability of viscoelastic free surface flows. Phys. Fluids 15 (6), 17021710.10.1063/1.1568340
Gugler, G., Beer, R. & Mauron, M. 2010 Coatability of viscoelastic liquid curtain. In Proceedings of the 15th International Coating Science and Technology Symposium, St. Paul, Minnesota.
Herrchen, M. & Öttinger, H. C. 1997 A detailed comparison of various fene dumbbell models. J. Non-Newtonian Fluid Mech. 68 (1), 1742.10.1016/S0377-0257(96)01498-X
Huang, D. C. & White, J. L. 1979 Extrudate swell from slit and capillary dies: an experimental and theoretical study. Polym. Engng Sci. 19 (9), 609616.10.1002/pen.760190904
Karim, A. M., Suszynski, W. J., Francis, L. F. & Carvalho, M. S. 2018a Effect of viscosity on liquid curtain stability. AIChE J. 64 (4), 14481457.10.1002/aic.16015
Karim, A.M., Suszynski, W.J., Griffith, W.B., Pujari, S., Francis, L.F. & Carvalho, M.S. 2018b Effect of viscoelasticity on stability of liquid curtain. J. Non-Newtonian Fluid Mech. 257, 8394.10.1016/j.jnnfm.2018.03.019
Kawale, D., Marques, E., Zitha, P. L., Kreutzer, M. T., Rossen, W. R. & Boukany, P. E. 2017 Elastic instabilities during the flow of hydrolyzed polyacrylamide solution in porous media: effect of pore-shape and salt. Soft Matt. 13 (4), 765775.
Kays, W. M., Crawford, M. E. & Weigand, B. 2005 Convective Heat and Mass Transfer, vol. 76. McGraw-Hill.
Keshavarz, B., Sharma, V., Houze, E. C., Koerner, M. R., Moore, J. R., Cotts, P. M., Threlfall-Holmes, P. & McKinley, G. H. 2015 Studying the effects of elongational properties on atomization of weakly viscoelastic solutions using Rayleigh Ohnesorge jetting extensional rheometry (ROJER). J. Non-Newtonian Fluid Mech. 222, 171189.10.1016/j.jnnfm.2014.11.004
Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheol. Acta 31 (3), 213263.10.1007/BF00366504
Larson, R. G. 1999 The Structure and Rheology of Complex Fluids (Topics in Chemical Engineering), vol. 86, p. 108. Oxford University Press.
Macosko, C. W. 1994 Rheology: Principles, Measurements, and Applications. Wiley-VCH.
Mathues, W., McIlroy, C., Harlen, O. G. & Clasen, C. 2015 Capillary breakup of suspensions near pinch-off. Phys. Fluids 27 (9), 093301.10.1063/1.4930011
McIlroy, C. & Harlen, O. G. 2014 Modelling capillary break-up of particulate suspensions. Phys. Fluids 26 (3), 033101.10.1063/1.4866789
McKinley, G. H. 2005 Visco-elasto-capillary thinning and break-up of complex fluids. Annu. Rheol. Rev. 3, 148.
McKinley, G. H., Raiford, W. P., Brown, R. A. & Armstrong, R. C. 1991 Nonlinear dynamics of viscoelastic flow in axisymmetric abrupt contractions. J. Fluid Mech. 223, 411456.10.1017/S0022112091001489
Miller, E., Clasen, C. & Rothstein, J. P. 2009 The effect of step-stretch parameters on capillary breakup extensional rheology (CaBER) measurements. Rheol. Acta 48 (6), 625639.10.1007/s00397-009-0357-9
Miyamoto, K. & Katagiri, Y. 1997 Curtain coating. In Liquid Film Coating, pp. 463494. Springer.10.1007/978-94-011-5342-3_13
Nigen, S. & Walters, K. 2002 Viscoelastic contraction flows: comparison of axisymmetric and planar configurations. J. Non-Newtonian Fluid Mech. 102 (2), 343359.10.1016/S0377-0257(01)00186-0
Oliveira, M. S., Yeh, R. & McKinley, G. H. 2006 Iterated stretching, extensional rheology and formation of beads-on-a-string structures in polymer solutions. J. Non-Newtonian Fluid Mech. 137 (1–3), 137148.10.1016/j.jnnfm.2006.01.014
Papanastasiou, T. C., Macosko, C. W., Scriven, L. E. & Chen, Z. 1987 Fiber spinning of viscoelastic liquid. AIChE J. 33 (5), 834842.10.1002/aic.690330516
Petrie, C. J. S. 1979 Elongational Flows. Pitman.
Purnode, B. & Crochet, M. J. 1996 Flows of polymer solutions through contractions. Part 1. Flows of polyacrylamide solutions through planar contractions. J. Non-Newtonian Fluid Mech. 65 (2–3), 269289.10.1016/0377-0257(96)01446-2
Ramos, J. I. 1996 Planar liquid sheets at low Reynolds numbers. Intl J. Numer. Meth. Fluids 22 (10), 961978.10.1002/(SICI)1097-0363(19960530)22:10<961::AID-FLD389>3.0.CO;2-D
Richardson, S. 1970 The die swell phenomenon. Rheol. Acta 9 (2), 193199.10.1007/BF01973479
Roche, J. S., Grand, N. L., Brunet, P., Lebon, L. & Limat, L. 2006 Pertubations on a liquid curtain near break-up: wakes and free edges. Phys. Fluids 18 (8), 082101.10.1063/1.2238867
Rodd, L. E., Scott, T. P., Boger, D. V., Cooper-White, J. J. & McKinley, G. H. 2005 The inertio-elastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries. J. Non-Newtonian Fluid Mech. 129 (1), 122.10.1016/j.jnnfm.2005.04.006
Rodd, L. E., Scott, T. P., Cooper-White, J. J., Boger, D. V. & McKinley, G. H. 2007 Role of the elasticity number in the entry flow of dilute polymer solutions in micro-fabricated contraction geometries. J. Non-Newtonian Fluid Mech. 143 (2-3), 170191.10.1016/j.jnnfm.2007.02.006
Rodd, L. E., Scott, T. P., Cooper-White, J. J. & McKinley, G. H. 2005 Capillary break-up rheometry of low-viscosity elastic fluids. Appl. Rheol. 15 (1), 1227.10.1515/arh-2005-0001
Rothstein, J. P. & McKinley, G. H. 1999 Extensional flow of a polystyrene boger fluid through a 4: 1: 4 axisymmetric contraction/expansion. J. Non-Newtonian Fluid Mech. 86 (1), 6188.10.1016/S0377-0257(98)00202-X
Rubinstein, M. & Colby, R. H. 2003 Polymer physics, vol. 23. Oxford University Press.
Satoh, N., Tomiyama, H. & Kajiwara, T. 2001 Viscoelastic simulation of film casting process for a polymer melt. Polym. Engng Sci. 41 (9), 15641579.10.1002/pen.10855
Sattler, R., Gier, S., Eggers, J. & Wagner, C. 2012 The final stages of capillary break-up of polymer solutions. Phys. Fluids 24 (2), 023101.10.1063/1.3684750
Sattler, R., Wagner, C. & Eggers, J. 2008 Blistering pattern and formation of nanofibers in capillary thinning of polymer solutions. Phys. Rev. Lett. 100 (16), 164502.10.1103/PhysRevLett.100.164502
Savva, N. & Bush, J. W. M. 2009 Viscous sheet retraction. J. Fluid Mech. 626, 211240.10.1017/S0022112009005795
Sevilla, A. 2011 The effect of viscous relaxation on the spatiotemporal stability of capillary jets. J. Fluid Mech. 684, 204226.10.1017/jfm.2011.297
Stelter, M., Brenn, G., Yarin, A. L., Singh, R. P. & Durst, F. 2002 Investigation of the elongational behavior of polymer solutions by means of an elongational rheometer. J. Rheol. 46 (2), 507527.10.1122/1.1445185
Sünderhauf, G., Raszillier, H. & Durst, F. 2002 The retraction of the edge of a planar liquid sheet. Phys. Fluids 14 (1), 198208.10.1063/1.1426387
Tanner, R. I. 1970 A theory of die-swell. J. Polym. Sci. B 8 (12), 20672078.
Tanner, R. I. 2000 Engineering Rheology, vol. 52. Oxford University Press.
Tanner, R. I. 2005 A theory of die-swell revisited. J. Non-Newtonian Fluid Mech. 129 (2), 8587.10.1016/j.jnnfm.2005.05.010
Taylor, G. 1959 The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets. Proc. R. Soc. Lond. A 313321.
Tirtaatmadja, V., McKinley, G. H. & Cooper-White, J. J. 2006 Drop formation and breakup of low viscosity elastic fluids: effects of molecular weight and concentration. Phys. Fluids 18 (4), 043101.10.1063/1.2190469
Villermaux, E. & Clanet, C. 2002 Life of a flapping liquid sheet. J. Fluid Mech. 462, 341363.10.1017/S0022112002008376
Virk, P. S. 1975 Drag reduction fundamentals. AIChE J. 21 (4), 625656.10.1002/aic.690210402
White, J. L. & Roman, J. F. 1976 Extrudate swell during the melt spinning of fibersinfluence of rheological properties and take-up force. J. Appl. Polym. Sci. 20 (4), 10051023.10.1002/app.1976.070200413
Wu, X. Y., Hunkeler, D., Hamielec, A. E., Pelton, R. H. & Woods, D. R. 1991 Molecular weight characterization of poly(acrylamide-co-sodium acrylate). I. Viscometry. J. Appl. Polym. Sci. 42 (7), 20812093.10.1002/app.1991.070420736
Zell, A., Gier, S., Rafai, S. & Wagner, C. 2010 Is there a relation between the relaxation time measured in caber experiments and the first normal stress coefficient? J. Non-Newtonian Fluid Mech. 165 (19), 12651274.10.1016/j.jnnfm.2010.06.010
Zhang, G., Zhou, J. S., Zhai, Y. A., Liu, F. Q. & Gao, G. 2008 Effect of salt solutions on chain structure of partially hydrolyzed polyacrylamide. J. Central South University of Technology 15 (1), 8083.10.1007/s11771-008-0319-x
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Viscoelastic liquid curtains: experimental results on the flow of a falling sheet of polymer solution

  • A. Gaillard (a1), M. Roché (a1), S. Lerouge (a1), C. Gay (a1), L. Lebon (a1) and L. Limat (a1)...

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