Comptes Rendus
Viscoelastic surface instabilities
Comptes Rendus. Physique, Volume 10 (2009) no. 8, pp. 712-727.

We review three different types of viscoelastic surface instabilities: The Rayleigh–Plateau, the Saffman–Taylor and the Faraday instabilities. These instabilities are classical examples of hydrodynamic surface instabilities. The addition of a small amount of polymer to pure water can alter its flow behavior drastically and the type of instability may change not only quantitatively but also qualitatively. We will show that some of the observed new phenomena can be explained by the use of simple rheological models that contain most of the underlying physical mechanisms leading to the instability. A quantitative description however is often only possible close to the onset of the instability or for weak deviations from Newtonian behavior. A complete theoretical description is still lacking when the system is driven far from equilibrium or for fluids with strong non-Newtonian behavior.

Nous discutons dans ce papier trois types d'instabilités interfaciales dans des liquides visco-élastiques : l'instabilité de Rayleigh–Plateau, l'instabilité de Saffman–Taylor et l'instabilité de Faraday. Ce sont toutes les trois des exemples typiques d'instabilités hydrodynamiques de surface. L'addition d'une faible quantité de polymères à de l'eau pure suffit à lui conférer un comportement fortement non-newtonien. Les instabilités dans ces solutions sont modifiées pas seulement de façon quantitative mais aussi dans leur nature. Nous montrons que des modèles rhéologiques simples peuvent expliquer l'origine des modifications observées sur les instabilités. Une description quantitative n'est en général possible uniquement près du seuil d'instabilité et uniquement pour des déviations faibles d'un comportement newtonien. Une description complète loin de l'équilibre et pour des propriétés non-newtoniennes fortes manque à ce jour.

Published online:
DOI: 10.1016/j.crhy.2009.10.017
Keywords: Pattern formation, Instability, Rheology, Polymer solution
Mot clés : Formation de motifs, Instabilité, Rhéologie, Solution de polymères

Anke Lindner 1; Christian Wagner 2

1 Laboratoire de physique et mécanique des milieux hétérogènes (PMMH), UMR 7636 CNRS – ESPCI, universités Paris 6 et 7, 10, rue Vauquelin, 75231 Paris cedex 05, France
2 Technische Physik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany
     author = {Anke Lindner and Christian Wagner},
     title = {Viscoelastic surface instabilities},
     journal = {Comptes Rendus. Physique},
     pages = {712--727},
     publisher = {Elsevier},
     volume = {10},
     number = {8},
     year = {2009},
     doi = {10.1016/j.crhy.2009.10.017},
     language = {en},
AU  - Anke Lindner
AU  - Christian Wagner
TI  - Viscoelastic surface instabilities
JO  - Comptes Rendus. Physique
PY  - 2009
SP  - 712
EP  - 727
VL  - 10
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crhy.2009.10.017
LA  - en
ID  - CRPHYS_2009__10_8_712_0
ER  - 
%0 Journal Article
%A Anke Lindner
%A Christian Wagner
%T Viscoelastic surface instabilities
%J Comptes Rendus. Physique
%D 2009
%P 712-727
%V 10
%N 8
%I Elsevier
%R 10.1016/j.crhy.2009.10.017
%G en
%F CRPHYS_2009__10_8_712_0
Anke Lindner; Christian Wagner. Viscoelastic surface instabilities. Comptes Rendus. Physique, Volume 10 (2009) no. 8, pp. 712-727. doi : 10.1016/j.crhy.2009.10.017.

[1] J. Eggers Nonlinear dynamics and breakup of free-surface flows, Rev. Modern Phys., Volume 69 (1997), pp. 865-929

[2] J. Eggers; E. Villermaux Physics of liquid jets, Rep. Progr. Phys., Volume 71 (2008), p. 036601

[3] M. Goldin; J. Yerushal; R. Pfeffer; R. Shinnar Breakup of a laminar jet of a viscoelastic fluid, J. Fluid Mech., Volume 38 (1969), p. 689

[4] A. Bazilevskii; S. Voronkov; V. Entov; A. Rozkhov On orientational effects of jets and threats of dilute polymer solutions, Dokl. Akad. Nauk SSSR, Volume 257 (1981), pp. 336-339

[5] M. Renardy How to integrate the upper convected Maxwell (UCM) stress near a singularity (and maybe elsewhere, too), J. Non-Newtonian Fluid Mech., Volume 52 (1994), pp. 91-95

[6] V.M. Entov; E.J. Hinch Effect of a spectrum of relaxation times on the capillary thinning of a filament of elastic liquid, J. Non-Newtonian Fluid Mech., Volume 72 (1997), pp. 31-54

[7] M. Stelter; G. Brenn; A.L. Yarin; R.P. Singh; F. Durst Validation and application of a novel elongational device for polymer solutions, J. Rheol., Volume 44 (2000), pp. 595-616

[8] Y. Amarouchene; D. Bonn; J. Meunier; H. Kellay Inhibition of the finite-time singularity during droplet fission of a polymeric fluid, Phys. Rev. Lett., Volume 86 (2001), pp. 3558-3561

[9] S. Anna; G. McKinley Elasto-capillary thinning and breakup of model elastic liquids, J. Rheol., Volume 45 (2001), pp. 115-138

[10] C. Wagner; Y. Amarouchene; D. Bonn; J. Eggers Droplet detachment and satellite bead formation in viscoelastic fluids, Phys. Rev. Lett., Volume 95 (2005), p. 164504

[11] V. Tirtaatmadja; G. McKinley; J. Cooper-White Drop formation and breakup of low viscosity elastic fluids: Effects of molecular weight and concentration, Phys. Fluids, Volume 18 (2006), p. 043101

[12] C. Clasen; J. Eggers; M. Fontelos; J. Li; G. McKinley The beads-on-string structure of viscoelastic threads, J. Fluid Mech., Volume 556 (2006), pp. 283-308

[13] P. Coussot; F. Gaulard Gravity flow instability of viscoplastic materials: The ketchup drip, Phys. Rev. E, Volume 72 (2005), p. 031409

[14] K. Niedzwiedz; O. Arnolds; N. Willenbacher; R. Brummer Capillary breakup extensional rheometry of yield stress fluids, Appl. Rheol., Volume 19 (2009), p. 41969

[15] R.J. Furbank; J.F. Morris Pendant drop thread dynamics of particle-laden liquids, Int. J. Multiphase Flow, Volume 33 (2007), pp. 448-468

[16] M. Faraday On the forms and states of fluids on vibrating elastic surfaces, Philos. Trans. R. Soc. London, Volume 52 (1831), p. 319

[17] W. Zhang; J. Vinals Square patterns and quasipatterns in weakly damped Faraday waves, Phys. Rev. E, Volume 53 (1996), p. R4283-R4286

[18] O. Lioubashevski; H. Arbell; J. Fineberg Dissipative solitary states in driven surface waves, Phys. Rev. Lett., Volume 76 (1996), pp. 3959-3962

[19] D. Binks; W. van de Water Nonlinear pattern formation of Faraday waves, Phys. Rev. Lett., Volume 78 (1997), pp. 4043-4046

[20] A. Kudrolli; B. Pier; J. Gollub Superlattice patterns in surface waves, Physica D, Volume 123 (1998), pp. 99-111

[21] C. Wagner; H. Müller; K. Knorr Crossover from a square to a hexagonal pattern in Faraday surface waves, Phys. Rev. E, Volume 62 (2000), p. R33-R36

[22] P.B. Umbanhowar; F. Melo; H.L. Swinney Localized excitations in a vertically vibrated granular layer, Nature, Volume 382 (1996), pp. 793-796

[23] F. Raynal; S. Kumar; S. Fauve Faraday instability with a polymer solution, Eur. Phys. J. B, Volume 9 (1999), pp. 175-178

[24] C. Wagner; H. Müller; K. Knorr Faraday waves on a viscoelastic liquid, Phys. Rev. Lett., Volume 83 (1999), pp. 308-311

[25] P. Ballesta; S. Manneville Signature of elasticity in the Faraday instability, Phys. Rev. E, Volume 71 (2005), p. 026308

[26] A.V. Kityk; C. Wagner Delay of disorder by diluted polymers, Europhys. Lett., Volume 75 (2006), pp. 441-447

[27] H. Müller; W. Zimmermann Faraday instability in a linear viscoelastic fluid, Europhys. Lett., Volume 45 (1999), pp. 169-174

[28] F.S. Merkt; R.D. Deegan; D.I. Goldman; E. Rericha; H.L. Swinney Persistent holes in a fluid, Phys. Rev. Lett., Volume 92 (2004), p. 184501

[29] P. Saffman; G. Taylor The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous liquid, Proc. R. Soc. A, Volume 245 (1958), pp. 312-329

[30] Y. Couder Viscous fingering as an archetype for growth patterns (G. Batchelor; H. Moffat; M. Worsert, eds.), Perspectives in Fluid Dynamics, Cambridge University Press, Cambridge, 2000, pp. 53-104

[31] Y. Couder Growth patterns: From stable curved fronts to fractal structures (R. Artuso et al., eds.), Chaos, Order and Patterns, Plenum Press, New York, 1991, pp. 203-227

[32] G. Homsy Viscous fingering in porous media, Annu. Rev. Fluid Mech., Volume 19 (1987), pp. 271-311

[33] D. Bensimon; L. Kadanoff; S. Liang; B. Shraiman; C. Tang Viscous flows in two dimensions, Rev. Modern Phys., Volume 58 (1986), pp. 977-999

[34] K. McCloud; J. Maher Experimental perturbations to Saffman–Taylor flow, Phys. Rep., Volume 260 (1995), pp. 139-185

[35] H. VanDamme Flow and interfacial instabilities in Newtonian and colloidal fluids (or the birth, life and death of a fractal) (D. Avnir, ed.), The Fractal Approach to Heterogeneous Chemistry, John Wiley and Sons Ltd., 1989

[36] A. Lindner; D. Bonn; E. Corvera Poiré; M. Ben Amar; J. Meunier Viscous fingering in non-Newtonian fluids, J. Fluid Mech., Volume 469 (2002), pp. 237-256

[37] H. Zhao; J. Maher Associating-polymer effects in a Hele–Shaw experiment, Phys. Rev. E, Volume 47 (1993), pp. 4278-4283

[38] O. Greffier; A. AlKawaji; J. Rouch; H. Kellay Observation of finite-time singularity in needle propagation in Hele–Shaw cells, Phys. Rev. Lett., Volume 81 (1998), pp. 3860-3863

[39] D. Bonn; H. Kellay; M. Ben Amar; J. Meunier Viscous finger widening with surfactants and polymers, Phys. Rev. Lett., Volume 75 (1995), pp. 2132-2135

[40] A. Lindner; P. Coussot; D. Bonn Viscous fingering in a yield stress fluid, Phys. Rev. Lett., Volume 85 (2000), pp. 314-317

[41] N. Puff; G. Debregeas; J.-M.d. Meglio; D. Higgins; D. Bonn; C. Wagner Stick–slip instability for viscous fingering in a gel, Europhys. Lett., Volume 58 (2002), pp. 524-529

[42] S. Park; D. Durian Viscous and elastic fingering instabilities in foam, Phys. Rev. Lett., Volume 72 (1994), pp. 3347-3350

[43] C. Chevalier; M. Ben Amar; D. Bonn; A. Lindner Inertial effects on Saffman–Taylor viscous fingering, J. Fluid Mech., Volume 552 (2006), pp. 83-97

[44] C. Chevalier; A. Lindner; M. Leroux; E. Clement Morphodynamics during air injection into a confined granular suspension, J. Non-Newtonian Fluid Mech., Volume 158 (2009), pp. 63-72

[45] O. Johnsen; C. Chevalier; A. Lindner; R. Toussaint; E. Clement; K.J. Maloy; E.G. Flekkoy; J. Schmittbuhl Decompaction and fluidization of a saturated and confined granular medium by injection of a viscous liquid or gas, Phys. Rev. E, Volume 78 (2008), p. 051302

[46] X. Cheng; L. Xu; A. Patterson; H.M. Jaeger; S.R. Nagel Towards the zero-surface-tension limit in granular fingering instability, Nat. Phys., Volume 4 (2008), pp. 234-237

[47] J. Nase; A. Lindner; C. Creton Pattern formation during deformation of a confined viscoelastic layer: From a viscous liquid to a soft elastic solid, Phys. Rev. Lett., Volume 101 (2008), p. 074503

[48] A. Gupta Stability of a visco-elastic liquid film flowing down an inclined plane, J. Fluid Mech., Volume 28 (1967) no. Part 1, pp. 17-76

[49] M.D. Graham Interfacial hoop stresses and instability of viscoelastic free surface flows, Phys. Fluids, Volume 15 (2003), pp. 1702-1710

[50] J.P. Tordella Fracture in the extrusion of amorphous polymers through capillaries, J. Appl. Phys., Volume 27 (1956), p. 454

[51] M.M. Denn Issues in viscoelastic fluid mechanics, Annu. Rev. Fluid Mech., Volume 22 (1990), p. 13

[52] V. Bertola; B. Meulenbroek; C. Wagner; C. Storm; A. Morozov; W. van Saarloos; D. Bonn Experimental evidence for an intrinsic route to polymer melt fracture phenomena: A nonlinear instability of viscoelastic Poiseuille flow, Phys. Rev. Lett., Volume 90 (2003), p. 114502

[53] L.T.N. Périnet; D. Juric Numerical simulation of Faraday waves, J. Fluid Mech., Volume 635 (2009), pp. 1-26

[54] J. Eggers Universal pinching of 3d axisymmetrical free-surface flow, Phys. Rev. Lett., Volume 71 (1993), p. 3458

[55] J. McLean; P. Saffman The effect of surface tension on the shape of fingers in a Hele–Shaw cell, J. Fluid Mech., Volume 102 (1981), pp. 455-469

[56] D. Hong; J. Langer Analytic theory of the selection mechanism in the Saffman–Taylor problem, Phys. Rev. Lett., Volume 56 (1986), pp. 2032-2035

[57] B. Shraiman Velocity selection in the Saffman–Taylor problem, Phys. Rev. Lett., Volume 56 (1986), pp. 2028-2031

[58] R. Combescot; T. Dombre; V. Hakim; Y. Pomeau Shape selection of Saffman–Taylor fingers, Phys. Rev. Lett., Volume 56 (1986), pp. 2036-2039

[59] B.H. Zimm Dynamics of polymer molecules in dilute solution, J. Chem. Phys., Volume 24 (1956), p. 269

[60] A. Lindner; J. Vermant; D. Bonn How to obtain the elongational viscosity of dilute polymer solutions?, Physica A: Statistical Mechanics and its Applications, Volume 319 (2003), pp. 125-133

[61] C. Clanet; J.C. Lasheras Transition from dripping to jetting, J. Fluid Mech., Volume 383 (1999), pp. 307-326

[62] A. Rothert; R. Richter; I. Rehberg Formation of a drop: Viscosity dependence of three flow regimes, New J. Phys., Volume 5 (2003)

[63] H. Chang; E. Demekhin; E. Kalaidin Iterated stretching of viscoelastic jets, Phys. Fluids, Volume 11 (1999), pp. 1717-1737

[64] M. Stelter; G. Brenn; A.L. Yarin; R.P. Singh; F. Durst Validation and application of a novel elongational device for polymer solutions, J. Rheol., Volume 44 (2001), pp. 595-616

[65] M. Oliveira; G. McKinley Iterated stretching and multiple beads-on-a-string phenomena in dilute solutions of highly extensible flexible polymers, Phys. Fluids, Volume 17 (2005), p. 071704

[66] M.S.N. Oliveira; R. Yeh; G.H. McKinley Iterated stretching, extensional rheology and formation of beads-on-a-string structures in polymer solutions, J. Non-Newtonian Fluid Mech., Volume 137 (2006), pp. 137-148

[67] R. Sattler; C. Wagner; J. Eggers Blistering pattern and formation of nanofibers in capillary thinning of polymer solutions, Phys. Rev. Lett., Volume 100 (2008), p. 164502

[68] B. Carroll Equilibrium conformations of liquid-drops on thin cylinders under forces of capillarity—a theory for the roll-up process, Langmuir, Volume 2 (1986), pp. 248-250

[69] D. James; J. Saringer Extensional flow of dilute polymer-solutions, J. Fluid Mech., Volume 97 (1980), pp. 655-676

[70] T. Kume; T. Hashimoto; T. Takahashi; G. Fuller Rheo-optical studies of shear-induced structures in semidilute polystyrene solutions, Macromolecules, Volume 30 (1997), pp. 7232-7236

[71] R. Sattler; A. Kityk; C. Wagner Molecular configurations in the droplet detachment process of a complex liquid, Phys. Rev. E, Volume 75 (2007), p. 051805

[72] R. Chuoke; P.v. Meurs; C.v.d. Poel The instability of slow, immiscible, viscous liquid–liquid displacements in permeable media, Pet. Trans. AIME, Volume 216 (1959), p. 188

[73] P. Tabeling; A. Libchaber Film draining and the Saffman–Taylor problem, Phys. Rev. A, Volume 33 (1986), pp. 794-796

[74] S. Nguyen, H. Henry, M. Plapp, private communication

[75] L. Kondic; M.J. Shelley; P. Palffy-Muhoray Non-Newtonian Hele–Shaw flow and the Saffman–Taylor instability, Phys. Rev. Lett., Volume 80 (1998), pp. 1433-1436

[76] A. Lindner; D. Bonn; J. Meunier Viscous fingering in a shear-thinning fluid, Phys. Fluids, Volume 12 (2000), pp. 256-261

[77] M. Rabaud; Y. Couder; N. Gerard Dynamics and stability of anomalous Saffman–Taylor fingers, Phys. Rev. A, Volume 37 (1988), pp. 935-947

[78] E. Corvera Poiré; M. Ben Amar Finger behavior of a shear thinning fluid in a Hele–Shaw cell, Phys. Rev. Lett., Volume 81 (1998), pp. 2048-2051

[79] L. Matthiesen, Annal. Phys. Chem. 5 (14) 107

[80] Rayleigh Numerical simulation of Faraday waves, Phil. Mag. J. Sci., Volume 5 (1883), p. 229

[81] T.B. Benjamin; F. Ursell Numerical simulation of Faraday waves, Proc. R. Soc. London Ser. A, Volume 225 (1954), p. 505

[82] K. Kumar; L. Tuckerman Parametric-instability of the interface between 2 fluids, J. Fluid Mech., Volume 279 (1994), pp. 49-68

[83] H. Muller; H. Wittmer; C. Wagner; J. Albers; K. Knorr Analytic stability theory for Faraday waves and the observation of the harmonic surface response, Phys. Rev. Lett., Volume 78 (1997), pp. 2357-2360

[84] H. Arbell; J. Fineberg Spatial and temporal dynamics of two interacting modes in parametrically driven surface waves, Phys. Rev. Lett., Volume 81 (1998), pp. 4384-4387

[85] W. Edwards; S. Fauve Patterns and quasi-patterns in the Faraday experiment, J. Fluid Mech., Volume 278 (1994), pp. 123-148

[86] A. Kityk; J. Embs; V. Mekhonoshin; C. Wagner Spatiotemporal characterization of interfacial Faraday waves by means of a light absorption technique, Phys. Rev. E, Volume 72 (2005), p. 036209

[87] P. Chen; J. Vinals Pattern selection in Faraday waves, Phys. Rev. Lett., Volume 79 (1997), pp. 2670-2673

Cited by Sources:

Comments - Policy