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.
Mots-clés : Formation de motifs, Instabilité, Rhéologie, Solution de polymères
Anke Lindner 1; Christian Wagner 2
@article{CRPHYS_2009__10_8_712_0, 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}, }
Anke Lindner; Christian Wagner. Viscoelastic surface instabilities. Comptes Rendus. Physique, Complex and biofluids, Volume 10 (2009) no. 8, pp. 712-727. doi : 10.1016/j.crhy.2009.10.017. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2009.10.017/
[1] Nonlinear dynamics and breakup of free-surface flows, Rev. Modern Phys., Volume 69 (1997), pp. 865-929
[2] Physics of liquid jets, Rep. Progr. Phys., Volume 71 (2008), p. 036601
[3] Breakup of a laminar jet of a viscoelastic fluid, J. Fluid Mech., Volume 38 (1969), p. 689
[4] On orientational effects of jets and threats of dilute polymer solutions, Dokl. Akad. Nauk SSSR, Volume 257 (1981), pp. 336-339
[5] 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] 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] Validation and application of a novel elongational device for polymer solutions, J. Rheol., Volume 44 (2000), pp. 595-616
[8] Inhibition of the finite-time singularity during droplet fission of a polymeric fluid, Phys. Rev. Lett., Volume 86 (2001), pp. 3558-3561
[9] Elasto-capillary thinning and breakup of model elastic liquids, J. Rheol., Volume 45 (2001), pp. 115-138
[10] Droplet detachment and satellite bead formation in viscoelastic fluids, Phys. Rev. Lett., Volume 95 (2005), p. 164504
[11] Drop formation and breakup of low viscosity elastic fluids: Effects of molecular weight and concentration, Phys. Fluids, Volume 18 (2006), p. 043101
[12] The beads-on-string structure of viscoelastic threads, J. Fluid Mech., Volume 556 (2006), pp. 283-308
[13] Gravity flow instability of viscoplastic materials: The ketchup drip, Phys. Rev. E, Volume 72 (2005), p. 031409
[14] Capillary breakup extensional rheometry of yield stress fluids, Appl. Rheol., Volume 19 (2009), p. 41969
[15] Pendant drop thread dynamics of particle-laden liquids, Int. J. Multiphase Flow, Volume 33 (2007), pp. 448-468
[16] On the forms and states of fluids on vibrating elastic surfaces, Philos. Trans. R. Soc. London, Volume 52 (1831), p. 319
[17] Square patterns and quasipatterns in weakly damped Faraday waves, Phys. Rev. E, Volume 53 (1996), p. R4283-R4286
[18] Dissipative solitary states in driven surface waves, Phys. Rev. Lett., Volume 76 (1996), pp. 3959-3962
[19] Nonlinear pattern formation of Faraday waves, Phys. Rev. Lett., Volume 78 (1997), pp. 4043-4046
[20] Superlattice patterns in surface waves, Physica D, Volume 123 (1998), pp. 99-111
[21] Crossover from a square to a hexagonal pattern in Faraday surface waves, Phys. Rev. E, Volume 62 (2000), p. R33-R36
[22] Localized excitations in a vertically vibrated granular layer, Nature, Volume 382 (1996), pp. 793-796
[23] Faraday instability with a polymer solution, Eur. Phys. J. B, Volume 9 (1999), pp. 175-178
[24] Faraday waves on a viscoelastic liquid, Phys. Rev. Lett., Volume 83 (1999), pp. 308-311
[25] Signature of elasticity in the Faraday instability, Phys. Rev. E, Volume 71 (2005), p. 026308
[26] Delay of disorder by diluted polymers, Europhys. Lett., Volume 75 (2006), pp. 441-447
[27] Faraday instability in a linear viscoelastic fluid, Europhys. Lett., Volume 45 (1999), pp. 169-174
[28] Persistent holes in a fluid, Phys. Rev. Lett., Volume 92 (2004), p. 184501
[29] 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] 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] 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] Viscous fingering in porous media, Annu. Rev. Fluid Mech., Volume 19 (1987), pp. 271-311
[33] Viscous flows in two dimensions, Rev. Modern Phys., Volume 58 (1986), pp. 977-999
[34] Experimental perturbations to Saffman–Taylor flow, Phys. Rep., Volume 260 (1995), pp. 139-185
[35] 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] Viscous fingering in non-Newtonian fluids, J. Fluid Mech., Volume 469 (2002), pp. 237-256
[37] Associating-polymer effects in a Hele–Shaw experiment, Phys. Rev. E, Volume 47 (1993), pp. 4278-4283
[38] Observation of finite-time singularity in needle propagation in Hele–Shaw cells, Phys. Rev. Lett., Volume 81 (1998), pp. 3860-3863
[39] Viscous finger widening with surfactants and polymers, Phys. Rev. Lett., Volume 75 (1995), pp. 2132-2135
[40] Viscous fingering in a yield stress fluid, Phys. Rev. Lett., Volume 85 (2000), pp. 314-317
[41] Stick–slip instability for viscous fingering in a gel, Europhys. Lett., Volume 58 (2002), pp. 524-529
[42] Viscous and elastic fingering instabilities in foam, Phys. Rev. Lett., Volume 72 (1994), pp. 3347-3350
[43] Inertial effects on Saffman–Taylor viscous fingering, J. Fluid Mech., Volume 552 (2006), pp. 83-97
[44] Morphodynamics during air injection into a confined granular suspension, J. Non-Newtonian Fluid Mech., Volume 158 (2009), pp. 63-72
[45] 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] Towards the zero-surface-tension limit in granular fingering instability, Nat. Phys., Volume 4 (2008), pp. 234-237
[47] 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] 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] Interfacial hoop stresses and instability of viscoelastic free surface flows, Phys. Fluids, Volume 15 (2003), pp. 1702-1710
[50] Fracture in the extrusion of amorphous polymers through capillaries, J. Appl. Phys., Volume 27 (1956), p. 454
[51] Issues in viscoelastic fluid mechanics, Annu. Rev. Fluid Mech., Volume 22 (1990), p. 13
[52] 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] Numerical simulation of Faraday waves, J. Fluid Mech., Volume 635 (2009), pp. 1-26
[54] Universal pinching of 3d axisymmetrical free-surface flow, Phys. Rev. Lett., Volume 71 (1993), p. 3458
[55] 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] Analytic theory of the selection mechanism in the Saffman–Taylor problem, Phys. Rev. Lett., Volume 56 (1986), pp. 2032-2035
[57] Velocity selection in the Saffman–Taylor problem, Phys. Rev. Lett., Volume 56 (1986), pp. 2028-2031
[58] Shape selection of Saffman–Taylor fingers, Phys. Rev. Lett., Volume 56 (1986), pp. 2036-2039
[59] Dynamics of polymer molecules in dilute solution, J. Chem. Phys., Volume 24 (1956), p. 269
[60] How to obtain the elongational viscosity of dilute polymer solutions?, Physica A: Statistical Mechanics and its Applications, Volume 319 (2003), pp. 125-133
[61] Transition from dripping to jetting, J. Fluid Mech., Volume 383 (1999), pp. 307-326
[62] Formation of a drop: Viscosity dependence of three flow regimes, New J. Phys., Volume 5 (2003)
[63] Iterated stretching of viscoelastic jets, Phys. Fluids, Volume 11 (1999), pp. 1717-1737
[64] Validation and application of a novel elongational device for polymer solutions, J. Rheol., Volume 44 (2001), pp. 595-616
[65] 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] 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] Blistering pattern and formation of nanofibers in capillary thinning of polymer solutions, Phys. Rev. Lett., Volume 100 (2008), p. 164502
[68] 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] Extensional flow of dilute polymer-solutions, J. Fluid Mech., Volume 97 (1980), pp. 655-676
[70] Rheo-optical studies of shear-induced structures in semidilute polystyrene solutions, Macromolecules, Volume 30 (1997), pp. 7232-7236
[71] Molecular configurations in the droplet detachment process of a complex liquid, Phys. Rev. E, Volume 75 (2007), p. 051805
[72] The instability of slow, immiscible, viscous liquid–liquid displacements in permeable media, Pet. Trans. AIME, Volume 216 (1959), p. 188
[73] 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] Non-Newtonian Hele–Shaw flow and the Saffman–Taylor instability, Phys. Rev. Lett., Volume 80 (1998), pp. 1433-1436
[76] Viscous fingering in a shear-thinning fluid, Phys. Fluids, Volume 12 (2000), pp. 256-261
[77] Dynamics and stability of anomalous Saffman–Taylor fingers, Phys. Rev. A, Volume 37 (1988), pp. 935-947
[78] 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] Numerical simulation of Faraday waves, Phil. Mag. J. Sci., Volume 5 (1883), p. 229
[81] Numerical simulation of Faraday waves, Proc. R. Soc. London Ser. A, Volume 225 (1954), p. 505
[82] Parametric-instability of the interface between 2 fluids, J. Fluid Mech., Volume 279 (1994), pp. 49-68
[83] Analytic stability theory for Faraday waves and the observation of the harmonic surface response, Phys. Rev. Lett., Volume 78 (1997), pp. 2357-2360
[84] Spatial and temporal dynamics of two interacting modes in parametrically driven surface waves, Phys. Rev. Lett., Volume 81 (1998), pp. 4384-4387
[85] Patterns and quasi-patterns in the Faraday experiment, J. Fluid Mech., Volume 278 (1994), pp. 123-148
[86] Spatiotemporal characterization of interfacial Faraday waves by means of a light absorption technique, Phys. Rev. E, Volume 72 (2005), p. 036209
[87] Pattern selection in Faraday waves, Phys. Rev. Lett., Volume 79 (1997), pp. 2670-2673
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