Shear banding of the fluid with a nonmonotonic dependence of flow stress upon strain rate
DOI:
https://doi.org/10.7242/1999-6691/2018.11.1.6Keywords:
non-monotonic flow curve, mesoscopic rheological model, pressure flow, flat channel with movable wall, analytical and numerical solutions, nonuniqueness, shear bandingAbstract
The problem of the pressure flow of a fluid in a flat channel with the counter motion of one of the walls is considered. The fluid is characterized a non-monotonic flow curve consisting of three segments: left segment (ascending branch), middle segment (descending branch) and right segment (ascending branch). The rheological properties of the fluid are described by a modified model of Vinogradov-Pokrovsky. The constants of the model are determined using the results of rheological tests of high-density polyethylene melt performed with a laser Doppler viscometer. All exact analytical solutions of this problem are obtained in parametric form. The profiles of velocity, effective viscosity and velocity gradient along the channel height are constructed for different parameters of the rheological model. It is shown that at the same prescribed stress field, in the range of shear rates corresponding to the descending branch of the flow curve, there are three solutions, of which one is unstable and not physically realizable and the other two are stable; which of them is realized depends on the loading prehistory. One of these solutions, corresponding to the left branch of the flow curve, is monotone, and the solution corresponding to the right branch of the curve demonstrates the stratification of the flow into strips with different physico-mechanical properties and at different strain rates. At the same time, the dependence of the effective viscosity on the strain rate is a monotonically decreasing function. The same problem is solved for a two-dimensional case by the finite element method using a weak Galerkin formulation. Comparison of the numerical results with the analytical solution shows that the results coincide with a sufficient degree of accuracy. In either case, as the counter pressure drop approaches zero, the limiting transition to the Couette flow is impossible.
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Cates M. E., Fielding S. M. Rheology of giant micelles, Phys., 2006, vol. 55, no. 7-8, pp.799–879. DOI
Olmsted P.D. Perspectives on shear banding in complex fluids, Acta, 2008, vol. 47, no. 3, pp. 283–300. DOI
Tapadia P., Wang S.-Q. Nonlinear flow behavior of entangled polymer solutions: Yieldlike entanglement-disentanglement transition. Macromolecules, 2004, vol. 37, no. 24, pp. 9083–9095. DOI
Ravindranath S., Wang S.-Q. Large amplitude oscillatory shear behavior of entangled polymer solutions: Particle tracking velocimetric investigation, Rheol., 2008, vol. 52, no. 2, pp. 341–358. DOI
Adams, J. M., Olmsted P. D. Nonmonotonic models are not necessary to obtain shear banding phenomena in entangled polymer solutions. Rev. Lett., 2009, vol. 102, no. 6, pp. 067801. DOI
Adams, J. M., Olmsted P. D. Adams and Olmsted reply, Rev. Lett., 2009, vol. 103, no. 21, pp. 219802. DOI
Ravindranath S., Wang S.-Q., Olechnowicz M., Quirk R. P. Banding in simple steady shear of entangled polymer solutions. Macromolecules, 2008, vol. 41, no. 7, pp. 2663–2670. DOI
Boukany, P. E., Wang S.-Q. Shear banding or not in entangled DNA solutions depending on the level of entanglement, Rheol., 2009, vol. 53, no. 1, pp. 73–83. DOI
V. Trusov, V.N. Ashikhmin, P.S. Volegov and A.I. Shveykin Constitutive relations and their application to the description of microstructure evolution, Fiz. mezomekh., vol. 12, no. 3, 2009, pp. 61-71.
Trusov P.V., Ashihmin V.N., Shveykin A.I. Dvuhurovnevaya model uprugoplasticheskogo deformirovaniya polikristallicheskih materialov [Two-level model of elastoplastic deformation of polycrystalline materials]. Mehanika kompozitsionnyih materialov i konstruktsiy, 2009, vol. 15. no. 3, pp.327-344
Trusov P.V., Shveykin A.I., Nechaeva E.S., Volegov P.S. Multilevel models of inelastic deformation of materials and their application for description of internal structure evolution // Phys Mesomech, 2012, vol. 15, no. 3-4, pр. 155-175. DOI
de Gennes P.G. Origin of internal viscosity in dilute polymer solution, Chem. Phys., 1977, vol. 66, no. 12, pp. 5825-5826. DOI
de Gennes P.G. Scaling Concepts in Polymer Physics. Cornell Univ. Press, Ithaca, N.Y., 1979, 319 p.
Doi M. Edwards S.F. Dynamics of concentrated polymer systems. Part 1. Brownian motion in the equilibrium state, Chem. Soc.: Faraday Trans. 2, 1978, vol. 74, pp.1789-1801. DOI
Doi M., Edwards S.F. The theory of polymer dynamics. Oxford University Press, Oxford, 1986. 391 p.
Marrucci G., Grizzuti N. Fast flows of concentrated polymers: predictions of the tube model on chain stretching, Gaz. Chim.Ital., 1988, vol. 118, pp.179-185.
Remmelgas J., Harrison G., Leal L.G. A differential constitutive equation for entangled polymer solutions, Non-Newtonian Fluid Mech., 1999, vol. 80, no. 2-3, pp. 115-134. DOI
Harrison G.M., Remmelgas J., Leal L.G. Comparison of dumbell-based theory and experiment for a dilute polymer solution in a corotating two-roll mill, Rheol., 1999, vol. 43, no. 1, pp. 197-218. DOI
Olbricht W.L., Rallison J.M., Leal L.G. Strong flow criteria based on microstructure deformation, Non-Newton. Fluid., 1982, no. 10, pp. 291-318.DOI
Bird R.B., Curtiss C.F., Armstrong R.C., Hassager O. Dynamics of Polymeric Liquids. Volume 2: Kinetic Theory. John Wiley & Sons, Inc., New York, 2nd Ed., 1987. 437 p.
Bird R.B., Dotson P.J., Johnson N.L. Polymer solution rheology based on a finitely extensible bead—spring chain model, Non-Newton. Fluid., 1980, vol. 7, no. 2-3, pp. 213-235. DOI
Volkov V.S., Vinogradov G.V. Theory of dilute polymer solutions in viscoelastic fluid with a single relaxation time, Non-Newton. Fluid Mech., 1984, vol. 15, no. 1, pp. 29-44. DOI
Volkov V.S., Vinogradov G.V. Relaxational interactions and viscoelasticity of polymer melts. Part I. Model development, Non-Newtonian Fluid Mech., 1985, vol. 18, no. 2, pp. 163-172. DOI
Pokrovskii V.N. Statisticheskaya Mekhanika Razbavlennykh Suspenzii (Statistical Mechanics of Dilute Suspensions, in Russian), Nauka, Moskow, 1978.
Pokrovskii V.N. Dynamics of weakly-coupled linear macromolecules, Sov. Phys. Uspekhi, 1992, vol. 35, no. 5, pp. 384-399. DOI
Pokrovskii V.N., Altukhov Yu.A., Pyshnograi G.V. The Mesoscopic Approach to the Dynamics of Polymer Melts: Consequences for the Constitutive Equation, Non-Newton. Fluid Mech., 1998, vol. 76, no. 1-3, pp.153-181. DOI
Altukhov Yu.A., Pokrovskii V.N., Pyshnograi G.V. On the Difference between Weakly and Strongly Entangled Linear Polymer, Non-Newton. Fluid Mech., 2004, vol. 121, no. 2-3, pp.73-86. DOI
Pyshnograi G.V., Gusev A S., Pokrovskii V.N. Constitutive Equations for Weakly Entangled Linear Polymers, // Non-Newton. Fluid Mech., 2009, vol. 163, no. 1-3. pp.17-28. DOI
Gusev А.S, Makarova., М.А., Pyshnograi G.V. Mesoscopic Equation of State of Polymer Systems and Description of the Dynamic Characteristics Based on It, Journal of Engineering Physics and Thermophysics, 2005, vol. 78, no. 5, pp. 892-898. DOI
Aristov N., Skul’skij O.I. Exact solution of the problem of flow of a polymer solution in a plane channel, J. Appl Mech. Techn. Phys., 2003, vol. 76, pp. 577-585. DOI
Skul’skij O.I., Kuznecova Yu.L. Reologicheskie modeli rastvorov polimerov. [Rheological models of polymer solutions]. Sb. nauch. trudov «Matematicheskoe modelirovanie sistem i processov» − PGTU, 2006, no. 14, pp. 178-188.
Kuznecova Ju.L., Skul’skij O.I. Issledovanie reologicheskih modelej rastvorov polimerov na reometricheskih techenijah. [Investigation of rheological models of polymer solutions on rheometric flows]. Matematicheskoe modelirovanie v estestvennyh naukah, 2013, no. 1, pp. 92-94.
Kuznecova Ju.L., Skul’skij O.I., Pyshnograj G.V. The flow of a nonlinear elastic viscous fluid in a flat channel under the action of a given pressure gradient. meh. splos. sred– Computational Continuum Mechanics, 2010, vol. 1, no. 2, pp. 55-69. DOI
Kuznecova Ju.L., Skul’skij O.I. Influence of interlacing of macromolecules on the simple shear flow of an elastic viscous liquid. meh. splos. sred– Computational Continuum Mechanics, 2013, vol. 6, no. 2, pp. 224-231. DOI
Kuznetsova J.L., Skul’skiy O. I. Verification of mesoscopic models of viscoelastic fluids with a non-monotonic flow curve. Korea-Aust. Rheol. J., 2016, vol. 28, no. 1, pp. 33-40. DOI
Robert L. Demay Y. Vergnes B. Stick-slip flow of high density polyethylene in a transparent slit die investigated by laser Doppler velocimetry, Rheol Acta, 2004, vol. 43, no. 1, pp.89-98. DOI
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