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Computer Modeling of Viscoelastic Flow

  • V. Legat
Chapter
  • 462 Downloads
Part of the NATO ASI Series book series (NSSE, volume 302)

Abstract

It is well known that the behaviour of all materials cannot be predicted by the classical Hookean elastic solids or Newtonian viscous liquids. Viscoelasticcoelastic effects, i.e. phenomena that cannot be explained on the basis of nonlinear purely-viscous or purely-elastic behaviour, can be important in polymer processing applications [1] [2] [3] [4]. Rheological non-linearities and geometrical singularities render analytical investigations difficult and lead to the numerical simulation of viscoelastic effects in complex geometries where simplifying assumptions cannot be made.

Keywords

Constitutive Equation Viscoelastic Fluid Weissenberg Number Newtonian Limit Maxwell Fluid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Lodge, A.S. (1969) Elastic liquids, Academic Press. LondonGoogle Scholar
  2. 2.
    Keunings, R. (1989) Simulation of Viscoelastic Fluid Flow Fundamentals of Computer Modeling for Polymer Processing, Ed. Tucker III, C.L., Carl Hanser Verlag, Munich, pp. 403–470Google Scholar
  3. 3.
    Boger, D.V. and Walters, K. (1993) Rheological Phenomena in Focus, Elsevier, AmsterdamzbMATHGoogle Scholar
  4. 4.
    Tanner, R.I. (1985) Engineering Rheology, Clarendon Press, OxfordzbMATHGoogle Scholar
  5. 5.
    Barnes, H.A., Hutton, J.F., and Walters, K. (1989) An Introduction to Rheology, Elsevier. AmsterdamzbMATHGoogle Scholar
  6. 6.
    Crochet, M.J., Davies, A.R. and Walters, K. (1984) Numerical Simulation of NonNewtonian Flows, Elsevier, AmsterdamGoogle Scholar
  7. 7.
    Crochet, M.J. (1989) Numerical Simulation of Viscoelastic Flow : a Review, Rubber Chemistry and Technology, Vol. no. 62, pp. 426–455CrossRefGoogle Scholar
  8. 8.
    Bird, R.B. Armstrong, R.C., and Hassager, O. (1987) Dynamics of Polymeric Liquids, Vol 1: Fluid Mechanics, 2nd ed., Wiley, New-YorkGoogle Scholar
  9. 9.
    Giesekus, H. (1982) A Simple Constitutive Equation for Polymer Fluids Based on the concept. of Deformation Dependent Tensorial Mobility, J. Non-Newtonian Fluid Mech., Vol. no. 11, pp. 69–109zbMATHCrossRefGoogle Scholar
  10. 10.
    Leonov, A. I. (1976) Nonequilibrium Thermodynamics and Rheology of Viscoelastic Polymer Media, Rheol. Acta, Vol. no. 15, pp. 85–98zbMATHCrossRefGoogle Scholar
  11. 11.
    Phan Thien, N. and Tanner, R.I. (1977) A New Constitutive Equation derived from Network Theory, J. Non-Newtonian Fluid Mech., Vol. no. 2, pp. 353–365zbMATHCrossRefGoogle Scholar
  12. 12.
    Apelian, I.R., Armstrong, R.C., and Brown, R.A. (1988) Impact of the Constitutive Equation and Singularity on the Calculation of stick-slip Flow : the Modified upper convected Maxwell Model, J. Non-Newtonian Fluid Mech., Vol. no. 27, pp. 299–321zbMATHCrossRefGoogle Scholar
  13. 13.
    Larson, R.G. (1988) Constitutive Equations for Polymer Melts and Solutions, Butterworths. BostonGoogle Scholar
  14. 14.
    Bird, R.B., Curtiss, C.F., Amstrong, R.C., and Hassager, O. (1987) Dynamics of Polymeric Liquids, Vol 2: Kinetic Theory, 2nd ed., Wiley, New-YorkGoogle Scholar
  15. 15.
    Chilcott, M. D. and Rallison, J. N.(1988) Creeping Flow of Dilute Polymer Solutions past Cylinders and Spheres, J. Non-Newtonian Fluid Mech., Vol. no. 29, pp. 381–432zbMATHCrossRefGoogle Scholar
  16. 16.
    Bernstein, B., Kearsley, E.A and Zapas, L. (1963) A Study of Stress Relaxation with Finite Strain, Trans. Soc. Rheol., Vol. no. 7, pp. 391–410zbMATHCrossRefGoogle Scholar
  17. 17.
    Zienkiewicz, O.C. and Morgan, K. (1983) Finite Elements and Approximations, Wiley, New-YorkGoogle Scholar
  18. 18.
    Zienkiewicz, O.C. and Taylor, R.L. (1991) The Finite Element Method (4th edition), McGraw-Hill, LondonGoogle Scholar
  19. 19.
    Kawahara, M. and Takeuchi, N. (1977) Mixed Finite Element Method for Analysis of Viscoelastic Fluid Flow, Comp. Fluids, Vol. no. 5, pp. 33–45ADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Crochet, M.J. and Keunings, R. (1980) Die Swell of a Maxwell Fluid : Numerical Prediction, J. Non-Newtonian Fluid ,ech., Vol. no. 7, pp. 199–212CrossRefGoogle Scholar
  21. 21.
    Ladyzhenskaya, O.A. (1969) The .Mfathematical Theory of Viscous Incompressible Flow. Gordon and Breach. New-YorkGoogle Scholar
  22. 22.
    Brezzi F. (1974) On the Existence. Uniqueness and Approximation of Saddle-point Problem Arising from Lagrange Multipliers, Revue Française d’Automatique Inform. Rech. Opér., Série Rouge Anal. Nurnér. 8, Vol. no. R-2, pp. 129–151MathSciNetzbMATHGoogle Scholar
  23. 23.
    Marchal, J.M. and Crochet, M.J. (1986) Hermitian Finite Element for Calculating Viscoelastic Flow, J. Non-Newtonian Fluid Mech., Vol. no. 20, pp. 187–207zbMATHCrossRefGoogle Scholar
  24. 24.
    Marchal, J.M. and Crochet, M.J. (1987) A New Mixed Finite Element for Calculating Viscoelastic Flow, J. Non-Newtonian Fluid Mech., Vol. no. 26, pp. 77–114zbMATHCrossRefGoogle Scholar
  25. 25.
    Brooks, A.N., and Hughes, T.J.R. (1982) Streamline Upwind/Petrov-Galerkin Formulation for Convection Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations, Comp. Meth. Appl. Mech. Engng., Vol. no. 32, pp. 199–259MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 26.
    Johnson, C. (1987) Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge Univ. Press, CambridgezbMATHGoogle Scholar
  27. 27.
    Crochet, M.J. and Legat, V. (1992) The consistent Streamline Petrov-Galerkin Method for Viscoelastic Flow Revisited, J. Non-Newtonian Fluid Mech., Vol. no. 42, pp. 283–299zbMATHGoogle Scholar
  28. 28.
    Debbaut, B. and Hocq, B. (1992) On the Numerical Simulation of Axisymmetric Swirling Flows of Differential Viscoelastic Liquids : the Rod Climbing Effect and the Quelleffekt, J. Non-Newtonian Fluid .MMech., Vol. no. 43, pp. 103–126zbMATHCrossRefGoogle Scholar
  29. 29.
    Renardy, M. (1985) Existence of Slow Steady Flows of Viscoelastic Fluids with Differential Constitutive Equations. Z. Angew. Math. u. Mech., Vol. no. 65, pp. 449–451MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    King, R.C., Apelian, M.N., Armstrong, R.C. and Brown, R.A. (1988) Numerically Stable Finite Element Techniques for Viscoelastic Calculations in Smooth and Singular Geometries, J. Non-Newtonian Fluid Mech., Vol. no. 29, pp. 147–216zbMATHCrossRefGoogle Scholar
  31. 31.
    Dupret, F. and Marchal, J.M. (1986) Sur le signe des valeurs propres du tenseur des extra-contraintes dans un écoulement de fluide de Maxwell, J. Méc. Théor. Appl., Vol. no. 5, pp. 403–427zbMATHGoogle Scholar
  32. 32.
    Dupret, F. and Marchal, J.M. (1986) Loss of Evolution in the Flow of Viscoelastic Fluids, J. Non-Newtonian Fluid Mech. Vol. no. 20. pp. 143–171zbMATHCrossRefGoogle Scholar
  33. 33.
    Rajagopalan, D., Armstrong, R.C. and Brown, R.A. (1990) Finite Element Methods for Calculation of Steady Viscoelastic Flow Using Constitutive Equations with a Newtonian Viscosity, J. Non-Newtonian Fluid Mech., Vol. no. 36, pp. 159–192zbMATHCrossRefGoogle Scholar
  34. 34.
    Debae, F., Legat, V. and Crochet, M.J. (1994) Practical Evaluation of Four Mixed Finite Element Methods for Viscoelastic Flow, J. of Rheology, Vol. no. 38, pp. 421–442ADSCrossRefGoogle Scholar
  35. 35.
    Baaijens, F.P.T. (1992) Numerical Analysis of Unsteady Viscoelastic Flow, Comp. Meth. Appl. Mech. Engng., Vol. no. 94, pp. 285–299ADSzbMATHCrossRefGoogle Scholar
  36. 36.
    Fortin, A. and Zine, A. (1992) An Improved GMRES Method for Solving Viscoelastic Fluid Flow Problem, J. Non-Newtonian Fluid Mech., Vol. no. 42, pp. 1–18zbMATHCrossRefGoogle Scholar
  37. 37.
    Fortin, A., Zine, A. and Agassant, J.M. (1992) Computing Viscoelastic Fluid Flow Problems at Low Cost, J. Non-Newtonian Fluid Mech., Vol. no. 45, pp. 209–229zbMATHCrossRefGoogle Scholar
  38. 38.
    Beris, A.N., Armstrong, R.C. and Brown, R.A. (1987) Spectral/Finite Element Calculations of the Flow of a Maxwell Fluid between Eccentric Rotating Cylinders, J. of Non-Newtonian Fluid Mech., Vol. no. 22, pp. 129–167zbMATHCrossRefGoogle Scholar
  39. 39.
    Souvaliotis, A. and Beris, A.N. (1992) Application of of Domain Decomposition Spectral Collocation Methods in Viscoelastic Flows Trough Model Porous Media, J. of Rheology, Vol. no. 36, pp. 1417–1453ADSCrossRefGoogle Scholar
  40. 40.
    Talwar, K.K. and Khomami, B. (1992) Application of Higher Order Finite Element Methods for Viscoelastic Flow in Porous Media, J. of Rheology, Vol. no. 36, pp. 1377–1416ADSCrossRefGoogle Scholar
  41. 41.
    Warichet, V. and Legat, V. (1994) An Adaptive hp Finite Element Method for Calculating Viscoelastic Fluids Flow. (in preparation)Google Scholar
  42. 42.
    Satrape, J.V. and Crochet, M.J. (1994) Numerical Simulation of the Motion of a Sphere in a Boger Fluid, J. Non-Newtonian Fluid Mech., in pressGoogle Scholar
  43. 43.
    Purnode, B. and Crochet, M.J. (1995) ,(in preparation)Google Scholar
  44. 44.
    Joseph D.D., Renardy, M. and Saut, J.C. (1985) Hyperbolicity and Change of Type in the Flow of Viscoelastic Fluids, Arch. Rational Mech. Anal., Vol. no. 87, pp. 213–251MathSciNetADSzbMATHCrossRefGoogle Scholar
  45. 45.
    Renardy, M. (1986) Inflow Boundary Conditions for Steady Flows of Viscoelastic Fluids with Differential Constitutive Laws, Mathematics Research Center Technical Summary Report2916, Univ. of Wisconsin, Madison, USA Google Scholar
  46. 46.
    Legat, V. and Marchai, J.M. (1992) On the Stability and the Accuracy of Fully Coupled Finite Element Techniques Used to Simulate the Flow of Differential Viscoelastic Fluids : a One-Dimensional Model, J. of Rheology, Vol. no. 36, pp. 1325–1348ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • V. Legat
    • 1
  1. 1.Applied Mechanics (CESAME)Université Catholique de LouvainLouvain-la-NeuveBelgium

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