Skip to main content
SpringerLink
Log in

Mathematical Results and Numerical Methods for Steady Incompressible Viscoelastic Fluid Flows

  • Chapter
Differential Equations and Nonlinear Mechanics

Part of the book series: Mathematics and Its Applications ((MAIA,volume 528))

  • 547 Accesses

Abstract

The aim of this paper is to present a short survey of recent mathematical results for some models of incompressible, homogeneous, viscoelastic non-Newtonian fluids, namely for the grade type Rivlin-Ericksen and Oldroyd type models. We address the questions of existence, uniqueness and asymptotic behavior of steady solutions, in several physically relevant flow geometries, by suitably decoupling the elliptic and hyperbolic parts in the systems of equations. Finally, we discuss a numerical scheme using a mixed finite element method for the simulation of the models in two-dimensional bounded domains.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Babuska, I., The finite element method with Lagrangian multipliers, Numer. Math., 20, 1973, 179–192.

    Article  MathSciNet  MATH  Google Scholar 

  2. Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRP, Anal. Numer., R2, 1974, 129–151.

    Google Scholar 

  3. Cioranescu, D. and Ouazar, E.H., Existence and uniqueness for fluids of second grade, Coll. France Sem., Pitman Res. Notes Math., 109, 1984, 178–197.

    Google Scholar 

  4. Cioranescu, D., Girault, V., Glowinski, R., and Scott, L.R., Some theoretical and numerical aspects of grade-two fluid models, Partial Differential Equations, CRC Research Notes Math., 406, 1999, Chapman & Hall, 99–110.

    Google Scholar 

  5. Criminale, W.O., Ericksen, J.L., and Filbey, G.I., Steady shear flow of non-Newtonian fluids, Arch. Rational Mech. Anal., 1, 1957, 410–417.

    MathSciNet  Google Scholar 

  6. Crochet, M.J., Davies, A.R., and Walters, K., Numerical Simulation of Non-Newtonian Flow, Elsevier, New York, 1984.

    MATH  Google Scholar 

  7. Dunn, J.E. and Fosdick, R.L., Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Rational Mech. Anal., 56, 1974, 191–252.

    Article  MathSciNet  MATH  Google Scholar 

  8. Finn, R., Estimates at infinity for stationary solutions of the Navier-Stokes equations, Bull. Math. Soc. Sci. Math. Phys. R.P. Roumaine (N.S.), 3, 1959, 387–418.

    MathSciNet  Google Scholar 

  9. Finn, R., On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal., 19, 1965, 363–406.

    MathSciNet  MATH  Google Scholar 

  10. Fortin, M. and Essellaoui, D., A finite element procedure for viscoelastic flows, Int. J. Numer. Meth. Fluid, 7, 1987, 1035–1052.

    Article  MATH  Google Scholar 

  11. Fortin, M. and Fortin, A., A new approach for the FEM simulation of viscoelastic flows, J. Non-Newtonian Fluid Mech., 32, 1989, 295–310.

    Article  MATH  Google Scholar 

  12. Fosdick, R.L. and Rajagopal, K.R., Uniqueness and drag for fluids of second-grade in steady motion, Int. J. Non-Linear Mechanics, 13, 1978, 131–137.

    Article  MATH  Google Scholar 

  13. Fosdick, R.L. and Rajagopal, K.R., Thermodynamics and stability of fluids of third grade, Proc. Roy. Soc. London, A339, 1980, 351–377.

    MathSciNet  Google Scholar 

  14. Fosdick, R.L. and Serrin, J., Rectilinear steady flow of simple fluids, Proc. Roy. Soc. London, A332, 1973, 311–333.

    MathSciNet  Google Scholar 

  15. Galdi, G.P., An introduction to the mathematical theory of the Navier-Stokes equations, Springer Tracts in Natural Philosophy, 38, 39, Springer, 1994.

    Google Scholar 

  16. Galdi, G.P., Grobbelaar-Van Dalsen, M., and Sauer, N., Existence and uniqueness of classical solutions of the equations of motion for second-grade fluids, Arch. Rational Mech. Anal., 124, 1993, 221–237.

    Article  MathSciNet  MATH  Google Scholar 

  17. Galdi, G.P., Sequeira, A., and Videman, J.H., Steady motions of a second-grade fluid in an exterior domain, Adv. Math. Sci. Appl., 7, 1997, 977–995.

    MathSciNet  MATH  Google Scholar 

  18. Girault, V. and Raviart, P.A., Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986.

    Book  MATH  Google Scholar 

  19. Guillopé, C. and Saut, J.-C., Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, M2AN, 24, 1990, 369–401.

    MATH  Google Scholar 

  20. Guillopé, C. and Saut, J.-C., Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Analysis, Theory, Methods & Applications, 15, 1990, 849–869.

    MATH  Google Scholar 

  21. Huigol, R.R., Continuum Mechanics of Viscoelastic Liquids, Hindustan Publishing Co., Delhi, 1975.

    Google Scholar 

  22. Ladyhenskaya, O.A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.

    Google Scholar 

  23. Ladyzhenskaya, O.A. and Solonnikov, V.A., Determination of the solutions of boundary value problems for stationary Stokes and Navier-Stokes equaitons having an unbounded Dirichlet integral, Zapiski Nauchn. Sem. LOMI, 96, 1980, 117–160. English Transi.: J. Soy. Math., 21, 1983, 728–761.

    Google Scholar 

  24. Lesaint, P. and Raviart, P.A., On a finite element method for solving the neutron transport equation, Mathematical Aspects of Finite Elements in Partial Differential Equations, C. Boor (ed.), Academic Press, 1974, 89–122.

    Google Scholar 

  25. Marchai, J.M. and Crochet, M.J., A new finite element for calculating viscoelastic flow, J. Non-Newtonian Fluid Mech., 26, 1987, 77–114.

    Article  Google Scholar 

  26. Najib, K. and Sandri, D., On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow, Numer. Math., 72, 1995, 223–238.

    Article  MathSciNet  MATH  Google Scholar 

  27. Nazarov, S.A., Weighted spaces with detached asymptotics in applications to the Navier-Stokes equations, Proceedings of the Sixth Winter School, Paseky, Czech Republic, 1999, Pitman Res. Notes Math., to appear.

    Google Scholar 

  28. Nazarov, S.A. and Pileckas, K., On steady Stokes and Navier-Stokes problems with zero velocity at infinity in a three-dimensional exterior domain, J. Math. Kyoto Univ., to appear.

    Google Scholar 

  29. Nazarov, S.A. and Plamenevskii, B.A., Elliptic Boundary Value Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter, 1994.

    Google Scholar 

  30. Nazarov, S.A., Sequeira, A., and Videman, J.H., Asymptotic behavior at infinity of three-dimensional steady viscoelastic flows, submitted.

    Google Scholar 

  31. Novotny, A., Sequeira, A., and Videman, J.H., Existence of three-dimensional flows of second-grade fluids past an obstacle, Nonlinear Analysis, Theory, Methods & Applications, 30, 1997, 3051–3058.

    MathSciNet  MATH  Google Scholar 

  32. Novotny, A., Sequeira, A., and Videman, J.H., Steady motions of viscoelastic fluids in 3-D exterior domains — Existence, uniqueness and asymptotic behavior, Arch. Rational Mech. Analysis, 149, 1999, 49–67.

    MathSciNet  MATH  Google Scholar 

  33. Oldroyd, J.G., On the formulation of Theological equations of state, Proc. Roy. Soc. London, A200, 1950, 523–541.

    MathSciNet  Google Scholar 

  34. Oldroyd, J.G., Non-Newtonian effects in steady motion of some idealized elasto-viscous liquids, Proc. Roy. Soc. London, A245, 1958, 278–297.

    MathSciNet  Google Scholar 

  35. Pileckas, K., Recent advances in the theory of Stokes and Navier-Stokes equations in domains with non-compact boundaries, Mathematical Theory in Fluid Mechanics, Pitman Res. Notes Math., 354, 1996, 30–85.

    MathSciNet  Google Scholar 

  36. Pileckas, K., Sequeira, A., and Videman, J.H., A note on steady flows of non-Newtonian fluids in channels and pipes, Magalhaes, L., Sanchez, L., and Rocha, C. (eds.), EQUADIFF-95 World Scientific, 1998, 458–467.

    Google Scholar 

  37. Pileckas, K., Sequeira, A., and Videman, J.H. Steady flows of viscoelastic fluids in domains with outlets to infinity, submitted.

    Google Scholar 

  38. Pokorny, M., Asymptotic Behavior of Solutions to Certain Partial Differential Equations Describing the Flow of Fluids in Unbounded Domains, Ph.D. Thesis, University of Toulon and Charles University of Prague, 1999.

    Google Scholar 

  39. Quarteroni, A. and Valli, A., Numerical Approximation of Partial Differential Equations, Springer-Verlag, Heidelberg, 1994.

    MATH  Google Scholar 

  40. Rajagopal, K.R., Mechanics of non-Newtonian fluids, Galdi, G.P. and Necas, J. (eds.), Recent Developments in Theoretical Fluid Mechanics, Pitman Res. Notes Math., 291, 1993, 129–162.

    Google Scholar 

  41. Renardy, M., Recent advances in the mathematical theory of steady flow of viscoelastic fluids, J. Non-Newtonian Fluid Mech., 29, 1988, 11–24.

    Article  MATH  Google Scholar 

  42. Rivlin, R.S., Solution of some problems in the exact theory of visco-elasticity, J. Rational Mech. Anal., 5, 1956, 179–188.

    MathSciNet  MATH  Google Scholar 

  43. Rivlin, R.S. and Ericksen, J.L., Stress-deformation relations for isotropic materials, J. Rational Mech. Anal., 4, 1955, 323–425.

    MathSciNet  MATH  Google Scholar 

  44. Schowalter, W.R., Mechanics of Non-Newtonian Fluids, Pergamon Press, 1978.

    Google Scholar 

  45. Sequeira, A. and Bata, M., A finite element approximation for the steady solution of a second-grade fluid model, J. Comp. Appl. Math., 111, 1999, 281–295.

    Article  MATH  Google Scholar 

  46. Solonnikov, V.A. and Pileckas, K., Certain spaces of solenoidal vectors and the boundary value problem for Navier-Stokes system of equations in domains with non-compact boundaries, Zapiski Nauchn. Sem. LOMI, 73, 1977, 136–151. English Transi.: J. Sov. Math., 34, No. 5, 1986, 2101 2111.

    Google Scholar 

  47. Solonnikov, V.A., Stokes and Navier-Stokes equations in domains with non-compact boundaries, College de France Seminars, 4, 1983, 240–349.

    MathSciNet  Google Scholar 

  48. Truesdell, C. and Noll, W., The Nonlinear Field Theories of Mechanics, 2nd edition, Springer, Berlin, 1992.

    Google Scholar 

  49. Videman, J.H., Mathematical Analysis of Viscoelastic Non-Newtonian Fluids, Ph.D. Thesis, Instituto Superior Técnico, Lisbon, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centro de Matemática Aplicada And Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Adélia Sequeira & Juha H. Videman

Authors
  1. Adélia Sequeira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juha H. Videman
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Central Florida, USA

    K. Vajravelu

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Kluwer Academic Publishers

About this chapter

Cite this chapter

Sequeira, A., Videman, J.H. (2001). Mathematical Results and Numerical Methods for Steady Incompressible Viscoelastic Fluid Flows. In: Vajravelu, K. (eds) Differential Equations and Nonlinear Mechanics. Mathematics and Its Applications, vol 528. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0277-3_23

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-0277-3_23

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-7923-6867-0

  • Online ISBN: 978-1-4613-0277-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation