Flows of Generalized Oldroyd-B Fluids in Curved Pipes

Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 80)

Abstract

The aim of this work is to present a numerical study of generalized Oldroyd-B flows with shear-thinning viscosity in a curved pipe of circular cross section and arbitrary curvature ratio. Flows are driven by a given pressure gradient and behavior of the solutions is discussed with respect to different rheologic and geometric flow parameters.

Keywords

Curved pipe finite elements fluids non-Newtonian. 

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References

  1. 1.
    N. Arada, M. Pires and A. Sequeira, Viscosity effects on flows of generalized Newtonian fluids through curved pipes, Computers and Mathematics with Applications, 53 (2007), 625–646.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    N. Arada, M. Pires and A. Sequeira, Numerical approximation of viscoelastic Oldroyd-B flows in curved pipes, RIMS, Kôkyûroku Bessatsu, B1 (2007), 43–70.MATHMathSciNetGoogle Scholar
  3. 3.
    N. Arada, M. Pires and A. Sequeira, Numerical simulations of shear-thinning Oldroyd-B fluids in curved pipes, IASMETransactions, issue 6 , Vol 2 (2005), 948–959.MathSciNetGoogle Scholar
  4. 4.
    A.A. Berger, L. Talbot and L.-S. Yao, Flow in curved pipes, Ann. Rev. Fluid Mech., 15 (1983), 461–512.CrossRefGoogle Scholar
  5. 5.
    R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, John Wiley & Sons, New York (1987).Google Scholar
  6. 6.
    S. Chien, S. Usami, R.J. Dellemback, M.I. Gregersen, Blood viscosity: influence of erythrocyte deformation, Science 157 (3790), 827–829, 1967.CrossRefGoogle Scholar
  7. 7.
    S. Chien, S. Usami, R.J. Dellemback, M.I. Gregersen, Blood viscosity: influence of erythrocyte aggregation, Science 157 (3790), 829–831, 1967.CrossRefGoogle Scholar
  8. 8.
    P. Daskopoulos and A.M. Lenhoff, Flow in curved ducts: bifurcation structure for stationary ducts, J. Fluid Mech., 203 (1989), 125–148.CrossRefMathSciNetGoogle Scholar
  9. 9.
    W.R. Dean, Note on the motion of fluid in curved pipe, Philos. Mag., 20, 208 (1927).Google Scholar
  10. 10.
    W.R. Dean, The streamline motion of fluid in curved pipe, Philos. Mag., 30, 673 (1928).Google Scholar
  11. 11.
    J. Eustice, Flow of water in curved pipes, Proc. R. Soc. Lond. A 84 (1910), 107–118.Google Scholar
  12. 12.
    J. Eustice, Experiments of streamline motion in curved pipes, Proc. R. Soc. Lond. A 85 (1911), 119–131.Google Scholar
  13. 13.
    Y. Fan, R.I. Tanner and N. Phan-Thien, Fully developed viscous and viscoelastic flows in curved pipes, J. Fluid Mech., 440 (2001), 327–357.MATHCrossRefGoogle Scholar
  14. 14.
    J.H. Grindley and A.H. Gibson, On the frictional resistance to the flow of air through a pipe, Proc. R. Soc. Lond. A 80 (1908), 114–139.MATHGoogle Scholar
  15. 15.
    D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer Verlag, New York (1990).MATHGoogle Scholar
  16. 16.
    R. Keunings, A Survey of Computational Rheology, In: Proceedings of the XIII International Congress on Rheology (D.M. Binding et al. ed.), British Soc. Rheol., 1 (2000), 7–14.Google Scholar
  17. 17.
    R. Owens and T.N. Phillips, Computational Rheology, Imperial College Press, London (2002).MATHCrossRefGoogle Scholar
  18. 18.
    T.J. Pedley, Fluid Mechanics of Large Blood Vessels, Cambridge University Press, Cambridge (1980).MATHGoogle Scholar
  19. 19.
    M. Pires, Mathematical and Numerical Analysis of Non-Newtonian Fluids in Curved Pipes, PhD thesis, IST, Lisbon (2005)Google Scholar
  20. 20.
    K.R. Rajagopal, Mechanics of non-Newtonian Fluids, Galdi,G.P. and J. Něcas (eds.) Recent Developments in Theoretical Fluid Mechanics, Pitman Research Notes in Mathematics 291 (1993), Longman Scientific an Technical, 129–162.Google Scholar
  21. 21.
    M. Renardy, Mathematical Analysis of Viscoelastic Flows, SIAM (2000).Google Scholar
  22. 22.
    M. Renardy, W.J. Hrusa and J.A. Nohel, Mathematical problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman 35 (1987).Google Scholar
  23. 23.
    A.M. Robertson, On viscous flow in curved pipes of non-uniform cross section, Inter. J. Numer. Meth. fluid., 22 (1996), 771–798.MATHCrossRefGoogle Scholar
  24. 24.
    A.M. Robertson and S.J. Muller, Flow of Oldroyd-B fluids in curved pipes of circular and annular cross-section, Int. J. Non-Linear Mechanics, 31 (1996), 1–20.MATHCrossRefGoogle Scholar
  25. 25.
    A.M. Robertson, A. Sequeira and M. Kameneva, Hemorheology, Hemodynamical Flows: Modeling, Analysis and Simulation, Series: Oberwolfach Seminars, G.P. Galdi, R. Rannacher, A.M. Robertson, S. Turek, Birkhäuser, 37 (2008), 63–120.MathSciNetGoogle Scholar
  26. 26.
    W.R. Schowalter, Mechanics of Non-Newtonian Fluids, Pergamon Press, New York (1978).Google Scholar
  27. 27.
    W.Y. Soh and S.A. Berger, Fully developed flow in a curved pipe of arbitrary curvature ratio, Int. J. Numer. Meth. Fluid., 7 (1987), 733–755.MATHCrossRefGoogle Scholar
  28. 28.
    G.B. Thurston, Viscoelasticity of human blood, Biophys. J., 12 (1972), 1205–1217.CrossRefGoogle Scholar
  29. 29.
    C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Encyclopedia of Physics, (ed. S. Flugge), vol. III / 3 (1965), Springer-Verlag.Google Scholar
  30. 30.
    G.S. Williams, C.W. Hubbell and G.H. Fenkell,  Experiments at Detroit, Mich. on the effect of curvature upon the flow of water in pipes, Trans. ASCE 47 (1902), 1–196.Google Scholar
  31. 31.
    Z.H. Yang and H.B. Keller, Multiple laminar flows through curved pipes, Appl. Numer. Math., 2 (1986), 257–271.MATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Centro de Investigação em Matemática e AplicaçõesUniversidade de ÉvoraÉvoraPortugal
  2. 2.Centro de Matemática e AplicaçõesInstituto Superior TécnicoLisboaPortugal

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