Advertisement

Numerical Study of the Significance of the Non-Newtonian Nature of Blood in Steady Flow Through a Stenosed Vessel

  • Tomáš Bodnár
  • Adélia SequeiraEmail author
Chapter

Abstract

In this paper we present a comparative numerical study of non-Newtonian shear-thinning and viscoelastic blood flow models through an idealized stenosis. Three-dimensional numerical simulations are performed using a finite volume semidiscretization in space, on structured grids, and a multistage Runge-Kutta scheme for time integration, to investigate the influence of combined effects of inertia, viscosity and viscoelasticity in this particular geometry. This work lays the foundation for future applications to pulsatile flows in stenosed vessels using constitutive models capturing the rheological response of blood, under relevant physiological conditions.

Keywords

Non-Newtonian fluids Blood rheology Stenosis Numerical simulations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anand, M., Rajagopal, K.R.: A shear-thinning viscoelastic fluid model for describing the flow of blood. Intern. J. Cardiovasc. Med. Sci. 4(2), 59–68 (2004)Google Scholar
  2. 2.
    Astarita, G., Marrucci, G.: Principles of Non-Newtonian Fluid Mechanics. McGraw Hill, London (1974)Google Scholar
  3. 3.
    Berger, S.A., Jou, L.D.: Flows in stenotic vessels. Ann. Rev. Fluid Mech. 32, 347–382 (2000)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bird, R., Armstrong, R., Hassager, O.: Dynamics of Polymeric Liquids, vol. 1, second edn. John Willey & Sons, New York (1987)Google Scholar
  5. 5.
    Bodnár, T., Sequeira, A.: Shear-thinning effects of blood flow past a formed clot. WSEAS Transactions on Fluid Mechanics 1(3), 207–214 (2006)Google Scholar
  6. 6.
    Bodnár, T., Sequeira, A.: Numerical simulation of the coagulation dynamics of blood. Comput. Math. Methods Med. 9(2), 83–104 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Boyer, F., Chupin, L., Fabrie, P.: Numerical study of viscoelastic mixtures through a cahn-hilliard flow model. Eur. J. Mech. B Fluids 23(5), 759–780 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Caro, C.G., Pedley, T.J., Schroter, R.C., Seed, W.A.: The Mechanics of the Circulation. Oxford University Press, Oxford (1978)Google Scholar
  9. 9.
    Charm, S.E., Kurland, G.S.: Blood Flow and Microcirculation. John Wiley & Sons, New York (1974)Google Scholar
  10. 10.
    Chien, S., Usami, S., Dellenback, R.J., Gregersen, M.I.: Blood viscosity: Influence of erythrocyte aggregation. Science 157(3790), 829–831 (1967)CrossRefGoogle Scholar
  11. 11.
    Chien, S., Usami, S., Dellenback, R.J., Gregersen, M.I.: Blood viscosity: Influence of erythrocyte deformation. Science 157(3790), 827–829 (1967)CrossRefGoogle Scholar
  12. 12.
    Chien, S., Usami, S., Dellenback, R.J., Gregersen, M.I.: Shear-dependent deformation of erythrocytes in rheology of human blood. Am. J. Physiol. 219, 136–142 (1970)Google Scholar
  13. 13.
    Ferry, J.D.: Viscoelastic Properties of Polymers. John Wiley & Sons, New York (1980)Google Scholar
  14. 14.
    Jameson, A.: Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. In: AIAA 10th Computational Fluid Dynamics Conference, Honolulu (1991). AIAA Paper 91-1596Google Scholar
  15. 15.
    Jameson, A., Schmidt, W., Turkel, E.: Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. In: AIAA 14th Fluid and Plasma Dynamic Conference, Palo Alto (1981). AIAA paper 81-1259Google Scholar
  16. 16.
    Joseph, D.D.: Fluid Dynamics of Viscoelastic Liquids. Springer-Verlag, New York (1990)Google Scholar
  17. 17.
    Leuprecht, A., Perktold, K.: Computer simulation of non-Newtonian effects of blood flow in large arteries. Computer Methods in Biomechanics and Biomechanical Engineering 4, 149–163 (2001)CrossRefGoogle Scholar
  18. 18.
    Lowe, D.: Clinical Blood Rheology, Vol.I, II. CRC Press, Boca Raton, Florida (1998)Google Scholar
  19. 19.
    Maxwell, J.C.: On the dynamical theory of gases. Phil. Trans. Roy. Soc. London A157, 26–78 (1866)Google Scholar
  20. 20.
    Nägele, S., Wittum, G.: On the influence of different stabilisation methods for the incompressible Navier–Stokes equations. J. Comp. Phys. 224, 100–116 (2007)zbMATHCrossRefGoogle Scholar
  21. 21.
    Owens, R.G.: A new microstructure-based constitutive model for human blood. J. Non-Newtonian Fluid Mech. 140, 57–70 (2006)zbMATHCrossRefGoogle Scholar
  22. 22.
    Picart, C., Piau, J.M., Galliard, H., Carpentier, P.: Human blood shear yield stress and its hematocrit dependence. J. Rheol. 42, 1–12 (1998)CrossRefGoogle Scholar
  23. 23.
    Rajagopal, K., Srinivasa, A.: A thermodynamic frame work for rate type fluid models. J. Non-Newtonian Fluid Mech. 80, 207–227 (2000)CrossRefGoogle Scholar
  24. 24.
    Robertson, A.M.: Review of relevant continuum mechanics. In: G. Galdi, R. Rannacher, A.M. Robertson, S. Turek (eds.) Hemodynamical Flows: Modeling, Analysis and Simulation (Oberwolfach Seminars), vol. 37, pp. 1–62. Birkhäuser Verlag, Basel, Boston (2008)Google Scholar
  25. 25.
    Robertson, A.M., Sequeira, A., Kameneva, M.V.: Hemorheology. In: G. Galdi, R. Rannacher, A.M. Robertson, S. Turek (eds.) Hemodynamical Flows: Modeling, Analysis and Simulation (Oberwolfach Seminars), vol. 37, pp. 63–120. Birkhäuser Verlag, Basel, Boston (2008)Google Scholar
  26. 26.
    Robertson, A.M., Sequeira, A., Owens, R.G.: Rheological models for blood. In: L. Formaggia, A. Quarteroni, A. Veneziani (eds.) Cardiovascular Mathematics. Modeling and simulation of the circulatory system (MS&A, Modeling, Simulations & Applications), vol. 1, pp. 211–241. Springer - Verlag, New York (2009)Google Scholar
  27. 27.
    Sankar, D.S., Hemalatha, K.: Pulsatile flow of herschel-bulkley fluid through catheterized arteries - a mathematical model. Appl. Math. Model. 31 (8), 1497–1517 (2007)zbMATHCrossRefGoogle Scholar
  28. 28.
    Thurston, G.B.: Viscoelasticity of human blood. Biophys. J. 12, 1205–1217 (1972)CrossRefGoogle Scholar
  29. 29.
    Thurston, G.B.: Non-Newtonian viscosity of human blood: Flow induced changes in microstructure. Biorheology 31(2), 179–192 (1994)MathSciNetGoogle Scholar
  30. 30.
    Vierendeels, J., Riemslagh, K., Dick, E.: A multigrid semi-implicit line-method for viscous incompressible and low-mach-number flows on high aspect ratio grids. J. Comput. Phys. 154, 310–344 (1999)zbMATHCrossRefGoogle Scholar
  31. 31.
    Vlastos, G., Lerche, D., Koch, B.: The superposition of steady on oscillatory shear and its effect on the viscoelasticity of human blood and a blood-like model fluid. Biorheology 34, 19–36 (1997)CrossRefGoogle Scholar
  32. 32.
    Walburn, F.J., Schneck, D.J.: A constitutive equation for whole human blood. Biorheology 13, 201–210 (1976)Google Scholar
  33. 33.
    Yeleswarapu, K.K., Kameneva, M.V., Rajagopal, K.R., Antaki, J.F.: The flow of blood in tubes: Theory and experiment. Mech. Res. Comm. 25, 257–262 (1998)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of Technical Mathematics, Faculty of Mechanical EngineeringCzech Technical UniversityPrague 2Czech Republic
  2. 2.Department of Mathematics and CEMAT, Instituto Superior TécnicoTechnical University of LisbonLisbonPortugal

Personalised recommendations