Advances in Computational Mathematics

, Volume 45, Issue 4, pp 2047–2063 | Cite as

Numerical modelling of generalized Newtonian fluids in bypass tube

  • Radka KeslerováEmail author
  • Hynek Řezníček
  • Tomáš Padělek


The following paper describes a numerical simulation of a complete bypass of a stenosed human artery. The considered geometry consists of the narrowed host tube and the bypass graft with a 45-degree angle of connection. Different diameters of the narrowing are tested. Blood is the fluid with shear rate–dependent viscosity; therefore, various rheology mathematical models for generalized Newtonian fluids are considered, namely Cross model, modified Cross model, Carreau model, and Carreau-Yasuda model. The fundamental system of equations is based on the system of generalized Navier-Stokes equations. Generalized Newtonian fluids flow in a bypass tube is numerically simulated by using a SIMPLE algorithm included in the open-source CFD tool, OpenFOAM. The aim of this work is to compare the numerical results for the different mathematical models of the viscosity with the changing diameter of the narrowed channel.


Generalized Newtonian fluids Generalized Navier-Stokes equations OpenFOAM Bypass 

Mathematics Subject Classification (2010)

65L06 65N08 76A05 76A10 76D05 


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Funding information

This work was supported by the grant agency of the Czech Technical University in Prague.


  1. 1.
  2. 2.
  3. 3.
  4. 4.
  5. 5.
  6. 6.
    On application of overset/chimera method for flow approximation over a vibrating body using OpenFOAM (2018)Google Scholar
  7. 7.
    Ali, D., Sen, S.: Permeability and fluid flow-induced wall shear stress of bone tissue scaffolds: Computational fluid dynamic analysis using Newtonian and non-Newtonian blood flow models. Comput. Biol. Med. 99, 201–208 (2018)CrossRefGoogle Scholar
  8. 8.
    Barth, T.: Aspects of unstructured grids and finite-volume solvers for the euler and navier-stokes equations. NASA, AGARD, Special course on unstructured grid methods for advection dominated flows pp. 6.1–6.61 (1992)Google Scholar
  9. 9.
    Baskurt, O., Meiselman, H.: Blood rheology and hemodynamics. In: Semin. Thromb. Hemos (2003)Google Scholar
  10. 10.
    Bodnár, T., Fasano, A., Sequeira, A.: Fluid-structure interaction and biomedical applications, chap. Mathematical models for blood coagulation. Springer, Heidelberg (2014)zbMATHGoogle Scholar
  11. 11.
    Cho, Y., Kensey, K.: Effects of the non-Newtonian viscosity of blood on flows in a diseased arterial vessel. part 1: Steady flows. Biorheology 28, 241–262 (1991)CrossRefGoogle Scholar
  12. 12.
    Jonášová, A.: Computational modelling of hemodynamics for non-invasive assessment of arterial bypass graft patency. Ph.D. thesis, University of West Bohemia, Pilsen, Czech Republic (2014)Google Scholar
  13. 13.
    Keslerová, R., Trdlička, D., Řezníček, H.: Numerical simulation of steady and unsteady flow for generalized Newtonian fluids. Journal of Physics, Conference Series 738, 1–6 (2016)CrossRefGoogle Scholar
  14. 14.
    Keynton, R., Rittgers, S., Shu, M.: The effect of angle and flow rate upon hemodynamics in distal vascular graft anastomoses: an in vitro model study. J. Biomech. Eng. 113, 458–463 (1991)CrossRefGoogle Scholar
  15. 15.
    Lee, D., Su, J., Liang, H.: A numerical simulation of steady flow fields in a bypass tube. J. Biomech. 34, 1407–1416 (2001)CrossRefGoogle Scholar
  16. 16.
    Leuprecht, A., Perktold, K.: Computer simulation of non-Newtonian effects on blood flow in large arteries. Comput. Methods Biomech. Biomed. Engin. 4, 149–163 (2001)CrossRefGoogle Scholar
  17. 17.
    LeVeque, R.: Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  18. 18.
    Moukalled, F., Mangani, L., Darwish, M.: The finite volume method in computational fluid dynamics. Springer, Heidelberg (2016)CrossRefzbMATHGoogle Scholar
  19. 19.
    Patankar, S.: Numerical heat transfer and fluid flow. Hemisphere Publishing Corporation (1980)Google Scholar
  20. 20.
    Patankar, S., Spalding, D.: A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transf. 15, 1787–1806 (1972)CrossRefzbMATHGoogle Scholar
  21. 21.
    Pereira, J., Serra e Moura, J., Ervilha, A., Pereira, J.: On the uncertainty quantification of blood flow viscosity models. Chem. Eng. Sci. 101, 253–265 (2013)CrossRefGoogle Scholar
  22. 22.
    Sequeira, A., Janela, J.: A portrait of state-of-the-art research at the Technical University of Lisbon, chap. An overview of some mathematical models of blood rheology. Springer, Amsterdam (2007)Google Scholar
  23. 23.
    Skiadopoulos, A., Neofytou, P., Housiadas, C.: Comparison of blood rheological models in patient specific cardiovascular system simulations. J. Hydrodyn. 29, 293–304 (2017)CrossRefGoogle Scholar
  24. 24.
    Vimmr, J., Jonášová, A.: Noninvasive assessment of carotid artery stenoses by the principle of multiscale modelling of non-Newtonian blood flow in patient-specific models. Appl. Math. Comput. 319, 598–616 (2018)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Xiang, J., Tremmel, M., Kolega, J., et al.: Newtonian viscosity model could overestimate wall shear stress in intracranial aneurysm domes and underestimate rupture risk. Journal of NeuroInterventional Surgery 4, 351–357 (2012)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Technical MathematicsFME CTU in PraguePraha 2Czech Republic
  2. 2.Department of Transport SystemsFTS CTU in PraguePraha 1Czech Republic

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