Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-27T21:07:33.895Z Has data issue: false hasContentIssue false
  • English
  • Français

Blood Flow Simulation Using Traceless Variant ofJohnson-Segalman Viscoelastic Model

Published online by Cambridge University Press:  24 September 2014

T. Bodnár ,
M. Pires  and
J. Janela
T. Bodnár
Affiliation:
Faculty of Mechanical Engineering, Czech Technical University in Prague Karlovo Náměstí 13, 121 35 Prague 2, Czech Republic
M. Pires
Affiliation:
Department of Mathematics and CIMA-UE, Évora University Rua Romão Ramalho, 7000-671, Évora, Portugal CEMAT, Instituto Superior Técnico, Universidade de Lisboa Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
J. Janela*
Affiliation:
Department of Mathematics and CEMAPRE, ISEG, Universidade de Lisboa Rua do Quelhas 6, 1200-781 Lisbon, Portugal
*
⋆Corresponding author. E-mail: [email protected]
Get access

Abstract

A traceless variant of the Johnson-Segalman viscoelastic model is presented and appliedto blood flow simulations. The viscoelastic extra stress tensor is decomposed into itstraceless (deviatoric) and spherical parts, leading to a reformulation of the classicalJohnson-Segalman model. The equivalence of the two models is established comparing modelpredictions for simple test cases. The new model is validated using several 2D benchmarkproblems, designed to reproduce difficulties that arise in the simulation of blood flow inblood vessels or medical devices. The structure and behaviour of the new model arediscussed and the future use of the new model in envisioned, both on the theoretical andnumerical perspectives.

Keywords

bloodviscoelasticJohnson-SegalmanOldroydtraceless tensor
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 6: Blood flows , 2014 , pp. 117 - 141
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anand, M., Rajagopal, K.R.. A shear-thinning viscoelastic fluid model for describing the flow of blood. Intern. Journal of Cardiovasc. Medicine and Sci., 4 no. 2, (2004), 59–68, 2004. Google Scholar
Arada, N., Pires, M., Sequeira, A.. Numerical simulations of a shear-thinning Oldroyd-B fluids in curved pipes. IASME Transactions, 2 no. 6 (2005), 948–959. Google Scholar
N. Arada, M. Pires, A. Sequeira.Numerical approximation of a viscoelastic Oldroyd-B flows in curved pipes. In Y. Giga, H. Kozono, H. Okamoto, and Y. Shibata, editors, Kyoto Conference on the Navier-Stokes Equations and their Applications, volume B1 of Kôkyûroku Bessatsu, RIMS, March 2007, 43–70.
Arada, N., Pires, M., Sequeira, A.. Viscosity effects on flows of generalized Newtonian fluids through curved pipes. Computers and Mathematics with Applications, 53 (2007), 625–646. CrossRefGoogle Scholar
R.B. Bird, R.C. Armstrong, O. Hassager. Dynamics of Polymeric Liquids, volume I. Fluid Mechanics. John Willey & Sons, second edition, 1987.
R.B. Bird, Ch.F. Curtis, R.C. Armstrong, O. Hassager. Dynamics of Polymeric Liquids, volume II. Kinetic Theory. John Willey & Sons, second edition, 1987.
Bodnár, T., Rajagopal, K.R., Sequeira, A.. Simulation of the three-dimensional flow of blood using a shear-thinning viscoelastic fluid model. Mathematical Modelling of Natural Phenomena, 6 no. 5 (2011), 1–24. CrossRefGoogle Scholar
T. Bodnár, A. Sequeira. Numerical study of the significance of the non-Newtonian nature of blood in steady flow through a stenosed vessel. In R. Rannacher and A. Sequeira, editors, Advances in Mathematical Fluid Mechanics, Springer Verlag, (2010), 83–104.
Bodnár, T., Sequeira, A., Prosi, M.. On the shear-thinning and viscoelastic effects of blood flow under various flow rates. Applied Mathematics and Computation, 217 no. 11 (2011), 5055–5067. CrossRefGoogle Scholar
J. Chen, X.-Y. Lu. Numerical investigation of the non-Newtonian pulsatile blood flow in a bifurcation model with a non-planar branch. Journal of Biomechanics, 2005.
B. Cockburn, G. Karniadakis, C.Shu. Discontinuous Galerkin Methods Theory, Computation and Applications, volume 11 of Lecture Notes in Computer Science and Engineering. Springer Verlag, 2000.
L. Dintenfass. Blood Viscosity, Hyperviscosity & Hyperviscoaemia. MTP Press Limited (Kluwer), 1985.
Ern, A., Guermond, J.. Discontinuous Galerkin methods for Friedrichs systems i. general theory. SIAM Journal of Numerical Analysis, 44 no. 2 (2006), 753–778. CrossRefGoogle Scholar
Español, P., Yuan, X.F., Ball, R.C.. Shear banding flow in the Johnson-Segalman fluid. Journal of Non-Newtonian Fluid Mechanics, 65 (1996), 93–109. CrossRefGoogle Scholar
Fyrillasa, M.M., Georgioua, G.C., Vlassopoulos, D.. Time-dependent plane poiseuille flow of a Johnson-Segalman fluid. Journal of Non-Newtonian Fluid Mechanics, 82 (1999), 105–123. CrossRefGoogle Scholar
G.P. Galdi, R. Rannacher, A.M. Robertson, S. Turek, editors. Hemodynamical Flows - Modeling, Analysis and Simulation, vol. 37 of Oberwolfach Seminars. Birkäuser, 2008.
Gambaruto, A., Janela, J., Moura, A., Sequeira, A.. Sensitivity of hemodynamics in a patient-specific cerebral aneurysm to vascular geometry and blood geometry. Mathematical biosciences and engineering, 8 no. 2 (2011), 409–423. Google Scholar
Gambaruto, A., Janela, J., Moura, A., Sequeira, A.. Shear-thinning effects of hemodynamics in patient-specific cerebral aneurysms. Mathematical biosciences and engineering, 10 no. 3 (2013), 649–665. CrossRefGoogle ScholarPubMed
Gijsen, F.J.H., van de Vosse, F.N., Janssen, J.D.. The influence of the non-Newtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model. Journal of Biomechanics, 32 (1999), 601–608. CrossRefGoogle Scholar
V. Girault, P.A. Raviart.Finite Element Approximation of the Navier Stokes Equations, volume 749 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1979.
F. Hecht. FreeFem++ v 3.23 documentation, June 2013. http://www.freefem.org/ff++.
Hron, J., Malek, J., Turek, S.. A numerical investigation of flows of shear-thinning fluids with applications to blood rheology. International Journal for Numerical Methods in Fluids, 32 (2000), 863–879. 3.0.CO;2-P>CrossRefGoogle Scholar
Kolkka, R.W., Malkus, D.S., Hansen, M.G., Ierley, G.R., Worthing, R.A.. Spurt phenomena of the Johnson-Segalman fluid and related models. Journal of Non-Newtonian Fluid Mechanics, 29 (1988), 303–335. CrossRefGoogle Scholar
P. Lesaint, P.A. Raviart.On a finite element method for solving the neutron transport equation. In Mathematical aspects of finite elements in partial differential equations, pages 89–123, New York, 1974. Academic Press. (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974).
Leuprecht, A., Perktold, K.. Computer simulation of non-Newtonian effects of blood flow in large arteries. Computer Methods in Biomechanics and Biomechanical Engineering, 4 (2001), 149–163. CrossRefGoogle ScholarPubMed
B.Q. Li. Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer. Springer Verlag, 2006.
Owens, R.G.. A new microstructure-based constitutive model for human blood. Journal of Non-Newtonian Fluid Mechanics, 140 (2006), (1–3), 57–70. CrossRefGoogle Scholar
Perktold, K., Hofer, M., Rappitsch, G., Loew, M., Kuban, B. D., Friedman, M. H.. Validated computation of physiologic flow in a realistic coronary artery branch. Journal of Biomechanics, 3 (1998), 217–228. Google Scholar
Perktold, K., Rappitsch, G.. Computed simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model. Journal of Biomechanics, 28 no. 7 (1995), 845–856. CrossRefGoogle Scholar
Picart, C., Piau, J.-M., Galliard, H., Carpentier, P.. Human blood shear yield stress and its hematocrit dependence. Journal of Rheology, 42 no. 1, 1998. CrossRefGoogle Scholar
Pires, M., Sequeira, A.. Flows of generalized Oldroyd-B fluids in curved pipes. Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 21–43. Google Scholar
Rao, I.J., Rajagopal, K.R.. Some simple flows of a Johnson-Segalman fluid. Acta Mechanica, 132 (1999), 209–219. CrossRefGoogle Scholar
Smith, K.D., Sequeira, A.. Micro-macro simulations of a shear-thinning viscoelastic kinetic model: applications to blood flow. Applicable Analysis, 90 no. 1 (2011), 227–252. CrossRefGoogle Scholar
Thurston, G.B.. Viscoelasticity of human blood. Biophysical Journal, 12 (1972), 1205–1217. CrossRefGoogle ScholarPubMed
Thurston, G.B.. Frequency and shear rate dependence of viscoelasticity of human blood. Biorheology, 10 (1973), 375–381. Google ScholarPubMed
Thurston, G.B.. Rheological parameters for the viscosity viscoelasticity and thixotropy of blood. Biorheology, 16 no. 3, (1979), 149–162. Google Scholar
Thurston, G.B. and Henderson, N.M.. Effects of flow geometry on blood viscoelasticity. Biorheology, 43 no. 6, (2006), 729–746. Google Scholar
Yeleswarapu, K.K., Kameneva, M.V., Rajagopal, K.R., Antaki, J.F.. The flow of blood in tubes: theory and experiments. Mechanics Research Communications, 25 no. 3, (1998), 257–262. CrossRefGoogle Scholar
Yilmaz, F., Gundogdu, M.Y.. A critical review on blood flow in large arteries; relevance to blood rheology, viscosity models, and physiologic conditions. Korea-Australia Rheology Journal, 20 no. 4, (2008), 197–211. Google Scholar