Abstract
In this paper we address some problems that arise when modelling the human cardiovascular system. On one hand, blood is a complex fluid and in many situations Newtonian models may not be capable of capturing important aspects of blood rheology, for example its shear-thinning viscosity, viscoelasticity or yield stress. On the other hand, the geometric complexity of the cardiovascular system does not permit the use of full three-dimensional (3D) models in large regions. We deal with these problems by using a relatively simple non-Newtonian model capturing the shear-thinning behaviour of blood in a confined region of interest, and coupling it with a zero dimensional (0D) model (also called lumped parameters model) accounting for the remaining circulatory system. More specifically, the 0D system emulates the global circulation, providing proper boundary conditions to the 3D model.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Burman, E., Fernández, M.: Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence. Numerische Mathematik 107(1), 39–77 (2007)
Fernández, M., Moura, A., Vergara, C.: Defective boundary conditions applied to multiscale analysis of blood flow. In: E. Cancès, J.F. Gerbeau (eds.) CEMRACS 2004 – Mathematics and applications to biology and medicine, Marseille, France, July 26 – September 3, 2004, vol. 14, pp. 89–100. ESAIM: Proceedings (2005)
Formaggia, L., Veneziani, A.: Reduced and multiscale models for the human cardiovascular system. Lecture notes VKI Lecture Series 2003–07, Brussels (2003)
Heywood, J., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Num. Meth. Fluids 22, 325–352 (1996)
Janela, J.: Mathematical and Numerical Modelling in Hemodynamics and Hemorheology. Ph.D. thesis, Instituto Superior Técnico (2008)
Janela, J., Sequeira, A.: High accuracy semi-analytical solutions for generalized Newtonian flows. In: Proc. of the Conf. on Topical problems in Fluid Mech., Inst. Thermomechanics, Prague (2007)
Kim, S., Cho, Y.I., Jeon, A.H., Hogenauer, B., Kensey, K.R.: A new method for blood viscosity measurements. J. non-Newtonian Fluid Mech. 94, 47–56 (2000)
Laganà, K., Dubini, G., Migliavacca, F., Pietrabissa, R., Pennati, G., Veneziani, A., Quarteroni, A.: Multiscale modelling as a tool to prescribe realistic boundary conditions for the study of surgical procedures. Biorheology 39, 359–364 (2002)
Moura, A.: The geometrical multiscale modelling of the cardiovascular system: coupling 3D and 1D models. Ph.D. thesis, Politecnico di Milano (2007)
Quarteroni, A., Formaggia., L.: Handbook of Numerical Analysis, vol. XII, chap. Mathematical modelling and numerical simulation of the cardiovascular system. Elsevier, Amsterdam (2002)
Quarteroni, A., Ragni, S., Veneziani, A.: Coupling between lumped and distributed models for blood flow problems. Comput. Vis. Sci. 4, 111–124 (2001)
Robertson, A., Sequeira, A., Kameneva, M.: Hemorheology. In: G.P. Galdi, R. Rannacher, A. Robertson, S. Turek (eds.) Haemodynamical Flows: Modelling Analysis and Simulation, Oberwolfach Seminars, vol. 37, pp. 63–120. Birkhauser Basel, Switzerland (2008)
Robertson, A., Sequeira, A., Owens, R.G.: Rheological models for blood. In: A. Quarteroni, L. Formaggia, A. Veneziani (eds.) Cardiovascular Mathematics: Modelling and simulation of the cardiovascular system, pp. 211–241. Springer-Verlag, Italia (2009)
Steinman, D., Vorp, D., Ethier, C.: Computational modeling of the arterial biomechanics: insights into pathogenesis and treatment of vascular disease. J. Vasc. Surg. 37, 1118–1128 (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Janela, J., Moura, A., Sequeira, A. (2010). Towards a Geometrical Multiscale Approach to Non-Newtonian Blood Flow Simulations. In: Rannacher, R., Sequeira, A. (eds) Advances in Mathematical Fluid Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04068-9_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-04068-9_18
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04067-2
Online ISBN: 978-3-642-04068-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)