Speaker: Anne M. Robertson
Title: On the use of Director Theories for Modeling Arterial Flows
Affiliation: University of Pittsburgh
Date: Friday, March 31, 2006.
Place and Time: Room 101 - Love Building, 3:35-4:30 pm.
Refreshments: Room 204 - Love Building, 3:00 pm.

Abstract. It is not possible, even with current computational power, to perform a complete unsteady three dimensional (3D) analysis of large sections of the circulatory system. Typically either (i) a small section of the vasculature, such as a single bifurcation, is analyzed in great detail, (solution to the full 2D or 3D governing equations) or (ii) large sections of the vasculature are studied using 1D or lumped parameter approximations of the full equations. In the first approach, inflow and outflow boundary conditions must be assumed for the arterial segment, effectively decoupling this segment from the remainder of the vascular system. The accuracy and applicability of the second approach are severely limited by the significant approximations introduced in the 1D and lumped parameter models. Recently, progress has been made in coupling these local and global equations. This issue is of great interest for modeling the human vascular system since it will allow the use of complex three dimensional local models where needed, while maintaining a coherent, sufficiently accurate and computationally cheaper description of the global system. A limiting factor in the success of the multiscale analysis is the inaccuracy of the lumped parameter and 1D models.

Part of the focus on multi-scale models in our group has been on the development of alternatives to the classical 1D models. Rather than using classical 1-D models of arterial systems, we have made use of a Cosserat type continuum theory for viscous fluid flow in pipes to develop models of segments of the arterial system [1]. This director theory has a number of advantages including (i) the theory is hierarchical making it possible to increase the capabilities of the model by including more directors; (ii) the wall shear stress enters independently as a dependent variable; and (iii) there is no need to make somewhat ad hoc approximations about the nonlinear convective terms [2,3]. In this talk, we focus on the rigid walled case, and discuss the well posedness of both the 9-Director and 1-D models. We then go on to compare the predictions of the nine director theory and 1-D models with analytical and numerical solutions to the full equations for specific benchmark problems relevant to arterial flow. The 9-director theory is shown to provide better results for the steady and unsteady benchmark flows considered here.

Motivated by results from the director theory, we revisited the classical 1D models to see if they can be improved. We found that by modifying the approximations for the nonlinear convective and unsteady acceleration terms, the classical 1-D theories can be substantially improved for the chosen benchmark problems over a wide range of Reynolds numbers. Building on these results, we have developed 1D models for fluids with non-constant viscosity, with particular emphasis on power-law fluids.

[1] A.M. Robertson and A. Sequeira, A Director Theory Approach for Modeling Blood Flow in the Arterial System: An Alternative to Classical 1D models, Mathematical Models and Methods in Applied Sciences. 15(6), p871-906, 2005
[2] A. E. Green and P.M.Naghdi, A direct theory of viscous fluid flow in pipes I. Basic general developments. Phil. Trans. R. Soc. Lond. A, 342:525-542, 1993.
[3] D. Caulk and P.M. Naghdi, Axisymmetric motion of viscous fluid flow inside a slender surface of revolution. J. Appl. Mech., 54:190-196, 1987.