Anne Robertson, Mechanical Engineering, University of Pittsburgh
"On the Use of Cosserat Theory for Modeling Arterial Flows"
It is not possible, even with current computational power, to perform a complete unsteady 3-D 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 2-D or 3-D governing equations) or (ii) large sections of the vasculature are studied using 1-D 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 1-D 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 1-D models. Here, rather than using either the classical 1-D models of arterial systems, we have made use of a directed continuum theory for viscous fluid flow in pipes (see, e.g. Green 1976, Green and Naghdi 1993, Caulk and Naghdi 1987) to develop models of segments of the arterial system. The Cosserat 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. 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 these steady benchmark flows. Significantly, the classical 1-D theories can be substantially improved by employing an alternative approximation for the nonlinear convective terms and modifying the unsteady acceleration term.
TUESDAY, October 21, 2003