Blood flow through axially symmetric sections of compliant vessels: New effective closed models

Due to a tremendous complexity of the human cardiovascular system it remains unfeasible to numerically simulate larger sections of the circulatory system using the full three-dimensional (viscous, incompressible Navier-Stokes) equations for blood flow in compliant vessels. Several "effective" one-dimensional models have been used to simplify the calculation in the axially symmetric sections. All of the one-dimensional models assume an ad hoc axial velocity profile to obtain a closed system of equations, and the Law of Laplace (the independent ring model) to model the vessel wall behavior. In this work we obtain an effective system of equations with the following two novel features: (1) the effective equations do not require an ad hoc closure assumption (the closure follows from the analysis of the original three-dimensional equations) and (2) the vessel wall is modeled as a nonlinearly elastic shell using the Koiter model or the nonlinear membrane model. The first novelty provides a higher-order accurate solution to the original three-dimensional problem, and the second allows deformations of the vessel wall that are not necessarily small. An efficient, fast ("real-time") numerical algorithm based on the coupled finite difference-finite element method has been obtained. Our numerical solutions show secondary flows in certain geometries that cannot be captured with one-dimensional models.